diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpvny" "b/data_all_eng_slimpj/shuffled/split2/finalzzpvny"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpvny"
@@ -0,0 +1,5 @@
+{"text":"\n\\section{Introduction}\n\\setcounter{equation}{0}\n\nThe nonlinear cubic-quintic Schr\\\"odinger equation is given by\n\\begin{equation}\\label{eqnlcqs}\niA_t=A_{xx}+|A|^2A+\\alpha|A|^4A,\n\\end{equation}\nwhere $A$ is a complex-valued function of the variables $(x,t)\\in\n{\\bf R}\\times{\\bf R}^+$.\nWhen $\\alpha=0$, the equation becomes the focusing\ncubic nonlinear Schr\\\"odinger equation, and is used to\ndescribe the propagation of the envelope of a light pulse in an optical\nfiber which has a Kerr-type nonlinear refractive index.\nFor short pulses and high input peak pulse power the \nrefractive index cannot be described by a Kerr-type nonlinearity, as\nthe index is then influenced by higher-order nonlinearities.\nIn materials with high nonlinear coefficients, such as \nsemiconductors, semiconductor-doped glasses, and organic polymers, \nthe saturation of the nonlinear \nrefractive-index change is no longer neglible at moderately \nhigh intensities and should be taken into account (\\cite{gatz:spi91}).\nEquation (\\ref{eqnlcqs}) is the correct model \nto describe the  propagation of the envelope of a light pulse \nin dispersive materials with either a saturable or higher-order\nrefraction index (\\cite{gatz:spi91}, \\cite{gatz:sca92}).\n\nEquation (\\ref{eqnlcqs}) cannot really be thought of as a \nsmall perturbation of the cubic nonlinear Schr\\\"odinger equation,\nas it has been shown that a physically realistic value for the\nparameter $\\alpha$ is $|\\alpha|\\sim 0.1$ (\\cite{herrmann:bbs92}).\nIt turns out that the most physically interesting behavior\noccurs when the nonlinearity is saturating, i.e., $\\alpha<0$,\nso for the rest of this paper it will be assumed that $\\alpha\\sim-0.1$\n(\\cite{angelis:stp94}, \\cite{gatz:spi91}, \\cite{gatz:spa92},\n\\cite{herrmann:bbs92}, \\cite{sombra:bpc92}).\nAn optical fiber which satisfies this condition can be\nconstructed, for example, by doping with two appropriate materials\n(\\cite{angelis:stp94}).\n\nOne of the more physically interesting phenomena\nassociated with the double-doped optical fiber is\nthe existence of bright solitary wave solutions \n($|A(x)|\\to0$ as $|x|\\to\\infty$) in which the peak amplitude becomes \na two-valued function of the pulse duration.\nThese solutions have two\ndifferent peak powers, and were proven to be stable \n(\\cite{boling:oso95}, \n\\cite{gatz:spi91}, \\cite{grillakis:sto87}, \\cite{herrmann:bbs92}).\nThe solution with the lower peak power corresponds to a perturbation\nof the one that exists for the cubic nonlinear Schr\\\"odinger\nequation, while the one with the larger peak power is due\nto the saturating nonlinearity.\n\nEquation (\\ref{eqnlcqs}) describes an idealized fiber; therefore,\nit is natural to consider the perturbative PDE\n\\begin{equation}\\label{eqperturbnlcqs}\niA_t=(1+i\\epsilon a_1)A_{xx}+i\\epsilon\\sigma A+(1+i\\epsilon d_1)|A|^2A+\n    (\\alpha+i\\epsilon d_2)|A|^4A,\n\\end{equation}\nwhere $0<\\epsilon\\ll1$ and the other parameters are real and of $O(1)$.\nThe parameter $a_1$ describes spectral filtering,\n$\\sigma$ describes the linear gain or loss due to the fiber, and\n$d_1$ and $d_2$ describe the nonlinear gain or loss due to\nthe fiber.\nSo that (\\ref{eqperturbnlcqs}) is a well-defined PDE for $\\epsilon>0$,\nit will henceforth be assumed that $a_1>0$.\n\nSolitary wave solutions to (\\ref{eqperturbnlcqs}) are found by\nsetting\n\\begin{equation}\\label{eqsolutionansatz}\nA(x,t)=A(x)e^{i\\mu t},\n\\end{equation}\nand then finding homoclinic solutions for the ODE\n\\begin{equation}\\label{eqperturbnlcqsodetemp}\n(1+i\\epsilon a_1)A''+(\\mu+i\\epsilon\\sigma)A+(1+i\\epsilon d_1)|A|^2A+\n    (\\alpha+i\\epsilon d_2)|A|^4A=0,\n\\end{equation}\nwhere $'=d\/dx$.\nMultiplying the above by $1-i\\epsilon a_1$ and setting \n\\[\nx=(1+\\epsilon^2a_1^2)\\hat{x}\n\\]\nyields the equivalent ODE\n\\begin{equation}\\label{eqperturbnlcqsodetemp2}\n\\begin{array}{l}\nA''+((\\mu+\\epsilon^2a_1\\sigma)+i\\epsilon(\\sigma-a_1\\mu))A+\n   ((1+\\epsilon^2a_1d_1)+i\\epsilon(d_1-a_1))|A|^2A \\\\\n\\quad\\quad\\quad+((\\alpha+\\epsilon^2a_1d_2)+i\\epsilon(d_2-a_1\\alpha))|A|^4A=0,\n\\end{array}\n\\end{equation}\nwhere now $'=d\/d\\hat{x}$.\n\nTo simplify matters, all the $O(\\epsilon^2)$ terms in the\nabove equation will be dropped.\nThis step is taken only so that the notational complexity is\nmade as simple as possible, and can clearly be done without\nany loss of generality, as for $\\epsilon$ small these terms are\nnegligible.\nUpon dropping these terms, the ODE that will be studied can\nfinally be written as\n\\begin{equation}\\label{eqperturbnlcqsode}\nA''+(\\mu+i\\epsilon(\\sigma-a_1\\mu))A+\n   (1+i\\epsilon(d_1-a_1))|A|^2A+\n   (\\alpha+i\\epsilon(d_2-a_1\\alpha))|A|^4A=0.\n\\end{equation}\n\nEquation (\\ref{eqperturbnlcqsode}) has been extensively studied by\nmany authors (\\cite{de:91}, \\cite{duan:fdw95}, \\cite{jkp:90}, \n\\cite{kapitula:sho95},\n\\cite{mn:90}, \\cite{marcq:eso94}, \\cite{saarloos:fps92}).\nThese papers have been concerned with finding various types of \nsolutions, including fronts (kinks), bright solitary waves, and\ndark solitary waves.\nThe methods employed have been both geometric (\\cite{de:91}, \n\\cite{duan:fdw95}, \\cite{jkp:90}, \\cite{kapitula:sho95}) and analytic\n(\\cite{mn:90}, \\cite{marcq:eso94}, \\cite{saarloos:fps92}).\n\nBright solitary waves exist when there are solutions to \n(\\ref{eqperturbnlcqsode}) which are homoclinic to $|A|=0$.\nWhen $\\epsilon=0$ with\n\\begin{equation}\\label{eqmurangebright}\n\\frac1{4\\alpha}<\\mu<\\frac3{16\\alpha},\n\\end{equation}\none can find a bright solitary wave with a peak power larger than\nthat associated with the cubic \nnonlinear Schr\\\"odinger equation.\nWhen $\\epsilon=0$ and $\\mu$ is in the range\n\\begin{equation}\\label{eqmurangedark}\n\\frac3{16\\alpha}<\\mu<0\n\\end{equation}\nthere exists dark solitary wave solutions.\nThe dark solitary waves are solutions to (\\ref{eqperturbnlcqsode}) which\nhave the property that $|A(x)|\\to A_0\\neq0$ as $|x|\\to\\infty$.\n\nIt can be hypothesized that for the perturbative \nPDE (\\ref{eqperturbnlcqs}) there\nmay be a competition between the bright and dark solitary waves for\n$\\mu$ sufficiently close to the critical value\n\\begin{equation}\\label{eqdefmustar}\n\\mu^*=\\frac3{16\\alpha}.\n\\end{equation}\nIt may be further hypothesized that for $\\mu$ sufficiently near $\\mu^*$ \nit may be possible to construct novel solutions by gluing together \nthe bright and dark solitary waves in some way.\nThe goal of this paper is to explore this possibility.\n\nDoelman \\cite{doelman:bth95} has recently considered the problem\nof finding $N$-circuit solutions.\nThese $N$-circuits are constructed by piecing together $N$ copies\nof a dark solitary wave, and exist as a solution to \n(\\ref{eqperturbnlcqsode}) for $\\epsilon$ sufficiently small and the\nparameters in a certain domain in parameter space.\nRecent work by De Bouard \\cite{bouard:ios95} has shown that the \ndark solitary waves are an unstable solution to (\\ref{eqnlcqs}),\ndue to the presence of a real positive eigenvalue for the operator\nobtained by linearizing about the wave.\nBy a result of Alexander and Jones \\cite{alexander:esa94},\nsince the original solitary wave has an unstable eigenvalue it\nis expected that the $N$-circuit solution will have at least\n$N$ unstable eigenvalues.\n\nIt was previously stated that the bright solitary wave is \na stable solution to (\\ref{eqnlcqs}).\nHowever, recent numerical work by Soto-Crespo et al \\cite{crespo:sot96}\nshows that this wave\nbecomes an unstable solution to (\\ref{eqperturbnlcqs}) for\n$\\epsilon$ nonzero.\nThe numerics suggest that this instability arises from the presence of a \nreal eigenvalue\nfor the linearized problem \nbifurcating out of the origin and into the right-half of the complex\nplane.\n\nDue to these results, when attempting to construct stable solitary\nwaves to (\\ref{eqperturbnlcqs}) one would hope to avoid using \neither the bright or dark waves which exist for (\\ref{eqnlcqs}).\nThis may by possible by considering the situation when \n$\\mu=\\mu^*$.\nWhen $\\epsilon=0$ and $\\mu$ is equal to this critical parameter, there exists\na front solution (kink) to (\\ref{eqperturbnlcqsode}) which has the \nasymptotic behavior\n\\[\n|A(x)|\\to\\left\\{\\begin{array}{ll}\n       0,\\quad&x\\to-\\infty \\\\\n       A_0,\\quad&x\\to\\infty.\n       \t       \\end{array}\\right.\n\\]\nFurthermore, since (\\ref{eqperturbnlcqsode}) is invariant under\n$x\\to-x$, there also exists a solution which satisfies\n\\[\n|A(x)|\\to\\left\\{\\begin{array}{ll}\n       A_0,\\quad&x\\to-\\infty \\\\\n       0,\\quad&x\\to\\infty.\n         \t\\end{array}\\right.\n\\]\nThus, in the ODE phase space there exists a heteroclinic loop,\nfrom which solitary waves may bifurcate as $\\epsilon$ is made\nnonzero.\nBy following the proof in De Bouard \\cite{bouard:ios95} it can\nbe conjectured that for the kink solution the linearized\noperator possesses no unstable eigenvalues.\nTherefore, it may be possible that any solitary waves \nbifurcating out of the heteroclinic loop may also not have\nany unstable eigenvalues.\n\nBased upon the above discussion, it will be of interest to study\nthe ODE (\\ref{eqperturbnlcqsode}) when $\\mu$ is (at least) $O(\\epsilon)$ \nclose to $\\mu^*$.\nIn particular, it will be of interest to study the dynamics\nof the ODE near the heteroclinic cycle for $\\epsilon$ nonzero.\nTowards this end, it will first be shown that the cycle\npersists for $\\epsilon$ nonzero for $\\mu=\\mu(\\epsilon)$, with\n$\\mu(\\epsilon)\\to\\mu^*$ as $\\epsilon\\to0$.\nAfter then fixing $\\epsilon$ the dynamics will be studied as\n$\\mu$ bifurcates from $\\mu(\\epsilon)$.\nIt is important to note that $\\mu$ is a natural bifurcation\nparameter in this problem, as it is a free parameter (recall\nequation (\\ref{eqsolutionansatz})).\n\nThis paper will be devoted to proving the following theorems.\nSet \n\\[\nA_0^2=-\\frac{3}{4\\alpha},\n\\]\nand define\n\\[\n\\begin{array}{lll}\n\\tilde{d}_1&=&d_1-a_1 \\\\\n\\tilde{d}_2&=&d_2-\\alpha a_1 \\\\\n\\tilde{\\sigma}&=&-(\\sigma+A_0^2d_1+A_0^4d_2)\/\\epsilon.\n\\end{array}\n\\]\nAssume that $\\tilde{\\sigma}=O(1)$ for all $\\epsilon>0$.\n\nBefore the main theorems can be stated, a preliminary lemma is\nneeded regarding the persistence of the kink solitary wave\nfor $\\epsilon\\neq0$.\n\n\\begin{lemma}\\label{lemgsw} For $0\\le\\epsilon\\ll1$ there exists\na $\\mu(\\epsilon)$, with $|\\mu(\\epsilon)-\\mu^*|=O(\\epsilon^2)$, such that\na kink solitary wave exists.\n\\end{lemma}\n\n\\begin{remark} Since (\\ref{eqperturbnlcqsode}) is\ninvariant under the transformation $x\\to-x$, the above lemma\nguarantees the existence of a heteroclinic cycle in the \nODE phase space for $\\epsilon$ nonzero.\n\\end{remark}\n\n\\begin{theorem}\\label{thmbsw} Let $0<\\epsilon\\ll1$, and assume that\n\\[\n(\\tilde{d}_1+\\frac32A_0^2\\tilde{d}_2)\\tilde{\\sigma}<0.\n\\]\nThere exists a $\\mu_h(\\epsilon)<\\mu(\\epsilon)$, with $\\mu(\\epsilon)-\\mu_h(\\epsilon)=\nO(e^{-c\/\\epsilon})$, such that a bright solitary wave exists.\nFurthermore, for each $N\\ge2$ there\nexists a bi-infinite sequence $\\{\\mu^N_k\\}$ such that when\n$\\mu=\\mu^N_k$ there is an $N$-pulse solution to (\\ref{eqperturbnlcqsode}).\nThe $N$-pulse is even in $x$.\nIn addition, \n\\[\n|\\mu^N_k-\\mu_h(\\epsilon)|=O(e^{-c|k|\/\\epsilon})\n\\]\nas $|k|\\to\\infty$.\n\\end{theorem}\n\n\\begin{remark} An $N$-pulse solution is constructed by\npiecing together $N$ copies of a bright solitary wave.\n\\end{remark}\n\n\\begin{remark} The work of Soto-Crespo et al\n\\cite{crespo:sot96} suggests that the 1-pulse solution\nis indeed stable as a solution to (\\ref{eqperturbnlcqs}).\n\\end{remark}\n\n\\begin{theorem}\\label{thmdsw} Suppose that $0<\\epsilon\\ll1$, and that\n$0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for some $n\\ge3$.\nFurther assume that the parameters satisfy\n\\[\n\\begin{array}{l}\n(\\tilde{d}_1+\\frac32A_0^2\\tilde{d}_2)\\tilde{\\sigma}<0 \\\\\n(\\tilde{d}_1+2A_0^2\\tilde{d}_2)\\tilde{\\sigma}<0 \\\\\n(\\tilde{d}_1+A_0^2\\tilde{d}_2)(\\tilde{d}_1+\\beta\\tilde{d}_2)>0,\n\\end{array}\n\\]\nwhere\n\\[\n\\beta=A_0^2+\\frac14A_0^2(-\\ln\\frac{\\eta}{A_0})^{-1}+O(\\eta^2)\n\\]\nfor $0<\\epsilon\\ll\\eta\\ll1$.\nThen there exists an $N(\\epsilon)>1$, with $N(\\epsilon)\\to\\infty$ as\n$\\epsilon\\to0$, such that $N$-circuit solutions exist for $1\\le N<N(\\epsilon)$.\n\\end{theorem}\n\n\\begin{remark} The suitable domain in parameter space is shown in\nFigure \\ref{fig_n_cirpar}.\nIn this figure the tildes have been dropped, $\\beta=\\alpha_2\/\\alpha_1$,\nand $x_0=A_0$.\n\\end{remark}\n\nThe rest of the paper is organized as follows.\nIn Section 2 the relevant three-dimensional ODE is set up, and the\nknown solution structure is discussed.\nIn Section 3 the relevant manifolds in the phase space are \ndescribed.\nThe bright, dark, and kink solitary waves are formed by\nintersecting the appropriate manifolds.\nThis section is the most technical, and can be skipped\non a first reading.\nSections 4 and 5 discuss the intersection of these manifolds.\nIn Section 6 the final proofs of the main theorems are given.\n\n\\vspace{3mm}\n\\noindent{\\bf Acknowledgements:} I would like to thank Alejandro\nAceves for introducing me to the cubic-quintic Schr\\\"odinger\nequation and some of the references pertaining to it.\n\n\n\\section{Setup of equations}\n\\setcounter{equation}{0}stable bright solitary\nwaves are possible for $\\mu$ near $\\mu^*$.\nThey do not consider the case of dark solitary waves in their\npaper, however.\n\nIn this section the first order system of ODEs that will be\nstudied will be derived.\nThe goal will be to write the system as compactly as possible so\nthat the notational burden of the succeeding sections will be\nminimized.\nDefine\n\\begin{equation}\\label{eqdeflambdaomega}\n\\begin{array}{lll}\n\\lambda(r)&=&\\mu^*+r^2+\\alpha r^4 \\\\\n\\omega(r)&=&\\tilde{d}_1r^2+\\tilde{d}_2r^4,\n\\end{array}\n\\end{equation}\nwhere\n\\begin{equation}\\label{eqdeftildedi}\n\\begin{array}{lll}\n\\tilde{d}_1&=&d_1-a_1 \\\\\n\\tilde{d}_2&=&d_2-a_1\\alpha,\n\\end{array}\n\\end{equation}\nand furthermore set\n\\begin{equation}\\label{eqdeftildesig}\n\\begin{array}{lll}\n\\tilde{\\sigma}&=&-(\\sigma-a_1\\mu) \\\\\n\\mu&=&\\mu^*-\\tilde{\\mu}\n\\end{array}\n\\end{equation}\n($\\mu^*$ is defined in (\\ref{eqdefmustar})).\nAfter setting \n\\[\nA(\\hat{x})=r(\\hat{x})e^{i\\int^{\\hat{x}}\\phi(s)\\,ds}\n\\]\nand substituting this expression into (\\ref{eqperturbnlcqsode})\none gets the third order system\n\\begin{equation}\\label{eqodetemp}\n\\begin{array}{lll}\nr'&=&ru \\\\\nu'&=&-u^2-\\lambda(r)+\\phi^2+\\tilde{\\mu} \\\\\n\\phi'&=&-2u\\phi+\\epsilon(\\omega(r)-\\tilde{\\sigma}).\n\\end{array}\n\\end{equation}\nIn the above equation the variable $u$ is defined by $u=r'\/r$.\n\nThe above equations will now be simplified notationally.\nFirst, redefine the variables $(\\hat{x},r,u,\\phi)$ by\n\\begin{equation}\\label{eqdefineodevariables}\n(t,x,y,z)=(\\hat{x},r,u,\\phi).\n\\end{equation}\nNext, remove the tildes from all the variables.\nThe ODE (\\ref{eqodetemp}) then becomes\n\\begin{equation}\\label{eqgloderedtemp}\n\\begin{array}{lll}\nx'&=&xy \\\\\ny'&=&-y^2+z^2-\\lambda(x)+\\mu \\\\\nz'&=&-2yz+\\epsilon(\\omega(x)-\\sigma),\n\\end{array}\n\\end{equation}\nwhere now $'=d\/dt$ and $\\omega(x)=d_1x^2+d_2x^4$.\nFor the subsequent analysis it will be useful to set\n\\[\n\\sigma=\\omega(x_0)+\\epsilon\\tilde\\sigma,\n\\]\nwhere $x_0$ is the largest positive root of $\\lambda(x)=0\\,(x_0^2=-3\/4\\alpha$).\nUpon substituting the above expression into (\\ref{eqgloderedtemp})\nand dropping the tilde, one obtains the equation\n\\begin{equation}\\label{eqglodered}\n\\begin{array}{lll}\nx'&=&xy \\\\\ny'&=&-y^2+z^2-\\lambda(x)+\\mu \\\\\nz'&=&-2yz+\\epsilon(\\omega(x)-\\omega(x_0)-\\epsilon\\sigma).\n\\end{array}\n\\end{equation}\n\nEquation (\\ref{eqglodered}) is the one which will be studied in \nthe subsequent sections.\nHowever, it is important to notice that the equation has the\nsymmetry given in the following proposition.\nThis symmetry will be used extensively in the upcoming\nsections.\n\n\\begin{prop}\\label{propsymmetry} Equation (\\ref{eqglodered}) \nremains invariant under\n\\[\n(t,x,y,z)\\to(-t,x,-y,-z).\n\\]\n\\end{prop}\n\nIn order to understand the behavior described in the succeeding \nsections, it is first necessary to understand the dynamics of\n(\\ref{eqglodered}) when $\\epsilon=0$.\nIn Figure \\ref{fig_z_0flow} the flow on the invariant (when $\\epsilon=0$)\nplane $\\{z=0\\}$ is shown for different values of $\\mu$.\nWhen $\\mu<0$ there exists a heteroclinic connection which connects\nthe invariant plane $\\{x=0\\}$ to itself.\nThis connection corresponds to a bright solitary wave.\nFor $\\mu=0$ there exists a connection between $x=0$ and $x=x_0$.\nThis solutions corresponds to a kink.\nDue to the symmetry described in Proposition \\ref{propsymmetry},\nthere also exists a connection between $x_0$ and zero.\nFinally, when $\\mu>0$ there exists a solution connecting $x_0$\nto itself, which corresponds to a dark solitary wave.\nAlthough it is not shown in the figure, this solution is\nactually embedded in the intersection of two two-dimensional\nmanifolds in the phase space.\n\nIn the upcoming sections it will occasionally be useful to have\nan analytic expression for the front (kink) which exists when \n$\\epsilon=\\mu=0$.\nThis will be especially true when attempting to gain information\non coefficients used in asymptotic expansions.\nFortunately, due to Marcq et al \\cite{marcq:eso94} such an\nexpression is available.\n\n\\begin{prop}\\label{propexactwave}  When $\\epsilon=\\mu=0$ there exists\na solution $(x(t),y(t),z(t))=(\\rho(t),u(t),\\phi(t))$ to \n(\\ref{eqglodered}) which is given by\n\\[\n\\begin{array}{lll}\n\\rho(t)&=&{\\displaystyle\\frac{x_0}{\\sqrt{2}}(1+\\tanh(\\frac{x_0}{2}t))^{1\/2}} \\\\\nu(t)&=&\\rho'(t)\/\\rho(t) \\\\\n\\phi(t)&=&0.\n\\end{array}\n\\]\nIn the above $x_0$ is the largest positive root of $\\lambda(x)=0$ and\nis given by\n\\[\nx_0^2=-\\frac{3}{4\\alpha}\n\\]\n(recall that $\\alpha\\sim-0.1$).\n\\end{prop}\n\nIn the subsequent sections the notation $p\\cdot t$ will be\nused to represent the trajectory of a point $p=(x,y,z)$ under\nthe flow generated by (\\ref{eqglodered}).\nUnder this convention, $p\\cdot0=p$.\nIn addition, given a point $p=(x,y,z)$, if it is of interest to\nonly study the behavior of one of the variables, say $x$, under\nthe flow, the notation $x\\cdot t$ will be used.\n\nWhen discussing the geometric objects in the following sections, the reader\nshould consult Figure \\ref{fig_geometry} to aid in the visualization.\n\n\n\n\n\n\n\n\\section{Geometric objects}\n\\setcounter{equation}{0}\n\nFor this section assume that $\\mu=\\mu(\\epsilon)$, with $\\mu(\\epsilon)=O(\\epsilon^2)$.\nThe validity of this assumption will be verified in Section 4.\n\n\\subsection{Flow near $\\{x=0\\}$}\n\nThe goal of this subsection is to characterized the flow near\nthe invariant plane $\\{x=0\\}$.\nFirst, there exists a pair of critical points, say\n$(0,y_\\pm(\\epsilon),z_\\pm(\\epsilon))$, which satisfy the algebraic equations\n\\begin{equation}\\label{eqdefypm}\n\\begin{array}{l}\n-y^2+z^2-\\lambda(0)+\\mu(\\epsilon)=0 \\\\\n-2yz-\\epsilon(\\omega(x_0)+\\epsilon\\sigma)=0.\n\\end{array}\n\\end{equation}\nIt is not difficult to check that\n\\[\ny_\\pm(0)=\\pm\\sqrt{-\\lambda(0)},\\quad z_\\pm(0)=0;\n\\]\nfurthermore, since $\\mu(\\epsilon)=O(\\epsilon^2)$,\n\\begin{equation}\\label{eqypmexpand}\n\\begin{array}{lll}\ny_\\pm(\\epsilon)&=&{\\displaystyle y_\\pm(0)+O(\\epsilon^2)} \\\\\nz_\\pm(\\epsilon)&=&{\\displaystyle-\\frac{\\omega(x_0)}{2y_\\pm(0)}\\epsilon+O(\\epsilon^2)}.\n\\end{array}\n\\end{equation}\nIt is also of interest to note that due to the symmetry described\nin Proposition \\ref{propsymmetry}, \n$y_-=-y_+<0,\\,z_-=-z_+$.\nFurthermore, upon recalling the expression for $x_0$ given in\nProposition \\ref{propexactwave}, it is not difficult to see\nthat $\\lambda(0)=-x_0^2\/4$, and hence $y_\\pm(0)=\\pm x_0\/2$.\n\nFor convenience, set\n\\[\n\\gamma_\\pm=y_\\pm(0),\\quad \\alpha_\\pm=-\\frac{\\omega(x_0)}{2y_\\pm(0)},\n\\]\nso that the critical points can be said to satisfy the asymptotic\nexpansion\n\\[\n\\begin{array}{lll}\ny_\\pm(\\epsilon)&=&\\gamma_\\pm+O(\\epsilon^2) \\\\\nz_\\pm(\\epsilon)&=&\\alpha_\\pm\\epsilon+O(\\epsilon^2).\n\\end{array}\n\\]\nLinearizing the vector field about the critical points\n$(0,y_\\pm,z_\\pm)$ yields the matrix\n\\begin{equation}\\label{eqdefApm}\nA_\\pm=\\left(\\begin{array}{ccc}\n   y_\\pm & 0 & 0 \\\\\n   0 & -2y_\\pm & 2z_\\pm \\\\\n   0 & -2z_\\pm & -2y_\\pm\n\t    \\end{array}\\right),\n\\end{equation}\nwhich has the eigenvalues\n\\begin{equation}\\label{eqdefApmevals}\n\\begin{array}{lll}\n\\lambda^1_\\pm&=&\\gamma_\\pm+O(\\epsilon^2) \\\\\n\\lambda^2_\\pm&=&-2\\gamma_\\pm+i\\alpha_\\pm\\epsilon+O(\\epsilon^2) \\\\\n\\lambda^3_\\pm&=&\\overline{\\lambda^2_\\pm}.\n\\end{array}\n\\end{equation}\n\nThus, the point $(0,y_+,z_+)$ has a one-dimensional unstable manifold,\n$W^u(0)$, coming out of the plane $\\{x=0\\}$ tangent to the vector\n$(1,0,0)$, and a two-dimensional stable manifold embedded in the\nplane $\\{x=0\\}$.\nIn addition, the point $(0,y_-,z_-)$ has a one-dimensional stable\nmanifold, $W^s(0)$, coming out of the invariant plane and tangent to\n$(1,0,0)$, and a two-dimensional unstable manifold embedded in\nthe invariant plane.\n\nGiven $\\delta>0$, set\n\\begin{equation}\\label{eqdefco}\n{\\cal C}_0=\\{(x,y,z)\\,:\\,0\\le x\\le\\delta,y=z=0\\}.\n\\end{equation}\nDue to the symmetry, this curve will be of paramount importance\nin subsequent sections.\nIn the following proposition the term $\\eta$ is assumed to be positive\nand small.\n\n\\begin{prop}\\label{propco} Let $p=(x,y,z)\\in{\\cal C}_0$ be\nsuch that $x=O(\\epsilon^n)$ for some $n\\ge1$.\nThen as long as $y_-+\\eta\\le y\\cdot t\\le y_+-\\eta,\\,z\\cdot t\\neq0$\nfor $t\\neq0$.\nFurthermore, $z\\cdot(-t)=-z\\cdot t$.\n\\end{prop}\n\n\\noindent{\\bf Proof: } By the nature of the function $\\omega(x)$ it\nis clear that for $x=O(\\epsilon^n),\\,\\omega(x)=O(\\epsilon^{2n})$.\nSince $x'=xy,\\,x\\cdot t=O(x\\cdot0)$ as long as \n$y_-+\\eta\\le y\\cdot t\\le y_+-\\eta$.\nThus, if $x\\cdot0=O(\\epsilon^n)$, then $x\\cdot t=O(\\epsilon^n)$ for $y\\cdot t$\nin the prescribed range. \nBy the above description of $\\omega(x)$ for $x$ small, this then\nimplies that for $x\\cdot0=O(\\epsilon^n)$,\n\\[\nz'=-2yz-\\epsilon(\\omega(x_0)+\\epsilon\\sigma)+O(\\epsilon^{2n+1}).\n\\]\nIf $n\\ge1$ it is now clear that trajectories can cross the plane\n$\\{z=0\\}$ at most once for $y_-+\\eta\\le y\\le y_+-\\eta$.\nThis proves the first part of the proposition.\nThe second part follows immediately from the symmetry inherent\nin (\\ref{eqglodered}).\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nIt is now of interest to determine the nature of the curve ${\\cal C}_0$\nas the flow carries it by the critical point $(0,y_+,z_+)$.\nUsing the matrix $A_+$ defined in (\\ref{eqdefApm}), the linear flow\nnear this critical point satisfies\n\\begin{equation}\\label{eqdeflinearcart}\n\\begin{array}{lll}\nx'&=&ax \\\\\ny'&=&-2ay+2\\epsilon bz \\\\\nz'&=&-2\\epsilon bz-2ay,\n\\end{array}\n\\end{equation}\nwhere\n\\[\na=\\gamma_++O(\\epsilon^2),\\quad b=\\alpha_++O(\\epsilon).\n\\]\nSetting $y=r\\sin\\theta,\\,z=r\\cos\\theta$ equation (\\ref{eqdeflinearcart}) can\nbe rewritten as\n\\begin{equation}\\label{eqdeflinearpolar}\n\\begin{array}{lll}\nx'&=&ax \\\\\nr'&=&-2ar \\\\\n\\theta'&=&\\epsilon b,\n\\end{array}\n\\end{equation}\nwhich has the solution\n\\begin{equation}\\label{eqdeflinearpolarsoln}\n\\begin{array}{lll}\nx(t)&=&x_0e^{at} \\\\\nr(t)&=&r_0e^{-2at} \\\\\n\\theta(t)&=&\\theta_0+\\epsilon bt.\n\\end{array}\n\\end{equation}\n\nFor each $p\\in{\\cal C}_0$ define $t_0(p)$ to be such that\n\\begin{equation}\\label{eqdeft0}\n{\\displaystyle t_0(p)=\\{\\inf_{t>0}\\,:\\,p\\cdot t\\cap\\{y_+-r=\\eta\\}\\neq\\emptyset\\}},\n\\end{equation}\nand set\n\\begin{equation}\\label{eqdefc0t0}\n{\\displaystyle{\\cal C}_0\\cdot t_0=\\bigcup_{p\\in{\\cal C}_0}p\\cdot t_0(p).}\n\\end{equation}\nAfter translating the critical point $(0,y_+,z_+)$ to the origin,\nthe curve ${\\cal C}_0\\cdot t_0$ can be written parametrically as\n\\[\n{\\cal C}_0\\cdot t_0=\\{(x,r,\\theta)\\,:\\,x=x_1(s),r=\\eta,\\theta=-\\pi\/2+\\theta_1(s)\\},\n\\]\nwhere $x_1(0)=0,\\,x_1'(s)>0,\\,x_1(s)\\le\\eta,$ and $\\theta_1(s)=O(\\epsilon)$.\n\nIt is of interest to determine the nature of ${\\cal C}_0\\cdot t_0$ as \nthe flow forces it to intersect the plane $\\{x=\\eta\\}$.\nGiven a point in ${\\cal C}_0\\cdot t_0$, the time of flight, $t_f$,\nis defined by $x(t_f)=\\eta$.\nSustitution of this expression into (\\ref{eqdeflinearpolarsoln})\nyields\n\\begin{equation}\\label{eqdeftf}\nt_f=\\frac1{a}\\ln\\frac{\\eta}{x_1(s)}.\n\\end{equation}\nThis expression for $t_f$ further yields that\n\\begin{equation}\\label{eqdefrtf}\n\\begin{array}{lll}\nr(t_f)&=&{\\displaystyle\\frac1{\\eta}x_1^2(s)} \\\\\n\\theta(t_f)&=&{\\displaystyle-\\frac{\\pi}2+\\theta_1(s)+\\epsilon\\frac{b}{a}\\ln\\frac{\\eta}{x_1(s)}}.\n\\end{array}\n\\end{equation}\nThe above curve is a logarithmic spiral centered upon the point at\nwhich $W^u(0)$ first intersects $\\{x=\\eta\\}$.\nIt is important to note that\n\\begin{equation}\\label{eqcospiral}\n\\begin{array}{lll}\nx_1(s)=O(\\epsilon^n)&\\Longrightarrow&{\\displaystyle\\theta(t_f)=-\\frac{\\pi}2+\\theta_1(s)+\n         O(n\\epsilon\\ln\\frac1{\\epsilon})} \\\\\nx_1(s)=O(e^{-c\/\\epsilon})&\\Longrightarrow&{\\displaystyle\\theta(t_f)=-\\frac{\\pi}2+\\theta_1(s)+\n         O(1)}.\n\\end{array}\n\\end{equation}\nThus, the spiralling effect is seen only for those points which\nare initially exponentially close to the plane $\\{x=0\\}$.\n\nDefine the transverse sections to $\\wu(0)$ and $\\ws(0)$ by\n\\begin{equation}\\label{eqdefbpm}\n\\begin{array}{lll}\nB^-_0&=&\\{(x,y,z)\\,:\\,x=\\eta,|y-y_-|\\le\\eta,|z-z_-|\\le\\eta\\} \\\\\nB^+_0&=&\\{(x,y,z)\\,:\\,x=\\eta,|y-y_+|\\le\\eta,|z-z_+|\\le\\eta\\},\n\\end{array}\n\\end{equation}\nwhere, as before, $\\eta>0$ is small.\nNow define\n\\begin{equation}\\label{eqdefmanint}\n\\begin{array}{lllll}\np^s_0&=&\\ws(0)\\capB^-_0&=&\\{(\\eta,y_-+O(\\eta),z_-+O(\\eta))\\} \\\\\np^u_0&=&\\wu(0)\\capB^+_0&=&\\{(\\eta,y_++O(\\eta),z_++O(\\eta))\\}, \n\\end{array}\n\\end{equation}\nwhere the above represents the first intersection of the manifold with\nthe set.\n\nAssume that $\\delta$ is sufficiently small so that for each $p\\in{\\cal C}_0$\nthere exists a $t>0$ such that $p\\cdot t\\capB^+_0\\neq\\emptyset$.\nFor each $p\\in{\\cal C}_0$ define\n\\begin{equation}\\label{eqdeftpo}\n{\\displaystylet^+_0(p)=\\{\\inf_{t>0}\\,:\\,p\\cdot t\\capB^+_0\\neq\\emptyset\\}},\n\\end{equation}\nand set $t^-_0(p)=-t^+_0(p)$.\nBy using a symmetry argument it can be shown that $p\\cdott^-_0(p)\\inB^-_0$.\nFinally, define the flow of ${\\cal C}_0$ as it intersects the sets\n$B^\\pm_0$ by\n\\begin{equation}\\label{eqdefcoflow}\n\\begin{array}{lll}\n{\\cal C}_0\\cdott^-_0&=&{\\displaystyle\\bigcup_{p\\in{\\cal C}_0}p\\cdott^-_0(p)} \\\\\n{\\cal C}_0\\cdott^+_0&=&{\\displaystyle\\bigcup_{p\\in{\\cal C}_0}p\\cdott^+_0(p)}.\n\\end{array}\n\\end{equation}\nWhen $\\epsilon=0$ these sets are given by\n\\begin{equation}\\label{eqcolocation}\n\\begin{array}{lll}\n{\\cal C}_0\\cdott^-_0&=&\\{(x,y,z)\\,:\\,x=\\eta,y_-\\le y\\le y_-+\\eta,z=0\\} \\\\\n{\\cal C}_0\\cdott^+_0&=&\\{(x,y,z)\\,:\\,x=\\eta,y_+-\\eta\\le y\\le y_+,z=0\\}.\n\\end{array}\n\\end{equation}\n\nThe blow-up sets will be defined next.\nThis set corresponds to points $p$ such that $|p\\cdot t|\\to\\infty$\nin finite time (\\cite{kapitula:sdo96}).\nAs a preliminary, set\n\\begin{equation}\\label{eqdefbpms}\n\\begin{array}{lll}\nB^-_{0,s}&=&\\{(x,y,z)\\,:\\,0\\le x\\le\\eta,y=y_--\\eta,|z-z_-|\\le\\eta\\} \\\\\nB^+_{0,s}&=&\\{(x,y,z)\\,:\\,0\\le x\\le\\eta,y=y_++\\eta,|z-z_+|\\le\\eta\\}.\n\\end{array}\n\\end{equation}\nFollowing Kapitula and Maier-Paape \\cite{kapitula:sdo96} there exist\nsets\n\\begin{equation}\\label{eqdefcpmi}\n\\begin{array}{lll}\n{\\cal C}_{-\\infty}&=&{\\displaystyle\\{p\\inB^-_{0,s}\\,:\\,\\exists\\,t(p)>0\\mbox{ such that }\\lim_{t\\to t^-(p)}\n       p\\cdot t=(0,-\\infty,0)\\}} \\\\\n{\\cal C}_{+\\infty}&=&{\\displaystyle\\{p\\inB^+_{0,s}\\,:\\,\\exists\\,t(p)<0\\mbox{ such that }\\lim_{t\\to t^+(p)}\n       p\\cdot t=(0,+\\infty,0)\\}}.\n\\end{array}\n\\end{equation}\nThe above sets are smooth curves, and symmetry considerations yield\nthat $(x,y,z)\\in{\\cal C}_{-\\infty}$ implies that $(x,-y,-z)\\in{\\cal C}_{+\\infty}$.\nFor $p\\in{\\cal C}_{+\\infty}$ set\n\\begin{equation}\\label{eqdeftpi}\nt_{+\\infty}(p)=\\{\\inf_{t>0}\\,:\\,p\\cdot t\\capB^+_0\\neq\\emptyset\\},\n\\end{equation}\nand for $p\\in{\\cal C}_{-\\infty}$ define\n\\begin{equation}\\label{eqdeftmi}\nt_{-\\infty}(p)=\\{\\sup_{t<0}\\,:\\,p\\cdot t\\capB^-_0\\neq\\emptyset\\}.\n\\end{equation}\nFinally, the flow of these sets will be given by\n\\begin{equation}\\label{eqdefcpmiflow}\n\\begin{array}{lll}\n{\\cal C}_{-\\infty}\\cdott_{-\\infty}&=&{\\displaystyle\\bigcup_{p\\in{\\cal C}_{-\\infty}}p\\cdott_{-\\infty}(p)} \\\\\n{\\cal C}_{+\\infty}\\cdott_{+\\infty}&=&{\\displaystyle\\bigcup_{p\\in{\\cal C}_{+\\infty}}p\\cdott_{+\\infty}(p)}.\n\\end{array}\n\\end{equation}\nWhen $\\epsilon=0$ these sets are given by\n\\begin{equation}\\label{eqcpilocation}\n\\begin{array}{lll}\n{\\cal C}_{-\\infty}\\cdott_{-\\infty}&=&\\{(x,y,z)\\,:\\,x=\\eta,y_--\\eta<y\\le y_-,z=0\\} \\\\\n{\\cal C}_{+\\infty}\\cdott_{+\\infty}&=&\\{(x,y,z)\\,:\\,x=\\eta,y_+<y\\le y_+-\\eta,z=0\\}.\n\\end{array}\n\\end{equation}\n\nThis subsection can now be concluded with the following lemma,\nwhose conclusion follows from the above discussion.\nThe fact that ${\\cal C}_{-\\infty}\\cdott_{-\\infty}$ and ${\\cal C}_{+\\infty}\\cdott_{+\\infty}$ are logarithmic\nspirals follows from the same argument as that which led to\nthe description of ${\\cal C}_0\\cdott^-_0$ and ${\\cal C}_0\\cdott^+_0$.\n\n\\begin{lemma}\\label{lemspirals} Suppose that $\\epsilon\\omega(x_0)\\neq0$.\nThen ${\\cal C}_0\\cdott^-_0$ and ${\\cal C}_{-\\infty}\\cdott_{-\\infty}$ are logarithmic spirals\ncentered on $p^s_0$, and the curves ${\\cal C}_0\\cdott^+_0$ and ${\\cal C}_{-\\infty}\\cdott_{+\\infty}$\nare logarithmic spirals centered on $p^u_0$.\nThe spiralling is seen only $O(e^{-c\/\\epsilon})$ close to the points\n$p^s_0$ and $p^u_0$; furthermore, their location is described within\n$O(\\epsilon)$ by equations (\\ref{eqcolocation}) and (\\ref{eqcpilocation}).\n\\end{lemma}\n\n\\begin{remark} By the symmetry described in Proposition \\ref{propsymmetry}\n it is necessarily true that\n$(\\eta,y,z)\\in{\\cal C}_0\\cdott^-_0$ implies that $(\\eta,-y,-z)\\in{\\cal C}_0\\cdott^+_0$,\nand $(\\eta,y,z)\\in{\\cal C}_{-\\infty}\\cdott_{-\\infty}$ implies that \n$(\\eta,-y,-z)\\in{\\cal C}_{+\\infty}\\cdott_{+\\infty}$.\n\\end{remark}\n\n\\subsection{Flow near $\\M_\\ep$}\n\nThe purpose of this subsection is to describe the flow generated\nby (\\ref{eqglodered}) near the point $(x_0,0,0)$, where $x_0$\nis the largest positive zero of $\\lambda(x)=0$.\n\nWhen $\\epsilon=0$ there exists a critical manifold $\\M_0$ which is given by\n\\begin{equation}\\label{eqm0}\n\\M_0=\\{(x,y,z)\\,:\\,y=0,\\,z^2-\\lambda(x)=0,\\,\\,4z^2+x\\lambda'(x)<0\\}.\n\\end{equation}\nThis manifold is normally hyperbolic, and therefore smoothly perturbs\nto a manifold $\\M_\\ep$ for $\\epsilon$ sufficiently small. \nFurthermore, the manifold $\\M_\\ep$ has a two-dimensional stable\nmanifold, $\\ws(\\Me)$, and a two-dimensional unstable manifold,\n$\\wu(\\Me)$, which are smooth perturbations of the center-stable\nand center-unstable manifolds which exist when $\\epsilon=0$\n(Fenichel \\cite{fe:73}, Jones \\cite{jones:gsp95}).\n\nAs in the previous section, assume that $\\mu=\\mu(\\epsilon)=O(\\epsilon^2)$.\nThe critical points for (\\ref{eqglodered}) which lie on $\\M_\\ep$ \nsatisfy $y=0$ and\n\\begin{equation}\\label{eqcriticalpts}\n\\begin{array}{l}\n\\lambda(x)-z^2+O(\\epsilon^2)=0 \\\\\n\\omega(x)-\\omega(x_0)-\\epsilon\\sigma=0.\n\\end{array}\n\\end{equation}\nA Taylor expansion about the point $x_0$ yields that\n\\begin{equation}\\label{eqcritpertom}\nx-x_0=\\epsilon\\frac1{\\omega'(x_0)}+O(\\epsilon^2).\n\\end{equation}\nSubstituting the expression in (\\ref{eqcritpertom}) into the first\nequation of (\\ref{eqcriticalpts}) and taking a Taylor expansion\nyields\n\\begin{equation}\\label{eqcritpertla}\n\\epsilon\\frac{\\lambda'(x_0)}{\\omega'(x_0)}-z^2+O(\\epsilon^2)=0,\n\\end{equation}\nfrom which one gets that\n\\begin{equation}\\label{eqzcritpts}\nz^2=\\epsilon\\frac{\\lambda'(x_0)}{\\omega'(x_0)}+O(\\epsilon^2).\n\\end{equation}\nThis in turn yields\n\\begin{equation}\nz=\\pm\\sqrt{\\lambda'(x_0)\\frac{\\sigma}{\\omega'(x_0)}}\\epsilon^{1\/2}+O(\\epsilon).\n\\end{equation}\nLetting $(x^*(\\epsilon),0,\\pm z^*(\\epsilon))\\in\\M_\\ep$ represent the critical points,\nit is now seen that these points have the asymptotic expansion\n\\begin{equation}\\label{eqdefxszs}\n\\begin{array}{lll}\nx^*(\\epsilon)&=&{\\displaystyle x_0+\\frac{\\sigma}{\\omega'(x_0)}\\epsilon+O(\\epsilon^2)} \\\\\nz^*(\\epsilon)&=&{\\displaystyle\\sqrt{\\lambda'(x_0)\\frac{\\sigma}{\\omega'(x_0)}}\\epsilon^{1\/2}+O(\\epsilon)}.\n\\end{array}\n\\end{equation}\nSince $\\lambda'(x_0)<0$, in order for the above expressions to make \nsense the following assumption must hold.\n\n\\begin{prop}\\label{asssig_1} If the parameters $d_1,d_2$, and $\\sigma$\nare chosen so that\n\\[\n\\omega'(x_0)\\sigma<0,\n\\]\ni.e.,\n\\[\n(d_1+2x_0^2d_2)\\sigma<0,\n\\]\nthen there exist critical points on $\\M_\\ep$ whose $x$ and $z$ components\nare given by (\\ref{eqdefxszs}).\n\\end{prop}\n\n\\input{slowman}\n\n\n\\section{$W^u(0)\\cap\\ws(\\Me)$ transversely}\n\\setcounter{equation}{0}\n\nAfter appending $\\mu'=0$ and $\\epsilon'=0$ to (\\ref{eqglodered})\none arrives at a three-dimensional manifold $\\wuo$ (abusing notation\nhere) and a\nfour-dimensional manifold $\\wsme$ in the five-dimensional phase\nspace.\nSince $x'>0$ along the relevant trajectory for $0<x<x_0$, by \ncontinuity this holds for $\\epsilon$ and $\\mu$ sufficiently small.\nAs such, for $0<x<x_0$ the manifolds can be parameterized in \nthe following manner:\n\\begin{equation}\\label{eqdeffolman}\n\\begin{array}{lll}\nW^u(0)&=&\\{(x,y,z)\\,:\\,y={\\cal U}^y(x,\\mu,\\epsilon),z={\\cal U}^z(x,\\mu,\\epsilon)\\} \\\\\n\\ws(\\Me)&=&\\{(x,y,z)\\,:\\,y={\\cal S}^y(x,z,\\mu,\\epsilon),|z|\\le\\tilde\\epsilon\\ll1\\}.\n\\end{array}\n\\end{equation}\nA heteroclinic orbit possibly exists when $\\wuo\\cap\\wsme$ nontrivially, or \nequivalently when the function\n\\begin{equation}\\label{eqdeffunintersect}\nF(x,\\mu,\\epsilon)={\\cal S}^y(x,{\\cal U}^z(x,\\mu,\\epsilon),\\mu,\\epsilon)-{\\cal U}^y(x,\\mu,\\epsilon)\n\\end{equation}\nhas a zero.\n\nIt is clear that $F(x,0,0)=0$.\nIf $F_\\mu(x,0,0)\\neq0$ for some value of $x\\in(0,x_0)$, then an application\nof the implicit function theorem yields that there exists a smooth\nfunction $\\mu=\\mu(\\epsilon)$ such that $F(x,\\mu(\\epsilon),\\epsilon)=0$.\nIn other words, the condition that $F_\\mu\\neq0$ gives that the manifolds\n$\\wuo$ and $\\wsme$ intersect transversely in $(x,y,z,\\mu)$-space, so that \ntheir intersection persists for small perturbation.\nFurthermore, the implicit function theorem yields that\n\\begin{equation}\\label{eqmuprime}\n\\mu'(0)=-\\frac{F_\\epsilon(x,0,0)}{F_\\mu(x,0,0)}.\n\\end{equation}\nThus, it is possible to get asymptotics for the function $\\mu(\\epsilon)$ by\nthoroughly understanding the manner in which the manifolds intersect.\n\nDifferentiating (\\ref{eqdeffunintersect}) yields that\n\\begin{equation}\\label{eqdeffmutemp}\nF_\\mu(x,0,0)=({\\cal S}^y_\\mu-{\\cal U}^y_\\mu+{\\cal S}^y_z{\\cal U}^z_\\mu)(x,0,0).\n\\end{equation}\nAn examination of (\\ref{eqglodered}) yields that when $\\epsilon=0$ it\nis unchanged under the transformation $z\\to-z$; therefore, \n\\begin{equation}\\label{eqsy=0}\n{\\cal S}^y_z(x,0,\\mu,0)=0.\n\\end{equation}\nThis gives the simplified expression of $F_\\mu$ to be\n\\begin{equation}\\label{eqdeffmu}\nF_\\mu(x,0,0)={\\cal S}^y_\\mu(x,0,0,0)-{\\cal U}^y_\\mu(x,0,0).\n\\end{equation}\nIn a similiar manner, it is seen that\n\\begin{equation}\\label{eqdeffep}\nF_\\epsilon(x,0,0)={\\cal S}^y_\\epsilon(x,0,0,0)-{\\cal U}^y_\\epsilon(x,0,0).\n\\end{equation}\n\nLet the heteroclinic orbit which exists for $\\mu=\\epsilon=0$ be denoted\nby $(\\rho(t),u(t),0)$.\nAssume that the wave has been translated so that $\\rho(0)=x_1$,\nwhere $0<x_1<x_0$ is the other root of $\\lambda(x)=0$. \nWhen linearizing (\\ref{eqglodered}) about this orbit the variational\nequations take the form\n\\begin{equation}\\label{eqodevar}\n\\begin{array}{lll}\n\\delta x'&=&u\\delta x+\\rho\\delta y \\\\\n\\delta y'&=&-\\lambda'(\\rho)\\delta x-2u\\delta y+\\delta\\mu \\\\\n\\delta z'&=&-2u\\delta z+(\\omega(\\rho)-\\omega(x_0))\\delta\\epsilon \\\\\n\\delta\\mu'&=&0 \\\\\n\\delta\\epsilon'&=&0.\n\\end{array}\n\\end{equation}\nEach of the coordinates $\\delta x_i$ of (\\ref{eqodevar}) ($x_i=x,y$, etc.)\nis a 1-form.\nFrom these 1-forms exterior products of forms of any degree $k$ can be \nconstructed.\nThe rest of the following discussion will only be concerned\nwith 2-forms $P_{x_ix_j}=\\delta x_i\\wedge\\delta x_j$; however, it can\nbe generalized to any $k$.\n\nEach 2-form associates to a 2-plane $T$ a number that is the\narea of the projection of a unit square of $T$ onto the coordinate planes\nof the two coordinates specified by $P_{x_ix_j}$.\nIn order to evaluate $P_{x_ix_j}$, the following rule is applied.\nLet $N$ represent a $k$-dimensional manifold, $k\\ge2$, and let $T_pN$\nrepresent the tangent space to $N$ at a point $p\\in N$.\nLet $\\{a_i(p)\\}$ represent a basis for $T_pN$.\nSuppose that $S(p)\\subset T_p(n)$ represents a two-dimensional subspace\nwhich is spanned by the vectors $\\{a_i(p),a_j(p)\\}$ for some\n$1\\le i,j\\le k$.\nThese vectors may be thought of as rows to a $2\\times k$ matrix, $A(p)$.\nIf $x_i$ and $x_j$ represent two of the $k$ coordinates of $T_pN$, then\n$P_{x_ix_j}$ is the determinant of the $2\\times2$ submatrix obtained by\nlooking at the $x_i^{\\scriptscriptstyle{\\mbox{th}}}$ and \n$x_j^{\\scriptscriptstyle{\\mbox{th}}}$ columns of  $A(p)$.\nFurthermore, the evolution equation for $P_{x_ix_j}$ is given by\nthe product rule, i.e., \n\\[\nP'_{x_ix_j}=\\delta x'_i\\wedge\\delta x_j+\\delta x_i\\wedge\\delta x'_j.\n\\]\n\n\\begin{lemma}\\label{lemuymu} For any $0<x<x_0,\\,{\\cal U}^y_\\mu(x,0,0)>0$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } Set $P_{xy}=\\delta x\\wedge\\delta y$ and $P_{x\\mu}=\\delta x\\wedge\\delta\\mu$.\nUsing the equations of variation (\\ref{eqodevar}) it is seen that\n\\[\nP_{xy}'=-uP_{xy}+uP_{x\\mu}.\n\\]\nTwo vectors which are tangent to the manifold $W^u(0)$ at a fixed $x$\nvalue are\n\\[\n\\xi_1=(\\rho',u',0,0,0)^T,\\quad\\xi_2=(0,{\\cal U}^y_\\mu,0,1,0)^T.\n\\]\nIn the above the fact that ${\\cal U}^z(x,\\mu,0)=0$ is implicitly used.\nWhen applied to these vectors\n\\[\nP_{x\\mu}(\\xi_1,\\xi_2)=\\rho',\n\\]\nso that the equation for $P_{xy}$ becomes\n\\[\nP_{xy}'=-uP_{xy}+\\rho u^2.\n\\]\nSince $u=\\rho'\/\\rho$, this equation has the solution\n\\[\n\\rho(t)P_{xy}(t)=\\int_{-\\infty}^t\\rho^2(s)u^2(s)\\,ds,\n\\]\nso that $P_{xy}(t)>0$ for all $t$.\nSince for a fixed $x$ value\n\\[\nP_{xy}(\\xi_1,\\xi_2)\\propto\\rho'{\\cal U}^y_\\mu(x,0,0),\n\\]\nthe fact that $\\rho'>0$ yields the conclusion of the lemma.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\n\\begin{lemma}\\label{lemsymu} For any $0<x<x_0,\\,{\\cal S}^y_\\mu(x,0,0,0)<0$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } The proof follows that of the previous lemma, and will therefore\nonly be sketched.\nSet\n\\[\n\\xi_1=(\\rho',u',0,0,0)^T,\\quad\\xi_2=(0,{\\cal S}^y_\\mu,0,1,0)^T.\n\\]\nThe equation for $P_{xy}$, when applied to these two vectors,\nbecomes\n\\[\nP_{xy}'=-uP_{xy}+\\rho u^2,\n\\]\nwhich has the solution\n\\[\n\\rho(t)P_{xy}(t)=-\\int_t^{\\infty}\\rho^2(s)u^2(s)\\,ds.\n\\]\nSince for any fixed $x$\n\\[\nP_{xy}(\\xi_1,\\xi_2)\\propto\\rho'{\\cal S}^y_\\mu(x,0,0,0),\n\\]\nthe conclusion immediately follows.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\n\\begin{cor}\\label{corfmu} For any $0<x<x_0,\\,F_\\mu(x,0,0)<0$.\n\\end{cor}\n\n\\noindent{\\bf Proof: } The result follows immediately from the above lemmas and\nequation (\\ref{eqdeffmu}).\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\n\\begin{lemma}\\label{lemuyep} ${\\cal U}^y_\\epsilon(x,0,0)=0$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } The proof follows that of Lemma \\ref{lemuymu}, and will \ntherefore again only be sketched.\nSet \n\\[\n\\xi_1=(\\rho',u',0,0,0)^T,\\quad\\xi_2=(0,{\\cal U}^y_\\epsilon,0,0,1)^T.\n\\]\nWhen applied to these two vectors, the equation for $P_{xy}$ is\n\\[\nP_{xy}'=-uP_{xy},\n\\]\nwhich has the solution\n\\[\nP_{xy}(t)=\\frac{C}{\\rho(t)}.\n\\]\nSince $\\lim_{t\\to-\\infty}P_{xy}(t)$ must be bounded, the fact that\n$\\rho(t)\\to0$ as $t\\to-\\infty$ implies that the constant $C$ must\nbe zero.\nSince for fixed $x$\n\\[\nP_{xy}(\\xi_1,\\xi_2)\\propto\\rho'{\\cal U}^y_\\epsilon(x,0,0),\n\\]\nthe conclusion of the lemma now follows.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nThe proof of the next lemma mimics those of the previous three lemmas, and\nwill therefore be left to the reader.\nAll that must be kept in mind is that the quantity $P_{xy}(t)$ must\napproach zero as $t\\to\\infty$, as $\\xi_1(t)\\to0$ and the vector\n$\\xi_2(t)$ remains bounded as $t\\to\\infty$.\n\n\\begin{lemma}\\label{lemsyep} ${\\cal S}^y_\\epsilon(x,0,0,0)=0$.\n\\end{lemma}\n\n\\begin{cor}\\label{corfep} $F_\\epsilon(x,0,0)=0$.\n\\end{cor} \n\n\\noindent{\\bf Proof: } The conclusion follows immediately from the previous two\nlemmas and equation (\\ref{eqdeffep}).\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nAs a consequence of Corollaries \\ref{corfmu} and \\ref{corfep}, this\nsection can be closed with the following theorem.\n\n\\begin{theorem}\\label{thmmanintersect} There exists a smooth function \n$\\mu=\\mu(\\epsilon)$ such\nthat when $0\\le\\epsilon\\ll1$ the manifolds $\\ws(\\Me)$ and $\\wu(0)$ have a\nnontrivial intersection.\nFurthermore, the function $\\mu(\\epsilon)$ satisfies\n\\[\n\\mu(0)=\\mu'(0)=0,\n\\]\nso that\n\\[\n\\mu(\\epsilon)=O(\\epsilon^2)\n\\]\n\\end{theorem}\n\n\\begin{remark} Due to the symmetries present in (\\ref{eqglodered}), when\n$\\mu=\\mu(\\epsilon)$ it is necessarily true that $W^s(0)\\cap\\wu(\\Me)$ is also\nnontrivial.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{$\\wu(\\Me)\\cap\\ws(\\Me)$ transversely}\n\\setcounter{equation}{0}\n\n\\subsection{Flow from $\\M_\\ep$ to $\\{x=\\eta\\}$}\n\nAssume that $\\mu=\\mu(\\epsilon)=\\mu_2\\epsilon^2+O(\\epsilon^3)$.\nFrom the results of Section 4 it is then known that \n$W^u(0)\\cap\\ws(\\Me)$ is nontrivial, as is $W^s(0)\\cap\\wu(\\Me)$.\nFurthermore, on $\\M_\\ep$ there is a connection between the\ncritical points $(x^*(\\epsilon),0,\\pm z^*(\\epsilon))$.\nSince there exists a connection between $(0,y_-,z_-)$ and\n$(0,y_+,z_+)$ in the plane $\\{x=0\\}$, it is clear that\nwhen $\\mu=\\mu(\\epsilon)$ there is a heteroclinic cycle in\nthe three-dimensional phase space.\nThe goal of this section is to explore the possible solution set\nas $\\mu$ is varied from $\\mu(\\epsilon)$.\n\nInitially, the focus will be on the intersection of $W^u(0)$ and\n$\\ws(\\Me)$; however, this is not a restriction in any way, as the\nsymmetry described in Proposition \\ref{propsymmetry} immediately\nyields similiar results for the intersection of $W^s(0)$ and\n$\\wu(\\Me)$.\nThe heteroclinic wave is singular; however, it can be written in a\nregular perturbation expansion for $0\\le x\\le x_0-\\nu$.\nTo set notation, for $0\\le x\\le x_0-\\nu$ write the wave\nas\n\\begin{equation}\\label{eqwaveexpand}\n\\begin{array}{lll}\n\\rho(t)&=&\\rho_0(t)+O(\\epsilon) \\\\\nu(t)&=&u_0(t)+O(\\epsilon) \\\\\n\\phi(t)&=&\\epsilon\\phi_1(t)+O(\\epsilon^2).\n\\end{array}\n\\end{equation}\nThe following lemma gives a characterization for $\\phi_1$.\n\n\\begin{lemma}\\label{lemphiexpand} For $t\\in(-\\infty,\\infty)$ the\nterm $\\phi_1$ is given by\n\\[\n\\phi_1(t)=\\beta_1(t)d_1+\\beta_2(t)d_2,\n\\]\nwhere \n\\[\n\\begin{array}{lll}\n\\beta_1(t)&=&-x_0 \\\\\n\\beta_2(t)&=&{\\displaystyle-\\frac14x_0^3(5+\\tanh(\\frac{x_0}{2}t))}.\n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{remark} Note that \n\\[\n\\frac{\\beta_1(t)}{\\beta_2(t)}>\\frac1{2x_0^2}\n\\]\nfor each $t$.\n\\end{remark}\n\n\\begin{remark} Although it is permissable to use $\\phi_1$ when\n$t\\in(-\\infty,T_\\nu]$, where $\\rho(T_\\nu)=x_0-\\nu$, note \nthat $\\beta_1(t)$ and $\\beta_2(t)$ are well-defined for all $t$.\n\\end{remark}\n\n\\noindent{\\bf Proof: } After substituting the expansion (\\ref{eqwaveexpand}) into\nthe ODE (\\ref{eqglodered}) and equating terms it is seen that\n$\\phi_1$ satisfies\n\\[\n\\phi_1'=-2u_0\\phi_1+\\omega(\\rho_0)-\\omega(x_0).\n\\]\nSince $u_0=\\rho_0'\/\\rho_0$, the above equation clearly has the\nsolution\n\\[\n\\rho_0^2(t)\\phi_1(t)=\\int_{-\\infty}^t\\rho_0^2(s)\n     (\\omega(\\rho_0(s))-\\omega(x_0))\\,ds.\n\\]\nThus, after using that $\\omega(a)=d_1a^2+d_2a^4$ one finds that\n\\[\n\\phi_1(t)=\\beta_1(t)d_1+\\beta_2(t)d_2,\n\\]\nwhere\n\\[\n\\beta_i(t)=\\frac1{\\rho_0^2(t)}\\int_{-\\infty}^t\\rho_0^2(s)\n     (\\rho_0^{2i}(s)-x_0^{2i})\\,ds.\n\\]\nSubstituting the expression for $\\rho_0(t)$ given in Proposition\n\\ref{propexactwave} into the above expression gives the \nresult of the lemma.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\begin{cor}\\label{corzlocation} When $\\epsilon\\neq0$ the function\n$\\phi(t)$ satisfies\n\\[\n\\begin{array}{lll}\n\\phi(T_\\eta)&=&-x_0(d_1+x_0^2d_2+O(\\eta))\\epsilon+O(\\epsilon^2) \\\\\n\\phi(T_\\nu)&=&{\\displaystyle-x_0(d_1+\\frac32x_0^2d_2+O(\\nu))\\epsilon+O(\\epsilon^2)}.\n\\end{array}\n\\]\n\\end{cor}\n\n\\noindent{\\bf Proof: } The result follows immediately from the expansion of the\nwave given in (\\ref{eqwaveexpand}) and the above lemma.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nIt is of interest to understand the behavior of $\\ws(\\Me)$ as it intersects\nthe section $B^+_0$.\nFor fixed $\\epsilon$ the intersection of $\\ws(\\Me)$ with $B^+_0$ forms a\ncurve $y={\\cal S}^y(\\eta,z,\\epsilon)$.\nSetting $T_\\eta$ to be such that $\\rho(T_\\eta)=\\eta$, it will be\ndesirable to determine ${\\cal S}^y_z(\\eta,\\phi(T_\\eta),\\epsilon)$.\nIn order to find this quantity, the tangent space to the manifold\nmust be tracked.\n\nIn the four-dimensional phase space the manifold $\\ws(\\Me)$ is \nthree-dimensional and can be written as the graph\n\\begin{equation}\\label{eqwsmegraph}\n\\ws(\\Me)=\\{(x,y,z,\\epsilon)\\,:\\,\\eta\\le x\\le x_0,y={\\cal S}^y(x,z,\\epsilon)\\}.\n\\end{equation}\nOver the underlying wave, for each fixed $x$ the manifold's tangent\nspace is spanned by the three vectors\n\\begin{equation}\\label{eqwsmetangent}\n\\begin{array}{lll}\n\\xi_1&=&(\\rho',u',\\phi',0)^T \\\\\n\\xi_2&=&(0,{\\cal S}^y_z,1,0)^T \\\\\n\\xi_3&=&(0,{\\cal S}^y_\\epsilon,0,1)^T.\n\\end{array}\n\\end{equation}\nSince $\\ws(\\Me)$ is a three-dimensional manifold, it will be \nnecessary to use three-forms to properly track its tangent\nspace.\nIn particular, the forms \n\\[\nP_{xy\\epsilon}=\\delta x\\wedge\\delta y\\wedge\\delta\\epsilon\n\\]\nand \n\\[\nP_{xz\\epsilon}=\\delta x\\wedge\\delta z\\wedge\\delta\\epsilon\n\\]\nwill be used.\nNote that for fixed $x$\n\\[\nP_{xy\\epsilon}(\\xi_1,\\xi_2,\\xi_3)=\\rho'{\\cal S}^y_z,\n\\]\nand in particular\n\\begin{equation}\\label{eqsyzratio}\n{\\cal S}^y_z(\\eta,\\phi(T_\\eta),\\epsilon)\\propto\\frac{P_{xy\\epsilon}(T_\\eta)}{\\rho'(T_\\eta)}.\n\\end{equation}\nThus, it will be desirable to compute $P_{xy\\epsilon}$ at $x=\\eta$.\n\nAfter linearizing about the wave for $\\epsilon\\neq0$ the variational\nequations become\n\\begin{equation}\\label{eqvariationalepneq0}\n\\begin{array}{lll}\n\\delta x'&=&u\\delta x+\\rho\\delta y \\\\\n\\delta y'&=&-\\lambda'(\\rho)\\delta x-2u\\delta y+2\\phi\\delta z+(2\\mu_2\\epsilon+O(\\epsilon^2))\\delta\\epsilon \\\\\n\\delta z'&=&\\epsilon\\omega'(\\rho)\\delta x-2\\phi\\delta y-2u\\delta z+\n           (\\omega(\\rho)-\\omega(x_0)-\\epsilon\\sigma)\\delta\\epsilon \\\\\n\\delta\\epsilon'&=&0.\n\\end{array}\n\\end{equation}\nThe ODE for $P_{xy\\epsilon}$ is then\n\\begin{equation}\\label{eqpxyep}\nP_{xy\\epsilon}'=-uP_{xy\\epsilon}+2\\phi P_{xz\\epsilon},\n\\end{equation}\nwhich, using the fact that $u=\\rho'\/\\rho$, has the solution\n\\begin{equation}\\label{eqpxyepsol}\n\\rho(t)P_{xy\\epsilon}(t)=(x_0-\\nu)P_{xy\\epsilon}(T_\\nu)\n    -2\\int_t^{T_\\nu}\\rho(s)\\phi(s)P_{xz\\epsilon}(s)\\,ds.\n\\end{equation}\n\nIt is clear that the evolution of $P_{xz\\epsilon}$ must be understood in\norder to understand the evolution of $P_{xy\\epsilon}$.\nTowards this end is the following proposition.\n\n\\begin{prop}\\label{propxi2} When $\\epsilon=0$,\n\\[\nP_{xz\\epsilon}(t)=x_0^2\\frac{u_0(t)}{\\rho_0(t)}.\n\\]\n\\end{prop}\n\n\\noindent{\\bf Proof: } In order to prove the proposition, it will first be shown\nthat the vector $\\xi_2$ is given by\n\\[\n\\xi_2(t)=(0,0,x_0^2\/\\rho_0^2(t),0)^T.\n\\]\nRecall that the vector $\\xi_2$ is formed by taking the\nderivative of the manifold $\\ws(\\Me)$ with respect to $z$.\nSince the flow has the symmetry $(x,y,z,t)\\to(x,y,-z,t)$ when\n$\\epsilon=0$, it is clear that the first two components of $\\xi_2$\nmust be zero.\nFrom the variational equations the third component satisfies the\nODE\n\\[\n\\delta z'=-2u_0\\delta z,\n\\]\nwhich has the solution\n\\[\n\\delta z(t)=C\/\\rho_0^2(t).\n\\]\nThe constant $C$ can be chosen so that \n\\[\n\\lim_{t\\to\\infty}\\delta z(t)=1,\n\\]\nfrom which arises the characterization of $\\xi_2(t)$.\n\nSince $\\xi_1(t)=(\\rho_0'(t),u_0'(t),0,0)$, the conclusion of\nthe proposition now immediately follows, as\n\\[\nP_{xz\\epsilon}(\\xi_1,\\xi_2,\\xi_3)=\\left|\n   \\begin{array}{ccc}\n     \\rho_0' & 0 & * \\\\\n     0 & x_0^2\/\\rho_0^2 & * \\\\\n     0 & 0 & 1\n   \\end{array}\\right|.\\quad\\quad\\rule{1.5mm}{1.5mm}\n\\]\n\n\\vspace{3mm}\nRecall that $T_\\nu$ and $T_\\eta$ are defined by\n\\[\n\\rho(T_\\eta)=\\eta,\\quad\\rho(T_\\nu)=x_0-\\nu.\n\\]\nFor $t\\in[T_\\eta,T_\\nu]$ the three-form $P_{xz\\epsilon}$ is given by\nthe regular perturbation expansion\n\\begin{equation}\\label{eqpxzep}\nP_{xz\\epsilon}(t)=x_0^2\\frac{u_0(t)}{\\rho_0(t)}+O(\\epsilon).\n\\end{equation}\nSubstitution of this expression into (\\ref{eqpxyepsol}) along\nwith the expansion for $\\phi$ given in Lemma \\ref{lemphiexpand}\nyields that\n\\begin{equation}\\label{eqpxyepsolexpand}\n\\rho_0(t)P_{xy\\epsilon}(t)=(x_0-\\nu)P_{xy\\epsilon}(T_\\nu)\n  -2\\epsilon x_0^2\\int_t^{T_\\nu}u_0(s)(\\beta_1(s)d_1+\\beta_2(s)d_2)\\,ds+O(\\epsilon^2).\n\\end{equation}\nIn order to finish the calculation, $P_{xy\\epsilon}(T_\\nu)$ must be\ndetermined.\n\nAs it has already been seen,\n\\[\nP_{xy\\epsilon}(T_\\nu)=\\rho'(T_\\nu){\\cal S}^y_z(x_0-\\nu,\\phi(T_\\nu),\\epsilon);\n\\]\nthus, the quantity ${\\cal S}^y_z(x_0-\\nu,\\phi(T_\\nu),\\epsilon)$ must be\ncalculated.\nSince $\\nu$ is small, this can be determined by understanding\nthe variation of the linear approximation to the manifold\n$\\ws(\\Me)$.\nSince $\\M_\\ep\\subset\\ws(\\Me)$ it is necessarily true that\n\\begin{equation}\\label{eqmesubsetwsme}\n{\\cal S}^y(\\M_x(z,\\epsilon),z,\\epsilon)=\\M_y(z,\\epsilon).\n\\end{equation}\nRecall that Lemma \\ref{lemdescribeme} states that the\nmanifold $\\M_\\ep$ satisfies\n\\[\n\\partial_z{\\cal M}_y(z,\\epsilon)=O(\\zeta^2).\n\\]\nAs such, due to (\\ref{eqmesubsetwsme}) the slow manifold has no effect \non $\\ws(\\Me)$, at least up to $O(\\epsilon)$.\nTherefore, in order to understand the variation of $\\ws(\\Me)$ it\nit enough to determine how the vectors which are tangent to\n$\\ws(\\Me)$ at $\\M_\\ep$ vary with respect to $z$ and $\\epsilon$.\nRecall Proposition \\ref{propevectsme}, which states that the tangent\nvectors to $\\ws(\\Me)$ at $\\M_\\ep$ are given by\n\\[\n(-1,-\\frac{v_2}{v_1},-\\frac{v_3}{v_1}),\n\\]\nwhere the quantities $v_i$ are defined in Proposition \\ref{propevectsme}.\nIn order to determine ${\\cal S}^y_z$, it is then enough to calculate\n$\\partial_z(v_2\/v_1)$, where these quantities are defined in\nProposition \\ref{propevectsme}.\nA simple calculation yields\n\\[\n-\\partial_z\\frac{v_2}{v_1}=\\epsilon\\frac{\\omega'(x_0)}{x_0\\lambda'(x_0)}+O(\\zeta^2),\n\\]\nfrom which it is seen that\n\\[\n{\\cal S}^y_z(x_0-\\nu,\\phi(T_\\nu),\\epsilon)=\\epsilon(\\frac{\\omega'(x_0)}{x_0\\lambda'(x_0)}+O(\\nu))\n   +O(\\zeta^2).\n\\]\nSince $\\rho'(T_\\nu)=O(\\nu)$ one then has that\n\\[\nP_{xy\\epsilon}(T_\\nu)=O(\\nu)\\epsilon+O(\\zeta^2)\n\\]\nUpon combining the above results, and using the fact that $\\beta_i(t)$ is\nbounded for $i=1,2$ and for all $t$, it is seen that\n\\begin{equation}\\label{eqpxyepfinal}\n\\rho_0(t)P_{xy\\epsilon}(t)=(\\alpha_1(t)d_1+\\alpha_2(t)d_2+O(\\nu))\\epsilon+O(\\epsilon^2),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eqdefalphai}\n\\alpha_i(t)=-2x_0^2\\int_t^\\infty u_0(s)\\beta_i(s)\\,ds\n\\end{equation}\nfor $i=1,2$.\n\n\\begin{prop}\\label{propalphai} When evaluated at $t=T_\\eta$,\n\\[\n\\begin{array}{lll}\n\\alpha_1(T_\\eta)&=&{\\displaystyle-2x_0^3\\ln\\frac{\\eta}{x_0}} \\\\\n\\alpha_2(T_\\eta)&=&{\\displaystyle-2x_0^5\\ln\\frac{\\eta}{x_0}+\\frac12x_0^5\n(1-\\frac{\\eta^2}{x_0^2})}.\n\\end{array}\n\\]\n\\end{prop}\n\n\\noindent{\\bf Proof: } The expression for $\\alpha_1(T_\\eta)$ only will be proved,\nas the proof for $\\alpha_2(T_\\eta)$ is similiar.\nFor the rest of this proof set $k=x_0\/2$.\n\nUsing the expression for $\\rho_0(t)$ given in Proposition\n\\ref{propexactwave} it is not difficult to see that\n\\[\n\\begin{array}{lll}\nu_0(t)&=&\\rho_0'(t)\/\\rho_0(t) \\\\\n{}&=&{\\displaystyle\\frac{x_0}{4}\\frac{\\mbox{sech}^2(kt)}{1+\\tanh(kt)}}.\n\\end{array}\n\\]\nSince, by Lemma \\ref{lemphiexpand}, $\\beta_1(t)=-x_0$, it is clear\nthat \n\\[\n\\begin{array}{lll}\n\\alpha_1(t)&=&{\\displaystyle x_0^3\\int_{kt}^\\infty\\frac{\\mbox{sech}^2(s)}{1+\\tanh(s)}\\,ds} \\\\\n{}&=&{\\displaystyle x_0^3(\\ln(2)-\\ln(\\frac{2\\rho_0^2(t)}{x_0^2})}.\n\\end{array}\n\\]\nThe last part of the above expression comes from the fact that \n\\[\n1+\\tanh(kt)=2\\rho_0^2(t)\/x_0^2.\n\\]\nSince $\\rho_0(T_\\eta)=\\eta$, upon simplification the expression for\n$\\alpha_1(T_\\eta)$ follows.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\begin{remark}\\label{remali} Note that\n\\[\n\\frac{\\alpha_2(T_\\eta)}{\\alpha_1(T_\\eta)}>x_0^2,\n\\]\nwith\n\\[\n\\frac{\\alpha_2(T_\\eta)}{\\alpha_1(T_\\eta)}=x_0^2+\n   \\frac14x_0^2(-\\ln\\frac{\\eta}{x_0})^{-1}+O(\\eta^2)\n\\]\nfor $0<\\eta\\ll1$.\n\\end{remark}\n\nNow that $\\alpha_i(T_\\eta)$ has been calculated for $i=1,2$, the following\nlemma has essentially been proved.\nAll that is now necessary is to recall (\\ref{eqsyzratio}), the \nfact that $0<\\rho'(T_\\eta)=O(\\eta)$, and the fact that \n$y_+=x_0\/2$.\n\n\\begin{lemma}\\label{lemsyz} The manifold $\\ws(\\Me)$ satisfies the \nestimate\n\\[\n{\\cal S}^y_z(\\eta,\\phi(T_\\eta),\\epsilon)\\propto\\frac1{\\eta^2}(\\alpha_1d_1\n+\\alpha_2d_2+O(\\nu))\\epsilon+O(\\epsilon^2),\n\\]\nwhere\n\\[\n\\begin{array}{lll}\n\\alpha_1&=&{\\displaystyle-4x_0^2\\ln\\frac{\\eta}{x_0}} \\\\\n\\alpha_2&=&{\\displaystyle-4x_0^4\\ln\\frac{\\eta}{x_0}+x_0^4\n(1-\\frac{\\eta^2}{x_0^2})}.\n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{remark} Note that $\\alpha_1\/\\alpha_2$ satisfies the same estimate\nas that given in Remark \\ref{remali}.\n\\end{remark}\n\n\\begin{cor}\\label{coruyz} The manifold $\\wu(\\Me)$ satisfies the estimate\n\\[\n{\\cal U}^y_z(\\eta,-\\phi(T_\\eta),\\epsilon)={\\cal S}^y_z(\\eta,\\phi(T_\\eta),\\epsilon).\n\\]\n\\end{cor}\n\n\\noindent{\\bf Proof: } The result follows immediately from the symmetry outlined\nin Proposition \\ref{propsymmetry}.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\subsection{Flow from $\\{x=\\eta\\}$ to $\\{y=y_-+\\eta\\}$}\n\nRecall the transversality argument given in Section 4.\nIt was shown how the manifolds $W^u(0)$ and $\\ws(\\Me)$ vary with\nrespect to $\\mu$ along the wave when $\\epsilon=0$.\nDue to the smoothness of the flow, these estimates persist for\n$\\epsilon$ and $\\mu$ sufficiently small.\nFurthermore, as a by-product of the symmetry discussed in Proposition\n\\ref{propsymmetry}, the manner in which $W^s(0)$ and $\\wu(\\Me)$ vary\nwith respect to $\\mu$ is also understood.\n\nSpecifically, the following is known.\nSet\n\\begin{equation}\\label{eqwumeandwso}\n\\begin{array}{rll}\n\\wu(\\Me)\\cap\\{x=\\eta\\}&=&\\{(y,z,\\mu,\\epsilon)\\,:\\,y={\\cal U}^y(z,\\mu,\\epsilon)\\} \\\\\nW^s(0)\\cap\\{x=\\eta\\}&=&\\{(y,z,\\mu,\\epsilon)\\,:\\,y={\\cal S}^y(\\mu,\\epsilon),\nz={\\cal S}^z(\\mu,\\epsilon)\\}.\n\\end{array}\n\\end{equation}\nAs a consequence of Proposition \\ref{propsymmetry} and Lemmas\n\\ref{lemuymu} and \\ref{lemsymu} it can be concluded that\n\\begin{equation}\\label{eqwumewsovary}\n{\\cal S}^y_\\mu(0,0)<0,\\quad{\\cal U}^y_\\mu(0,0,0)>0;\n\\end{equation}\nfurthermore, these estimates hold for $\\epsilon$ and $\\mu$ sufficiently\nsmall.\nWhen $\\mu=\\mu(\\epsilon)$ the manifolds intersect, so that\n\\begin{equation}\\label{eqwumewsointersect}\n{\\cal S}^y(\\mu,\\epsilon)={\\cal U}^y({\\cal S}^z(\\mu,\\epsilon),\\mu,\\epsilon).\n\\end{equation}\nThus, when $\\epsilon$ is fixed, by the estimate (\\ref{eqwumewsovary}) \nfor $\\mu>\\mu(\\epsilon)$ the manifold $W^s(0)$ is below $\\wu(\\Me)$, while for \n$\\mu<\\mu(\\epsilon)$ the configuration is reversed.\n\nRecall Lemma \\ref{lemspirals}, which gives a description of\nthe curves ${\\cal C}_0\\cdot t_0^\\pm$ and ${\\cal C}_{\\pm\\infty}\\cdot t_{\\pm\\infty}$.\nSpecifically, it is of interest here to contemplate the nature of\n${\\cal C}_0\\cdott^-_0$ and ${\\cal C}_{-\\infty}\\cdott_{-\\infty}$, both of which are contained\nin $B^-_0$ and centered upon\n\\begin{equation}\\label{eqdefpso}\np^s_0=W^s(0)\\capB^-_0.\n\\end{equation}\nThe set\n\\begin{equation}\\label{eqdefcmo}\n{\\cal C}^-_0={\\cal C}_{-\\infty}\\cdott_{-\\infty}\\cupp^s_0\\cup{\\cal C}_0\\cdott^-_0\n\\end{equation}\ndefineds a curve in $B^-_0$; in fact, for $\\epsilon\\omega(x_0)\\neq0$ it\nis a two-armed logarithmic spiral centered upon $p^s_0$.\nWhen $\\mu=\\mu(\\epsilon),\\,\\wu(\\Me)$ intersects ${\\cal C}^-_0$ at $p^s_0$; indeed,\nthe intersection is transverse.\nThus, as $\\mu$ is varied from $\\mu(\\epsilon),\\,\\wu(\\Me)$ continues to\nintersect ${\\cal C}^-_0$; however, the intersection no longer occurs at\n$p^s_0$.\nThis implies that $\\wu(\\Me)$ intersects either ${\\cal C}_{-\\infty}\\cdott_{-\\infty}$ or\n${\\cal C}_0\\cdott^-_0$.\nFrom (\\ref{eqwumewsovary}) and Lemma \\ref{lemspirals} it can be\nconcluded that if $|\\mu-\\mu(\\epsilon)|$ is $O(\\epsilon^n)$ for $n\\ge3$ but\nis not exponentially small in $\\epsilon$, then \n\\begin{equation}\\label{eqwumeintersect}\n\\begin{array}{lll}\n\\mu>\\mu(\\epsilon)&\\Longrightarrow&\\wu(\\Me)\\cap{\\cal C}_0\\cdott^-_0\\neq\\emptyset \\\\\n\\mu<\\mu(\\epsilon)&\\Longrightarrow&\\wu(\\Me)\\cap{\\cal C}_{-\\infty}\\cdott_{-\\infty}\\neq\\emptyset\n\\end{array}\n\\end{equation}\n(see Figure \\ref{fig_wume}).\nFurthermore, there exists only one point in each of the\nintersections. \nIf $0<|\\mu-\\mu(\\epsilon)|\\le O(e^{-c\/\\epsilon})$, then there exists a\nfinite number of points in both $\\wu(\\Me)\\cap{\\cal C}_0\\cdott^-_0$ and\n$\\wu(\\Me)\\cap{\\cal C}_{-\\infty}\\cdott_{-\\infty}$, with the number of points increasing\nto infinity as $\\mu\\to\\mu(\\epsilon)$.\n\nFor the rest of this subsection it will be assumed that\n$|\\mu-\\mu(\\epsilon)|$ is not exponentially small in $\\epsilon$.\nThis is done to clarify the following arguments.\nIn addition, it will be assumed that $\\mu\\ge\\mu(\\epsilon)$;\nhowever, this is not a necessary restriction and the below\narguments can be modified to draw conclusions if $\\mu<\\mu(\\epsilon)$.\n\nSince $\\mu>\\mu(\\epsilon)$, by (\\ref{eqwumeintersect}) $\\wu(\\Me)\\cap{\\cal C}_0\\cdott^-_0$\nis nonempty, which immediately implies that $\\wu(\\Me)\\cap{\\cal C}_0$ is\nnonempty.\nDue to symmetry outlined in Proposition \\ref{propsymmetry} this\nyields not only that $\\ws(\\Me)\\cap{\\cal C}_0$ is nonempty, but\n\\[\n\\wu(\\Me)\\cap{\\cal C}_0=\\ws(\\Me)\\cap{\\cal C}_0\\subset\\wu(\\Me)\\cap\\ws(\\Me).\n\\]\nThus, there is a connection between $(x^*(\\epsilon),0,z^*(\\epsilon))$ and\n$(x^*(\\epsilon),0,-z^*(\\epsilon))$ which passes ``near'' (to be made more\nprecise later) the plane $\\{x=0\\}$.\nThe goal of the rest of this subsection is to understand the\nnature in which $\\wu(\\Me)$ and $\\ws(\\Me)$ intersect at ${\\cal C}_0$.\n\nNow that the orientation of the manifolds $\\wu(\\Me)$ and $\\ws(\\Me)$ is\nunderstood as they intersect the plane $\\{x=\\eta\\}$, it is desirable\nto understand their behavior under the flow as they pass near\nthe critical points $(0,y_-,z_-)$ and $(0,y_+,z_+)$,\nrespectively.\nThe goal of this subsection is to show that under a suitable\nrestriction the passage near these critical points does not\neffect their orientation.\nIt will be sufficient to track the manifold $\\wu(\\Me)$, as the\nsymmetry present in (\\ref{eqglodered}) allows one to then draw an\nimmediate conclusion regarding the behavior of $\\ws(\\Me)$.\n\nFirst, suppose that $\\mu$ is such that\n\\begin{equation}\\label{eqmurestrict}\n0<O(\\epsilon^{n+1})\\le|\\mu-\\mu(\\epsilon)|\\le O(\\epsilon^n)\n\\end{equation}\nfor some $n\\ge3$.\nAs it will be seen, this restriction on $\\mu$ will guarantee that\nthe spiralling action near the critical points on $\\{x=0\\}$ will be \nunseen by $\\wu(\\Me)$ as it passes near $\\{x=0\\}$.\nSince $n\\ge2$, the perturbation of $\\mu$ given in (\\ref{eqmurestrict})\nwill not effect the asymptotic estimates given in Lemma \\ref{lemsyz}\nand Corollary \\ref{coruyz}.\nUsing the asymptotics for ${\\cal U}^y$, it will be desirous to look at\nthe image of the curve\n\\begin{equation}\\label{equycurve}\nx=\\eta,\\quad y=y_0+\\epsilon cz,\n\\end{equation}\nwhere $|z|\\le\\eta,\\,y_0=O(\\epsilon^n),$ and\n\\[\n\\mbox{sgn}\\,(c)=\\mbox{sgn}\\,({\\cal U}^y_z).\n\\]\nThe point $y_0$ describes the difference between $\\mu$ and \n$\\mu(\\epsilon)$, and by supposition is therefore positive.\n\nIt will be advantageous to look at the flow near the critical point\nin polar coordinates.\nUpon setting\n\\[\ny=r\\sin\\theta,\\quad z=r\\cos\\theta\n\\]\nthe curve (\\ref{equycurve}) can be parametrically described as\n\\begin{equation}\\label{equypolarcurve}\n\\begin{array}{lll}\nr_0(s)&=&\\sqrt{y_0^2+2c\\epsilon s+(1+\\epsilon^2c^2)s^2}\\\\\n\\theta_0(s)&=&{\\displaystyle\\mbox{Tan}^{-1}(\\frac{y_0+\\epsilon cs}{s})},\n\\end{array}\n\\end{equation}\nwhere $|s|\\le\\eta$.\nThe linear flow near $(0,y_-,z_-)$ satisfies the ODE\n\\begin{equation}\\label{eqzerolinear}\n\\begin{array}{lll}\nx'&=&-ax \\\\\nr'&=&2ar \\\\\n\\theta'&=&\\epsilon b,\n\\end{array}\n\\end{equation}\nwhere\n\\[\n\\begin{array}{lll}\na&=&\\sqrt{-\\lambda(0)}+O(\\epsilon^2)\\\\\nb&=&{\\displaystyle\\frac{\\omega(x_0)}{\\sqrt{-\\lambda(0)}}+O(\\epsilon)}.\n\\end{array}\n\\]\nThus, upon using (\\ref{equypolarcurve}) as an initial condition \nto (\\ref{eqzerolinear}) it is seen that the behavior of ${\\cal U}^y_z$\nis well approximated by the solution formula\n\\begin{equation}\\label{equypolarflow}\n\\begin{array}{lll}\nx(t)&=&\\eta e^{-at}\\\\\nr(t)&=&r_0(s)e^{2at}\\\\\n\\theta(t)&=&\\theta_0(s)+\\epsilon bt.\n\\end{array}\n\\end{equation}\n\nIt is now of interest to determine the nature of (\\ref{equypolarcurve})\nas the flow forces it to intersect the cylinder $\\{r=\\eta\\}$.\nThe time of flight, $t_f$, is defined by $r(t_f)=\\eta$.\nUsing (\\ref{equypolarflow}), this yields that\n\\[\nt_f=\\frac{1}{2a}\\ln\\frac{\\eta}{r_0(s)}.\n\\]\nThus, under the linear flow (\\ref{equypolarflow}) there is the\nmapping from $\\{x=\\eta\\}$ to $\\{r=\\eta\\}$ given by\n\\begin{equation}\\label{equymap}\n\\left(\\begin{array}{c}\n  \\eta \\\\ r_0(s) \\\\ \\theta_0(s)\n      \\end{array}\\right)\\to\n\\left(\\begin{array}{c}\n  \\sqrt{\\eta}\\sqrt{r_0(s)} \\\\ \\eta \\\\\n  {\\displaystyle\\theta_0(s)+\\epsilon\\frac{b}{2a}\\ln\\frac{\\eta}{r_0(s)}}\n      \\end{array}\\right)\n\\end{equation}\nfor $|s|\\le\\eta$.\nSet $x(s)$ to be the first component and $\\theta(s)$ to be the third\ncomponent of the vector on the right hand side of the above \nequation.\nGiven the asymptotic expansion for ${\\cal U}^y_z$, it is then of interest\nto perform an expansion of $x(s)$ and $\\theta(s)$ around $s=0$.\nThis step is taken because it is only of interest to determine\nthe behavior of the curve near where $\\wu(\\Me)$ intersects ${\\cal C}_0\\cdott^-_0$.\nFirst, a Taylor expansion yields that\n\\[\n\\begin{array}{lll}\nr_0(s)&=&y_0+\\epsilon cs+O(w^2) \\\\\n\\theta_0(s)&=&{\\displaystyle\\frac{\\pi}2-\\frac1{y_0}s+O(w^2)},\n\\end{array}\n\\]\nwhere $w=\\epsilon s\/y_0$ and it has been used that $y_0>0$.\nPlugging this expansion into (\\ref{equymap}) and performing\nanother Taylor expansion then gives that\n\\begin{equation}\\label{equypolartaylor}\n\\begin{array}{lll}\nx(s)&=&{\\displaystyle\\sqrt{\\eta}(\\sqrt{y_0}+\\frac{\\epsilon c}{2\\sqrt{y_0}}s\n         +O(w^2))} \\\\\n\\theta(s)&=&{\\displaystyle\\frac{\\pi}2+\\epsilon\\frac{b}{2a}\\ln\\frac{\\eta}{y_0}\n         -\\frac1{y_0}(1+\\epsilon^2\\frac{ac}{2b})s+O(w^2)}.\n\\end{array}\n\\end{equation}\nIn rectangular coordinates the right hand side of (\\ref{equymap})\nis given by $x(s),\\,y(s)=\\eta\\sin\\theta(s),$ and $z(s)=\\eta\\cos\\theta(s)$.\nUsing the expansion (\\ref{equypolartaylor}) it is then seen that \n\\begin{equation}\\label{equyrecttaylor}\n\\begin{array}{llllll}\nx(0)&=&\\sqrt{\\eta}\\sqrt{y_0},\\quad&\nx'(0)&=&{\\displaystyle\\sqrt{\\eta}\\frac{\\epsilon c}{2\\sqrt{y_0}}} \\\\\ny(0)&=&\\eta\\sin\\theta(0),\\quad&\ny'(0)&=&{\\displaystyle-\\eta(\\frac1{y_0}(1+\\epsilon^2\\frac{ac}{2b}))\\cos\\theta(0)} \\\\\nz(0)&=&\\eta\\cos\\theta(0),\\quad&\nz'(0)&=&{\\displaystyle\\eta(\\frac1{y_0}(1+\\epsilon^2\\frac{ac}{2b}))\\sin\\theta(0)}.\n\\end{array}\n\\end{equation}\n\nIt is clear that the curve given by the right hand side of \n(\\ref{equymap}) can be parameterized by $z$ for $s$ sufficiently\nnear zero.\nUsing the expansion coefficients given in (\\ref{equyrecttaylor}) it\nis then seen that $x=x(z)$ satisfies the relation\n\\begin{equation}\\label{equyderiv}\n{\\displaystyle\\frac{dx}{dz}=\\frac{\\epsilon c\\sqrt{y_0}}{2\\sqrt{\\eta} \n(1+\\epsilon^2\\frac{ac}{2b}))\\sin\\theta(0)}}.\n\\end{equation}\nSince\n\\[\n\\theta(0)=\\frac{\\pi}2+\\epsilon\\frac{b}{2a}\\ln\\frac{\\eta}{y_0},\n\\]\nthis means that for $y_0=O(\\epsilon^n)$,\n\\[\n\\theta(0)=\\frac{\\pi}2+O(n\\epsilon\\ln\\frac1{\\epsilon}).\n\\]\nSubstituting this estimate into (\\ref{equyderiv}) and applying \n(\\ref{equyrecttaylor}) yields the following lemma.\n\n\\begin{lemma}\\label{lemuyzb0} Suppose that $\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for\nsome $n\\ge2$.\nThe intersection of $\\wu(\\Me)$ with the plane $\\{y=y_-+\\eta\\}$ is a \nparametric curve given by $(x,z)=(x(s),z(s))$.\nThe curve satisfies the estimates\n\\[\n\\begin{array}{ll}\na.\\quad&x(0)=O(\\epsilon^{n\/2}) \\\\\nb.\\quad&{\\displaystyle\\left.\\frac{dx}{dz}\\right|_{s=0}=O(\\epsilon^{(n+2)\/2})} \\\\\nc.\\quad&{\\displaystyle\\mbox{sgn}\\,(\\frac{dx}{dz})=\n           \\mbox{sgn}\\,({\\cal U}^y_z(\\eta,-\\phi(T_\\eta),\\mu(\\epsilon),\\epsilon))}.\n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{remark} If $\\wu(\\Me)$ intersects the plane $\\{y=y_--\\eta\\}$, then\nthe conclusion of the lemma holds, with part c. being changed to\n\\[\n{\\displaystyle\\mbox{sgn}\\,(\\frac{dx}{dz})=\n          -\\mbox{sgn}\\,({\\cal U}^y_z(\\eta,-\\phi(T_\\eta),\\mu(\\epsilon),\\epsilon))}.\n\\]\n\\end{remark}\n\nAs an immediate consequence of the symmetry of the ODE one gets\nthe following corollary.\n\n\\begin{cor}\\label{corsyzb0} Suppose that $\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for\nsome $n\\ge2$.\nThe intersection of $\\ws(\\Me)$ with the plane $\\{y=y_+-\\eta\\}$ is a \nparametric curve given by $(x,z)=(x(s),z(s))$.\nThe curve satisfies the estimates\n\\[\n\\begin{array}{ll}\na.\\quad&x(0)=O(\\epsilon^{n\/2}) \\\\\nb.\\quad&{\\displaystyle\\left.\\frac{dx}{dz}\\right|_{s=0}=O(\\epsilon^{(n+2)\/2})} \\\\\nc.\\quad&{\\displaystyle\\mbox{sgn}\\,(\\frac{dx}{dz})=\n           \\mbox{sgn}\\,({\\cal S}^y_z(\\eta,\\phi(T_\\eta),\\mu(\\epsilon),\\epsilon))}.\n\\end{array}\n\\]\n\\end{cor} \n\n\\begin{remark} If $\\wu(\\Me)$ intersects the plane $\\{y=y_++\\eta\\}$, then\nthe conclusion of the lemma holds, with part c. being changed to\n\\[\n{\\displaystyle\\mbox{sgn}\\,(\\frac{dx}{dz})=\n          -\\mbox{sgn}\\,({\\cal U}^y_z(\\eta,\\phi(T_\\eta),\\mu(\\epsilon),\\epsilon))}.\n\\]\n\\end{remark}\n\n\n\\subsection{Flow from $\\{y=y_-+\\eta\\}$ to $\\{y=0\\}$}\n\nNow that the passage of the manifold $\\wu(\\Me)$ near the critical point\n$(0,y_-,z_-)$ is understood, it is necessary to understand its\nbehavior under the flow near $\\{x=0\\}$.\nThe previous lemma states that the manifold is $O(\\epsilon^{n\/2})$ near\n$\\{x=0\\}$ when it intersects the plane $\\{y=y_-+\\eta\\}$.\nSince $x'=xy$ and $y<0$ in the region of interest, it is clear\nthat manifold stays within $O(\\epsilon^{n\/2})$ of $\\{x=0\\}$ until \nit intersects the plane $\\{y=0\\}$.\nThus, it is expected that the flow on $\\{x=0\\}$ dominates the\nbehavior of the manifold.\nThe goal of this subsection is to make this intuition rigorous.\n\nAfter the manifold $\\wu(\\Me)$ has intersected the plane $\\{y=y_-+\\eta\\}$\nit can be written as the graph\n\\begin{equation}\\label{eqdefux}\n\\wu(\\Me)=\\{(x,y,z)\\,:\\,x={\\cal U}^x(y,z,\\epsilon),y_-+\\eta\\le y\\le y_+-\\eta,|z|\\le\\eta\\}.\n\\end{equation}\nThe previous lemma gives an expression for ${\\cal U}^x_z$ along the wave when\n$y=y_-+\\eta$.\nThe goal is to calculate ${\\cal U}^x_z(0,0,\\epsilon)$.\nIn order to accomplish this, it will be necessary to once again use\ndifferential forms.\n\nThe vector to be tracked has the initial condition\n\\[\n\\xi=({\\cal U}^x_z(y_-+\\eta,z,\\epsilon),0,1)^T,\n\\]\nwhere the $z$-coordinate is taken to be such that the vector is\nover the underlying solution.\nThe variational equations are\n\\begin{equation}\\label{eqvarnear0}\n\\begin{array}{lll}\n\\delta x'&=&u\\delta x+\\rho\\delta y \\\\\n\\delta y'&=&-\\lambda'(\\rho)\\delta x-2u\\delta y+2\\phi\\delta z \\\\\n\\delta z'&=&\\epsilon\\omega'(\\rho)\\delta x-2\\phi\\delta y-2u\\delta z.\n\\end{array}\n\\end{equation}\nProjectivize the equations by setting \n\\begin{equation}\\label{eqdefproject}\na=\\delta x\/\\delta z,\\quad b=\\delta y\/\\delta z.\n\\end{equation}\nIt is not difficult to check that $\\delta z(\\xi)=O(1)$ for \n$y_-+\\eta\\le y\\le 0$, so that the above quantities are\nwell-defined in the region of interest.\nThe equations for $a$ and $b$ are given by\n\\begin{equation}\\label{eqprojectodes}\n\\begin{array}{lll}\na'&=&(-u+2\\phi b)a+\\rho b-\\epsilon\\omega'(\\rho)a^2 \\\\\nb'&=&-\\lambda'(\\rho)a+2\\phi(1+b^2)-\\epsilon\\omega'(\\rho)ab,\n\\end{array}\n\\end{equation}\nwith the initial conditions being\n\\[\n\\begin{array}{l}\na(0)={\\cal U}^x_z(u\\cdot0,\\phi\\cdot0,\\epsilon) \\\\\nb(0)=0.\n\\end{array}\n\\]\n\nWithout loss of generality, assume that the $z$-coordinate\nof $W^u(0)\\capB^-_0,\\,-\\phi(T_\\eta),$ is such that $-\\phi(T_\\eta)>0$.\nBy Corollary \\ref{corzlocation}, this implies that \n\\begin{equation}\\label{eqfirstd1d2restrict}\nd_1+x_0^2d_2>0.\n\\end{equation}\nLet $(\\rho,u,\\phi)\\cdot t\\in\\wu(\\Me)\\cap\\ws(\\Me)$ represent the orbit\nconnecting $(x^*(\\epsilon),0,\\pm z^*(\\epsilon))$, and let it be normalized so\nthat $u\\cdot0=y_-+\\eta$.  \nIt is known that this orbit intersects ${\\cal C}_0$, as $\\wu(\\Me)\\cap{\\cal C}_0\\cdott^-_0$\nis nontrivial.\nTherefore, there is a $T_0>0$ such that $u\\cdot T_0=\\phi\\cdot T_0=0$.\nIn addition, there exists a $T_1<0$ such that $\\rho\\cdot T_1=\\eta$.\n\nThe claim is that $\\phi\\cdot t>0$ for $t\\in[T_1,T_0]$.\nTo see that this is true, consider the following argument.\nThe $z$-coordinate of $\\wu(\\Me)\\cap{\\cal C}_0\\cdott^-_0,\\,\\phi\\cdot T_1$, is\nwithin $O(\\epsilon^3),\\,n\\ge3$, of $-\\phi(T_\\eta)$; therefore, since\n$-\\phi(T_\\eta)>0$ is $O(\\epsilon),\\,\\phi\\cdot T_1$ is also positive.\nFollowing the argument of the previous subsection, it is not \ndifficult to see that $\\phi\\cdot t>0$ for $t\\in[T_1,0]$.\nSince $\\phi\\cdot0>0$, as a consequence of Proposition \\ref{propco}\nit is necessarily true that $\\phi\\cdot t>0$ for $t\\in[0,T_0]$\n(recall that, by Lemma \\ref{lemuyzb0}, $\\rho\\cdot0=O(\\epsilon^{n\/2})$).\n\nAs a consequence of Corollary \\ref{coruyz} and Lemma \\ref{lemuyzb0} it\nis known that \n\\[\n\\begin{array}{l}\n\\mbox{sgn}\\,({\\cal U}^x_z(u\\cdot 0,\\phi\\cdot 0,\\epsilon))=\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2) \\\\\n{\\cal U}^x_z(u\\cdot 0,\\phi\\cdot 0,\\epsilon)=O(\\epsilon^{(n+2)\/2}).\n\\end{array}\n\\]\nIn what follows, assume that\n\\begin{equation}\\label{eqsecondd1d2restrict}\n\\alpha_1d_1+\\alpha_2d_2>0.\n\\end{equation}\nSince $\\alpha_1\/\\alpha_2$ is close to $1\/x_0^2$ (for an exact description\nsee Lemma \\ref{lemsyz}), this is not that much more a restriction than\nthat already imposed by (\\ref{eqfirstd1d2restrict}).\n\nUnder this hypothesis, it is then being assumed that $a(0)>0$, with\n$a(0)=O(\\epsilon^{(n+2)\/2})$.\nIt is of interest to calculate $a(T_0)$, as\n\\begin{equation}\\label{eqat0}\n{\\cal U}^x_z(0,0,\\epsilon)\\propto a(T_0).\n\\end{equation}\nTowards this end is the following lemma.\n\n\\begin{lemma}\\label{lemagrowthestimate} $a(t)\\le O(\\epsilon^{n\/2})$ for\n$t\\in[0,T_0]$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } First, since $\\rho'=\\rho u$,\n\\[\n\\rho(t)=\\rho(0)e^{\\int_0^tu(s)\\,ds},\n\\]\nso that $\\rho(t)=O(\\rho(0))=O(\\epsilon^{n\/2})$ for $t\\in[0,T_0]$.\nIn addition, since $\\delta z(\\xi)$ is $O(1)$ for $y_-+\\eta\\le y\\le0$, and\n$\\delta x(\\xi)$ and $\\delta y(\\xi)$ satisfy the linear ODE (\\ref{eqvarnear0}),\nit is necessarily true that $a$ and $b$ are $O(1)$ for $t\\in[0,T_0]$.\nThese two facts will be used extensively in what follows.\n\nSet\n\\[\n\\begin{array}{lll}\nf(t)&=&{\\displaystyle e^{\\int_0^tu(s)-2\\phi(s)b(s)\\,ds}} \\\\\n    &=&{\\displaystyle\\frac{\\rho(t)}{\\rho(0)}e^{-2\\int_0^t\\phi(s)b(s)\\,ds}}.\n\\end{array}\n\\]\nSince $b(t)=O(1)$ and $\\rho(t)=O(\\rho(0))$, it is clear that $f(t)=O(1)$ for \n$t\\in[0,T_0]$.\nSolving (\\ref{eqprojectodes}), it is seen that $a(t)$ is given by\nthe integral formula\n\\begin{equation}\\label{eqasol}\nf(t)a(t)=a(0)+\\int_0^tf(s)\\rho(s)b(s)\\,ds-\n             \\epsilon\\int_0^tf(s)\\omega'(\\rho(s))a^2(s)\\,ds.\n\\end{equation}\nUsing the estimates on $f$ and $\\rho$, and assuming that $a$ and\n$b$ are $O(1)$, it can then be concluded that\n\\[\nf(t)a(t)=a(0)+O(\\epsilon^{n\/2})\\cdot t+O(\\epsilon^{(n+2)\/2})\\cdot t.\n\\]\nSince $a(0)=O(\\epsilon^{(n+2)\/2})$, the conclusion follows.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nIt will be desirable to revise the above estimate.\nIn order to do so, the following lemma is needed.\n\n\\begin{lemma}\\label{lembgrowthestimate} $b(t)=O(\\epsilon)$ for\n$t\\in[0,T_0]$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } Since $b(0)=0$, the function $b(t)$ satisfies the integral\nequation\n\\begin{equation}\\label{eqbsolution}\nb(t)=2\\int_0^t\\phi(s)(1+b^2(s))\\,ds-\\int_0^t(\\lambda'(\\rho(s))a(s)\n+\\epsilon\\omega'(\\rho(s))a(s)b(s))\\,ds.\n\\end{equation}\nAssuming that $b(t)=O(1)$, which was justified in the proof\nof the above lemma, using the conclusion of the above\nlemma, and using the fact that $\\phi(t)=O(\\epsilon)$, one gets\nthat\n\\[\nb(t)=O(\\epsilon)\\cdot t+O(\\epsilon^{n})\\cdot t.\n\\]\nSince $T_0=O(1)$, the conclusion now follows.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nA more careful examination of (\\ref{eqbsolution}) yields the\nfollowing result for $b(t)$, which in turn can be used to improve upon\nthe estimate made upon $a(t)$ in Lemma \\ref{lemagrowthestimate}.\nUsing the estimates provided in the above two lemmas, it is clear\nthat \n\\[\nb'=2\\phi(1+b^2)+O(\\epsilon^n),\n\\]\nwhich can be solved to get\n\\[\nb(t)=\\tan(\\int_0^t2\\phi(s)\\,ds)+O(\\epsilon^n)\\cdot t.\n\\]\nSubstituting this expression into (\\ref{eqasol}), and again using\nthe above two lemmas, yields\n\\[\nf(t)a(t)=a(0)+\\int_0^tf(s)\\rho(s)\\tan(\\int_0^s2\\phi(r)\\,dr)\\,ds\n+O(\\epsilon^{3n\/2})\\cdot t.\n\\]\nNote that the second term on the right-hand side of the above\nequation is $O(\\epsilon^{(n+2)\/2})$, as is $a(0)$.\nFurthermore, this term is positive, as $\\phi(t)>0$ for \n$t\\in[0,T_0]$.\nSince, by supposition, $a(0)>0$, it is now clear that\n\\[\nf(T_0)a(T_0)>a(0),\n\\]\nand furthermore, $a(T_0)=O(a(0))$.\n\nUsing (\\ref{eqat0}), the proof of the following lemma is now\ncomplete.\nWhile the lemma was proved only for the case of both $\\phi\\cdot0>0$ and\n$a(0)>0$, it can easily be modified in the event that both quantities\nare negative.\n\n\\begin{lemma}\\label{lemuxzy=0} Suppose that $0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$\nfor some $n\\ge3$.\nFurther suppose that\n\\[\n\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2)=\\mbox{sgn}\\,(d_1+x_0^2d_2),\n\\]\nwhere the $\\alpha_i$'s are defined in Lemma \\ref{lemsyz}.\nThen \n\\[\n\\begin{array}{l}\n\\mbox{sgn}\\,({\\cal U}^x_z(0,0,\\epsilon))=\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2) \\\\\n{\\cal U}^x_z(0,0,\\epsilon)=O(\\epsilon^{(n+2)\/2}).\n\\end{array}\n\\]\n\\end{lemma}\n\nBy the symmetries present in (\\ref{eqglodered}) one arrives at\nthe following corollary.\n\n\\begin{cor}\\label{corsxzy=0} Let the hypotheses of Lemma \\ref{lemuxzy=0}\nbe satisfied.\nThen\n\\[\n\\mbox{sgn}\\,({\\cal S}^x_z(0,0,\\epsilon))=-\\mbox{sgn}\\,({\\cal U}^x_z(0,0,\\epsilon)).\n\\]\n\\end{cor}\n\nUsing the time of flight estimates given in Subsection 5.2, one finally\ngets the following theorem.\n\n\\begin{theorem}\\label{thmwumewsmeintersect} Suppose that \n$0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for some $n\\ge3$.\nFurther suppose that the parameters $d_1$ and $d_2$ are chosen\nso that \n\\[\n\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2)=\\mbox{sgn}\\,(d_1+x_0^2d_2),\n\\]\nor\nwhere the $\\alpha_i$'s are defined in Lemma \\ref{lemsyz}.\nThen the manifolds $\\wu(\\Me)$ and $\\ws(\\Me)$ intersect transversely \nat $\\{y=0\\}$,\nwith the transversality being $O(\\epsilon^{(n+2)\/2})$.\nFurthermore, the time the resultant trajectory spends in the region\n$0\\le x\\le\\eta$ is given by\n\\[\nT_f=O(2n\\ln\\frac1{\\epsilon}).\n\\]\n\\end{theorem}\n\n\\subsection{Completion of argument}\n\nUsing the notation of the previous section, near the invariant plane\n$\\{x=0\\}$ write\n\\[\n\\begin{array}{lll}\n\\wu(\\Me)=\\{(x,y,z,\\epsilon)\\,:\\,x={\\cal U}^x(y,z,\\epsilon),y_-+\\eta\\le y\\le y_+-\\eta\\} \\\\\n\\ws(\\Me)=\\{(x,y,z,\\epsilon)\\,:\\,x={\\cal S}^x(y,z,\\epsilon),y_-+\\eta\\le y\\le y_+-\\eta\\}.\n\\end{array}\n\\]\nBy Theorem \\ref{thmwumewsmeintersect} it is known that if\n$0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for $n\\ge3$ and if the pair $(d_1,d_2)$ satisfy\nthe hypotheses of Lemma \\ref{lemuxzy=0}, then there exists a\n$(\\tilde{x}(\\epsilon),0,0)\\in{\\cal C}_0,\\,\\tilde{x}(\\epsilon)=O(\\epsilon^{n\/2})$, such that\n\\[\n\\begin{array}{l}\n{\\cal U}^x(0,0,\\epsilon)={\\cal S}^x(0,0,\\epsilon) \\\\\n\\mbox{sgn}\\,(({\\cal U}^x_z-{\\cal S}^x_z)(0,0,\\epsilon))=\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2) \\\\\n({\\cal U}^x_z-{\\cal S}^x_z)(0,0,\\epsilon)=O(\\epsilon^{(n+2)\/2}).\n\\end{array}\n\\]\nSince the manifolds intersect transversely at $(\\tilde{x}(\\epsilon),0,0)$, \nthere is an orbit, \n$(\\rho(t),u(t),\\phi(t))$, which connects the two critical\npoints $(x^*(\\epsilon),0,\\pm z^*(\\epsilon))\\in\\M_\\ep$.\nLet this trajectory by translated so that $(\\rho(0),u(0),\\phi(0))=\n(\\tilde{x}(\\epsilon),0,0)\\in{\\cal C}_0$.\n\nSince the manifolds $\\wu(\\Me)$ and $\\ws(\\Me)$ intersect transversely\nat $\\{y=0\\}$, they intersect transversely everywhere along\nthe orbit.\nThe next goal is to understand the transversality as $\\wu(\\Me)$ intersects\n$B^+_{x_0}$.\nDefine\n\\begin{equation}\\label{eqdeftangentvectors}\n\\begin{array}{lll}\n\\xi_1&=&(\\rho'(0),u'(0),\\phi'(0))^T \\\\\n\\xi_2&=&({\\cal U}^x_z(\\tilde{x}(\\epsilon),0,1)^T \\\\\n\\xi_3&=&({\\cal S}^x_z(\\tilde{x}(\\epsilon),0,1)^T.\n\\end{array}\n\\end{equation}\nIt is clear that $\\xi_2\\cdot t\\in T\\wu(\\Me)$ and $\\xi_3\\cdot t\\in T\\ws(\\Me)$\nfor all $t$, and that $\\xi_1\\cdot t\\in T\\wu(\\Me)\\cap T\\ws(\\Me)$ for all\n$t$.\nThe three-form $P_{xyz}$ will be used to gain an understanding\nas to how the flow carries these vectors.\nUsing (\\ref{eqvarnear0}), the three-form $P_{xyz}$ satisfies\nAbel's formula\n\\[\nP_{xyz}'=-3uP_{xyz},\n\\]\nwhich, using the fact that $u=\\rho'\/\\rho$, has the solution\n\\begin{equation}\\label{eqpxyzsolvetemp}\nP_{xyz}(t)=\\frac{\\rho^3(0)}{\\rho^3(t)}P_{xyz}(0).\n\\end{equation}\nWhen applied to the tangent vectors $\\xi_i$ defined in\n(\\ref{eqdeftangentvectors}),\n\\begin{equation}\\label{eqpxyz0}\nP_{xyz}(0)=(\\lambda(0)+O(\\epsilon^2))({\\cal U}^x_z-{\\cal S}^x_z)(0,0,\\epsilon);\n\\end{equation}\nthus, after substituting into (\\ref{eqpxyzsolvetemp}) one gets\n\\begin{equation}\\label{eqpxyzsolve}\nP_{xyz}(t)=\\frac{\\rho^3(0)}{\\rho^3(t)}\n     (\\lambda(0)+O(\\epsilon^2))({\\cal U}^x_z-{\\cal S}^x_z)(0,0,\\epsilon).\n\\end{equation}\n\nNow define $\\tilde{T}_\\nu$ such that $\\rho(\\tilde{T}_\\nu)=x_0-\\nu$.\nAs they intersect the section $\\{x=x_0-\\nu\\}$ the manifolds \n$\\wu(\\Me)$ and $\\ws(\\Me)$ are given by the curves\n\\[\n\\begin{array}{lll}\n\\wu(\\Me)\\capB^+_{x_0}&=&\\{(y,z,\\epsilon)\\,:\\,y={\\cal U}^y(z,\\epsilon)\\} \\\\\n\\ws(\\Me)\\capB^+_{x_0}&=&\\{(y,z,\\epsilon)\\,:\\,y={\\cal S}^y(z,\\epsilon)\\}.\n\\end{array}\n\\]\nIn addition, there exists a $\\tilde{z}(\\epsilon)$ such that\n\\[\n{\\cal U}^y(\\tilde{z}(\\epsilon),\\epsilon)={\\cal S}^y(\\tilde{z}(\\epsilon),\\epsilon).\n\\]\n\nIn order to understand the transversality of the manifolds\nat $\\{x=x_0-\\nu\\}$ it is necessary to compute\n$({\\cal U}^y_z-{\\cal S}^y_z)(\\tilde{z}(\\epsilon),\\epsilon)$.\nTowards this end, set\n\\[\n\\begin{array}{lll}\n\\tilde{\\xi}_2&=&(0,{\\cal U}^y_z(\\tilde{z}(\\epsilon),\\epsilon),1)^T \\\\\n\\tilde{\\xi}_3&=&(0,{\\cal S}^y_z(\\tilde{z}(\\epsilon),\\epsilon),1)^T.\n\\end{array}\n\\]\nIt is clear that\n\\begin{equation}\\label{eqpxyzpropto}\n\\begin{array}{lll}\nP_{xyz}(\\tilde{T}_\\nu)&\\propto&\n    P_{xyz}(\\xi_1\\cdot\\tilde{T}_\\nu,\\tilde{\\xi}_2,\\tilde{\\xi}_3) \\\\\n{}&=&\\rho'(\\tilde{T}_\\nu)({\\cal U}^y_z-{\\cal S}^y_z)(\\tilde{z}(\\epsilon),\\epsilon).\n\\end{array}\n\\end{equation}\nSince $\\rho'(\\tilde{T}_\\nu)>0$ is $O(\\nu)$, upon substituting\n(\\ref{eqpxyzsolve}) into (\\ref{eqpxyzpropto}) one gets\n\\begin{equation}\\label{eqtransverseatx0}\n\\begin{array}{lll}\n({\\cal U}^y_z-{\\cal S}^y_z)(\\tilde{z}(\\epsilon),\\epsilon)&\\propto&\n    \\rho^3(0)\\lambda(0)({\\cal U}^x_z-{\\cal S}^x_z)(0,0,\\epsilon) \\\\\n{}&=&A\\epsilon^{2n+1}+O(\\epsilon^{2n+2}),\n\\end{array}\n\\end{equation}\nwhere $\\mbox{sgn}\\,(A)=-\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2)$ (recall that $\\lambda(0)<0$).\nThe argument for the following lemma is now complete.\n\n\\begin{lemma}\\label{lemtransverseatx0} Let $0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ \nfor some $n\\ge3$.\nThere exists a $\\tilde{z}(\\epsilon)$, with $|\\tilde{z}(\\epsilon)-\\phi(T_\\nu)|=\nO(\\epsilon^{n-1})$, such that\n\\[\n\\begin{array}{ll}\n1.\\quad&{\\cal U}^y(\\tilde{z}(\\epsilon),\\epsilon)={\\cal S}^y(\\tilde{z}(\\epsilon),\\epsilon) \\\\\n2.\\quad&\\mbox{sgn}\\,(({\\cal U}^y_z-{\\cal S}^y_z)(\\tilde{z}(\\epsilon),\\epsilon))=\n       -\\mbox{sgn}\\,(\\alpha_1d_1+\\alpha_2d_2) \\\\\n3.\\quad&({\\cal U}^y_z-{\\cal S}^y_z)(\\tilde{z}(\\epsilon),\\epsilon)=O(\\epsilon^{2n+1}).\n\\end{array}\n\\]\n\\end{lemma}\n\n\\begin{remark} To paraphrase, the manifold $\\wu(\\Me)$ intersects $\\ws(\\Me)$\ntransversely at the section $\\{x=x_0-\\nu\\}$, with the transversality being\n$O(\\epsilon^{2n+1})$ for $0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$.\n\\end{remark}\n \n\n\n\n\n\n\n\n\n\n\n\\section{Existence of solitary waves}\n\\setcounter{equation}{0}\n\n\\subsection{Existence of bright solitary waves}\n\nA bright solitary wave is characterized by\n\\[\n\\lim_{t\\to\\pm\\infty}x(t)=0;\n\\]\nhence, in order to have such a wave it is necessary that\n$W^u(0)\\capW^s(0)$ be nontrivial.\nDue to the symmetry described in Proposition \\ref{propsymmetry},\nit is enough to show that $W^u(0)\\cap{\\cal C}_{x_0}\\cdott_{x_0}$, and hence\n$W^u(0)\\cap{\\cal C}_{x_0}$.\n\nRecall Lemma \\ref{lemimagecxo}, which states that ${\\cal C}_{x_0}\\cdott_{x_0}$ is\n$C^1-O(e^{-c\/\\epsilon})$ close to $\\ws(\\Me)$ for points $p\\in{\\cal C}_{x_0}$ which take\n$O(1\/\\epsilon)$ time to exit $B_{x_0}$.\nUnder a time reversal, the flow on $\\M_\\ep$ for $z=O(\\epsilon)$ satisfies\n\\[\nz'=\\epsilon^2\\sigma+O(\\epsilon^3).\n\\]\nTherefore, ${\\cal C}_{x_0}\\cdott_{x_0}$ will be close to $\\ws(\\Me)$ in a region where\n$z=O(\\epsilon)$ with $\\mbox{sgn}\\,(z)=\\mbox{sgn}\\,(\\sigma)$.\n\nNow recall Corollary \\ref{corzlocation}, which states that when\n$\\mu=\\mu(\\epsilon)$ that $\\phi(T_\\nu)$, the $z$-coordinate of $W^u(0)\\capB^+_{x_0}$,\nis given by\n\\[\n\\phi(T_\\nu)=-x_0(d_1+\\frac32x_0^2d_2+O(\\nu))\\epsilon+O(\\epsilon^2).\n\\]\nThus, by the comments of the previous paragraph, if \n\\[\n(d_1+\\frac32x_0^2d_2)\\sigma<0,\n\\]\nthen $W^u(0)\\cap\\ws(\\Me)$ is $C^1$-$O(e^{-c\/\\epsilon})$ close to the curve\n${\\cal C}_{x_0}\\cdott_{x_0}$ (see Figures \\ref{fig_wsme} and \\ref{fig_projflow}).\nUsing the facts that ${\\cal C}_{x_0}\\cdott_{x_0}$ is $C^1$-$O(e^{-c\/\\epsilon})$ close\nto $\\ws(\\Me)$ and that $W^u(0)$ intersects $\\ws(\\Me)$ transversely yield\nthe following lemma.\n\n\\begin{lemma}\\label{lem1brightwave} Suppose that\n\\[\n(d_1+\\frac32x_0^2d_2)\\sigma<0\n\\]\n(see Figure \\ref{fig_hompar}).\nThere exists a $\\mu_h(\\epsilon)<\\mu(\\epsilon)$, with $\\mu(\\epsilon)-\\mu_h(\\epsilon)=\nO(e^{-c\/\\epsilon})$, such that when $\\mu=\\mu_h(\\epsilon)$, then $W^u(0)\\capW^s(0)$\nis nontrivial.\n\\end{lemma}\n\n\\begin{remark} The fact that $\\mu(\\epsilon)-\\mu_h(\\epsilon)>0$ is a \nconsequence of Lemmas \\ref{lemuymu} and \\ref{lemsymu}.\n\\end{remark}\n\nThe solution described in the above lemma can be thought of as a\n1-pulse solution.\nIt is characterized by the fact $W^u(0)$ intersects the plane\n$\\{y=0\\}$ at exactly one point.\nAn $N$-pulse solution is characterized by both \n$W^u(0)\\capW^s(0)$ being nontrivial and $W^u(0)$ intersecting $\\{y=0\\}$ at\n$N$ distinct points.\nGiven the 1-pulse solution, it is natural to inquire as to the\nexistence of $N$-pulses.\nFortunately, this question has been studied in the work of\nKapitula and Maier-Paape \\cite{kapitula:sdo96}.\nIn order to quote the results stated in that paper, the next\nlemma must first be proved.\n\n\\begin{lemma}\\label{lemwoutransverse} Let $0<\\epsilon\\ll1$ be \nfixed.\nLet $z_h(\\mu)$ represent the $z$-coordinate of $W^u(0)\\cap\\{y=0\\}$.\nThen\n\\[\n\\begin{array}{ll}\n1.\\quad&z_h(\\mu_h(\\epsilon))=0 \\\\\n2.\\quad&{\\displaystyle\\frac{d}{d\\mu}z_h(\\mu_h(\\epsilon))\\neq0}.\n\\end{array}\n\\]\n\\end{lemma}\n\n\\noindent{\\bf Proof: } The first part of the conclusion follows immediately from\nthe fact that when $\\mu=\\mu_h(\\epsilon),\\,W^u(0)\\cap{\\cal C}_{x_0}\\neq\\emptyset$.\n\nFor fixed $\\epsilon$ the set $W^u(0)\\capB^+_{x_0}$ yields a curve parameterized\nby $\\mu$, i.e.,\n\\[\nW^u(0)\\capB^+_{x_0}=\\{(y,z,\\mu)\\,:\\,y=y^u(\\mu), z=z^u(\\mu)\\}.\n\\]\nDue to Lemma \\ref{lemuymu} it is known that\n\\[\n\\frac{d}{d\\mu}y^u(0)>0,\n\\]\nso that \n\\begin{equation}\\label{eqyumu}\n\\frac{d}{d\\mu}y^u(\\mu_h(\\epsilon))>0,\n\\end{equation}\nas $\\mu_h(\\epsilon)=O(\\mu(\\epsilon))=O(\\epsilon^2)$.\n\nLet $p=(x_0-\\nu,y^u(\\mu_h(\\epsilon)),z^u(\\mu_h(\\epsilon)))\\inW^u(0)\\capB^+_{x_0}$, \nand let $T_h>0$ be such that $p\\cdot T_h\\in{\\cal C}_{x_0}$.\nSince $z'=-\\epsilon^2\\sigma+O(\\epsilon^3)$ and $z^u(\\mu_h(\\epsilon))=O(\\epsilon)$, it\nis necessarily true that $T_h=O(1\/\\epsilon)$.\nTherefore, as a consequence of the Exchange Lemma with Exponentially\nSmall Error \\cite{jones:tim96}, the transversality described\nby (\\ref{eqyumu}) gets transferred into a transversality condition\nin the slow direction.\nSince the slow direction near $\\M_\\ep$ is described by $z$, this means\nthat \n\\[\n\\frac{d}{d\\mu}z_h(\\mu_h(\\epsilon))\\neq0.\\quad\\quad\\rule{1.5mm}{1.5mm}\n\\]\n\n\\vspace{3mm}\nWith the above lemma in hand, it is now possible to state Theorem \n1.7 in Kapitula and Maier-Paape \\cite{kapitula:sdo96}.\n\n\\begin{theorem}\\label{thmNbrightwaves} Let $0<\\epsilon\\ll1$, and suppose that\n\\[\n(d_1+\\frac32x_0^2d_2)\\sigma<0.\n\\]\nFor each $N\\ge2$ there\nexists a bi-infinite sequence $\\{\\mu^N_k\\}$ such that when\n$\\mu=\\mu^N_k$ there is an $N$-pulse solution to (\\ref{eqglodered}).\nThe $N$-pulse is such that $x(t)$ is even in $t$.\nFurthermore,\n\\[\n|\\mu^N_k-\\mu_h(\\epsilon)|=O(e^{-c|k|\/\\epsilon})\n\\]\nas $|k|\\to\\infty$.\n\\end{theorem}\n\n\\begin{remark} The estimate on $|\\mu^N_k-\\mu_h(\\epsilon)|$ is not\nexplicitly provided in Theorem 1.7 of Kapitula and Maier-Paape;\nhowever, it is implicit in the proof of that theorem.\n\\end{remark}\n\n\\begin{remark} Although one can discuss the existence of $N$-pulses\nwhich are odd in $t$, this will not be done here.\nFor a more complete description of the dynamical behavior for\n$\\mu$ near $\\mu_h(\\epsilon)$, the interested reader should consult\n\\cite{kapitula:sdo96}.\n\\end{remark}\n\n\\subsection{Existence of dark solitary waves}\n\nA dark solitary wave is characterized by\n\\[\n\\lim_{t\\to\\pm\\infty}(x,y,z)\\cdot t\\in\\M_\\ep;\n\\]\nhence, in order to have such a wave it is necessary that both\n$\\wu(\\Me)\\cap\\ws(\\Me)\\neq\\emptyset$ and there exist critical points\nin $\\M_\\ep$.\nBy Proposition \\ref{asssig_1}, the existence of the critical points\nis guaranteed if\n\\begin{equation}\\label{eqd1d2cond1}\n(d_1+2x_0^2d_2)\\sigma<0.\n\\end{equation}\nBy Lemma \\ref{lemtransverseatx0}, $\\wu(\\Me)\\cap\\ws(\\Me)\\neq\\emptyset$ if\n\\begin{equation}\\label{eqd1d2cond2}\n\\begin{array}{l}\n0<\\mu-\\mu(\\epsilon)=O(\\epsilon^n),\\quad n\\ge3 \\\\\n(d_1+x_0^2d_2)(\\alpha_1d_1+\\alpha_2d_2)>0,\n\\end{array}\n\\end{equation}\nwhere, by Lemma \\ref{lemsyz}, the coefficients $\\alpha_i$ satisfy\n\\[\n\\frac{\\alpha_2}{\\alpha_1}=x_0^2+\\frac14x_0^2(-\\ln\\frac{\\eta}{x_0})^{-1}\n+O(\\eta^2).\n\\]\nFurthermore, the manifolds intersect transversely, with the \ntransversality being $O(\\epsilon^{2n+1})$ (Lemma \\ref{lemtransverseatx0}).\n\nAn $N$-circuit solution is a dark solitary wave whose trajectory\npasses near $\\{x=0\\}\\,N$ times.\nIf (\\ref{eqd1d2cond1}) and (\\ref{eqd1d2cond2}) are satisfied, then\nfor $0<\\epsilon\\ll1$ there is a 1-circuit solution.\nThe goal is to show that there exist $N$-circuits for $2\\le N<N(\\epsilon)$,\nwhere $N(\\epsilon)\\to\\infty$ as $\\epsilon\\to0$.\nThis will be accomplished through the use of the Exchange Lemma with\nExponentially Small Error (ELESE) (Jones et al \\cite{jones:tim96}).\n\nIn order for the ELESE to apply, it must first be shown that\n$\\wu(\\Me)\\cap\\ws(\\Me)$ transversely.\nThis task was accomplished in the series of lemmas leading to Lemma\n\\ref{lemtransverseatx0}.\nThe ELESE then applies to all those points in\n\\[\n{\\cal C}^+_u=\\wu(\\Me)\\capB^+_{x_0}\n\\]\nwhich take $O(1\/\\epsilon)$ time to exit $B_{x_0}$.\n\nSet\n\\[\n{\\cal C}^+_s=\\ws(\\Me)B^+_{x_0}.\n\\]\nThe $z$-coordinate of ${\\cal C}^+_u\\cap{\\cal C}^+_s$, say $\\hat{z}$, is given by\n\\[\n\\hat{z}=-x_0(d_1+\\frac32x_0^2d_2+O(\\nu))\\epsilon+O(\\epsilon^2),\n\\]\ni.e., $|\\hat{z}-\\phi(T_\\nu)|=O(\\epsilon^2)$.\nThis is due to the assumption that $\\mu-\\mu(\\epsilon)=O(\\epsilon^n)$ for\nsome $n\\ge3$.\nSet\n\\[\n{\\cal C}^-_u=\\wu(\\Me)\\capB^-_{x_0},\\quad{\\cal C}^-_s=\\ws(\\Me)\\capB^-_{x_0}.\n\\]\nDue to the symmetry inherent in the ODE, not only is\n${\\cal C}^-_u\\cap{\\cal C}^-_s\\neq\\emptyset$, the $z$-coordinate of the\nintersection is given by $-\\hat{z}$.\nNow, recall that for $|z|=O(\\epsilon)$ the flow on $\\M_\\ep$ is given by\n\\[\nz'=-\\epsilon^2\\sigma+O(\\epsilon^3)\n\\]\n(equation \\ref{eqmeflowzsmall}).\nThus, if\n\\begin{equation}\\label{eqd1d2cond3}\n(d_1+\\frac32x_0^2d_2)\\sigma<0\n\\end{equation}\nthe flow on $\\M_\\ep$ is from $\\hat{z}$ to $-\\hat{z}$; otherwise, it\nis in the opposite direction.\n\nAssume that (\\ref{eqd1d2cond3}) holds.\nSince $z'=O(\\epsilon^2)$, it takes a time of $O(1\/\\epsilon)$ for trajectories\nto traverse from $\\hat{z}$ to $-\\hat{z}$.\nLet ${\\cal C}^+_u\\cdot T^+_u$ represent the intersection\nof the curve ${\\cal C}^+_u$ with the section $B^-_{x_0}$ as it is carried by the flow \ngenerated by the ODE (\\ref{eqglodered}).\nBy the ELESE the curve ${\\cal C}^+_u\\cdot T^+_u$ will be $C^1$-$O(e^{-c\/\\epsilon})$\nclose to ${\\cal C}^-_u$ at $-\\hat{z}$.\nSince ${\\cal C}^-_u\\cap{\\cal C}^-_s$ transversely at $-\\hat{z}$, with the order\nof transversality being $O(\\epsilon^{2n+1})$, this implies that \n${\\cal C}^+_u\\cdot T^+_u\\cap{\\cal C}^-_s$ transversely exponentially close to $-\\hat{z}$.\nIn other words, upon passing through $B_{x_0}$ the manifold\n$\\wu(\\Me)$ becomes oriented so that it once again intersects\n$\\ws(\\Me)$ transversely.\nGiven the existence of a 1-circuit solution, the existence of a\n2-circuit has now been proven.\n\nThe above argument can be repeated to show the existence of $N$-circuit\nsolutions for each $N\\ge2$.\nGiven an $\\epsilon>0$, the maximal $N$, say $N(\\epsilon)$, will be such that\n$N(\\epsilon)\\to\\infty$ as $\\epsilon\\to0$.\nThis is due to the fact that upon passage through $B_{x_0}$, the curve\n${\\cal C}^+_u\\cdot T^+_u$ is  $C^1$-$O(e^{-c\/\\epsilon})$\nclose to ${\\cal C}^-_u$ at $-\\hat{z}$.\nThe following theorem has now been proven.\n\n\\begin{theorem}\\label{thmNdarkwaves} Suppose that\n$0<\\epsilon\\ll1$, and suppose that (\\ref{eqd1d2cond1}), (\\ref{eqd1d2cond2}),\nand (\\ref{eqd1d2cond3}) hold (see Figure \\ref{fig_n_cirpar}).\nThen there exists an $N(\\epsilon)>1$, with $N(\\epsilon)\\to\\infty$ as\n$\\epsilon\\to0$, such that $N$-circuit solutions exist for $1\\le N<N(\\epsilon)$.\n\\end{theorem}\n\n\\subsection{Characterization of $\\M_\\ep$}\n\nThe next goal is to get an asymptotic description of $\\M_\\ep$ \nwithin $O(\\epsilon)$ of $\\{z=0\\}$.\nA preliminary lemma is first needed.\nIn the following, let the curve ${\\cal C}_{x_0}$ be given by\n\\begin{equation}\\label{eqdefcxo}\n{\\cal C}_{x_0}=\\{(x,y,z)\\,:\\,y=z=0,|x-x_0|\\le\\nu\\}.\n\\end{equation}\n\n\\begin{lemma}\\label{lemcxome} There exists an $\\epsilon_0>0$ such\nthat if $0\\le\\epsilon<\\epsilon_0$, then ${\\cal C}_{x_0}\\cap\\M_\\ep\\neq\\emptyset$.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } Clearly, $\\M_\\ep$ intersects the plane $\\{z=0\\}$ nontrivially.\nSet\n\\[\n\\{(x_\\epsilon,y_\\epsilon,0)\\}=\\M_\\ep\\cap\\{z=0\\}.\n\\]\nIn order to prove the lemma, it must be shown that $y_\\epsilon=0$ for $\\epsilon\\neq0$.\nBy the definition of $\\M_0$ and the fact that $\\M_\\ep$ is a smooth perturbation\nof $\\M_0$ it is clear that \n\\[\n{\\displaystyle\\lim_{\\epsilon\\to0}y_\\epsilon=0.}\n\\]\n\nIn a sufficiently small neighborhood of $\\M_\\ep$ the manifolds $\\ws(\\Me)$ and\n$\\wu(\\Me)$ satisfy the relation\n\\[\n\\ws(\\Me)\\cap\\wu(\\Me)=\\M_\\ep.\n\\]\nThis immediately yields that\n\\begin{equation}\\label{eqxepyep}\n\\ws(\\Me)\\cap\\wu(\\Me)\\cap\\{z=0\\}=\\{(x_\\epsilon,y_\\epsilon,0)\\}.\n\\end{equation}\nSuppose without loss of generality that\n\\[\n{\\displaystyle\\lim_{t\\to\\pm\\infty}(x_\\epsilon,y_\\epsilon,0)\\cdot t=(x^*,0,\\pm z^*)}.\n\\]\nBy the symmetry of the ODE, if $(x_\\epsilon,y_\\epsilon,0)\\cdot t$ is a \nsolution, then so is $(x_\\epsilon,-y_\\epsilon,0)\\cdot(-t)$.\nThis immediately yields that\n\\[\n{\\displaystyle\\lim_{t\\to\\pm\\infty}(x_\\epsilon,-y_\\epsilon,0)\\cdot t=(x^*,0,\\mp z^*)},\n\\]\nso that\n\\[\n\\{(x_\\epsilon,-y_\\epsilon,0)\\}\\subset\\ws(\\Me)\\cap\\wu(\\Me)\\cap\\{z=0\\}.\n\\]\nBut equation (\\ref{eqxepyep}) then gives\n\\[\n(x_\\epsilon,y_\\epsilon,0)=(x_\\epsilon,-y_\\epsilon,0)\n\\]\nso that $y_\\epsilon=0$.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\nFor the rest of this subsection set $\\zeta=|z|+\\epsilon$.\nThe slow manifold $\\M_\\ep$ is given by the graph\n\\begin{equation}\\label{eqdefme}\n\\M_\\ep=\\{(x,y,z,e)\\,:\\,x=\\M_x(z,\\epsilon),y=\\M_y(z,\\epsilon)\\}.\n\\end{equation}\nBy the definition of $\\M_0$ the function $\\M_y$ satisfies\nthe relation $\\M_y(z,0)=0$.\nFurthermore, the conclusion of Lemma \\ref{lemcxome} yields that\n$\\M_y(0,\\epsilon)=0$.\nThus, a Taylor expansion of $\\M_y$ gives\n\\begin{equation}\\label{eqdefmy}\n\\M_y(z,\\epsilon)=\\epsilon z\\tilde\\M_y(z,e),\n\\end{equation}\nwhere the function $\\tilde\\M_y$ is uniformly bounded for $\\zeta$\nsufficiently small.\nThe following lemma gives a description of the functions $\\M_x$ and\n$\\tilde\\M_y$.\n\n\\begin{lemma}\\label{lemdescribeme} The functions $\\M_x$ and $\\M_y$ \ncomprising the graph of $\\M_\\ep$ have the Taylor expansion\n\\[\n\\begin{array}{lll}\n\\M_x(z,\\epsilon)&=&{\\displaystyle x_0+\\frac1{\\lambda'(x_0)}z^2+\\tilde{x}\\epsilon^2+O(\\zeta^3)}\\\\\n\\M_y(z,\\epsilon)&=&{\\displaystyle\\epsilon z(-\\frac{2\\sigma}{x_0\\lambda'(x_0)}\\epsilon+O(\\zeta^2))},\n\\end{array}\n\\]\nwhere $\\tilde x$ is an $O(1)$ constant.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } The basic idea of the proof is to write out a Taylor expansion\nfor both $\\M_x$ and $\\M_y$ and to use the fact that the manifold $\\M_\\ep$\nis invariant under the flow.\nAfter using (\\ref{eqdefmy}) the functions can be expanded as\n\\[\n\\begin{array}{lll}\n\\M_x(z,\\epsilon)&=&x_0+x_1z+x_2\\epsilon+x_3z^2+x_4\\epsilon z+x_5\\epsilon^2+O(\\zeta^3) \\\\\n\\M_y(z,\\epsilon)&=&\\epsilon z(y_0+y_1z+y_2\\epsilon+O(\\zeta^2)),\n\\end{array}\n\\]\nwhere $x_0$ is such that $\\lambda(x_0)=0$.\nIn order to prove the lemma, it must be shown that\n\\[\nx_1=x_2=x_4=y_0=y_1=0\n\\]\nand \n\\[\nx_3=\\frac1{\\lambda'(x_0)},\\,y_2=-\\frac{2\\sigma x_3}{x_0}.\n\\]\n\nWhen $\\epsilon=0$ the function $\\M_x$ satisfies the relation\n\\[\nz^2-\\lambda(\\M_x(z,0))=0.\n\\]\nTaking a Taylor expansion of $\\lambda(x)$ about $x_0$ and equating\ncoefficients quickly yields the desired relation for $x_1$ and\n$x_3$.\n\nIn order to show that $x_2=0$, consider the following \nargument.\nThe flow on $\\M_\\ep$ satisfies\n\\begin{equation}\\label{eqmegeneralflow}\nz'=-2\\M_y(z,\\epsilon)z+\\epsilon(\\omega(\\M_x(z,\\epsilon))-\\omega(x_0)-\\epsilon\\sigma),\n\\end{equation}\nso that when $z=0$\n\\[\nz'=\\epsilon((x_2\\omega'(x_0)-\\sigma)\\epsilon+O(\\epsilon^2)).\n\\]\nThe above expression is arrived at upon taking a Taylor expansion\nof $\\omega(x)$ about $x_0$ and using the expansion for $\\M_x(0,\\epsilon)$.\nNote that $z'=O(\\epsilon^2)$.\nNow, the fact that on $\\M_\\ep,\\,y=\\M_y(z,\\epsilon)$ implies that\n\\[\ny'=z'\\partial_z\\M_y(z,\\epsilon).\n\\]\nUsing the Taylor expansion for $\\M_y$ and the fact that $z'=O(\\epsilon^2)$\nyields that when $z=0,\\,y'=O(\\epsilon^3)$.\nBut when $z=0$,\n\\[\ny'=-\\M_y^2(0,\\epsilon)-\\lambda(\\M_x(0,\\epsilon))+\\mu(\\epsilon).\n\\]\nSince $\\mu(\\epsilon)=O(\\epsilon^2)$, after taking a Taylor expansion of \n$\\lambda(x)$ about $x_0$ one gets that\n\\[\ny'=-x_2\\lambda'(x_0)\\epsilon+O(\\epsilon^2).\n\\]\nThus, the fact that $y'=O(\\epsilon^3)$ necessarily gives that $x_2=0$.\n\nUsing the above results and equation (\\ref{eqmegeneralflow}) gives\nthat the equation for $z$ on $\\M_\\ep$ is\n\\[\nz'=-\\sigma\\epsilon^2+(x_3\\omega'(x_0)-2y_0)\\epsilon z^2+x_4\\omega'(x_0)\\epsilon^2z\n  +x_5\\omega'(x_0)\\epsilon^3+O(\\zeta^4).\n\\]\nSince $x'=xy$, on $\\M_\\ep$\n\\begin{equation}\\label{eqmxprime}\n\\M_x'=\\M_x\\M_y.\n\\end{equation}\nBut\n\\[\n\\begin{array}{lll}\n\\M_x'&=&z'\\partial_z\\M_x \\\\\n{ } &=&-2\\sigma x_3\\epsilon^2z-\\sigma x_4\\epsilon^3+O(\\zeta^4)\n\\end{array}\n\\]\nand\n\\[\n\\M_x\\M_y=x_0y_0\\epsilon z+x_0y_1\\epsilon z^2+x_0y_2\\epsilon^2z+O(\\zeta^4),\n\\]\nso that (\\ref{eqmxprime}) becomes\n\\[\n-2\\sigma x_3\\epsilon^2z-\\sigma x_4\\epsilon^3+O(\\zeta^4)=\n    x_0y_0\\epsilon z+x_0y_1\\epsilon z^2+x_0y_2\\epsilon^2z+O(\\zeta^4).\n\\]\nUpon equating coefficients the desired result is reached.\n\\quad\\rule{1.5mm}{1.5mm}\n\n\\vspace{3mm}\n\\subsection{Flow near $\\M_\\ep$}\n\nRecall that the flow on $\\M_\\ep$ is given by (\\ref{eqmegeneralflow}).\nUsing the result of Lemma \\ref{lemdescribeme} and Taylor expanding\n$\\omega(x)$ around $x_0$ then yields that on $\\M_\\ep$ the flow is\ngiven by\n\\begin{equation}\\label{eqmeflowestimate}\n{\\displaystyle z'=\\epsilon(\\frac{\\omega'(x_0)}{\\lambda'(x_0)}z^2-\\sigma\\epsilon+O(\\epsilon^2))+O(\\zeta^4)}.\n\\end{equation}\nNote that the above equation implies that for $|z|=O(\\epsilon)$,\n\\begin{equation}\\label{eqmeflowzsmall}\nz'=-\\epsilon^2\\sigma+O(\\epsilon^3).\n\\end{equation}\nThis estimate will turn out to be crucial for the estimates that follow.\n\nIn order to determine the nature of the flow near $\\M_\\ep$, it will\nbe desirous to use Fenichel coordinates (\\cite{jones:gsp95},\n\\cite{jones:tif93}, \\cite{jones:tim96}).\nBefore doing so, it will first be beneficial to understand the\nmatrix arrived at upon linearizing about $\\M_\\ep$.\nSpecifically, the manner in which the eigenvalues and eigenvectors\nvary as a function of $z$ and $\\epsilon$ must be understood.\n\nLinearizing the flow (\\ref{eqglodered})\nabout a point $(x,y,z)\\in\\M_\\ep$ yields the matrix\n\\begin{equation}\\label{eqmelinear}\nA_{\\M_\\ep}=\\left(\\begin{array}{rrr}\n   y & x & 0 \\\\\n   -\\lambda'(x) & -2y & 2z \\\\\n   \\epsilon\\omega'(x) & -2z & -2y,\n      \\end{array}\\right)\n\\end{equation}\nwhich has the characteristic equation\n\\begin{equation}\\label{eqmechareq} \nP(\\gamma,z,\\epsilon)=(\\gamma-y)((\\gamma+2y)^2+4z^2)+x(\\lambda'(x)(\\gamma+2y)-2\\epsilon\\omega'(x)z)=0.\n\\end{equation}\nThe above equation has two solutions which are $O(1)$, say $\\gamma^\\pm(z,\\epsilon)$,\nwith $\\gamma^+=-\\gamma^-$ and $\\gamma^-(0,0)=-\\sqrt{-x_0\\lambda'(x_0)}$.\n\nIn the following calculations the result of Lemma \\ref{lemdescribeme}\nwill always be implicitly used.\nIt is easy to check that \n\\[\nP_\\gamma(\\gamma^-,0,0)=-2x_0\\lambda'(x_0)\n\\]\nand \n\\[\nP_z(\\gamma^-,0,0)=P_\\epsilon(0,0)=0,\n\\]\nso that \n\\[\n\\gamma^-_z(0,0)=\\gamma^-_\\epsilon(0,0)=0.\n\\]\nThe above follows immediately from the fact that\n\\[\n\\gamma^-_z=-P_z\/P_\\gamma,\\quad\\gamma^-_\\epsilon=-P_\\epsilon\/P_\\gamma.\n\\]\n\nTaking second derivatives gives \n\\[\\begin{array}{lll}\nP_{zz}(\\gamma^-,0,0)&=&-8\\sqrt{-x_0\\lambda'(x_0)} \\\\\nP_{z\\epsilon}(\\gamma^-,0,0)&=&-2x_0\\omega'(x_0).\n  \\end{array}\n\\]\nSince\n\\[\n\\gamma_{zz}=-P_{zz}\/P_\\gamma,\\quad\\gamma_{z\\epsilon}=-P_{z\\epsilon}\/P_\\gamma,\n\\]\nthis immediately gives that\n\\[\\begin{array}{lll}\n\\gamma^-_{zz}(0,0)&=&{\\displaystyle\\frac4{\\sqrt{-x_0\\lambda'(x_0)}}} \\\\\n\\gamma^-_{z\\epsilon}(0,0)&=&{\\displaystyle-\\frac{\\omega'(x_0)}{\\lambda'(x_0)}}.\n  \\end{array}\n\\]\nThe proof of the following proposition is now complete. \n\n\\begin{prop}\\label{propevalsme} The $O(1)$ eigenvalues $\\gamma^\\pm$ \nof $A_{\\M_\\ep}$ satisfy\n\\[\\begin{array}{ll}\n1)\\quad&\\gamma^+(z,\\epsilon)=-\\gamma^-(z,\\epsilon) \\\\\n2)\\quad&{\\displaystyle\\gamma^-(z,\\epsilon)=-\\sqrt{-x_0\\lambda'(x_0)}+\n                  \\frac2{\\sqrt{-x_0\\lambda'(x_0)}}z^2- \n                  \\frac{\\omega'(x_0)}{\\lambda'(x_0)}\\epsilon z+\n                  \\tilde{\\gamma}\\epsilon^2+O(\\zeta^3)},\n  \\end{array}\n\\]\nwhere $\\tilde{\\gamma}$ is an $O(1)$ constant.\n\\end{prop}\n\nAlthough the following information is not necessary at the\nmoment, it will be useful at a later time.\nThe eigenvector ${\\bf v}_{\\gamma^-}$ of $A_{\\M_\\ep}$ associated with\nthe eigenvalue $\\gamma^-$ is given by\n\\begin{equation}\\label{eqevector}\n{\\displaystyle{\\bf v}_{\\gamma^-}=(x,\\gamma^--y,\\frac{-2(\\gamma^--y)z+\\epsilon x\\omega'(x)}{\\gamma^-+2y})^T}.\n\\end{equation}\nLet ${\\bf v}_{\\gamma^-}=(v_1,v_2,v_3)^T$.\nAfter using the expansions given in Lemma \\ref{lemdescribeme} and \nProposition \\ref{propevalsme} the following proposition is realized.\n\n\\begin{prop}\\label{propevectsme} The components of the eigenvector \n${\\bf v}_{\\gamma_-}$ are given by the expansions\n\\[\n\\begin{array}{lll}\nv_1&=&{\\displaystyle x_0+\\frac1{\\lambda'(x_0)}z^2+\\tilde{x}\\epsilon^2+O(\\zeta^3)}\\\\\nv_2&=&{\\displaystyle-\\sqrt{-x_0\\lambda'(x_0)}+\n                  \\frac2{\\sqrt{-x_0\\lambda'(x_0)}}z^2- \n                  \\frac{\\omega'(x_0)}{\\lambda'(x_0)}\\epsilon z+\n                  \\tilde{\\gamma}\\epsilon^2+O(\\zeta^3)} \\\\\nv_3&=&{\\displaystyle-2z-\\frac{x_0\\omega'(x_0)}{\\sqrt{-x_0\\lambda'(x_0)}}\\epsilon+O(\\zeta^2)},\n\\end{array}\n\\]\nwhere $\\tilde{x}$ and $\\tilde{\\gamma}$ are $O(1)$ constants.\n\\end{prop}\n\n\\begin{remark} Due to the symmetry of the ODE, the eigenvector \nassociated with $\\gamma^+$ is such that the second and third \ncomponents are the negative of those given above.\n\\end{remark}\n\n\\begin{remark}\\label{remstrongstable}\nWhen the above expressions are evaluated at the critical points\n$(x^*(\\epsilon),0,\\pm z^*(\\epsilon))$, it is of interest to note\nthat the third component, $v_3$, satisfies the estimate\n\\[\nv_3=-2(\\pm z^*(\\epsilon))+O(\\epsilon).\n\\]\nThe above expression is valid because it is known that \n$z^*(\\epsilon)=O(\\epsilon^{1\/2})$.\n\\end{remark}\n\nRecall that for $|z|=O(\\epsilon)$ that the flow on $\\M_\\ep$ satisfies\n\\[\nz'=-\\epsilon^2\\sigma+O(\\epsilon^3).\n\\]\nThus, in Fenichel coordinates the equations near $\\M_\\ep$ for \n$|z|=O(\\epsilon)$ may be written as\n\\begin{equation}\\label{eqfenichel}\n\\begin{array}{lll}\na'&=&\\Lambda(a,b,z,\\epsilon)a \\\\\nb'&=&\\Gamma(a,b,z,\\epsilon)b \\\\\nz'&=&\\epsilon^2(-\\sigma+h(a,b,z,\\epsilon)ab).\n\\end{array}\n\\end{equation}\nIn the above equation the set $a=0$ refers to $\\ws(\\Me)$, the set\n$b=0$ refers to $\\wu(\\Me)$, and the function $h$ is uniformly \nbounded.\nUsing the eigenvalue expansion in Proposition \\ref{propevalsme}\none can say that for $|z|=O(\\epsilon)$,\n\\[\n\\Lambda(0,0,z,\\epsilon)=\\sqrt{-x_0\\lambda'(x_0)}+O(\\epsilon^2),\\quad\n\\Gamma(0,0,z,\\epsilon)=-\\Lambda(0,0,z,\\epsilon).\n\\]\nAfter taking a Taylor expansion the equations (\\ref{eqfenichel})\nmay then be rewritten as\n\\begin{equation}\\label{eqfenichel2}\n\\begin{array}{lll}\na'&=&\\sqrt{-x_0\\lambda'(x_0)}\\,a+O((|a|+|b|+\\epsilon)^2) \\\\\nb'&=&-\\sqrt{-x_0\\lambda'(x_0)}\\,b+O((|a|+|b|+\\epsilon)^2)\\\\\nz'&=&\\epsilon^2(-\\sigma+h(a,b,z,\\epsilon)ab).\n\\end{array}\n\\end{equation}\n\nDefine the set $B_{x_0}$ by\n\\begin{equation}\\label{eqdefbxo}\nB_{x_0}=\\{(a,b,z)\\,:\\,|a|+|b|\\le\\nu,|z|\\le O(\\epsilon)\\}.\n\\end{equation}\nFor $\\epsilon$ and $\\nu$ sufficiently small it is clear that one can, without\nloss of generality, use the linear equations\n\\begin{equation}\\label{eqfenichellinear1}\n\\begin{array}{lll}\na'&=&\\sqrt{-x_0\\lambda'(x_0)}\\,a \\\\\nb'&=&-\\sqrt{-x_0\\lambda'(x_0)}\\,b \\\\\nz'&=&-\\epsilon^2\\sigma\n\\end{array}\n\\end{equation}\nwhen discussing the flow in $B_{x_0}$.\nFor convenience, in the rest of this subsection $\\gamma=\\sqrt{-x_0\\lambda'(x_0)}$,\nso that (\\ref{eqfenichellinear1}) can be rewritten as\n\\begin{equation}\\label{eqfenichellinear}\n\\begin{array}{lll}\na'&=&\\gamma a \\\\\nb'&=&-\\gamma b \\\\\nz'&=&-\\epsilon^2\\sigma.\n\\end{array}\n\\end{equation}\n\nNow that the equations for the flow near $\\M_\\ep$ have been determined,\nit will be of interest to determine the nature of the set ${\\cal C}_{x_0}$\nas it exits $B_{x_0}$ under the influence of the flow.\nTo be precise, for each $p\\in{\\cal C}_{x_0}$ define $t_{x_0}$ by\n\\[\n{\\displaystyle t_{x_0}(p)=\\{\\sup_{t<0}\\,:\\,p\\cdot t\\inB^+_{x_0}\\}},\n\\]\nand set\n\\begin{equation}\\label{eqdefcxoflow}\n{\\displaystyle{\\cal C}_{x_0}\\cdott_{x_0}=\\bigcup_{p\\in{\\cal C}_{x_0}}p\\cdot t_{x_0}(p).}\n\\end{equation}\nThe set ${\\cal C}_{x_0}\\cdott_{x_0}$ will be close to the set $a=0$, i.e.,\nclose to $\\ws(\\Me)$, as it exits $B^+_{x_0}$, where\n\\begin{equation}\\label{eqdefbpxo}\nB^+_{x_0}=\\{(x,y,z)\\,:\\,x=x_0-\\nu,|y-(\\sqrt{-\\lambda'(x_0)\/x_0})\\nu|\\le\\nu,\n        |z|\\le\\nu\\}\n\\end{equation}\nThe following lemma gives a determination as to how close.\n\n\\begin{lemma}\\label{lemimagecxo} Let $p\\in{\\cal C}_{x_0}$ be such that \n$t_{x_0}(p)=O(1\/\\epsilon)$.\nThe curve ${\\cal C}_{x_0}\\cdott_{x_0}$ is $C^1-O(e^{-c\/\\epsilon})$ close to\n$a=0$ in a neighborhood of $p\\cdott_{x_0}(p)$,\nwhere $c>0$ is some constant.\n\\end{lemma}\n\n\\noindent{\\bf Proof: } In order to prove the lemma the time-reversed flow for\n(\\ref{eqfenichellinear}) must be considered.\nAfter such a time reversal, the solution to (\\ref{eqfenichellinear}) is\ngiven by\n\\begin{equation}\\label{eqfenproof}\n\\begin{array}{lll}\na(t)&=&a_0e^{-\\gamma t} \\\\\nb(t)&=&b_0e^{\\gamma t} \\\\\nz(t)&=&z_0+\\epsilon^2\\sigma t.\n\\end{array}\n\\end{equation}\nIn Fenichel coordinates the curve ${\\cal C}_{x_0}$ can be parametrically represented \nas $(a_0(s),b_0(s),z_0(s))=(s,s,0)$ with $0\\le s\\le\\nu$.\nSubstituting this set of initial conditions into (\\ref{eqfenproof}) yields \nthat the flow of ${\\cal C}_{x_0}$ is governed by\n\\begin{equation}\\label{eqfenproof2}\n\\begin{array}{lll}\na(t)&=&se^{-\\gamma t} \\\\\nb(t)&=&se^{\\gamma t} \\\\\nz(t)&=&\\epsilon^2\\sigma t.\n\\end{array}\n\\end{equation}\n\nA solution to (\\ref{eqfenproof2}) leaves $B_{x_0}$ when $b(t)=\\nu$.\nSolving this equation yields that the time-of-flight, $t_f$, is\ngiven by\n\\[\nt_f=\\frac1{\\gamma}\\ln\\frac{\\nu}{s}.\n\\]\nSubstituting $t_f$ into (\\ref{eqfenproof2}) yields\n\\begin{equation}\\label{eqfenproof3}\n\\begin{array}{lll}\na(t_f)&=&{\\displaystyle\\frac{s^2}{\\nu}} \\\\\nz(t_f)&=&{\\displaystyle-\\epsilon^2\\frac1{\\gamma}\\ln\\frac{\\nu}{s}}.\n\\end{array}\n\\end{equation}\n\nThe above equation is a parametric representation of a curve on\nthe section $B^+_{x_0}$.\nIt can be rewritten as\n\\[\na(z)=\\nu e^{\\beta z\/\\epsilon^2},\n\\]\nwhere \n\\[\n\\beta=2\\gamma\/\\sigma,\\quad\\sigma z<0.\n\\]\nThe conclusion of the lemma is now clear, as when $z=O(\\epsilon)$ with \n$\\sigma z<0$,\n\\[\na(z)=O(e^{-c\/\\epsilon}),\\quad a'(z)=O(\\frac{e^{-c\/\\epsilon}}{\\epsilon^2}),\\quad\\quad\\rule{1.5mm}{1.5mm}.\n\\]\n\n\\vspace{3mm}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#2} \\label{sec:#1} \\input{#1}}\n\n\\newcommand{\\textsc{Mixer}\\xspace}{\\textsc{Mixer}\\xspace}\n\\newcommand{\\textsc{Utility}\\xspace}{\\textsc{Utility}\\xspace}\n\\newcommand{\\textsc{Bleu}\\xspace}{\\textsc{Bleu}\\xspace}\n\\newcommand{\\textsc{Rouge}\\xspace}{\\textsc{Rouge}\\xspace}\n\\newcommand{\\textsc{Meteor}\\xspace}{\\textsc{Meteor}\\xspace}\n\\newcommand{\\textsc{Diversity}\\xspace}{\\textsc{Diversity}\\xspace}\n\\newcommand{\\textsc{Reinforce}\\xspace}{\\textsc{Reinforce}\\xspace}\n\\newcommand{\\mathbb{U}}{\\mathbb{U}}\n\n\\renewcommand\\cite\\citep\n\n\\title{Answer-based Adversarial Training for\\\\Generating Clarification Questions}\n\n\\author{Sudha Rao\\Thanks{This research performed when the author was still at University of Maryland, College Park.} \\\\\nMicrosoft Research, Redmond \\\\\n{\\tt Sudha.Rao@microsoft.com} \\\\ \\And\nHal Daum\\'e III \\\\\nUniversity of Maryland, College Park \\\\\nMicrosoft Research, New York City\\\\\n{\\tt me@hal3.name} }\n\n\n\n\\date{}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nWe present an approach for generating clarification questions with the goal of eliciting new information that would make the given textual context more complete. \nWe propose that modeling hypothetical answers (to clarification questions) as latent variables can guide our approach into generating more useful clarification questions.\nWe develop a Generative Adversarial Network (GAN) where the generator is a sequence-to-sequence model and the discriminator is a utility function that models the value of updating the context with the answer to the clarification question. We evaluate on two datasets, using both automatic metrics and human judgments of usefulness, specificity and relevance, showing that our approach outperforms both a retrieval-based model and ablations that exclude the utility model and the adversarial training.\n\\end{abstract}\n\n\\includesection{intro}{Introduction}\n\\includesection{model}{Training a Clarification Question Generator}\n\\includesection{experiments}{Experimental Results}\n\\includesection{related}{Related Work}\n\\includesection{conclusion}{Conclusion}\n\n\\section*{Acknowledgments}\n\nWe thank the three anonymous reviewers for their helpful comments and suggestions. We also thank the members of the Computational Linguistics and Information Processing (CLIP) lab at University of Maryland for helpful discussions. This work was supported by NSF grant IIS-1618193. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the view of the sponsors.\n\n\n\n\\subsection{Datasets} \\label{sec:datasets}\n\nWe evaluate our model on two datasets. \n\n\\textbf{Amazon.}\nIn this dataset, \\emph{context} is a product description on amazon.com combined with the product title, \\emph{question} is a clarification question asked to the product and \\emph{answer} is the seller's (or other users') reply to the question. \nTo obtain these data triples, we combine the Amazon question-answering dataset \\cite{mcauley2016addressing} with the Amazon reviews dataset  \\cite{mcauley2015image}.\nWe show results on the \\texttt{Home \\& Kitchen} category of this dataset since it contains a large number of questions and is relatively easier for human-based evaluation. \nIt consists of $19,119$ training, $2,435$ tune and $2,305$ test examples (product descriptions), with 3 to 10 questions (average: 7) per description.\n\n\\textbf{Stack Exchange.}\nIn this dataset, \\emph{context} is a post on stackexchange.com combined with the title, \\emph{question} is a clarification question asked in the comments section of the post and \n\\emph{answer} is either the update made to the post in response to the question or the author's reply to the question in the comments section.\n\\citet{rao2018learning} curated a dataset of $61,681$ training, $7,710$ tune and $7,709$ test such triples from three related subdomains on stackexchage.com (askubuntu, unix and superuser).\nAdditionally, for 500 instances each from the tune and the test set, their dataset includes 1 to 6 other questions identified as valid questions by expert human annotators from a pool of candidate questions. \n\n\\subsection{Baselines and Ablated Models}\n\nWe compare three variants (ablations) of our proposed approach, together with an information retrieval baseline:\n\n\\textbf{GAN-Utility} is our full model which is a \\textsc{Utility}\\xspace calculator based GAN training (\\autoref{sec:gan}) including the \\textsc{Utility}\\xspace discriminator and the \\textsc{Mixer}\\xspace question generator.\\footnote{Experimental details are in \\autoref{sec:appendix-exp-details}.}\n\n\\textbf{Max-Utility} is our reinforcement learning baseline where the pretrained question generator model is further trained to optimize the  \\textsc{Utility}\\xspace reward (\\autoref{sec:mixer}) without the adversarial training.\n\n\\textbf{MLE} is the question generator model pretrained on context, question pairs using maximum likelihood objective (\\autoref{sec:seq2seq}).\n\n\\textbf{Lucene}\\footnote{\\url{https:\/\/lucene.apache.org\/}} is our information retrieval baseline similar to the Lucene baseline described in \\citet{rao2018learning}. \nGiven a context in the test set, we use Lucene, which is a TF-IDF based document ranker, to retrieve top 10 contexts that are most similar to the given context in the train set. \nWe randomly choose a question from the human written questions paired with these 10 contexts in the train set to construct our Lucene baseline\\footnote{For the Amazon dataset, we ignore questions asked to products of the same brand as the given product since Amazon replicates questions across same brand allowing the true question to be included in that set.}. \n\n\\begin{toappendix}\n\n\n \\section{Experimental Details}\\label{sec:appendix-exp-details}\n \n  In this section, we describe the details of our experimental setup. \n  \n We preprocess all inputs (context, question and answers) using tokenization and lowercasing. \n We set the max length of context to be 100, question to be 20 and answer to be 20. \n We test with context length 150 and 200 and find that the automatic metric results are similar as that of context length 100 but the experiments take much longer. Hence, we set the max context length to be 100 for all our experiments. \n Similarity, we find that an increased length of question and answer yields similar results with increased experimentation time. \n \n Our sequence-to-sequence model (Section 2.1) operates on word embeddings which are pretrained on in domain data using Glove \\citep{pennington2014glove}.\n As frequently used in previous work on neural network modeling, we use an embeddings of size 200 and a vocabulary with cut off frequency set to 10. \n During train time, we use teacher forcing \\citep{williams1989learning}. \n During test time, we use beam search decoding with beam size 5.\n \n We use a hidden layer of size two for both the encoder and decoder recurrent neural network models with size of hidden unit set to 100. \n We use a dropout of 0.5 and learning ratio of 0.0001. \n In the \\textsc{Mixer}\\xspace model, we start with $\\Delta = T$ and decrease it by 2 for every epoch (we found decreasing $\\Delta$ to 0 is ineffective for our task, hence we stop at 2).\n\n\\end{toappendix}\n\n\\subsection{Evaluation Metrics} \\label{sec:methodology}\n\nWe evaluate initially with automated evaluation metrics, and then more substantially with crowdsourced human judgments.\n\n\\subsubsection{Automatic Metrics}\n\\textbf{Diversity}, which calculates the proportion of unique trigrams in the output to measure the diversity as commonly used to evaluate dialogue generation \\citep{li2016deep}.\\\\\n\\textbf{\\textsc{Bleu}\\xspace} \\cite{papineni2002bleu} \\footnote{\\small\\url{https:\/\/github.com\/moses-smt\/mosesdecoder\/blob\/master\/scripts\/generic\/multi-bleu.perl}}, which evaluates n-gram precision between the output and the references. \\\\\n\\textbf{\\textsc{Meteor}\\xspace} \\cite{banerjee2005meteor}, which is similar to \\textsc{Bleu}\\xspace but includes stemmed and synonym matches to measure similarity between the output and the references.\n\n\\subsubsection{Human Judgements}\n\nWe use Figure-Eight\\footnote{\\url{https:\/\/www.figure-eight.com}}, a crowdsourcing platform, to collect human judgements. \nEach judgement\\footnote{We paid crowdworkers 5 cents per judgment and collected five judgments per question.} consists of showing the crowdworker a context and a generated question and asking them to evaluate the question along following axes:\n\n\\noindent \\textbf{Relevance}: We ask \\textit{``Is the question on topic?\"}\n and let workers choose from: Yes (1) and No (0)\n\n\\noindent \\textbf{Grammaticality}: We ask \\textit{``Is the question grammatical?\"} and let workers choose from: Yes (1) and No (0)\n\n\\noindent \\textbf{Seeking new information}: We ask \\textit{``Does the question ask for new information currently not included in the description?''} and let workers choose from: Yes (1) and No (0)\n\n\\noindent \\textbf{Specificity}: We ask \\textit{``How specific is the question?''} and let workers choose from: \n \\begin{description}[noitemsep,nolistsep]\n \\item 4: Specific pretty much only to this product (or same product from different manufacturer)\n \\item 3: Specific to this and other very similar products\n \\item 2: Generic enough to be applicable to many other products of this type\n \\item 1: Generic enough to be applicable to any product under Home and Kitchen\n \\item 0: N\/A (Not applicable) i.e. Question is not on topic OR  is incomprehensible\n \\end{description}\n\n\\noindent \\textbf{Usefulness}: We ask \\textit{``How useful is the question to a potential buyer (or a current user) of the product?''} and let workers choose from:\n \\begin{description}[noitemsep,nolistsep]\n \\item 4: Useful enough to be included in the product description\n \\item 3: Useful to a large number of potential buyers (or current users)\n \\item 2: Useful to a small number of potential buyers (or current users)\n \\item 1: Useful only to the person asking the question\n \\item 0: N\/A (Not applicable) i.e. Question is not on topic OR is incomprehensible  OR is not seeking new information\n \\end{description}\n\n\\subsubsection{Inter-annotator Agreement}\n\n\n\\autoref{tab:agreement} shows the inter-annotator agreement (reported by Figure-Eight as confidence\\footnote{\\small\\url{https:\/\/success.figure-eight.com\/hc\/en-us\/articles\/201855939-How-to-Calculate-a-Confidence-Score}}) on each of the above five criteria.\nAgreement on \\textit{Relevance}, \\textit{Grammaticality} and \\textit{Seeking new information} is high. This is not surprising given that these criteria are not very subjective.\nOn the other hand, the agreement on usefulness and specificity is quite moderate since these judgments can be very subjective. \n\n\n\\begin{table}[t]\n\\begin{tabular}{l c}\n\\bf Criteria & \\bf Agreement \\\\\n\\midrule\nRelevance & 0.92 \\\\\nGrammaticality & 0.92 \\\\\nSeeking new information & 0.84 \\\\\nUsefulness & 0.65 \\\\\nSpecificity & 0.72 \\\\\n\\end{tabular}\n\\caption{Inter-annotator agreement on the five criteria used in human-based evaluation.}\\label{tab:agreement}\n\\vspace{-1.1em}\n\\end{table}\n\nSince the inter-annotator agreement on the usefulness criteria was particularly low, in order to reduce the subjectivity involved in the fine grained annotation,\nwe convert the range [0-4] to a more coarse binary range [0-1] by mapping the scores 4 and 3 to \\textbf{1} and the scores 2, 1 and 0 to \\textbf{0}. \n\n\\subsection{Automatic Metric Results} \\label{sec:results}\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{l  c c c  |  c c c}\n  \\toprule\n& \\multicolumn{3}{c|}{\\texttt{Amazon}} & \\multicolumn{3}{c}{\\texttt{StackExchange}} \\\\\nModel & \\textsc{Diversity}\\xspace & \\textsc{Bleu}\\xspace & \\textsc{Meteor}\\xspace & \\textsc{Diversity}\\xspace & \\textsc{Bleu}\\xspace & \\textsc{Meteor}\\xspace \\\\\n\\midrule\nReference & 0.6934 & --- & --- & 0.7509 & --- & ---  \\\\\nLucene & 0.6289 & 4.26 & 10.85 & 0.7453 & 1.63 & 7.96  \\\\\n\\midrule\nMLE & 0.1059 & \\bf 17.02 & 12.72 &  0.2183 & 3.49 & 8.49  \\\\\nMax-Utility & 0.1214 & 16.77 & 12.69 &  \\bf 0.2508 & 3.89 & 8.79  \\\\\nGAN-Utility & \\bf 0.1296 & 15.20 & \\bf 12.82 &  0.2256 & \\bf 4.26 & \\bf 8.99 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{\\textsc{Diversity}\\xspace as measured by the proportion of unique trigrams in model outputs. Bigrams and unigrams follow similar trends. \\textsc{Bleu}\\xspace and \\textsc{Meteor}\\xspace scores using up to 10 references for the Amazon dataset and up to six references for the StackExchange dataset. Numbers in bold are the highest among the models. All results for Amazon are on the entire test set whereas for StackExchange they are on the 500 instances of the test set that have multiple references.}\\label{tab:results}\n\\end{table*}\n\n\n\\autoref{tab:results} shows the results on the two datasets when evaluated according to automatic metrics. \n\nIn the Amazon dataset, GAN-Utility outperforms all ablations on \\textsc{Diversity}\\xspace, suggesting that it produces more diverse outputs.\nLucene, on the other hand, has the highest \\textsc{Diversity}\\xspace since it consists of human written questions, which tend to be more diverse because they are much longer compared to model generated questions.\nThis comes at the cost of lower match with the reference as visible in the \\textsc{Bleu}\\xspace and \\textsc{Meteor}\\xspace scores. \nIn terms of \\textsc{Bleu}\\xspace and \\textsc{Meteor}\\xspace, there is inconsistency.\nAlthough GAN-Utility outperforms all baselines according to \\textsc{Meteor}\\xspace, the fully ablated MLE model has a higher \\textsc{Bleu}\\xspace score. \nThis is because \\textsc{Bleu}\\xspace score looks for exact n-gram matches and since MLE produces more generic outputs, it is much more likely that it will match one of 10 references compared to the specific\/diverse outputs of GAN-Utility, since one of those ten is highly likely to itself be generic.\n\nIn the StackExchange dataset GAN-Utility outperforms all ablations on both \\textsc{Bleu}\\xspace and \\textsc{Meteor}\\xspace. \nUnlike in the Amazon dataset, MLE does not outperform GAN-Utility in \\textsc{Bleu}\\xspace. This is because the MLE outputs in this dataset are not as generic as in the amazon dataset due to the highly technical nature of contexts in StackExchange.  \nAs in the Amazon dataset, GAN-Utility outperforms MLE on \\textsc{Diversity}\\xspace. Interestingly, the Max-Utility ablation achieves a higher \\textsc{Diversity}\\xspace score than GAN-Utility. \nOn manual analysis we find that Max-Utility produces longer outputs compared to GAN-Utility but at the cost of being less grammatical.\n\n\\subsection{Human Judgements Analysis} \\label{sec:analysis}\n\n\\autoref{tab:analysis-results} shows the numeric results of human-based evaluation performed on the reference and the system outputs on 300 random samples from the test set of the Amazon dataset.\\footnote{We could not ask crowdworkers evaluate the StackExchange data due to its highly technical nature.}\nAll approaches produce relevant and grammatical questions.\nAll models are all equally good at seeking new information, but are weaker than Lucene, which performs better at seeking new information but at the cost of much lower specificity and lower usefulness.\n\nOur full model, GAN-Utility, performs significantly better at the usefulness criteria showing that the adversarial training approach generates more useful questions. \nInterestingly, all our models produce questions that are more useful than Lucene and Reference, largely because Lucene and Reference tend to ask questions that are more often useful only to the person asking the question, making them less useful for potential other buyers (see \\autoref{fig:usefulness-analysis}). \nGAN-Utility also performs significantly better at generating questions that are more specific to the product (see details in \\autoref{fig:specificity-analysis}), which aligns with the higher \\textsc{Diversity}\\xspace score obtained by GAN-Utility under automatic metric evaluation. \n\n\\autoref{tab:example-outputs} contains example outputs from different models along with their usefulness and specificity scores. \nMLE generates questions such as \\textit{``is it waterproof?''} and \\textit{``what is the wattage?''}, which are applicable to many other products.\nWhereas our GAN-Utility model generates more specific question such as \\textit{``is this shower curtain mildew resistant?''}. \n\\autoref{sec:appendix-analysis} includes further analysis of system outputs on both Amazon and Stack Exchange datasets.\n\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{lccccc}\n\\toprule\nModel & Relevant {\\tiny [0-1]} & Grammatical {\\tiny [0-1]} & New Info {\\tiny [0-1]} & Useful {\\tiny [0-1]} & Specific {\\tiny [0-4]}   \\\\\n\\midrule\nReference &  0.96 & 0.99 & 0.93 & 0.72  & 3.38 \\\\\n\\midrule\nLucene & \\bf 0.90 & \\bf 0.99 & \\bf 0.95 & 0.68 & 2.87 \\\\\nMLE & \\bf 0.92 & \\bf 0.96  & 0.85 & 0.91 & 3.05  \\\\\nMax-Utility & \\bf 0.93 & \\bf 0.96 & 0.88 &  0.91 & 3.29  \\\\\nGAN-Utility & \\bf 0.94 & \\bf 0.96  & 0.87 & \\bf 0.96 & \\bf 3.52 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Results of human judgments on model generated questions on 300 sample Home \\& Kitchen product descriptions. Numeric range corresponds to the options described in \\autoref{sec:methodology}.\nThe difference between the bold and the non-bold numbers is statistically significant with p \\textless 0.05. Reference is excluded in the significance calculation.}\\label{tab:analysis-results}\n\\end{table*}\n\n\n\\begin{figure*}[t]\n    \\begin{minipage}[b]{0.48\\textwidth}\n    \\includegraphics[width=\\textwidth]{usefulness_analysis}\n    \\caption{Human judgements on the usefulness criteria.}\\label{fig:usefulness-analysis}\n    \\end{minipage}\n\t\\hspace{4mm}\n\t\\begin{minipage}[b]{0.48\\textwidth}\n\t\\includegraphics[width=\\textwidth]{specificity_analysis}\n    \\caption{Human judgements on the specificity criteria.}\\label{fig:specificity-analysis}\n\t\\end{minipage}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{tabular}{l |  l | c c}\n\\toprule\nTitle & Raining Cats and Dogs Vinyl Bathroom \\textbf{Shower Curtain} & &  \\\\\n\\midrule\nProduct & This adorable shower curtain measures 70 by 72  & & \\\\\nDescription & inches and is sure to make a great gift!  & & \\\\\n\\midrule\n& & Usefulness {\\tiny [0-4]} & Specificity {\\tiny [0-4]}  \\\\\nReference & does the vinyl smells? & 3 & 4 \\\\\nLucene & other than home sweet home , what other sayings on the shower curtain ? & 2 & 4 \\\\\nMLE & is it waterproof ? & 4 & 2 \\\\\nMax-Utility  & is this shower curtain mildew ? & 0 & 0 \\\\\nGAN-Utility & is this shower curtain mildew resistant ?  & 4 & 4 \\\\\n\\bottomrule\n& & & \\\\\n\n\nTitle & PURSONIC HF200 Pedestal \\textbf{Bladeless Fan \\& Humidifier} All-in-one & &  \\\\\n\\midrule\nProduct & The first bladeless fan to incoporate a humidifier! ,  & & \\\\\nDescription & This product operates solely as a fan, a humidifier or both simultaneously. & & \\\\\n& Atomizing function via ultrasonic. 5.5L tank lasts up to 12 hours. & & \\\\ \n\\midrule\n& & Usefulness {\\tiny [0-4]} & Specificity {\\tiny [0-4]}  \\\\\nReference & i can not get the humidifier to work & 1 & 2\\\\\nLucene & does it come with the vent kit  & 3 & 3 \\\\\nMLE & what is the wattage of this fan ? & 4 & 2\\\\\nMax-Utility & is this battery operated ? & 3 & 2 \\\\\nGAN-Utility & does this fan have an automatic shut off ? & 4 & 4 \\\\\n\\bottomrule\n\n\\end{tabular}\n\\caption{Example outputs from each of the systems for two product descriptions along with the usefulness and the specificity score given by human annotators.}\\label{tab:example-outputs}\n\\end{table*}\n\n\\begin{toappendix}\n\n\\section{Analysis of System Outputs}\\label{sec:appendix-analysis}\n\n\\subsection{Amazon Dataset}\n\n\\autoref{tab:amazon-example-outputs} shows the system generated questions for three product descriptions in the Amazon dataset.\n\nIn the first example, the product is a shower curtain. \nThe Reference question is specific and highly useful.\nLucene, on the other hand, picks a moderately specific (``how to clean it?'') but useful question. \nMLE model generates a generic but useful ``is it waterproof?''.\nMax-Utility generates comparatively a much longer question but in doing so loses out on relevance. \nThis behavior of generating two unrelated sentences is observed quite a few times in both Max-Utility and GAN-Utility models. \nThis suggests that these models, in trying to be very specific, end up losing out on relevance.\nIn the same example, GAN-Utility also generates a fairly long question which, although awkwardly phrase, is quite specific and useful. \n\nIn the second example, the product is a Duvet Cover Set. \nBoth Reference and Lucene questions here are examples of questions that are pretty much useful only to the person asking the question. \nWe find many such questions in both Reference and Lucene outputs which is the main reason for the comparatively lower usefulness scores for their outputs. \nAll three of our models generate irrelevant questions since the product description explicitly says that the set is full size. \n\nIn the last example, the product is a set of mopping clothes.\nReference question is quite specific but has low usefulness.\nLucene picks an irrelevant question.\nMLE and Max-Utility generate highly specific and useful questions.\nGAN-Utility generates an ungrammatical question by repeating the last word many times. \nWe observe this behavior quite a few times in the outputs of both Max-Utility and GAN-Utility models suggesting that our sequence-to-sequence models are not very good at maintaining long range dependencies.\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{tabular}{l l c c}\n\\toprule\n\nTitle & Extra Wide \\textbf{Shower Curtain} for a Clawfoot Tub\/opaque with Magnets & &  \\\\\n\\midrule\nProduct  & Approximately 180\" Wide 70\" Long.  & & \\\\\n& Extra Wide to keep the water and heat in. & & \\\\\nDescription & Opaque in color to let the light in. & & \\\\\n& Can be used as a liner or the curtain itself. & & \\\\\n\\midrule\n& & Usefulness {\\tiny [0-4]} & Specificity {\\tiny [0-4]}\\\\\nReference & does this curtain come with hooks? & 4 & 4\\\\\nLucene & how to clean it ? i had it over 10 months now . & 3 & 3 \\\\\n& and some mildew grow on it  & & \\\\\nMLE  & is it waterproof ? & 1 & 2 \\\\\nMax-Utility  & what is the diameter of the bottom of the shower ?   & 0 & 0 \\\\\n & i have a kohler shower curtain . & & \\\\\nGAN-Utility & does this curtain have suction feet on the bottom of the shower curtain ? & 3 & 4  \\\\\n\\bottomrule\n& & & \\\\\n\nTitle & Chezmoi Collection 7-piece Chic Ruched White \\textbf{Duvet Cover Set}, & &  \\\\\n& Full Size (with Throw Pillows)   & & \\\\\n\\midrule\nProduct & Create a world of romance with the elegantly-styled, & & \\\\\nDescription & ruching,and luxurious all white duvet cover set.   & & \\\\\n\\midrule\n& & Usefulness {\\tiny [0-4]} & Specificity {\\tiny [0-4]}  \\\\\nReference & How long will it take to ship this to my pin code? & 1 & 1\\\\\nLucene & and can you use the duvet as is ?   & 1 &  4\\\\\n& if not what shall i purchase to put in it for winter or fall ? & & \\\\\nMLE & what are the dimensions of the king size ? & 0 & 0\\\\\nMax-Utility  & what are the dimensions of the king size ? & 0 & 0 \\\\\nGAN-Utility & does the king size come with a duvet cover & 0 & 0 \\\\\n& or do you have to buy a king duvet ?  & & \\\\\n\\bottomrule\n& & & \\\\\nTitle & Microfiber 3-Pack, \\textbf{Pro-Clean Mopping Cloths} for & &  \\\\\n& Braava Floor Mopping Robot  & & \\\\\n\\midrule\nProduct &  Braavas textured Pro-Clean microfiber mopping cloths remove dirt and& & \\\\\nDescription &  hair from your floors. The cloths can be washed and used hundreds   & & \\\\\n& of times. They are compatible with all Braava models,  including the & & \\\\\n& Pro-Clean Reservoir Pad. Each cloth is easy to attach and remove from  & & \\\\\n&  the magnetic cleaning pad. & & \\\\\n\\midrule\n& & Usefulness {\\tiny [0-4]} & Specificity {\\tiny [0-4]}  \\\\\nReference & do i have to use a new cloth every time i want to clean my floor? & 2 & 4\\\\\n&  \\$5\/\\$6 seems expensive per clean & & \\\\\nLucene & do they remove pet odor ?  & 0  & 0 \\\\\nMLE & will these work with the scooba ? & 3 & 3\\\\\nMax-Utility  & do these cloths work on hardwood floors ? & 3 & 4 \\\\\nGAN-Utility & will this work with the scooba mop mop mop mop mop mop mop & 0 & 0 \\\\\n\\bottomrule\n\n\n\\end{tabular}\n\\caption{Example outputs from each of the systems for three product descriptions from the Home \\& Kitchen category of the Amazon dataset. }\\label{tab:amazon-example-outputs}\n\\end{table*}\n\n\\subsection{Stack Exchange Dataset}\n\n\\autoref{tab:se-example-outputs} includes system outputs for three posts from the Stack Exchange dataset.\n\nThe first example is of a post where someone describes their issue of not being able to recover from their boot. \nReference and Lucene questions are useful.\nMLE generates a generic question that is not very useful.\nMax-Utility generates a useful question but has slight ungrammaticality in it.\nGAN-Utility, on the other hand, generates a specific and an useful question.\n\nIn the second example, again Reference and Lucene questions are useful.\nMLE generates a generic question.\nMax-Utility and GAN-Utility both generate fairly specific question but contain unknown tokens. \nThe Stack Exchange dataset contains several technical terms leading to a long tail in the vocabulary.\nOwing to this, we find that both Max-Utility and GAN-Utility models generate many instances of questions with unknown tokens.\n\nIn the third example, the Reference question is very generic. Lucene asks a relevant question.\nMLE again generates a generic question.\nBoth Max-Utility and GAN-Utility generate specific and relevant questions. \n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{tabular}{l l }\n\\toprule\nTitle & how can i recover my boot ?\\\\\n\\midrule\nPost & since last week i am trying to recover my boot . \\\\\n& after the last update for ubuntu 12.04 i lost it and nobody could help me , \\\\\n& i used boot-repair but there are problems with dependences , which ca n't be fix .\\\\\n& i will be very grateful if somebody could help me . \\\\\n\\midrule\nReference &  what happens when you try to boot ? \\\\\nLucene & can you get into bios ?   \\\\\nMLE & how much ram do you have ?   \\\\\nMax-Utility  & do you have a swap partition partition ? \\\\\nGAN-Utility & what happens when you try to boot into safe mode ?  \\\\\n\\bottomrule\n& \\\\\nTitle & packages have unmet dependencies when trying to install anything \\\\\n\\midrule\nPost & i 'm running ubuntu 14.04 lts . ive recently run into this problem with several applications , \\\\\n& it seems to happen to anything i need to update or install . i just gave up the first 2 or 3 times this \\\\\n& happened after trying a few solutions to no avail , but now i 'm having the same issue with steam \\\\\n& trying to update , which i use quite a lot . ive looked through dozens of posts about similar issues \\\\\n& and tried a lot of solutions and nothing seems to work.  \\\\\nReference & sudo dpkg -reconfigure all ? ?\\\\\nLucene & if you use the graphical package manager , does n't add the required packages automatically ?\\\\\nMLE & how long did you wait ? \\\\\nMax-Utility & can you post the output of `apt-cache policy UNK ?\\\\\nGAN-Utility & can you post a screenshot of the output of `sudo apt-get install UNK \\\\\n\\bottomrule\n& \\\\\nTitle & full lubuntu installation on usb ( uefi capable )  \\\\\n\\midrule\nPost &  i want to do a full lubuntu installation on a usb stick that can be booted in uefi mode. \\\\\n& i do not want persistent live usb but a full lubuntu installation ( which happens to live on a usb stick ) \\\\\n& and that can boot fromanyuefi-capable computer ...\\\\\n\\midrule\nReference & hello and welcome on askubuntu . could you please clarify what you want ?\\\\\nLucene & so , ubuntu was installed to the pen drive ? \\\\\nMLE & which version of ubuntu ? \\\\\nMax-Utility & do you have a live cd or usb stick ? \\\\\nGAN-Utility & what is the model of the usb stick ? \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Example outputs from each of the systems for three posts of the Stack Exchange dataset. }\\label{tab:se-example-outputs}\n\\end{table*}\n\\end{toappendix}\n\n\n\n\n\\subsection{Sequence-to-sequence Model for Question Generation}\\label{sec:seq2seq}\nWe use a standard attention based sequence-to-sequence model \\cite{luong2015effective} for our question generator. Given an input sequence (context) $c=(c_1,c_2, ..., c_N)$, this model generates an output sequence (question) $q =(q_1, q_2, ..., q_T)$. The architecture of this model is an encoder-decoder with attention. The encoder is a recurrent neural network (RNN) operating over the input word embeddings to compute a source context representation $\\tilde{c}$. The decoder uses this source representation to generate the target sequence one word at a time:\n\\begin{equation}\n\\begin{split}\n  p(q | \\tilde{c}) & = \\prod_{t=1}^{T} p(q_t | q_1, q_2, ..., q_{t-1}, \\tilde{c_t}) \\\\\n  & = \\prod_{t=1}^T \\textit{softmax}(W_s{\\tilde h}_t) \\quad;\\quad \\\\\n  \\textrm{where } {\\tilde h}_t & = \\text{tanh}(W_c[\\tilde{c_t}; h_t])\n\\end{split}\n\\label{eq:output-prob}\n\\end{equation}\nIn \\autoref{eq:output-prob}, ${\\tilde h}_t$ is the attentional hidden state of the RNN at time $t$ and $W_s$ and $W_c$ are parameters of the model.\\footnote{Details are in \\autoref{sec:appendix-seq2seq}.}\nThe predicted token $q_t$ is the token in the vocabulary that is assigned the highest probability using the softmax function. \nThe standard training objective for sequence-to-sequence model is to maximize the log-likelihood of all $(c, q)$ pairs in the training data $D$ which is equivalent to minimizing the following loss, \n\\begin{equation}\nL_{\\text{mle}}(D) =  - \\sum_{(c,q) \\in D} \\sum_{t=1}^{T} \\log p(q_t | q_1, ..., q_{t-1}, \\tilde{c_t})\n\\end{equation}\n\n\\begin{figure*}[!t]\n\\centering\n \\includegraphics[scale=0.23]{model}\n \\caption{Overview of our GAN-based clarification question generation model (refer preamble of~\\autoref{sec:model})}\\label{fig:model}\n\\end{figure*}\n\n\\begin{toappendix}\n\n\\section{Sequence-to-sequence model details}\\label{sec:appendix-seq2seq}\n\nIn this section, we describe some of the details of the attention based sequence-to-sequence model introduced in Section 2.1 of the main paper. \nIn equation 1, ${\\tilde h}_t$ is the attentional hidden state of the RNN at time $t$ obtained by concatenating the target hidden state $h_t$ and the source-side context vector $\\tilde c_t$,\n and $W_s$ is a linear transformation that maps $h_t$ to an output vocabulary-sized vector.\nEach attentional hidden state ${\\tilde h}_t$ depends on a distinct input context vector $\\tilde c_t$  computed using a global attention mechanism over the input hidden states as: \n\\begin{align}\n  \\tilde c_t &= \\sum_{n=1}^{N} a_{nt} h_n  \\\\\n  a_{nt} & = \\text{align}(h_n, h_t) \\\\\n  & = {\\exp\\Big[ h_t^T W_a h_n \\Big]} \\Big\/ {\\sum_{n'}\\exp\\Big[ h_t^T W_a h_{n'} \\Big] }\n\\end{align}\nThe attention weights $a_{nt}$ is calculated based on the alignment  score between the source hidden state $h_n$ and the current target hidden state $h_t$.\n\n\\end{toappendix}\n\n\n\\subsection{Training the Generator to Optimize \\textsc{Utility}\\xspace}\\label{sec:mixer}\n\nTraining sequence-to-sequence models for the task of clarification question generation (with context as input and question as output) \nusing maximum likelihood objective unfortunately leads to the generation of highly generic questions, such as \\textit{``What are the dimensions?''} when asking questions about home appliances.\nRecently, \\citet{rao2018learning} observed that the usefulness of a question can be better measured as the \\emph{utility} that would be obtained if the context were updated with the answer to the proposed question. \nFollowing this observation, we first use a pretrained answer generator (\\autoref{sec:pretraining}) to generate an answer given a context and a question. \nWe then use a pretrained \\textsc{Utility}\\xspace calculator (\\autoref{sec:utility} ) to predict the likelihood that the generated answer would increase the utility of the context by adding useful information to it.\nFinally, we train our question generator to optimize this \\textsc{Utility}\\xspace based reward. \n\nSimilar to optimizing metrics like \\textsc{Bleu}\\xspace and \\textsc{Rouge}\\xspace, this \\textsc{Utility}\\xspace calculator also operates on discrete text outputs, which makes optimization difficult due to non-differentiability. \nA successful recent approach dealing with the non-differentiability while also retaining some advantages of maximum likelihood training is the Mixed Incremental Cross-Entropy Reinforce \\citep{ranzato2015sequence} algorithm (\\textsc{Mixer}\\xspace).\nIn \\textsc{Mixer}\\xspace, the overall loss $L$ is differentiated as in \\textsc{Reinforce}\\xspace \\citep{williams1992simple}:\n\\begin{equation}\n\\begin{split}\nL(\\theta) & = - \\mathbb{E}_{q^s \\sim p_{\\theta}}  r(q^s) \\quad;\\quad \\\\\n\\nabla_{\\theta} L(\\theta) & = - \\mathbb{E}_{q^s \\sim p_{\\theta}} r(q^s) \\nabla_{\\theta} \\log p_{\\theta} (q^s)\n\\end{split}\n \\label{eq:r}\n\\end{equation}\nwhere $q^s$ is a random output sample according to the model $p_\\theta$ and $\\theta$ are the parameters of the network.\nThe expected gradient is then approximated using a single sample $q^s = (q^s_1, q^s_2, ..., q^s_T)$ from the model distribution ($p_{\\theta}$).\nIn \\textsc{Reinforce}\\xspace, the policy is initialized randomly, which can cause long convergence times.\nTo solve this, \\textsc{Mixer}\\xspace starts by optimizing maximum likelihood for the initial $\\Delta$ time steps, and slowly shifts to optimizing the expected reward from \\autoref{eq:r} for the remaining $(T - \\Delta)$ time steps.\n\n\n\nIn our model, for the initial $\\Delta$ time steps, we minimize $L_{\\text{mle}}$ and for the remaining steps, we minimize the following \\textsc{Utility}\\xspace-based loss: \n\\begin{equation}\n\\small\n\\begin{split}\nL_{\\text{max-utility}} =  - (r(q^p) - r(q^b)) \\sum_{t=1}^{T} \\log p(q_t | q_1, ..., q_{t-1}, \\tilde{c_t})\n\\end{split}\n\\end{equation}\nwhere $r(q^p)$ is the \\textsc{Utility}\\xspace based reward on the predicted question and $r(q^b)$ is a baseline reward introduced to reduce the high variance otherwise observed when using \\textsc{Reinforce}\\xspace.\nTo estimate this baseline reward, we take the idea from the self-critical training approach \\citet{rennie2017self} where the baseline is estimated using the reward obtained by the current model under greedy decoding during test time. We find that this approach for baseline estimation stabilizes our model better than the approach used in \\textsc{Mixer}\\xspace. \n\n\n\n\\subsection{Estimating \\textsc{Utility}\\xspace from Data}\\label{sec:utility}\n\nGiven a (context, question, answer) triple, \\citet{rao2018learning} introduce a utility calculator $\\textsc{Utility}\\xspace(c, q, a)$ to calculate the value of updating a context $c$ with the answer $a$ to a clarification question $q$.\nThey use the utility calculator to estimate the probability that an \\emph{answer} would be a meaningful addition to a context.\nThey treat this as a binary classification problem where the positive instances are the true (context, question, answer) triples in the dataset whereas the negative instances are contexts paired with a random (question, answer) from the dataset. \nFollowing  \\citet{rao2018learning}, we model our \\textsc{Utility}\\xspace calculator by first embedding the words in $c$ and then using an LSTM (long-short term memory) \\citep{hochreiter1997long} to generate a neural representation $\\bar{c}$ of the context by averaging the output of each of the hidden states. \nSimilarly, we obtain neural representations $\\bar{q}$ and $\\bar{a}$ of $q$ and $a$ respectively using a question and an answer LSTM models. \nFinally, we use a feed forward neural network $F_{\\textsc{Utility}\\xspace} (\\bar{c}, \\bar{q}, \\bar{a})$ to predict the usefulness of the question.\n\n\\subsection{\\textsc{Utility}\\xspace GAN for Clarification Question Generation}\\label{sec:gan}\n\nThe \\textsc{Utility}\\xspace calculator trained on true vs random samples from real data (as described in the previous section) can be a weak reward signal for questions generated by a model due to the large discrepancy between the true data and the model's outputs.\nIn order to strengthen the reward signal, we reinterpret the \\textsc{Utility}\\xspace calculator (coupled with the answer generator) as a discriminator in an adversarial learning setting.\nThat is, instead of taking the \\textsc{Utility}\\xspace calculator to be a fixed model that outputs the expected quality of a (question, answer) pair, we additionally optimize it to distinguish between true (question, answer) pairs and model-generated ones.\nThis reinterpretation turns our model into a form of a generative adversarial network (GAN) \\citep{goodfellow2014generative}.\n\nGAN is a training procedure for ``generative'' models that can be interpreted as a game between a generator and a discriminator.\nThe generator is a model $g \\in \\mathcal{G}$ that produces outputs (in our case, questions).\nThe discriminator is another model $d \\in \\mathcal{D}$ that attempts to classify between true outputs and model-generated outputs.\nThe goal of the generator is to generate data such that it can fool the discriminator; the goal of the discriminator is to be able to successfully distinguish between real and generated data. In the process of trying to fool the discriminator, the generator produces data that is as close as possible to the real data distribution.\nGenerically, the GAN objective is:\n\\begin{equation}\n\\small\n\\begin{split}\n  L_{\\text{GAN}}(\\mathcal{D}, \\mathcal{G}) =\n  \\max_{d \\in \\mathcal{D}} \\min_{g \\in \\mathcal{G}} & \\mathbb{E}_{x \\sim \\hat p} \\log d(x) + \\\\\n                                            &  \\mathbb{E}_{z \\sim p_z} \\log(1 - d(g(z)))\n\\end{split}                                      \n\\end{equation}\nwhere $x$ is sampled from the true data distribution $\\hat p$, and $z$ is sampled from a prior defined on input noise variables $p_{z}$.\n\nAlthough GANs have been successfully used for image tasks, training GANs for text generation is challenging due to the discrete nature of outputs in text. The discrete outputs from the generator make it difficult to pass the gradient update from the discriminator to the generator. Recently, \\citet{yu2017seqgan} proposed a sequence GAN model for text generation to overcome this issue. They treat their generator as an agent and use the discriminator as a reward function to update the generative model using reinforcement learning techniques.\nOur GAN-based approach is inspired by this sequence GAN model with two main modifications: a) We use \\textsc{Mixer}\\xspace algorithm as our generator (\\autoref{sec:mixer}) instead of a purely policy gradient approach; and b) We use \\textsc{Utility}\\xspace calculator (\\autoref{sec:utility}) as our discriminator instead of a convolutional neural network (CNN).\n\nTheoretically, the discriminator should be trained using (context, true question, true answer) triples as positive instances and (context, generated question, generated answer) triples as the negative instances.\nHowever, we find that training a discriminator using such positive instances makes it very strong since the generator would have to not only generate real looking questions but also generate real looking answers to fool the discriminator. \nSince our main goal is question generation and since we use answers only as latent variables, we instead use (context, true question, \\emph{generated answer}) as our positive instances where we use the pretrained answer generator to get the \\emph{generated answer} for the true question.\nFormally, our objective function is:\n\\begin{equation}\n\\footnotesize\n\\begin{split}\n  L_{\\text{GAN-U}}(\\mathcal{U}, \\mathcal{M}) = &\n  \\max_{u \\in \\mathcal{U}} \\min_{m \\in \\mathcal{M}} \\mathbb{E}_{q \\sim \\hat p} \\log u(c, q, \\mathcal{A}(c, q)) + \\\\\n                                             & \\mathbb{E}_{c \\sim \\hat p} \\log(1 - u(c, m(c), \\mathcal{A}(c, m(c))))\n\\end{split}                                      \n\\end{equation}\n\\normalsize\nwhere $\\mathcal{U}$ is the \\textsc{Utility}\\xspace discriminator, $\\mathcal{M}$ is the \\textsc{Mixer}\\xspace generator, $\\hat p$ is our data of (context, question, answer) triples and $\\mathcal{A}$ is the answer generator. \n\n\n\\subsection{Pretraining}\\label{sec:pretraining}\n\n\\textbf{Question Generator.} We pretrain our question generator using the sequence-to-sequence model (\\autoref{sec:seq2seq}) to maximize the log-likelihood of all (context, question) pairs in the training data. \nParameters of this model are updated during adversarial training. \n\n\\textbf{Answer Generator.} We pretrain our answer generator using the sequence-to-sequence model (\\autoref{sec:seq2seq}) to maximize the log-likelihood of all ([context+question], answer) pairs in the training data. \nParameters of this model are kept fixed during the adversarial training.\\footnote{We leave the experimentation of updating parameters of answer generator during adversarial training to future work.} \n\n\\textbf{Discriminator.} In our \\textsc{Utility}\\xspace GAN model (\\autoref{sec:gan}), the discriminator is trained to differentiate between true and generated questions. \nHowever, since we want to guide our \\textsc{Utility}\\xspace based discriminator to also differentiate between true (``good'') and random (``bad'') questions, \nwe pretrain our discriminator in the same way we trained our \\textsc{Utility}\\xspace calculator.\nFor positive instances, we use a context and its true question, answer from the training data and for negative instances, we use the same context but randomly sample a question from the training data (and use the answer paired with that random question). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGraphene is a two-dimensional (2D) layer of carbon atoms ordered\ninto a honeycomb lattice as shown in Fig.~\\ref{graphene_lattice}.\nIt is a material with a host of unusual properties\n\\cite{Geim2007,neto_etal,ISI:000224756700045,chakraborty_review,\nchem_review_graphene,review_mag_yazaev,\ncresti_disord_review,beenakker_colloq,sarma_review,mucciolo_review_j_phys}\nincluding (among others): Dirac\nspectrum of low-lying quasiparticles \\cite{neto_etal}, large\nmean-free-path \\cite{ISI:000224756700045}, and high electron\nmobility\n\\cite{high-mobility,du_high-mobility}.\n\nBesides its purely fundamental importance, researchers view\ngraphene as a promising new material for electronic\n\\cite{electronic_device}, chemical \\cite{chem_sensing}, or\nelectromechanical \\cite{electromechanical} applications, where\ngraphene's unique properties may be of substantial benefit. Unlike\n3D matter, whose bulk is hidden from direct observation and\ninfluence, graphene's ``bulk'', its 2D surface, is always exposed,\nand its structure may be inspected or modified with greater ease.\nFurthermore, the Dirac energy dispersion in 2D implies that\ngraphene is a gapless semiconductor, whose density of states\nvanishes linearly when approaching the Fermi energy. As such, it\nis ``a bridge material'' separating the worlds of semiconductors\n(with an energy gap between the valence and conducting bands) and\nmetals, with a finite density of electronic states at the Fermi\nenergy. Depending on the operating regime, graphene can be pushed\nin either direction. For example, it is possible to open a gap in\na sample with the help of chemical modifications\n\\cite{graphane_2003,sofo_graphane},\nor lateral confinement\n\\cite{lattice_distortion,han_experiment_gap,chen_experiment_gap}.\nAlternatively, one can make graphene metallic, e.g., by chemical\ndoping \\cite{chem_doping}. Some graphene samples have\nspatially-varying electronic properties, due to local\nmodifications on the sample. The long electronic mean-free-path, which can\nbe of the order of micrometer, implies that electronic signals can\ntravel unimpeded large distances through a device. These features\nmight be very useful in applications.\n\nThe unusual properties of graphene motivated significant research\nefforts. The field grows very fast: the ISI web site reports that by\nOctober 2010 there were more than 5,000 publications with the word\n``graphene'' in their titles. Clearly, this is an enormous volume of\nscientific work, of which our brief review covers only a very small\nfraction. Its scope is very limited in several respects. As it is obvious\nfrom the title, we direct our attention to mesoscopic graphene systems, a\ntopic at the boundary between fundamental and applied research.\nFurthermore, we mainly discuss the electronic aspects of graphene\nmesoscopic systems, especially those which may be relevant for possible\nelectronic or spintronic applications, for example, charge\/spin transport\nand confinement, and control over them. Lattice properties are dealt with\nonly when the lattice affects the electrons significantly. Several topics\nare deliberately omitted due to space constraints; these include: quantum\nHall effect, thermal transport phenomena, phonons, and mechanical\nproperties of graphene.\n\nThe review is organized as follows. In Sec.~\\ref{graphene} we\ndiscuss the most basic electron properties of an infinite graphene\nsheet. The physics of graphene edges is reviewed in\nSec.~\\ref{edge}. Sections~\\ref{nanoribbon}, \\ref{qdot}, and\n\\ref{pnj} focus on nanoribbons, quantum dots, as well as {\\it\npn}-junctions and {\\it pnp}-structures, respectively.\nSec.~\\ref{barrier} discusses the barriers created by the combined\napplication of magnetic and\/or electric fields. Conclusions are\npresented in Sec.~\\ref{conclusions}. The main part of the review\nis kept non-technical for it to be accessible by a general reader.\nMore involved discussions are relegated to Appendices.\n\\begin{figure}[btp]\n\\centering \\leavevmode \\epsfxsize=12.5cm\n\\epsfbox{graphene_no_xy.eps} \\caption[] {\\label{graphene_lattice}\n(Color online) Geometry of the graphene lattice showing: primitive lattice\nvectors ${\\bf a}_{1,2}$, diatomic lattice unit cell (dotted-line\nrhombus), and vectors ${\\bm \\delta}_{1,2,3}$, connecting the\nnearest neighbors. Red (black) circles correspond to the\n${\\cal A}$ (${\\cal B}$) sublattice. Three different types of edges\n(zigzag, armchair, and Klein edge) are shown. The Klein and zigzag\nedge violate the symmetry between the sublattices (the atoms at\nthe edge sites belong exclusively to sublattice ${\\cal A}$: they are all\nred), while the armchair edge does not (it has both black and red atoms). }\n\\end{figure}\n\n\n\n\\input{basic.tex}\n\n\\input{edge.tex}\n\n\\input{nribb.tex}\n\n\\input{dot.tex}\n\n\\input{pn_arxiv.tex}\n\n\\input{qwire.tex}\n\n\n\\hspace{4in}\n\n\\section{Conclusions}\n\\label{conclusions}\n\nGraphene is a material with many interesting features which make\nit an attractive candidate for microelectronic and micromechanical\napplications. In the above pages we very briefly outlined several\nideas and notions driving current graphene mesoscopic research:\nedge states, geometric quantization, quasi-bound states, Coulomb\nblockade, Klein tunneling through {\\it pn}-junctions, etc. Some of\nthem, like Klein tunneling, are unique to graphene. Others, e.g.,\nCoulomb blockade and geometric quantization, have a much longer\nhistory. Yet, even in the latter case, the peculiar properties of\ngraphene give rise to new features, for example, a much larger\nenergy scale for the confinement inside a quantum dot.\n\nAlthough, many of the theoretical studies of graphene mesoscopic\nsystems are done in the single-electron approximation, the use of\nmany-body techniques are often warranted. Indeed, the\nsingle-electron approximation could introduce qualitative errors,\nwhich may be corrected only if proper many-body effects are\naccounted.\n\nNumerous theoretical proposals have not been explored\nexperimentally. For instance, the realization of nanoribbons with\natomically-sharp edges remains a distant possibility. Many\nsuggested devices impose stringent conditions on samples, in terms\nof purity and regularity of the sample geometry. Some of these\nproposals are stimulating various current experiments. An\nimportant direction in this quest is to control and to understand\nthe disorder, which may enter through many routes: as foreign\natoms adsorbed on samples or chemically attached to the edges, as\nimperfections of the sample edges, as random elastic deformations,\nor as bulk defects of varied nature. On the other hand, disorder\nis not always an enemy, since its use may be beneficial under\ncertain circumstances.\n\nGraphene studies are still in their infancy, and it is too early\nto guess which of its unusual features will be more useful for\napplications. Yet, fabrication of several prototypic\nmicroelectronic devices, like field-effect transistors, biosensor,\nand integrated circuit, have been reported. In addition, graphene\npresents an excellent playground for fundamental condensed matter\nresearch, exciting enthusiasm of both experimentalists and\ntheorists in numerous subfields of condensed matter physics.\n\n\\section*{Acknowledgements}\n\n\nWe are grateful to L.A.~Openov who provided a high-resolution file\nused to create Fig.~\\ref{gg_interface}. We are grateful for the\nsupport provided by the grant RFBR-JSPS 09-02-92114. A.V.R. is\npartially supported by the grant RFBR 09-02-00248. G.G.\nacknowledges support from the Japan Society for the Promotion of\nScience (JSPS). F.N. acknowledges partial support from the\nNational Security Agency (NSA), Laboratory Physical Sciences\n(LPS), Army Research Office (ARO), DARPA, Air Force Office of\nScientific Research (AFOSR), and National Science Foundation (NSF)\ngrant No.~0726909, Grant-in-Aid for Scientific Research (S), MEXT\nKakenhi on Quantum Cybernetics, and Funding Program for Innovative\nR\\&D on S\\&T (FIRST).\n\n\n\\newpage\n\n\\section{Basic physics of a graphene sheet}\n\\label{graphene}\n\nFor completeness, in this section we quickly remind the reader the basic\nsingle-electron properties of a graphene sheet. A more detailed\npresentation can be found in\nAppendix~\\ref{appendix::basic}.\nIt is common to describe a graphene sample in terms of a tight-binding\nmodel on the honeycomb lattice. Lattice parameters for graphene, as well as\nsome other microscopic characteristics, are given in\nTable~\\ref{glance}.\n\\begin{table}\n\\begin{tabular}{||c|c||}\n\\hline \\hline\n\\quad    Graphene parameters \\quad & Value \\\\\n\\hline\\hline\n    C-C bond length, $a_0$ &  1.4 \\AA  \\\\\n\\hline\n    Lattice constant & 2.46 \\AA \\\\\n\\hline\n    Hopping amplitudes: &   \\\\\n    nearest neighbor, $t$   & 2.8 eV    \\\\\n    next-nearest, $t'$ & 0.1 eV \\\\\n    third-nearest, $t''$ & 0.07 eV  \\\\\n\\hline\n    Fermi velocity, $v_{\\rm F}$ & \\quad $1.1 \\times 10^6 \\textrm{m\/s}$\n\\quad\n    \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{Graphene parameters at a glance.}\n\\label{glance}\n\\end{table}\nHoneycomb lattice can be split into two sublattices, denoted by\n${\\cal A}$\nand\n${\\cal B}$.\nThe Hamiltonian of an electron hopping on a graphene sheet is given by\n\\begin{eqnarray}\nH =\n-t\n\\sum_{{\\bf R} \\in {\\cal A}}\n\\sum_{i=1,2,3}\nc^\\dagger_{\\bf R}\nc^{\\vphantom{\\dagger}}_{{\\bf R} + {\\bm \\delta}_i}\n+\n{\\rm H.c.},\n\\label{H\n\\end{eqnarray}\nwhere\n${\\bf R}$\nruns over sublattice\n${\\cal A}$,\nand $t = 2.8$ eV is the nearest-neighbor hopping amplitude. The vectors\n${\\bm \\delta}_i$\n($i=1,2,3$)\nconnect the nearest neighbors (see\nFig.~\\ref{graphene_lattice}\nshowing the geometry of the graphene lattice). When necessary, $H$ can be\naugmented by interaction or longer-range hopping terms (see\nTable~\\ref{glance}\nfor values of the longer-range hopping amplitudes).\n\nSince there are two atoms in graphene's unit cell, it is convenient to\ndescribe the single-electron wave function of graphene as a two-component\nspinor $\\Psi$. This introduces an isospin quantum number. For every\nmomentum ${\\bf k}$ lying within the Brillouin zone,\nFig.~\\ref{bz},\nthe Hamiltonian $H$ has two eigenvalues\n$\\varepsilon_{{\\bf k} \\pm}$,\nwhich have the same magnitude and opposite signs. The eigenvalue\n$\\varepsilon_{{\\bf k} +} > 0 $\n($\\varepsilon_{{\\bf k} -} < 0$) corresponds to the conduction\n(valence) band of graphene.\n\n\\begin{figure}[btp]\n\\centering \\leavevmode \\epsfxsize=7.5cm\n\\epsfbox{bz_rev.eps}\n\\caption[]\n{\\label{bz}\n(Color online) The Brillouin zone of graphene is a perfect hexagon. The\nDirac cones are located at the corners of the Brillouin zone. The six cones\ncan be split into two equivalence classes (cones within the same class are\nconnected by dashed lines). These classes are commonly referred to as $K$\nand $K'$.\n}\n\\end{figure}\n\nThe functions\n$\\varepsilon_{{\\bf k} \\pm}$\nvanish at the six corners of the Brillouin zone:\n${\\bf K}_{1,2} = (0, \\pm 4\\pi\/(3\\sqrt{3}a_0))$\nand\n${\\bf K}_{3,4,5,6}\n=\n(\\pm 2\\pi \/ (3 a_0), \\pm 2\\pi\/(3\\sqrt{3}a_0))$.\nHere the symbol $a_0$ denotes the carbon-carbon bond length. Near\n${\\bf K}_i$,\n$i={1,\\ldots, 6}$,\nthe dispersion surface can be approximated by two cones with a common apex\n\\begin{eqnarray}\n\\varepsilon_{{\\bf k} \\pm} \\propto \\pm |{\\bf k} - {\\bf K}_i|.\n\\label{wd_disp\n\\end{eqnarray}\nThe conduction and\nvalence bands touch each other at the cones' apex.\n\nOf the six cones only two can be chosen to be independent: the remaining\nfour are connected to these two by a reciprocal lattice vector. Thus, the\ncones\n${\\bf K}_{1, \\ldots, 6}$\ncan be split into two equivalence classes. These classes are commonly\ndenoted by $K$ and $K'$, and referred to as `valleys'.\n\nWhen graphene is not doped, its Fermi level passes through the cone\napexes. In such a situation, if one is interested in the low-energy\ndescription, only the states near the cones must be accounted. For states\nwith energies near the cone apexes, it is possible to use the following\nWeyl-Dirac equations\n\\begin{eqnarray}\n\\label{dirac\nE\\Psi_{1,2}\n&=&\nH \\Psi_{1,2},\n\\\\\nH\n&=&\n- i \\hbar v_{\\rm F}\n(\\sigma_y \\partial_x\n\\pm\n\\sigma_x \\partial_y)\n=\n\\hbar v_{\\rm F}\n\\left(\n\\matrix{\n    0&  -\\partial_x \\pm i\\partial_y \\cr\n    \\partial_x \\pm i\\partial_y &    0\n    }\n\\right),\n\\end{eqnarray}\nwhich have dispersion as in\nEq.~(\\ref{wd_disp}).\nThese equations become invalid away from the cones. The spinor wave\nfunction\n$\\Psi_1$ ($\\Psi_2$)\ncorresponds to the electron states near the cone $K$ ($K'$). The plus\n(minus) sign in Eq.~(\\ref{dirac}) corresponds to $K$ ($K'$). The low-energy\nphysics of electrons in graphene is equivalent to four species of\ntwo-dimensional massless Dirac electrons: two different spin directions and\ntwo cones, $K$ and $K'$, giving overall fourfold degeneracy.\n\n\nPristine undoped graphene is a gapless semiconductor. This means\nthat its density of states does not have a gap, but vanishes\nlinearly when the energy approaches the apexes. Sometimes it is\ndesirable to open a gap in the graphene spectrum. As shown in\nTable~\\ref{energy_gap} this can be achieved by employing various\nmechanical, electronic, and\/or chemical methods. In particular, in\nmonolayer graphene the gap can be induced by substrate or strain\nengineering~\\cite{Ni2008,low_gap_strain}, as well as by deposition\nor adsorption of molecules on the graphene layer, such as, for\nexample, water and ammonia~\\cite{Ribeiro2008}. Based on numerical\nstudies, the value of the energy gap can range from a few meV to\nhundreds of meV. Most importantly it can be larger than room\ntemperature as required for graphene-based transistors. In bilayer\ngraphene the gap can be induced and continuously tuned, for\ninstance, chemically through selective doping~\\cite{Ohta2006}, or\neven electrically by applying gate voltages~\\cite{Zhang2009}. The\nfact that graphene's band structure can be controlled externally\nand with rather simple processes is a nontrivial result which\nreveals the potential of graphene for nanotechnology.\n\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{||c|c|c||}\n    \\hline\n    \\hline\n    \\multicolumn{3}{||c||}{\\textbf{Inducing an energy gap in graphene}}\\\\\n     \\hline\n     \\hline\n  Method & \\quad Gap in monolayer (meV) \\quad & \\quad Gap in bilayer (meV) \\quad \\\\\n    \\hline\n    \\hline\n \\quad \\quad Nanoribbons$^{*}$ (width $\\sim$ 15 nm) ~\\cite{han_experiment_gap} \\quad \\quad & 200 & \\\\\n BN-\\textit{h} \/ Cu(111) substrate ~\\cite{Giovannetti2007} &   53 \/ 11 &\\\\\n SiC substrate$^{*}$ ~\\cite{Zhou2007} & 260  & \\\\\n External square superlattice ~\\cite{Tiwari2009}&  65 & \\\\\n Strain engineering ~\\cite{Ni2008} &  300  & \\\\\n Adsorption of molecules~\\cite{Berashevich2009} & 2$\\times 10^{3}$ &  \\\\\n Graphene covered by H$_{2}$O \/ NH$_{3}$ ~\\cite{Ribeiro2008} & 18 \/ 11  & 30 \/ 42 \\\\\n Nanoribbons$^{*}$ (width $\\sim$ 30 nm) ~\\cite{Szafranek2010}&  & 50 \\\\\n Electrical gates$^{*}$ ~\\cite{Zhang2009}  & & 250 \\\\\n Selective doping$^{*}$ (potassium) ~\\cite{Ohta2006} & & 100 \\\\\n Electric field effect$^{*}$ ~\\cite{Castro2007} &  &  150  \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{Brief summary of possible methods to induce an energy gap\nin monolayer and bilayer graphene. Asterisks ($^*$) indicate\nexperimental demonstrations; otherwise the value of the gap is a\ntheoretical prediction. In some cases, the energy gap is tunable\nand its exact value critically depends on the details of the\nspecific method. Here, `BN-\\textit{h}' denotes boron nitride in\nthe hexagonal configuration.}\\label{energy_gap}\n\\end{table}\n\n\n\\section{Quantum dots}\\label{qdot}\n\nQuantum dots formed in semiconductor heterostructures have been\nstudied extensively because they are considered promising\ncandidates for applications in optoelectronics on the nanometer\nscale~\\cite{Wiel2003,Reimann2002,Michler2009,Buluta2009,Buluta2010}.\nFor instance, dots might be used in detectors, diodes, memory and\nlaser devices. Furthermore, single-electron transport devices\nwhich make use of quantum dots could be employed as transistors,\nand spin-based dot devices might be useful for quantum logic\ngates. Electrons confined in usual semiconductor dots, with a\ntypical size of a few hundreds of nanometers, are described by the\nSchr\\\"odinger equation and most of their electronic properties are\nnow well-understood and have been experimentally studied by many\nresearch groups.\n\n\nThe physics of graphene quantum dots is very different from that\nin usual semiconductor dots. The reason is twofold: $(i)$ charge\ncarriers in graphene are massless and obey the relativistic 2D\nWeyl-Dirac equation~(\\ref{dirac}), and $(ii)$ the different\nconfigurations of the carbon atoms at the boundaries of the dot\naffect significantly the dot properties. There are two basic\nmethods of defining a graphene quantum dot. In the first method,\nthe dots are defined by the actual geometry of the graphene layer\nand they are usually referred to as graphene islands. In the\nsecond method, the dots are defined through the application of\nelectric and magnetic fields. Of course quantum dots can also be\ndefined by combining these two methods and recently some other\nideas have been put forward for dot formation, which for example\ninclude the application of strain to the graphene sheet, a\nspectral gap opening, and even chemical techniques.\n\n\n\\subsection{Geometry-induced dots and graphene islands}\n\nIt is now possible to mechanically cut (i.e., etch) a graphene\nflake into various shapes of a few tens of nanometers, which can\nconfine electrons and thus act as quantum dots. These\ngeometry-induced dots or graphene islands have well-defined\ndiscrete energy levels whose spectrum depends on the size, shape\nand the edge type of the dot. Further, disorder and interaction\neffects are also important for the electronic properties of any\nrealistic graphene system.\n\n\nA range of typical dot geometries including triangular, hexagonal,\nrectangular and circular have been studied numerically, mainly\nwithin the tight-binding and DFT\nmodels~\\cite{Ezawa2007,Viana-Gomes2009,Fernandez-Rossier2007,Tang2008,\nBhowmick2008,akola_trigonal,heiskanen_trigonal,Kim2010}. In some\ncases exact analytical solutions are also\npossible~\\cite{potasz_exact_trigonal,ezawa_exact_trigonal,Rozhkov2010}.\nFor example, for a triangular armchair dot, exact tight-binding\neigenfunctions and eigenenergies were obtained, and a technique\nfor matrix element calculation was developed~\\cite{Rozhkov2010}\n(see Fig.~\\ref{exactstate}). Quantum dots with arbitrary shapes\nhave also been examined~\\cite{Wimmer2010}.\n\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=12.cm \\epsfbox{NEW_tqd_41_41_39.eps}\\caption{(Color\nonline) The Schr\\\"odinger equation for the tight-binding\nHamiltonian on a triangular armchair quantum dot can be solved\nexactly~\\cite{Rozhkov2010}. Moreover, the algebraic structure of\nthe wave functions found is sufficiently simple to allow for\nanalytical expressions for some matrix elements. In this figure\nthe exact probability density for certain electron eigenstate on a\ntriangular quantum dot is plotted.}\\label{exactstate}\n\\end{center}\n\\end{figure}\n\n\nAn important feature of the nano-islands is the appearance of\ndegenerate zero-energy states that are mostly localized at the\nedges, as predicted for triangular and circular zigzag dots, as\nwell as rectangular\ndots~\\cite{Ezawa2007,Viana-Gomes2009,Fernandez-Rossier2007,Tang2008,Bhowmick2008,Kim2010}.\nFor triangular dots there is a sublattice imbalance, i.e.,\n$N_{Z}$=$N_{A}-N_{B}$$\\neq$0, where $N_{A}$ $(N_{B})$ is the\nnumber of carbon atoms of sublattice $\\mathcal{A}$\n$(\\mathcal{B})$, and this condition is sufficient in order to have\n$N_{Z}$ zero-energy states~\\cite{Fernandez-Rossier2007}. The\nnumber of these states is proportional to the size of the edges\nwhich, in principle, can be made quite large.\n\n\nNanostructures with degenerate zero-energy states are useful for\napplications, since the electrons inside such structures may order\nmagnetically. Magnetism is a consequence of the Coulomb\ninteraction and Hund's rule. For example, the ground state of\nrectangular dots can support antiferromagnetic ordering whereby\nthe magnetic moments are localised at the zigzag edges with\nopposite orientation (for edge magnetism, see also the previous\nsection). As shown in Ref.~\\cite{Tang2008} for rectangular dots\nthere is a critical minimum width between the zigzag edges that\ngives rise to magnetic ordering. If the width is smaller, then the\nstate is nonmagnetic. On the other hand, triangular zigzag dots\nfavor ferromagnetic ordering (see Table~\\ref{island1}).\n\n\nInterestingly, external uniaxial strain on square dots enhances\nthe magnetization and leads to a spatial displacement (drift) of\nthe magnetization from the zigzag to the armchair\nedges~\\cite{Viana-Gomes2009}. A magnetization enhancement of 100\\%\nwas predicted for a strain on the order of 20\\%, which might be\npossible to induce by mechanical methods~\\cite{mohiuddin2009}.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=7.50cm,angle=0]{805825JCP2a}\n\\includegraphics[height=7.50cm,angle=0]{805825JCP2c}\n\\includegraphics[height=7.50cm,angle=0]{805825JCP2b}\n\\includegraphics[height=7.50cm,angle=0]{805825JCP2d}\n\\caption{(Color online) Magnetic properties of graphene dots were\ninvestigated numerically in Ref.~\\cite{Bhowmick2008} within the\nframework of a mean field theory of the Hubbard model. Expectation\nvalue of the spin magnetization $S^{z}$ in four different dots.\nThe size of the arrows is proportional to the magnitude of the\nmagnetic moment. In this figure, $N$, $N_{m}$ and $N_{n}$ denote\nthe size of the dot in the perfect hexagon configuration, the\nnumber of armchair edges per vertical side, and the number of\narmchair edges per slanted edge, respectively. The magnetic\nmoments are much larger at the zigzag edges than at the armchair\nedges and the internal sites. Reprinted with permission from S.\nBhowmick and V. B. Shenoy, Journal of Chemical Physics {\\bf128},\n244717 (2008). Copyright 2008 American Institute of\nPhysics.}\\label{nanodot}\n\\end{center}\n\\end{figure}\n\nMoreover, it was theoretically predicted that the edge-state\nmagnetism is robust to impurities and edge-defects, and survives\neven to irregular structures as long as there are three to four\nrepeat units of zigzag edges~\\cite{Bhowmick2008}. A similar\nconclusion was reached in\nRef.~\\cite{kumazaki_magnetic_edge_states}. Some examples of\nmagnetic structures for different dots are presented in\nFig.~\\ref{nanodot}. The results are derived from the mean field\nanalysis of the Hubbard model~\\cite{Bhowmick2008}.\n\n\nAnother promising property, which may be used in memory devices,\nis that the spin relaxation time can be long enough, as shown for\ntriangular dots. In general, the spin relaxation time increases\nwith the interaction strength and system size, but remains long\neven for a relatively small system~\\cite{Ezawa2007}.\n\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{||c|c|c||}\n\\hline\\hline \\multicolumn{3}{||c||}{\\textbf{Graphene dots or\nislands (theoretical studies)}}\\cr \\hline\\hline\n   Type & \\quad Zero-energy edge states \\quad \\quad & Magnetic ordering  \\cr\n    \\hline\\hline\n  Triangular   &  Yes & Ferromagnetic \\cr\n  Hexagonal  &  No  &  No \\cr\n\\quad \\quad Parallelogram \\quad \\quad &  No & No \\cr\n Rectangular &  Yes & \\quad \\quad Antiferromagnetic \\quad \\quad \\cr\n   \\hline\n   \\hline\n\\end{tabular}\n\\caption{Graphene dots have attracted considerable theoretical\ninterest. Various geometries were examined and degenerate\nzero-energy edge states were predicted for triangular zigzag and\nrectangular dots (the rectangular dot is defined by two zigzag and\ntwo armchair edges). These states are mainly localized at the\nzigzag edges having only a small amplitude at the centre of the\ndot. Edge states are absent at arcmhair edges. The existence of\nsuch states leads to magnetic ordering that critically depends on\nthe specific geometry. For rectangular dots there exists a minimum\nwidth between the two zigzag edges for stable antiferromagnetic\nordering~\\cite{Tang2008}. The magnetic properties of the\ngeometry-induced dots are robust to defects and impurities of the\nedges, can survive to irregular structures~\\cite{Bhowmick2008},\nand can be tuned by the application of an external\nstrain~\\cite{Viana-Gomes2009}.}\\label{island1}\n\\end{table}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=10.0cm,angle=0]{graphite.eps}\n\\caption{(Color online) Gated graphite quantum dots were\nfabricated and low-temperature electrical transport measurements\nwere performed~\\cite{Bunch2005}. (a) Current versus gate voltage\n$V_{g}$ with source-drain bias $V_{sd}=10$ $\\mu$V at temperature\n$T\\sim 100$ mK. Coulomb oscillations are observed with a period in\ngate voltage of $\\Delta V_{g}=1.5$ mV. (b) The differential\nconductance ($dI\/dV_{sd}$) is plotted as a color scale versus gate\nvoltage ($V_{g}$) and source-drain bias $V_{sd}$. Blue (red)\nsignifies low (high) conductance. The charging energy of the dot\nis equal to the maximum height of the diamonds: $\\Delta\nV_{sd}=0.06$ mV. Reprinted with permission from J. S. Bunch, Y.\nYaish, M. Brink, K. Bolotin, and P. L. McEuen, Nano Letters\n{\\bf5}, 287 (2005). Copyright 2005 American Chemical\nSociety.}\\label{graphitedot}\n\\end{center}\n\\end{figure}\n\n\nReference~\\cite{Bunch2005} reported low-temperature electrical\ntransport measurements on gated quasi-2D graphite quantum dots.\nThese were the first measurements on mesoscopic samples of\ngraphite which consists of many stacked layers of graphene held\ntogether by weak van der Waals forces. Coulomb charging phenomena\nwere demonstrated with the help of data in Fig.~\\ref{graphitedot},\nwhere the electrical current through the dot as a function of gate\nvoltage and source-drain bias is plotted. More recent experiments\nprobed the energy spectrum of quantum dots formed in a single\nlayer of\ngraphene~\\cite{Ponomarenko2008,Stampfer2008a,Stampfer2008}. An\nall-graphene single-electron transistor, exhibiting\nCoulomb-blockade behaviour, was operational well-above\nliquid-helium temperatures~\\cite{Ponomarenko2008}. The\nCoulomb-blockade peaks are (nearly) periodic as a function of gate\nvoltage for large islands ($>100$ nm), and nonperiodic for small\nones ($<100$ nm). The distance between the peaks is proportional\nto the sum of charging and confinement energies. The former, being\ntypically constant for a specific dot geometry, dominates for\nlarge islands~\\cite{Ponomarenko2008}. For small islands the size\nquantization becomes important, and the confinement energy\nprevails, leading to nonperiodic peaks (see Table~\\ref{island2}).\nThe energy-level statistics of graphene islands was also probed,\nand it was shown to agree well with the theory of chaotic Dirac\nbilliards~\\cite{Ponomarenko2008}.\n\n\nCoulomb-blockade measurements on a graphene island ($\\sim 200$ nm)\nwith an integrated charge detector were also\nreported~\\cite{Guttinger2008}. A nanoribbon placed 60 nm from the\nisland acts as a detector, which enhanced the resolution of single\ncharging events on the island. In addition, tunable double quantum\ndots were fabricated whereby the coupling to the leads and the\ninterdot coupling were tuned by graphene in-plane\ngates~\\cite{Molitor2009}. Spin spectroscopy has also been\ninvestigated in graphene dots~\\cite{Guttinger2010}.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=9.50cm,angle=0]{guttinger}\n\\caption{(Color online) Graphene quantum dot device with charge\ndetector and transport measurement data~\\cite{Guttinger2008}. (a)\nScanning force micrograph of the measured device. The central\nisland, that acts as the quantum dot (QD), is connected to source\n(S) and drain (D) contacts via two narrow constrictions. The\ndiameter of the dot is 200 nm and the constrictions are 35 nm\nwide. The charge detector (CD) is a graphene nanoribbon and the\nlateral gates B1, B2 and PG are used to tune the device. (b)\nCurrent as a function of back-gate voltage of the QD (upper panel)\nand CD (lower panel). The source-drain voltage is 500 $\\mu$V and\nthe temperature is 1.7 K. The inset shows typical Coulomb-blockade\nfeatures as expected in a dot device. (c) Differential conductance\nis plotted as a color scale versus source-drain bias and PG gate\nvoltage, for a back-gate voltage of 2 V. The charging energy was\nestimated to be about 4.3 meV. Reprinted with permission from J.\nGuttinger, C. Stampfer, S. Hellmuller, F. Molitor, T. Ihn, and K.\nEnsslin, Applied Physics Letters {\\bf93}, 212102 (2008). Copyright\n2008 American Institute of Physics.}\\label{chargedetector}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{||c|c|c||}\n\\hline\\hline \\multicolumn{3}{||c||}{\\textbf{Graphene dots or\nislands (experimental studies)}}\\cr \\hline\\hline\n \\quad  Size of island ($D$) \\quad & \\quad Coulomb peaks \\quad \\quad & Energy scale  \\cr\n    \\hline\\hline\n  $<$ 100 nm   &  Nonperiodic & \\quad Confinement $\\sim v_{F}h\/2D\\approx41$ meV ($D=$ 50 nm) \\quad \\cr\n  $>$ 100 nm  &  Periodic  & Charging $\\approx$ 3 meV ($D\\approx$ 250 nm) \\cr\n   \\hline\n   \\hline\n\\end{tabular}\n\\caption{Graphene islands were investigated experimentally via\nelectrical transport measurements, and single-electron transport\nwas demonstrated~\\cite{Ponomarenko2008}. When the diameter of the\nisland is large ($D>100$ nm) the Coulomb peaks in the conductance,\nas a function of back-gate voltage, are periodic and their\nposition is determined mainly by the characteristic charging\nenergy. For small-diameter islands ($D<100$ nm), the position of\nthe peaks is nonperiodic and the dominant energy scale is the\nconfinement energy on the order of $E_{D}\\sim v_{F}h\/2D$ ($h$ is\nPlanck's constant). This is much larger than the corresponding\nenergy $E_{S}\\sim h^{2}\/8 mD^2$ of Schr\\\"odinger electrons with\neffective mass $m$; $E_{D}\/E_{S}\\sim40$ for $D=100$ nm, and $m$ is\nthe effective mass for GaAs. Stable, robust and conductive dot\nislands as small as 15 nm were fabricated, showing the potential\nof graphene for nanoelectronics.}\\label{island2}\n\\end{table}\n\n\n\n\n\\subsection{Field-induced dots}\n\nCharge confinement within the ``bulk'' graphene sheet is tricky\ndue to the Klein tunneling effect~\\cite{Katsnelson2006}. In case\nof normal incidence, this allows for perfect transmission of\nmassless relativistic particles through high and wide potential\nbarriers. A key point is that in the barrier region the states of\nmassless Weyl-Dirac particles have an oscillatory character, even\nat energies lower than the potential height, as happens exactly\noutside the barrier. This is completely different from\nSchr\\\"odinger particles with non-zero mass, for which the states\nin the barrier region decay exponentially and therefore perfect\ntransmission is not feasible. Experimentally, the Klein tunneling\nwas demonstrated in graphene through electrical transport\nmeasurements in steep potential barriers generated by metallic\ngates~\\cite{Stander2009}.\n\n\nBecause of the Klein tunneling, an electrostatic potential minimum\nin graphene leads to quasi-bound states, i.e., resonant states\n(see Table~\\ref{states} for a classification of the dot states)\nand therefore it is inadequate to confine\nelectrons~\\cite{Chen2007,Matulis2008,Hewageegana2008}.\nNevertheless, the finite lifetime of the states, characterizing\nthe trapping time of an electron in the dot region, can be\nrelatively long. It depends on the potential profile and\neigenenergy. A smooth potential and a large angular momentum\nenhance the lifetime of the quasi-bound\nstates~\\cite{Chen2007,Hewageegana2008}, as happens also with\neigenenergies close to the maximum of the potential\nbarrier~\\cite{Matulis2008}. Figure~\\ref{densitymatulis} shows the\neffect of the quasi-bound states on the local density of states of\na circular dot.\n\n\nThe electrostatic confinement of electrons in graphene dots was\nalso examined through the dependence of the conductance on the dot\narea, which is tunable with a metal gate~\\cite{Bardarson2009}.\nBoth disc-shaped dots, in which the classical dynamics is regular,\nand stadium-shaped dots where the classical dynamics is chaotic\nwere studied. Confinement can be achieved only in the former when\nthe corresponding Weyl-Dirac equation is separable.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=9.50cm,angle=0]{matulis.eps}\n\\caption{(Color online) The physics of quasi-bound states in\ncircular graphene dots was examined theoretically by solving the\nWeyl-Dirac equation~\\cite{Matulis2008}. The figure shows the local\ndensity of states as a function of energy in the case of a barrier\nheight of $V=12$ (dimensionless units) for three angular momentum\nnumbers: $m=0$ (solid curve), $m=1$ (dotted curve), and $m=2$\n(dashed curve). The peaks become narrower as the momentum\nincreases, within a specific energy range, thus the lifetime of\nthe corresponding states becomes longer. Notice the very narrow\npeak when the energy is close to the barrier height (for $m=0$),\nas a consequence of the total reflection of the wavefunction at\nthe dot edge. Reprinted figure with permission from A. Matulis and\nF. M. Peeters, Physical Review B {\\bf77}, 115423 (2008). Copyright\n(2008) by the American Physical Society.}\\label{densitymatulis}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=10.0cm,angle=0]{Picture4.eps}\n\\caption{(Color online) Klein tunneling in a circular graphene\nquantum dot in a uniform magnetic field. The dot is defined by the\nelectrostatic potential $V(r)$, which vanishes at the centre\n(\\textit{O}) of the dot and asymptotically rises to a constant\nvalue, while the magnetic field $B$ is perpendicular to the\ngraphene sheet. (a) For $B=0$ and small angular momentum, the\nradial probability distribution $\\rho(r)$, for one of the spinor\ncomponents, has a large amplitude near the centre of the dot and\noscillates inside the barrier region because of the Klein\ntunneling. In this case, which is unique to graphene, the quantum\nstate is quasi-bound and has an oscillatory asymptotic character.\n(b) When the magnetic field is nonzero, the Klein tunneling is\npartially suppressed. At large $r$ the oscillatory behavior is\nreplaced by exponential decay, indicating that the state is bound.\nAs in (a), this case is also unique to graphene. (c) With\nincreasing magnetic field, the Klein tunneling is completely\nsuppressed and the probability distribution decays exponentially\ninside the electrostatic barrier. The state is now bound near the\ncentre of the dot and such a state can be seen in both graphene\ndots and usual semiconductor dots, e.g., GaAs. For the latter,\nthis state can be seen even when $B=0$. A magnetic-field-induced\nconfinement-deconfinement transition in a graphene dot due to the\nKlein tunneling was theoretically examined in\nRef.~\\cite{Giavaras2009}.}\\label{dotklein}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=7.50cm,angle=0]{peterpotential.eps}\n\\includegraphics[height=8.0cm,angle=270]{maksym_dot}\n\\includegraphics[height=8.0cm,angle=270]{maksym_antidot}\n\\caption{A graphene dot or an antidot can be formed with the help\nof a uniform magnetic field applied perpendicular to the graphene\nsheet~\\cite{Maksym2010}. (top) The electrostatic potential profile\nof an antidot (solid line) and a dot (dashed line) along the\nradial direction. Energy spectra as a function of the applied\nmagnetic field of dot (left) and antidot (right). The dashed lines\nshow the Landau levels of an ideal graphene sheet. The potential\nis adjusted so that the confined dot (antidot) states lie in the\ngap between Landau level 0 and Landau level $-$1 ($+$1). The\nsymmetry between the energy levels ($E\\rightarrow-E$) of the two\nsystems is a direct consequence of the Dirac cone band-structure\nin graphene.}\\label{dotmaksym}\n\\end{center}\n\\end{figure}\n\n\n\nThe application of a magnetic field can completely suppress the\nKlein tunneling, leading to bound\nstates~\\cite{DeMartino2007,Masir2009,Giavaras2009,Giavaras2010a}.\nThus graphene quantum dots can be formed with the help of a\nnonuniform magnetic field, whereby the field is zero within a disc\narea defining the spatial region of the dot, and nonzero outside\nthe dot~\\cite{DeMartino2007}. The combination of an electrostatic\npotential and a vector magnetic potential allows the\nconfined-deconfined character of the dot states to be tuned at\nwill~\\cite{Giavaras2009}. Most interestingly, it allows graphene\ndots to be formed in a uniform magnetic field using standard gate\nelectrodes as in common semiconductors~\\cite{Giavaras2009}. Then\nthe quantum states can be tuned with the strength of the magnetic\nfield and this property allows the Klein tunneling mechanism to be\nprobed experimentally in graphene dots. A dot design suitable for\nthis experiment was suggested in Ref.~\\cite{Giavaras2009}. The\nconcept of defining a magnetic graphene dot was further developed\ntheoretically in Ref.~\\cite{Maksym2010}. In particular, in a\nstrong magnetic field the electrostatic potential of the dot is\nadjusted so that the confined dot states lie in the gap between\nLandau level 0 and Landau level $-1$ (see Fig.~\\ref{dotmaksym}).\nThis ensures that the dot states are energetically isolated in a\nregion of low density of states and thus they can be probed using\nstandard charge-sensing measurements as in a GaAs dot. Numerical\nestimates showed that a typical spacing between the dot levels is\n$\\sim2$ meV at a magnetic field of 5 T. In addition,\nRef.~\\cite{Maksym2010} considered how this basic idea can be\nextended to a graphene antidot for which the levels of the\nconfined states lie in the region between Landau level 0 and\nLandau level +1. For a confined state with energy $E$ in the dot,\nthere is a corresponding confined state with energy $-E$ in the\nantidot. This is a unique property of graphene due to the symmetry\nof the Dirac cone. The physics of graphene antidots in a magnetic\nfield was also examined in Ref.~\\cite{Park2010}.\n\n\n\n\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{||c|c|c||}\n\\hline\\hline \\multicolumn{3}{||c||}{\\textbf{Quantum dot\nstates}}\\cr \\hline\\hline\n \\quad  Type  \\quad & \\quad Lifetime \\quad \\quad & \\quad Spatial asymptotic behaviour of wavefunction \\quad \\cr\n  (other used names)  & & \\cr\n    \\hline\\hline\n  Bound   &   Infinite & \\quad  Exponential decay \\quad \\cr\n (confined, stable)  & &  \\cr\n  Quasi-bound  &  Finite  & Oscillatory \\cr\n \\quad (deconfined, resonant) \\quad &  &  \\cr\n   \\hline\n   \\hline\n\\end{tabular}\n\\caption{The quantum states of a dot, formed within the ``bulk''\ngraphene sheet, can be either bound or quasi-bound. Because of the\nKlein tunneling, both types of states can have a large amplitude\nin the barrier region of the quantum dot, though their asymptotic\nbehaviour is different. Exponential decay is characteristic of\nbound states, whereas oscillatory behaviour is characteristic of\nquasi-bound states. As shown in Ref.~\\cite{Giavaras2009} the type\nof states can be tuned with an electrostatic potential and a\nuniform magnetic field perpendicular to the graphene sheet (see\nalso Table~\\ref{circular}).}\\label{states}\n\\end{table}\n\n\n\nFigure~\\ref{dotklein} illustrates the magnetic field-induced\nsuppression of the Klein tunneling for an electron inside a dot,\nand Table~\\ref{circular} summarizes the general conditions for\nconfinement in a circular quantum dot, as derived in\nRef.~\\cite{Giavaras2009}. The physics of electrostatic barriers in\nthe presence of uniform and nonuniform magnetic fields is analyzed\nin Sec.~\\ref{barrier}.\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{||c|c|c||}\n    \\hline\n    \\hline\n    \\multicolumn{3}{||c||}{\\textbf{Circular graphene dot: $V=V_{0}r^{s}$, $A_{\\theta}=A_{0}r^{t}$}}\\\\\n     \\hline\\hline\n \\quad Asymptotically \\quad & \\quad $s$, $t$ \\quad & Dot states \\quad \\\\\n    \\hline\n     \\hline\n$A_{\\theta}<V$    & \\quad $t<s$  \\qquad & \\quad Quasi-bound for all $V_{0}$, $A_{0}$ \\quad\\\\\n$A_{\\theta}\\sim V$& \\quad $t=s$  \\qquad & \\quad Quasi-bound for $V_{0}> v_{F} eA_{0}$\\quad \\quad\\\\\n$A_{\\theta}\\sim V$& \\quad $t=s$  \\qquad & \\quad Bound for $V_{0}< v_{F} eA_{0}$ \\quad \\\\\n$A_{\\theta}>V$    & \\quad $t>s$  \\qquad & \\quad Bound for all  $V_{0}$, $A_{0}$ \\quad \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{Confinement of electrons in a circular graphene dot is\nconditional because of the Klein tunneling~\\cite{Giavaras2009}.\nConsider a graphene dot defined by the electrostatic potential\n$V=V_{0}r^{s}$ and the magnetic vector potential\n$\\mathbf{A}=(0,A_{\\theta},0)$, with $A_{\\theta}=A_{0}r^{t}$ and\n$s$, $t>0$. In such a situation, if asymptotically $A_{\\theta}<V$,\nthen the quantum states are quasi-bound; if $A_{\\theta}>V$, they\nare bound. In the special case $A_{\\theta}\\sim V$, the states are\nbound only when $V_{0}<v_{F}eA_{0}$ ($e$ is the absolute value of\nthe electron charge). This suggests that the states can be tuned\nat will with an electric or magnetic field. A simple limit occurs\nfor a constant potential ($V=$ const.) and uniform magnetic field\n$B$ perpendicular to the graphene sheet ($t=1$,\n$A_{0}=\\frac{B}{2}$). Then the states are always bound and\ncorrespond to the well-known Landau levels.}\\label{circular}\n\\end{table}\n\n\n\\subsection{Nanoribbons of graphene and dots}\n\n\nAs discussed in the previous section, the most commonly studied\ngraphene nanoribbons are either of armchair or zigzag type. A\nquantum dot defined by an external electrostatic (parabolic)\npotential along the nanoribbon was investigated theoretically in\nRef.~\\cite{Silvestrov2007}. Quasi-bound states inside nanoribbons\nwith either zigzag or armchair edges, in both metallic and\nsemiconducting samples, were predicted. The lifetime of such\nstates can be made long enough by increasing the characteristic\nlength of the external potential, for example, with a gate\nelectrode. Further, the dependence of the conductance on the gate\nvoltage was found to be sensitive to the type of edges as\nillustrated in Fig.~\\ref{ribbondot}. The proposed device can\noperate either as a quantum dot or as a point contact. For the\nformer, the conductance displays resonances at negative Fermi\nenergy, whereas for the latter it has a step-like behaviour at\npositive Fermi energy. The conductance steps in the semiconducting\nsystem are twice smaller than those in the metallic system.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=8.0cm,angle=0]{silvestr}\n\\caption{Theoretical studies showed that a graphene quantum dot\ncan be formed by an external electrostatic potential in a\nnanoribbon system~\\cite{Silvestrov2007}. The conductance through\nthe dot as a function of the Fermi energy depends on whether the\nsystem is semiconducting (solid line) or metallic (dashed line).\nIn both cases, the nanoribbons have armchair edges. For\n$\\varepsilon<$ 0 the device operates as a quantum dot and the\nconductance exhibits resonances. For $\\varepsilon>$ 0 the device\noperates as a point contact and the conductance exhibits steps.\nFor $\\varepsilon>$ 0 the conductance steps in the semiconducting\nsystem are twice smaller than those in the metallic. Reprinted\nfigure with permission from P. G. Silvestrov and K. B. Efetov,\nPhysical Review Letters {\\bf98}, 016802 (2007). Copyright (2007)\nby the American Physical Society.}\\label{ribbondot}\n\\end{center}\n\\end{figure}\n\n\nSemiconducting nanoribbons with armchair edges were proposed for\nthe formation of spin qubits in graphene\ndots~\\cite{Trauzettel2007,Recher2010}. Confinement in one\ndirection is achieved naturally by the nanoribbon and in the\nsecond direction electrically by gate voltages. In this set-up,\nthe valley degeneracy is lifted, thus allowing Heisenberg spin\nexchange coupling in tunnel-coupled dots. Such graphene dots can\nbe coupled over long distances as a consequence of the\nrelativistic nature of electrons in graphene, exhibiting Klein\ntunneling.\n\n\nThe electrostatic confinement of electrons in graphene nanoribbons\nas well as the Coulomb-blockade effect were experimentally\ndemonstrated~\\cite{Liu2009}. In particular, electrons are confined\nbetween gate-induced $pn$-junctions acting as barriers. However,\neven when no $pn$-junctions are formed, the electrons are still\nconfined, though in a larger area due to strong disorder.\n\n\n\\subsection{More dots}\n\nTunable quantum dots which take advantage of a gap in the energy\ndispersion were also proposed in both monolayer and bilayer\ngraphene~\\cite{Recher2009,Pereira2007,Giavaras2010b,Giavaras2011}.\nExperimentally, the gap can be introduced via a chemical and\/or\nelectrical technique~\\cite{Zhang2009,Ohta2006} (see also\nTable~\\ref{energy_gap}), and it allows dots to be formed\nelectrostatically in a quite similar manner as in common\nsemiconductors. A finite gap introduces a mass term in the\nWeyl-Dirac Hamiltonian. Then for a dot-confining potential with a\nfinite asymptotic value, the gap gives rises to an energy range\nwithin which the Klein tunneling is suppressed, leading to the\nformation of bound states. In this energy range, which is directly\nproportional to the value of the gap, hole states do not exist and\ntherefore the electron states decay exponentially. Moreover, for\nquantum dots formed in the gapped sample the valley degeneracy is\nlifted by a uniform magnetic field. This property might be\nattractive in order to define spin and valley\nqubits~\\cite{Recher2009}.\n\n\nIt was also shown theoretically that a spatially modulated Dirac\ngap in the graphene sheet can lead to confined states with\ndiscrete energy levels, thus giving rise to a dot. The basic\nadvantage of this proposal is that the dot is formed without\napplying external electric and\/or magnetic\nfields~\\cite{Giavaras2010b}. Thus magnetic fields can be used to\nmanipulate the spin states without affecting the confinement of\nthe corresponding orbital states. The properties of a Dirac\ngap-induced graphene dot in the presence of an electrostatic\nquantum well potential were studied in Ref.~\\cite{Giavaras2011}.\nIt was shown that confined states which are induced thanks to the\nspatially modulated Dirac gap couple to the states induced by the\npotential. The resulting hybridised states are localised in a\nregion which can be tuned with the potential strength; an effect\nwhich involves Klein tunneling. Numerical calculations of the\nlocal density of states suggest that this effect could be\nprobed~\\cite{Giavaras2011}.\n\n\nStrain engineering is another proposal in order to generate\nconfinement in a sheet of graphene~\\cite{Pereira2009}. Local\npatterning of the substrate induces in-plane strain in the\ngraphene lattice, anisotropically changing the hopping amplitude\nbetween the carbon atoms. As a result, in the continuum\napproximation the quasi-particles are described by an effective\nWeyl-Dirac equation in the presence of a gauge field. It turns out\nthat this field can act in a rather similar manner as a magnetic\nfield and therefore it can lead to confined quantum states. A\nnoteworthy advantage of this proposal~\\cite{Pereira2009} is that\npatterning can be made directly on the substrate, hence protecting\nthe graphene layer from possible damage.\n\n\nVacancy clusters in the graphane sheet were also suggested for dot\nformation. In particular, DFT and tight binding calculations\nshowed that cluster of hydrogen vacancies can serve as quantum\ndots. The stability as well as the shape and size of these dots\ndepend crucially on the graphene\/graphane interface energy and the\ndegree of aromaticity~\\cite{Singh2010}.\n\n\n\\subsection{Quantum rings}\n\n\nQuantum rings in graphene have also attracted some interest,\nmainly because these types of devices allow the investigation of\nphase-coherence phenomena, as it is now well-known from studies in\nusual semiconductor systems.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=11.0cm,angle=0]{russoring.eps}\n\\caption{ (Color online) The Aharonov-Bohm (AB) effect was\nexperimentally demonstrated in a graphene ring\ndevice~\\cite{Russo2008}. (a)-(c) AB conductance oscillations\nversus applied magnetic field, for a back-gate voltage of 30 V and\ntemperature of 150 mK. For $B\\sim$ 3 T, an increase of the AB\namplitude is observed. Panel (a) shows a magnified view of the\nyellow region in (c), while panel (b) expands the blue part in\n(c). Reprinted figure with permission from S. Russo, J B Oostinga,\nD. Wehenkel, H. B. Heersche, S. S. Sobhani, L. M. K. Vandersypen,\nand A. F. Morpurgo, Physical Review B {\\bf77}, 085413 (2008).\nCopyright (2008) by the American Physical Society.}\n\\end{center}\n\\end{figure}\n\n\nIn a graphene ring device, conductance oscillations versus\nmagnetic field were reported as a consequence of the Aharonov-Bohm\neffect~\\cite{Russo2008}. The amplitude of the oscillations\nincreases at high magnetic field in the regime where the cyclotron\ndiameter becomes comparable to the width of the arms of the ring.\nFor temperatures below 1~K the extracted phase-coherence length is\ncomparable to or larger than the diameter of the ring, which is\napproximately 1 $\\mu$m.\n\n\nTheoretical investigations of graphene rings showed that the\nvalley-induced orbital degeneracy is lifted, as a result of the\nring confinement and the applied magnetic field~\\cite{Recher2007}.\nThis lifting has observable consequences on the persistent current\nand the ring conductance. An interesting finding is that the\ndegeneracy can be controlled with the induced Aharonov-Bohm flux,\nand this can be achieved irrespective of the magnitude (weak or\nstrong) of the intervalley scattering.\n\n\nAnother theoretical work showed that both electrons and holes can\nbe confined in electrostatically formed quantum rings in bilayer\ngraphene~\\cite{Zarenia2009}. There are two main advantages in this\nproposal. First, bound states can be created owing to a\nposition-dependent energy gap that suppresses the Klein tunneling.\nSecond, the ring parameters can be tuned by external fields.\n\n\nThe role of Coulomb-induced electron-electron interactions and\ntheir interplay with the valley polarization in a graphene quantum\nring were also examined~\\cite{Abergel2008}.\nIn a few-electron ring, the interactions have a direct signature\non the fractional nature of the Aharonov-Bohm oscillations in the\npersistent current and the absorption spectrum, and therefore they\ncould be observed.\n\n\\section{Edges of graphene samples}\n\\label{edge}\n\nThe characteristics of a mesoscopic device depend substantially on its\nedges. Therefore, it is important to study the electron behavior near the\ngraphene edge.\n\n\\subsection{Edge-stability issues}\n\nTwo kinds of edges are often discussed in the literature: zigzag and\narmchair. A form of the zigzag is the Klein edge\n\\cite{klein_edge}.\nAll three types are shown in\nFig.~\\ref{graphene_lattice}.\nThey are the most symmetric variants of edges in graphene. More complicated\nedges were also studied\n\\cite{complicated_edges,nakada-fujita_nribb_edge_st,\ndirac_boundary_cond_graphene,tkachev_zigzag_states_finite_length,\ngan_edge_stability}.\n\nOf course, in a laboratory sample some of these edge types may be unstable\nchemically or undergo reconstruction. The possibility of the edge\nreconstruction has been addressed in several publications. Most\nimportantly, it appears that the pristine zigzag edge is unstable:\nrecently, it was predicted on the basis of density-functional theory (DFT)\ncalculations\n\\cite{koskinen_reczag2}\nthat it might undergo reconstruction at room temperature, and become a {\\it\nreczag}\n(short for `reconstructed zigzag'\n\\cite{koskinen_reczag},\nsee Fig.~\\ref{reczag}).\nThis kind of edge is often called `ZZ 57'; namely, it is a version of zigzag\n(thus ZZ) edge, in which the edge hexagons are replaced by pentagons and\nheptagons (hence the 5 and 7). Experimental data supporting the existence\nof the reczag edge were presented in\n\\cite{koskinen_reczag}.\nA similar conclusion was reached in\n\\cite{gan_edge_stability}:\nthe energy of the zigzag edge is substantially higher than the energy of\nthe reczag.\nIn Refs.~\\cite{wassmann_edge_stab,wassmann_phys_stat_sol}\nthe non-hydrogenated zigzag edge was not listed among stable configurations.\n\\begin{figure}[btp]\n\\centering \\leavevmode \\epsfxsize=10.5cm\n\\epsfbox{reczag.eps}\n\\caption[]\n{\\label{reczag}\nReczag, or ZZ 57, edge of a graphene sheet,\nfrom \\cite{koskinen_reczag2}.\nIn the right panel the edge unit cell is shown. It consists of a pentagon\nand a heptagon. The latter polygons are the reason why this edge type is\ncalled `57'. Numbers in the right panel are the bond lengths in \\AA.\nReprinted figure with permission from \nP. Koskinen, S. Malola, and H. Hakkinen, Phys. Rev. Lett. {\\bf 101},\n115502 (2008).\nCopyright (2008) by the American Physical Society.\n}\n\\end{figure}\n\nThe conclusions of Ref.~\\cite{koskinen_reczag2}, regarding the\nrelative stability of zigzag and reczag, were challenged in\nRef.~\\cite{magnet_p-orbit}, where the experiment of\nRef.~\\cite{ritter_lyding}, proving the existence of the zigzag\nedge in a laboratory sample, was quoted. The experimental\ndemonstration of the zigzag edge stability was also reported in\nRef.~\\cite{girit_zigzag_stab}. Results of Ref.~\\cite{mol_dyn} are\nalso in disagreement with Ref.~\\cite{koskinen_reczag2}. However,\nthe authors of Ref.~\\cite{mol_dyn} were unsure whether their\nmolecular dynamics simulations can provide a reliable answer to\nthe question of the edge stability.\n\nThe chemical stability was also investigated. It was pointed out in\nRefs.~\\cite{wassmann_edge_stab,wassmann_phys_stat_sol}\non the basis of DFT calculations that the reczag and armchair are stable\nonly when the concentration of hydrogen in the surrounding media is very\nsmall. If this is not the case, other types of edges, with hydrogen atoms\nattached, are stabilized (see\nFig.~\\ref{wass_table}).\nThe results of DFT are consistent with Clar's theory of the aromatic sextet\n\\cite{clar_poly,clar_sextet}.\n\nThe DFT calculations of\nRef.~\\cite{gan_edge_stability}\ndemonstrated that the energy of zigzag, armchair, reczag, and more\ncomplicated regular edges always decreases upon monohydrogenation. This\nagrees with\nRef.~\\cite{wassmann_edge_stab}:\nwhen enough hydrogen is present in the surrounding media, hydrogenation of\nthe edge occurs.\n\nThese results suggest that the edge stability is a complicated problem in\ngraphene. The edge stability depends on the orientation of the edge and is\naffected by the chemical environment.\n\\begin{figure*}\n\\centering\n\\leavevmode\n\\epsfxsize=17cm\n\\epsfbox{wass_table.eps}\n\\caption[]\n{\\label{wass_table}\nStable types of graphene edges, from\nRef.~\\cite{wassmann_edge_stab}.\nRow A of the figure shows the five most stable configurations of the\ngraphene edge with and without hydrogen attached to the unsaturated bonds.\nThere, hydrogen is represented by small black circles. Rows B and C show other\nstable armchair and zigzag terminations. The DFT calculations reported in\nRef.~\\cite{wassmann_edge_stab}\npredict that monohydrogenated zigzag denoted as z$_1$ in the figure and\nreczag denoted as z(57)$_{00}$ are stable only at extremely low hydrogen\nconcentrations. At standard atmospheric conditions, a$_{11}$, z$_{211}$,\nand a$_{22}$ are the most stable types of edges. Note that the pristine\nzigzag edge (studied in numerous papers) is {\\it not} listed as a stable\nconfiguration. More complicated types of the graphene terminations are\npresented in panel~D. In panel~E the representation of the benzenoid\naromatic carbon ring as a superposition of two degenerate Kekule\nconfigurations is shown.\nReprinted figure with permission from \nT. Wassmann, A. P. Seitsonen, A. M. Saitta, M. Lazzeri, and F. Mauri,\nPhys. Rev. Lett. {\\bf 101}, 096402 (2008).\nCopyright (2008) by the American Physical Society.\n}\n\\end{figure*}\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline\\hline\n\\multicolumn{5}{||c||}{\\textbf{Graphene edges}} \\cr\n\\hline\\hline\n          &  Zigzag &  Armchair &\\quad Klein \\quad \\quad &  Reczag \\cr\n\\hline\\hline\n\\quad Stability \\cite{koskinen_reczag2} \\quad \\quad\n         & Unstable & Stable    &        & Stable  \\cr\n          &  (to reczag)&       &        &         \\cr\n\\hline\nEdge states & Yes       & No    & Yes    &         \\cr\n\\hline\n\\quad Magnetism \\cite{wassmann_edge_stab} \\quad \\quad & Ferromagnetic &\n                                                    No   &        & No \\cr\n\\hline\nStress \\cite{edge_stress}\n         &  \\quad Compression \\quad \\quad& \\quad Compression \\quad \\quad &   & \\quad\nTension (weak)\\quad \\quad \\cr\n\\hline\\hline\n\\end{tabular}\n\\caption{ Different properties of graphene edges. In addition to the three\ntypes presented in\nFig.~\\ref{graphene_lattice}\n(zigzag, armchair, and Klein edges), a {\\it reconstructed zigzag} (reczag)\nedge\n\\cite{koskinen_reczag}\nis now included in this comparison.\n}\n\\label{compare_edges}\n\\end{table}\n\n\\subsection{Electrons near edges}\n\nThe simplest way to describe an electron near the edge is to\nresort to the Weyl-Dirac equation (\\ref{dirac}) with appropriate\nboundary conditions. One has to keep in mind that the realistic\nboundary condition depends on a variety of factors: the\norientation of the edge, deformation of the chemical bonds near\nthe edge, edge reconstruction, and possible chemical\nfunctionalization of the unsaturated bonds. Theoretical studies of\nthese conditions were performed in several papers\n\\cite{dirac_boundary_cond_nanotube,dirac_boundary_cond_graphene,\ncomplicated_edges,volkov_zagorodnev_bc,volkov_zagorodnev_bc2}.\n\nThe physics of electrons near the armchair edge is simple: the edge\nalways acts as a reflector of the incident electron current. The scattering\nis affected by details of the edge structure, such as C-C bond lengths near\nthe edge and non-carbon radicals attached to the edge. Some additional\ndetails are provided in\nAppendix~\\ref{appendix::armchair}.\n\nThe physics of Klein and zigzag edges, however, is quite different. These\nedges bind electrons. When the nearest-neighbor hopping Hamiltonian is used\nto describe graphene, the bound eigenstates (edge states) form a\ndispersionless band at the zero of energy (see\nAppendix~\\ref{appendix::zigzag}).\nThese edge states can be observed experimentally as a peak in the local\ndensity of states\n\\cite{niimi_ldos,kobayashi_ldos,ritter_lyding}.\nFor example,\nFig.~\\ref{zigzag_dos}\nshows scanning tunneling microscopy data from\nRef.~\\cite{kobayashi_ldos}.\nThere, the edge states are seen near the zigzag edge as a stripe of bright\nspots extending along the edge.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{zigzag_edge.eps}\n\\caption[]\n{\\label{zigzag_dos}\n(Color online) Scanning tunneling microscope image of different graphene\nterminations, from\nRef.~\\cite{kobayashi_ldos}.\n(a) The termination of a graphene sample is investigated using a scanning\ntunneling microscope. Fragments of both zigzag and armchair types are\nidentified. The edge states near the zigzag edge are clearly visible as\nstripes of bright spots stretching along the zigzag edge. No edge states are\npresent near the armchair termination. (b) A typical dependence of the\ndifferential conductance near the zigzag edge is plotted. The peak near\nthe zero voltage\n($V_{\\rm s} = 0$)\ncorresponds to the edge states.\nReprinted figure with permission from \nY. Kobayashi, K.-i. Fukui, T. Enoki, K. Kusakabe, and Y. Kaburagi,\nPhys. Rev. B {\\bf 71}, 193406 (2005).\nCopyright (2005) by the American Physical Society.\n}\n\\end{figure}\n\nThe dispersionless band of the bound states is unstable with respect to\ndifferent perturbations of $H$. For example, the inclusion of longer-range\nhopping makes the band disperse\n(Refs.~\\cite{el_prop_disordered_graphene,sasaki_nnn,sasaki_gague_nnn,\ndecomposition,volkov_zagorodnev_bc,volkov_zagorodnev_bc2}\nand Appendix~\\ref{appendix::zigzag}).\n\nThe most interesting way of lifting the degeneracy of edge states is by\nadding electron-electron interactions to $H$. It was predicted quite some\ntime ago\n\\cite{kusakabe_magnetizm_edge_state}\nthat magnetic correlations develop at a zigzag edge as a result of the\ninteraction. This effect was investigated in several papers\n\\cite{yazayev_magnetic_edge_states,\nkumazaki_magnetic_edge_states,\nsasaki_magnetic_edge_states,\nwunsch_zigzag_mag}\n\nFor example, a detailed DFT study was reported in\n\\cite{yazayev_magnetic_edge_states}.\nIt predicts that an isolated graphene zigzag edge is a ferromagnet with\nmagnetic moment $m$ of 0.3 of the Bohr magneton per unit cell of the zigzag\nedge.  This may be understood qualitatively as follows. The spin coupling\nbetween nearest-neighbor carbon atoms is antiferromagnetic. Thus, different\nsublattices have opposite magnetic momenta. In the bulk, this would lead to\ncancellation of the total moment. Near the edge, however, the electron\ndensity at the exposed row is higher that at rows located deeper into the\nbulk. Most importantly, all sites at the zigzag edge belong to the same\nsublattice; therefore, they have the same magnetic moment. This leads to a\nlocal imbalance of the total magnetic moment, which is seen as edge\nferromagnetism.\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c||}\n\\hline\\hline\n\\multicolumn{4}{||c||}{\\textbf{Zigzag edge magnetism}}\\cr\n\\hline\\hline\n$M$,        &  $\\xi(T)$         &  $\\xi(T)$        & \\quad Anisotropy\\quad \\quad \\cr\n\\quad per unit cell\\quad \\quad\n        &\\quad $T = 300$ K \\quad\\quad\n                &\\quad $T < 10$ K\\quad \\quad &           \\cr\n\\hline\\hline\n$0.3\\ \\mu_{\\rm B}$\n            & $\\sim 1$ nm          &  $\\sim 1$ $\\mu$m     & Ising, $10^{-4}$ \\cr\n\\hline\\hline\n\\end{tabular}\n\\caption{Summary of the magnetic properties of the graphene zigzag edge, as\nreported\nin Ref.~\\cite{yazayev_magnetic_edge_states}.\nWhen the electron-electron interaction is taken into consideration, the\nedge-state degeneracy is lifted through the magnetization of the\nelectrons near the edge. This creates a one-dimensional magnetic\nsystem. The magnetic momentum $M$ of such system is 0.3 of the Bohr's\nmagneton per zigzag edge unit cell. The correlation length $\\xi$ at room\ntemperature is rather short, suggesting that it would be difficult\nto utilize the pristine zigzag edge in a spintronic device\noperating at room temperature. However, below $T_x = 10$\\ K, a crossover to\nIsing-like magnetic correlations occurs, and the correlation length\nincreases exponentially upon approaching $T=0$. It was proposed\n\\cite{yazayev_magnetic_edge_states}\nthat $\\xi$ could be as large as a micrometer.\n}\n\\label{edge_magnetism}\n\\end{table}\n\nReference~\\cite{yazayev_magnetic_edge_states}\nreports many properties of the edge ferromagnetism\n(see also Table~\\ref{edge_magnetism}):\nspin-wave dispersion\n($E = \\kappa q^2$, where $\\kappa = 320 \\,{\\rm meV \\AA^2}$),\nstiffness\n($D = 2\\kappa\/m = 2100 \\,{\\rm meV \\AA^2}$),\nmagnetic anisotropy\n($\\sim 10^{-4}$),\nand crossover temperature between the Heisenberg and the Ising regimes\n($T_x \\sim 10$\\,K).\nThe spin correlation length at room temperature is estimated to be of the\norder of one nanometer. At temperatures below $T_x$ it increases\nexponentially as the temperature decreases. The effects of edge disorder on\nthe magnetic properties of the zigzag edge were also investigated.\n\nIn Ref.~\\cite{kumazaki_magnetic_edge_states}\nmagnetic properties of small fragments of zigzag edge were studied using\nthe Hubbard model. Such model is relevant for systems with rough edges,\nconsisting of alternating fragments of different terminations. It is\ndemonstrated that a very short, of the order of three lattice constants,\nzigzag sequence is sufficient to generate a local magnetic moment.\n\nSome other interacting effects have also been studied. For example, the\ninfluence of the long-range Coulomb interaction and doping on the edge\nmagnetism is discussed in\nRef.~\\cite{wunsch_zigzag_mag}\nusing the Hubbard model with Coulomb interactions.\nThe interaction of edge states with phonons was studied in\nRef.~\\cite{sasaki_phonons}.\n\n\\subsection{Graphene\/graphane interface}\n\\label{interface}\n\nGraphane\n\\cite{sofo_graphane,elias_graphane}\nis a hydrogenated sheet of graphene. Unlike graphene, graphane is a\nsemiconductor with a gap of the order of few eVs. The graphene\/graphane\ninterface, shown in\nFig.~\\ref{gg_interface},\ncan be viewed as a type of graphene edge: low-lying electron states in\ngraphene decay exponentially inside the gapped media of graphane.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{gg_interface.eps} \\caption[] {\\label{gg_interface}\nGraphene\/graphane interface, as studied in\nRef.~\\cite{openov_gr_interface}.\nLarger dark balls correspond to carbon atoms, while the smaller\nlight balls correspond to hydrogen. The lower part of the sample\nthat is shown is graphene, while the higher part is graphane. In\nbulk graphane every carbon atom has a hydrogen atom attached to\nit. }\n\\end{figure}\nAs demonstrated by molecular dynamics simulations\n\\cite{openov_gr_interface},\nthe interface remains almost atomically sharp even at sufficiently high\ntemperatures, which is an extremely attractive feature since it reduces\nscattering and simplifies the theoretical description.\n\nDepending on the orientation of the interface relative to the\ncrystallographic axis of graphene, one can distinguish a zigzag-type\ninterface (as in\nFig.~\\ref{gg_interface}),\nor an armchair-type (at the right angle to the zigzag). The zigzag\ninterface supports edge states whose electronic and magnetic properties\nwere investigated in\nRef.~\\cite{schmidt_gg_interface}.\n\n\nBesides hydrogenation, graphene may be subjected to fluorination\nin order to produce fluoridated graphene\n\\cite{cheng_flouridation_2010,nair_fluorographene,robinson_fluoridation,\nwithers_fluoridation,xiang_hydro_fluo}. Like graphene, the latter\nis a semiconductor with a gap of the order of a few eV. A similar\nconversion occurs upon functionalization of graphene by\nnitrophenyl \\cite{nitrophenyl}. The properties of the interface\nbetween pure graphene and the functionalized material must be\nsimilar to the properties of the graphene\/graphane interface.\n\n\n\\subsection{Fabrication of high-quality edges}\n\nMost of the theoretical work so far has assumed that the edges of the\nnanostructures are atomically perfect. Needless to say, this is not easy to\nrealize experimentally. However, recently, substantial progress in the area\nof high-quality edge fabrication has been achieved\n(e.g., \\cite{sharp_edges,current_edge_rect}).\n\nIn Ref.~\\cite{sharp_edges}\na chemical method of deriving narrow graphene stripes with sharp edges was\nreported. A graphene sample was placed in a solvent and subjected to\nsonification. Strips with sharp edges and widths varying from 50~nm to\nsub-10~nm were extracted from the solution. The strips produced were\nused to fabricate a field-effect transistor-like device.\n\nIn Ref.~\\cite{current_edge_rect}\nit was experimentally demonstrated that during Joule heating of the\ngraphene sample with disordered edges carbon atoms at the edge were\nvaporized, and sharp edges were stabilized. Model calculations shown that\nthe edge defects were healed through point defect annealing and edge\nreconstruction. This process was modelled in\nRef.~\\cite{engelund_reconstruction}.\nThese findings suggest that many theoretical predictions dependent on the\nedge quality could be tested experimentally.\n\n\\subsection{Armchair edge}\n\\label{appendix::armchair}\n\nThe physics of the armchair edge is simpler than that of the zigzag edge\nbecause the latter supports zero-energy localized states, while the former\ndoes not. The easiest way to describe an electron near an armchair edge is\nto use the Weyl-Dirac equation\n(\\ref{dirac})\nwith the appropriate boundary condition. The general problem of the\nboundary condition for the Weyl-Dirac equation is investigated in\nRefs.~\\cite{dirac_boundary_cond_nanotube,dirac_boundary_cond_graphene,\ncomplicated_edges,volkov_zagorodnev_bc,volkov_zagorodnev_bc2}.\nHere we use a simple explicit form of the boundary condition suitable for\narmchair edge\n\\cite{brey_fertig_dirac_eq_nb}.\nNamely, we demand that our spinor wave function \nEq.~(\\ref{spinor_decomp})\nvanishes at the edge, which we assume to be located at\n$y = 0$\n\\begin{eqnarray}\n\\Psi_1 ({\\bf R})|_{y = 0}\n=\n- \\Psi_2 ({\\bf R})|_{y = 0}.\n\\end{eqnarray}\nFor an infinite half-plane, the solution of the Weyl-Dirac equations with\nthis boundary condition is equal to\n\\begin{eqnarray}\n\\Psi_{1\\pm} ({\\bf R}) \n=\n\\frac{1}{\\sqrt{2}} \n\\left(\n\t\\matrix{\n\t\t1 \n\t\t\\cr\n\t\t\\mp(i k_x + k_y)\/{k}\n\t}\n\\right)\n\\exp(- i k_x x - i k_y y ),\n\\label{wd_solution1\n\\\\\n\\Psi_{2\\pm} ({\\bf R}) \n=\n-\\frac{1}{\\sqrt{2}} \n\\left(\n\t\\matrix{\n\t\t1 \n\t\t\\cr\n\t\t\\mp(i k_x + k_y)\/{k}\n\t}\n\\right)\n\\exp(- i k_x x + i k_y y ).\n\\label{wd_solution2\n\\end{eqnarray} \nThe eigenfunctions \n$\\Psi_{1,2+}$\n($\\Psi_{1,2-}$)\ncorrespond to positive (negative) eigenvalues. The total wave function is\nto be constructed according to\nEq.~(\\ref{spinor_decomp}).\n\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{aatoms.eps}\n\\caption[]\n{\\label{aux}\nIntroduction of auxiliary atoms (hatched circles) at an armchair edge.\n}\n\\end{figure}\nHowever, it is not necessary to use the approximate description in terms of\nthe Weyl-Dirac equation. The tight-binding model may be solved near the\narmchair edge as well. To construct such a solution, note first that the\natoms at the very edge are special: they have only two nearest neighbors,\nwhile the atoms in the bulk have three. This means that\nEqs.~(\\ref{sch_a},\\ref{sch_b}) \nmust be modified to account for this fact. It is more convenient, however,\nto introduce auxiliary rows of carbon atoms (Fig.~\\ref{aux}) and demand\nthat the wave function vanishes on these additional atoms. Since there is\nno wave function density on the auxiliary atoms, they do not contribute to\nthe Schr\\\"odinger equation for the physical atoms at the edge. This\nconstruction is used in several papers (e.g.,\nRef.~\\cite{gunlycke_nanoribbon_gap,our_nanoribbon_paper_2009}).\nThe solution of Eqs.~(\\ref{sch_a}) and (\\ref{sch_b}) is\n\\begin{eqnarray}\n\\Psi_{\\bf R} \n=\n\\Psi_{{\\bf k} \\pm} \\exp( -i k_x x ) \\sin [k_y(y+\\sqrt{3}a_0\/2)].\n\\label{tb_solution\n\\end{eqnarray}\nThis wave function vanishes at \n$y = - \\sqrt{3}a_0\/2$,\nwhich is where the auxiliary atoms are located. At the physical edge of\nthe nanoribbon \n($y = 0$),\nhowever, the electron density remains non-zero. Note that momentum\ncomponents\n$k_{x,y}$\nin \nEq.~(\\ref{tb_solution})\nis measured from the center of the Brillouin zone. However, in\nEq.~(\\ref{wd_solution1})\nand\nEq.~(\\ref{wd_solution2})\nthey are measured from the Dirac cones. When this subtlety is accounted it\nis easy to demonstrate that both solutions have the identical\ndependence on\n${\\bf R}$.\nThey differ only in the value of the spinor part \n$\\Psi_{\\bf k}$.\nThis is because \nEq.~(\\ref{wd_solution1})\nand\nEq.~(\\ref{wd_solution2})\nare only approximations which are accurate at small energy only.\n\nThe discussion presented above assumes that the properties of the\ncarbon-carbon bonds near the edge remain the same. In a real sample this\nassumption is only an approximation. A variety of perturbations may be\npresent near the edge. For example, \nRef.~\\cite{density-func_zig_armch}\ninvestigates the edge structure with the help of density-functional\nmethods. It is found that the carbon-carbon bonds at armchair edges are\nshorter than in the bulk (1.26\\,\\AA\\ versus 1.4\\,\\AA). This means that\nthe effective electron hopping at the edge \n$t_{\\rm e}$\ndiffers from its value $t$ in the bulk.\n\nAlso, non-carbon radicals may be attached to the unpaired chemical bond of\ncarbon atoms at the armchair edge. This means that, depending on the\ndescription, either the Hamiltonian near the edge, or the boundary\nconditions have to be modified. Fortunately, these alterations are small,\nand may be treated as weak corrections.\n\n\\subsection{Zigzag edge}\n\\label{appendix::zigzag}\n\nThe physics of the zigzag edge is richer than the physics of the armchair\nedge due to the zero-energy states localized at the edge. When graphene is\ndescribed by the Hamiltonian (\\ref{H}), which only contains\nnearest-neighbor hopping, these states are dispersionless and\nmacroscopically degenerate. \n\nThe simplest way to detect the presence of edge states is to use the\nWeyl-Dirac equation with the boundary condition appropriate for the zigzag\nedge\n\\cite{dirac_boundary_cond_nanotube,dirac_boundary_cond_graphene,\ncomplicated_edges}.\nFor the purpose of demonstrating the presence of edge states the simplest\nversion of the boundary condition is used here\n\\cite{brey_fertig_dirac_eq_nb}.\nThe zigzag edge, unlike the armchair edge, consists of atoms belonging to\nthe same sublattice. This is clearly seen in \nFig.~\\ref{graphene_lattice}:\nall atoms at the left edge belong to \n${\\cal A}$\nsublattice (red). The right zigzag edge has all its atoms on the\n${\\cal B}$\nsublattice. Extending the left edge by a column of the auxiliary atoms, we\ndemand that the wave function vanishes on them (assume that the auxiliary\natoms are located at $x=0$):\n\\begin{eqnarray}\n\\psi^{\\cal B}|_{x=0} = 0.\n\\label{bc_zigzag\n\\end{eqnarray}\nThen the following spinor functions are the solutions of the Weyl-Dirac\nequation:\nwhen $k_y > 0$\n\\begin{eqnarray}\n\\Psi_1 ({\\bf R})\n=\n\\left(\n\t\\matrix{\n\t\t\t1 \\cr\n\t\t\t0\n\t\t}\n\\right)\n\\exp(-k_y x + i k_y y),\n\\quad \n\\Psi_2 = 0;\n\\label{es_dirac1\n\\end{eqnarray}\nand when\n$k_y < 0$\n\\begin{eqnarray}  \n\\Psi_1 = 0,\n\\quad\n\\Psi_2 ({\\bf R})\n=\n\\left(\n\t\\matrix{\n\t\t\t1 \\cr\n\t\t\t0\n\t\t}\n\\right)\n\\exp(-|k_y| x + i k_y y).\n\\label{es_dirac2\n\\end{eqnarray}\nBoth solutions decay for large values of $x$ and correspond to the zero\neigenvalue. The difference between them is that\nEq.~(\\ref{es_dirac1})\ndescribes the solution near \n${\\bf K}_1$,\nwhile \nEq.~(\\ref{es_dirac2})\nnear \n${\\bf K}_2$.\nThe wave function \nEq.~(\\ref{spinor_decomp})\nconstructed from\nEqs.~(\\ref{es_dirac1}) and (\\ref{es_dirac2})\nis equal to\n\\begin{eqnarray}\n\\Psi ({\\bf R})\n=\n\\left(\n\t\\matrix{\n\t\t\t1\\cr\n\t\t\t0\n\t\t}\n\\right)\n\\exp ( - i {\\bf K}_1 \\cdot {\\bf R} - k_y x + i k_y y)\n+\n\\left(\n\t\\matrix{\n\t\t\t1\\cr\n\t\t\t0\n\t\t}\n\\right)\n\\exp ( - i {\\bf K}_2 \\cdot {\\bf R} - k'_y x - i k'_y y),\n\\label{es_dirac_full\n\\end{eqnarray}\nwhere both\n$k_y$ and $k'_y$\nare positive and small compared to \n$1\/a_0$. \nThe quantity $k_y$ sets a parameter with dimension of length\n\\begin{eqnarray}\n\\lambda_{\\rm edge} = \\frac{1}{|k_y|}\n\\end{eqnarray}\nwhich characterizes how deeply the edge states extend into the bulk of\ngraphene.\n\nSince the Weyl-Dirac equation is only an approximation of the tight-binding\nHamiltonian (\\ref{H}), we cannot reliably use \nEq.~(\\ref{es_dirac_full}) \nfor large values of \n$k_y$ and $k'_y$.\nHowever, the tight-binding problem may be solved exactly\n\\cite{nakada-fujita_nribb_edge_st}\nto discover that there is a degenerate manifold of states labeled by the\nmomentum $k_y$, which stretches from \n${\\bf K}_1$\nto\n${\\bf K}_2$.\n\n\\subsubsection{Effect of the longer-range hopping}\n\\label{tb_es}\n\nThe degeneracy of edge states is purely accidental property of Hamiltonian\n(\\ref{H}).\nIt disappears when other terms are added to $H$. For example, in\nRefs.~\\cite{el_prop_disordered_graphene,sasaki_nnn,sasaki_gague_nnn,\ndecomposition}\nthe effect of next-to-nearest-neighbor hopping on edge states is studied both\nanalytically and numerically. In\nRef.~\\cite{decomposition}\nthe authors made three keen observations: $(i)$ the next-to-nearest-neighbor\nhopping effectively induces a shift in the local potential; $(ii)$ the\nshift depends on site's position, namely, near the edge it is not the same\nas in the bulk; and $(iii)$ this spacial variation of the potential induces\nfinite dispersion for an otherwise dispersionless edge states. To prove\nthat $(iii)$ holds true, consider the following argument. The states with\nlarge \n$\\lambda_{\\rm edge}$\nare insensitive to the potential variation (they ``feel\" the potential\naveraged over large \n$\\lambda_{\\rm edge}$),\nhowever, those with small \n$\\lambda_{\\rm edge}$\nare affected strongly; since \n$\\lambda_{\\rm edge}$\ndepends on \n$|k_y|$,\nthe edge states acquire the dispersion.\n\nThis discussion suggests that, if we were to describe this phenomena with\nthe help of Weyl-Dirac equation, the boundary conditions,\nEq.~(\\ref{bc_zigzag}),\nmust be modified to account for the effect of the potential modulation\nnear the boundary. Indeed, it is demonstrated in\nRef.~\\cite{volkov_zagorodnev_bc2}\nthat one can generalize\nEq.~(\\ref{bc_zigzag})\nand reproduce the findings of\nRef.~\\cite{decomposition},\nat least near the apexes of the Dirac cones.\n\n\\section{Graphene nanoribbons}\n\\label{nanoribbon}\n\nNanoribbons, which are strips of graphene, are among the most studied\nmesoscopic graphene structures. There are several reasons for this. First,\nthey demonstrate unusual physical properties, for example, edge states,\nwhich might be used in future spintronics applications. Second, nanoribbons\nare easy to produce and demonstrate excellent transport properties.\n\nThird, they have an energy gap in their single-electron spectrum. This gap\nis a consequence of the electron confinement, and it is inversely\nproportional to the width of a nanoribbon\n\\cite{han_experiment_gap,chen_experiment_gap}.\nThis suggests that, at least in principle, a nanoribbon with a desired\nvalue of the gap may be fabricated. Finite gap and high mobility are both\nvery useful for the design of field-effect transistors (FET): they allow\nfor large on\/off ratios, small losses, and high operating frequencies. For\nexample, a nanoribbon-based FET realized in\nRef.~\\cite{wang_room_t_fet}\ndemonstrated an on\/off ratio of about 10$^6$ at room temperature. Other\nnanoribbon-FET devices were described\nin Ref.~\\cite{Liao2010,fet_nribbon}.\n(Carbon nanotubes also have a gapped spectrum. However, their fabrication\nprocess is much more involved).\n\nNanoribbons are usually classified by their type of edge; for instance,\nthere are zigzag and armchair nanoribbons. Nanoribbons may also have\ndisordered\n\\cite{cresti_disord_review}\nor\nmore complicated regular types of edges\n\\cite{barone_semicond_nanoribbon}.\n\n\n\\subsection{Zigzag nanoribbons}\n\\label{zigzag_nanoribbon}\n\nMany interesting properties of zigzag nanoribbons are related to the\npresence of edge states in the nanoribbon electron spectrum. These states\nmay be derived, together with other low-lying states, with the help of\nEq.~(\\ref{dirac})\nsupplemented by a boundary condition suitable for the zigzag edge\n\\cite{brey_fertig_dirac_eq_nb}.\nThe resultant spectrum is shown in\nFig.~\\ref{nanoribbon_spectr}a.\nTwo almost dispersionless branches connecting the Dirac cones, $K$ and\n$K'$, correspond to the edge states. They are analogous to the edge states\ndiscussed in\nSec.~\\ref{edge}.\n\nSimilar conclusions about edge states may be reached using first-principle\ncalculations. These reveal that the zigzag nanoribbon is a\nsemiconductor with a width-dependent gap\n\\cite{son_half_metal,pisani_monohydrogen,yang_armchair_gw}.\nThe lowest energy states are edge states.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{nribb_spectr.eps}\n\\caption[]\n{\\label{nanoribbon_spectr}\n(Color online) Single-electron energy spectrum of a nanoribbon calculated\nwith the help of the Weyl-Dirac\nequation~(\\ref{dirac}),\nfrom\nRef.~\\cite{brey_fertig_dirac_eq_nb}.\nDue to the finite width of a nanoribbon, the transverse momentum is\nquantized. Therefore, the nanoribbon's spectrum consists of a number of\nbranches corresponding to different values of the quantized momentum. Since\nthe graphene lattice is anisotropic, the spectrum of a zigzag nanoribbon\n[panel (a)] differs in several respects from the spectrum of an armchair\nnanoribbon [panels (b) and (c)]. For the zigzag nanoribbon the remnants of\ntwo Dirac cones, $K$ and $K'$, are visible. The almost flat branch\nconnecting $K$ and $K'$ corresponds to the edge states. It acquires a very\nweak dispersion due to interference of the edge states (exponentially)\nlocalized at the opposite edges of the nanoribbon. For the armchair\nnanoribbon both $K$ and $K'$ `coalesce' together. According to the\ncalculations of\nRef.~\\cite{brey_fertig_dirac_eq_nb},\nthe armchair nanoribbon may be either a semiconductor with a small gap\n[panel (b)], or a gapless metal [panel (c)]. The size of the gap depends on\nthe nanoribbon's width. However, more elaborate treatments accounting for\nthe electron-electron interaction\n\\cite{sandler_coulomb_gap,sols_coulomb_blockade},\nelectron-lattice interaction\n\\cite{lattice_distortion,son_gap,our_nanoribbon_paper_2009},\nor longer-range hopping\n\\cite{white_gap_longer_range,gunlycke_nanoribbon_gap}\nproved that the gap is always non-zero.\nReprinted figure with permission from \nL.~Brey and H.~Fertig, Phys. Rev. B {\\bf 73}, 235411 (2006).\nCopyright (2006) by the American Physical Society.\n}\n\\end{figure}\n\n\\subsubsection{Edge magnetism}\n\nEdge states are responsible for magnetism in zigzag nanoribbons. Edge\nmagnetism is an interesting feature with potential spintronic applications;\nsince the edge-state branch is both magnetized and able to carry current,\nit can be used to couple spin magnetization and current. This property may\nbe used to control the magnetization with current or vice versa.\n\nThe magnetism of a nanoribbon with pristine zigzag edges was\ninvestigated theoretically in 1996\n\\cite{fujita_nanoribbon_hubbard}.\nIt is quite similar to the magnetism of an isolated zigzag edge: each edge\nof the nanoribbon has a finite magnetization, which is induced due to the\ninstability of a nearly-flat edge-state band. There is a non-zero coupling\nbetween the magnetizations of the two edges.\nIn Ref.~\\cite{fujita_nanoribbon_hubbard}\nit was shown that this coupling is antiferromagnetic, i.e., the\nmagnetization vectors at the opposite edges are antiparallel. Such result\nis easy to understand. Consider the following two statements: $(i)$\nrepulsive interactions between electrons on a half-filled bipartite lattice\ninduce an antiferromagnetic correlation between sublattices; $(ii)$ in case\nof the zigzag nanoribbon, one of its edges always terminates in atoms of\n${\\cal A}$\nsublattice, the opposite edge terminates in atoms of\n${\\cal B}$.\nAs a consequence of $(i)$ and $(ii)$, the local antiferromagnetic tendency\nis translated into weak inter-edge antiferromagnetic interactions.\n\nSince the pristine zigzag edge is likely to be chemically unstable\n\\cite{wassmann_edge_stab,wassmann_phys_stat_sol},\nit is therefore important to study nanoribbons with non-carbon atoms or\nfunctional groups attached to the edges. The case of a zigzag nanoribbon\nwith monohydrogenated edges was discussed in\nRef.~\\cite{pisani_monohydrogen}.\nSuch nanoribbons support edge states and have an edge ferromagnetic moment\nwith antiferromagnetic coupling between the edges. This is consistent with\nthe results of\nRef.~\\cite{wassmann_edge_stab},\nwhere an extensive list of various edge types and their properties was\npresented. However, not all versions of functionalized or reconstructed\nzigzag edge support magnetism.\n\nA chemical way to produce a nanoribbon with finite magnetic moment was\nproposed in\nRef.~\\cite{kusakabe_ferromag}:\nsince the opposite edges of the zigzag nanoribbon have opposite magnetic\nmomenta, a disparity (e.g., a non-equivalent chemical functionalization)\nbetween the two edges may create a nanoribbon with non-zero magnetization.\nLocal-spin-density calculations reported\nin Ref.~\\cite{kusakabe_ferromag}\nproved that fact for the nanoribbon whose one edge is monohydrogenated,\nwhile the other is the dihydrogenated (see\nFig.~\\ref{kusakabe_ferro}).\nThis result can be easily understood qualitatively. The monohydrogenated\nedge is ferromagnetic, while the dihydrogenated is non-magnetic\n\\cite{wassmann_edge_stab}.\nThus, the whole system is ferromagnetic.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{kusakabe_ferro.eps}\n\\caption[]\n{\\label{kusakabe_ferro}\nZigzag nanoribbon with disparity between edges, from\nRef.~\\cite{kusakabe_ferromag}.\nThe nanoribbon edges are parallel to $y$-axis. The left edge is\nmonohydrogenated, while the right edge is dihydrogenated. Here, filled\ncircles are carbon atoms, empty circles are hydrogen atoms. Since the\nopposite edges of a pristine zigzag nanoribbon have opposite magnetic\nmomenta, a disparity between the two edges may induce a non-zero\nmagnetization of the nanoribbon. Indeed, a local-spin-density approximation\nstudy~\\cite{kusakabe_ferromag}\nrevealed that the nanoribbon in the figure possesses a finite magnetic\nmoment. This result can be easily understood qualitatively. The\nmonohydrogenated edge is ferromagnetic, while the dihydrogenated is\nnon-magnetic. Thus, the whole system is ferromagnetic.\nReprinted figure with permission from \nK. Kusakabe and M. Maruyama, Phys. Rev. B {\\bf 67}, 092406 (2003).\nCopyright (2003) by the American Physical Society.\n}\n\\end{figure}\n\nReference~\\cite{cervantes_func_mag}\npresented a very detailed DFT study of the effect the edge\nfunctionalization exerts on the zigzag nanoribbon's magnetism. The main\nfocus there was the monohydrogenated zigzag nanoribbon, where some of the\nhydrogen atoms were replaced by other radicals. When the edges have\nnon-identical chemical structure (i.e., one edge is purely hydrogenated,\nthe other has some of its hydrogens substituted), it was determined that a\nfinite magnetization may be generated. The effect is particularly strong\nfor the oxygen substitution. This result is consistent with\nRef.~\\cite{kusakabe_ferromag}.\nIt was also reported that such nanoribbon has different band gaps for\ndifferent spin orientations. This state can be described as a\nspin-selective semiconductor.\n\nIn addition, it was shown in\nRef.~\\cite{cervantes_func_mag}\nthat not only the magnetic properties but the band gap of a zigzag\nnanoribbon is sensitive to the chemical functionalization of the edges as\nwell. For example, doping with oxygen may close the gap, provided that its\nconcentration is sufficiently high. A variety of other effects dependent on\nthe edge chemistry was also discussed.\n\n\n\\subsubsection{Half-metallicity}\n\nA half-metal is a conductor whose charge carriers are fully spin-polarized.\nThis is a very desirable property with potential applications to\nspintronics because it can be used to create a fully spin-polarized\ncurrent.\nReference~\\cite{son_half_metal}\nsuggested that the application of a transverse electric field to a zigzag\nnanoribbon closes the gap for one spin orientation. The gap for another\nspin orientation is increased even more (see Fig.~\\ref{half_metal}).\nA similar conclusion was reached in\nRef.~\\cite{Hod2008}.\nNote that this idea is also based on inducing a disparity between the two\nedges: this time the transverse field is the agent producing the\ndisparity.\n\n\\begin{figure}[btp] \\centering \\leavevmode \\epsfxsize=12.5cm\n\\epsfbox{halfm.eps} \\caption[] {\\label{half_metal} (Color online)\nZigzag nanoribbon without (a) and with (b) a transverse electric field,\nfrom~Ref.\\cite{son_half_metal}.\nSymbols $\\alpha$ ($\\beta$), in red (blue), represent the\nspin-up (spin-down) orientations of the edge band electrons. In the lower\npart of panel (a) both orientations are present: spin-up is on the left\nedge (L) of the nanoribbon, and spin-down is on the right edge (R). The\nletter `M' stands for `middle' of the nanoribbon. In the upper part of (a)\nthe energies of different edge states are plotted. On the left edge, the\nspin-up states are filled ($E<0$), and the spin-down are empty. On the\nright edge, the situation is reversed. Panel (b) shows what happens when a\ntransverse electric field is applied. The electrostatic potential pushes\ndown the states on the left edge, and pushes up the states on the right\nedge. As a result, the density of states at the Fermi energy is zero\n(finite) for spin-up (spin-down) electrons. The system becomes half-metal:\nit is a conductor (metal), since there is a finite density of\ncurrent-carrying states at the Fermi energy, yet, these states correspond\nonly to one spin polarization (spin-down).\nReprinted by permission from Macmillan Publishers Ltd: \n\\href{http:\/\/www.nature.com\/nature\/index.html}{Nature},\nY.-W. Son, M. L. Cohen, and S. G. Louie, Nature {\\bf 444}, 347 (2006),\ncopyright 2006.\n}\n\\end{figure}\n\nThe proposal\n\\cite{son_half_metal}\nsummarized in the previous paragraph was disputed in\nRef.~\\cite{rudberg_half_semicond},\nwhere it was argued that, when the transverse field is applied, two spin\npolarizations have different gap values. However, both gaps are finite for\nany value of the electric field, producing a spin-selective semiconductor,\nand not half-metal.\nReference~\\cite{rudberg_half_semicond}\nattributed the discrepancy to the artifacts of the computational technique\nof\nRef.~\\cite{son_half_metal}.\n\nA suitable functionalization can enhance the half-metallic features of\nthe zigzag nanoribbon\n\\cite{hod_half_metal_func}.\nAlso,\nRef.~\\cite{dutta_half_metal_chemical_mod} explored a wider range\nof chemical modifications of zigzag nanoribbons in search for\nrobust half-metallicity. The results in\nRef.~\\cite{hod_half_metal_func,dutta_half_metal_chemical_mod}\nsuggested that the chemical modifications may by a powerful tool for the\ncontrol of the zigzag nanoribbon's half-metallic properties.\n\n\\subsection{Armchair nanoribbons}\n\\label{armchair nanoribbon}\n\nThe electron properties of the armchair nanoribbon are simpler than those of the\nzigzag nanoribbon. Both tight-binding model and Weyl-Dirac equation\ncalculations show that graphene armchair nanoribbons with pristine edges\nmay be either in a semiconducting (finite gap,\nFig.~\\ref{nanoribbon_spectr}b),\nor metallic (zero gap,\nFig.~\\ref{nanoribbon_spectr}c),\nstate with the gap oscillating as a function of the nanoribbon's width. A\nmore accurate numerical study\n\\cite{lattice_distortion}\nfrom 1997, which allowed for deformation of the carbon-carbon bonds\ndangling at the edges, proved that the metallic state is unstable: the\ndangling bonds deform, inducing a finite gap in the electron spectrum.\nThus, according to\nRef.~\\cite{lattice_distortion},\nthe armchair nanoribbon is always a semiconductor (see\nFig.~\\ref{armchair_gap}).\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{armchair_gap.eps}\n\\caption[]\n{\\label{armchair_gap}\nArmchair nanoribbon's gap as a function of the nanoribbon's width, from\nRef.~\\cite{lattice_distortion}.\nThe tight-binding calculation (dashed line) predicts that the gap of the\narmchair nanoribbon vanishes periodically as a function of the nanoribbon's\nwidth. However, the numerical calculations within a more elaborate model,\nwhich allows for deformation of the carbon-carbon bonds, prove that the gap\nis non-zero (albeit very close to it) for any width (solid line), although,\nthe dependence on the width remains oscillatory. An analytical demonstration\nof the instability of the zero-gap state is given in\nRef.~\\cite{our_nanoribbon_paper_2009}.\nThis instability occurs because the increase of the elastic energy due to\nthe lattice deformation is smaller than the decrease of the electron\nkinetic energy due to the gap opening.\nFigure is reprinted from:\nM. Fujita, M. Igami, and K. Nakada, J. Phys. Soc. Jpn. {\\bf 66}, 1864 (1997).\n}\n\\end{figure}\n\nThe above line of reasoning was generalized in\nRef.~\\cite{our_nanoribbon_paper_2009}, where it was shown\nanalytically that the metallic state of an armchair nanoribbon is\ngenerically unstable: the edge bond instability is only one\npossibility. It was also discussed\n\\cite{our_nanoribbon_paper_2009}\nhow the electron gap of a {\\it finite-length} armchair nanoribbon can be\neffectively closed with the help of chemical modifications of the\nnanoribbon's edge.\n\nA gap may also be generated by electron-electron interactions\n\\cite{sandler_coulomb_gap},\nor longer-range hopping\n\\cite{white_gap_longer_range,gunlycke_nanoribbon_gap}.\nIn several papers, the electronic gap was determined with the help\nof first-principles techniques.\nRefs.~\\cite{barone_semicond_nanoribbon,son_gap} reported DFT\ncalculations of the gap for the armchair nanoribbon with different\nwidths, with both pristine and monohydrogenated armchair edges. The\nresults of Ref.~\\cite{barone_semicond_nanoribbon} for the gap are\nsummarized in Table~\\ref{armchair_edge}.\nReferences~\\cite{han_experiment_gap,chen_experiment_gap}\nreported the experimental measurement of the gap. The experimental value\nfor the gap was found to be consistent with the results of DFT\ncalculations. However,\nRef.~\\cite{yang_armchair_gw}\nclaimed that DFT underestimates the gap, and the use of the so-called\n{\\it GW} approximation \\cite{mahan} is more appropriate. {\\it GW} values of\nthe gap are significantly higher than DFT values.\n\nThe effect of the edge functionalization on the spectral gap was studied in\nRef.~\\cite{cervantes_func_mag}.\nThe gap was found to be robust against functionalization. This is different\nfrom the case of zigzag nanoribbons whose gap is very sensitive to the\nchemical structure of the edges (see\nsubsection~\\ref{zigzag_nanoribbon}).\n\nThus, it is expected on the basis of theoretical studies that an armchair\nnanoribbon is a semiconductor with a width-dependent gap, whose value is\nrather insensitive to the edge chemical structure.\n\n\\begin{table}\n\\begin{tabular}{||c|c|c|c||}\n\\hline\\hline \\multicolumn{4}{||c||}\n{\\textbf{Armchair nanoribbon energy gap}} \\cr\n\\hline\\hline \n\\quad Energy gap [eV]   \\quad\\quad   \n\t&  \\quad Bulk semiconductors with similar values of the gap \\quad \\quad\n\t\t& \\quad Width [nm]  \\quad\\quad  \n\t\t\t& \\quad Width [$\\sqrt{3}a_0$] \\quad \\quad\\cr\n\\hline\\hline\n0.7         & Ge, InN &  2--3          &  8--12               \\cr\n\\hline\nfrom 1.1 to 1.4     &Si, InP, GaAs &  1--2          &  4--8                \\cr\n\\hline\\hline\n\\end{tabular}\n\\caption{Gap values for the armchair nanoribbons of different widths found\nusing the density functional theory,\nfrom Ref.~\\cite{barone_semicond_nanoribbon}.\nThe quantity\n$\\sqrt{3} a_0$\nrepresents the lattice constant. The gap of the armchair nanoribbon\noscillates with its width; however, the gap is never zero. The details of\nthe oscillations depend on the edge functionalization (for example, in\nRef.~\\cite{barone_semicond_nanoribbon},\nboth pristine and monohydrogenated armchair nanoribbons are discussed). The\nlargest values of the gap are quite insensitive to functionalization. In\norder to have a gap value similar to known bulk semiconductors, a very\nnarrow armchair nanoribbon must be used. These results were reconsidered\nin Ref.~\\cite{yang_armchair_gw},\nwhere it was claimed that density functional theory underestimates the gap,\nand that the use of the so-called {\\it GW} approximation\n\\cite{mahan}\nis more appropriate. Within the {\\it GW} framework, the values of the gap\nsubstantially increase.}\n\\label{armchair_edge}\n\\end{table}\n\n\\subsection{Nanoroads}\n\\label{nanoroads}\n\nSo far we have assumed that graphene nanoribbons are formed by cutting a\npiece of graphene into a narrow strip. Another way to define a nanoribbon\nwas proposed in\nRef.~\\cite{singh_nanoroad}:\nto sculpture a graphene nanoribbon by removing hydrogen atoms along a\nnarrow strip inside a wider graphane sample, as shown in\nFig.~\\ref{nanoroad}.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{nanoroad.eps}\n\\caption[]\n{\\label{nanoroad}\n(Color online)\nNanoroads (i.e., graphene nanoribbons bounded by two graphene\/graphane\ninterfaces), from\nRef.~\\cite{singh_nanoroad}.\nSmall dark balls are carbon atoms, while the large blue balls are hydrogen\natoms. The armchair nanoroad is on panel~(a), while the zigzag nanoroad is\non panel~(b). The quantities\n$N_a = 1, \\ldots, N$\nand\n$N_z = 1, \\ldots, N$\ncharacterize the width of the nanoroads. For the nanoroad on panel~(a)\n$N=13$,\nwhile\n$N=6$\nfor the nanoroad on panel~(b).\nReprinted with permission from:\nK. Singh and B. I. Yakobson, Nano Lett. {\\bf 9}, 1540 (2009).\nCopyright 2009 American Chemical Society.\n}\n\\end{figure}\nIn such a case, a nanoribbon, called nanoroad in\nRef.~\\cite{singh_nanoroad},\nis bound by two graphene\/graphane interfaces, which are discussed in\nsubsection~\\ref{interface}.\n\nAs it was established in\nRef.~\\cite{openov_gr_interface},\nthe graphene\/graphane interface remains almost atomically sharp even at\nhigh temperatures. This makes nanoroads a promising candidate to observe\nballistic transport.\n\nThe magnetic and electronic properties of graphene nanoroads, including the\neffects of spin-orbit coupling, were discussed in\nRefs.~\\cite{singh_nanoroad,tozzini_nanoroad,Xiang2009,schmidt_nanoroad}.\nIn\nRef.~\\cite{singh_nanoroad}\nthe electronic structure of nanoroads was studied using DFT. It was found\nthat the armchair-type nanoroads are semiconducting. As for zigzag\nnanoribbons, when they are wide enough they demonstrate edge magnetism.\nZigzag nanoribbons are semiconducting in the antiferromagnetic state and\nmetallic in the ferromagnetic state.\n\nSeveral other works also used DFT to analyze nanoroad properties.  In\nRef.~\\cite{tozzini_nanoroad},\nthe zigzag nanoroad stability and electronic structure were investigated,\nand it was established that even extremely narrow zigzag nanoroads are\nstable. Narrow nanoroads are always semiconducting due to Peierls\ninstability,\nwhich opens a gap. A similar mechanism is responsible for the gap in\npolyacetylene.\nReference~\\cite{Xiang2009}\nstudied the adsorption of hydrogen on a graphene nanoribbon, also\nformulating the rules governing such adsorption. It was proposed to use\nsuch process in order to create narrow nanoroads.\n\n\nReference~\\cite{schmidt_nanoroad} showed that, due to enhanced\nspin-orbit coupling at the interface, a nanoroad might be used to\nconvert spin polarization into valley polarization and vice versa.\nSuch a device can operate at temperatures of about 1\\,K.\n\n\nThe theoretical research summarized in this subsection indicates that\nnanoroads may be an attractive alternative to usual nanoribbons, able to\nsustain ballistic propagation of electrons, and also exhibit unusual spin\nfeatures.\n\n\\subsection{Transport properties of nanoribbons}\n\\label{nribb_transport}\n\nFor applications, such as FET\n\\cite{Liao2010,wang_room_t_fet},\nthe transport properties of graphene nanoribbons must be investigated.\nA study of the conductance through a pristine nanoribbon within the\nframework of the Landauer formalism was presented in\nRef.~\\cite{peres_landauer}\n(see also the review\n\\cite{peres_review}).\nIn such a case, the nanoribbon's conductance is quantized: it changes in\ndiscrete steps when the gate voltage is varied. For zigzag nanoribbons it\nwas found that, when the gate potential is tuned to the charge neutrality\npoint (i.e., the Fermi energy is at the apex of the Dirac cone), the\nconductance is finite due to the edge states. These are the only\ncurrent-carrying modes under such conditions.\nReference~\\cite{peres_landauer}\nstudied the transport through metallic (zero-gap) armchair nanoribbons.\nSince it is understood now that, strictly speaking, all armchair\nnanoribbons are semiconducting (see\nsubsection~\\ref{armchair nanoribbon}),\nthe results obtained for such objects are valid as long as the gap may be\nneglected, e.g., when the temperature exceeds the gap. Another study of\nelectron transport through a disorder-free short-and-wide nanoribbon was\npresented in\nRef.~\\cite{tworzydlo_landauer}.\nIts findings were compared well with the experiments in\nRef.~\\cite{miao_nribb_transp_exp}.\nIt was concluded in\nRef.~\\cite{miao_nribb_transp_exp}\nthat the electron propagation in graphene is ballistic up to lengths of the\norder of 1\\,$\\mu$m. \n\nHowever, in a typical experimental situation ballistic propagation can be\nspoiled both by edge disorder\n\\cite{areshkin_transport,evaldsson_edge_disorder_nribb,gunlycke_edge_disorder,\nmucciolo_transport}\nand bulk disorder\n\\cite{areshkin_transport,mucciolo_transport}.\nFor bulk disorder, it was found that electron transport is rather\ninsensitive to long-range disorder. Yet, when the disorder becomes\nshort-range, it leads to Anderson localization and to destruction of the\nballistic propagation\n\\cite{areshkin_transport,mucciolo_transport}.\n\nIf a nanoribbon is sufficiently narrow, the edge disorder is\nimportant\n\\cite{areshkin_transport,evaldsson_edge_disorder_nribb,gunlycke_edge_disorder,\nmucciolo_transport}. The armchair nanoribbons are more sensitive\nto edge disorder than the zigzag nanoribbons\n\\cite{areshkin_transport,mucciolo_transport}.\nReference~\\cite{evaldsson_edge_disorder_nribb} reported that when\nedge disorder is present, the difference between transport\nproperties of the armchair and zigzag nanoribbons disappears.\nThese results suggest that to experimentally produce a ballistic\nnanoribbon one has to overcome very stringent limitations on the\nedge purity \\cite{mucciolo_transport}. However, for a\nsemiconducting armchair nanoribbon of finite length one can\noptimize the width so that the localization effects are, to some\nextent, masked \\cite{gunlycke_edge_disorder}.\n\n\nDoping \\cite{biel_transport} and edge functionalization\n\\cite{gunlycke_edge_funct,our_nanoribbon_paper_2009} affect the\ntransport as well. For example, if a finite-length armchair\nnanoribbon has a gap induced by the edge-bond deformations (see\nsubsection~\\ref{armchair nanoribbon}), then a suitably chosen\nchemical disorder at the edges may actually increase the\nconductivity by effectively closing the gap\n\\cite{our_nanoribbon_paper_2009}. Namely, if the edges are\nfunctionalized with two different kinds of radicals randomly\ndistributed along the length of the ribbon, then the term of the\nHamiltonian responsible for the opening of the gap becomes\ndisordered, and the gap closes when this term vanishes on average.\nClearly, one has to counteract the Anderson localization in such a\nsituation. Fortunately, this phenomenon is not important for a\nnanoribbon of sufficiently short length. Transport through this\nnanoribbon is effectively metallic\n\\cite{our_nanoribbon_paper_2009}.\n\n\nReference~\\cite{sols_coulomb_blockade} pointed out that for nanoribbons\nwith very corrugated edges the interaction effects can seriously affect the\ncharge transport. According to\nRef.~\\cite{sols_coulomb_blockade}, a nanoribbon with corrugated\nedges can be viewed as a series of weakly coupled quantum dots\ndefined by the random geometry of the edges. Electron transport\nthrough such system is limited by the Coulomb blockade effect in\nthese dots.\n\n\nRecent experiments\n\\cite{molitor_transport_gap,stampfer_transport_exp,Todd2009,\ngallagher_transport_gap} on gated nanoribbons discovered that\nthere is a Fermi energy interval where the conductance is\nsuppressed, see \nFig.~\\ref{tr_gap}.\nThis phenomenon is called the transport gap. It was pointed out in\nRef.~\\cite{stampfer_transport_exp}\nthat the size of the experimentally observed transport gap is too big to be\nconsistent with conclusions of simple one-particle nanoribbon models, where\nthe gap is generated due to the transverse quantization. To develop a more\nrealistic description, the effect of both disorder and interactions must be\naccounted, in addition to the transverse quantization. It was proposed that\nthe transport occurs through tunneling between consecutive ``charge\npuddles\", which are the areas with non-zero charge induced by external\nthe disorder potential.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{tr_gap_todd.eps}\n\\caption[]\n{\\label{tr_gap}\n(Color online)\nGraphene nanoribbon (a) and the nanoribbon's electrical conductance data\n(b,c), from\nRef.~\\cite{Todd2009}.\nPanel~(a) shows a nanoribbon defined by etching a graphene sheet. The\ndarker area is graphene, while the lighter areas inside the red dashed\nlines are graphene-free. In panel~(b) the differential conductivity is\nplotted as a function of the back-gate voltage and source-drain voltage.\nNote that for small source-drain voltage the conductivity remains\nsuppressed (dark area) for small back-gate voltages, and grows when the\nlatter exceeds a certain value. This is a manifestation of the transport\ngap. In panel~(c) the same data is plotted in a smaller back-gate voltage\nwindow. The plot has the characteristic shape of overlapping noisy\n``Coulomb diamonds\", suggesting that the transport occurs through several\n``charge puddles\" acting as quantum dots.\nReprinted with permission from:\nK. Todd, H. T. Chou, S. Amasha, and D. Goldhaber-Gordon, Nano Lett. {\\bf 9},\n416 (2009).\nCopyright 2009 American Chemical Society.\n}\n\\end{figure}\n\n\nOur discussion shows that transport properties of the nanoribbons are\naffected most prominently by bulk and edge disorder, transverse\nquantization, edge type, and interactions.\n\n\\section{Graphene {\\it pn}-junctions and {\\it pnp}-structures}\n\\label{pnj}\n\nSeveral graphene-based field-effect devices have been realized in\nlaboratories.\nReference~\\cite{electronic_device}\nreported the fabrication of a FET made of graphene which operates at a\nrecord-breaking frequency of 100 GHz. In\nRef.~\\cite{fed_paper}\na room-temperature-operated switch demonstrating an on\/off ratio exceeding\n10$^6$ was described. Also, the implementation of a digital integrated\ncircuit was reported in\nRef.~\\cite{integr_circuit}.\nThe microcircuit consists of two transistors and performs the logical\ninversion operation. A graphene FET used as a biosensor was described in\nRef.~\\cite{fet_biosensor}.\n\nAll such devices are characterized by a spatially inhomogeneous Fermi level\ninside the graphene sample. There are several basic types of such systems:\ninterfaces separating regions with different concentrations of the charge\ncarriers\n({$pp^\\prime$}-junctions,\n{$nn^\\prime$}-junctions),\nor regions with carriers of opposite signs\n({\\it pn}-junctions),\nor series of such interfaces\n({\\it pnp}-structure,\n{$pp^\\prime p$}-structure,\netc.). Transport properties of these systems is an important subject of\nboth theoretical and experimental investigations. These studies are\nreviewed below.\n\n\\subsection{{\\it pn}-junction}\n\\label{junction}\n\nIf two planar electrostatic gates separated from the graphene sample by an\ninsulating layer are charged in such a way that the chemical potential at\n$x>0$ is shifted above the electroneutrality level and at $x<0$ below the\nelectroneutrality level, a\n{\\it pn}-junction\nis formed at $x=0$, see\nFig.~\\ref{pn-junction}.\nThe simplest model of the electrostatically-defined {\\it pn}-junction was studied in\nRefs.~\\cite{cheianov_np_simple,cayssol_pnp}.\n\\begin{figure}[btp]\n\\centering\n\\leavevmode\n\\epsfxsize=8.5cm\n\\epsfbox{pn_graphene_1.eps} \\caption[] {\\label{pn-junction} (Color\nonline) The {\\it pn}-junction studied in\nRef.~\\cite{zhang_screening}.\nThere are two gate electrodes in this device. The first one is the\nsemi-infinite gate on the left side. It controls the density drop\n$\\rho_2 - \\rho_1$\nacross the junction. The second electrode is an infinite back gate above\nthe sheet (not shown). It fixes the density $\\rho_2$ at far right. Lines\nwith the arrows show trajectories of an electron ($-$) and a hole ($+$).\nThe electron current in $n$-region is converted into hole current in\n$p$-region. Note that the direction of the incident electron current in\n$n$-region and the direction of the hole current in $p$-region are\nsymmetric with respect to $y$-axis reflection (the same type of refraction\nis shown in\nFig.~\\ref{meta_focus}). \nReprinted figure with permission from \nL.M.~Zhang and M.M.~Fogler, Phys. Rev. Lett. {\\bf 100}, 116804 (2008).\nCopyright (2008) by the American Physical Society.\n}\n\\end{figure}\n\nIn Ref.~\\cite{cheianov_np_simple} the current transmission through\na {\\it pn}-junction was investigated. When the current approaches the {\\it pn}-junction at the\nright angle, it passes through with no reflection. This is a\nmanifestation of the Klein tunneling. Otherwise, the reflected\ncurrent appears. The primary source of reflection in such a system\nis a classically-forbidden strip near the center of the junction,\nwhich can be crossed only by quantum tunnelling. The\nstrip's width $l$ depends on the incidence angle $\\theta$ [for\nnormal incidence $l(\\theta=0)=0$, hence, Klein tunneling]. In the\nmodel of Ref.~\\cite{cheianov_np_simple}, parameters of the\npotential barrier under which the particle has to tunnel depend on\nthe geometry of the {\\it pn}-junction and the gates' potentials.\n\nThe main finding of Ref.~\\cite{cheianov_np_simple} was that the\ncurrent transmission through such {\\it pn}-junction is very sensitive to the\nangle $\\theta$: for normal incidence the transmission is perfect,\nbut it exponentially quickly deteriorates when $\\theta$ grows. This allows\none to create very collimated beams of current. Additionally, the\nselectivity to $\\theta$ can be used to detect the magnetic field: since the\nmagnetic field bends the trajectory of a charged particle, then, in a\nproperly designed device, a particle hits the interface at a\nmagnetic-field-dependent angle. As a result, the transmission becomes\nsensitive to the field. Different devices utilizing properties of the\ngraphene\n{\\it pn}-junction\nwere proposed in\nRef.~\\cite{cheianov_np_simple}.\n\nThe treatment of\nRef.~\\cite{cheianov_np_simple}\nwas re-examined in\nRef.~\\cite{zhang_screening}.\nIt was noted there that the non-linear charge screening affects the\n{\\it pn}-junction\ncharacteristics. When the non-linear screening is accounted, significant\ndeviations from the findings of\nRef.~\\cite{cheianov_np_simple},\nwhich neglects many-body effects, are discovered. The results in\nRef.~\\cite{zhang_screening}\nindicate that the interaction significantly reduces the {\\it pn}-junction\nresistance.\n\nA more general study, including not only the electron-electron\ninteraction, but also the disorder, was presented in\nRef.~\\cite{fogler_pn_disorder}. It was shown that, depending on\nthe junction's parameters, it may be in either of three regimes:\n$(i)$ ballistic, where the {\\it pn}-junction resistance is\ndominated by the ballistic contribution, $(ii)$ diffusive, where\nthe resistance is dominated by the diffusive contribution, and\n$(iii)$ the crossover regime, when both ballistic and diffusive\ncontributions are comparable. In Ref.~\\cite{fogler_pn_disorder}\nseveral experimental {\\it pn}-junctions\n\\cite{Lemme2007,huard_2007,Oezyilmaz2007,Williams2007} were\nanalyzed trying to find junctions in the ballistic regime. It was\nconcluded that the considered experimental systems satisfy the\nconditions for ballistic propagation only marginally at best. It\nwas suggested that higher mobility or a larger carrier\nconcentration gradient near the junction is required to create a\nballistic {\\it pn}-junction.\n\n\nThe ballistic {\\it pn}-junction has attracted considerable attention due to\nits unusual electron-refraction properties. In\nRef.~\\cite{graphene_veselago}\nit was observed that, under certain conditions, the electron beam passing\nthrough a graphene {\\it pn}-junction experiences refraction in such a\nmanner that the refraction angle equals to {\\it minus} the angle of\nincidence, see\nFig.~\\ref{meta_focus}.\n\\begin{figure}[btp]\n\\centering \\leavevmode \\epsfxsize=10.5cm\n\\epsfbox{refraction.eps}\n\\caption[]\n{\\label{meta_focus}\n(Color online) {\\it pn}-junction as an electron-focusing device, as\ndescribed \nin~\\cite{graphene_veselago}.\nIf the gates' voltages are such that the concentration of electrons in the\n$n$-region and the concentration of holes in the $p$-region are the same, a\nparticle hitting the {\\it pn}-junction interface at the incidence angle\n$\\theta_c$ would be converted into a hole on the other side of\nthe junction propagating at the refraction angle\n$\\theta_v = - \\theta_c$.\nUnder such conditions, a current emanating from a point source is focused\ninto a small spot, ``the image\", on the other side of the\n{\\it pn}-junction\n\\cite{graphene_veselago}.\n}\n\\end{figure}\nIf this is the case, then current emanating from a point source on one side\nof the {\\it pn}-junction is focused into ``a point image\" on the other side\nof the junction. This situation is similar to the refraction of light at\nthe interface with a metamaterial whose refraction index is minus unity.\nThe focusing properties of the {\\it pn}-junction with circular geometry\nwere investigated in\nRef.~\\cite{graphene_caustics}.\n\nHowever, this ability to focus the electrical current is easy to spoil.\nRef.~\\cite{graphene_veselago}\npointed out that at the level of the geometrical optics the focus is\nperfect only if the density of holes in the\n$p$-electrode\nis the same as the electron's density in the\n$n$-electrode.\nIn Ref.~\\cite{fogler_pn_disorder}\nit was shown that disorder destroys the focus as well. Finally, since the\ntransmission of a {\\it pn}-junction decays quickly as the incidence angle\ndeviates from $\\pi\/2$, only a small fraction of the initial current is able\nto pass through the junction to form ``the image\".\n\n\nWhen a graphene\n{\\it pn}-junction\nis placed in a non-uniform magnetic field, it acquires new interesting\nfeatures. This type of devices is discussed in\nsection~\\ref{barrier}.\n\n\\subsection{Doping graphene by contact with metals}\n\\label{metal_contact}\n\nIn addition to electrostatic doping, it is possible to change the charge\ndensity in graphene by making contact with a metal electrode. In such a\ncase, depending on the electrode's material, the electrons either leave the\ngraphene sample to the electrode or flow into the graphene from the\nelectrode.\n\nJunctions created with the help of this kind of doping were\ninvestigated experimentally in Ref.~\\cite{huard_contact_doping}.\nMaterials for the metallic electrodes were chosen in such a way as to dope\nthe graphene with holes. Then, depending on the voltage of the back gate,\neither $pp^\\prime$- or $pn$-junctions were formed.\n\nThe graphene-metal interface was investigated theoretically in\nRefs.~\\cite{blanter_martin,robinson_metal,golizadeh_metal}. Charge\ntransfer between the metal electrode and the graphene sample was\nstudied in\nRefs.~\\cite{PhysRevLett.101.026803,PhysRevB.79.195425}\nwith the help of DFT. According to\nRef.~\\cite{PhysRevLett.101.026803}\nthe Fermi energy shift inside the graphene sample is a monotonous function\nof the metal work-function, as one should expect. However, when the\nwork-function of the metal coincides with that of graphene, the graphene\nsample is not neutral, as one naively might expect, but rather it is\npredicted that the sample is $n$-doped. This happens because of the\nchemical interaction between the metal and graphene.\n\n\n\\subsection{{\\it pnp}-structure}\n\\label{pnp}\n\nThe theory of electronic transport in clean $pnp$-structures was presented\nin Ref.~\\cite{cayssol_pnp}.\nThere, they demonstrated that the conductance of ballistic\n{\\it pnp}-structure exhibits oscillations (`Fabry-P\\'erot' resonances) as a\nfunction of the carrier concentration in the middle ($n$) area of the\n{\\it pnp}-structure. These resonances are due to quasi-bound electron\nstates in the $n$-region of the {\\it pnp}-structure.\n\nA more general numerical study, which accounts for interaction and\nimpurities, was performed in\nRef.~\\cite{rossi_pnp}.\nIt reported a crossover from ballistic to diffusive regime when the\nmean-free-path becomes comparable to the length of the middle\nregion. The disorder wipes out the `Fabry-P\\'erot' resonances.\nHowever, it is conceivable that these survive under a small concentration of\nimpurities, and, thus, could be seen experimentally.\n\nA phenomenon analogous to the `Fabry-P\\'erot' resonances was\ndiscussed in\nRef.~\\cite{bliokh_superlatt,arovas_superlattice},\nwhere the transmission\nthrough several junctions connected in series was studied. Because\nof electron wave function interference, the transport through such\nstructure demonstrates a non-monotonous dependence on the current\nincidence angle and the distance between the junctions.\n\nElectrostatically-defined $npn$- and $pnp$-structures were realized\nexperimentally\n\\cite{young_fabry_perot,velasco_fabry_perot,gorbachev_air_bridge}.\nFor example,\nFig.~\\ref{pnp_experiment}\nshows a scanning electron microscope image of a\n$pnp$-structure\nfrom\nRef.~\\cite{young_fabry_perot}.\nThe experimental observation of `Fabry-P\\'erot' oscillations in\n$pnp$-structures\nwas reported in\nRef.~\\cite{young_fabry_perot,velasco_fabry_perot}.\nIn\nRef.~\\cite{gorbachev_air_bridge}\nexperimental data were analyzed within the theoretical framework of\nRef.~\\cite{cheianov_np_simple}.\nReference~\\cite{gorbachev_air_bridge}\nconcluded that, in the fabricated\n{\\it pnp}-structure,\nthe individual\n$pn$-junctions\nare ballistic, and that the fabrication of a ballistic graphene\n{\\it pnp}-structure\nis feasible.\n\n\\begin{figure}[btp]\n\\centering \\leavevmode \\epsfxsize=10.5cm\n\\epsfbox{pnp_fig.eps}\n\\caption[]\n{\\label{pnp_experiment}\n(Color online) Experimental realization of a graphene\n$pnp$-structure,\nfrom\nRef.~\\cite{young_fabry_perot}.\nPanel~(a) shows the scanning electron microscope image of the structure.\nLarge purple rectangle is the graphene sheet. Two bulk yellow electrodes\n(source and drain) and one narrow blue electrode (top gate) are placed on\ntop of the graphene. The back gate beneath the structure is not visible.\nThe inset presents an enlarged view of the top gate. Panel~(b) shows a\nschematic diagram of the same setup.\nReprinted by permission from Macmillan Publishers Ltd:  \n\\href{http:\/\/www.nature.com\/nphys\/index.html}{Nature Physics},\nA. F. Young and P. Kim, Nat. Phys. {\\bf 5}, 222 (2009),\ncopyright (2009).\n}\n\\end{figure}\n\nIn the previous subsection we discussed the peculiar electron refraction at\nthe $pn$-interface.\nIn Ref.~\\cite{graphene_veselago}\nseveral possible applications of this effect were proposed, among which the\nmost known is the so-called ``electron Veselago lens\".\nThe latter device is a ballistic graphene $npn$- or $pnp$-structure in\nwhich both junctions are tuned to operate in such a manner that the\nelectronic current emitted from a point current source in the left\n$n$-electrode\ntravels through two junctions and would be focused into a point (image of\nthe source) in the right\n$n$-electrode.\nA similar phenomenon was predicted by Veselago\n\\cite{veselago_original}\nin the optics of materials with a negative-refractive index: the\nelectromagnetic rays emitted from a point source are focused upon passing\nthrough a slab of such material. This slab is called the Veselago lens. It\nis an analog of the $npn$-structure under discussion.\n\nHowever, the above analogy is incomplete. The ``superresolution\", the most\nadvantageous property of the optical Veselago lens\n\\cite{pendry,lagarkov_kissel}\n(see also\nRef.~\\cite{veselago_uspekhi,bliokh_meta,kats_2d_lensing,graphene_optics}),\nis absent for the graphene device\n\\cite{Yampol'skii2008}.\nAlso, since the current transmission decays quickly for non-normal\nincidence, the graphene lens is very opaque. This might make its\napplication problematic.\n\n\nOur discussion in this section suggests that graphene\n{\\it pn}-junctions\nand\n{\\it pnp}-structures,\ndue to their interesting properties, may, in principle, be used for current\ncontrol and magnetic field sensing applications, provided that a way to\nattenuate the effects of the disorder is found.\nIf a graphene\n{\\it pnp}-structure\nis placed into a non-homogeneous magnetic field, it may, under certain\ncondition, act as a electron waveguide. The relevant discussion can be\nfound in Sec.~\\ref{wguide}.\n\n\\section{Quantum barriers, wires, and waveguides}\n\\label{barrier}\n\nIn this section, we address  charge transport  for designing tunable charge-conducting\nelements. Unlike quantum dots (Section V), where electrons are bound in a closed space,\nin graphene-based quantum wires and waveguides the charged particles should be confined\nonly in \\textit{one} direction and be freely propagating in another one, as in\n$pnp$-structures.\n\nThere are several methods of charge confinement in graphene (see\nSections IV-VI above). Typically, this is achieved either $(i)$\nchemically (by binding graphene atoms to foreign atoms: e.g.,\noxygen \\cite{Jia-AnYan2009}, fluoride\n\\cite{cheng_flouridation_2010, withers_fluoridation,\nxiang_hydro_fluo}, hydrogen \\cite{Novoselov2009,\nxiang_hydro_fluo}, or aryl groups \\cite{nitrophenyl}); $(ii)$\nmechanically (either cutting or bending graphene sheets\n\\cite{Pereira2009, Guinea2010, Pereira2010}, or by creating\ninhomogeneous spatial strain distributions \\cite{Pereira2010,\nPereira2009a, Levy2010, low_gap_strain}); $(iii)$ thermally\n\\cite{du_high-mobility}, and $(iv)$ electronically (by applying\nelectromagnetic fields).\n\nEach of these methods has its advantages and disadvantages. The first two are very\neffective; however, they are rather difficult to control, in the sense that any tuning\n(change of parameters) requires a reconstruction, either chemical or geometrical, of the\nwhole graphene sample, which usually cannot be done quickly.  Thus, these methods are not\nvery suitable in designing \\textit{tunable} electronic devices.\n\nHere we concentrate on confining electrons using  electromagnetic fields. This approach\nis more flexible than chemical and mechanical approaches. Not only electromagnetic fields\nare easy to control, but being tailored properly, they enable the creation of\ngraphene-based tunable elements, including quantum wires and waveguides with unique\nproperties, such as unidirectional conductivity, robustness to disorder, etc. The\ndiversity of methods and approaches makes it increasingly difficult to summarize of the\ncurrent state-of-the-art in this area, and calls for a systematic classification by both\nmethods and results. An attempt of such classification is a goal of this chapter.\n\nManipulating charge carriers by \\textit{electric fields} (i.e., adding scalar potentials\nof different shapes to the Dirac equation) is a very popular approach (see, e.g., the\nreviews in Refs.~\\cite{neto_etal, Peres2009}). This method has provided interesting and\nsurprising outcomes (i.e., \\cite{Titov2007, chakraborty_review, Peres2009, Geim2009}).\nUnfortunately, it turned out to be impossible to repel or localize electrons in all\ndirections and at all energies by only using an electric field \\cite{beenakker_colloq}.\nThere is always a channel in any electric-field barrier where the charge can escape\nthrough.\n\nThe charge-confining and current-guiding capabilities produced by \\textit{magnetic\nbarriers} are well known and have already opened certain possibilities for practical\napplications (e.g., \\cite{Ghosh2009}). However, when it comes to designing fast-tunable\nelectronic devices (switches, filters, etc.) a difficulty emerges:  most of the existing\nmagnetic-barrier technologies usually involve the deposition, either on top or beneath\nthe graphene sheet, of a pattern of magnetic material, which reproduces the desired\nmagnetic field distribution in the sample. Any subsequent change of parameters would\nrequire building a new setup, creating formidable (if surmountable) obstacles for\nharnessing magnetic barriers as elements of fast-acting electronic devices. Grathene\nstructures based on the effective magnetic field created by applying inhomogeneous\npressure or strain to the graphene sheet \\cite{Pereira2010, Pereira2009a, Levy2010}might\nalso be useful for applications.\n\nAn efficient way around this problem is the simultaneous use of inhomogeneous magnetic\nand electric fields. The proper combination of these two not only preserves (or even\nimproves) the necessary transport properties of graphene samples, but makes them easy to\ncontrol by tuning the spatial distribution of  the electric potential, for a fixed\nmagnetic field.\n\nHereafter, we focus on low-energy excitations when the\ninter-valley scattering \\cite{Akhmerov2008} is negligible and\nquasiparticles can be considered as massless Dirac fermions. To\nsimplify the presentation, all spin-related effects are also\nneglected below.\n\nThe building block of all charge-confining elements is a field-induced barrier (i.e., a\nreflecting wall). To introduce this in the most general way, consider two graphene\nhalf-planes, $x<x_{0}-l\/2$ and $x>x_{0}+l\/2$, subject to different stationary electric\n$V_{i}$ and magnetic $\\bm{A}_{i}$\npotentials ($i=1,2$), as shown in Fig. \\ref{Fig_0}. Here we assume that $%\n\\bm{A}\\equiv A(x)\\widehat{\\bm{y}}$, which means that the magnetic field $%\n\\bm{B}=B(x)\\widehat{\\bm{z}}$. When $|x-x_{0}|<l\/2$, the fields and\npotentials are $x$-dependent, providing the transition from\n$\\{V_{1},{A}_{1},\\bm{B}_{1}\\} $ to $\\{V_{2},{A}_{2},\\bm{B}_{2}\\}$\n(see Fig.~\\ref{Fig_0}). If the width $l$ is large,  compared to\nthe graphene lattice spacing $a,$ and much smaller than the Fermi\nwavelength $\\lambda _{F}$, $a\\ll l\\ll \\lambda _{F}$, then the\nsmooth profile can be replaced in the calculations by a step\nfunction as shown by the dotted line in Fig.~\\ref{Fig_0}. When\n$B=0$ we have \\textit{pn} junction, which is discussed in\nSection~VI.\n\n\n\n\n\\begin{figure}[tbh]\n\\centering \\scalebox{0.8}{\\includegraphics{Fig_0.eps}} \\caption{ (Color online) Depending\non the problem being considered in this section, the function $P(x)$ is either one of the\npotentials $V(x)$, $A(x)$, or the magnetic field $B(x)$. Here, $P(x)$ is shown as a\ncontinuous green line, while the red dashed line repeats the step-like approximation.}\n\\label{Fig_0}\n\\end{figure}\n\n\\subsection{Magnetic barriers: ${B}\\neq 0,$ $V=0$}\\label{NewA}\n\nAs it was demonstrated theoretically in Refs.~\\cite{DeMartino2007, DeMartino2007a,\nDell'Anna2009}, charge carriers in graphene could be confined in space by applying a\nproperly tailored static inhomogeneous magnetic field perpendicular to the graphene plane\n($x,y$). There are numerous unusual and intriguing features in this phenomenon, but at\nits basic level, this happens\nbecause the trajectory of the quasiparticle incoming from  free space $%\nx<0 $ bends inside the magnetic barrier (Fig.~\\ref{Fig_0}) and\neventually exits backward, independently of its cyclotron radius,\nthus making the wall a perfect reflector for Dirac electrons of\nany energy \\cite{Masir2008}. Obviously, a barrier of finite width\n$d$ reflects only charges with energies bellow a threshold, namely\nthose whose cyclotron radius is smaller than $d$. Below we\nconsider two types of these magnetic barriers.\n\n\\subsubsection{Step-wise vector potential $A$: $\\protect\\delta$-function-like magnetic field $B$}\\label{NewA1}\n\nThis basic magnetic field profile, $B(x)=B_{0}\\,\\ell_{B_0}\\,\\delta (x)$ (here\n$\\ell_{B_0}=\\sqrt{\\hbar c\/eB_0}$ is the magnetic length), already displays an important\nadvantage of magnetic barriers over electric ones: there is a range of energies for which\nit is a perfect mirror for Dirac electrons for all angles of incidence including\nperpendicular \\cite{Ghosh2009}. Outside this energy range, the transparency is\nangular-dependent, and the analog of the total internal reflection (TIR) of light takes\nplace when the angle of incidence exceeds some critical angle $\\phi _{{\\rm TIR}}$, which\ndepends on the applied magnetic field. The resulting charge refraction is not of the\nclassical Snell's law type, and is\nasymmetric with respect to the transformation of the angle of incidence $%\n\\phi \\rightarrow -\\phi $ \\cite{Ghosh2009}.\n\nRather counter-intuitive is also the fact that a ``point-like'' magnetic field,\n$B(x)\\propto\\delta (x)$, creates a bound (in the $x$-direction) state at zero energy.\nThis state is a ``linear'' analog of two-dimensional surface waves: it is exponentially\nlocalized in the $x$-direction around the line $x=0$, and propagates along the $y$-axis\nwith  wave numbers $\\pm $ $k_{y}$. The longitudinal wave number $k_{y}$ is enclosed in\nthe finite interval $( -\\ell_{B_0}^{-1}\/2,\\,\\ell_{B_0}^{-1}\/2) $, and the $x $-dependence\nof the wave function is asymmetric, so that the rates of the exponential decay at $x<0$\nand $x>0$ are different, depending on the sign of $k_{y}$ (direction of propagation)\n\\cite{Masir2008}. The possibilities to experimentally produce highly localized magnetic\nfields have been discussed in Ref.~\\cite{Ghosh2009}, where charge transport in the\npresence of various arrangements of $\\delta $-type magnetic barriers was studied in terms\nof ``electron optics''.\n\n\n\\subsubsection{Step-wise magnetic field $B$: piece-wise linear vector\npotential $A$}\\label{NewA2}\n\nThe transport properties of Dirac electrons in a step-wise magnetic field [i.e.,\n$B(x)=B_1\\theta (-x)+$ $B_2\\theta (x)$] are different and of greater variety than in the\nstep-wise potential considered in the previous subsection \\ref{NewA1}. While in the later\ncase, the system is obviously invariant with respect to a shift of the $A$-step [namely,\n$A(x)\\rightarrow A(x)+\\mathrm{const}$], for a step-wise $B$ this is not true: if $B_1$\nand $B_2$ are  parallel ($\\gamma=B_1\/B_2>0)$ or antiparallel $(\\gamma<0)$ makes a\nsignificant difference.\n\nTo better understand  the behavior of Dirac electrons in nonhomogeneous magnetic fields,\nit is illuminating to compare it with the conventional two-dimensional electron gas.\nFundamental differences  already exist in\nthe uniform field. In contrast with the classical Landau levels $\\mathcal{%\nE}_{n}$ in infinite space, $\\mathcal{E}_{n}\\propto (n+1\/2)$,\n$n\\geq 0,$ the\nquantization of the eigen-energy of the Dirac equation produces $\\mathcal{E%\n}_{n}\\propto \\mathrm{sign}(n)\\sqrt{|n|}$, with $-\\infty \\leq n\\leq \\infty $. Positive\nvalues of $n$, $n>0$, are associated with electron-like charge carriers, while $n<0$\ncorresponds to holes. The eigenstates with $n\\neq 0$ are similar to the states of the\nconventional two-dimensional electron gas, whereas the zero-energy state $n=0$ possesses\ndifferent properties. Bound states associated with this $n=0$ Landau level have different\nfeatures than the states associated with $n\\neq 0$.\n\n\nThe energy in homogeneous magnetic fields does not depend on the\nwave number $k$, therefore the group velocity is zero,\n$v_{g}=d\\mathcal{E}_{n}\/dk=0$, and the states\ncarry no current. A dispersion, and therefore a non-zero group velocity $%\nv_{y}=d\\mathcal{E}_{n}\/dk_{y}\\neq 0$ can appear either for states localized near the\nboundary of a finite sample (so-called edge states), or due to the spatial inhomogeneity\nof the magnetic field, $\\nabla_x B\\neq 0$, which creates bound states localized in the\n$x$-direction and propagating along the $y$-axis with its drift velocity proportional to\n$\\nabla B\\times \\mathbf{B}$ (denoted, by analogy with edge states, as \\textit{magnetic}\nedge states \\cite{Muller1992}).\n\nFor the particular case $B_{1}=0$ (free half-space for $x<0$, constant magnetic field\n$B_{2}>0$ at $x>0$), there is an infinite number of bound (in the $x$-direction)\ndispersive states labeled by the Landau-level index $n$, whose energies are proportional\nto $\\mathrm{sign}(n)\\sqrt{|n|}$. These states are localized as functions of $x$, centered\naround points whose locations depend on the wave number $k_{y}$. Remarkably, these\nlocalized states exist only with one sign of the wave number $k_{y}$, either positive or\nnegative, depending on the orientation of the magnetic field $B_{2}.$ Since the group\nvelocity has opposite signs for $+n$ and $-n,$ the direction of the charge flow created\nby electrons and holes is the same, and therefore any bound $n$-state carries a finite\nunidirectional current along the $y$-axis \\cite{Ghosh2009}. Therefore, a particle in such\na state never undergoes backscattering; hence it is practically insensitive to disorder\nand Anderson localization never takes place, no matter how strong the disorder. There is\nalso a bound state with $\\mathcal{E}=0$, when $k_{x}=ik_{y}$. However, this state is\ndispersionless and does not carry any current.\n\nWhen $B_{1}\\neq 0$ \\cite{Park2008} and it is parallel to $B_{2}$ ($\\gamma >0$), the\nLandau levels of Dirac quasiparticles at large positive $k_{y}$ are localized in an\neffective potential well around $x=-k_{y}\\ell_B^{2}$. With $k_{y}$ decreasing to negative\nvalues, the\ndimensionless energy levels gradually change to $\\mathrm{sign}(n)\\sqrt{%\n\\gamma |n|}$ at large negative $n$ and shift in space to $%\nx=-k_{y}\\ell_B^{2}\/\\gamma $. It is important to note that the\ndirections of the drift (signs of $d\\mathcal{E}_{n}\/dk_{y}$) are\nopposite for electron- and hole-like particles ($\\pm n$), thus\nproviding a non-zero total current. In the vicinity of $k_{y}=0,$\nthe corresponding states become localized at $x=0$. Similarly to\n$B_{1}=0$ case, the magnetic barrier with $\\gamma\n>0$ supports bound states, which create unidirectional conductivity at $%\nn\\neq 0$, while zero-energy solutions carry no currents.\n\nWhen $\\gamma <0$ (for antiparallel $B_{1}$ and $B_{2}$) \\cite{Park2008}, the effective\npotential at large positive $k_{y}$ has two minima (two connected harmonic wells),\nlocated far away from the boundary $x=0$, so that the states are localized in each well.\nAs $k_{y}$ moves to negative values, the effective potential shifts toward $x=0$ and\ntransforms into a single non-harmonic potential well. The eigen-energies with $n\\neq 0$\ncorrespond to states which support non-zero unidirectional current following classical,\nso-called snake, orbits \\cite{Muller1992} confined to a narrow one-dimensional channel\ncentered at the line $x=0$ where the magnetic field changes its sign. It is shown in\n\\cite{Oroszlany2008, Ghosh2008} that in a symmetric graphene sample of a finite width $L$\n($-L\/2\\leq x\\leq L\/2$) this current is compensated by real edge states localized close to\nthe sample boundaries. The states with $n=0$ exhibit both electron and hole features,\nwhich is highly unusual and is unique for Dirac quasi-particles in graphene.\n\n\n\\subsection{Combined magneto-electric barriers: $B\\neq 0$, $V\\neq 0$}\\label{NewB}\n\nIn principle, the charge-confining and guiding capabilities of magnetic walls presented\nabove open up certain possibilities for practical applications. However, as it was\nmentioned before, the parameters of the magnetic barriers cannot be changed fast enough,\nwhich makes it problematic to use them as elements of fast-acting electronic devices.\n\n\nTo overcome this problem, it is convenient to use a combination of inhomogeneous magnetic\nand electric fields, which enables the efficient control of the transport properties of\ngraphene samples by tuning the electric potential without changing the parameters of the\nmagnetic field.\n\nTo introduce a basic setup combining magnetic and electric fields, we now consider a\nsingle magneto-electric barrier produced by superimposing a scalar potential $V$ of the\nsame step-like shape on the magnetic structure with a $\\delta$-like magnetic field [i.e.,\na step-wise vector potential $A(x)=A_1\\theta (-x)+$ $A_2\\theta (x)$]. This system\npossesses unique properties that make it different from other types of barriers. Graphene\nsubject to mutually perpendicular electric and magnetic fields supports states which are\nlocalized near the barrier. These current-carrying states (surface waves) correspond to\nquasiparticles moving along the barrier only in one direction \\cite{Bliokh2010}. This\ndirection, as well as the value of the quasiparticle velocity, are easily controlled by\nthe electrostatic potential. These states correspond to the classical drift of charged\nparticles in crossed electric and magnetic fields. They exist if and only if the drift\nvelocity $v_{d}=cE\/B$ is smaller than the Fermi velocity $v_{F}$ [here $E\\simeq\n(V_{2}-V_{1})\/l$ and $B\\simeq (A_{1}-A_{2})\/l$ are the electric and magnetic fields in a\nbarrier of  finite length $l$]. The absence of counter-propagating states prevents the\nbackscattering induced by either irregularities in graphene \\cite{Titov2007,\nbliokh_superlatt} or by the fluctuations of the magnetic field.\n\nFor potential applications, the important feature of a single magneto-electric barrier is\nthat the transport (electric current) across or along this structure can be controlled by\nmanipulating only the electric potentials $V_{1}$ and $\\mathit{V}_{2}$. In particular:\n\n--- The transmission and reflection coefficients across the junction\nbetween two areas with different values $V_{1},A_{1}$ and $V_{2},A_{2}$ (Fig.\n\\ref{Fig_0}), and the angle of refraction (i.e., the direction of the transmitted\ncurrent) depend on the electric potentials. Specifically, tuning $V_{1}$ and\/or $V_{2}$\ncan change the angle of incidence where the barrier is totally transparent, and thus the\nKlein tunneling can be observed;\n\n--- When the inequality\n\\begin{equation}\n\\left\\vert V_{1}-\\mathcal{E}\\right\\vert +\\left\\vert V_{2}-\\mathcal{E}%\n\\right\\vert <\\left\\vert A_{1}-A_{2}\\right\\vert  \\label{1}\n\\end{equation}%\nholds (here $\\mathcal{E}$ is the energy of the quasiparticles; all units are\ndimensionless), the step is a perfect reflector for electrons at all angles of incidence\nand the junction is locked for the electric current.\n\n--- If\n\\begin{equation}\n\\left\\vert V_{1}-V_{2}\\right\\vert<\\left\\vert A_{1}-A_{2}\n\\right\\vert,  \\label{2}\n\\end{equation}%\na wave (current) exists which propagates unidirectionally along\nthe barrier (in the $y$-direction) with the dimensionless group\nvelocity $\\nu=v_d\/v_F<1$ and is exponentially localized in the\n$x$-direction.\n\n\\subsection{Waveguide with electrically-tuned parameters}\n\\label{wguide}\n\nWhile one barrier forms a wire, two such barriers constitute a\nwaveguide. This waveguide supports modes that are similar to the\nelectromagnetic eigenmodes of a dielectric waveguide and likewise\nhave quantized transverse wave numbers (The analogy between the\ntransport of Dirac electrons in graphene and light propagation in\ndielectrics is described in Refs.~\\cite{graphene_veselago,\nbliokh_superlatt, Bliokh2010, graphene_optics}). However, along\nwith them there is another set of waves that is appropriate to\ncall ``extraordinary'' \\cite{Bliokh2010}. They are formed by two\ncoupled surface waves propagating along the waveguide walls\n(barriers). There is an energy gap where only extraordinary modes\nexist. Decreasing the spacing between the barriers broadens this\ngap. The extraordinary modes are also stable against\nbackscattering.\n\nAn important feature of field-induced waveguides in graphene, which is favorable for the\ncreation of tunable electronic devices is that the transport properties of these\nstructures are strongly dependent  on the parameters of the barriers. These parameters\nare the potentials $A_{l,r}$ and $V_{l,r}$ of the left and right semiplanes,\nrespectively, surrounding the central region where the potentials are equal to zero,\n$A_c=V_c=0$. In particular:\n\n--- When\n\\[A_{l}=A_{r} \\hspace{3mm}{\\rm and}  \\hspace{3mm} V_{l}=V_{r}\\]\nand   the inequalities~(\\ref{1}) and (\\ref{2}) are valid for both barriers (waveguide\nwalls), then the extraordinary modes are unidirectional. This makes them immune to\nbackscattering, and therefore robust against $y$-dependent disorder.\\\n\n --- If\n \\[A_{l}=-A_{r}  \\hspace{3mm} {\\rm and}  \\hspace{3mm} V_{l}=V_{r},\\]\nthen the surface waves ``attached'' to the barriers propagate along the $y$-axis in\nopposite directions and the extraordinary modes are bidirectional. Nevertheless, the\nbackscattering also does not affect the total current, due to the spatial separation of\nthe charge fluxes with opposite directions.\n\n --- When \\[A_{l}=-A_{r}  \\hspace{3mm} {\\rm and} \\hspace{3mm} V_{l}=-V_{r},\\]\nthe spectrum of the extraordinary modes is independent of the distance between the\nbarriers, and therefore there is no cutoff energy for them. This means that extraordinary\nmodes can penetrate through an \\textit{arbitrary} narrow part of the waveguide.\n\n\nIn Tables~\\ref{Tab1} and \\ref{Tab2}, the barriers and waveguides of the above mentioned\ntypes and combinations of fields are categorized according to the following features:\n\n\\begin{itemize}\n\\item Their ability to reflect all incident current (perfect wall)\n\n\\item Their ability to support bounded electron states\n\n\\item The type of spectrum of the propagating modes (either\ncontinuous or discrete)\n\n\\item The directionality of the current (either uni- or in\nbidirectional)\n\n\\item  A separate column lists some other distinctive features because graphene in\nmagnetic fields has unusual properties.\n\\end{itemize}\n\nIn summary, changing the electric potential (with the magnetic field unchanged) one can\nswitch on and off the current through the barrier and create\/destroy a unidirectional\nquantum wire along the barrier. Moreover, changing the electric potential one can create\nwaveguides with unique, exotic transport properties.\n\n\n\n\\begin{table}\n\\begin{tabular}{||p{4.7cm}||p{0.9cm}|p{1.1cm}|p{3.8cm}|p{5.5cm}||}\n\\hline \\centering\\textbf{Barrier type} & \\textbf{Per\\-fect wall} &\n\\textbf{Bound sta\\-tes} & \\centering\\textbf{Directionality of the\nbound state current} & {\\textbf{Comments}} \\\\ \\hline\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Tab1a.eps}} & No & No & No bound state current.\n & When $\\min\\{V_i\\}<\\mathcal{E}<\\max\\{V_i\\}$, the optics analogy of the\nbarrier is the interface between two dielectrics with opposite signs of the refraction\nindexes. Otherwise, this is the interface between usual dielectric media. Total internal\nrefraction is\npossible. Cannot be opaque for all angles of incidence (i.e., Klein tunneling). \\\\\n\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Tab1b.eps}}\\, & \\,Yes\\, &\n\\,Yes\\, &\n One bound state with zero energy ($\\mathcal{E}=0$) and zero group velocity ($v_g=0$)\nalong the barrier, and, therefore, carries no current. This bound state is associated\nwith the Landau level with $n=0$.\\, & In a certain range of energies the barrier is\nopaque for all angles of incidence. This barrier is similar to the barrier generated by a\ngraphene sheet strain. The difference is that the strain generates an effective vector\npotentials jump (effective magnetic fields) with opposite (due to the time-reversal\nsymmetry) signs\nin two valleys, whereas the real magnetic field has the same sign in both valleys. \\, \\\\\n\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Tab1c.eps}}\\, & \\,Yes\\, & \\,Yes\\, & Bi-directional\nconductivity when both $B_1$ and $B_2$ are non-zero; otherwise uni-directional\nconductivity along the barrier. & When $\\gamma=B_1\/B_2>0$, the bound state is similar to\nthe classical electrodynamics  state with $\\vec{\\nabla}B\\times\\vec{B}$ drift. When\n$\\gamma<0$ and Landau-level index $n\\neq 0$, the state is similar to a snake state\n(charged particle motion along the $B=0$ line). Can be opaque for all angles of\nincidence.\n\\\\ \\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Tab1d.eps}}\\, & \\,Yes\\, & \\,Yes\\, & Confined\nuni-directional state with linear spectrum $d\\mathcal{E}\/dk_y=v_d$ when\n$|v_d|=c|(V_2-V_1)\/(A_1-A_2)|<v_F $. In a certain range of energies, the barrier is\nopaque for all angles of incidence. Easily controlled by the electric potential. & The\nclassical electrodynamics analogy of the bound state is the charged particle drift in\ncrossed\nelectric and magnetic fields. The drift velocity is $v_d=cE\/B=c(V_2-V_1)\/(A_1-A_2)$ \\\\\n\\hline \\,\\scalebox{0.4}{\\includegraphics{Tab1e.eps}}\\, & \\,Yes\\, & \\,Yes\\, & Same as the\nmagnetic field barrier shown above. & Same as the magnetic field barrier shown above. \\\\\n\\hline\n\\end{tabular}%\n\\caption{Basic properties of electric and\/or magnetic barriers in\ngraphene. The ability to reflect all incident electric currents\n(or the ability to be a perfect reflecting wall) is indicated in\nthe second column. The presence or absence of bound states is\nlisted in the third column. Additional physical properties are\nsummarized in the remaining columns.}\n\\label{Tab1}%\n\\end{table}\n\n\\newpage\n\n\\begin{table}\n\\begin{tabular}{||p{5.0cm}||p{4.0cm}|p{5.0cm}||}\n\\hline\n\\centering\\textbf{Waveguide type} & \\centering\\textbf{Guided modes} & %\n\\textbf{Comments}  \\\\ \\hline\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Table2a.eps}}\\, & Discrete set of bulk waves. & Optics\nanalogy: this is a dielectric waveguide based on  total internal reflection.\n \\\\ \\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Table2b.eps}}\\, & Discrete set of bulk waves and one\nsurface mode with zero group velocity along the waveguide. & The bulk waves are similar\nto dielectric waveguide modes. The surface mode is localized near the waveguide walls\nand consists of two coupled surface modes associated with the barriers (the waveguide walls).  \\\\\n\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Table2c.eps}}\\, & Discrete set of bulk waves when\n$\\sigma=B_l\/B_r=\\pm1$ and $B_c=0$. The mode is localized near the boundaries when\n$\\sigma=1$ and $B_c<0$. & The discrete set of bulk waves is similar to the eigenmodes of\na dielectric slab surrounded by a medium with negative permittivity\/permeability. It\nalways has a  snake state with noncompensated current. When $\\sigma=1$ and $B_c<0$, there\nare counter-propagating currents near the walls (two snake states near the walls).  \\\\\n\\hline\n\n\\,\\scalebox{0.4}{\\includegraphics{Table2d.eps}}\\, & Discrete set of bulk waves and two\nsurface modes. & Bulk waves are similar to the eigenmodes of dielectric waveguides. When\nthe spacing between the barriers is rather small, there are only surface modes.\nThe propagation direction of surface modes is controlled by the electric potentials.  \\\\\n\\hline\n\n\\end{tabular}%\n\\caption{Basic properties of waveguides in graphene. These are\nproduced by various configurations of the applied electric and\nmagnetic fields. Note that the convention used here for dotted and\ndashed lines is different from the one used in the previous\nTable.} \\label{Tab2}%\n\\end{table}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThermal leptogenesis \\cite{FY} is an attractive and minimal mechanism to\ngenerate the baryon asymmetry of the Universe\n$n_\\mathrm{B} \/n_\\gamma \n\\,\\approx\\, (6.10\\,\\pm\\,0.21)\\,\\times\\,10^{-10}$,\nor, normalised to the entropy density, \n$Y_B\\,\\approx\\, (0.87\\,\\pm\\,0.03)\\,\\times\\,10^{-10}$ \\cite{Spergel:2006hy}.\nA lepton asymmetry is dynamically generated\nand then  converted into a baryon asymmetry\ndue to $(B+L)$-violating sphaleron interactions \\cite{kuzmin}\nwhich exist in the Standard Model (SM) and its minimal supersymmetric extension, the MSSM.\nLeptogenesis  can be implemented within  the \n``Seesaw''(type I) model \n\\cite{seesaw}, consisting\nof the SM (MSSM)\nplus  three   right-handed (RH) Majorana neutrinos (and their superpartners)\nwith a hierarchical spectrum. In the simplest case,  the \nlightest of the RH neutrinos\nis produced by thermal scattering after inflation,  and subsequently\ndecays out-of-equilibrium in a lepton number and CP-violating way, \nthus satisfying Sakharov's constraints  \\cite{sakharov}. \n\n\nThe asymmetry\nis commonly calculated by solving a Boltzmann equation for the\ntotal lepton asymmetry (flavour-independent approximation) \nand for the abundance of the \nlightest RH neutrino \\cite{lept,ogen,work}. However, this \nflavour-independent treatment is\nrigorously correct only when the interactions mediated by \ncharged lepton Yukawa couplings are out of equilibrium.\nThe impact of  flavour in thermal leptogenesis has been first addressed in\nRef.~\\cite{barbieri} and then \nstudied in  detail \n\\cite{davidsonetal,nardietal,Abada:2006ea}, including the\nquantum oscillations\/correlations of the asymmetries in lepton flavour space \n\\cite{davidsonetal}. It was shown that \nthe Boltzmann equations describing the asymmetries in flavour space \nhave  additional terms which can  significantly affect the result for the\nfinal baryon asymmetry.\nThis is because\nleptogenesis involves  the  production and \ndestruction of right-handed neutrinos, \nand  of a  lepton asymmetry  that is distributed among\n{\\it distinguishable} flavours.  The processes which\nwash out lepton number\nare {\\it flavour-dependent}, e.g., the inverse decays\nfrom electrons can destroy the lepton asymmetry carried by,\nand only by,  the electrons.\nThe  asymmetries  in each flavour\nare  therefore washed out differently,\nand will appear with different weights in the final formula\nfor the baryon asymmetry.  This is physically inequivalent\nto the treatment of washout in the flavour-independent approximation,\nwhere indistinguishable \nleptons propagate between decays and inverse decays,\nso inverse decays  from all flavours are taken to wash out asymmetries\nin any flavour. Flavour-dependent\neffects in leptogenesis \nhave been studied in the two right-handed neutrino model \n\\cite{Abada:2006ea}, where they have been shown to be \nrelevant. \nIt remains to be seen how important these effects are in more general \nneutrino mass models with three right-handed neutrinos.\n\nThe latest experimental data on neutrino oscillations\nis consistent with (approximate) tri-bimaximal\nmixing \\cite{tribi}. The fact that the tri-bimaximal neutrino\nmixing matrix involves\nsquare roots of simple ratios motivates models in which the mixing\nangles are independent of the mass eigenvalues. One such class of\nmodels are seesaw models with sequential dominance (SD) of\nright-handed neutrinos \\cite{King:1998jw}.\nIn SD, a neutrino mass hierarchy is shown to\nresult from\nhaving one of the right-handed neutrinos give the dominant\ncontribution to the\nseesaw mechanism, while a second right-handed neutrino gives the\nleading sub-dominant contribution, leading to a neutrino mass\nmatrix with naturally small determinant.\nIn a basis where the right-handed neutrino mass matrix is\ndiagonal, the atmospheric and solar neutrino mixing angles are determined in\nterms of ratios\nof Yukawa couplings involving the dominant and subdominant\nright-handed neutrinos, respectively. If these Yukawa couplings\nare simply related in some way, then it is possible for simple\nneutrino mixing angle relations, such as appear in tri-bimaximal\nneutrino mixing, to emerge in a simple and natural way,\nindependently of the neutrino mass eigenvalues.\nIf SD is combined with\nvacuum alignment in flavour models then tri-bimaximal\nneutrino mixing can be readily achieved \n\\cite{King:2005bj,deMedeirosVarzielas:2005ax}.\nThe SD mechanism has been widely applied to a large variety \nof unified flavour models \ninvolving three right-handed neutrinos \\cite{Antusch:2004gf},\nand it is therefore of interest to see how important the effects\nof flavour-dependent leptogenesis are in this more general framework.\n\nIn this paper we study thermal leptogenesis in classes of neutrino mass models \nbased on the seesaw mechanism with sequential\ndominance, taking into account lepton flavour in the Boltzmann equations. \nWe generalize the issue of \nincluding flavour effects in the Boltzmann equations to the \nsupersymmetric case. Within this class of model\nwe investigate constraints and expectations on the individual decay\nasymmetries and washout parameters from the present data on neutrino masses \nand mixings.\nFlavour effects are shown to have important consequences for the estimation \nof the produced baryon asymmetry in this class of models.\nFlavour-independent leptogenesis has previously been considered\nfor sequential dominance models in \\cite{Hirsch:2001dg}.\nThe present analysis clearly goes well beyond the previous\nanalysis by considering the important flavour-dependent effects.\nAlso the link between the leptogenesis phase and the MNS\nneutrino oscillation phase has been explored in sequential dominance in\n\\cite{King:2002qh}. Here we shall revisit this link in the light\nof flavour effects and obtain new links between the cosmology and\nneutrino oscillation physics.\n\nThe remainder of the paper is set out as follows: In\nSec.~\\ref{Sec:FML}, we review the treatment of flavour in the\nBoltzmann equations and generalize it to the case of the\nMSSM. Sec.~\\ref{SD} contains a review of sequential dominance (SD) in\nthe seesaw mechanism. In Sec.~\\ref{Sec:FlavourLGinSD}, we investigate\nleptogenesis in neutrino mass models with SD, taking into account\nlepton flavour in the Boltzmann equations. Generic properties of the\ndecay asymmetries and washout parameters are derived in\nSec.~\\ref{Sec:DecayAndWashout} and the link between the leptonic CP\nviolating phase $\\delta$, observable in neutrino oscillations, and the\nCP violation required for leptogenesis is established in\nSec.~\\ref{Sec:MNSLGlink}. Sec.~\\ref{examples} contains examples to\nillustrate our results. In Sec.~\\ref{concl} we conclude.\n\n\n\n\n\n\n\\section{Flavour Matters in Leptogenesis}\\label{Sec:FML}\n\\subsection{Temperatures where Flavour Matters}\nIn the SM extended by right-handed (singlet) neutrinos $N_i$ ($i=1,2,3$) with Majorana masses $M_i$, the additional terms of the Lagrangian are given by\n\\begin{eqnarray}\\label{Eq:L}\n \\mathcal{L} & = &\n- (\\lambda_\\nu)_{\\alpha i}({\\ell}^{\\alpha}\\cdot\n H)\\,  N^{i} \n - \\frac{1}{2} \\overline{N^{i}} (M_\\mathrm{RR})_{ij}\n N^{\\ChargeC j}\\;+\\;\\text{H.c.}\\;,\n\\end{eqnarray}\nwhere the dot indicates the $\\text{SU} (2)_\\mathrm{L}$-invariant product, \n$(\\ell^{\\alpha}\\cdot H) := \\ell_a^{\\alpha}(i\\tau_2)^{ab} ({H})_b$, \nwith $\\tau_A$ $(A\\in \\{1,2,3\\})$ being the Pauli matrices. \n$\\ell_\\alpha$ ($\\alpha=e,\\mu,\\tau$) are the lepton \nSU(2)$_\\mathrm{L}$-doublets and $H$ is the Higgs field which develops a vacuum expectation value (vev) of $\\<H^0\\> \\equiv v_\\mathrm{u} = 175$ GeV in its neutral component after electroweak symmetry breaking. We will work in a basis where the charged lepton Yukawa matrix and the mass matrix of the right-handed neutrinos are diagonal, i.e.\\ $\\lambda_e = \\mbox{diag}(y_e,y_\\mu,y_\\tau)$ and $M_\\mathrm{RR}= \\mbox{diag}(M_1,M_2,M_3)$, respectively. We will assume a hierarchical spectrum of right-handed neutrino masses, $M_1 \\ll M_2 \\ll M_3$, in the following. \n\nIn the SM, the flavour-independent formulae\nto describe leptogenesis  are only appropriate when \nthe dynamics takes place at \ntemperatures larger than about $10^{12}$ GeV, before the charged lepton Yukawa couplings\ncome into equilibrium, estimating the interaction rate for a Yukawa coupling $y_\\alpha$ as $\\Gamma_\\tau \\approx 5 \\times 10^{-3} \\,y_\\alpha^2 \\,T$ \\cite{Cline:1993bd}.  However, if leptogenesis occurs at smaller \ntemperatures $T\\sim M_1$, where $M_1$ is the mass of the lightest RH neutrino,\nthen one has to distinguish two possible cases. If \n $10^5 \\: \\mbox{GeV} \\ll M_1 \\ll 10^{9} \\: \\mbox{GeV}$,  \nthen charged $\\mu$ and $\\tau$ Yukawa couplings are in thermal equilibrium \nand all flavours in the Boltzmann equations are to be \ntreated separately. For \n$10^9 \\: \\mbox{GeV} \\ll M_1 \\ll 10^{12} \\: \\mbox{GeV}$, only the $\\tau$ Yukawa coupling is in equilibrium and is treated separately in the Boltzmann equations, while\nthe $e$ and $\\mu$ flavours are indistinguishable.\n\nIn the MSSM, extended by singlet superfields $\\SuperField{N}^\\mathrm{C}_i$ ($i=1,2,3$) containing the right-handed neutrinos $N^i$ as fermionic components, we use a notation analogous to the SM. \nThe additional terms of the superpotential are given by  \n\\begin{eqnarray}\\label{Eq:W}\n \\mathcal{W} & = &\n(\\lambda_\\nu)_{\\alpha \\Ng}(\\SuperField{\\ell}^{\\alpha}\\cdot\n \\SuperField{H}_\\mathrm{u})\\,  \\SuperField{N}^{\\mathrm{C} j} \n + \\frac{1}{2} \\SuperField{N}^{\\mathrm{C} i} (M_\\mathrm{RR})_{ij}\n \\SuperField{N}^{\\mathrm{C} j}\\;,\n\\end{eqnarray}\nwhere hats denote superfields. In the MSSM, the vev of the Higgs field $H_\\mathrm{u}$, which couples to the right-handed neutrinos, is given by $\\<H_\\mathrm{u}^0\\> \\equiv v_\\mathrm{u} = \\sin (\\beta) \\times 175$ GeV, with $\\tan \\beta$ defined as usual as the ratio of the vevs of the Higgs fields which couple to up-type quarks (and right-handed neutrinos) and down-type quarks (and charged leptons). \n\nIn the MSSM, the flavour-independent formulae can only be applied for temperatures larger than $(1+\\tan^2 \\beta)\\times 10^{12}$ GeV, since \nthe squared charged lepton Yukawa couplings in the MSSM are multiplied by this factor. Consequently, \ncharged $\\mu$ and $\\tau$ lepton Yukawa couplings are in thermal equilibrium for\n $(1+\\tan^2 \\beta)\\times 10^5 \\: \\mbox{GeV} \\ll M_1 \\ll (1+\\tan^2 \\beta)\\times 10^{9} \\: \\mbox{GeV}$  \nand all flavours in the Boltzmann equations are to be \ntreated separately. For \n$(1+\\tan^2 \\beta)\\times 10^9 \\: \\mbox{GeV} \\ll M_1 \\ll (1+\\tan^2 \\beta)\\times 10^{12} \\: \\mbox{GeV}$, only the $\\tau$ Yukawa coupling is in equilibrium and only the $\\tau$ flavour is treated separately in the Boltzmann equations, while the $e$ and $\\mu$ flavours are indistinguishable.\n\nIn what follows we will concern ourselves mainly with the regime where all flavours in the Boltzmann equations are to be treated separately, i.e.\\ \n$10^5 \\: \\mbox{GeV} \\ll M_1 \\ll 10^{9} \\: \\mbox{GeV}$ in the SM and \n $(1+\\tan^2 \\beta)\\times 10^5 \\: \\mbox{GeV} \\ll M_1 \\ll (1+\\tan^2 \\beta)\\times 10^{9} \\: \\mbox{GeV}$ in the MSSM. We will comment on the other two regimes and point out the differences between flavour-independent approximation and the flavour-dependent treatment, where lepton flavour is taken into account correctly in the Boltzmann equations. We start with the SM and then turn to the MSSM. \n\n\\subsection{The Boltzmann Equations in the SM}\\label{SM}\nIn the regime where all lepton flavours are to be treated seperately, the Boltzmann equations in the SM are given by \n\\begin{eqnarray}\n\\label{1a}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\frac{\\mathrm{d} Y_{N_1}}{\\mathrm{d} z} \\!\\!&=&\\!\\! \n  \\frac{-z}{s H(M_1)} (\\gamma_D + \\gamma_{\\mathrm{S},\\Delta L = 1})\n\\left(  \\frac{Y_{N_1}}{Y^\\mathrm{eq}_{N_1}} - 1  \\right)\\!,\\\\\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\frac{\\mathrm{d} Y_{\\Delta_\\alpha}}{\\mathrm{d} z} \\!\\!&=&\\!\\! \n \\frac{-z}{s H(M_1)}  \\!\\left[\n\\varepsilon_{1,\\alpha}   (\\gamma_D + \\gamma_{\\mathrm{S},\\Delta L = 1}) \\left(  \\frac{Y_{N_1}}{Y^\\mathrm{eq}_{N_1}} - 1  \\right) -\n(\\frac{\\gamma^{\\alpha}_D}{2} + \\gamma^{\\alpha}_{\\mathrm{W},\\Delta L = 1})\\, \\frac{\\sum_\\beta A_{\\alpha\\beta} Y_{\\Delta_\\beta}}{Y^\\mathrm{eq}_\\ell}\n\\right]\\!,\n\\label{2a}\n\\end{eqnarray} \nwhere there is no sum over $\\alpha$ in the last term on the right-side of Eq.~(\\ref{2a}) and where $z = M_1\/T$ with $T$ being the temperature. \n$Y_{N_1}$ is the density of the lightest right-handed neutrino $N_1$ with mass $M_1$.  \n$Y_{\\Delta_\\alpha}$ are defined as $Y_{\\Delta_\\alpha}\\equiv Y_B\/3 - Y_{L_\\alpha}$, where  \n$Y_{L_\\alpha}$ are the total lepton number densities for the flavours $\\alpha = e,\\mu,\\tau$ and where $Y_B$ is the total baryon density. \nIt is appropriate to solve the Boltzmann equations for $Y_{\\Delta_\\alpha}$ instead of for the number densities $Y_{\\alpha}$ of the lepton doublets $\\ell_\\alpha$, since $\\Delta_\\alpha \\equiv B\/3 - L_\\alpha$ is conserved by sphalerons and by the other SM interactions. \nFor all number densities $Y$, normalization to the entropy density $s$ is understood. \n$Y^\\mathrm{eq}_{N_1}$ and $Y^\\mathrm{eq}_\\ell$ stand for the corresponding equilibrium number densities.  \n$\\gamma_D$ is the thermally averaged total decay rate of $N_1$ and \n$\\gamma_{\\mathrm{S},\\Delta L = 1}$ represents the rates for the $\\Delta L = 1$ scattering processes in the thermal bath. Notice, in particular, that\n$\\gamma_{\\mathrm{S},\\Delta L = 1}$ contributes to the asymmetry, as was \nrecently pointed out in \n\\cite{Abada:2006ea}. \n\nThe corresponding flavour-dependent rates for washout processes involving the lepton flavour $\\alpha$ are $\\gamma^{\\alpha}_D$ (from inverse decays involving leptons $\\ell_\\alpha$) and $\\gamma^{\\alpha}_{\\mathrm{W},\\Delta L = 1}$. For brevity, we have not \ndisplayed further contributions from $\\Delta L = 2$ scatterings, which \ncan be neglected under conditions we will specify below. \n$\\varepsilon_{1,\\alpha}$ is the decay asymmetry of $N_1$ and $H (T)$ is the Hubble parameter.\n\nThe matrix $A$, which appears in the washout term, is defined by $Y_\\alpha = \\sum_\\beta A_{\\alpha\\beta} \\,Y_{\\Delta_\\beta}$. The values of its elements depend on which interactions, in addition to the weak and strong sphalerons, are in thermal equilibrium at the temperatures where leptogenesis takes place. \nBelow $10^9$ GeV in the SM, $A$ is given by \\cite{Abada:2006ea}\n\\begin{eqnarray}\nA^\\mathrm{SM} =  \n\\begin{pmatrix}\n-151\/179 & 20\/179 & 20\/179 \\\\\n25\/358 & -344\/537 & 14\/537 \\\\\n25\/358 & 14\/537 & -344\/537\n\\end{pmatrix}.\n\\end{eqnarray} \nBetween $10^9$ and $10^{12}$ GeV in the SM, regarding the leptons only the interaction mediated by the $\\tau$ Yukawa coupling is in equilibrium, and the lepton asymmetries and $B\/3 - L_\\alpha$ asymmetries in the $e$ and $\\mu$ flavour can be combined to $Y_2 \\equiv Y_{e + \\mu}$ and $Y_{\\Delta_2} \\equiv Y_{\\Delta_e+\\Delta_\\mu}$. In this temperature range, $A$ is given by \\cite{Abada:2006ea}\n\\begin{eqnarray}\nA^\\mathrm{SM} =  \n\\begin{pmatrix}\n-920\/589 & 120\/589  \\\\\n30\/589 & -390\/589  \\\\ \n\\end{pmatrix}.\n\\end{eqnarray} \nAbove $10^{12}$ GeV in the SM we recover the flavour-independent treatment, where all asymmetries can be combined to $Y_{\\Delta} \\equiv Y_{\\Delta_e + \\Delta_\\mu +  \\Delta_\\tau}$, and $A$ is given by $A^\\mathrm{SM} = - 1$. \n\n\n\n\nEqs.~(\\ref{1a}) and (\\ref{2a})\ncan be safely used in the range  of temperatures in which the lepton Yukawa\nreactions for each flavour are fully in equilibrium. Indeed, for values\nof $M_1$ close to $10^9$ GeV in the SM \nthe reactions induced by  the muon \nYukawa coupling are about to be in equilibrium and the quantum oscillations\nof the asymmetries $Y_{e\\mu}$ might not have been dumped fast enough\nto be neglected. \nEqs.~(\\ref{2a}) may be generalized to include quantum oscillations,  \nfollowing Ref.~\\cite{davidsonetal}. \nHowever, preliminary numerical simulations have shown that \nthe off diagonal terms change the final diagonal lepton asymmetries \nby factors of order unity and therefore, from now on, we will safely\nneglect them and restrict \nourselves to Eqs.~(\\ref{1a}) and (\\ref{2a}). \n\n\nOne can implement the relevant parameters for connecting leptogenesis to neutrino models directly in the Boltzmann equations, following \\cite{Abada:2006ea}. Let us therefore introduce, in addition to the decay asymmetries $\\varepsilon_{1,\\alpha}$, the parameters $K_{\\alpha}$, which control the washout processes for the asymmetry in an individual lepton flavour $\\alpha$ and $K$, which controls the source of RH neutrinos in the thermal bath, as\n\\begin{eqnarray}\\label{Eq:Kaa}\nK_{\\alpha} \\equiv \\frac{\\Gamma_{N_1 \\ell_\\alpha} +\\Gamma_{N_1 \\overline\\ell_\\alpha}}{H(M_1)}\\;,\\quad \nK \\equiv \\sum_\\alpha  K_\\alpha\\;,\\quad \nK_\\alpha = K\n\\frac{(\\lambda_{\\nu}^{\\dagger})_{1\\alpha}(\\lambda_{\\nu})_{\\alpha 1}}\n{(\\lambda_{\\nu}^{\\dagger}\\lambda_{\\nu})_{11}}\\;.\n\\end{eqnarray}\n$H(M_1)$ is the Hubble parameter at $T=M_1$, given by \n$H(M_1) \\approx 1.66 \\sqrt{g_*} M_1^2\/M_p$ with $g^\\mathrm{SM}_*=106.75$ being the effective number of degrees of freedom in the SM, $\\lambda_{\\nu}$ denotes the neutrino Yukawa matrix (using left-right notation) and $\\Gamma_{N_1 \\ell_\\alpha}$ ($\\Gamma_{N_1 \\overline\\ell_\\alpha}$) is the decay rate of $N_1$ into Higgs and lepton doublet $\\ell_\\alpha$ (or conjugate final states, respectively).\nThe thermally averaged decay rate $\\gamma_D$ is then given in terms of $\\Gamma_{N_1 \\ell_\\alpha}$ by\n\\begin{eqnarray}\n\\gamma_D (z) = \\sum_\\alpha \\gamma_D^\\alpha \\; , \\quad \n\\gamma^\\alpha_D (z) = s \\,Y^\\mathrm{eq}_{N_1} \\,\\frac{K_1 (z)}{K_2 (z)}  \\, (\\Gamma_{N_1 \\ell_\\alpha}+\\Gamma_{N_1 \\overline\\ell_\\alpha}) \\; ,\n\\end{eqnarray} \nwhere $K_1$ and $K_2$ are the modified Bessel functions of the second kind. This allows to replace\n\\begin{eqnarray}\n\\frac{\\gamma_D (z)}{s H(M_1)} = K\\,\\frac{K_1 (z)}{K_2 (z)}  \\,Y^\\mathrm{eq}_{N_1}\\; , \\quad \n\\frac{\\gamma^\\alpha_D (z)}{s H(M_1)} = K_{\\alpha}\\,\\frac{K_1 (z)}{K_2 (z)}  \\,Y^\\mathrm{eq}_{N_1}\\; , \\;\n\\end{eqnarray} \nin Eqs.~(\\ref{1a}) and (\\ref{2a}). Defining in addition two functions $f_1$ and $f_2$ by  \n\\begin{eqnarray}\n\\gamma_D + \\gamma_{\\mathrm{S},\\Delta L = 1} \\equiv\n\\gamma_D f_1 \\; , \\quad\n\\frac{\\gamma^{\\alpha}_D}{2}  + \\gamma^{\\alpha}_{\\mathrm{W},\\Delta L = 1} \\equiv\n\\gamma^{\\alpha}_D f_2\\; ,\n\\end{eqnarray}\n we can re-write the Boltzmann equations with correct flavour treatment in a simplified form as follows \\cite{Abada:2006ea}:  \n\\begin{eqnarray}\n\\label{1}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\frac{\\mathrm{d} Y_{N_1}}{\\mathrm{d} z} \\!\\!&=&\\!\\! \n-\\,  K\\,z \\,\\frac{K_1 (z)}{K_2 (z)}\\, f_1 (z) \\,(Y_{N_1} - Y^\\mathrm{eq}_{N_1}) \\; ,\\\\\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\frac{\\mathrm{d} Y_{\\Delta_\\alpha}}{\\mathrm{d} z} \\!\\!&=&\\!\\! \n-\\,\\varepsilon_{1,\\alpha}\\, K\\,z \\,\\frac{K_1 (z)}{K_2 (z)}\\,  f_1 (z) \\,(Y_{N_1} - Y^\\mathrm{eq}_{N_1}) +\nK_{\\alpha} \\, z\\, \\frac{K_1 (z)}{K_2 (z)} \\, f_2 (z) \\,Y^\\mathrm{eq}_{N_1}\\,\n \\frac{\\sum_\\beta A_{\\alpha\\beta} Y_{\\Delta_\\beta}}{Y^\\mathrm{eq}_\\ell} \\!\\; .\n\\label{2}\n\\end{eqnarray}\nThe  function $f_1(z)$ accounts for the \npresence of $\\Delta L=1$ scatterings \nand $f_2(z)$ accounts for scatterings in the washout\nterm of the asymmetry \\cite{lept,ogen}.  \nIn our numerical computations we only include processes mediated by neutrino and top Yukawa couplings, following Ref.~\\cite{ogen}. This means, we neglect $\\Delta L = 1$ scatterings involving gauge bosons \\cite{Pilaftsis:2003gt,lept} and thermal corrections \\cite{lept}, but we take into account corrections from renormalization group running between electroweak scale and $M_1$ \\cite{barbieri,Antusch:2003kp}. We also neglect $\\Delta L=2$ scatterings, which is a good approximation as long as \n$K_\\alpha \\gg 10 \\times M_1 \/ (10^{14} \\:\\mbox{GeV})$ \n\\cite{Abada:2006ea}. \nFinally, according to the usual assumptions for computing the damping rates in the \nBoltzmann equations, i.e.\\ that elastic scattering rates are fast and that the phase space densities for both, fermions and scalars, can be approximated as \n$f(E_i,T)=(n_{i}\/n^{\\mathrm{eq}}_{i}) e^{-E_i\/T}$, where \n$n_i^\\mathrm{eq} = \\tfrac{g_i}{2\\pi} T m_i^2 K_2 (m_i\/T)$ with $g_i$ being the number of degrees of freedom, we use  \n\\begin{eqnarray}\n Y^{\\mathrm{eq}}_{\\ell} \\approx \\frac{45 }{ \\pi^4\n g_* } \\; , \\quad\n Y^{\\mathrm{eq}}_{N_1}(z) \\approx \\frac{45}{ 2 \\pi^4\n g_* } \\,z^2\\, K_2 (z)\\;.\n\\end{eqnarray}\t\n\nThe final lepton asymmetry in each flavour is governed by three sets of   \nparameters, which can be computed within a neutrino model: \n$\\varepsilon_{1,\\alpha},K_{\\alpha}$ \nand $K = \\sum_\\alpha K_{\\alpha}$. \n$\\varepsilon_{1,\\alpha}$ are the decay asymmetries of the lightest right-handed neutrino $N_1$ into Higgs $H_\\mathrm{u}$ and lepton doublet $\\ell_\\alpha$, defined as\n\\begin{eqnarray}\\label{Eq:DefEps1a}\n\\varepsilon_{1,\\alpha} = \n\\frac{\n\\Gamma_{N_1 \\ell_\\alpha} - \\Gamma_{N_1 \\overline \\ell_\\alpha}\n}{\n\\sum_\\alpha (\\Gamma_{N_1 \\ell_\\alpha} + \\Gamma_{N_1 \\overline \\ell_\\alpha})\n}\\; ,\n\\end{eqnarray} \nwith the decay rates \n$\\Gamma_{N_1 \\ell_\\alpha} = \\Gamma (N_1\\rightarrow H_\\mathrm{u} \\ell_\\alpha)$ and\n$\\Gamma_{N_1 \\overline \\ell_\\alpha} = \\Gamma (N_1\\rightarrow H^*_\\mathrm{u} \\overline \\ell_\\alpha)$.\nSU(2)$_\\mathrm{L}$-indices in the final state, not displayed explicitly, are summed over. \nIn the SM, the tree-level decay rates are   \n\\begin{eqnarray}\\label{Eq:treeSM}\n\\Gamma^\\mathrm{SM}_{N_1 \\ell_\\alpha} +\\Gamma^\\mathrm{SM}_{N_1 \\overline \\ell_\\alpha} = M_1\\,\\frac{(\\lambda_{\\nu}^{\\dagger})_{1\\alpha}(\\lambda_{\\nu})_{\\alpha 1}}{8 \\pi}\\; .\n\\end{eqnarray} \nThe decay asymmetry (arising at one-loop order) is \\cite{Covi:1996wh,earlier}\n\\begin{equation}\n\\epsilon^\\mathrm{SM}_{1,\\alpha}=\\frac{1}{8\\pi}\n\\frac{\\sum_{J=2,3}\\mathrm{Im}\\left[\n(\\lambda_{\\nu}^{\\dagger})_{1\\alpha}[\\lambda_{\\nu}^{\\dagger}\\lambda_{\\nu}]_{1J}(\\lambda_{\\nu}^T)_{J\\alpha}\n\\right]}\n{(\\lambda_{\\nu}^{\\dagger}\\lambda_{\\nu})_{11}}\\,\ng^\\mathrm{SM}\\left(\\frac{M_J^2}{M_1^2}\\right) , \\label{eq:epsaa}\n\\end{equation}\nwith the loop function $g$ in the SM given by \n\\begin{eqnarray}\ng^\\mathrm{SM}(x) = \\sqrt{x} \\left[\n\\frac{1}{1-x} + 1 - (1+x)\\ln\\left(\\frac{1+x}{x}\\right)\n\\right] \\stackrel{x \\gg 1}{\\longrightarrow} - \\frac{3}{2 \\sqrt{x}}\\;.\n\\end{eqnarray}\nAlternatively to $K_\\alpha$ and $K$ defined in Eq.~(\\ref{Eq:Kaa}), the parameters $\\widetilde m_{1,\\alpha}$ and \n$\\widetilde m_1$ will be used in following, which we define as\n\\begin{eqnarray}\\label{Eq:mtildeaa}\\label{eq:mtildeaa}\n\\widetilde{m}_{1,\\alpha } \\equiv (\\lambda_{\\nu}^{\\dagger})_{1 \\alpha}(\\lambda_{\\nu})_{\\alpha 1}\\frac{v_{\\rm u}^2}{M_1}\n \\;  , \\quad\n\\widetilde{m}_1 \\equiv \\sum_\\alpha \\widetilde{m}_{1,\\alpha }\\; ,\n\\end{eqnarray} \nwith $v_\\mathrm{u}=175\\:\\mbox{GeV}$. They are related to $K_\\alpha$ and $K$ by\n\\begin{eqnarray}\\label{Eq:mStar}\nK = \\frac{\\widetilde{m}_1}{m^*}\\, ,\\;\\: \\mbox{or equivalently}\\;\\:\nK_\\alpha = \\frac{\\widetilde{m}_{1,\\alpha}}{m^*},\\;\\: \\mbox{with} \\;\\:\nm_\\mathrm{SM}^* \\approx 1.08 \\times 10^{-3} \\ {\\rm eV}\n\\, .\n\\end{eqnarray}\n\n\\subsection{The Boltzmann Equations in the MSSM}\\label{MSSM}\nIn the MSSM, the density $Y_{\\widetilde N_1}$ of right-handed sneutrinos as well as the densities $Y_{\\widetilde \\alpha}$ of the slepton doublets have to be included in the Boltzmann equations.\nDenoting the total (particle and sparticle) $B\/3 - L_\\alpha$ asymmetries \nas $\\hat Y_{\\Delta_\\alpha}$, the simplified Boltzmann equations are given by \n\\begin{eqnarray}\n\\label{1s}\n\\frac{\\mathrm{d} Y_{N_1}}{\\mathrm{d} z} &=& \n- \\, 2 K\\,z \\,\\frac{K_1 (z)}{K_2 (z)}\\, f_1 (z) \\,(Y_{N_1} - Y^\\mathrm{eq}_{N_1}) \\; ,\\\\\n\\label{2s}\\frac{\\mathrm{d} Y_{\\widetilde N_1}}{\\mathrm{d} z} &=& \n- \\, 2 K\\,z \\,\\frac{K_1 (z)}{K_2 (z)}\\, f_1 (z) \\,(Y_{\\widetilde N_1} - Y^\\mathrm{eq}_{\\widetilde N_1}) \\; ,\\\\\n\\label{3s}  \\frac{\\mathrm{d} \\hat Y_{\\Delta_\\alpha}}{\\mathrm{d} z} &=& \n-\\;(\\varepsilon_{1,\\alpha}+\\varepsilon_{1,\\widetilde \\alpha}) \\,K\\, z \\, \n\\frac{K_1 (z)}{K_2 (z)}\\,  f_1 (z) \\,(Y_{N_1} - Y^\\mathrm{eq}_{N_1})\\nonumber \\\\\n&&-\\;\n(\\varepsilon_{\\widetilde 1,\\alpha}+\\varepsilon_{\\widetilde 1,\\widetilde \\alpha}) \\,K\\, z \\, \n\\frac{K_1 (z)}{K_2 (z)}\\,  f_1 (z) \\,(Y_{\\widetilde N_1} - Y^\\mathrm{eq}_{\\widetilde N_1})\n\\nonumber \\\\\n&&+\\;\nK_{\\alpha} \\, z\\, \\frac{K_1 (z)}{K_2 (z)} \\, f_2 (z) \\,\n \\frac{\\sum_\\beta  A_{\\alpha\\beta} \\hat Y_{\\Delta_\\beta}}{\\hat Y^\\mathrm{eq}_{\\alpha}} \\,(Y^\\mathrm{eq}_{N_1}+Y^\\mathrm{eq}_{\\widetilde N_1})\n\\;  .\n\\end{eqnarray}\nThe matrix $A$ is defined via the relation\n$\\hat Y_{\\alpha} = \\sum_\\beta A_{\\alpha\\beta} \\, \\hat Y_{\\Delta_\\alpha}$, with  \n$\\hat Y_\\alpha \\equiv Y_\\alpha + Y_{\\widetilde \\alpha}$ being the combined densities for lepton and slepton doublets. \n\n\nTo obtain Eqs.~(\\ref{1s}) - (\\ref{3s}) we have made use of the fact that the tree level decay rates satisfy\n\\begin{eqnarray}\\label{Eq:treeMSSM}\n\\Gamma_{N_1 \\ell_\\alpha} + \\Gamma_{N_1 \\overline \\ell_\\alpha}\n\\,=\\,\n\\Gamma_{N_1 \\widetilde \\ell_\\alpha} + \\Gamma_{N_1 \n\\widetilde{\\ell}_\\alpha^*}\n\\,=\\,\n\\Gamma_{\\widetilde N_1^* \\ell_\\alpha}\n\\,=\\,\n\\Gamma_{\\widetilde N_1 \\overline \\ell_\\alpha}\n\\,=\\,\n\\Gamma_{\\widetilde N_1 \\widetilde \\ell_\\alpha}\n\\,=\\,\n\\Gamma_{\\widetilde N_1^* \\widetilde{\\ell}_\\alpha^*}\\; ,\n\\end{eqnarray}\nwith $\\Gamma_{N_1 \\ell_\\alpha} \\!+\\! \\Gamma_{N_1 \\overline \\ell_\\alpha}$ given in Eq.~(\\ref{Eq:treeSM}), \nleading to the identities\n\\begin{eqnarray}\nK_{\\alpha} \n=\\! \\frac{\\Gamma_{N_1 \\ell_\\alpha} \\!\\!+\\!\\Gamma_{N_1 \\overline\\ell_\\alpha}}{H(M_1)}\n\\!=\\! \\frac{\\Gamma_{N_1 \\widetilde \\ell_\\alpha} \\!\\!+\\! \\Gamma_{N_1 \n\\widetilde{\\ell}_\\alpha^*}}{H(M_1)} \n= \\frac{\\Gamma_{\\widetilde N_1^* \\ell_\\alpha}}{H(M_1)}\n= \\frac{\\Gamma_{\\widetilde N_1 \\overline \\ell_\\alpha}}{H(M_1)}\n= \\frac{\\Gamma_{\\widetilde N_1 \\widetilde \\ell_\\alpha}}{H(M_1)}\n= \\frac{\\Gamma_{\\widetilde N_1^* \\widetilde{\\ell}_\\alpha^*}}{H(M_1)} ,\n\\end{eqnarray}\nwith $K$, $K_{\\alpha}$ (and $\\widetilde m_1$, $\\widetilde m_{1,\\alpha}$) being defined  analogously to the SM case (c.f.\\ Eqs.~(\\ref{Eq:Kaa}) and (\\ref{Eq:mtildeaa})).\nFrom Eq.~(\\ref{Eq:mStar}), we find \n\\begin{eqnarray}\nm_\\mathrm{MSSM}^*\\approx \\sin^2(\\beta) \\times 1.58 \\times 10^{-3} \\ {\\rm eV}  \\; ,\n\\end{eqnarray}\nusing $g^{\\mathrm{MSSM}}_*=228.75$ for computing $H(M_1)$. $m_\\mathrm{MSSM}^*$ relates $K$, $K_{\\alpha}$ to $\\widetilde m_1$, $\\widetilde m_{1,\\alpha}$ in the MSSM by the analogous of Eq.~(\\ref{Eq:mStar}).\nIn this conventions, the functions $f_1$ and $f_2$ in the Boltzmann equations are approximately unchanged (apart from an obvious modification in one of the scattering terms, which has only small effects). \nNote that $v_u = \\sin (\\beta) \\times 175$ GeV in the MSSM. \nIn Eqs.~(\\ref{1s}) and (\\ref{2s}), processes with particles and superpartners are combined, leading to the additional factor of $2$ on the right-side of the equations.\nIgnoring supersymmetry breaking, the right-handed neutrinos and sneutrinos\n have equal mass $M_{1}$. \nWith the usual \napproximation of taking Boltzmann statistics for both, fermions and scalars, \nwe have used $Y^{\\mathrm{eq}}_{\\widetilde \\ell} \\approx Y^{\\mathrm{eq}}_{\\ell}$ to combine the washout terms for leptons and sleptons in the last term in Eq.~(\\ref{3s}). Correspondingly, for the  \ndensity $Y^\\mathrm{eq}_{\\widetilde N_1}$ of the right-handed sneutrinos we use   \n\\begin{eqnarray}\n Y^{\\mathrm{eq}}_{\\widetilde N_1}(z) \\approx \\frac{45}{ 2 \\pi^4\n g_* } \\,z^2\\, K_2 (z)\\;.\n\\end{eqnarray}\t\n\n\n$\\varepsilon_{1,\\alpha}$, $\\varepsilon_{1,\\widetilde \\alpha}$, $\\varepsilon_{\\widetilde 1,\\alpha}$ and $\\varepsilon_{\\widetilde 1,\\widetilde \\alpha}$ are the decay asymmetries for the decay of  \n neutrino into Higgs and lepton,\n neutrino into Higgsino and slepton,\n sneutrino into Higgsino and lepton, and \n sneutrino into Higgs and slepton, respectively,  \ndefined by \n\\begin{eqnarray}\\label{Eq:EpsMSSM_def}\n\\varepsilon_{1,\\alpha} \\!\\!\\!&=&\\!\\!\\!\n\\frac{\n\\Gamma_{N_1 \\ell_\\alpha} - \\Gamma_{N_1 \\overline \\ell_\\alpha}\n}{\n\\sum_\\alpha (\\Gamma_{N_1 \\ell_\\alpha} + \\Gamma_{N_1 \\overline \\ell_\\alpha})\n}\\; , \\quad\n\\varepsilon_{1,\\widetilde \\alpha} =\n\\frac{\n\\Gamma_{N_1 \\widetilde \\ell_\\alpha} - \\Gamma_{N_1\n\\widetilde{\\ell}_\\alpha^*}\n}{\n\\sum_\\alpha (\\Gamma_{N_1 \\widetilde \\ell_\\alpha} + \\Gamma_{N_1\n\\widetilde{\\ell}_\\alpha^*})\n}\\; , \\nonumber  \\\\\n\\varepsilon_{\\widetilde 1,\\alpha} \\!\\!\\!&=&\\!\\!\\!\n\\frac{\n\\Gamma_{\\widetilde N^*_1 \\ell_\\alpha} - \\Gamma_{\\widetilde N_1 \\overline\n  \\ell_\\alpha}\n}{\n\\sum_\\alpha (\\Gamma_{\\widetilde N^*_1 \\ell_\\alpha} +\n\\Gamma_{\\widetilde N_1 \\overline \\ell_\\alpha})\n}\\; , \\quad\n\\varepsilon_{\\widetilde 1,\\widetilde \\alpha} =\n\\frac{\n\\Gamma_{\\widetilde N_1 \\widetilde \\ell_\\alpha} - \\Gamma_{\\widetilde N^*_1\n\\widetilde{\\ell}_\\alpha^*}\n}{\n\\sum_\\alpha (\\Gamma_{\\widetilde N_1 \\widetilde \\ell_\\alpha} +\n\\Gamma_{\\widetilde N^*_1\n\\widetilde{\\ell}_\\alpha^*})\n}\\;.\n\\end{eqnarray}\nIn the MSSM, the four decay asymmetries are equal,\n$\\varepsilon^\\mathrm{MSSM}_{1,\\alpha} = \n\\varepsilon^\\mathrm{MSSM}_{1,\\widetilde \\alpha} = \n\\varepsilon^\\mathrm{MSSM}_{\\widetilde 1,\\alpha} = \n\\varepsilon^\\mathrm{MSSM}_{\\widetilde 1,\\widetilde \\alpha}$, \nand given by \\cite{Covi:1996wh,earlier}\n\\begin{eqnarray}\n\\varepsilon^\\mathrm{MSSM}_{1,\\alpha} = \n\\frac{1}{8\\pi}\n\\frac{\\sum_{J=2,3}\\mathrm{Im}\\left[\n(\\lambda_{\\nu}^{\\dagger})_{1\\alpha}[\\lambda_{\\nu}^{\\dagger}\\lambda_{\\nu}]_{1J}(\\lambda_{\\nu}^T)_{J\\alpha}\n\\right]}\n{(\\lambda_{\\nu}^{\\dagger}\\lambda_{\\nu})_{11}}\\,\ng^\\mathrm{MSSM}\\left(\\frac{M_J^2}{M_1^2}\\right) , \n\\end{eqnarray}\nwith\n\\begin{eqnarray}\ng^\\mathrm{MSSM}(x) =\\sqrt{x} \\left[\n\\frac{2}{1-x} - \\ln\\left(\\frac{1+x}{x}\\right)\\right]\n\\stackrel{x \\gg 1}{\\longrightarrow} - \\frac{3}{\\sqrt{x}}  \\; .\n\\end{eqnarray}\nThe matrix $A$ depends on which MSSM interactions are in thermal equilibrium at the temperatures where leptogenesis takes place. \nBelow $(1+\\tan^2 \\beta)\\times 10^9$ GeV, where the Boltzmann equations are solved for the individual asymmetries $\\hat Y_{\\Delta_e}$, $\\hat Y_{\\Delta_\\mu}$ and $\\hat Y_{\\Delta_\\tau}$, $A$ is given by\n\\begin{eqnarray}\nA^\\mathrm{MSSM} =  \n\\begin{pmatrix}\n-93\/110 & 6\/55 & 6\/55 \\\\\n3\/40 & -19\/30 & 1\/30 \\\\\n3\/40 & 1\/30 & -19\/30\n\\end{pmatrix}.\n\\end{eqnarray} \nBetween $(1+\\tan^2 \\beta)\\times 10^9$ and $(1+\\tan^2 \\beta)\\times 10^{12}$ GeV, where the relevant flavour-dependent asymmetries are $\\hat Y_{\\Delta_2} \\equiv \\hat Y_{\\Delta_e + \\Delta_\\mu}$ and $\\hat Y_{\\Delta_\\tau}$, we find \n\\begin{eqnarray}\nA^\\mathrm{MSSM} =  \n\\begin{pmatrix}\n-541\/761 & 152\/761  \\\\\n46\/761   & -494\/761  \\\\ \n\\end{pmatrix},\n\\end{eqnarray}    \nand above $(1+\\tan^2 \\beta)\\times 10^{12}$ GeV, we recover the flavour-independent treatment with $A^\\mathrm{MSSM} = - 1$. \n\n\n\\subsection{Solving the Boltzmann Equations in the SM and MSSM}\nSolving the Boltzmann equations, for $z$ from $0$ to $\\infty$, in the\nSM or in the MSSM yields the final $B\/3 - L_\\alpha$ asymmetries $Y^\\mathrm{SM}_{\\Delta_\\alpha}$ or\n$\\hat Y^\\mathrm{MSSM}_{\\Delta_\\alpha}$  \nin the individual flavours. \nIt is convenient to parameterize the produced asymmetries in terms of \nan efficiency factor $\\eta_{\\alpha}$ which, in the approximation that the small off-diagonal elements of $A$ are neglected, is a function\nof $A_{\\alpha\\alpha}K_{\\alpha}$ (no sum over $\\alpha$) and $K$, i.e.\\ $\\eta_{\\alpha} = \\eta (A_{\\alpha\\alpha}K_{\\alpha},K)$, as\n\\begin{eqnarray}\\label{Eq:eta_aa}\nY^\\mathrm{SM}_{\\Delta_\\alpha} &=& \\eta^\\mathrm{SM}_{\\alpha} \\,\\varepsilon^\\mathrm{SM}_{1,\\alpha} \\,\nY^{\\mathrm{eq}}_{N_1}(z\\ll 1)\\;, \\\\\n\\label{Eq:eta_aa_MSSM} \\hat Y^\\mathrm{MSSM}_{\\Delta_\\alpha} &=& \\eta^\\mathrm{MSSM}_{\\alpha} \\,\n\\varepsilon^\\mathrm{MSSM}_{1,\\alpha}\\, \\left[ \nY^{\\mathrm{eq}}_{N_1}(z\\ll 1) +Y^{\\mathrm{eq}}_{\\widetilde N_1}(z\\ll 1)\n\\right] ,\\label{Eq:eta_aa_MSSM}\n\\end{eqnarray}\ngeneralizing the notation of \\cite{lept} to the flavour-dependent treatment. \nBeyond the approximations of Secs.~\\ref{SM} and \\ref{MSSM}, the efficiency factors also depend on $M_1$, and they depend on $\\tan \\beta$ in the MSSM.\n  $Y^{\\mathrm{eq}}_{N_1}(z\\ll 1)$ and \n $Y^{\\mathrm{eq}}_{\\widetilde N_1}(z\\ll 1)$ are the \n number densities of the neutrino and sneutrino at $T \\gg M_{1}$, \n if they were in thermal \n equilibrium, normalized with respect to the entropy\n density. In the Boltzmann approximation, they are given by\n\\begin{eqnarray}\n Y^{\\mathrm{eq}}_{N_1}(z\\ll 1) \\;\\approx \\;Y^{\\mathrm{eq}}_{\\widetilde N_1}(z\\ll 1) \\;\\approx\\; \\frac{45}{ \\pi^4\n g_* }.\n \\end{eqnarray}\nEqs.~(\\ref{Eq:eta_aa}) and (\\ref{Eq:eta_aa_MSSM}) define the flavour-dependent efficiency factor in the SM and in the MSSM, and it can be computed by means of the Boltzmann equations \\cite{davidsonetal,Abada:2006ea} and its MSSM generalizations in Eqs.~(\\ref{1s}) - (\\ref{3s}), where lepton flavour is taken into account correctly. \nThe equilibrium number densities in Eqs.~(\\ref{Eq:eta_aa}) and (\\ref{Eq:eta_aa_MSSM}) serve as a normalization.  \nA thermal population $N_1$ (and $\\widetilde N_1$) decaying completely out of equilibrium (without washout effects) would lead to $\\eta_\\alpha = 1$.\nOf course, $K\/K_{\\alpha} \\ge 1$ always holds by definition \nbut $K$ can be significantly larger than $K_{\\alpha}$. $\\eta(A_{\\alpha\\alpha}K_{\\alpha},K)$ is shown as a function of $A_{\\alpha\\alpha}K_\\alpha$ in the MSSM for fixed values of $K\/|A_{\\alpha\\alpha}K_{\\alpha}| = 2,5$ and $100$ in Fig.~\\ref{fig:eta}. In the SM, $\\eta$ has the same qualitative (and a similar quantitative) behavior. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=1,angle=0]{eta_MSSM.eps}\n \\caption{\\label{fig:eta}\nFlavour-dependent efficiency factor $\\eta(A_{\\alpha\\alpha}K_{\\alpha},K)$ in the MSSM as a function of $A_{\\alpha\\alpha}K_{\\alpha}$, for fixed values of $K\/|A_{\\alpha\\alpha}K_{\\alpha}| = 2,5$ and $100$, obtained from solving the flavour-dependent Boltzmann equations \\cite{Abada:2006ea} generalized to the MSSM (with $\\tan \\beta = 50$, as an example), as displayed in Eqs.~(\\ref{1s}) - (\\ref{3s}). \nFor larger $K\/|A_{\\alpha\\alpha}K_{\\alpha}|$ the plot looks virtually like for $K\/|A_{\\alpha\\alpha}K_{\\alpha}| = 100$. More relevant than the differences in the flavour-dependent efficiency factors for different $K\/|A_{\\alpha\\alpha}K_{\\alpha}|$ is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factor. As explained in the text, this can change the amount of the produced baryon asymmetry dramatically, compared to the flavour-independent approximation \n\\cite{davidsonetal,nardietal,Abada:2006ea}. }\n\\end{figure}\n\n\nWhat is more relevant than the differences in the flavour-dependent efficiency factors (c.f.\\ Fig.~\\ref{fig:eta}) is that the total baryon asymmetry is the sum of each individual lepton asymmetries, which is weighted by the corresponding efficiency factor \\cite{davidsonetal,nardietal,Abada:2006ea}. Therefore,\nupon summing over the lepton asymmetries, the total baryon number is\ngenerically not proportional to the sum over the\nCP asymmetries, \n$\\varepsilon_1 = \\sum_\\alpha \\varepsilon_{1,\\alpha}$\nas in the flavour-independent \napproximation where the lepton flavour is \nneglected in the Boltzmann equations. In Eq.~(\\ref{2}), this corresponds \nto replacing $Y_{\\Delta_\\alpha}$ by the ``one-single'' flavour \n$Y_\\Delta= Y_{\\Delta_e + \\Delta_\\mu + \\Delta_\\tau}$ (the total lepton asymmetry), \nthe flavour \ndependent decay asymmetries by $\\varepsilon_1$, the washout \nparameters by \n$K = \\sum_\\alpha K_{\\alpha}$, and the matrix $A$ by $-1$. \nThe produced asymmetry in the flavour-independent approximation is then given by\n$Y = \\varepsilon_1 \\,\\eta^\\mathrm{ind} (K)\\,Y^\\mathrm{eq}_{N_1}(z\\ll 1)$, \nwhere the flavour-independent efficiency factor $\\eta^\\mathrm{ind} (K)$ is related to \n$\\eta (A_{\\alpha\\alpha}K_{\\alpha},K)$ in Eq.~(\\ref{Eq:eta_aa}) by \n$\\eta^\\mathrm{ind} (x) = \\eta(-x,x)$. \nIn other words, in the flavour-independent approximation the total baryon asymmetry is a function of\n$\\left(\\sum_\\alpha \\varepsilon_{1,\\alpha}\\right)\\times \\eta^\\mathrm{ind}\\ (\\sum_\\beta \nK_{\\beta})$. In the correct flavour treatment\nthe baryon asymmetry is a function of $\\sum_\\alpha \n\\varepsilon_{1,\\alpha}\\eta\\left(A_{\\alpha\\alpha}K_{\\alpha},K\\right)$.\nIf $N_1$ decays about equally to all flavours and produces\nabout the equal asymmetry in all flavours, then (neglecting the effects of $A$ for the moment) one expects that the \nflavour-independent approximation underestimates the asymmetry by roughly a \nfactor of three for the case of \nstrong washout and overestimates by a factor of three for the case of weak wash out. \nThe approximate factor of three can be understood with the analytic approximations presented in \\cite{Abada:2006ea}, \nfrom which we can see that in the case of weak (strong) washout for all flavours, the efficiencies are roughly (inverse) proportional to $K_{\\alpha}$.\nHowever, as we will see, there are situations in the SD \nmodels where the difference is much more dramatic.\n\n\n\nLet us note at this point how to generalize the above discussion for the range \n$10^9 \\: \\mbox{GeV} \\ll M_1 \\ll 10^{12} \\: \\mbox{GeV}$ in the SM and  \n$(1+\\tan^2 \\beta)\\times 10^9 \\: \\mbox{GeV} \\ll M_1 \\ll (1+\\tan^2 \\beta)\\times 10^{12} \\: \\mbox{GeV}$ in the MSSM, where only the $\\tau$ is in equilibrium and is treated separately in the Boltzmann equations, while the $e$ and $\\mu$ flavours are indistinguishable. In this case, following \\cite{davidsonetal,Abada:2006ea}, one can combine the asymmetries for the $e$ and $\\mu$ flavours to a combined density $Y_{\\Delta_2} = Y_{\\Delta_e + \\Delta_\\mu}$ in the Boltzmann equations, where $K_\\alpha$ is substituted by $K_2 = K_e + K_\\mu$. The corresponding decay asymmetry is $\\varepsilon_{1,2} = \\varepsilon_{1,e} + \\varepsilon_{1,\\mu}$. \nAbove $10^{12} \\: \\mbox{GeV}$ in the SM and above $(1+\\tan^2 \\beta)\\times 10^{12} \\: \\mbox{GeV}$ in the MSSM, we can combine all asymmetries for the $e,\\mu$ and $\\tau$  flavours to a combined density $Y_\\Delta = Y_{\\Delta_e + \\Delta_\\mu + \\Delta_\\tau}$ in the Boltzmann equations and substitute $K_\\alpha$ by $K = K_e + K_\\mu + K_\\tau$. The decay asymmetry then reduces to the flavour-independent one, $\\varepsilon_{1} = \\varepsilon_{1,e} + \\varepsilon_{1,\\mu} + \\varepsilon_{1,\\tau}$.\n\n\n\nThe produced lepton asymmetries are partly converted into a final baryon asymmetry by \nsphalerons. For all temperature ranges, the produced baryon asymmetry (normalized to the entropy density) can be computed from the densities $Y^\\mathrm{SM}_{\\Delta_\\alpha}$ and $\\hat Y^\\mathrm{MSSM}_{\\Delta_\\alpha}$ in the SM and MSSM, respectively, as\n\\begin{eqnarray}\\label{Eq:YB3f}\nY^\\mathrm{SM}_B &=& \\frac{12}{37} \\sum_\\alpha Y^\\mathrm{SM}_{\\Delta_\\alpha} \\; ,\\\\\nY^\\mathrm{MSSM}_B &=& \\frac{10}{31} \\sum_\\alpha \\hat Y^\\mathrm{MSSM}_{\\Delta_\\alpha}\\;. \n\\end{eqnarray} \n\n\n\n\\section{Sequential Dominance}\\label{SD}\n\nTo understand how sequential dominance works, we begin by\nwriting the right-handed neutrino Majorana mass matrix $M_{\\mathrm{RR}}$ in\na diagonal basis as\n\\begin{equation}\nM_{\\mathrm{RR}}=\n\\begin{pmatrix}\nM_A & 0 & 0 \\\\\n0 & M_B & 0 \\\\\n0 & 0 & M_C%\n\\end{pmatrix}.\n\\end{equation}\nNote that, as stated earlier, we work in a basis where the charged lepton Yukawa matrix \nis diagonal. \nIn this basis we write the neutrino (Dirac) Yukawa matrix $\\lambda_{\\nu}$ in\nterms of $(1,3)$ column vectors $A_i,$ $B_i,$ $C_i$ as\n\\begin{equation}\n\\lambda_{\\nu }=\n\\begin{pmatrix}\nA & B & C\n\\end{pmatrix},\n  \\label{Yukawa}\n\\end{equation}\nusing left-right convention as in Eqs.~(\\ref{Eq:L}) and (\\ref{Eq:W}). \nThe Dirac neutrino mass matrix is then given by $m_{\\mathrm{LR}}^{\\nu}=\\lambda_{\\nu}v_{\\mathrm{\nu}}$. The term for the light neutrino masses in the effective Lagrangian (after electroweak symmetry breking), resulting from integrating out the massive right\nhanded neutrinos, is\n\\begin{equation}\n\\mathcal{L}^\\nu_{eff} = \\frac{(\\nu_{i}^{T} A_{i})(A^{T}_{j} \\nu_{j})}{M_A}+\\frac{(\\nu_{i}^{T} B_{i})(B^{T}_{j} \\nu_{j})}{M_B}\n+\\frac{(\\nu_{i}^{T} C_{i})(C^{T}_{j} \\nu_{j})}{M_C}  \\label{leff}\n\\end{equation}\nwhere $\\nu _{i}$ ($i=1,2,3$) are the left-handed neutrino fields.\nSequential dominance then corresponds to the third\nterm being negligible, the second term subdominant and the first term\ndominant:\n\\begin{equation}\\label{SDcond}\n\\frac{A_{i}A_{j}}{M_A} \\gg\n\\frac{B_{i}B_{j}}{M_B} \\gg\n\\frac{C_{i}C_{j}}{M_C} \\, .\n\\label{SD1}\n\\end{equation}\nIn addition, we shall shortly see that small $\\theta_{13}$ \nand almost maximal $\\theta_{23}$ require that \n\\begin{equation}\n|A_1|\\ll |A_2|\\approx |A_2|.\n\\label{SD2}\n\\end{equation}\nWithout loss of generality, then, we shall label the dominant\nright-handed neutrino and Yukawa couplings as $A$, the subdominant\nones as $B$, and the almost decoupled (subsubdominant) ones as $C$. \nNote that the mass ordering of right-handed neutrinos is \nnot yet specified. Again without loss of generality we shall \norder the right-handed neutrino masses as $M_1<M_2<M_3$,\nand subsequently identify $M_A,M_B,M_C$ with $M_1,M_2,M_3$\nin all possible ways. The following results for the masses and mixing\nangles are independent of the mass ordering of the right-handed neutrinos.\n\nWriting \n$A_\\alpha = |A_\\alpha| e^{i \\phi_{A_1}}$, \n$B_\\alpha = |B_\\alpha| e^{i \\phi_{B_1}}$, \n$C_\\alpha = |C_\\alpha| e^{i \\phi_{C_1}}$ and working in the mass basis\nof the charged leptons, under the SD condition Eq.~(\\ref{SD1}),\nwe obtain for the lepton mixing angles \n\\cite{King:1998jw}:\n\\begin{subequations}\\label{anglesSD}\\begin{eqnarray}\n\\label{Eq:t23} \n\\tan \\theta_{23} &\\approx& \\frac{|A_2|}{|A_3|}\\;, \\\\\n\\label{Eq:t12}\n\\tan \\theta_{12} &\\approx& \n\\frac{|B_1|}{c_{23}|B_2|\\cos \\tilde{\\phi}_2 - \ns_{23}|B_3|\\cos \\tilde{\\phi}_3  } \\;,\\\\\n\\label{Eq:t13}\n\\theta_{13} &\\approx& \ne^{i (\\tilde{\\phi} + \\phi_{B_1} - \\phi_{A_2})}\n\\frac{|B_1| (A_2^*B_2 + A_3^*B_3) }{\\left[|A_2|^2 + |A_3|^2\\right]^{3\/2} }\n\\frac{M_A}{M_B} \n+\\frac{e^{i (\\tilde{\\phi} + \\phi_{A_1} - \\phi_{A_2})} |A_1|}\n{\\sqrt{|A_2|^2 + |A_3|^2}} ,\n\\end{eqnarray}\\end{subequations}\nand for the masses\n\\begin{subequations}\\label{massesSD}\\begin{eqnarray}\n\\label{Eq:m3} m_3 &\\approx& \\frac{(|A_2|^2 + |A_3|^2)v^2}{M_A}\\;, \\\\\n\\label{Eq:m2} m_2 &\\approx& \\frac{|B_1|^2 v^2}{s^2_{12} M_B}\\;, \\\\\n\\label{Eq:m1}m_1 &\\approx& {\\cal O}(|C|^2 v^2\/M_C) \\;. \n\\end{eqnarray}\\end{subequations}\nWe would like to note that tri-bimaximal mixing \\cite{tribi} corresponds to the choice\n\\cite{King:2005bj}:\n\\begin{eqnarray}\n|A_{1}| &=&0,  \\label{tribicondsd} \\\\\n\\text{\\ }|A_{2}| &=&|A_{3}|,  \\label{tribicondse} \\\\\n|B_{1}| &=&|B_{2}|=|B_{3}|,  \\label{tribicondsa} \\\\\nA^{\\dagger }B &=&0.  \\label{zero}\n\\end{eqnarray} \nThis is called constrained sequential dominance (CSD) \\cite{King:2005bj}.\n\n\nLet us now turn to the issue of leptonic CP phases in SD. In\nparticular we will consider the MNS phase $\\delta$,\nrelevant for neutrino oscillations,\nand then study in Sec.~\\ref{Sec:MNSLGlink} how it is linked to CP violation required for leptogenesis. \nAs in \\cite{King:2002qh}\nthe MNS phase $\\delta$ is fixed by the requirement that we have already \nimposed in Eq.~(\\ref{Eq:t12}) that $\\tan(\\theta_{12})$ is real and positive,\n\\begin{eqnarray}\n\\lefteqn{ \\label{real12} c_{23}|B_2|\\sin \\tilde{\\phi}_2 \\;\\approx\\; s_{23}|B_3|\\sin \\tilde{\\phi}_3 \\; ,}  \\\\\n\\lefteqn{\\label{pos12} c_{23}|B_2|\\cos \\tilde{\\phi}_2 - s_{23}|B_3|\\cos \\tilde{\\phi}_3 \\;>\\; 0\\; ,}\n\\end{eqnarray} \nwhere\n\\begin{eqnarray}\n\\tilde{\\phi}_2 & \\equiv & \\phi_{B_2}-\\phi_{B_1}-\\tilde{\\phi}+\\delta\\; ,\n\\nonumber \\\\\n\\tilde{\\phi}_3 & \\equiv & \\phi_{B_3}-\\phi_{B_1}+\\phi_{A_2}-\\phi_{A_3}\n-\\tilde{\\phi}+\\delta \\; .\n\\label{tildephi23}\n\\end{eqnarray}\nThe phase $\\tilde{\\phi}$ is fixed by the requirement (not yet imposed\nin Eq.~(\\ref{Eq:t13}))\nthat the angle $\\theta_{13}$ is real and positive.\nIn general this condition is rather complicated since the expression\nfor $\\theta_{13}$ is a sum of two terms.\nHowever if, for example, $A_1=0$ then $\\tilde{\\phi}$ is fixed by:\n\\begin{equation}\n\\tilde{\\phi}\\approx \\phi_{A_2}-\\phi_{B_1}-\\zeta\n\\label{tildephi}\n\\end{equation}\nwhere \n\\begin{equation}\n\\zeta = \\arg\\left(A_2^*B_2 + A_3^*B_3  \\right).\n\\label{eta}\n\\end{equation}\nEq.~(\\ref{eta}) may be expressed as \n\\begin{equation}\n\\tan \\zeta \\approx \\frac{|B_2|s_{23}s_2+|B_3|c_{23}s_3}\n{|B_2|s_{23}c_2+|B_3|c_{23}c_3}\\,.\n\\label{taneta}\n\\end{equation}\nInserting $\\tilde{\\phi}$ of Eq.~(\\ref{tildephi}) into \nEqs.~(\\ref{real12}), (\\ref{tildephi23}), we obtain a relation\nwhich can be expressed as\n\\begin{equation}\n\\tan (\\zeta +\\delta) \\approx \\frac{|B_2|c_{23}s_2-|B_3|s_{23}s_3}\n{-|B_2|c_{23}c_2+|B_3|s_{23}c_3}\\,.\n\\label{tanetadelta}\n\\end{equation}\nIn Eqs.~(\\ref{taneta}), (\\ref{tanetadelta}) we have written\n$s_i=\\sin \\zeta_i$, $c_i=\\cos \\zeta_i$, where we have defined\n\\begin{equation}\n\\zeta_2\\equiv \\phi_{B_2}-\\phi_{A_2}\\;, \\ \\ \\zeta_3\\equiv\n\\phi_{B_3}-\\phi_{A_3}\\;,\n\\label{eta23}\n\\end{equation}\nwhich are invariant under a charged lepton phase transformation.\nThe reason why the seesaw parameters only involve two invariant\nphases rather than the usual six, is due to the SD assumption\nin Eq.~(\\ref{SD1})\nthat has the effect of effectively decoupling the right-handed neutrino \nof mass $M_C$ from the seesaw mechanism, which removes three phases,\ntogether with the further assumption (in this case) of\n$A_1=0$, which removes another phase.\n\n\n\n\n\n\n\n\\section{Flavour Matters in Leptogenesis with Sequential Dominance}\\label{Sec:FlavourLGinSD}\n\n\nLet us now consider leptogenesis in neutrino mass models with sequential dominance (SD), taking into account lepton flavour in the Boltzmann equations. In SD, there are three classes of models with different characteristic predictions for leptogenesis. They differ by the role of the lightest right-handed neutrino in SD, which can either be the dominant one $M_1 = M_A$, the subdominant one $M_1 = M_B$ or the subsubdominant one $M_1 = M_C$ (which, in the SD limit, only contributes to $m_1$ but has a negligible effect on the neutrino mixing angles and CP phases). \nThe possible form of the neutrino Yukawa matrix $\\lambda_\\nu$ is then given by\n\\begin{eqnarray}\n\\lambda_\\nu &=& (A,B,C)\\; \\mbox{or} \\;\\lambda_\\nu = (A,C,B)\\;,\\;\\, \\mbox{for $M_1 = M_A$,}\\\\\n\\lambda_\\nu &=& (B,A,C)\\; \\mbox{or} \\;\\lambda_\\nu = (B,C,A)\\;,\\;\\, \\mbox{for $M_1 = M_B$,}\\\\\n\\lambda_\\nu &=& (C,A,B)\\; \\mbox{or} \\;\\lambda_\\nu = (C,B,A)\\;,\\;\\, \\mbox{for $M_1 = M_C$,}\n\\end{eqnarray}\nusing the notation of Sec.~\\ref{SD}, where we have ordered the columns\naccording to $M_{RR}=\\mbox{diag}(M_1,M_2,M_3)$ where $M_1<M_2<M_3$. \nThe flavour specific decay asymmetries $\\varepsilon_{1,\\alpha}$ and washout parameters $\\widetilde m_{1,\\alpha}$,  calculated from Eqs.~(\\ref{eq:epsaa}) and (\\ref{eq:mtildeaa}), \nare given in Tab.~\\ref{tab:DecayAsym} for the three classes of SD.   \n  \n\\begin{table} \n  \\centering \n  \\begin{eqnarray*} \n  \\begin{array}{|c||c|c|} \n \\hline \n\\mbox{Type of SD}\n&\n\\varepsilon_{1,\\alpha}   \\vphantom{\\frac{f^f}{f^f}}\n&\n\\widetilde m_{1,\\alpha}\n\\\\\n\\hline\\hline\nM_1 = M_A\n&\n\\displaystyle -\\frac{3 M_1}{16 \\pi } \\left\\{ \\frac{\\mathrm{Im}\\left[ A_\\alpha^* B_\\alpha  (A^\\dagger B) \\right]}{ M_B\\,(A^\\dagger A)}\n+\n\\frac{\\mathrm{Im}\\left[ A_\\alpha^* C_\\alpha  (A^\\dagger C) \\right]}{ M_C\\,(A^\\dagger A)}\n\\right\\}\\! \n                        \\vphantom{\\frac{\\frac{f^f}{f}}{\\frac{f}{f}}}\n&\n\\displaystyle \\frac{|A_\\alpha|^2 v_\\mathrm{u}^2}{M_1}\n\\\\\n\\hline\nM_1 = M_B\n&\n\\displaystyle  -\\frac{3 M_1}{16 \\pi } \\left\\{\\frac{\\mathrm{Im}\\left[ B_\\alpha^* A_\\alpha  (B^\\dagger A) \\right]}{ M_A\\,(B^\\dagger B)}\n+\n\\frac{\\mathrm{Im}\\left[ B_\\alpha^* C_\\alpha  (B^\\dagger C) \\right]}{ M_C\\,(B^\\dagger B)}\n\\right\\}\\!   \n                       \\vphantom{\\frac{\\frac{f^f}{f}}{\\frac{f}{f}}}\n& \n\\displaystyle \\frac{|B_\\alpha|^2 v_\\mathrm{u}^2}{M_1}\n\\\\\n\\hline \nM_1 = M_C\n&\n\\displaystyle  -\\frac{3 M_1}{16 \\pi } \\left\\{\n\\frac{\\mathrm{Im}\\left[ C_\\alpha^* A_\\alpha (C^\\dagger A) \\right]}{ M_A (C^\\dagger C)} \n+\n\\frac{\\mathrm{Im}\\left[ C_\\alpha^* B_\\alpha (C^\\dagger B) \\right]}{ M_B (C^\\dagger C)}\n\\right\\}\\! \n                       \\vphantom{\\frac{\\frac{f^f}{f}}{\\frac{f}{f_f}}}\n&\n\\displaystyle \\frac{|C_\\alpha|^2 v_\\mathrm{u}^2}{M_1}\n\\\\\n\\hline \n\\end{array} \n\\end{eqnarray*} \n\\caption{\n\\label{tab:DecayAsym} Flavour specific decay asymmetries $\\varepsilon_{1,\\alpha}$ for the decay of the lightest right-handed neutrino with mass $M_1$ (in the limit $M_1 \\ll M_2 , M_3$) and washout parameters $\\widetilde m_{1,\\alpha}$ in classes of\nmodels with sequential right-handed neutrino dominance (SD) in the SM. The MSSM decay asymmetries are a factor of $2$ larger, in the considered limit.} \n\\end{table} \n\n\n\n\\subsection{Decay Asymmetries and Washout Parameters}\n\\label{Sec:DecayAndWashout}\nWe can now derive theoretical expectations and constraints on the\nflavour specific decay asymmetries and washout parameters from\nrequiring consistency with the present neutrino data in the classes of\nneutrino mass models discussed above. The experimental results from\nneutrino oscillations indicate nearly maximal mixing $\\theta_{23}\n\\approx 45^\\circ$, large, but non-maximal mixing $\\theta_{12} \\approx\n33^\\circ$ and small mixing $\\theta_{13} \\lesssim 13^\\circ$, as well as\nsmall neutrino mass squared differences $\\Delta m^2_{31} = m_3^2 -\nm_1^2 \\approx 2.2 \\times 10^{-3}$ eV$^2$ and $\\Delta m^2_{21} = m_2^2 -\nm_1^2 \\approx 8.1 \\times 10^{-5}$ eV$^2$ \\cite{Maltoni:2004ei}. In\nneutrino mass models satisfying SD, the neutrino Yukawa couplings and\nthe masses of the right-handed neutrinos are related to the neutrino\nmasses and mixings by Eqs.~(\\ref{anglesSD}) and (\\ref{massesSD}).\n\nWe will focus first on the case where all three flavours are treated separately in the Boltzmann equations. As discussed in Sec.~\\ref{Sec:FML}, this is the case for  \ntemperatures below $10^{9} \\: \\mbox{GeV}$ in the SM and below $(1+\\tan^2 \\beta)\\times 10^{9} \\: \\mbox{GeV}$ in the MSSM. Typical examples where this case is relevant are thus   unified models of flavour with large $\\tan \\beta$. \nWe will then comment on the two remaining cases, where (i) the $e$ and $\\mu$ flavour are combined in the Boltzmann equations and only the $\\tau$ is treated independently and (ii) where all flavours are combined and the treatment becomes flavour-independent, in Sec.~\\ref{Sec:2otherCases}. \n\n\\subsubsection{Properties of the Decay Asymmetries $\\varepsilon_{1,\\alpha}$}\\label{Sec:Decay}\n\nThe decay asymmetries $\\varepsilon_{1,\\alpha}$ of\nTab.~\\ref{tab:DecayAsym} contain contributions from the two heavier\nright-handed neutrinos with masses $M_2$ and $M_3$. Using the SD\nconditions of Eq.~(\\ref{SDcond}), we see that the largest contributions to\nthe decay asymmetries stem from the terms in Tab.~\\ref{tab:DecayAsym}\ncontaining $M_A$, followed by the contributions containing $M_B$, and\nfinally terms containing $M_C$ give the smallest contributions which\nvanish in the limit $m_1 \\to 0$. In the following, we will neglect the\ncontributions to the decay asymmetry containing $M_C$, keeping in mind\nthat they may be important when the leading contributions are\nsuppressed, for instance due to a configuration of complex phases or\ndue to a particular structure of the neutrino Yukawa couplings $A_i$\nand $B_i$.  In the following, we will compare the flavour specific\ndecay asymmetries with the maximally possible decay asymmetry\n\\begin{eqnarray}\n\\varepsilon^{\\mathrm{max}} = \\frac{3}{16 \\pi} \\frac{M_1}{v_\\mathrm{u}^2} m_3 \\; ,\n\\end{eqnarray} \nwhich provides an upper bound on the decay asymmetry $\\varepsilon_1$ in the SM in the flavour-independent approximation. In the MSSM, the maximal decay asymmetry is a factor of 2 larger. The flavour specific decay asymmetries can be constrained in the three cases as follows: \n\n\\begin{itemize}\n\\item In the case $M_1 = M_A$, assuming that the \nsub-dominant right-handed neutrino provides the leading contribution to the decay asymmetry and that the term containing $M_C$ can be neglected, the flavour-specific  asymmetries are given by\n\\begin{eqnarray}\n\\varepsilon_{1,\\alpha} \n\\approx -\\frac{3 M_1}{16 \\pi } \n\\frac{\\mathrm{Im}\\left[ A_\\alpha^* B_\\alpha  (A^\\dagger B) \\right]}{ M_B\\,(A^\\dagger A)}\\;.\n\\end{eqnarray}\nUsing $|A_1| \\ll |A_2| \\approx |A_3|$, as suggested by and Eqs.~(\\ref{Eq:t13}) and\n(\\ref{Eq:t23}) and the smallness of $\\theta_{13}$ together with the approximate maximality of $\\theta_{23}$, we can constrain $\\varepsilon_{1,\\beta}$ with $\\beta\\in\\{\\mu,\\tau\\}$ and $\\varepsilon_{1,e}$ as follows: \n\\begin{eqnarray}\n|\\varepsilon_{1,\\beta}| \n&\\approx& \\left|\\frac{3 M_1}{16 \\pi } \n\\frac{\\mathrm{Im}\\left[ A_\\beta^* B_\\beta  (A^\\dagger B) \\right]}{ M_B\\,(A^\\dagger A)}\\right|\n\\;\\le\\;\n\\left|\\frac{3 M_1}{16 \\pi } \n\\frac{\\left[\\, |A_3| |B_\\beta| |A_3|  (|B_2| + |B_3|) \\right]}{ M_B\\,2 |A_3|^2}\\right|\\nonumber \\\\\n&\\le& \n\\left|\\frac{3 M_1}{16 \\pi } \\frac{1}{2}\n\\frac{\\left[ \\, |B_\\beta|   (|B_2| + |B_3|) \\right]}{ M_B} \\right|\n\\;\\approx\\; \\varepsilon^{\\mathrm{max}} \\label{eq:MAepsbb} \\, \n{\\cal O}\\!\\left(\\frac{m_2}{m_3}\\right), \\\\\n|\\varepsilon_{1,e}| \n&\\lesssim& \\varepsilon^{\\mathrm{max}}  \\, {\\cal O}\\! \\left(\\frac{m_2}{m_3} \\right)\\, \\frac{|A_1|}{|A_3|} \\left|\\frac{\\left[ \\, |B_1|   (|B_2| + |B_3|) \\right]}{ M_B} \\right|\n\\;\\ll\\;   \\varepsilon^{\\mathrm{max}} \\, {\\cal O}\\! \\left(\\frac{m_2}{m_3} \\right)\\label{eq:MAepsee} .\n\\end{eqnarray}\nThe estimate in the last step of Eqs.~(\\ref{eq:MAepsbb}) and\n(\\ref{eq:MAepsee}) is based on the SD conditions in\nEq.~(\\ref{SD1}) and on Eq.~(\\ref{Eq:t12}). The latter states\nthat in order to generate large mixing $\\theta_{12} \\approx 33^\\circ$,\n$|B_2|$ and\/or $|B_3|$ have to be of the same order as $|B_1|$, which\nin turn is related to $m_2$ by Eq.~(\\ref{Eq:m2}).  We conclude that\n$|\\varepsilon_{1,\\mu}|$ and $|\\varepsilon_{1,\\tau}|$ are somewhat\nsuppressed compared to the maximally possible value\n$\\varepsilon^{\\mathrm{max}}$ as was found for $\\varepsilon_1 = \\sum_\\alpha \\varepsilon_{1,\\alpha}$ in the flavour-independent treatment\nin \\cite{Hirsch:2001dg}.  The value of $|\\varepsilon_{1,e}|$\ngiven in Eq.~(\\ref{eq:MAepsee}) is suppressed compared to\n$|\\varepsilon_{1,\\mu}|$ and $|\\varepsilon_{1,\\tau}|$ due to $|A_1|\n\\ll |A_2|,|A_3|$ and it is related to the mixing angle $\\theta_{13}$,\nas can be seen from Eq.\\ (\\ref{Eq:t13}).\n\n\\item In the case $M_1 = M_B$, using again $|A_1| \\ll| A_2| \\approx |A_3|$, we find that we can constrain the decay asymmetries \n$\\varepsilon_{1,\\beta}$, with $\\beta \\in \\{\\mu,\\tau\\}$, and $\\varepsilon_{1,e}$ as\n\\begin{eqnarray}\n|\\varepsilon_{1,\\beta}| \n&\\approx& \\left| \n\\frac{3 M_1}{16 \\pi } \n\\frac{\\mathrm{Im}\\left[ B_\\beta^* A_\\beta  (B^\\dagger A) \\right]}{ M_A\\,(B^\\dagger B)}\n\\right|\n\\;\\le\\;\n\\left|\\frac{3 M_1}{16 \\pi } \n\\frac{\\left[ \\,|B_\\beta| \\,|A_3|^2  (|B_2| + |B_3|) \\right]}{M_A\\,(B^\\dagger B)}\\right|\\nonumber \\\\\n&=& \n \\varepsilon^{\\mathrm{max}}\\,\\frac{1}{2} \\,\n\\left|\\frac{\\left[ \\, |B_\\beta|   (|B_2| + |B_3|) \\right]}{(B^\\dagger B)} \\right|\n\\;\\le\\; \\varepsilon^{\\mathrm{max}}  \\label{eq:MBepsbb}\\\\\n|\\varepsilon_{1,e}| \n&\\lesssim&  \\varepsilon^{\\mathrm{max}}  \\,\\frac{1}{2}\\, \\frac{|A_1|}{|A_3|} \n\\left|\\frac{\\left[ \\, |B_1|   (|B_2| + |B_3|) \\right]}{(B^\\dagger B)} \\right|\n\\;\\ll\\; \\varepsilon^{\\mathrm{max}} \\label{eq:MBepsee}\\;.\n\\end{eqnarray}\nFor $|\\varepsilon_{1,\\beta}|$, the maximal value $\\varepsilon^{\\mathrm{max}}$ can be nearly saturated, whereas $|\\varepsilon_{1,e}|$ is suppressed due to $|A_1| \\ll| A_2| ,|A_3|$ as in the case $M_1=M_A$. \n\n\n\\item In the case $M_1 = M_C$, the decay asymmetries\n$\\varepsilon_{1,\\alpha}$ obtain contributions from the right-handed\nneutrinos with masses $M_A$ and $M_B$.  Using analogous estimates than\nfor the case $M_1 = M_B$, we can show that the contribution from the\nright-handed neutrino with mass $M_A$ allows decay asymmetries\n$|\\varepsilon_{1,\\mu}|$ and $|\\varepsilon_{1,\\tau}|$ close to the\nupper bound $\\varepsilon^{\\mathrm{max}}$ similar to the result for $\\varepsilon_1$ in the\nflavour-independent treatment in \\cite{Hirsch:2001dg},\nbut due to $|A_1| \\ll |A_2|\n\\approx |A_3|$ the decay asymmetry $\\varepsilon_{1,e}$ is suppressed.\nFrom $M_B$, using estimates as for the case $M_1 = M_A$, all decay\nasymmetries can get contributions $|\\varepsilon_{1,\\alpha}| \\lesssim\n{\\cal O}(m_2\/m_3) \\,\\varepsilon^{\\mathrm{max}}$. We will return to the\ncase $M_1 = M_C$ in Sec.~\\ref{Sec:CSDinMC}, where we discuss the\nspecific example of tri-bimaximal mixing via constrained sequential\ndominance (CSD). There we will also present explicit formulae for the\ndecay asymmetries which illustrate the above statements.\n\\end{itemize}\n\n\\noindent The above derived constraints on the decay asymmetries $\\varepsilon_{1,\\alpha}$ in the classes of models under consideration are summarized in Tab.~\\ref{tab:PropertiesEpsSD}.\n\n\\begin{table} \n  \\centering \n  \\begin{eqnarray*} \n  \\begin{array}{|c||c|c|c|} \n \\hline \n\\mbox{Type of SD}\n&\n\\mbox{relation for}\\;  \\varepsilon_{1,e}  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{relation for}\\;  \\varepsilon_{1,\\mu}\n&\n\\mbox{relation for}\\;  \\varepsilon_{1,\\tau}\n\\\\\n\\hline\\hline\nM_1 = M_A  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\ll      {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n& \n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n\\\\\n\\hline \nM_1 = M_B  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\ll  \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim \\varepsilon^{\\mathrm{max}}\n\\\\\n\\hline\nM_1 = M_C  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim  \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim  \\varepsilon^{\\mathrm{max}}\n\\\\\n\\hline \n\\end{array} \n\\end{eqnarray*} \n\\caption{\n\\label{tab:PropertiesEpsSD} Constraints and theoretical expectations on the flavour specific decay asymmetries from consistency with low energy neutrino observables in classes of models with SD in the SM, in the limit $M_1 \\ll M_2 , M_3$. \n$\\varepsilon^{\\mathrm{max}}$, the upper bound on the decay asymmetry, is given by $\\frac{3}{16 \\pi} \\frac{M_1}{v_\\mathrm{u}^2} m_3$. In the MSSM, the decay asymmetries are a factor of $2$ larger.} \n\\end{table} \n\n\n\\subsubsection{Properties of the Washout Parameters $\\widetilde m_{1,\\alpha}$}\\label{Sec:Washout}\n\n\\begin{itemize}\n\\item \nIn the case $M_1 = M_A$, comparing the formula for $m_3$ in Eq.\\\n(\\ref{Eq:m3}) with the formulae for the washout parameters $\\widetilde\nm_{1,\\alpha}$ in Tab.\\ \\ref{tab:DecayAsym} yields\n\\begin{equation}\n\\widetilde m_{1,\\mu} = \\widetilde m_{1,\\tau} \\approx \\frac{1}{2} m_3 \\;.\n\\end{equation}\nSince for hierarchical light neutrino masses $m_3 \\approx \\sqrt{\\Delta\nm^2_{31}} \\approx 0.05$ eV, we find that for these flavours we are\nconstrained to be in the strong washout regime, where $\\widetilde\nm_{1,\\alpha} \\gg m^*$ (c.f.~Fig.\\ \\ref{fig:eta}).  On the contrary,\n$\\widetilde m_{1,e}$ is significantly smaller since $|A_1| \\ll\n|A_2|,|A_3|$ and $\\widetilde m_{1,e}$ is thus typically in the weak\n(optimal) region for $\\widetilde m_{1,e} \\ll m^*$ ($\\widetilde\nm_{1,e}\\approx m^*$).\n\nNote that in the flavour-independent approximation, the washout parameter would be given by $\\widetilde m_1 = \\sum_\\alpha \\widetilde m_{1,\\alpha} \\approx m_3$. The fact that we expect weak (or optimal) washout for $\\widetilde m_{e}$ thus provides an important difference between flavour-dependent leptogenesis compared to the flavour-independent approximation. We will analyze this situation in more detail in Sec.~\\ref{A1andLSD}. \n\n\\item In the case $M_1 = M_B$, from comparing the formula for $m_2$ in \nEq.\\ (\\ref{Eq:m2}) with the formula for the washout parameter $\\widetilde  m_{1,e}$ in Tab.\\ \\ref{tab:DecayAsym}, we infer that \n\\begin{eqnarray}\n\\widetilde m_{1,e} \\approx  s_{12}^2 m_2 \\; .\n\\end{eqnarray}\nFurthermore, from Eq.\\ (\\ref{Eq:t12}) and from the SD conditions  Eq.~(\\ref{SDcond}) it follows that $B_2$ and\/or $B_3$ are of the order of $B_1$ in order to generate large neutrino mixing $\\theta_{12}$, leading to the expectations \n\\begin{eqnarray}\n\\widetilde m_{1,\\mu} \\;\\: \\mbox{and\/or} \\;\\: \\widetilde m_{1,\\tau} = {\\cal O}(m_2) \\; .\n\\end{eqnarray}\n\n\\item In the case $M_1 = M_C$, Eq.\\ (\\ref{Eq:m1}) implies that  \nthe washout parameters $\\widetilde m_{1,\\alpha}$ are typically $\\lesssim {\\cal O}(m_1)$ and therefore, with a hierarchical neutrino mass spectrum $m_1 \\ll m_2 < m_3$, they are in general in the weak (or optimal) washout region. \n\\end{itemize}\n\n\\noindent A summary of the above derived constraints on the washout parameters $\\widetilde m_{1,\\alpha}$ in the classes of models with sequential dominance is given in  Tab.~\\ref{tab:FlavouredWashout}.\n\n\\begin{table} \n  \\centering \n  \\begin{eqnarray*} \n  \\begin{array}{|c||c|c|c|} \n \\hline \n\\mbox{Type of SD}\n&\n\\mbox{washout by}\\;\\widetilde m_{1,e}  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{washout by}\\;\\widetilde m_{1,\\mu}\n&\n\\mbox{washout by}\\;\\widetilde m_{1,\\tau}\n\\\\\n\\hline\\hline \nM_1 = M_A\n&\n\\mbox{weak\/optimal}     \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{strong, $\\widetilde m_{1,\\mu} \\approx \\tfrac{1}{2} m_3$}\n&\n\\mbox{strong, $\\widetilde m_{1,\\tau} \\approx \\tfrac{1}{2} m_3$}\n\\\\\n\\hline\nM_1 = M_B\n&\n\\mbox{optimal, $\\widetilde m_{1,e} \\approx s_{12}^2 m_2$}     \\vphantom{\\frac{f^f}{f^f}}\n& \n\\lesssim {\\cal O}(m_2)^*\n&\n\\lesssim {\\cal O}(m_2)^*\n\\\\\n\\hline \nM_1 = M_C\n&\n\\mbox{weak\/optimal}     \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{weak\/optimal}\n&\n\\mbox{weak\/optimal}\n\\\\\n\\hline \n\\end{array} \n\\end{eqnarray*} \n\\caption{\n\\label{tab:FlavouredWashout} Flavour specific type of washout for the considered classes of models. (${}^*$At least one out of $\\widetilde m_{1,\\mu},\\widetilde m_{1,\\tau}$ has to be ${\\cal O}(m_2)$ for $M_1 = M_B$.) } \n\\end{table} \n\n\n\\subsubsection{Remaining Cases: Only $\\boldsymbol{y_\\tau}$ in Equilibrium and the Flavour-Independent Case}\\label{Sec:2otherCases}\nAs discussed in Sec.~\\ref{Sec:FML}, all three flavours are only treated differently for \ntemperatures smaller than $10^9 \\: \\mbox{GeV}$ in the SM and larger than $(1+\\tan^2 \\beta)\\times 10^9 \\: \\mbox{GeV}$ in the MSSM. \n\nIn the temperature ranges \n$10^9 \\: \\mbox{GeV} \\ll M_1 \\ll 10^{12} \\: \\mbox{GeV}$ in the SM and \n$(1+\\tan^2 \\beta)\\times 10^9 \\: \\mbox{GeV} \\ll M_1 \\ll (1+\\tan^2 \\beta)\\times 10^{12} \\: \\mbox{GeV}$ in the MSSM, only the $y_\\tau$ is in equilibrium and thus only the $\\tau$ flavour is treated separately in the Boltzmann equations while the $e$ and $\\mu$ flavours are indistinguishable. In this temperature range, one can use the combined decay asymmetry and washout parameter\n\\begin{eqnarray}\n\\varepsilon_{1,2} &\\equiv& \\varepsilon_{1,e}+\\varepsilon_{1,\\mu} \\; ,\\\\\n\\widetilde m_{1,2}&\\equiv& \\widetilde m_{1,e} + \\widetilde m_{1,\\mu}\\; ,\n\\end{eqnarray}\ninstead of $\\varepsilon_{1,e},\\varepsilon_{1,\\mu}$ and $\\widetilde m_{1,e},\\widetilde m_{1,\\mu}$ and solve the Boltzmann equations for $Y_{\\Delta_2} \\equiv Y_{\\Delta_e + \\Delta_\\mu}$ and $Y_{\\Delta_\\tau}$.\n\nAbove $10^{12} \\: \\mbox{GeV}$ in the SM and \n$(1+\\tan^2 \\beta)\\times 10^{12} \\: \\mbox{GeV}$ in the MSSM, all flavours are combined to the flavour-independent decay asymmetry and washout parameter \n\\begin{eqnarray}\n\\varepsilon_{1} &\\equiv& \\varepsilon_{1,e}+\\varepsilon_{1,\\mu}+\\varepsilon_{1,\\tau} \\; ,\\\\\n\\widetilde m_{1}&\\equiv& \\widetilde m_{1,e} + \\widetilde m_{1,\\mu} + \\widetilde m_{1,\\tau}\n\\end{eqnarray}\nand one can solve the Boltzmann equations for the total asymmetry $Y_\\Delta \\equiv Y_{\\Delta_e + \\Delta_\\mu + \\Delta_\\tau}$. \n\nThe properties of the decay asymmetries and washout parameters in these two cases can be discussed analogously to Secs.~\\ref{Sec:Decay} and \\ref{Sec:Washout}. \nThe results are presented in Tabs.~\\ref{tab:CaseTau} and \\ref{tab:CaseNoYeinEq}.\n\n\\begin{table}[t] \n  \\centering \n  \\begin{eqnarray*} \n  \\begin{array}{|c||c|c||c|c|} \n \\hline \n\\mbox{Type of SD}\n&\n  \\varepsilon_{1,2}  \\vphantom{\\frac{f^f}{f^f}}\n&\n  \\varepsilon_{1,\\tau}\n&\n\\mbox{washout by}\\;\\widetilde m_{1,2}  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{washout by}\\;\\widetilde m_{1,\\tau}\n\\\\\n\\hline\\hline\nM_1 = M_A  \\vphantom{\\frac{f^f}{f^f}}\n& \n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n&\n\\mbox{$\\widetilde m_{1,2} \\approx \\tfrac{1}{2} m_3$}\\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{$\\widetilde m_{1,\\tau} \\approx \\tfrac{1}{2} m_3$}\n\\\\\n\\hline \nM_1 = M_B  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim \\varepsilon^{\\mathrm{max}}\n& \n\\lesssim {\\cal O}(m_2) \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim {\\cal O}(m_2) \n\\\\\n\\hline\nM_1 = M_C  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim  \\varepsilon^{\\mathrm{max}}\n&\n\\lesssim  \\varepsilon^{\\mathrm{max}}\n&\n\\mbox{weak\/optimal}     \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{weak\/optimal}\n\\\\\n\\hline \n\\end{array} \n\\end{eqnarray*} \n\\caption{\n\\label{tab:CaseTau} Constraints and theoretical expectations on the flavour specific decay asymmetries and the flavour specific type of washout for the case where only the tau Yukawa coupling $y_\\tau$ is in thermal equilibrium. For the $e$ and $\\mu$ flavours, the combined quantities $\\varepsilon_{1,2} = \\varepsilon_{1,e}+\\varepsilon_{1,\\mu}$ and $\\widetilde m_{1,2} = \\widetilde m_{1,e} + \\widetilde m_{1,\\mu}$ are considered.} \n\\end{table} \n\n\n\\begin{table}[t] \n  \\centering \n  \\begin{eqnarray*} \n  \\begin{array}{|c||c||c|} \n \\hline \n\\mbox{Type of SD}\n&\n\\mbox{relation for}\\;  \\varepsilon_{1}  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\mbox{washout by}\\;\\widetilde m_{1}  \\vphantom{\\frac{f^f}{f^f}}\n\\\\\n\\hline\\hline\nM_1 = M_A  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim {\\cal O}(\\tfrac{m_2}{m_3}) \\varepsilon^{\\mathrm{max}}\n&\n\\mbox{strong, $\\widetilde m_{1} = m_3$}\\vphantom{\\frac{f^f}{f^f}}\n\\\\\n\\hline \nM_1 = M_B  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim \\varepsilon^{\\mathrm{max}}\n& \n\\lesssim {\\cal O}(m_2) \\vphantom{\\frac{f^f}{f^f}}\n\\\\\n\\hline\nM_1 = M_C  \\vphantom{\\frac{f^f}{f^f}}\n&\n\\lesssim  \\varepsilon^{\\mathrm{max}}\n&\n\\mbox{weak\/optimal}     \\vphantom{\\frac{f^f}{f^f}}\n\\\\\n\\hline \n\\end{array} \n\\end{eqnarray*} \n\\caption{\n\\label{tab:CaseNoYeinEq} Constraints and theoretical expectations on the flavour specific decay asymmetries and the flavour specific type of washout for the case where no charged lepton Yukawa coupling is in equilibrium. All flavours are combined effectively to a single flavour in the Boltzmann equations and the relevant quantities are the flavour-independent ones, $\\varepsilon_{1} = \\varepsilon_{1,e} +\\varepsilon_{1,\\mu} +\\varepsilon_{1,\\tau}$ and $\\widetilde m_{1}\\equiv \\widetilde m_{1,e} + \\widetilde m_{1,\\mu} + \\widetilde m_{1,\\tau}$.} \n\\end{table} \n\n\n\\subsection{Leptonic CP Violation and the MNS - Leptogenesis Link}\\label{Sec:MNSLGlink}\nIn \\cite{King:2002qh} it was shown how, in the case of SD where the\nlightest right-handed neutrino was the dominant one, there was a link\nbetween the MNS Dirac CP violating phase $\\delta$ which appears in\nneutrino oscillations and the phase which is relevant for\nleptogenesis.  However the analysis in \\cite{King:2002qh} was based on\nthe flavour-independent formulation of leptogenesis, and it is\ntherefore interesting to revisit this analysis in the light of\nflavour-dependent leptogenesis.\n\nEqs.~(\\ref{tanetadelta}) and (\\ref{eta23}) show explicitly that  \nthe MNS Dirac CP violating phase $\\delta$ is a function\nof only two seesaw phases $\\zeta_2,\\ \\zeta_3$,\nas in \\cite{King:2002qh}.\nSince these are the only remaining seesaw phases it must also\nbe the case that leptogenesis must depend also on these\nsame two phases. This opens up the possibility of a link\nbetween the leptogenesis phase and the oscillation MNS phase $\\delta$,\nas in \\cite{King:2002qh}. In the flavour-independent treatment\nof leptogenesis in \\cite{King:2002qh} the relevant leptogenesis\nphase was $\\zeta$ and the leptogenesis-MNS link was \ngiven by Eqs.~(\\ref{taneta}), (\\ref{tanetadelta}). \nHowever, in the present case, flavour-dependent\nleptogenesis will depend on both phases\n$\\zeta_2,\\ \\zeta_3$, as we now discuss.\n\n\\begin{itemize}\n\\item\nFor the case $M_1=M_A$, assuming that $A_1=0$,\nwe have \n\\begin{eqnarray}\n\\varepsilon_{1,\\alpha} \n\\approx -\\frac{3 M_1}{16 \\pi } \n\\frac{\\mathrm{Im}\\left[ A_\\alpha^* B_\\alpha  (A^\\dagger B) \\right]}{ M_B\\,(A^\\dagger A)}\\;.\n\\end{eqnarray}\nFrom this result it is clear that \n\\begin{eqnarray}\n\\varepsilon_{1,e} & = & 0,\\nonumber \\\\ \n\\varepsilon_{1,\\mu} & \\propto & \n+|A_2||B_2|\\sin (\\zeta + \\zeta_2),\\nonumber \\\\ \n\\varepsilon_{1,\\tau} & \\propto & \n+|A_3||B_3|\\sin (\\zeta + \\zeta_3).\n\\end{eqnarray}\nFrom the previous section we see that \n\\begin{equation}\nY_B \\propto Y_{\\Delta_\\mu} + Y_{\\Delta_\\tau}\n\\end{equation}\nwith $Y_{\\alpha}$ given in Eq.~(\\ref{Eq:eta_aa}).\nSince the efficiency factor $\\eta_{\\mu}$ depends on \n$\\tilde{m}_{1,\\mu}\\approx |A_2|^2v^2\/M_A$\nand \n$\\eta_{\\tau}$ depends on \n$\\tilde{m}_{1,\\tau}\\approx |A_3|^2v^2\/M_A$,\nwith $|A_2|\\approx |A_3|$ we see that,\n$\\eta_{\\mu}\\approx \\eta_{\\tau}$,\nand\n\\begin{equation}\nY_B\\propto |B_2|\\sin (\\zeta + \\zeta_2)\n+ |B_3|\\sin (\\zeta + \\zeta_3),\n\\label{YB1}\n\\end{equation}\nwhich shows explicitly how leptogenesis depends on the\ntwo phases $\\zeta_2,\\ \\zeta_3$ in this case.\n\n\\item\nFor the case $M_1=M_B$, assuming that $A_1=0$,\nwe have \n\\begin{eqnarray}\n\\varepsilon_{1,\\alpha} \n\\approx -\\frac{3 M_1}{16 \\pi } \n\\frac{\\mathrm{Im}\\left[ B_\\alpha^* A_\\alpha  (B^\\dagger A) \\right]}{ M_B\\,(B^\\dagger B)}\\;.\n\\end{eqnarray}\nFrom this result it is clear that \n\\begin{eqnarray}\n\\varepsilon_{1,e} & = & 0,\\nonumber \\\\ \n\\varepsilon_{1,\\mu} & \\propto & \n-|A_2||B_2|\\sin (\\zeta + \\zeta_2),\\nonumber \\\\ \n\\varepsilon_{1,\\tau} & \\propto & \n-|A_3||B_3|\\sin (\\zeta + \\zeta_3).\n\\end{eqnarray}\nSince now $\\eta_{\\mu}$ depends on the washout parameter  \n$\\tilde{m}_{1,\\mu}\\approx |B_2|^2 v^2\/M_B$\nand \n$\\eta_{\\tau}$ depends on \n$\\tilde{m}_{1,\\tau}\\approx |B_3|^2 v^2\/M_B$,\nthe two efficiencies are in general unequal in this case,\nand with $|A_2|\\approx |A_3|$ we see that,\n\\begin{equation}\nY_B\\propto -\\eta_{\\mu}|B_2|\\sin (\\zeta + \\zeta_2)\n-\\eta_{\\tau} |B_3|\\sin (\\zeta + \\zeta_3),\n\\label{YB2}\n\\end{equation}\nwhich shows explicitly how leptogenesis depends on the\ntwo phases $\\zeta_2,\\ \\zeta_3$ in this case.\n\n\\item\nClearly, for the case $M_1=M_C$, there is no leptogenesis-MNS\nlink since leptogenesis will depend on phases associated\nwith the right-handed neutrino of mass $M_C$, which does not\ncontribute significantly to the seesaw mechanism, and therefore\nthese phases will not significantly contribute to the MNS phases.\n\n\\end{itemize}\n\nThe leptogenesis-MNS link in the flavour-dependent case is therefore\nrather similar to the link in the flavour-independent case,\nnamely the connection between leptogenesis and $\\delta$\nis made via the remaining two seesaw phases $\\zeta_2,\\ \\zeta_3$ which are \nresponsible for both leptogenesis and $\\delta$.\nThese two phases fix $\\delta$ as in \nEqs.~(\\ref{tanetadelta}), (\\ref{eta23}), \nwhich are valid for all types of SD,\nincluding the cases $M_1=M_A$ and $M_1=M_B$.\nHowever the two phases $\\zeta_2,\\ \\zeta_3$ contribute to \nleptogenesis differently for the cases $M_1=M_A$ and $M_1=M_B$,\nas shown in Eqs.~(\\ref{YB1}) and (\\ref{YB2}), where both these results\ndiffer from the flavour-independent result in which leptogenesis\nis a simple function of $\\zeta$.\n\nIn order to obtain a more precise leptogenesis-MNS link,\none can reduce the number of phases still further by assuming\nan additional zero Yukawa coupling, in addition to assuming that\n$A_1=0$. For example if we additionally assume that $B_3=0$, \nwhich implies that $\\varepsilon_{1,\\tau}=0$, \nthen this removes the phase $\\zeta_3$, and we find that\n$\\zeta = \\zeta_2$.\nFrom Eqs.~(\\ref{real12}) - (\\ref{eta}), we obtain for the \nMNS phase $\\delta$ \\cite{King:2002qh}: \n\\begin{eqnarray}\n\\delta = - 2 \\zeta \\; .\\label{deltaeta2}\n\\end{eqnarray}\nUnder the same assumption that $B_3=0$ (as well as $A_1=0$),\nfor the case $M_1=M_A$ we see that \n\\begin{equation}\nY_B \\propto +\\sin (2\\zeta ),\n\\label{YB11}\n\\end{equation}\nwhile for the case $M_1=M_B$ we see that \n\\begin{equation}\nY_B \\propto -\\sin (2\\zeta ),\n\\label{YB21}\n\\end{equation}\nwhere in both cases leptogenesis depends on a single\nphase $\\zeta$, which is directly related to the \nMNS phase $\\delta$ in Eq.~(\\ref{deltaeta2}). \nThe sign of the CP violating phase $\\delta$ measurable in \nhigh precision neutrino oscillation experiments, or more precisely the sign of \n$\\sin(\\delta)$, therefore will\ndepend on whether $M_1=M_A$ or $M_1=M_B$.\nIf $M_1=M_A$ then $\\sin(\\delta)$ must be negative since baryon asymmetry\nof the universe in Eq.~(\\ref{YB11}) must be positive.\nIf $M_1=M_B$ then $\\sin(\\delta)$ must be positive since baryon asymmetry\nof the universe in Eq.~(\\ref{YB21}) must be positive.\nThus the measured sign of the oscillation phase $\\delta$ \nis capable of distinguishing between the \ntwo types of SD, namely $M_1=M_A$ or $M_1=M_B$,\nunder the assumption of the two texture zeroes $A_1 = 0$ and $B_3 = 0$.\\footnote{A different possible choice of two texture zeros, which can be discussed analogously, is $A_1 = B_2 = 0$. In this case, the relation between $\\delta$ and $\\zeta$ would be given by $\\delta = - 2 \\zeta + \\pi$, leading to the prediction of the opposite sign of $\\sin(\\delta)$ from the observed baryon asymmetry.}\n\n\\subsection{Examples}\\label{examples}\n\n\\subsubsection{The Role of $A_1$ in the Case $M_1 = M_A$}\\label{A1andLSD}\nAs we have remarked in the discussion of the flavour specific washout\nfactors $\\widetilde m_{1,\\alpha}$, one particularly interesting\ndifference between flavour-dependent treatment and flavour-independent\napproximation is the possibility to have an optimal washout parameter\n$\\widetilde m_{1,e}$ and strong washout in the flavours $\\mu$ and\n$\\tau$ since in the flavour-independent \ntreatment this leads to a strong total washout.  \nWe will therefore discuss the role of $A_1$ in the\ncase $M_1 = M_A$, sometimes also called light sequential dominance\n(LSD), in more detail.\n\nLet us start with the special case is $A_1 = 0$, where due to $|A_2| \\approx |A_3|$ required by $\\theta_{23} \\approx 45^\\circ$ (c.f.\\ Eq.~(\\ref{Eq:t23})) the \nwashout parameters satisfy\n\\begin{eqnarray}\n\\widetilde m_{1,\\mu} \\approx \\widetilde m_{1,\\tau} \\approx \\frac{1}{2} m_3\\; , \\;\\,\n\\mbox{leading to} \\; \\eta_{\\mu} \\approx \\eta_{\\tau}\\approx \n\\eta \\left(A_{\\tau\\tau}\\frac{1}{2} \\frac{m_3}{m^*},\\frac{m_3}{m^*}\\right).\n\\end{eqnarray}\nWe can now estimate for the baryon-to-entropy density in the MSSM, using Eqs.~(\\ref{Eq:eta_aa_MSSM}) and (\\ref{Eq:YB3f}),\n\\begin{eqnarray}\nY_B \\approx - \\frac{10}{31}\\,\\eta\\left(A_{\\tau\\tau}\\frac{1}{2} \\frac{m_3}{m^*},\\frac{m_3}{m^*}\\right) \\, ( \\varepsilon_{1,\\mu} + \\varepsilon_{1,\\tau}) \\, \n\\left[ Y^{\\mathrm{eq}}_{N_1}(z\\gg1) + Y^{\\mathrm{eq}}_{\\widetilde N_1}(z\\gg1)\\right] .\n\\end{eqnarray} \nApart from minor modifications in the efficiency factor, the\nflavour-dependent treatment is thus similar to the flavour-independent\napproximation in this case \\cite{Hirsch:2001dg}.\n   \nThe importance of a flavour-dependent treatment becomes clear if we\nallow for a non-zero $A_1 \\ll A_2 \\approx A_3$. As we have discussed\nabove, $\\widetilde m_{1,e}$ can now be optimal whereas $\\widetilde\nm_{1,\\mu}$ and $\\widetilde m_{1,\\tau}$ correspond to a strong\nwashout. In the flavour-independent approximation, the total washout\nparameter $\\widetilde m_1 = \\sum_\\alpha \\widetilde m_{1,\\alpha}\n\\approx m_3$ implies strong washout.\nIn Fig.~\\ref{fig:A1}, the flavour-dependent treatment is compared to the flavour-independent approximation in an example with non-zero $A_1$. Dashed lines correspond to the flavour-independent approximation and solid lines stand for the same examples with  \nflavour-effects included. \nFor non-zero $A_1\/A_2$ above roughly $0.05$, the produced baryon asymmetry is significantly enhanced in the flavour-dependent case (more than two orders of magnitude in this example for $A_1\/A_2 = 0.2$) because this induces an efficiency in the $e$-flavour which is close to optimal. \nIn the flavour-independent approximation, the common efficiency factor  $\\eta^\\mathrm{ind}(\\widetilde m_1\/m^*)$ is much lower due to the strong washout. \n\n\\vspace{7mm}\n\\begin{figure}\n \\centering\n \\includegraphics[scale=1.0,angle=0]{A1_MSSM.eps}\n \\caption{\\label{fig:A1}\nExample in the MSSM (with large $\\tan \\beta = 50$), which illustrates the difference between flavour-dependent treatment and  flavour-independent approximation for the case $M_1 = M_A$ if $A_1$ is non-zero. We have chosen $\\phi_{A_1} = - \\tfrac{\\pi}{2}, \\phi_{B_3} = \\pi, \\phi_{A_2} = \\phi_{A_3} = \\phi_{B_3} = \\phi_{C_i} = 0, B_3 = 0, C_i = 0$. The other parameters are determined from the experimental data assuming SD. For the parameters under consideration, $\\theta_{13}$ remains below the present experimental upper bound. \n }\n\\end{figure}\n\n\n\\subsubsection{Leptogenesis and Tri-Bimaximal Mixing}\nAs discussed in Sec.~\\ref{SD}, tri-bimaximal mixing can be realized in\nSD in a natural way via vacuum alignment and is sometimes referred to\nas constrained sequential dominance (CSD) \\cite{King:2005bj}. \nAs can be seen directly\nfrom Eq.~(\\ref{zero}), which states that $A^\\dagger B = B^\\dagger A =\n0$, together with the formulae for the decay asymmetries\n$\\varepsilon_{1,\\alpha}$ (c.f.\\ Tab.~\\ref{tab:DecayAsym}), in CSD the MSSM \ndecay asymmetries reduce to\n\\begin{eqnarray}\n\\varepsilon_{1,\\alpha} &=& -\\frac{3 M_1}{8 \\pi } \n\\frac{\\mathrm{Im}\\left[ A_\\alpha^* C_\\alpha  (A^\\dagger C) \\right]}{ M_C\\,(A^\\dagger A)}\n\\;\\lesssim\\; \\varepsilon^{\\mathrm{max}}  \\, \n{\\cal O}\\!\\left(\\frac{m_1}{m_3}\\right)\n\\quad \\mbox{for $M_1 = M_A$, and} \\label{epsCSDmm}\\\\\n\\varepsilon_{1,\\alpha} &=& -\\frac{3 M_1}{8 \\pi } \n\\frac{\\mathrm{Im}\\left[ B_\\alpha^* C_\\alpha  (B^\\dagger C) \\right]}{ M_C\\,(B^\\dagger B)}\n\\;\\lesssim\\; \\varepsilon^{\\mathrm{max}}  \\, \n{\\cal O}\\!\\left(\\frac{m_1}{m_3}\\right)\n\\quad \\mbox{for $M_1 = M_B$.} \\label{epsCSDtt}\n\\end{eqnarray}\nFor $M_1 = M_A$ and $M_1 = M_B$, the decay asymmetries are thus suppressed by $m_1\/m_3$.\n\nWe would like to remark at this point that in realistic models, tri-bimaximal mixing is typically realized for the neutrino mass matrix in a basis where the charged lepton mass matrix is not entirely diagonal. However, including such charged lepton corrections do not change the above conclusions since $A^\\dagger B = B^\\dagger A = 0$ remains unchanged by a change to the basis where the charged lepton mass matrix is diagonal, even though the lepton mixing is no longer exactly tri-bimaximal. In this respect, the conclusions are the same in leptogenesis with correct flavour-dependent treatment and in the flavour-independent approximation.\\footnote{Note that in the R-matrix parameterization with $R=\\mathbbm{1}$, where all mixings and phases in $\\lambda_\\nu$ are introduced via the leptonic mixing matrix $U_\\mathrm{MNS}$, all flavour-specific decay asymmetries also vanish exactly as in the flavour-independent, since they are proportional to $(\\lambda_\\nu^\\dagger \\lambda_\\nu)_{1J}$ which is zero for $1 \\not= J$.}   \nIn general, however, we stress that charged lepton corrections \\cite{Antusch:2005kw} do have an effect on leptogenesis in the flavour-dependent treatment, in contrast to the flavour-independent approximation.  \n \nIn contrast to the cases $M_1 = M_A$ and $M_1 = M_B$, tri-bimaximal mixing via CSD has excellent prospects for leptogenesis in the classes of models with \n$M_1 = M_C$, as we now discuss.\n\n\\subsubsection{Tri-Bimaximal Mixing and CP phases in the Case $M_1=M_C$} \n\\label{Sec:CSDinMC}\nTo be explicit, let us consider the specific choice $A=(0,a,-a)$,\n$B=(b,b,b)$ with real $a$ and $b$ and furthermore a texture zero in\nthe (1,1)-element of $\\lambda_\\nu$, i.e. $C$ of the form\n$C=(0,C_2,C_3)$. Then, $\\varepsilon_{1,e}=0$ and the remaining two\ndecay asymmetries in the MSSM simplify to\n\\begin{eqnarray}\n\\varepsilon_{1,\\mu} &=& -\\frac{3 M_1}{8 \\pi } \\left\\{\n\\frac{\\mathrm{Im}\\left[ C_2^* a^2 (C_2^* - C_3^*) \\right]}{ M_A (C^\\dagger C)} \n+\n\\frac{\\mathrm{Im}\\left[ C_2^* b^2 (C_2^* + C_3^*) \\right]}{ M_B (C^\\dagger C)}\n\\right\\} \\nonumber \\\\\n&\\approx&\n-\\frac{3 M_1}{8 \\pi v^2} \\left\\{\n\\frac{\\mathrm{Im}\\left[ C_2^* (C_2^* - C_3^*) \\right]}{ (C^\\dagger C)} \n\\, \\frac{m_3}{2}\n+\n\\frac{\\mathrm{Im}\\left[ C_2^* (C_2^* + C_3^*) \\right]}{ (C^\\dagger C)} \n\\,s_{12}^2 m_2\n\\right\\}, \\label{eq:emmCSDMC}\\\\\n\\varepsilon_{1,\\tau} &=& -\\frac{3 M_1}{8 \\pi } \\left\\{\n\\frac{\\mathrm{Im}\\left[- C_3^* a^2 (C_2^* - C_3^*) \\right]}{ M_A (C^\\dagger C)} \n+\n\\frac{\\mathrm{Im}\\left[ C_3^* b^2 (C_2^* + C_3^*) \\right]}{ M_B (C^\\dagger C)}\n\\right\\} \\nonumber \\\\\n&=&\n-\\frac{3 M_1}{8 \\pi v^2} \\left\\{\n\\frac{\\mathrm{Im}\\left[- C_3^* (C_2^* - C_3^*) \\right]}{ (C^\\dagger C)} \n\\, \\frac{m_3}{2} \n+\n\\frac{\\mathrm{Im}\\left[ C_3^* (C_2^* + C_3^*) \\right]}{ (C^\\dagger C)} \n\\,s_{12}^2 m_2\n\\right\\}\\label{eq:ettCSDMC}.\n\\end{eqnarray}  \nOne can see that the two terms from the dominant (subdominant) RH neutrinos can easily  contribute $\\approx \\varepsilon^\\mathrm{max}$ ($\\approx \\varepsilon^\\mathrm{max} s_{12}^2 m_2\/m_3$) to the decay asymmetries. This is nearly the optimal case for leptogenesis with hierarchical neutrinos. Furthermore, the washout parameters $\\widetilde m_{1,\\alpha}$ are ${\\cal O}(m_1)$, which allows optimal efficiencies $\\eta_{\\alpha}$ if $m_1 \\approx m^* \\approx 10^{-3}$ eV.  \n\nLet us now discuss under which conditions flavour effects are\nimportant, or play only a minor role. Similar to the discussion in\nSec.~\\ref{A1andLSD}, when the relevant washout parameters are equal,\nas this would in our example be the case for $|C_2| = |C_3|$, the\nsituation is similar to the flavour-independent approximation. On the\nother hand, it is clear that if the washout parameters differ for the\n$\\mu$ and $\\tau$ flavour, the produced asymmetry in the\nflavour-dependent treatment will differ in general from the\nflavour-independent approximation.\n\nIn addition, the values of the decay asymmetries depend strongly on the complex phases, in our simplified example on the two phases $\\phi_{C_2}$ and $\\phi_{C_3}$ of $C_2$ and $C_3$, respectively. If $|C_2| \\not= |C_3|$ and $\\phi_{C_2} \\not= \\phi_{C_3}$, the differences between flavour-dependent treatment and flavour-independent approximation can be even more dramatic: \nTo illustrate this, let us focus on the dominant terms proportional to $m_3$ in $\\varepsilon_{1,\\mu}$ and $\\varepsilon_{1,\\tau}$ given in Eqs.~(\\ref{eq:emmCSDMC}) and (\\ref{eq:ettCSDMC}), which are proportional to \n\\begin{eqnarray}\n\\varepsilon_{1,\\mu} &\\propto& \\mathrm{Im}\\left[ C_2^* (C_2^* - C_3^*) \\right], \\\\\n\\varepsilon_{1,\\tau} &\\propto& \\mathrm{Im}\\left[- C_3^* (C_2^* - C_3^*) \\right].\n\\end{eqnarray}\nOn the other hand, in the flavour-independent approximation, the leading contribution to the decay asymmetry $\\varepsilon_1 = \\sum_\\alpha \\varepsilon_{1,\\alpha} = \\varepsilon_{1,\\mu}+ \\varepsilon_{1,\\tau}$ is proportional to \n\\begin{eqnarray}\n\\varepsilon_1 \n&\\propto& \\mathrm{Im}\\left[ (C_2^* - C_3^*)^2\\right]. \n\\end{eqnarray}\nWe can now imagine the situation that $\\mathrm{Im}\\left[ (C_2^* - C_3^*)^2\\right]$ vanishes exactly, which would correspond to a choice of phases $\\phi_{C_2}$ and $\\phi_{C_3}$ such that $\\mbox{Arg}(C_2 - C_3) \\in \\{0,\\tfrac{\\pi}{2}\\}$ (mod $\\pi$). \nObviously, the phases $\\phi_{C_2}$ and $\\phi_{C_3}$ can be chosen to satisfy this (for given $|C_2| \\not= |C_3|$) also with non-trivial $\\phi_{C_2},\\phi_{C_3} \\notin \\{0,\\tfrac{\\pi}{2}\\}$, for instance with \n\\begin{eqnarray}\\label{Eq:phasesCSD}\n\\frac{\\sin(\\phi_{C_2})}{\\sin(\\phi_{C_3})} = \\frac{|C_3|}{|C_2|},\n\\end{eqnarray} \nwhich implies $\\mathrm{Im}\\left[(C_2^* - C_3^*)\\right]=0$.\nThen, we have a situation where \n\\begin{equation}\n\\varepsilon_{1,\\mu} = - \\varepsilon_{1,\\tau} =: \\,\\hat \\varepsilon\n\\end{equation}\n are non-zero, whereas the corresponding (leading) contribution to $\\varepsilon_1$ vanishes exactly. Since the washout parameters are different, the leading contribution to the decay asymmetries, using Eqs.~(\\ref{Eq:eta_aa_MSSM}) and (\\ref{Eq:YB3f}) in the MSSM, can thus generates a baryon asymmetry of\n\\begin{eqnarray}\\label{Eq:CSDexample}\nY_B \\approx - \\frac{10}{31}\\, \\,(\\eta_{\\mu} - \\eta_{\\tau}) \\,\\, \\hat \\varepsilon \\, \\left[ Y^{\\mathrm{eq}}_{N_1}(z\\gg1) + Y^{\\mathrm{eq}}_{\\widetilde N_1}(z\\gg1)\\right]\\;, \n\\end{eqnarray} \n proportional to the difference of the efficiencies $\\eta_{\\mu}=\\eta(A_{\\mu\\mu}\\widetilde m_{1,\\mu}\/m^*,(\\sum_\\alpha \\widetilde m_{1,\\alpha})\/m^*)$ and $\\eta_{\\tau}=\\eta(A_{\\tau\\tau}\\widetilde m_{1,\\tau}\/m^*,(\\sum_\\alpha \\widetilde m_{1,\\alpha})\/m^*)$.\nLet us consider the MSSM with $\\tan \\beta = 50$ as an example, such that all flavours are to be treated independently for temperatures below about $2.5 \\times 10^{12}$ GeV. \nThe flavour-specific washout parameters $\\widetilde m_{1,\\mu}$ and $\\widetilde m_{1,\\tau}$ are given by $C_2^2 v^2_\\mathrm{u}\/M_1$ and $C_3^2 v^2_\\mathrm{u}\/M_1$, respectively (c.f.\\ Tab.~\\ref{tab:DecayAsym}). For instance, $\\widetilde m_{1,\\tau} \\approx m^*$ ($K_\\tau \\approx 1$) could lead to a nearly optimal efficiency factor $\\eta_{\\tau}$, while $C_2 \\ll C_3$, and correspondingly $\\widetilde m_{1,\\mu}\\ll m^*$ ($K_\\mu \\ll 1$), could imply a very low efficiency $\\eta_\\mu \\ll \\eta_{\\tau}$ (c.f.\\ Fig.~\\ref{fig:eta}). \nAccording to Eq.~(\\ref{Eq:CSDexample}), in the correct flavour-dependent treatment a baryon asymmetry can be generated, while in the flavour-independent approximation (in leading order in SD) it would vanish. This situation is illustrated in Fig.~\\ref{fig:CSD}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[scale=1,angle=0]{CSD_phases.eps}\n \\caption{\\label{fig:CSD}\nEvolution of the flavour specific $B\/3 - L_\\alpha$ asymmetries $Y_{\\Delta_\\mu}$ and $Y_{\\Delta_\\tau}$ as a function of $z=M_1\/T$ for the example of tri-bimaximal mixing via CSD, with $M_1 = M_C$ and the choice $C_1 = 0$. \nComplex phases $\\phi_{C_2}$ and $\\phi_{C_3}$ are chosen as in Eq.~(\\ref{Eq:phasesCSD}), leading to $\\varepsilon_{1,\\tau} = - \\varepsilon_{1,\\mu}$. \nWhile the flavour-independent approximation yields zero lepton asymmetry $Y_\\mathrm{ind}$, the two flavour-specific asymmetries evolve differently for different flavour-specific washout parameters $K_\\mu = \\widetilde m_{1,\\mu}\/m^*$ and $K_\\tau = \\widetilde m_{1,\\tau}\/m^*$, allowing a final baryon asymmetry to be produced (c.f.\\ Eq.~(\\ref{Eq:CSDexample})). \nThis situation can be realized, for example, with  \n$M_1 = M_C = 1.85 \\times 10^{11}$ GeV, \n$|C_1|=0, \n|C_2|=1.69 \\times 10^{-4}, |C_3|=3.09 \\times 10^{-3},\n\\phi_{C_2}= - \\pi\/2, \\phi_{C_3} = 0.028$ and\n$M_A = 5 \\times 10^{12}$ GeV, \n$|A_1|=0,|A_2|=|A_3|=0.063, \\phi_{A_2}=0, \\phi_{A_3}=0$. \nThe subdominant SD contribution to the decay asymmetries vanishes for $B_2 = 0, \\phi_{B_3} =  \\phi_{C_3}$. \nThe decay asymmetries are given by $\\epsilon_{1,\\tau} \\approx - 10^{-6}, \\epsilon_{1,\\mu} \\approx 10^{-6}$, and the washout parameters are\n$K_\\tau \\approx 1, K_\\mu \\approx 3 \\times 10^{-3}$. \n}\n\\end{figure}\n\n\n\\section{Summary and Conclusions}\\label{concl}\nWe have studied thermal leptogenesis in a class of neutrino mass models\nbased on the see-saw mechanism and SD, taking into account lepton\nflavour in the Boltzmann equations in both the SM and MSSM.\nSD models are a very well motivated class of models which have been\nwidely applied to unified flavour models, and thus are quite\nrepresentative of the general class of three right-handed neutrino\nmodels where the importance of flavour-dependent leptogenesis\neffects has hitherto not been evaluated.  \n\nIn general, flavour-dependent effects of leptogenesis are relevant \nat low enough temperatures (set by the lightest right-handed neutrino mass)\nsuch that at least one charged lepton flavour is in thermal equilibrium. \nWhen this condition is\nmet, flavour-dependent effects are important when the efficiency\nfactors $\\eta_{\\alpha}$ differ significantly for the distinguishable\nflavours. The efficiency factors $\\eta_{\\alpha}$ depend\non $\\tilde{m}_{1,\\alpha}$ which in SD are determined from the\nmagnitudes of the Yukawa couplings of the lightest right-handed\nneutrino to the different lepton flavours, as indicated in Tab.~\\ref{tab:DecayAsym}. \n\nIf the dominant right-handed neutrino is\nthe lightest one, then only $\\eta_{e}$ may differ,\nsince $|A_2|\\approx |A_3|$ implies that $\\eta_{\\mu}\\approx \\eta_{\\tau}$, \nand flavour-dependent effects are only important when $A_1\\neq 0$,\ncorresponding to non-zero texture element in\nthe (1,1)-element of the neutrino Yukawa matrix $\\lambda_\\nu$. We have studied this case\nnumerically and shown that dramatic effects can result.\nMoreover, we have also seen that, even with $A_1=0$,\nflavour-dependent effects are relevant for the leptogenesis-MNS\nlink since the decay asymmetries $\\epsilon_{1,\\mu},\\epsilon_{1,\\tau}$\nmay differ due to $B_2\\neq B_3$. \nAn extreme example of this is when there are two zero textures in the dominant and leading subdominant parts of $\\lambda_\\nu$, for instance $A_1=B_3=0$, in which case the cosmological CP violating phase for leptogenesis is directly related to the CP violating phase $\\delta$ of the MNS matrix observable in neutrino oscillations.\nFurthermore, the baryon asymmetry of the universe under this conditions determines the sign of the MNS phase $\\delta$ (i.e.\\ the sign of $\\sin(\\delta)$) for the case where the lightest right-handed neutrino is the dominant one. We have shown that this phase\nis opposite to the case where the lightest right-handed neutrino\nis the leading subdominant one, enabling the two types of SD to \nbe distinguished experimentally.\n\nFor the case that the lightest right-handed\nneutrino is the almost decoupled one, there is no leptogenesis-MNS link.\nNevertheless this case is amongst the most promising for thermal\nleptogenesis since the efficiencies may be optimal.\nAlthough this is also true in the flavour-independent treatment,\nflavour-dependent effects are expected to play \nan important role when the efficiences are unequal,\ncorresponding to the magnitudes of the Yukawa couplings $|C_{\\alpha}|$\nbeing unequal. Unlike the previous case,\nthis can be achieved with a zero texture in\nthe (1,1)-element of $\\lambda_\\nu$ corresponding to \n$C_1=0$, since here we can have $|C_2|\\neq |C_3|$ since \nthese\ncouplings are unconstrained by neutrino phenomenology,\nand so in general may be quite different.\nThus very large flavour-dependent effects can be present\nin this case, an extreme example being when \n$\\epsilon_{1,\\mu}$ and  $\\epsilon_{1,\\tau}$ \nare equal and opposite which would lead to a zero \nresult in the flavour-independent case, but a \nnon-zero result in the flavour-dependent case due to the\ndifferent efficiency factors.\n\nIn conclusion, we have seen that in many \ncases flavour-dependent effects may be important for \nleptogenesis in SD models involving three right-handed neutrinos\ndue to the efficiencies of the distinguishable flavours being different.\nThe effects range from factors of three, up to differences\nof a few orders of magnitude, or in extreme cases having a large non-zero \nflavour-dependent result for cases where the flavour-independent\nresult gives zero. In other cases \nthe flavour-dependence of leptogenesis is not important\nfor the asymmetry, due to the efficiencies of distinguishable\nflavours being equal, as in the case when the lightest right-handed\nneutrino dominates the see-saw mechanism, and there is a \nzero texture in the (1,1)-element of $\\lambda_\\nu$.\nHowever even in this case the relationship\nbetween leptogenesis phases and the MNS phase is affected\nby flavour-dependent effects due to the lepton asymmetries\nbeing unequal. We conclude that flavour-dependent effects \ncannot be ignored when dealing with three right-handed neutrino\nmodels.\n\n\n\\section*{Acknowledgements}\nThe work of S.~Antusch was supported by the EU 6$^\\text{th}$\nFramework Program MRTN-CT-2004-503369 ``The Quest for Unification:\nTheory Confronts Experiment''. S.~F.~King would like to acknowledge \na CERN Scientific Associateship. \n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe notion of probabilistic frames was first  introduced in \\cite{EhlerRTF} in the setting of probability measures on the unit sphere, and was later generalized to probability measures  on $\\mathbb R^d$ in \\cite{EhlOko2012}. In essence, this theory is a generalization of the theory of finite frames which has seen a wealth of activities in recent year, \\cite{CasKut2013, chri2003, KovChe1, KovChe2, OkoudjouFiniteFrame}.\n\n\\subsection{Review of finite frame theory}\\label{subsec1.1}\nBefore we give the definition and some elementary properties of probabilistic frames, we recall that a set $\\Phi=\\{\\varphi_i\\}_{i=1}^N \\subset \\mathbb R^d$ is a frame for $\\mathbb R^d$ if and only if there exist $0<A\\leq B<\\infty$ such that $$A\\|x\\|^2\\leq \\sum_{i=1}^N\\ip{x}{\\varphi_i}^2\\leq B\\|x\\|^2\\qquad \\forall\\, x\\in \\mathbb R^2.$$\nThe  frame $\\Phi$ is a \\emph{tight frame} if we can choose $A = B$.  Furthermore, if $A=B=1$, $\\Phi$ is called a Parseval frame. In the sequel the set of frames for $\\mathbb R^d$ with $N$ vectors will be denoted by $\\mathcal{F}(N,d)$, and simply  $\\mathcal{F}$ when the context is clear. The subset of frames with frame bounds $0<A\\leq B<\\infty$ will be denoted $\\mathcal{F}_{A, B}(N,d),$ or simply $\\mathcal{F}_{A,B}$.  We equip the set  $\\mathcal{F}(N, d)$  with the metric\n\\begin{equation}\\label{frame-metric}\nd(\\Phi, \\Psi)= \\sqrt{\\sum_{i=1}^{N}\\|\\varphi_i - \\psi_i\\|^2}=\\sqrt{\\sum_{i=1}^d\\|R_i-P_i\\|^2}\n\\end{equation} where $\\Phi=\\{\\varphi_i\\}_{i=1}^{N}, \\Psi=\\{\\psi_i\\}_{i=1}^{N }) \\in \\mathcal{F}(M,d),$   $\\{R_i\\}_{i=1}^d, \\{P_i\\}_{i=1}^d \\subset \\mathbb R^N$ denote the rows of $\\Phi$, and those of $\\Psi$, respectively.\n\n\n\nLet $\\Phi=\\{\\varphi_i\\}_{i=1}^N$ be a frame for $\\mathbb R^d$. Throughout the paper we shall abuse notation and denote the  \\emph{synthesis matrix} of the frame by   $\\Phi$,  the $d\\times N$ whose $i^{th}$ column is $\\varphi_i.$ The matrix $$S:=S_{\\Phi}=\\Phi \\Phi^{T}=\\sum_{i=1}^N \\ip{\\cdot}{\\varphi_i}\\varphi_i$$ is the \\emph{frame matrix}. It is known that $\\Phi=\\{\\varphi_i\\}_{i=1}^N$ is a frame for $\\mathbb R^d$ if and only if $S$ is a positive definite matrix. Moreover, the smallest eigenvalue of $S$ is the optimal lower frame bound, while its largest eigenvalue is the optimal upper frame bound. $\\Phi$ is a tight frame if and only if $S$ is a multiple of the $d\\times d$ identity matrix. In particular, $\\Phi$ is a Parseval frame if and only if $S=I$.\n\nIf $\\Phi$ is a frame, then $S$ is positive definite and thus invertible. Consequently,  $$\\Phi^{\\dag}=\\{\\varphi_i^\\dag\\}_{i=1}^N= \\{S^{-1\/2}\\varphi_i\\}_{i=1}^N$$ is a Parseval frame, leading to following reconstruction formula:\n\n$$x=\\sum_{i=1}^N\\ip{x}{\\varphi_i^{\\dag}}\\varphi_i=\\sum_{i=1}^N\\ip{x}{\\varphi_i} \\varphi_i^{\\dag}\\, \\forall\\, x\\in \\mathbb R^d.$$\n\n\n\nIn addition,  $\\Phi^\\dag$ is the unique Parseval frame which solves the following problem \\cite[Theorem 3.1]{CasKut07}:\n\n\\begin{equation}\\label{min-clos-pf}\n\\min\\{d(\\Phi, \\Psi)^2=\\sum_{i=1}^N\\|\\varphi_i-\\psi_i\\|^2: \\Psi=\\{\\psi_i\\}_{i=1}^N\\subset \\mathbb R^d, \\, \\, {\\textrm Parseval \\, frame}\\}.\n\\end{equation}\nTo be specific,\n\n\\begin{theorem}\\label{uniqueness_prob_1}\\cite[Theorem 3.1]{CasKut07}\n\nIf $\\Phi=\\{\\varphi_i\\}_{i=1}^N$ is a frame for $\\mathbb R^d$,  then $\\Phi^{\\dag}=\\{\\varphi_i^\\dag\\}_{i=1}^N= \\{S^{-1\/2}\\varphi_i\\}_{i=1}^N$ is the unique solution to~\\eqref{min-clos-pf}.\n\\end{theorem}\n\n\nIn Section~\\ref{sec2}, and for the sake of completeness, we give a new and simple proof of this result and we refer to  \\cite{Balan99, BodCas10, CahillCasazza13} for related results.\n\n\n\n\\subsection{Probabilistic frames}\\label{subsec1.2}\nThe main goal of this paper is to characterize the minimizers of an optimal problem analog of ~\\eqref{min-clos-pf} for probabilistic frames.  To motivate the definition of a probabilistic frame, we note that given a  frame $\\Phi = \\{\\varphi_i\\}_{i=1}^N \\subset \\mathbb R^{d}$, then the discrete probability  measure $$\\mu_{\\Phi}=\\tfrac{1}{N}\\sum_{k=1}^N\\delta_{\\varphi_k}$$ has the property that its support ($\\{\\varphi_k\\}_{k=1}^N$) spans $\\mathbb R^d$ and that it has finite second moment, i.e., $$\\int_{\\mathbb R^d}\\|x\\|^2d\\mu_{\\Phi}(x)=\\tfrac{1}{N}\\sum_{k=1}^N\\|\\varphi_k\\|^2< \\infty.$$  The  probability measure $\\mu_\\Phi$ is an example of a probabilistic frame that was introduced in \\cite{EhlerRTF, EhlOko2012}.\n\n\nMore specifically, a Borel probability measure $\\mu$   is a \\emph{probabilistic frame} if there exist $0<A\\leq B < \\infty$ such that for all $x\\in\\mathbb R^d$ we have\n\\begin{equation}\\label{pfineq}\n A\\|x\\|^2 \\leq \\int_{\\mathbb R^{d}} |\\langle x,y\\rangle |^2 d\\mu (y) \\leq B\\|x\\|^2.\n \\end{equation}\nThe constants $A$ and $B$ are called \\emph{lower and upper probabilistic frame bounds}, respectively.\nWhen $A=B,$ $\\mu$ is called a \\emph{tight probabilistic frame}. In particular, when $A=B=1$, $\\mu$ is called a \\emph{Parseval probabilistic frame}.\n\nA special class of probabilistic frames that will be considered in the sequel consists of discrete measures $\\mu_{\\Phi, w}=  \\sum_{i=1}^{N}w_i\\delta_{\\varphi_i}$ where  $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\subset \\mathbb R^d$, and $w=\\{w_i\\}_{i=1}^N\\subset [0, \\infty)$ is a set of  weights such that $\\sum_{i=1}^Nw_i=1$. A probability measure such as $\\mu_{\\Phi, w}$ will be termed  \\emph{finite probabilistic frame}, if and only if  it is a probabilistic frame for $\\mathbb R^d$. When the context is clear we will simply write $\\mu$ for $\\mu_{\\Phi, w}$.  We shall also identify a finite probabilistic frame $\\mu_{\\Phi, w}$ with the frame $\\Phi_w=\\{\\sqrt{w_i}\\varphi_i\\}_{i=1}^{N}$, as both have the same frame bounds. We refer to the surveys \\cite{EhlOko2013, KOPrecondPF2016} for an overview of the theory of probabilistic frames.\n\n\n\nWe shall prove an analog of Theorem~\\ref{uniqueness_prob_1} by endowing the set of probabilistic frames with  the Wasserstein metric.\nLet $\\mathcal{P}:=\\mathcal{P}(\\mathcal{B},\\mathbb R^d)$ denote the collection of probability measures on $\\mathbb R^d$ with respect to the Borel $\\sigma$-algebra $\\mathcal{B}$.\nLet $$\n\\mathcal{P}_{2}:= \\mathcal{P}_{2}(\\mathbb R^{d})=\\bigg\\{ \\mu \\in \\mathcal{P}: M_{2}^{2}(\\mu):=\\int_{\\mathbb R^{d}}\\nm{x}{}^{2}d\\mu(x) < \\infty\\bigg\\}$$\nbe the set of all probability measures with finite second moments.  For $\\mu, \\nu \\in \\mathcal{P}_2$, let\n$\\Gamma(\\mu, \\nu)$ be the set of all Borel probability measures  $\\gamma$ on $\\mathbb R^d \\times \\mathbb R^d$ whose marginals are $\\mu$ and $\\nu$, respectively, i.e., $\\gamma(A\\times \\mathbb R^d)=\\mu(A)$ and $\\gamma(\\mathbb R^d \\times B) = \\nu(B)$ for all Borel subset $A, B$ in $\\mathbb R^d$. The space\n$\\mathcal{P}_{2}$ is equipped with the $2$-\\emph{Wasserstein metric}  given by\n\\begin{equation}\\label{wmetric}\nW_{2}^{2}(\\mu, \\nu):=\\min\\bigg\\{\\int_{\\mathbb R^d \\times \\mathbb R^d}\\nm{x-y}{}^{2}d\\gamma(x, y), \\gamma \\in \\Gamma(\\mu, \\nu)\\bigg\\}.\n\\end{equation}\n The minimum defined by~\\eqref{wmetric} is achieved at a  measure $\\gamma_0 \\in \\Gamma(\\mu, \\nu)$, that is:\n$$W_{2}^{2}(\\mu, \\nu)=\\int_{\\mathbb R^d \\times \\mathbb R^d}\\nm{x-y}{}^{2}d\\gamma_0(x, y). $$\n  We refer to \\cite[Chapter 7]{AGS2005}, and  \\cite[Chapter 6]{Villani2009} for more details on the Wasserstein spaces.\n\n\n\\subsection{Our contributions}\\label{subsec1.3} The investigation of probabilistic frames is still at its initial stage. For example, in \\cite{WCKO16} the authors introduced the notion of transport duals and used the setting of the Wasserstein metric to investigate the properties of such probabilistic frames. In particular, this setting offers the flexibility to find (non-discrete) probabilistic frames which are duals to a given probabilistic frame.  Transport duals are the probabilistic analogues of alternate duals in frame theory \\cite{chri2003, Larsen}. The main contribution of this paper (Theorem~\\ref{maintheorem}) is to investigate the properties of the canonical Parseval probabilistic frame associated to a given  probabilistic frame, see Section~\\ref{sec2} for definitions. To prove this result we approximate a given probabilistic frame with one that is compactly supported and whose frame bounds are controlled in a precise way (Theorem~\\ref{density-dpf}).  In the process of proving our main result, we prove a number of results that are of interest on their own right. For example, in Section~\\ref{sec2} we  establish a number of new results about the canonical Parseval frame $\\Phi^{\\dag}$  associated to a frame $\\Phi$.\n\n\n\\section{Optimal Parseval probabilistic frames}\\label{sec2}\nBefore proving our main result in Section~\\ref{subsec2.3}, we revisit  the canonical Parseval frame $\\Phi^{\\dag}$ associated to a given frame $\\Phi=\\{\\varphi_k\\}_{k=1}^N\\subset \\mathbb R^d$. In particular, Section~\\ref{subsec2.1} considers the continuity  properties of the map $F(\\Phi)=\\Phi^\\dag$. In Section~\\ref{subsec2.2} we show how a probabilistic frame can be approximated in the $2$-Wasserstein metric  by a sequence of finite frames whose bounds are controlled by those of the initial probabilistic frame. While such approximation for  probability measures in the $2$-Wasserstein metric is well known \\cite[Theorem 6.18]{Villani2009}, our key contribution here is the control of the frame bounds of the approximating sequence.\n\n\n\\subsection{Continuity properties of the canonical Parseval frame}\\label{subsec2.1}\n\nIn this section we revisited the canonical Parseval frame $\\Phi^{\\dag}$ associated to a given frame $\\Phi=\\{\\varphi_k\\}_{k=1}^N\\subset \\mathbb R^d$.\nFirst, we give a new and elementary proof of Theorem~\\ref{uniqueness_prob_1}.\n\n\n\n\\begin{proof}{Proof of Theorem~\\ref{uniqueness_prob_1}}\nWe first note that a frame $\\Psi\\subset \\mathbb R^d$ is Parseval if the rows of its synthesis matrix are orthonormal. Furthermore, $\\Psi \\subset \\mathbb R^d$ is a Parseval frame if and only if $U\\Psi$ is a Parseval frame for any $d\\times d$ orthogonal matrix $U$.\n\nNow let   $\\Phi=\\{\\varphi_i\\}_{i=1}^N$ be a frame for $\\mathbb R^d$.  Write $S=\\Phi\\Phi^T= UDU^T$ for some orthogonal matrix $U$. Observe that $U^T\\Phi$ is the matrix of $\\Phi$ written with respect to the orthonormal basis given by the rows of $U^T$. In addition, the rows of $U^T\\Phi$ are pairwise  orthogonal. Let $\\Psi=\\{\\psi_i\\}_{i=1}^N\\subset \\mathbb R^d$ be any Parseval frame, then\n\n\n$$d^2(\\Phi, \\Psi)=d^2(U^T\\Phi, U^T\\Psi)=\\sum_{i=1}^d\\|R_i-P_i\\|^2,$$ where $\\{R_i\\}_{i=1}^d\\subset \\mathbb R^N$ and $\\{P_i\\}_{i=1}^d\\subset \\mathbb R^N$  denote respectively  the rows of $U^T\\Phi$  and $U^T\\Psi$.\nConsequently, finding $$\\min\\{d(\\Phi, \\Psi)^2=\\sum_{i=1}^N\\|\\varphi_i-\\psi_i\\|^2: \\Psi=\\{\\psi_i\\}_{i=1}^N\\subset \\mathbb R^d, \\, \\, {\\textrm Parseval \\, frame}\\}$$ is equivalent to finding\n$$\\min \\{\\sum_{i=1}^d\\|R_i-P_i\\|^2: \\{P_i\\}_{i=1}^N\\subset \\mathbb R^N, \\, \\, {\\textrm orthonormal \\, set }\\}$$  where $\\{R_i\\}_{i=1}^d$ form an orthogonal set of vectors in $\\mathbb R^N$.\n\nBut $\\Phi^\\dag=\\{S^{-1\/2}\\varphi_i\\}_{i=1}^N$  is a Parseval frame, so its rows form an orthonormal set in $\\mathbb R^N$. Consequently,  $\\Phi^\\dag$  is  a solution to~\\eqref{min-clos-pf}.  The uniqueness follows by observing that the (unique) closest orthonormal set to a given orthogonal vectors $\\{u_i\\}_{i=1}^d\\subset \\mathbb R^N$ is $\\{\\tfrac{u_i}{\\|u_i\\|}\\}_{i=1}^d.$\n\nConsequently, $$\\min\\{d(\\Phi, \\Psi)^2=\\sum_{i=1}^N\\|\\varphi_i-\\psi_i\\|^2: \\Psi=\\{\\psi_i\\}_{i=1}^N\\subset \\mathbb R^d, \\, \\, {\\textrm Parseval \\, frame}\\}=\\sum_{k=1}^d(1-\\lambda_k^{-1\/2})^2 $$ where $\\{\\lambda_k\\}_{k=1}^d\\subset (0, \\infty)$ are the eigenvalues of $S=\\Phi\\Phi^T$.\n\n\\end{proof}\n\nIn the remaining part of  section we study the continuity properties of the functions that maps a given frame to its canonical Parseval frame. This map  $$F: \\mathcal{F}(N, d) \\rightarrow \\mathcal{F}(N,d)$$ given by\n\\begin{equation}\\label{func-F}\nF(\\Phi)=F(\\{\\varphi_i\\}_{i=1}^{N}) = S_{\\Phi}^{-1\/2}(\\{\\varphi_i\\}_{i=1}^{N})=\\{S_{\\Phi}^{-1\/2}\\varphi_i\\}_{i=1}^N.\n\\end{equation} In fact, our results show that for $0< A\\leq B$, $F$ is uniformly continuous on $\\mathcal{F}_{A, B}$, the set of frames with frame bounds between $A$ and $B$. More specifically,\n\n\\begin{theorem}\\label{cont-F}\nLet $0<A\\leq B <\\infty$, and  $\\delta > 0$ be given. Then there  exists  $\\epsilon > 0$ such that given any frame  $\\Phi=\\{\\varphi_i\\}_{i=1}^N$,  with frame bounds between $A$ and $B$, and $N:=N_{\\Phi}\\geq 2$, for any  frame $\\Psi=\\{\\psi_i\\}_{i=1}^N$ such that $d(\\Phi, \\Psi)< \\epsilon$ we have $d(F(\\Phi), F(\\Psi))< \\delta.$\n\\end{theorem}\n\nBefore proving this theorem, we establish a number of preliminary results and make the following remark that will be used in the sequel.\n\n\n\n\\begin{remark}\\label{rowsframe}\n Let $\\Phi=\\{\\varphi_i\\}_{i=1}^{N}\\in \\mathcal{F}(N, d)$ be a frame. Then,  $S=\\Phi \\Phi^{T}= ODO^{T}$ where $O$ is a $d\\times d$ orthogonal matrix and $D$ is a positive definite diagonal matrix. Fix the orthonormal basis of $\\mathbb R^d$ whose columns form the matrix $O$ and write each frame vector $\\varphi_i$ in this basis. The synthesis matrix of the frame $\\Phi$ in the basis $O$ is $$[\\Phi]_{O}=O^{T}\\Phi.$$ Let $\\{R_i\\}_{i=1}^d$ be the rows of $[\\Phi]_{O}$. We shall refer to $\\{R_i\\}_{i=1}^d$ as simply the rows of  $\\Phi$.\n\\end{remark}\n\n\n\n\\begin{lemma}\\label{rows-prop} Let $\\Phi=\\{\\varphi_i\\}_{i=1}^{N} \\in \\mathcal{F}(N, d)$. Denote by $\\{R_i\\}_{i=1}^{d}$ the rows of $\\Phi$  as described by Remark~\\ref{rowsframe}.  Let $\\epsilon>0$ and $ \\Psi=\\{\\psi_i\\}_{i=1}^{N} \\in \\mathcal{F}(N, d)$ be such that $d(\\Phi, \\Psi) <  \\epsilon $. Denote by $\\{P_i\\}_{i=1}^d$ the rows of $\\Psi$ when written in the orthonormal basis $O$.\nThen\n\\begin{enumerate}\n\\item[(a)]  $\\big|\\|\\ R_i\\|-\\|P_i\\|\\big|< \\epsilon.$ Furthermore, $\\sqrt{A}-\\epsilon <\\|P_i\\|<\\sqrt{B}+\\epsilon$ for each $i=1, 2, \\hdots, d.$\n\\item[(b)] $$d(\\Phi,F(\\Phi)) \\geq \\sqrt{\\sum_{i=1}^{d}\\|R_i - \\dfrac{R_i}\n{\\|R_i\\|}\\|^2}.$$\n\\item[(c)] For each $i\\in \\{1, 2\\, \\hdots, d\\}$ we have\n$$\\bigg\\|\\dfrac{P_i}{\\|P_i\\|} - \\dfrac{R_i}{\\|R_i\\|}\\bigg\\| <  \\dfrac{2\\epsilon}{\\sqrt{A}}.$$\n\\item[(d)] For each $i\\in \\{1, 2\\, \\hdots, d\\}$ we have $$0\\leq \\|P_i-\\tfrac{R_i}{\\|R_i\\|}\\|^2-\\|P_i-\\tfrac{P_i}{\\|P_i\\|}\\|^2\\leq \\tfrac{4\\epsilon}{\\sqrt{A}}c+\\tfrac{4\\epsilon^2}{A},$$ where $c=max(1-\\sqrt{A}+\\epsilon, \\sqrt{B} + \\epsilon - 1)$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\n\n\\begin{proof}\n\\begin{enumerate}\n\\item[(a)] This is trivial so we omit it.\n\\item[(b)] This follows immediately from the fact that the rows of a Parseval frame are an orthonormal set when written with respect to any orthonormal basis and $\\dfrac{R_i}{\\|R_i\\|}$ is the closest unit norm vector  to $R_i$.\n\\item[(c)]\nSince, $d(\\Phi, \\Psi)<\\epsilon,$ we know that $\\big|\\|P_i\\| - \\|R_i\\|\\big| < \\epsilon$. Hence\n$$ \\bigg\\|\\dfrac{P_i}{\\|P_i\\|}\\cdot\\|R_i\\| - R_i\\bigg\\| \\leq \\bigg\\|\\dfrac{P_i}{\\|P_i\\|}\\cdot\\|R_i\\| - P_i\\bigg\\| +\\|P_i-R_i\\|= \\big|\\|P_i\\| - \\|R_i\\|\\big|+\\|P_i-R_i\\|< 2\\epsilon.$$\nThe result follows by recalling that $\\|R_i\\|\\geq \\sqrt{A}$.\n\\item[(d)]  It is clear that $\\|P_i -\\dfrac{P_i}{\\|P_i\\|}\\| =|\\|P_i\\|-1| \\leq max(1-\\sqrt{A}+\\epsilon, \\sqrt{B} + \\epsilon - 1) = c$.\nBy part (c) we know that  $\\|\\dfrac{P_i}{\\|P_i\\|} - \\dfrac{R_i}{\\|R_i\\|}\\| <  \\dfrac{2\\epsilon}{\\sqrt{A}}$. Using the fact hat $\\dfrac{P_i}{\\|P_i\\|}$ is the closest unit norm vector to $P_i$,  we see that\n$$\\|P_i - \\dfrac{P_i}{\\|P_i\\|}\\| \\leq \\|P_i - \\dfrac{R_i}{\\|R_i\\|}\\| \\leq \\|P_i - \\dfrac{P_i}{\\|P_i\\|}\\| + \\dfrac{2\\epsilon}{\\sqrt{A}}.$$\nThe result follows by squaring the last inequality.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\nFinally, we have the following technical lemma, that contains the key argument in the proof of Theorem~\\ref{cont-F}.\n\n\n\\begin{lemma}\\label{contradiction-unifcont} Given $0<A\\leq B<0$, fix $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\in \\mathcal{F}_{A,B}$. Let $\\epsilon, \\delta>0$ be such that $\\dfrac{\\delta}{\\sqrt{d}} - \\dfrac{2\\epsilon}{\\sqrt{A}}>0$ and $\\sqrt{A} - \\epsilon>0$.  Let $\\Psi=\\{\\psi_i\\}_{i=1}^N$ be such that  $d(\\Phi, \\Psi) <  \\epsilon$, and $d(S_{\\Phi}^{-1\/2}\\Phi, S^{-1\/2}_{\\Psi}\\Psi) =d(\\Phi^{\\dag}, \\Psi^{\\dag})> \\delta$.  Then,\n$$\\sum_{i=1}^{d}(\\|P_i - R_{i}'\\|^2 - \\|P_i - \\dfrac{P_i}{\\|P_{i}\\|}\\|^2) \\geq min(Cd'^2,C^2), $$ where\n$d' = \\dfrac{\\delta}{\\sqrt{d}} - \\dfrac{2\\epsilon}{\\sqrt{A}}$, $C = \\min(\\sqrt{A} - \\epsilon, 1)$, and $\\{R_i'\\}_{i=1}^d\\subset \\mathbb R^d$ is the set of the rows of $S_{\\Psi}^{-1\/2}\\Psi$.\n\\end{lemma}\n\n\n\\begin{proof} \nWe first show  that there exists $k$ then $$\\|P_k- R_{k}'\\|^2 - \\|P_k - \\dfrac{P_k}{\\|P_{k}\\|}\\|^2 \\geq min(\\|R'_{k} - \\frac{P_k}{\\|P_k\\|}\\|^2 \\cdot min(\\|P_k\\|,1),\\|P_k\\|^2).$$\n\n\n\nSince $d(S_{\\Phi}^{-1\/2} \\Phi,S_{\\Psi}^{-1\/2} \\Psi) \\geq \\delta$,  then $\\|\\dfrac{R_k}{\\|R_k\\|} - R_{k}'\\| \\geq \\dfrac{\\delta}{\\sqrt{d}}$ for some $k$. By Lemma~\\ref{rows-prop} we know that   $\\|\\dfrac{P_k}{\\|P_k\\|} - \\dfrac{R_k}{\\|R_k\\|}\\| <  \\dfrac{2\\epsilon}{\\sqrt{A}}$. It follows from the triangle inequality that\n $$\\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\| \\geq \\dfrac{\\delta}{\\sqrt{d}} - \\dfrac{2\\epsilon}{\\sqrt{A}} = d'.$$\n\nSuppose that $C =\\min(\\sqrt{A} - \\epsilon, 1)= 1$, or equivalently,  $\\sqrt{A} - \\epsilon \\geq 1$. Hence, by Lemma~\\ref{rows-prop} we have $\\|P_i\\|\\geq 1$ for each for all $i$.\n\n\nSince the angle $\\widehat{R_k'\\tfrac{P_k}{\\|P_k\\|}P_i}> \\pi\/2$, it follows that   $$\\|P_k - R_{k}'\\|^2 > \\|P_k - \\dfrac{P_k}{\\|P_k\\|}\\|^2 + \\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\|^2.$$ But since,  $\\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\| \\geq \\dfrac{\\delta}{\\sqrt{d}} - \\dfrac{2\\epsilon}{\\sqrt{A}}$,  we conclude that\n$$  \\|P_k- R_{k}'\\|^2 - \\|P_k - \\dfrac{P_k}{\\|P_k\\|}\\|^2>   \\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\|^2 \\geq d'^2 = Cd'^2$$ and we are done.\n\n\n\n\nAssume now $C=\\sqrt{A}-\\epsilon < 1$  and $\\|P_k\\|+\\eta \\leq 1$, where $\\eta$ is defined in Figure~\\ref{fig:figure1}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=.35]{Ball1.png}\n\\end{center}\n\\caption{  $Q$ is  the orthogonal projection of $R'_k$ onto  $P_k$, and $\\eta= \\|Q - \\frac{P_k}{\\|P_k\\|}\\|$. }\\label{fig:figure1}\n\\end{figure}\n\n\n\nThen,\n\\begin{align*}\n\\bigg\\|P_k - R_{k}'\\|^2 - \\|P_k - \\dfrac{P_k}{\\|P_{k}\\|}\\bigg\\|^2 &=\n(1 - ( \\|P_k\\|+ \\eta))^2 + 2\\eta - \\eta^2 - (1-\\|P_k\\|)^2 \\\\\n&= 2\\eta \\|P_k\\| \\\\\n&= \\bigg\\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\bigg\\|^{2}\\|P_k\\|.\n\\end{align*} The  the conclusion follows  from  $\\|\\dfrac{P_k}{\\|P_k\\|} - R_{k}'\\|^{2} \\geq d'^2$.\n\nNow assume $\\|P_k\\|+\\eta > 1$ and $\\eta \\leq 1$, where $\\eta$ is defined in Figure~\\ref{fig:figure2}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=.35]{Ball2.png}\n\\end{center}\n\\caption{$Q$ is  the orthogonal projection of $R'_k$ onto  $P_k$, and $\\eta= \\|Q - \\frac{P_k}{\\|P_k\\|}\\|$.}\\label{fig:figure2}\n\\end{figure}\n\n\n\n\n\n$$\\bigg\\|P_k - R_{k}'\\|^2 - \\|P_k - \\dfrac{P_k}{\\|P_{k}\\|}\\bigg\\|^2 = ((\\|P_k\\| + \\eta) - 1)^2 + 2\\eta - \\eta^2 - (1-\\|P_k\\|)^2 = 2\\eta \\|P_k\\|$$ and the rest of the proof is similar to the one given above.\n\n\n\n\n\n\nIf $\\eta > 1$ where where $\\eta$ is defined in Figure~\\ref{fig:figure3}, then the angle $\\angle P_{k}0R'_{k} > \\frac{\\pi}{2}$ hence $\\|P_k - R_{k}'\\|^2 > \\|P_k\\|^2 + 1$. We know $\\|P_k - \\dfrac{P_k}{\\|P_{k}\\|}\\|^2 \\leq 1$ hence $$\\|P_k - R_{k}'\\|^2 - \\|P_k - \\dfrac{P_k}{\\|P_{k}\\|}\\|^2 > \\|P_k\\|^2 \\geq C^2$$\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[scale=.35]{Ball3.png}\n\\end{center}\n\\caption{$Q$ is  the orthogonal projection of $R'_k$ onto  $P_k$, and $\\eta= \\|Q - \\frac{P_k}{\\|P_k\\|}\\|$.}\\label{fig:figure3}\n\\end{figure}\n\n\n\n\n\\end{proof}\nWe are now ready to prove Theorem~\\ref{cont-F}.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{cont-F}]\nAssume by way of contradiction that there exists $\\delta > 0$ such that for all $\\epsilon > 0$ there exist $\\Phi_{\\epsilon}=\\{\\varphi_{i,\\epsilon}\\}_{i=1}^{N_{\\epsilon}}, \\in \\mathcal{F}_{A, B}.$ and\n$\\Psi_\\epsilon=\\{\\psi_{i,\\epsilon}\\}_{i=1}^{N_{\\epsilon}}$\n\n\n\n\n\n such that $$d(\\Phi_{\\epsilon}, \\Psi_{\\epsilon})< \\epsilon$$ and $$d(S_{\\Phi_{\\epsilon}}^{-1\/2}\\Phi_{\\epsilon}, S_{\\Psi_{\\epsilon}}^{-1\/2}\\Psi_{\\epsilon}) > \\delta.$$  Furthermore, choose  $\\epsilon$ small enough so that $\\dfrac{\\delta}{\\sqrt{d}} - \\dfrac{2\\epsilon}{\\sqrt{A}}>0$ and $\\sqrt{A} - \\epsilon>0$ and\n  $$\\sum_{i=1}^{d}(\\|P_i - \\dfrac{R_i}{\\|R_i\\|}\\|^2 - \\|P_i - \\dfrac{P_i}{\\|P_i\\|}\\|^2) <  min(Cd'^2,C^2)$$ where $C$ and $d'^2$ are as in Lemma~\\ref{contradiction-unifcont}(such $\\epsilon$ exists by Lemma~\\ref{rows-prop}).\n\n  Hence $$\\sum_{i=1}^{d}(\\|P_i - \\dfrac{R_i}{\\|R_i\\|}\\|^2 - \\|P_i - \\dfrac{P_i}{\\|P_i\\|}\\|^2) <  \\sum_{i=1}^{d}(\\|P_i - R_{i}'\\|^2 - \\|P_i - \\dfrac{P_i}{\\|P_{i}\\|}\\|^2)$$ Consequently,  $\\sum_{i=1}^{d}\\|P_i - \\dfrac{R_i}{\\|R_i\\|}\\|^2 <  \\sum_{i=1}^{d}\\|P_i - R_{i}'\\|^2$ contradicting that $R_{i}'$ are the rows of the closest Parseval frame to $\\Psi_{\\epsilon}=\\{\\psi_{i,\\epsilon}\\}_{i=1}^{N_{\\epsilon}}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Approximation of probabilistic frames in the $2-$ Wasserstein metric}\\label{subsec2.2}\nIn this section we prove some of the technical results needed to establish our main result.\nThe key idea is that a probabilistic frame $\\mu$ with frame bounds $A, B$ can be approximated in the Wasserstein metric by a finite probabilistic frame whose bounds are arbitrarily close to $A, B$.   We prove this statement in Proposition~\\ref{density-dpf} and point out that it is a refinement of a well-known result, e.g.,  \\cite[Theorem 6.18]{Villani2009}. But first, we prove a few new results about finite probabilistic frames that are of interest in their own right.  In particular, Lemma~\\ref{frame_modi} will be a very useful technical tool that we shall often use. It shows that given a finite frame we may replace any frame vector by a finite number of new vectors so as to leave unchanged the frame operator. More specifically,\n\n\\begin{lemma}\\label{frame_modi}\nGiven a frame $\\Phi=\\{\\varphi_{i}\\}_{i=1}^N$ with frame operator $S_\\Phi$. Fix $i\\in \\{1, 2, \\hdots, N\\}$ and consider the new set of vectors $$\\Phi_i=\\{\\varphi_k\\}_{k=1, k\\neq i}^N \\cup\\{a_{j}\\varphi_i\\}_{j=1}^{p}= \\{\\varphi_k'\\}_{k=1}^{N+p-1} $$ where $\\sum_{j=1}^{p}a_{j}^{2} = 1$. Then, $\\Phi_i\\in \\mathcal{F}(N+p-1, d)$, that is, $\\Phi_i$ is  a frame for $\\mathbb R^d$ and its frame operator is $S_{\\Phi}$. Furthermore, $$\\sum_{k=1}^{N}\\|\\varphi_k- \\varphi_k^{\\dag}\\|^2 = \\sum_{k=1}^{N+p-1}\\|\\varphi_k' -  \\varphi_k^{' \\dag}   \\|^2$$ where $\\varphi_k^{\\dag}=S^{-1\/2}\\varphi_k$ and $\\varphi_k^{'\\dag}=S^{-1\/2}\\varphi'_k$\n\\end{lemma}\n\n\n\n\\begin{proof} It is easy to see that for each $x\\in \\mathbb R^d$ we have\n$$\\sum_{k=1}^{N}|\\ip{x}{\\varphi_k}|^2 = \\sum_{k=1}^{i-1}|\\ip{x}{\\varphi_k}|^2 + \\sum_{j=1}^{p}a_j^2|\\ip{x}{\\varphi_i}|^2 + \\sum_{k=i+1}^{N}|\\ip{x}{\\varphi_k}|^2 .$$\n\\end{proof}\n\n\n\n\nWe now use Lemma~\\ref{frame_modi} and Theorem~\\ref{uniqueness_prob_1} to find the closest Parseval frame to a finite probabilistic frame in the $2$-Wasserstein metric.\n\n\\begin{proposition}\\label{dists-to-parseval}\nLet  $\\mu_{\\Phi, w}$ be a finite probabilistic frame with bounds $A$ and $B$, where $\\Phi=\\{\\varphi_{i}\\}_{i=1}^N \\subset \\mathbb R^{d}$ and  $w=\\{w_i\\}_{i=1}^N \\subset [0, \\infty)$. Then the closest finite Parseval probabilistic frame to $\\Phi$ is $\\Phi^\\dag=\\{S^{-1\/2}\\varphi_i\\}_{i=1}^N$ and it satisfies\n\n$$W_{2}(\\mu_{\\Phi, w}, \\mu_{\\Phi^{\\dag}, w})=\\sqrt{\\sum_{i=1}^{N}w_{i}\\|\\varphi_i - \\tilde{\\varphi}_i\\|^2} \\leq  \\sqrt{d\\, \\max((\\sqrt{A} - 1)^2,(\\sqrt{B} - 1)^2)}$$\n where $\\tilde{\\varphi}_i=S^{-1\/2}\\varphi_i$.\n\\end{proposition}\n\n\n\n\\begin{proof}\nWe first prove that $W_{2}^2(\\mu_{\\Phi, w}, \\mu_{\\Phi^{\\dag}, w}) \\leq  d\\, \\max((\\sqrt{A} - 1)^2,(\\sqrt{B} - 1)^2).$\n\n\n\nLet  $\\Phi_w=\\{\\sqrt{w_i}\\varphi_{i}\\}_{i=1}^N$. Let $S=\\Phi_w\\Phi^T_w=ODO^T$ be the frame operator of $\\Phi_w$. Consider the columns of $O$ as an orthonormal basis for $\\mathbb R^d$. Writing the vectors $\\sqrt{w}_k\\varphi_k$ with respect to this basis leads to  $\\Phi_w'=O^T\\Phi_w$ where  \\[\\Phi_w = \\left( \\begin{array}{ccc}\n| & ... & | \\\\\n\\sqrt{w_1}\\varphi_1 & ... & \\sqrt{w_n}\\varphi_m \\\\\n| & ... & | \\end{array} \\right)\\]\nLet $\\{P_{k, w}\\}_{k=1}^d$ and $\\{R_{k, w}\\}_{k=1}^d$  respectively denote the rows of $\\Phi_w'$ and  $\\Phi_w$. Notice that $$\\sqrt{A}\\leq \\|P_{k,w}\\|\\leq \\sqrt{B}, \\quad \\forall\\, k=1, 2, \\hdots, d.$$ It is easily seen that\n\n$$\\min_{u\\in \\mathbb R^d, \\|u\\|=1}\\|P_{k,w}-u\\|^2=\\|P_{k,w}-\\tfrac{P_{k,w}}{\\|P_{k,w}\\|}\\|^2=|\\|P_{k,w}\\|-1|^2\\leq \\max((\\sqrt{A} - 1)^2,(\\sqrt{B} - 1)^2).$$ But by construction, $\\ip{P_{k,w}}{P_{\\ell,w}}=0$ for $k\\neq \\ell$, and $ \\tfrac{P_{k,w}}{\\|P_{k,w}\\|}=\\lambda_k^{-1\/2}P_{k,w}$ where $\\lambda_k$ is the $k^{th}$ eigenvalue of $S$. Consequently, $\\{\\lambda_k^{-1\/2}P_{k,w}\\}_{k=1}^d$ represents the rows of the canonical tight frame $S^{-1\/2}\\Phi_w$ written in the orthonormal basis $O$. Therefore,\n\n$$d(\\Phi_w, S^{-1\/2}\\Phi_w)^2=\\sum_{k=1}^d\\|P_{k,w}-\\lambda_k^{-1\/2}P_{k,w}\\|^2\\leq d \\max((\\sqrt{A} - 1)^2,(\\sqrt{B} - 1)^2).$$\n\n\nClearly,\n$$W_{2}^2(\\mu_{\\Phi, w}, \\mu_{S^{-1\/2}\\Phi, w}) \\leq \\sum_{i=1}^{N}w_{i}\\|\\varphi_{i} - S^{-1\/2}\\varphi_{i}\\|^2=d(\\Phi_w, S^{-1\/2}\\Phi_w)^2\\leq d \\max((\\sqrt{A} - 1)^2,(\\sqrt{B} - 1)^2) .$$\nSuppose there exists a finite probabilistic Parseval frame $\\mu_{\\Psi, v}$ where $\\Psi =\\{\\psi_i\\}_{i=1}^{M}\\subset \\mathbb R^d$, $v=\\{v_i\\}_{i=1}^M\\subset [0, \\infty)$ such that $$W_{2}^2(\\mu_{\\Phi, w}, \\mu_{\\Psi, v}) < \\sum_{i=1}^{N}w_{i}\\|\\varphi_{i} - S^{-1\/2}\\varphi_{i}\\|^2.$$\nLet $\\gamma \\in \\Gamma (\\mu_{\\Phi, w}, \\mu_{\\Psi, v})$ be such that $$W_2^2(\\mu_{\\Phi, w}, \\mu_{\\Psi, v}) =\\iint_{\\mathbb R^{2d}} \\|x-y\\|^2d\\gamma(x,y).$$\nNote that $\\gamma$ is a discrete measure with $\\gamma(x,y)=\\sum_{i, j}w'_{i,j}\\delta_{\\varphi_{i}}(x)\\delta_{\\psi_{i}}(y)$ with\n $\\sum_{j}w'_{i,j} = w_i$ and $\\sum_{i}w'_{i,j} = v_j$.\n\nFurthermore, by assumption $$W_{2}^2(\\mu_{\\Phi, w}, \\mu_{\\Psi, v})=\\sum_{i,j}w'_{i,j}\\|\\varphi_i - \\psi_j\\|^2 <  \\sum_{i=1}^{N}w_{i}\\|\\varphi_{i} - S^{-1\/2}\\varphi_{i}\\|^2.$$\nNotice since $\\sum_{i}w'_{i,j} = v_j$ the frame $\\Psi'=\\{\\sqrt{w'_{i,j}}\\psi_j\\}_{i,j}$ is a Parseval frame. Since $\\sum_{j}w'_{i,j} = w_i$, it easy to see that $\\sum_{j} \\tfrac{w'_{i,j}}{w_i}   =1$.  We now  use Lemma~\\ref{frame_modi}. For each $i,$ replace  $\\sqrt{w_i}\\varphi_i$ with $\\{\\sqrt{w_{i,j}'}\\varphi_i\\}_{j}$. This results in a frame $\\Phi'=\\{\\sqrt{w_{i,j}'}\\varphi_i\\}_{i, j}$. Consequently, $d(\\Phi', \\Psi')=d(\\Phi_w, \\Psi_v)<d(\\Phi_w, \\Phi_w^{\\dag})$ where $\\Psi_v$ is a Parseval frame. This is a contradiction.\n\\end{proof}\n\n\n\n\nThe next result is one of our key technical results. It allows us to approximate a probabilistic frame in the $2$-Wasserstein metric with a compactly supported finite probabilistic frame whose bounds are controlled by those of the original probabilistic frame.\n\n\n\n\n\n\n\\begin{theorem}\\label{density-dpf}\nLet $\\mu$ be a  probabilistic frame with frame bounds $A$ and $B$, and $\\epsilon > 0$. Then,  there exists a finite probabilistic $\\mu_{\\Phi}$ with frame bounds $A', B'$ such that $A'\\geq A-\\epsilon$, $B'\\leq B+\\epsilon$ and $$\\|\\mu - \\mu_{\\Phi}\\|_{W_2}<\\epsilon.$$\n\\end{theorem}\n\n\nTo establish this result we first prove the following two Lemmas.\n\n\n\n\\begin{lemma}\\label{prop-2}\nLet $\\mu$ be a probabilistic frame with frame bound $A$ and $B$. Given $\\epsilon>0$, there exists a probabilistic frames $\\nu$ with compact support and frame bounds $A', B'$ such that\n\\begin{enumerate}\n\\item[(a)] $W_2^2(\\mu, \\nu)< \\epsilon$,\n\\item[(b)] $A'\\geq A-\\epsilon$, and $B'=B$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\begin{proof}\n\\begin{enumerate}\n\\item[(a)] Let $\\mu$ be a probabilistic frame with frame bound $A$ and $B$. Given $\\epsilon>0$, there exists  $R_1>0$ such that $$\\int_{\\mathbb R^{d}\\setminus B(0, R_1)} \\|x\\|^{2}d\\mu(x) < \\epsilon.$$\n\nLet $\\nu$ be the measure defined for each Borel set $A\\subset \\mathbb R^d$ by  $$\\nu(A) = \\mu(A\\bigcap B(0,R_1)+ \\mu(\\mathbb R^d\\setminus B(0,R_1))\\delta_0.$$  Clearly,  $\\nu$ is a probabilistic measure with compact support.\n\n We consider the marginal $\\gamma$ of $\\mu$ and $\\nu$ defined for each Borel sets $A, B\\subset \\mathbb R^d$   by\n \\begin{equation*}\n\\gamma(A\\times B)= \\left\\{ \\begin{array} {r@{\\quad {\\textrm if} \\quad}l}\n\\mu(A\\bigcap B(0,R_1) \\bigcap B) + \\mu(A\\bigcap B^{c}(0,R_1) & 0\\in B\\\\\n\\mu(A\\bigcap B(0,R_1) \\bigcap B) & 0\\not\\in B \\end{array}\\right.\n\\end{equation*}\nSince $\\nu $ is supported in $B(0,R_1)$\n \\begin{align*}\n \\iint_{\\mathbb R^{2d}}\\|x-y\\|^{2}d\\gamma(x,y) &= \\iint_{\\mathbb R^d \\times B(0,R_1)}\\|x-y\\|^{2}d\\gamma(x,y)\\\\\n &=\\iint_{B(0,R_1) \\times B(0,R_1)}\\|x-y\\|^{2}d\\gamma(x,y)  \\\\\n &+ \\iint_{B^{c}(0,R_1)\\times B(0,R_1)}\\|x-y\\|^{2}d\\gamma(x,y).\n \\end{align*}\n However,  we know\n$$\\int_{B(0,R_1)\\times B(0,R_1)}\\|x-y\\|^{2}d\\gamma(x,y)=0$$ since, when restricted to $B(0,R_1)\\times B(0,R_1)$, $\\gamma$ is supported only on the diagonal where $\\|x-y\\| = 0.$ Moreover,\n\\begin{align*}\n\\int_{B^{c}(0,R_1)\\times B(0,R_1)}\\|x-y\\|^{2}d\\gamma(x,y) &=\\iint_{B^{c}(0,R_1) \\times B(0,R_1)\\setminus \\{0\\}}\\|x-y\\|^{2}d\\gamma(x,y)\\\\\n& + \\iint_{B^{c}(0,R_1)\\times \\{0\\}}\\|x-y\\|^{2}d\\gamma(x,y)\\\\\n&=0+\\iint_{B^{c}(0,R_1)\\times \\{0\\}}\\|x-y\\|^{2}d\\gamma(x,y)\\\\\n&<\\epsilon.\n\\end{align*}\nTherefore, $W_2^2(\\mu, \\nu)<\\epsilon.$\n\n \\item[(b)] The upper bound $B$ is obtained trivially as $\\nu$ is $\\mu$ restricted to $B(0,R_1)$.\n\nFor  $x\\in \\mathbb R^d$ we have $\\int|\\left\\langle  x,y\\right\\rangle|^{2}d\\nu(y) = \\int_{B(0,R_1)}|\\left\\langle  x,y\\right\\rangle|^{2}d\\mu(y).$ From the fact that $\\int_{\\mathbb R^d \\setminus B(0,R_1)}\\|x\\|^2d\\mu(x) \\leq \\epsilon$ it follows that $$\\int_{\\mathbb R^d \\setminus B(0,R_1)}|\\left\\langle  x,y\\right\\rangle|^{2}d\\mu(y)\\leq \\|x\\|^2\\epsilon.$$\n\\end{enumerate}\n\\end{proof}\n\n\n\nSuppose that $\\mu$ is a probabilistic frame supported in a ball $B(0, R)$. Let $r>0$ and consider $Q=[0, r)^d$. Choose points $\\{c_k\\}_{k=1}^M\\subset \\mathbb R^d$ with $c_1=0$ such that $B(0, R)=\\cup_{k=0}^M Q_k$ where $Q_k=c_k+Q$. Observe that $Q_k\\cap Q_\\ell=\\emptyset$ whenever $k\\neq \\ell$. Let $\\mu_{1,Q}=\\sum_{k=1}^M\\mu(Q_k)\\delta_{c_k}$.\n\nNext partition each cube $Q_k$ uniformly into cube of size $r\/2$ and construct the probability measure $\\mu_{2,Q}$ as above. Iterate this process to construct a sequence of probability measures $\\mu_{n, Q}$.\n\n\\begin{lemma}\\label{prop-3} Let $\\mu$ be a probabilistic frame with bounds $A$ and $B$, which supported in a ball $B(0, R)$. For  $r>0$  let $\\{\\mu_{n,Q}\\}_{n=1}^{\\infty}$ be a sequence of probability measures as constructed above. Then, $$lim_{n\\to \\infty}W_2(\\mu, \\mu_{n,Q})=0.$$ Furthermore, there exists $N$ such that for all $n\\geq N$, $\\mu_{n, Q}$ is a finite probabilistic\nframe whose bounds are arbitrarily close to those of $\\mu$.\n\\end{lemma}\n\n\n\\begin{proof}\nLet $d=\\max_{x\\in Q_k}\\|x-c_k\\|$.  Given, $x\\in Q_k$, $x=c_k + a_k$, where $\\|a_k\\|\\leq d$.\n\nFor any $x\\in \\mathbb R^d$,\n\\begin{align*}\n\\bigg|\\int_{B(0, R)}\\ip{x}{y}^2d\\mu(y)-\\sum_{k=1}^M\\ip{x}{c_k}^2\\mu(Q_k)\\bigg|&= \\bigg|\\sum_{k=1}^M\\int_{Q_k}\\ip{x}{y}^2 d\\mu(y)-\\sum_{k=1}^M\\ip{x}{c_k}^2\\mu(Q_k)\\bigg|\\\\\n&=\\bigg|\\sum_{k=1}^M\\int_{Q_k}(\\ip{x}{y}^2-\\ip{x}{c_k}^2)d\\mu(y)\\bigg|\\\\\n&\\leq \\sum_{k=1}^M\\int_{Q_k}\\big|\\ip{x}{y}^2-\\ip{x}{c_k}^2\\big|d\\mu(y)\\\\\n&=\\sum_{k=1}^M\\int_{Q_k}|\\ip{x}{c_k+a_k}^2-\\ip{x}{c_k}^2|d\\mu(y)\\\\\n&=\\sum_{k=1}^M\\int_{Q_k}|\\ip{x}{a_k}^2+2\\ip{x}{c_k}\\ip{x}{a_k}|d\\mu(y)\\\\\n&\\leq \\|x\\|^2\\sum_{k=1}^M\\mu(Q_k)(\\|a_k\\|^2+2\\|c_k\\|\\|a_k\\|)\\\\\n&\\leq (d^2+2d(R+d))\\|x\\|^2.\n\\end{align*}\n\nNote that by the iterative construction of $\\mu_{n, Q}$ we get that for each $x\\in \\mathbb R^d$ $$\\bigg|\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu(y)-\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu_{n, Q}(y)\\bigg|\\leq (d_{n}^2+2d_{n}(R+d_{n}))\\|x\\|^2$$ where $\\lim_{n\\to \\infty}d_n=0$. It follows that given $\\epsilon>0$, we can find $N>1$ such that for all $n\\geq N$,\n\n$$\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu_{n, Q}(y)>\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu(y)-\\epsilon\\|x\\|^2>\\|x\\|^2(A-\\epsilon)$$ which concludes that $\\mu_{n, Q}$ is a a finite probabilistic frame whose lower bound is at least $A-\\epsilon$. Furthermore, $$\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu_{n, Q}(y)<\\int_{\\mathbb R^{d}} \\ip{x}{y}^2d\\mu(y)+\\epsilon\\|x\\|^2\\leq \\|x\\|^2(B+\\epsilon)$$ which implies that the upper frame bound  $\\mu_{n, Q}$ is at most $B+\\epsilon$.\n\nNext, fix $n\\geq N$ and let $\\gamma_n(x,y)$ be the measure on $\\mathbb R^d\\times \\mathbb R^d$ be defined  for any Borel sets  $ A, B \\subset \\mathbb R^d$  by:\n\n$$\\gamma_n(A\\times B)=\\sum_{k: c_k\\in B}\\mu(A\\cap Q_k)=\\sum_{k=1}^M\\mu_{|_{Q_k}}\\times \\delta_{c_{k}}(A\\times B)$$ where $A, B$  $c_k$ denoting the centers of the cubes $Q_{k}$. It is easy to see  that  $\\gamma_n \\in \\Gamma(\\mu, \\mu_{n, Q})$\nand so\n\\begin{align*}\nW_2^2(\\mu, \\mu_{n, Q})&\\leq \\iint\\|x-y\\|^2d\\gamma_n(x, y)\\\\\n&=\\sum_{k=1}^{M}\\iint \\|x-y\\|^2d(\\mu_{|_{Q_{k}}}\\times \\delta_{c_{k}})(x,y)\\\\\n&=\\sum_{k=1}^M \\int_{Q_k}\\|x-c_k\\|^2d\\mu(x)\\\\\n&\\leq \\sum_{k=1}^M\\mu(Q_k)\\int_{Q_k}d_n^2d\\mu(x)\\\\\n&\\leq d_n^2\n\\end{align*}\nand the result follows from the fact that $\\lim_{n\\to \\infty}d_n=0$.\n\n\\end{proof}\n\n\n\nWe can now give a proof  of  Theorem~\\ref{density-dpf}.\n\\begin{proof}[Proof of Theorem~\\ref{density-dpf}]\nLet $\\mu$ be a  probabilistic frame with frame bounds $A$ and $B$, and $\\epsilon > 0$.  By Lemma~\\ref{prop-2} let  $\\nu$ be a compactly supported probabilistic frame with  frame bounds between $A-\\epsilon\/2$ and $B$ and such that $W_2(\\mu, \\nu)< \\epsilon\/2$.\n\nBy Lemma~\\ref{prop-3}  we know there exists a finite probabilistic frame $\\mu_{\\Phi, w}$ whose frame bounds are within $\\epsilon\/2$ of that of $\\nu$ and such that $W_2(\\nu, \\mu_{\\Phi, w})< \\epsilon\/2 $.  Consequently, $W_2(\\mu, \\mu_{\\Phi, w})<\\epsilon$ which concludes the proof.\n\\end{proof}\n\n\n\n\\begin{cor}\\label{appro-prob}\nLet $\\mu$ be a probabilistic Parseval frame and $\\epsilon > 0.$ Then, there exists a finite Parseval probabilistic frame $\\mu_{\\Phi, w}$ with $$W_{2}(\\mu, \\mu_{\\Phi, w})< \\epsilon.$$\n\\end{cor}\n\\begin{proof}\nThis follows from Proposition~\\ref{dists-to-parseval} and Theorem~\\ref{density-dpf}.\n\\end{proof}\n\\begin{remark}\nSince the set of finite Parseval frames is dense in the set of all Parseval frames in the Wasserstein metric, by Proposition 2.6 since there is no finite Parseval frame closer to $\\Phi$  than $\\Phi^\\dag=\\{S^{-1\/2}\\varphi_i\\}_{i=1}^N$, there are no Parseval frame closer to $\\Phi$  than $\\Phi^\\dag$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The closest  Parseval frame  in the $2-$Wasserstein distance}\\label{subsec2.3}\n\nIn this section we prove and state of our main result, Theorem~\\ref{maintheorem}. We recall that if $\\mu$ is a probabilistic frame for $\\mathbb R^d$, then its  probabilistic frame operator (equivalently, the matrix of second moments associated to $\\mu$)\n\n$$S_\\mu:\\mathbb R^d\\rightarrow \\mathbb R^d,\\qquad S_\\mu (x) = \\int_{\\mathbb R^d} \\ip{ x}{y} y d\\mu(y)$$ is positive definite, and thus $S_\\mu^{-1\/2}$ exists. We define the push-forward of $\\mu$ through $S_{\\mu}^{-1\/2}$  by $$\\mu^{\\dagger} (B) =\\mu(S^{1\/2} B)$$ for each Borel set in $\\mathbb R^d$. Alternatively, if $f$ is a continuous bounded function on $\\mathbb R^d$, $$\\int_{\\mathbb R^d} f(y)d \\mu^{\\dag}(y) = \\int_{\\mathbb R^d} f(S_{\\mu}^{-1\/2}y) d\\mu(y).$$\n\nIt then follows that\n$$\nx=S_{\\mu}^{-1\/2}S_{\\mu} S_{\\mu}^{-1\/2}(x)=   \\int_{\\mathbb R^d}   \\ip{S_{\\mu}^{-1\/2} x}{y} \\, S_{\\mu}^{-1\/2}y \\, d\\mu(y)=   \\int_{\\mathbb R^d} \\ip{x}{ y}\\, y \\, d\\mu^{\\dagger} (y) $$ implying that $\\mu^{\\dag}$ is a Parseval probabilistic frame \\cite{EhlOko2013, KOPrecondPF2016}. In particular, $S_{\\mu^{\\dag}}=I$ where $I$ is the identity matrix on $\\mathbb R^d$. As was the case with the canonical Parseval frame $\\Phi^{\\dag}$ of a given frame $\\Phi$, $\\mu^{\\dag}$ is the (unique) closest  Parseval probabilistic frame to $\\mu$.\n\n\n\\begin{theorem}\\label{maintheorem}\nLet $\\mu$ be a probabilistic frame on $\\mathbb R^d$ with probabilistic frame operator $S_\\mu$. Then $\\mu^{\\dag}$ is the (unique) closest probabilistic Parseval frame to $\\mu$ in the $2-$Wasserstein metric, that is\n\\begin{equation}\\label{optimum}\n\\mu^{\\dag}=\\textrm{arg} \\min  W_{2}^{2}(\\mu, \\nu)\n\\end{equation}\nwhere $\\nu $ ranges over all Parseval probabilistic frames.\n\\end{theorem}\n\n\n\n\nBefore proving this theorem, we need to establish a few preliminary results. We start by extending Theorem~\\ref{cont-F} to finite probabilistic frames in the Wasserstein metric. In particular, this extension allows use to deal with finite probabilistic frames of different cardinalities.\n\n\n\n\n\n\n\\begin{theorem}\\label{continuityFPF}\nLet $0<A\\leq B <\\infty$, and  $\\delta > 0$ be given. Then there  exists  $\\epsilon > 0$ such that given any finite probabilistic frame $\\mu_{\\Phi, w}=\\sum_{i=1}^Nw_i\\delta_{\\varphi_i}$ with   frame bounds between $A$ and $B$, $N:=N_{\\Phi}\\geq 2$,  $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\subset \\mathbb R^d$, and weights $w=\\{w_i\\}_{i=1}^N \\subset [0, \\infty)$, for any  finite probabilistic frame $\\mu_{\\Psi, \\eta}=\\sum_{i=1}^M\\eta_i\\delta_{\\psi_i},$ $M:=M_{\\Psi}\\geq 2$, where $\\Psi=\\{\\psi_i\\}_{i=1}^M\\subset \\mathbb R^d$, and weights $\\eta=\\{\\eta_i\\}_{i=1}^N \\subset [0, \\infty)$  if  $W_2(\\mu_{\\Phi, w}, \\mu_{\\Psi, \\eta})< \\epsilon,$  then we have $$W_2(F(\\mu_{\\Phi, w}), F(\\mu_{\\Psi, \\eta})) < \\delta.$$\n\\end{theorem}\n\n\n\n\n\n\n\\begin{proof} Fix $\\delta>0$.  By Theorem~\\ref{cont-F} we know that there exists $\\epsilon$ such that given a frame $X=\\{x_i\\}_{i=1}^M$ ($M\\geq 2$ is arbitrary) with frame bounds between $A$ and $B$, and  $Y=\\{y_{i}\\}_{i=1}^M$   is a frame such that $$d(X, Y)=\\sqrt{\\sum_{i=1}^{M}\\|x_{i} - y_{i}\\|^2} <  \\epsilon$$ then $$d(F(X), F(Y))=d(S^{-1\/2}_{X}X, S^{-1\/2}_{Y}Y) <  \\delta.$$\n\n\nLet $\\mu_{\\Phi, w}=\\sum_{i=1}^N w_i\\delta_{\\varphi_i}$ be a finite probabilistic frame with   frame bounds between $A$ and $B$, $N \\geq 2$,  $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\subset \\mathbb R^d$, and weights $w=\\{w_i\\}_{i=1}^N \\subset [0, \\infty)$. Then by Theorem~\\ref{dists-to-parseval}, $\\mu_{\\Phi^{\\dag}, w}$ where $\\Phi^{\\dag}=\\{S^{-1\/2}_{\\Phi}\\varphi_i\\}_{i=1}^N$ is the closest Parseval frame to $\\mu_{\\Phi, w}$.\n\n\n\n\n\n\n\nLet $\\mu_{\\Psi, v}$ where $\\Psi=\\{\\psi_i\\}_{i=1}^M$, $M\\geq 2$ such that $W_2(\\mu_{\\Phi, w}, \\mu_{\\Psi, \\eta})< \\epsilon$. Choose $\\gamma \\in \\Gamma(\\mu_{\\Phi, w}, \\mu_{\\Psi, v})$  such that\n$$W_2(\\mu_{\\Phi, w}, \\mu_{\\Psi, \\eta})^2=\\iint_{\\mathbb R^d \\times \\mathbb R^d}\\|x-y\\|^2d\\gamma(x,y)< \\epsilon^2.$$ Identify $\\gamma$ with $\\{w_{i,j}\\}_{i, j=1}^{N, M}$. Then,\n\n$$W_2(\\mu_{\\Phi, w}, \\mu_{\\Psi, \\eta})^2=\\iint_{\\mathbb R^d \\times \\mathbb R^d}\\|x-y\\|^2d\\gamma(x,y)=\\sum_{i=1}^M\\sum_{j=1}^Nw_{i,j}\\|\\varphi_i-\\psi_j\\|^2< \\epsilon^2.$$\n\nObserve that $\\Phi'=\\{\\sqrt{w_{i,j}}\\varphi_i\\}_{i, j=1}^{M,N}$ is a frame whose frame bounds are the same as  those for $\\mu_{\\Phi,w}$. Similarly, $\\Psi'=\\{\\sqrt{w_{i,j}}\\psi_j\\}_{i, j=1}^{M,N}$ is a frame whose frame bounds are the same as  those for $\\mu_{\\Psi,\\eta}.$ Furthermore,\n\n$$d(\\Phi', \\Psi')=W_2(\\mu_{\\Phi, w}, \\mu_{\\Psi, \\eta})<\\epsilon$$ which implies that\n\n\n$$d(F(\\Phi'), F(\\Psi'))^2= \\sum_{i, j=1}^{M,N}\\|S^{-1\/2}_{\\Phi}(\\sqrt{w_{i,j}}\\varphi_i) - S^{-1\/2}_{\\Psi}(\\sqrt{w_{i,j}}\\psi_j)\\|^2<  \\delta^2.$$\n\nHowever,\n$$\\sum_{i,j}\\|S^{-1\/2}_{\\Phi}(\\sqrt{w_{i,j}}\\varphi_i) - S^{-1\/2}_{\\Psi}(\\sqrt{w_{i,j}}\\psi_j)\\|^2 = \\sum_{i,j}w_{i,j}\\|S^{-1\/2}_{\\Phi}\\varphi_i - S^{-1\/2}_{\\Psi}\\psi_j\\|^2$$\n\n\nBut since $w_{i,j} = \\gamma(\\{\\varphi_i\\}, \\{\\psi_j\\})$ we have $\\sum_{j}w_{i,j} = w_i$ and $\\sum_{i}w_{i,j} = v_j$ we see that\n$$W^2_2(F(\\mu_{\\Phi, w}), F(\\mu_{\\Psi, \\eta}))=W_2^2(\\mu_{\\Phi^{\\dag}, w}, \\mu_{\\Psi^{\\dag}, v}) \\leq \\sum_{i,j}w_{i,j}\\|S^{-1\/2}_{\\Phi}\\varphi_i - S^{-1\/2}_{\\Psi}\\psi_j\\|^2.$$\n\\end{proof}\n\nLet $DPF(A, B)$ denote the set of all discrete (finite)  probabilistic frames in $\\mathbb R^d$ whose lower frame bounds are less than or equal to $A$ and whose upper bounds are greater or equals to $B$. It follows from the above result that $F$ is uniformly continuous from $DPF(A, B)$ into itself when equipped with the Wasserstein metric. Consequently, we can prove the following result.\n\n\n\\begin{proposition}\\label{well-defined-F}\nLet $\\mu$ be a probabilistic frame with frame bounds $A$ and $B$. Let $\\mu_k:=\\mu_{\\Phi_{k}, w_k}$, where  $\\Phi_{k}:=\\Phi_{k, w_{k}}=\\{\\varphi_{k}\\}_{k=1}^{N_k}$ and $\\nu_k:=\\mu_{\\Psi_{k}, v_{k}}$, where $\\Psi_{k}:=\\Psi_{k, v_{k}}=\\{\\psi_{k}\\}_{k=1}^{M_k}$ be two sequences of finite probabilistic frames in $\\mathbb R^d$ such that $\\lim_{k\\to \\infty}W_{2}(\\mu, \\mu_{\\Phi_{k}})=\\lim_{k\\to \\infty}W_2(\\mu, \\mu_{\\Psi_{k}})=0$. Furthermore, suppose that the frame bounds of $\\mu_{\\Phi_{k}}$ are between $A\/2$ and $B+A\/2$. Then $$\\lim_{k\\to \\infty}F(\\mu_{\\Phi_{k}})=\\lim_{k\\to \\infty}F(\\mu_{\\Psi_{k}}).$$\n\\end{proposition}\n\n\n\\begin{proof}\nTheorem~\\ref{density-dpf} ensures the existence of the finite probabilistic frames $\\mu_{\\Phi_{k}}$.\n\nLet $\\delta>0$ be given. By Theorem~\\ref{continuityFPF} there exists $\\epsilon >0$ such that for any finite probabilistic frame $\\nu$ and any $k\\geq 1$, $$W_2(\\mu_{\\Phi_{k}}, \\nu)< \\epsilon \\implies W_2(F(\\nu), F(\\mu_{\\Phi_{k}}))< \\delta.$$\n\nChoose $N_\\epsilon>1$ such that  for all $k>N_{\\epsilon}$, $W_{2}(\\mu,\\mu_{\\Phi_{k}}) < \\frac{\\epsilon}{2}$ and $W_{2}(\\mu,\\mu_{\\Psi_{k}}) < \\frac{\\epsilon}{2}$. Thus, for $k\\geq N_\\epsilon$,  $W_{2}(\\mu_{\\Phi_{k}},\\mu_{\\Psi_{k}}) < \\epsilon$, which implies that for all $k\\geq N_{\\epsilon}$, $W_{2}(F(\\mu_{\\Phi_{k}}),F(\\mu_{\\Psi_{k}})) < \\delta$. It easily follows that $\\lim_{k\\to \\infty}F(\\mu_{\\Phi_{k}})=\\lim_{k\\to \\infty}F(\\mu_{\\Psi_{k}})$.\n\n\\end{proof}\nWe can now use this proposition to extend the definition of the map $F$ to all probabilistic frames. Let $\\mu$ be a probabilistic frame with bounds $0<A\\leq B<\\infty.$ Let $\\{\\mu_{\\Phi_{k}}\\}_{k=1}^{\\infty}$ be a sequence of finite probabilistic frames with bounds between $A\/2$ and $B+A\/2$ such that $lim_{k\\to \\infty}W_2(\\mu_{\\Phi_{k}}, \\mu)=0$. Then, $$F(\\mu)=\\lim_{k\\to \\infty}F(\\mu_{\\Phi_{k}})$$ is well-defined. Before proving Theorem~\\ref{maintheorem} we first identify the minimizer of ~\\eqref{optimum} with $F(\\mu)$.\n\n\\begin{theorem}\\label{optimizer1}\nLet $\\mu$ be a probabilistic frame on $\\mathbb R^d$ with probabilistic frame operator $S_\\mu$. Then $F(\\mu)$ is the unique closest probabilistic Parseval frame to $\\mu$ in the $2-$Wasserstein metric, that is $F(\\mu)$ is the unique solution to~\\eqref{optimum}.\n\\end{theorem}\n\n\n\n\n\n\n\n\\begin{proof} Set $Q=\\min  W_{2}(\\mu, \\nu)$\nwhere $\\nu $ ranges over all Parseval probabilistic frames.\n\nLet  $\\delta>0$, and  $\\mu$ be a probabilistic frame wth frame bounds $A$ and $B$. By Theorem~\\ref{density-dpf}, there exists a  sequence of finite probabilistic frame $\\mu_{\\Phi_{k}}$ with frame bounds between $\\frac{A}{2}$ and $B + \\frac{A}{2}$ where $\\Phi_{k}:=\\Phi_{k, w(k)}=\\{\\varphi_{k}\\}_{k=1}^{N_k} \\subset \\mathbb R^d$, $w(k)=\\{w_n\\}_{n=1}^{N_k}\\subset (0, \\infty)$, and $N_k\\geq 2$ such that $\\lim_{k\\to \\infty}W_2(\\mu, \\mu_{\\Phi_{k}})=0$.\n\nObserve that for all $k\\geq 1$, $$W_{2}(\\mu, F(\\mu_{\\Phi_{k}}))\\leq W_2(\\mu, F(\\mu)) + W_2(F(\\mu), F(\\mu_{\\Phi_{k}})).$$ Choose $\\epsilon>0$ as in Theorem~\\ref{continuityFPF} and pick $K\\geq 1$ such that $W_2(\\mu, \\mu_{\\Phi_{K}})< \\epsilon.$ Thus, $W_2(F(\\mu), F(\\mu_{\\Phi_{K}}))< \\delta.$ Consequently,\n\n$$W_{2}(\\mu, F(\\mu_{\\Phi_{K}}))\\leq W_2(\\mu, F(\\mu)) + W_2(F(\\mu), F(\\mu_{\\Phi_{K}}))<  W_2(\\mu, F(\\mu)) + \\delta.$$  Since  $F(\\mu_{\\Phi_{K}})$ is a Parseval frame we conclude that $F(\\mu)$ minimizes~\\eqref{optimum}.\n\n\n\nWe now prove that $F(\\mu)$ is the unique minimizer of  ~\\eqref{optimum} by considering three cases.\n\n\\noindent {\\bf Case 1.} If $\\mu$ is a finite frame $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\subset \\mathbb R^d$, it is known that $S^{-1\/2}\\Phi$ is the (unique) closest  Parseval frame to $\\Phi$, see Theorem~\\ref{uniqueness_prob_1}, and \\cite[Theorem 3.1]{CasKut07}.\n\n\\noindent {\\bf Case 2.} If $\\mu=\\mu_{\\Phi, w}$, where $\\Phi=\\{\\varphi_i\\}_{i=1}^N\\subset \\mathbb R^d$, and $w=\\{w_i\\}_{i=1}^N \\subset [0, \\infty)$. Then, $\\mu_{\\Phi^{\\dag}, w}$ where $\\Phi^{\\dag}=S^{-1\/2}\\Phi$ is the unique closest  Parseval probabilistic frame to $\\Phi$. Indeed, we already know that $\\mu_{\\Phi^{\\dag}, w}$ achieves the minimum distance Proposition~\\ref{dists-to-parseval}. We now prove that it is unique. We argue by contradiction and assume that there exists  another Parseval probabilistic frame $\\nu$ that achieves this distance.\n\nFirst, we assume  that $\\nu=\\mu_{\\kappa, v}$ where  $\\kappa =$ $\\{\\kappa'_i\\}_{i=1}^{M}\\subset \\mathbb R^d$ with weights $v=\\{v_i\\}_{i=1}^M\\subset [0, \\infty)$. Let $\\gamma \\in \\Gamma(\\mu, \\nu)$ such that $$W_2(\\mu, \\nu)^2=\\iint\\|x-y\\|^2d\\gamma(x,y).$$  For all $i$, $j$ let $w_{i,j} = \\gamma(\\varphi_i,\\kappa'_j)$. Let $Q = \\sum_{i=1}^{N}w_i\\|\\varphi_i - \\varphi_i^{\\dag}\\|^2$, where $\\varphi_i^{\\dag}=S^{-1\/2}\\varphi_i$. Since $\\kappa$ also achieved this distance we clearly have $Q = \\sum_{i,j}w_{i,j}\\|\\varphi_i - \\kappa'_j\\|^2$.\n\nWe now use Lemma~\\ref{frame_modi}. For each $i$, we replace the vector $\\varphi_i$ and its weight $w_i$ by $M$ copies of itself (i.e.,  $\\varphi_i$ ) each weighted by $w_{i,j}$. Apply the same procedure to  $\\Phi^{\\dag}$, and to $\\kappa$, except that for the latter we break each vector $\\kappa_j'$ into $N$ copies of itself with weights $w_{i,j}$.  Denote by $F_1, F_2,$ and $F_3$ the three resulting frames. We note that the vectors in each of these frames can be considered to have weight $1$.\n\nIt follows  from Theorem~\\ref{uniqueness_prob_1} that the finite frame $F_3=\\{\\sqrt{w_{i,j}}\\kappa'_j\\}_{i,j}$ is the (unique) closest Parseval frame to $F_1=\\{\\sqrt{w_{i,j}}\\varphi_i\\}_{i,j}$, which we also know is $F_2=\\{\\sqrt{w_{i,j}}\\varphi_i^{\\dag}\\}_{i,j}$. Therefore,  $\\mu_{\\kappa, v}=\\mu_{\\Phi^{\\dag}, w}$.\n\nNext, we assume that $\\nu$ is not discrete. Choose  a sequence of finite Parseval frames $\\{\\nu_n\\}_{n}^\\infty$  such that\n$\\lim_{n \\to \\infty}W_2(\\nu_n, \\nu)=0$.  Hence, $$Q=W_{2}(\\mu, F(\\mu))=W_2(\\mu_{\\Phi, w}, \\nu)=\\lim_{n \\to \\infty}W_2(\\mu_{\\Phi, w}, \\nu_n).$$ We now prove that $$\\lim_{n\\to \\infty}W_2(\\nu_n, \\mu_{\\Phi^{\\dag}, w})=0.$$\n\n\nLet $\\delta > 0$ and choose $N\\geq 1$  such that for all $n>N$\n$$W_2(\\nu_n, \\mu_{\\Phi, w})< Q + \\delta. $$\n\n\nSuppose by contradiction that $\\lim_{n\\to \\infty}W_2(\\nu_n, \\mu_{\\Phi^{\\dag}, w})> 0$. Thus, there is $\\epsilon>0$ such for all $k\\geq 1$, there exists $n>\\max(k, N)$ such that $$W_2(\\nu_n, \\mu_{\\Phi^{\\dag}, w})>\\epsilon.$$\n\nFor  $n$ given above,  let $\\gamma_n \\in \\Gamma(\\nu_n, \\mu_{\\Phi, w})$ be such that $$W_2^2(\\nu_n, \\mu_{\\Phi, w})=\\iint_{\\mathbb R^d}\\|x-y\\|^2d\\gamma_n(x,y).$$\nSince $\\nu_n$ is a finite probabilistic frame we may assume further that  $\\nu_n=\\mu_{u_{n}, v}$ where $u_n= \\{\\psi_i\\}_{i=1}^M\\subset \\mathbb R^d$ and $v=\\{v_i\\}_{i=1}^M \\subset [0, \\infty)$. For the sake of simplicity in notations, we omit the dependence of both $\\psi_i$ and $v_i$ on $n$.    Let $w_{n,j,k} = \\gamma_{n}(\\varphi_j,\\psi_k)$.\n\nNow consider the finite frames $\\{u'_j\\}_{j}=\\{\\sqrt{w_{n,j,k}}\\psi_k\\}_{j,k}$ and $\\Phi' =$ $\\{\\sqrt{w_{n,j,k}}\\varphi_j\\}_{j,k}$.\n\nNote that $W_2(\\mu_{\\Phi'}, \\mu_{\\Phi'^{\\dag}})=Q$.  Now we consider the rows of these frames written with respect to the eigenbasis of the frame operator $S:=S_{\\Phi'}$ of  $\\Phi'$.\n\n\nBecause, $W_2(\\nu_n, \\mu_{\\Phi^{\\dag}, w})>\\epsilon,$ then  $\\sum_{j,k}w_{n,j,k}\\|\\psi_k - S^{-1\/2}\\varphi_j\\|^2 > \\epsilon$.\n\nUsing this and Lemma~\\ref{contradiction-unifcont} we have the following estimates:\n$$\nW_2^2(\\mu_{\\Phi, w}, \\nu_{n})\\geq W_2^2(\\mu_{\\Phi, w}, \\nu)+ \\min( \\tfrac{\\epsilon^2}{d} \\cdot M,M^2)$$ where $A$ is the lower frame bound of $\\Phi$ and $M = \\min(1,\\sqrt{A})$.\n\nConsequently,\n\n$$W_2^2(\\mu_{\\Phi, w}, \\nu_{n})-   Q^2\\geq  \\min( \\tfrac{\\epsilon^2}{d} \\cdot M,M^2)>0 .$$\nBut,  this contradicts the fact that $Q=W_2(\\mu_{\\Phi, w}, \\nu)=\\lim_{n \\to \\infty}W_2(\\nu_{\\Phi, w},\\nu_n).$  Hence, $\\lim_{n\\to \\infty}W_2(\\nu_n, \\mu_{S^{-1\/2}\\Phi, w})= 0$, and $\\nu=\\mu_{\\Phi^{\\dag}, w}.$\n\n\n\n\\noindent {\\bf Case 3:} Next, we suppose that $\\mu$ is non discrete probabilistic frame with frame bounds $A,$ and $B$. Let $\\{\\mu_n\\}_{n=1}^\\infty=\\{\\mu_{\\Phi_{n}, w(n)}\\}_{n=1}^\\infty$ be a sequence of finite probabilistic frames with bounds between $A\/2$ and $B+A\/2$ such that $\\lim_{n\\to \\infty}W_2(\\mu_n, \\mu)=0$. Then $F(\\mu)=\\lim_{n\\to \\infty}F(\\mu_{n})$ is such that $Q=W_{2}(F(\\mu), \\mu).$ Suppose there exists another Parseval frame $\\nu$  such that $Q=W_2(\\nu, \\mu)$. Choose a sequence of finite Parseval $\\{\\nu_n\\}_{n=1}^\\infty$  such that $\\lim_{n\\to \\infty}\\nu_n=\\nu.$\n\nObserve that $Q=\\lim_{n\\to \\infty}W_2(\\mu_n, F(\\mu_n))=\\lim_{n\\to \\infty}W_2(\\nu_n, \\mu_n)$.\nWrite  $\\Phi_n  = \\{\\varphi_{n,j}\\}_{j=1}^{M}$ and $w(n)=\\{w_{j}\\}_{j=1}^M$, where for simplicity we omit the dependence of $M$ on $n$. Similarly,  $\\{\\nu_n\\}_{n=1}^{\\infty}= \\{\\psi_{n,j}\\}_{j=i}^{M'}$  with weights $v(n)= \\{v_{j}\\}_{j=1}^{M'}$.\n\n\n Let $\\gamma_n\\in \\Gamma(\\mu_n, \\nu_n)$  be such that $$W_2^2(\\mu_{n},\\nu_n) =\\iint \\|x-y\\|^2d\\gamma_n(x,y).$$  Set $$w_{j,k} = \\gamma_n(\\varphi_{n,j},\\psi_{n,k})$$\nWe know that\n\\begin{align*}\nW_2^2(\\mu_n, F(\\mu_n)) &= \\sum_{j= 1}^{M}w_j\\|\\varphi_{n,j} - \\varphi^{\\dag}_{n,j}\\|^2 = \\sum_{j,k}w_{j,k}\\|\\varphi_{n,j} - \\varphi^{\\dag}_{n,j}\\|^2 \\\\\n&=\\sum_{j,k}\\|\\sqrt{w_{j,k}}\\varphi_{n,j} - \\sqrt{w_{j,k}}\\varphi^{\\dag}_{n,j}\\|^2\n\\end{align*}\n\n\nWe also know that\n$$\nW_2^2(\\mu_{n},\\nu_n)  =  \\sum_{j,k}w_{j,k}\\|\\varphi_{n,j} - \\psi_{n,k}\\|^2 =\\sum_{j,k}\\|\\sqrt{w_{j,k}}\\varphi_{n,j} - \\sqrt{w_{j,k}}\\psi_{n,k}\\|^2 $$\n\nSuppose that $\\lim_{n\\to \\infty} W_2(F(\\mu_n), \\nu_n)>0$. Thus, there exists $\\epsilon>0$ and and integer $n>1$ such that $W_2(F(\\mu_n),\\nu_n) > \\epsilon$. Consequently,\n$$\\epsilon< \\sum_{j,k}w_{j,k}\\|\\varphi^{\\dag}_{n,j} - \\psi_{n,k}\\|^2 = \\sum_{j,k}\\|\\sqrt{w_{j,k}}\\varphi^{\\dag}_{n,j} - \\sqrt{w_{j,k}}\\psi_{n,k}\\|^2 $$\n\nHence  $$d(\\Phi'^{\\dag }_n, \\Psi_n')>\\epsilon$$ where $\\Psi'_n = \\{\\sqrt{w_{j,k}}\\psi_{n,k}\\}$.\n\n\n\nBy the same argument as in Lemma~\\ref{contradiction-unifcont}  we conclude that  $W_2^2(\\mu_{n},\\nu_n)- W_2^2(\\mu_n, F(\\mu_n))\\geq min(M \\frac{\\epsilon^2}{d},M^2).$ where $M = \\min(1,\\sqrt{\\frac{A}{2}})$\n\nThis contradicts the fact that Since $\\lim_{n\\to \\infty}W_2(\\mu_n,\\nu_n) = Q = \\lim_{n\\to \\infty}W_2(\\mu_n,F(\\mu_n))$. Thus $\\lim_{n\\to \\infty} W_2(F(\\mu_n), \\nu_n)=0$ and so $F(\\mu)=\\nu$.\n\n\n\\end{proof}\n\n\n\n\nBy Proposition~\\ref{well-defined-F} it follows that given a probabilistic frame $\\mu$ and any sequence $\\Phi_{k}:=\\Phi_{k, w_{k}}=\\{\\varphi_{k}\\}_{k=1}^{N_k}$ of finite probabilistic frames in $\\mathbb R^d$ such that $\\lim_{k\\to \\infty} W_{2}(\\mu, \\mu_{\\Phi_{k}})=0$, then $F(\\mu)=\\lim_{k\\to \\infty}F(\\mu_{\\Phi_{k}}).$ Furthermore, it is proved in \\cite{WCKO17} that if $\\{\\mu_n\\}_{n\\geq 1} \\subset \\mathcal{P}_2$ converges in the Wassertein metric to $\\mu \\in \\mathcal{P}_2$, then $$\\|S_{\\mu}-S_{\\mu_{n}}\\|\\leq CW_2(\\mu_{n}, \\mu).$$\n\n\n\n\n\n\nAll that is needed to prove Theorem~\\ref{maintheorem} is to show that $F(\\mu)=\\mu^{\\dag}$.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{maintheorem}]\nLet $\\mu$ be a probabilistic frame with bounds $A, B$. Let $0< \\epsilon<A\/2$ and choose a compactly supported probabilistic frame $\\nu_{\\epsilon}$ as in Lemma~\\ref{prop-2}. In particular $\\nu_{\\epsilon}$ is supported on $B(0,R_{\\epsilon})$ with frame bounds between $A\/2$ and $B+A\/2$, where $R_{\\epsilon}>0$ is such that $$\\int_{\\mathbb R^d\\setminus B(0,R_{\\epsilon})}\\|x\\|^2dx< \\epsilon\/3.$$  \n\nChoose a finite probabilistic frame $\\mu_{\\epsilon}$ with bounds between $\\frac{A}{2}$ and $B + \\frac{A}{2}$ such that $W_{2}(\\mu_{\\epsilon}, \\nu_{\\epsilon}) < \\frac{\\epsilon}{3}$. By taking  a sequence $\\{\\epsilon_n\\}_{n=1}^{\\infty}\\subset [0, \\infty)$  with $\\lim_{n\\to \\infty}\\epsilon_n=0$, we can pick $\\{\\mu_n\\}_{n\\geq 1}:=\\{\\mu_{\\epsilon_n}\\}_{n\\geq 1}$ such that  $\\lim_{n\\to \\infty}W_{2}(\\mu_{n}, \\mu)=0$. Consequently,  $\\lim_{n\\to \\infty}S_{\\mu_{n}}=S_{\\mu}$, and $\\lim_{n\\to \\infty}S^{-1\/2}_{\\mu_{n}}=S^{-1\/2}_{\\mu}$ in the operator norm. \n\nWe recall that $\\lim_{n\\to \\infty}W_{2}(\\mu_{n}, \\mu)=0$ is equivalent to\n\n\n$$\n\\lim_{n\\to \\infty}\\int f\\, d\\mu_{n}(x) =\\int f\\, d\\mu(x) \\\\\n$$ for all continuous function $f$ such that $|f(x)|\\leq C(1+\\|x-x_0\\|^2)$ for some $x_0\\in \\mathbb R^d$ \\cite[Theorem  6.9]{Villani2009}\n\nWe know that $\\lim_{n\\to \\infty} F(\\mu_n)=\\lim_{n\\to \\infty} \\mu_{n}^{\\dag}=F(\\mu)$ in the Wasserstein metric. We would like to show that $\\lim_{n\\to \\infty} F(\\mu_n)=\\lim_{n\\to \\infty}\\mu_{n}^{\\dag}=\\mu^{\\dag}$.\n\nWe  show that for all continuous function $f$ such that $|f(x)|\\leq C(1+\\|x-x_0\\|^2)$ for some $x_0\\in \\mathbb R^d$ $$\\lim_{n\\to \\infty}\\int f\\, d\\mu_{n}^{\\dag}(x) =\\int f\\, d\\mu^{\\dag}(x).$$\n\n\n\\begin{align*}\n|\\int f\\, d\\mu_{n}^{\\dag}(x) -\\int f\\, d\\mu^{\\dag}(x)|& = |\\int f(S^{-1\/2}_{\\mu_{n}} x)\\, d\\mu_{n}(x) -\\int f(S^{-1\/2}_{\\mu} x)\\, d\\mu(x)|\\\\\n&\\leq \\int |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|\\, d\\mu_{n}(x) +\\\\\n&|\\int f(S^{-1\/2}_{\\mu} x)\\, d\\mu_{n}(x)  -\\int f(S^{-1\/2}_{\\mu} x)\\, d\\mu(x)|\\\\\n\\end{align*}\n\n\nLet $f$ be continuous with $|f(x)|\\leq C(1+\\|x-x_0\\|^2)$ for some $x_0\\in \\mathbb R^d$. Then, $f(S^{-1\/2}_\\mu)$ is continuous and satisfies  $$|f(S^{-1\/2}_\\mu x)|\\leq C(1+\\|x_0-S^{-1\/2}_\\mu x\\|^2)\\leq C(1+\\|S^{-1\/2}_\\mu\\|^2\\|x-S^{1\/2}_{\\mu}x_0\\|^2)\\leq C' (1+\\|x-S^{1\/2}_{\\mu}x_0\\|^2)).$$ Consequently, we can find $N_1$ such that for all $n\\geq N_1$, $$|\\int f(S^{-1\/2}_{\\mu} x)\\, d\\mu_{n}(x)  -\\int f(S^{-1\/2}_{\\mu} x)\\, d\\mu(x)|< \\epsilon\/3.$$\n\nSince $f$ is continuous, there exists  $\\delta>0$ such that for all $x, y \\in B(0,R')$, $\\|x-y\\|< \\delta $ implies that $|f(x)-f(y)|< \\epsilon\/3$, where $R'>0$ is chosen  so as to guarantee that for large $n$, and $x\\in B(0,R)$, $S^{-1\/2}_{\\mu_{n}}x, S^{-1\/2}_{\\mu}x \\in B(0, R)$. Since,  $\\lim_{n\\to \\infty}S^{-1\/2}_{\\mu_{n}}=S^{-1\/2}_{\\mu}$ , there exists $N_2$ such that for all $n\\geq N_2$,\n$$\\|S^{-1\/2}_{\\mu_{n}}x-S^{-1\/2}_{\\mu}x\\|\\leq \\|S^{-1\/2}_{\\mu_{n}}-S^{-1\/2}_{\\mu}\\| \\|x\\|\\leq R \\|S^{-1\/2}_{\\mu_{n}}-S^{-1\/2}_{\\mu}\\|< \\delta.$$\n\n\nTherefore, for $n\\geq N_2$, $ |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|< \\epsilon\/3$ for all $x\\in B(0, R)$. Consequently,\n\\begin{align*}\n\\int |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|\\, d\\mu_{n}(x) &= \\int_{B(0,R)} |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|\\, d\\mu_{n}(x)\\\\&+\\int_{\\mathbb R^d\\setminus B(0,R)} |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|\\, d\\mu_{n}(x) \\\\\n&<  \\epsilon\/3+ \\int_{\\mathbb R^d\\setminus B(0,R)} |f(S^{-1\/2}_{\\mu_{n}} x) - f(S^{-1\/2}_{\\mu} x)|\\, d\\mu_{n}(x)\\\\\n&< \\epsilon\/3 + M \\int_{\\mathbb R^d\\setminus B(0,R)} \\|x\\|^2\\, d\\mu_{n}(x)\\\\\n&<2\\epsilon\/3\n\\end{align*}\nwhere $M>0$ is a constant that depends only on $f$, and $\\mu$.\n\nIt follows that for all $n\\geq \\max(N_1, N_2),$  we have\n$$|\\int f\\, d\\mu_{n}^{\\dag}(x) -\\int f\\, d\\mu^{\\dag}(x)|< \\epsilon$$ which implies that $\\lim_{n\\to \\infty}\\int f\\, d\\mu_{n}^{\\dag}(x) =\\int f\\, d\\mu^{\\dag}(x).$\n\n\n\n\\end{proof}\n\n\\section*{Acknowledgment}\nBoth authors were partially supported by ARO grant W911NF1610008. K.~A.~Okoudjou  was also partially supported by a grant from the Simons Foundation $\\# 319197$. This material is based upon work supported by the National Science Foundation under Grant No.~DMS-1440140 while K.~A.~Okoudjou was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. \n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}