diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzccyc" "b/data_all_eng_slimpj/shuffled/split2/finalzzccyc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzccyc" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn this note we consider non-spurious solutions by using a critical point\ntheory to the following Dirichlet problem \n\\begin{equation}\n\\begin{array}{l}\n\\ddot{x}\\left( t\\right) =f\\left( t,x\\left( t\\right) \\right) \\\\ \nx\\left( 0\\right) =x\\left( 1\\right) =\n\\end{array}\n\\label{par}\n\\end{equation}\nwhere $f:\\left[ 0,1\\right] \\times \\mathbb{R} \\rightarrow \\mathbb{R}$ is a\njointly continuous function. Further we will make precise what is meant by\nthe solutions to (\\ref{par}).\n\nThe existence of non-spurious solutions is very important for the\napplications since in such a case one can approximate solutions to (\\ref{par\n) with a sequence of solutions to a suitably chosen family of discrete\nproblems and one is sure that this approximation converges to the solution\nof the original problem, see \\cite{kelly}. There are many ways in which a\nboundary value problem can be discretized and the existence and multiplicity\ntheory on difference equations is very vast, see for example \\cite{agrawal}, \n\\cite{candito1}, \\cite{guojde}, \\cite{MRT}. However, as underlined by\nAgarwal, \\cite{agarvalpaper}, there are no clear relations between\ncontinuous problems and their discretization which means that both problems\ncan be solvable, but the approximation approaches nothing but the solution\nto the continuous problem or else, the discrete problem is solvable and the\ncontinuous one is not or the other way round. Let us recall his examples:\n\n\\begin{example}\nThe continuous problem $\\ddot{x}(t) + \\frac{\\pi^2}{n^2}x(t)=0$, $x(0)=x(n)=0$\nhas an infinite number of solutions $x(t)= c \\sin \\frac{\\pi t}{n}$ ($c$ is\narbitrary) whereas its discrete analogue $\\Delta^2x(k)+\\frac{\\pi^2}{n^2\nx(k)=0$, $x(0)=x(n)=0$ has only one solution $x(k)\\equiv 0$. The problem \n\\ddot{x}(t)+\\frac{\\pi^2}{4n^2}x(t)=0$, $x(0)=0$, $x(n)=1$ has only one\nsolution $x(t)=\\sin\\frac{\\pi t}{2n}$, and its discrete analogue $\\Delta^2\nx(k)+ \\frac{\\pi^2}{4n^2}x(k)=0$, $x(0)=0$, $x(n)=1$ also has one solution.\nThe continuous problem $\\ddot{x}(t)+4\\sin^2\\frac{\\pi}{2n}x(t)=0$, $x(0)=0$, \nx(n)=\\varepsilon \\neq 0$ has only one solution $x(t)= \\varepsilon \\frac{\\sin\n(2 \\sin\\frac{\\pi}{2n})t]}{\\sin[(2 \\sin\\frac{\\pi}{2n})n]}$, whereas its\ndiscrete analogue $\\Delta^2x(k)+4\\sin^2\\frac{\\pi}{2n}x(k)=0$, $x(0)=0$, \nx(n)=\\varepsilon \\neq 0$ has no solution.\n\\end{example}\n\nThus, the nature of the solution changes when a continuous boundary value\nproblem is being discretized. Moreover, two-point boundary value problems\ninvolving derivatives lead to multipoint problems in the discrete case.\n\nThe above remarks and examples show that steal it is important to consider\nboth continuous and discrete problems simultaneously and investigate\nrelation between solutions which is the key factor especially when the\nexistence part follows by standard techniques.\n\nThere have been some research in this case addressing mainly problems whose\nsolutions where obtained by the fixed point theorems and the method of lower\nand upper solutions, \\cite{rech1}, \\cite{rachunkowa2}, \\cite{thomsontisdell\n. In this submission we are aiming at using critical point theory method,\nnamely the direct method of the calculus of variations (see for example \\cit\n{Ma} for a nice introduction to this topic) in order to show that in this\nsetting one can also obtain suitable convergence results. The advance over\nworks mentioned is that we can have better growth conditions imposed on $f$\nat the expense of not putting derivative of $x$ in $f$. As expected we will\nhave to get the uniqueness of solutions for the associated discrete problem,\nwhich is not always easy to be obtained, see \\cite{uni2}.\n\nIn \\cite{kelly} following \\cite{gaines}, it is suggested which family of\ndifference equations for $n\\in \\mathbb{N}$ is to be chosen when\napproximating problem (\\ref{par}). For $a$, $b$ such that $a0$\nindependent of $n$ and such that \n\\begin{equation}\nn|\\Delta x^{n}(k-1)|\\leq Q\\text{ and }|x^{n}(k)|\\leq N \\label{ewa1}\n\\end{equation\nfor all $k\\in \\mathbb{N}(0,n)$ and all $n\\geq n_{0}$, where $n_{0}$ is fixed\n(and arbitrarily large). Lemma 9.2. from \\cite{kelly} says that for some\nsubsequence $x^{n_{m}}=\\left( x^{n_{m}}(k)\\right) $ of $x^{n}$ it holds \n\\begin{equation}\n\\lim_{m\\rightarrow \\infty }\\max_{0\\leq k\\leq n_{m}}\\left\\vert\nx^{n_{m}}\\left( k\\right) -x\\left( \\frac{k}{n_{m}}\\right) \\right\\vert =0.\n\\label{ewa2}\n\\end{equation\nIn other words, this means that the suitable chosen discretization\napproaches the given continuos boundary value problem. Such solutions to\ndiscrete BVPs are called non-spurious in contrast to spurious ones which\neither diverge or else converge to anything else but the solution to a given\ncontinuous Dirichlet problem.\n\n\\section{Non spurious solutions for (\\protect\\ref{par})}\n\n\\subsection{The continuous problem}\n\nIn the existence part we apply variational methods. This means that with\nproblem under consideration we must associate the Euler action functional,\nprove that this functional is weakly lower semicontinuous in a suitable\nfunction space, coercive and at least G\\^{a}teaux differentiable. Given this\nthree conditions one knows that at least a weak solution to problem under\nconsideration exists whose regularity can further be improved with known\ntools. Such scheme, commonly used within the critical point theory is well\ndescribed in the first chapters of \\cite{Ma}.\n\nThe solutions to (\\ref{par}) will be investigated in the space \nH_{0}^{1}\\left( 0,1\\right) $ consisting of absolutely continuous functions\nsatisfying the boundary conditions and with a.e. derivative being integrable\nwith square. Such a solution is called a weak one, i.e. a function $x\\in\nH_{0}^{1}\\left( 0,1\\right) $ is a weak $H_{0}^{1}\\left( 0,1\\right) $\nsolution to (\\ref{par}), if \n\\begin{equation*}\n\\int_{0}^{1}\\dot{x}\\left( t\\right) \\dot{v}\\left( t\\right)\ndt+\\int_{0}^{1}f\\left( t,x\\left( t\\right) \\right) v\\left( t\\right) dt=0\n\\end{equation*\nfor all $v\\in H_{0}^{1}\\left( 0,1\\right) $. The classical solution to (\\re\n{par}) is then defined as a function $x:$ $\\left[ 0,1\\right] \\rightarrow \n\\mathbb{R}$ belonging to $H_{0}^{1}\\left( 0,1\\right) $ such that $\\ddot{x}$\nexists a.e. and $\\ddot{x}\\in L^{1}\\left( 0,\\pi \\right) $. Since $f$ is\njointly continuous, then it is known from the Fundamental Theorem of the\nCalculus of Variations, see \\cite{Ma}, that $x$ is in fact twice\ndifferentiable with classical continuous second derivative. Thus $x\\in\nH_{0}^{1}\\left( 0,1\\right) \\cap C^{2}\\left( 0,1\\right) $.\n\nLet $F\\left( t,x\\right) =\\int_{0}^{x}f\\left( t,s\\right) ds$ for $\\left(\nt,x\\right) \\in \\left[ 0,1\\right] \\times \\mathbb{R}$. We link solutions to \n\\ref{par}) with critical points to a $C^{1}$ functional $J:H_{0}^{1}\\left(\n0,1\\right) \\rightarrow \\mathbb{R}$ given by \n\\begin{equation*}\nJ\\left( x\\right) =\\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right)\ndt+\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt.\n\\end{equation*\nLet us examine $J$ for a while. Due to the continuity of $f$ functional $J$\nis well defined. Recall that the norm in $H_{0}^{1}\\left( 0,1\\right) $ reads \n\\begin{equation*}\n\\left\\Vert x\\right\\Vert =\\sqrt{\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt}.\n\\end{equation*\nThen we see $\\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt=\\frac{1}{2\n\\left\\Vert x\\right\\Vert ^{2}$ is a $C^{1}$ functional by standard facts. Its\nderivative is a functional on $H_{0}^{1}\\left( 0,1\\right) $ which reads \n\\begin{equation*}\nv\\rightarrow \\int_{0}^{1}\\dot{x}\\left( t\\right) \\dot{v}\\left( t\\right) dt.\n\\end{equation*\nConcerning the nonlinear part we see that for any fixed $v\\in\nH_{0}^{1}\\left( 0,1\\right) $ (which is continuous of course) function \n\\varepsilon \\rightarrow \\int_{0}^{1}F\\left( t,x\\left( t\\right) +\\varepsilon\nv\\left( t\\right) \\right) dt$ (where the integral we can treat as the Riemann\none) due to the Leibnitz differentiation formula under integral sign is \nC^{1}$ and the derivative of $\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right)\ndt$ is a functional on $H_{0}^{1}\\left( 0,1\\right) $ which reads \n\\begin{equation*}\nv\\rightarrow \\int_{0}^{1}f\\left( t,x\\left( t\\right) \\right) v\\left( t\\right)\ndt\n\\end{equation*\nif we recall that $F\\left( t,x\\right) =\\int_{0}^{x}f\\left( t,s\\right) ds$.\nSince the above is obviously continuous in $x$ uniformly in $v$ form unit\nsphere, we see that $J$ is in fact $C^{1}.$\n\nRecall also Poincar\\'{e} inequality $\\int_{0}^{1}x^{2}\\left( t\\right) dt\\leq \n\\frac{1}{\\pi ^{2}}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt$ and Sobolev's\none $\\max_{t\\in \\left[ 0,1\\right] }\\left\\vert x\\left( t\\right) \\right\\vert\n\\leq \\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt$.\n\nWe sum up the assumptions on the nonlinear term in (\\ref{par}) since in\norder to get the above mentioned observations continuity of $f$ is\nsufficient. We assume that\\newline\n\\textit{\\textbf{H1}} $f:\\left[ 0,1\\right] \\times \\mathbb{R} \\rightarrow \n\\mathbb{R}$ is a continuous function such that $f\\left( t,0\\right) \\neq 0$\nfor $t\\in \\left[ 0,1\\right]$;\\newline\n\\textit{\\textbf{H2}} $f$ is nondecreasing in $x$ for all $t\\in \\left[ 0,\n\\right] $\n\n\\begin{proposition}\nAssume that \\textbf{H1} and \\textbf{H2} are satisfied. Then problem (\\re\n{par}) has exactly one nontrivial solution.\n\\end{proposition}\n\n\\begin{proof}\nFirstly, we consider the existence part. Note that by Weierstrass Theorem\nthere exists $c>0$ such that \n\\begin{equation*}\n\\left\\vert f\\left( t,0\\right) \\right\\vert \\leq c\\text{ for all }t\\in \\left[\n0,1\\right] .\n\\end{equation*\nSince $f$ is nondecreasing in $x$ \\textit{\\textbf{H2}} it follows that $F$\nis convex. Since $F\\left( t,0\\right) =0$ for all $t\\in \\left[ 0,1\\right] $\nwe obtain from the well known inequality \n\\begin{equation}\nF(t,x)=F(t,x)-F(t,0)\\geq f\\left( t,0\\right) x\\geq -\\left\\vert f\\left(\nt,0\\right) x\\right\\vert \\label{aaa}\n\\end{equation\nvalid for any $x$ and all for all $t\\in \\left[ 0,1\\right] $. We observe that\nfrom (\\ref{aaa}) we get \n\\begin{equation}\nF\\left( t,x\\right) \\geq -c\\left\\vert x\\right\\vert \\text{ for all }t\\in \\left[\n0,1\\right] \\text{ and all }x\\in \\mathbb{R}. \\label{estimF}\n\\end{equation\nHence for any $x\\in H_{0}^{1}\\left( 0,1\\right) $ we see by Schwartz and\nPoincar\\'{e} inequality \n\\begin{equation*}\n\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt\\geq\n-c\\int_{0}^{1}\\left\\vert x\\left( t\\right) \\right\\vert dt\\geq -\\frac{c}{\\pi \n\\left\\Vert x\\right\\Vert .\n\\end{equation*\nTherefor\n\\begin{equation}\n\\begin{array}{l}\nJ\\left( x\\right) \\geq \\frac{1}{2}\\left\\Vert x\\right\\Vert ^{2}-\\left\\vert\nc\\right\\vert \\left\\Vert x\\right\\Vert \n\\end{array}\n\\label{Ju_}\n\\end{equation\nHence from (\\ref{Ju_}) we obtain that $J$ is coercive. Note that $\\frac{1}{2\n\\left\\Vert x\\right\\Vert ^{2}$ is obviously w.l.s.c. on $H_{0}^{1}\\left(\n0,1\\right) $. Next, by the Arzela-Ascoli Theorem and Lebesgue Dominated\nConvergence, see these arguments in full detail in \\cite{Ma} in the proof of\nTheorem 1.1 we see that $x\\rightarrow \\int_{0}^{1}F\\left( t,x\\left( t\\right)\n\\right) dt$ is weakly continuous. Thus $J$ is weakly l.s.c. as a sum of a\nw.l.s.c. and weakly continuous functionals. Since $J$ is $C^{1}$ and convex\nfunctional it has exactly one argument of a minimum which is necessarily a\ncritical point and thus a solution to (\\ref{par}). Putting $x=0$ in (\\re\n{par}) one see that we have a contradiction, so any solution is nontrivial.\n\\end{proof}\n\nIn order to get the existence of nontrivial solution to (\\ref{par}) it would\nsuffice to assume that $f\\left( t_{0},0\\right) \\neq 0$ for some $t_{0}\\in\n\\left[ 0,1\\right] $ but since we need to impose same conditions on discrete\nproblem it is apparent that our assumption is more reasonable. Moreover,\nthere is another way to prove the weak lower semincontinuity of $J$, namely\nshow that $J$ is continuous. Then it is weakly l.s.c. since it is convex.\nHowever, in proving continuity of $J$ on $H_{0}^{1}\\left( 0,1\\right) $ one\nuses the same arguments.\n\n\\subsection{The discrete problem}\n\nNow we turn the discretization of (\\ref{par}), i.e. to problem (\\ref{diffequ\n).\nconsidered in the $n$-dimensional Hilbert space $E$ consisting of functions \nx:\\mathbb{N}(0,n)\\rightarrow \\mathbb{R}$ such that $x(0)=x(n)=0$. Space $E$\nis considered with the following nor\n\\begin{equation}\n\\left\\Vert x\\right\\Vert =\\left( \\sum\\limits_{k=1}^{n}|{\\Delta \nx(k-1)|^{2}\\right) ^{\\frac{1}{2}}. \\label{norm_operator}\n\\end{equation\nWe can also consider $E$ with the following norm \n\\begin{equation*}\n\\left\\Vert u\\right\\Vert _{0}=\\left( \\sum\\limits_{k=1}^{n}|u(k)|^{2}\\right) ^\n\\frac{1}{2}}.\n\\end{equation*\nSince $E$ is finite dimensional there exist constants $c_{b}=\\frac{1}{2}$ \\\nand $c_{a}=\\left( \\left( n-1\\right) n\\right) ^{1\/2}$ such that \n\\begin{equation}\nc_{b}\\left\\Vert u\\right\\Vert \\leq \\left\\Vert u\\right\\Vert _{0}\\leq\nc_{a}\\left\\Vert u\\right\\Vert \\text{ for all }u\\in E. \\label{c_a_c_b}\n\\end{equation\nSolutions to (\\ref{diffequ}) correspond in a $1-1$ manner to the critical\npoints to the following $C^{1}$ functional $\\mathcal{I}:E\\rightarrow \\mathbb\nR}$ \n\\begin{equation*}\n\\mathcal{I}(x)=\\sum\\limits_{k=1}^{n}\\tfrac{1}{2}|\\Delta x(k-1)|^{2}+\\frac{1}\nn^{2}}\\sum\\limits_{k=1}^{n-1}F(\\frac{k}{n},x(k))\n\\end{equation*\nwith $F$ defined as before. This means that \n\\begin{equation*}\n\\frac{d}{dx}\\mathcal{I}(x)=0\\text{ if and only if }x\\text{ satisfies (\\re\n{diffequ}).}\n\\end{equation*\nNow we do not need to introduce the notion of the weak solution that is why\nwe have only one type of variational solution. We know that by the discrete\nSchwartz Inequality by (\\ref{estimF}) and by (\\ref{c_a_c_b}) \n\\begin{equation}\n\\begin{array}{l}\n\\mathcal{I}(x)\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\frac{1}{n^{2}}\\left\\vert\nc\\right\\vert \\sqrt{n}\\left( \\sum\\limits_{k=1}^{n-1}\\left\\vert x\\left(\nk\\right) \\right\\vert ^{2}\\right) ^{1\/2} \\\\ \n\\\\ \n\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert c\\right\\vert \\frac{\\sqrt{n-1}}{\n}\\left\\Vert x\\right\\Vert \\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert\nc\\right\\vert \\left\\Vert x\\right\\Vert \n\\end{array}\n\\label{relcoer}\n\\end{equation\nHence $\\mathcal{I}(x)\\rightarrow +\\infty $ as $\\Vert x\\Vert \\rightarrow\n+\\infty $ and we are in position to formulate the following\n\n\\begin{proposition}\n\\label{solvability_diff_equ}Assume that \\textbf{H1}, \\textbf{H2} hold. Then\nproblem (\\ref{diffequ}) has exactly one nontrivial solution.\n\\end{proposition}\n\n\\subsection{Main result}\n\n\\begin{theorem}\n\\label{first convergence theorem}Assume that conditions \\textbf{H1}, \\textbf\nH2} are satisfied. Then there exists $x\\in H_{0}^{1}\\left( 0,1\\right) \\cap\nC^{2}\\left( 0,1\\right) $ which solves uniquely (\\ref{par}) and for each \nn\\in \\mathbb{N}$ there exists $x^{n}$ which solves uniquely (\\ref{diffequ}).\nMoreover, there exists a subsequence $x^{n_{m}}$ of $x^{n}$ such that\ninequalities (\\ref{ewa2}) are satisfied.\n\\end{theorem}\n\n\\begin{proof}\nWe need to show that there exist two constants independent of $n$ such that\ninequalities (\\ref{ewa1}) hold.\nwhere $n_{0}$ is fixed. Then Lemma 9.2. from \\cite{kelly} provides the\nassertion of the theorem. In our argument we use some observations used in\nthe investigation of continuous dependence on parameters for ODE, see \\cit\n{LedzewiczWalczak}. Fix $n$. By Proposition \\ref{solvability_diff_equ},\nthere exists $x^{n}$ solving uniquely (\\ref{diffequ}) and which is an\nargument of a minimum to $\\mathcal{I}$ such that it holds that $\\mathcal{I\n(x^{n})\\leq \\mathcal{I}(0)=0$. Thus relation (\\ref{relcoer}) leads to the\ninequality \n\\begin{equation*}\n\\frac{1}{2}\\Vert x^{n}\\Vert\\leq \\left\\vert c\\right\\vert \\frac{\\sqrt{n-1}}{n}.\n\\end{equation*}\nSince $\\max_{k\\in \\mathbb{N}(0,n)}\\left\\vert x^{n}\\left( k\\right)\n\\right\\vert \\leq \\frac{\\sqrt{n+1}}{2}\\left\\Vert x^{n}\\right\\Vert$ we get\nthat for all $k\\in \\mathbb{N}(0,n)$ \n\\begin{equation*}\n\\left\\vert x^{n}\\left( k\\right) \\right\\vert \\leq 2\\left\\vert c\\right\\vert \n\\frac{\\sqrt{n-1}}{n}\\frac{\\sqrt{n+1}}{2}\\leq \\left\\vert c\\right\\vert =N.\n\\end{equation*}\nBy Lemma 9.3 in \\cite{kelly} we now obtain that there is a constant $Q$ such\nthat condition \n\\begin{equation*}\nn\\vert \\Delta x^{n}(k-1) \\vert \\leq Q \\text{ and }\\vert x^{n}(k) \\vert \\leq N\n\\end{equation*}\nfor all $k\\in \\mathbb{N}(0,n)$ and all $n\\in \\mathbb{N}$ is satisfied. This\nmeans that the application of Lemma 9.2 from \\cite{kelly} finishes the proof.\n\\end{proof}\n\n\\section{Final comments and examples}\n\nIn this section we provide the examples of nonlinear terms satisfying our\nassumptions and we will investigate the possibility of replacing the\nconvexity assumption imposed on $F$ with some weaker requirement as well as\nwe comment on exisiting results in the literature.\n\nConcerning the examples of nonlinear terms any nondecreasing $f$ is of order\nbounded or unbounded, see \n\n\\begin{enumerate}\n\\item[a)] $f\\left( t,x\\right) =g\\left( t\\right) \\exp \\left( x-t^{2}\\right)$;\n\n\\item[b)] $f\\left( t,x\\right) =g\\left( t\\right) \\arctan \\left( x\\right)$;\n\n\\item[c)] $f\\left( t,x\\right) =g\\left( t\\right) x^{3}+\\exp \\left(\nx-t^{2}\\right)$,\n\\end{enumerate}\n\nwhere $g$ is any lower bounded continuous function with positive values. \n\nIn view of remarks contained in \\cite{Ma} functional $J$ can be written \n\\begin{equation*}\nJ\\left( x\\right) =\\left( \\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right)\ndt-\\frac{a}{2\\pi }\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) \n\\end{equation*\n\\begin{equation*}\n+\\left( \\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt+\\frac{a}{2\\pi \n\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) .\n\\end{equation*\nThen functional \n\\begin{equation*}\nx\\rightarrow \\left( \\frac{1}{2}\\int_{0}^{1}\\dot{x}^{2}\\left( t\\right) dt\n\\frac{a}{2\\pi }\\int_{0}^{1}x^{2}\\left( t\\right) dt\\right) \n\\end{equation*\nis strictly convex as long as $a\\in \\left( 0,1\\right) $. Note that the first\neigenvalue of the differential operator $-\\frac{d^{2}}{dt^{2}}$ with\nDirichlet boundary conditions on $\\left[ 0,1\\right] $ is $\\frac{1}{\\pi }$\n(note this is the best constant in Poincar\\'{e} inequality). Hence we can\nrelax convexity assumption $F$ by assuming that \n\\begin{equation*}\nx\\rightarrow F\\left( t,x\\right) +\\frac{a}{2\\pi }x^{2}\n\\end{equation*\nis convex for any $t\\in \\left[ 0,1\\right] $. Then $F_{1}\\left( t,x\\right)\n=F\\left( t,x\\right) +\\frac{a}{2\\pi }x^{2}$ satisfies (\\ref{estimF}). \n\nThe natural question arises if similar procedure is possible as far as the\ndiscrete problem (\\ref{diffequ}) is concerned. However there is one big\nproblem here since the first eigenvalue for $-\\Delta ^{2}$ reads $\\lambda\n_{1}=2-2\\cos \\left( \\frac{\\pi }{n+1}\\right) $ and of course $\\lambda\n_{1}\\rightarrow 0$ as $n\\rightarrow \\infty $. This means that the above idea\nwould not work, since we cannot find $a$ for all $n$ idependent of $n$ (for\neach $n$ such $a=a\\left( n\\right) $ exists ). \n\nA comparison with existing results is also in order. The only papers\nconcerning the existence of non-spurious solutions are \\cite{rech1}, \\cit\n{rachunkowa2}, \\cite{thomsontisdell} which follow ideas developed in \\cit\n{gaines} and which were mentioned already in the Introduction. We not only\nuse different methods, namely critical point theory, but also we are not\nlimited as far as the growth is concerned since in sources mentioned $f$ is\nsublinear. However, we could not incorporate the derivative of $x$ into the\nnonlinear term. This is not possible by variational approach but could be\nmade possible by connecting variational methods with Banach contraction\nprinciple and it shows that the research concerning the existence of\nnon-spurious solutions with critical point approach can be further developed.\n\nWe cannot use sublinear growth as in sources mentioned since it does not\nprovide the inequality \n\\begin{equation}\nF\\left( t,x\\right) -F\\left( t,0\\right) \\geq f\\left( t,0\\right) x\\text{ for\nall }t\\in \\left[ 0,1\\right] \\text{ and all }x\\in \\mathbb{R}. \\label{ccccc}\n\\end{equation\nWith our approach inequality (\\ref{ccccc}) is essential in proving the\nrequired estimations which lead to the existence of non-spurious solutions.\nThis is shown by the below remarks where direct calculations are performed.\n\nThe relevant growth condition reads\\newline\n\\textit{\\textbf{H2a} There exist constants} $a,b>0$ and $\\gamma \\in \\left[\n0,1\\right) $ such that \n\\begin{equation}\nf\\left( t,x\\right) \\leq a+b\\left\\vert x\\right\\vert ^{\\gamma }\\text{ for all \nt\\in \\left[ 0,1\\right] \\text{ and all }x\\in \\mathbb{R}.\n\\label{cond_unboud_below}\n\\end{equation\nBy (\\ref{cond_unboud_below}) for all $t\\in \\left[ 0,1\\right] $ and all $x\\in \n\\mathbb{R}$ it holds \n\\begin{equation*}\nF\\left( t,x\\right) \\leq a\\left\\vert x\\right\\vert +\\frac{b}{\\gamma +1\n\\left\\vert x\\right\\vert ^{\\gamma +1}.\n\\end{equation*\nSince $F\\left( t,x\\right) \\geq -\\left\\vert F\\left( t,x\\right) \\right\\vert $\nwe see by Schwartz, Holder and Poincar\\'{e} inequality for any $x\\in\nH_{0}^{1}\\left( 0,1\\right) $ \n\\begin{equation*}\n\\int_{0}^{1}F\\left( t,x\\left( t\\right) \\right) dt\\geq -c_{1}\\left\\Vert\nx\\right\\Vert -c_{2}\\left\\Vert x\\right\\Vert ^{\\gamma +1},\n\\end{equation*\nwhere $c_{1}=a$ and $c_{2}>0$ (the exact value of $c_{2}$ is not important\nsince $\\gamma +1<2$ and functional $J$ is coercive disregarding of the value\nof $c_{2}$. Then problem (\\ref{par}) has at least one solution by the direct\nmethod of the calculus of variations.\n\nIn order to consider problem (\\ref{diffequ}) we need to perform exact\ncalculations since in this case, in view of the convergence Theorem \\re\n{first convergence theorem}, the precise values of constants are of utmost\nimportance. In case of \\textit{\\textbf{H2a}} from H\\\"{o}lder's inequality\nand (\\ref{c_a_c_b}) we get \n\\begin{equation*}\n\\begin{array}{ll}\n\\sum\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1} & =\\su\n\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1} \\cdot 1 \\\\ \n& \\\\ \n& \\leq \\left( \\sum\\limits_{k=1}^{n-1}\\vert u(k)\\vert^{\\gamma +1}\\vert^{\\frac\n2}{\\gamma +1} }\\right) ^{\\frac{\\gamma +1}{2}}\\left(\n\\sum\\limits_{k=1}^{n-1}\\vert1\\vert^{\\frac{1}{1- \\frac{\\gamma +1}{2}}}\\right)\n^{1-\\frac{2}{\\gamma +1}} \\\\ \n& \\\\ \n& =\\left( n-1\\right) ^{\\frac{1-\\gamma }{2}}\\left\\Vert u\\right\\Vert\n_{0}^{\\gamma +1}\\leq \\left( \\left( n-1\\right) n\\right) ^{\\frac{\\gamma +1}{2\n}\\left( n-1\\right) ^{\\frac{1-\\gamma }{2}}\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\\\ \n& \\\\ \n& =\\left( n-1\\right) n^{\\frac{\\gamma +1}{2}}\\left\\Vert u\\right\\Vert ^{\\gamma\n+1}\\leq \\left( n-1\\right) n\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\end{array\n\\end{equation*}\nThus \n\\begin{equation*}\n\\frac{1}{n^{2}}\\frac{b}{\\gamma +1}\\sum\\limits_{k=1}^{n-1}\\vert\nu(k)\\vert^{\\gamma +1}\\leq \\frac{b}{\\gamma +1}n^{\\frac{\\gamma -1}{2\n}\\left\\Vert u\\right\\Vert ^{\\gamma +1}\n\\end{equation*}\n\nHence by the above calculations and (\\ref{relcoer}) we get for any $x\\in E$ \n\\begin{equation}\n\\mathcal{I}(x)\\geq \\frac{1}{2}\\Vert x\\Vert ^{2}-\\left\\vert a\\right\\vert \n\\frac{\\sqrt{n-1}}{n}\\left\\Vert x\\right\\Vert -\\frac{b}{\\gamma +1}n^{\\frac\n\\gamma -1}{2}}\\left\\Vert x\\right\\Vert ^{\\gamma +1}. \\label{rel_add_coer}\n\\end{equation\nThus $\\mathcal{I}(x)\\rightarrow +\\infty $ as $\\Vert x\\Vert \\rightarrow\n+\\infty .$ By Lemma 9.2. from \\cite{kelly}, we need to show that (\\ref{ewa1\n) holds. Fix $n$. Since $\\mathcal{I}(x^{n})\\leq \\mathcal{I}(0)=0$, the\nrelation (\\ref{rel_add_coer}) leads to the inequality \n\\begin{equation}\n\\frac{1}{2}\\Vert x^{n}\\Vert \\leq \\left\\vert a\\right\\vert \\frac{\\sqrt{n-1}}{n\n+\\frac{b}{\\gamma +1}n^{\\frac{\\gamma -1}{2}}\\left\\Vert x^{n}\\right\\Vert\n^{\\gamma }. \\label{ineq}\n\\end{equation\nSince $\\gamma <1$ we see $n^{\\frac{\\gamma -1}{2}}\\rightarrow 0$. Thus there\nis some $n_{0}$ that for all $n\\geq n_{0}$ it holds $\\frac{b}{\\gamma +1}n^\n\\frac{\\gamma -1}{2}}<\\frac{1}{4}$. Take $n\\geq n_{0}$. Let us consider two\ncases, namely $\\left\\Vert x^{n}\\right\\Vert \\leq 1$ and $\\left\\Vert\nx^{n}\\right\\Vert >1$. In case $\\left\\Vert x^{n}\\right\\Vert >1$ we get from \n\\ref{ineq}) that \n\\begin{equation*}\n\\frac{1}{2}\\Vert x^{n}\\Vert \\leq \\left\\vert a\\right\\vert \\frac{\\sqrt{n-1}}{n\n+\\frac{1}{4}\\left\\Vert x^{n}\\right\\Vert \n\\end{equation*\nRecall $\\max_{k\\in \\mathbb{N}(0,n)}\\left\\vert x^{n}\\left( k\\right)\n\\right\\vert \\leq \\frac{\\sqrt{n+1}}{2}\\left\\Vert x^{n}\\right\\Vert $ we get\nthat for all $k\\in \\mathbb{N}(0,n)$ \n\\begin{equation*}\n\\left\\vert x\\left( k\\right) \\right\\vert \\leq 4\\left\\vert a\\right\\vert \\frac\n\\sqrt{n-1}}{n}\\frac{\\sqrt{n+1}}{2}\\leq 2\\left\\vert a\\right\\vert =N.\n\\end{equation*\nFor the case $\\left\\Vert x^{n}\\right\\Vert \\leq 1$ we however we cannot\nproceed without (\\ref{ccccc}). The reason is what while on space $E$\ndisregarding of $n$ the sequence is norm bounded by $1$ (uniformely in $n$)\nin norm given by (\\ref{norm_operator}), this is not the case with the\nmax-norm where it is unbounded as $n\\rightarrow \\infty $.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nThis paper considers estimation of parameters of distributions whose domain is a particular non-Euclidean geometry: a topological space divided into $M$ equivalence classes by actions of a finite spherical symmetry group. A well known example of a finite spherical symmetry group is the point group in 3 dimensions describing the soccer ball, or football, with truncated icosahedral symmetry that also corresponds to the symmetry of the Carbon-60 molecule. This paper formulates a general approach to parameter estimation in distributions defined over such domains. We use a restricted finite mixture representation introduced in~\\cite{chen_parameter_2015} for probability distributions that are invariant to actions of any topological group. This representation has the property that the number of mixture components is equal to the order of the group, the distributions in the mixture are all parameterized by the same parameters, and the mixture coefficients are all equal. This is practically significant since many reliable algorithms have been developed for parameter estimation when samples come from finite mixture distributions~\\cite{dempster_maximum_1977,sohn_efficient_2011}.\n\nWe apply the representation to an important problem in materials science: analysis of mean orientation in polycrystals. \nCrystal orientation characterizes properties of materials including electrical conductivity and thermal conductivity. Polycrystalline materials are composed of grains of varying size and orientation, where each grain contains crystal forms with similar orientations. The quality of the material is mainly determined by the grain structure i.e. the arrangement of the grains, their orientations, as well as the distribution of the precipitates. Thus accurate estimation of crystal orientation of the grains is useful for predicting how materials fail and what modes of failure are more likely to occur \\cite{de_graef_structure_2007}.\n\nThe mean orientation of the grain, characterized for example by its Euler angles, can only be specified modulo a set of angular rotations determined by the symmetry group associated with the specific type of crystal, e.g. hexagonal, cubic. This multiplicity of equivalent Euler angles complicates the development of reliable mean orientation estimators. The problem becomes even harder when the orientations are sampled from a region encompassing more than one grain such that the orientations cluster over different mean directions. In such a case, we would like to identify whether the orientations are multi-modally distributed and also estimate the mean direction for each cluster. \n\nIn our previous work~\\cite{chen_parameter_2015}, we introduced the finite mixture of Von Mises-Fisher (VMF) distribution for observations that are invariant to actions of a spherical symmetry group. We applied the expectation maximization (EM) maximum likelihood (ML) algorithm, called EM-VMF, to estimate the group-invariant parameters of this distribution. In this paper, we develop a hyperbolic representation simplification of the EM-VMF algorithm that reduces the computation time by a factor of $2$. We also introduce a new group invariant distribution for spherical symmetry groups, called the $\\mathcal{G}$-invariant Watson distribution, which like VMF is a density parameterized by location (angle mean) and scale (angle concentration) over the $p$-dimensional sphere. An EM algorithm is presented for estimation of the parameters, called the EM-Watson algorithm. Furthermore, mixture-of-$\\mathcal{G}$-invariant Watson (mGIW) and von Mises-Fisher (mGIV) distributions are introduced to perform clustering on the $\\mathcal{G}$-invariant sphere. An EM algorithm is presented for estimation of the parameters of the mGIW and mGIV distributions. We illustrate how the Generalized Likelihood Ratio Test (GLRT) can be used to detect the presence of multiple modes in a sample and how it can be combined with the EM algorithm for mGIW and mGIV distributions to cluster multiple orientations on the sphere. \n\nThe performance of the proposed EM orientation estimators is evaluated by simulation and compared to other estimators. The EM orientation estimators are then illustrated on Electron Backscatter Diffraction EBSD data collected from a Nickel alloy whose crystal form induces the $m\\overline{3}m$~\\cite{newnham_properties_2004} cubic point symmetry group. We establish that the EM orientation estimators result in significantly improved estimates of the mean direction in addition to providing an accurate estimate of concentration about the mean. Furthermore, with the extended mixture models, we are able to identify and cluster multi-modally distributed samples more accurately than the K-means algorithm.\n\nThe paper is organized as follows. Section~\\ref{sec:group-invariant} describes group invariant random variables and gives the mixture representation for their densities. Section \\ref{sec:spherical_symmetry_group} specializes to random variables invariant relative to actions of the spherical symmetry group and develops the $\\mathcal G$-invariant VMF and Watson distributions along with EM-ML parameter estimator. The clustering methods based on the $\\mathcal{G}$-invariant distributions along with the GLRT are elaborated in Section \\ref{sec:clustering_spherical_symmetry_group}. The crystallography application and data simulation are presented in Section~\\ref{sec:app_crystal_orientation_estimation} and the experiment results are shown in Section \\ref{sec:experiment}. Section~\\ref{sec:conclusion} has concluding remarks.\n\n\\section{Group-invariant random variables}\n\\label{sec:group-invariant}\n\\def\\bfx{{\\mathbf x}}\nConsider a finite topological group $\\mathcal G=\\{G_1, \\ldots, G_M\\}$ of $M$ distinct actions on a topological space $\\mathcal X$, $G_i: \\mathcal X \\rightarrow \\mathcal X$ and a binary operation \"*\" defining the action composition $G_i * G_j$, denoted $G_i G_j$. $\\mathcal G$ has the properties that composition of multiple actions is associative, for every action there exists an inverse action, and there exists an identity action \\cite{birkhoff_brief_1963}. A real valued function $f(\\bx)$ on $\\mathcal X$ is said to be invariant under $\\mathcal G$ if: $f(G\\bx)=f(\\bx)$ for $G\\in \\mathcal G$. Let $\\bX$ be a random variable defined on $\\mathcal X$. We have the following theorem for the probability density $f(\\bx)$ of $\\bX$.\n\\begin{theorem}\n\\label{thm:1}\nThe density function $f: \\mathcal X\\rightarrow \\Reals$ is invariant under $\\mathcal G$ if and only if\n\\begin{eqnarray}\n\\label{eq:thm1_representation}\n\\!\\begin{aligned}\n\\exists\\ &h: \\mathcal X\\rightarrow \\Reals\\ s.t. \\\\\n&f(\\bx)= \\frac{1}{M} \\sum_{i=1}^M h(G_i\\bx).\n\\end{aligned}\n\\end{eqnarray}\n\\end{theorem}\nThis theorem is a slight generalization of \\cite[Thm. 2.1]{chen_parameter_2015} in that the density $h(.)$ is not necessarily the same as $f(.)$. The proof is analogous to that of \\cite[Thm. 2.1]{chen_parameter_2015}.\n\nTheorem \\ref{thm:1} says that any density $f(\\bx)$ that is invariant under group $\\mathcal G$ can be represented as a finite mixture of a function and its translates $h(G_i\\bx)$ under the group's actions $G_i \\in \\mathcal G$. As pointed out in~\\cite{chen_parameter_2015}, Thm.~\\ref{thm:1} has important implications on $\\mathcal G$-invariant density estimation and parameter estimation. In particular it can be used to construct maximum likelihood estimators for parametric densities. Let $h(\\bx;\\btheta)$ be a density on $\\mathcal X$ that is parameterized by a parameter $\\btheta$ in a parameter space $\\Theta$. We extend $h(\\bx;\\btheta)$ to a $\\mathcal G$-invariant density $f$ by using Thm. \\ref{thm:1}, obtaining:\n\\begin{eqnarray}\nf(\\bx;\\btheta)=\\frac{1}{M} \\sum_{i=1}^M h_i(\\bx;\\btheta),\n\\label{eq:SSG}\n\\end{eqnarray}\nwhere $h_i(\\bx;\\btheta)=h(G_i\\bx;\\btheta)$. This density is of the form of a finite mixture of densities $h_i(\\bx;\\btheta)$ of known parametric form where the mixture coefficients are all identical and equal to $1\/M$. Maximum likelihood (ML) estimation of the parameter $\\btheta$ from an i.i.d. sample $\\{\\bx_i\\}_{i=1}^n$ from any $\\mathcal G$-invariant density $f$ can now be performed using finite mixture model methods \\cite{mclachlan_finite_2004} such as the Expectation-Maximization (EM) algorithm~\\cite{dempster_maximum_1977} or the restricted Boltzman machine (RBM) \\cite{sohn_efficient_2011}.\n\n\\section{ML within a Spherical Symmetry Group}\n\\label{sec:spherical_symmetry_group}\nAs in~\\cite{chen_parameter_2015} we specialize Thm.~\\ref{thm:1} to estimation of parameters for the case that the probability density is on a sphere and is invariant to actions in a spherical symmetry group. In Section~\\ref{sec:app_crystal_orientation_estimation} this will be applied to a crystallography example under spherical distribution likelihood models for the mean crystal orientation. In general, the measured and mean orientations can be represented by Euler angles~\\cite{eberly_euler_2008}, Rodrigues Vectors~\\cite{rodrigues_lois_1840}, or Quaternions~\\cite{altmann_rotations_2005}. As in~\\cite{chen_parameter_2015}, we use the quaternion representation to enable orientations to be modeled by spherical distributions since the quaternion representation is a $4$D vector on the $3$-sphere $S^3$, i.e. $\\bq = (q_1, q_2, q_3, q_4)$ such that $\\|\\bq\\| = 1$.\n\nAny of the aforementioned orientation representations have inherent ambiguity due to crystal symmetries. For example, if the crystal has cubic symmetry, its orientation is only uniquely defined up to a 24-fold set of proper rotations of the cube about its symmetry axes.\nThese actions form a point symmetry group, called $432$, a sub-group of $m\\overline{3}m$. In quaternion space, since each orientation corresponds to two quaternions with different sign $\\{\\bq,-\\bq\\}$, these rotations reflections, and inversions can be represented as a spherical symmetry group $\\mathcal{G}$ of quaternionic matrices $\\{\\bP_1,\\ldots,\\bP_{M}\\}$, with sign symmetry such that $\\bP_i=-\\bP_{i-M\/2}\\ \\forall M\/2 0, \\\\\n\\hat\\bmu &= \\bt_p, \\hat{\\kappa} < 0.\n\\end{split}\n\\end{equation}\n\nSimilarly by fixing $\\bmu$ and setting to zero the derivative of (\\ref{eq:Watson_Mstep_Dev}) with respect to $\\kappa$, we have:\n\\begin{equation}\n\\label{eq:Watson_Tp_func}\n\\begin{split}\n&Y_p(\\kappa) = \\frac{\\bbM'(\\frac{1}{2},\\frac{p}{2},\\kappa)}{\\bbM(\\frac{1}{2},\\frac{p}{2},\\kappa)} = \\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,m}(\\bmu^T\\bP_m^T\\bx_i)^2}{n} \\\\\n\\Rightarrow& \\hat{\\kappa} = Y_p^{-1}\\left( \\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,m}(\\bmu^T\\bP_m^T\\bx_i)^2}{n}\\right),\n\\end{split}\n\\end{equation}\nThe final estimates of $\\bmu$ and $\\kappa$ are obtained by checking both cases ($\\hat{\\kappa}>0$, $\\hat{\\kappa}<0$) and choosing the one which is consistent for (\\ref{eq:Watson_mu_est})(\\ref{eq:Watson_Tp_func}).\n\n\\section{Clustering with a Spherical Symmetry Group}\n\\label{sec:clustering_spherical_symmetry_group}\nIn this section we extend the parameter estimation problem to the situation where there are multiple group-invariant distributions with different parameters that govern the samples. This problem arises, for example, in poly-crystaline materials when estimating the mean crystal orientation over a region containing more than one grain (perhaps undetected). This problem can be solved by first applying some standard clustering methods, e.g. K-means\\cite{hartigan_algorithm_1979}, and then estimating the parameters for each cluster. However, clustering methods based on the distance relation between the samples are complicated by the presence of spherical symmetry because it is necessary to distinguish modes that are due only to symmetry from those that distinguish different clusters. Therefore, we propose a model-based clustering algorithm which accommodates symmetry to handle this problem.\n\nConsider the situation where the samples $\\{\\bx_i\\}_{i=1}^n$ follow a mixture of $\\mathcal{G}$-invariant density functions. For the VMF distribution, the mixture density has the following form:\n\\begin{equation}\n\\label{eq:mixture_of_Ginv_VMF}\ng_v(\\bx;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}) = \\sum_{c=1}^C\\alpha_c\\left(\\sum_{m=1}^M \\frac{1}{M} \\phi(\\bx;\\bP_m\\bmu_c,\\kappa_c)\\right),\n\\end{equation}\nwhere $C$ is the number of clusters assumed to be fixed a priori, $\\bmu_c,\\kappa_c$ are the parameters for the $c$-th cluster and $\\alpha_c$ are the mixing coefficients where $\\sum_{c=1}^C\\alpha_c=1$ and $\\alpha_c>0$ for all $c$. The parameters of (\\ref{eq:mixture_of_Ginv_VMF}) can be estimated by the EM algorithm:\n\nE-step:\n\\begin{equation}\n\\label{eq:mVMF_mClusters_Estep}\nr_{i,c,m}=\\frac{\\alpha_c\\phi(\\bx_i; \\bP_m\\bmu_c,\\kappa_c)}{\\sum_{h=1}^C\\alpha_{h}\\sum_{l=1}^M\\phi(\\bx_i;\\bP_{l}\\mu_{h}, \\kappa_{h})}\n\\end{equation} \n\nM-step:\n\\begin{align}\n\\alpha_c&=\\sum_{i=1}^n\\sum_{m=1}^Mr_{i,c,m}, \\hat{\\bmu}_c=\\frac{\\bgamma_c}{\\|\\bgamma_c\\|}, \\hat{\\kappa}_c=A_p^{-1}\\left(\\frac{\\|\\bgamma_c\\|}{n\\alpha_c}\\right),\\\\\n\\label{eq:mVMF_mClusters_Mstep}\n\\bgamma_c&=\\sum_{i=1}^n\\sum_{m=1}^M r_{i,c,m}\\bP_m^T\\bx_i,\n\\end{align}\nwhere $r_{i,c,m}$ is the probability of sample $\\bx_i$ belonging to the $c$-th cluster and the $m$-th symmetric component.\n\nFor the Watson distribution, the mixture of $\\mathcal{G}$-invariant Watson density is\n\\begin{equation}\n\\label{eq:mixture_of_Ginv_Watson}\ng_w(\\bx;\\{\\bmu_c,\\kappa_c, \\alpha_c\\}) = \\sum_{c=1}^C\\alpha_c\\left(\\sum_{m=1}^{M'} \\frac{1}{M'}W_p(\\bx; \\bP_m\\bmu_c,\\kappa_c)\\right)\n\\end{equation}\n\nThe E-step is similar to (\\ref{eq:mVMF_mClusters_Estep}) with $\\phi$ replaced by $W_p$ function. The M-step can be computed with a similar approach as in Section~\\ref{sec:ginv_Watson_dist} with the following modifications:\n\\begin{align}\n\\label{eq:Watson_mClusters_Mstep_hatT}\n\\tilde{T_c}&=\\frac{1}{n\\alpha_c}\\sum_{i=1}^n\\sum_{m=1}^{M'} r_{i,c,m}(\\bP_m^T\\bx_i\\bx_i^T\\bP_m), \\\\\n\\label{eq:Watson_mClusters_Mstep_kappa}\n\\hat{\\kappa}_c&=Y_p^{-1}\\left(\\frac{\\sum_{i=1}^n\\sum_{m=1}^{M'}r_{i,c,m}(\\bmu_c^T\\bP_m^T\\bx_i)^2}{n\\alpha_c}\\right),\n\\end{align}\nwhere $\\alpha_c=\\sum_{i=1}^n\\sum_{m=1}^Mr_{i,c,m}$.\n\\subsection{Multi-modality Tests on $\\mathcal{G}$-invariant Spherical Distributions}\nGiven sample set $\\{\\bx_i\\}_{i=1}^n$ on $S^{p-1}$, the objective is to determine whether the $n$ samples are drawn from one single distribution or a mixture of $C$ distributions. For polycrystalline materials, the result of this determination can be used to discover undetected grains within a region. We propose to use a multi-modal hypothesis test based on the $\\mathcal{G}$-invariant distributions to solve this problem. The two hypotheses are $H_0$: The samples are from a single $\\mathcal{G}$-invariant distribution $f(\\bx;\\{\\bmu,\\kappa\\})$; and $H_1$: The samples are from a mixture of $C$ distributions $g(\\bx;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C)$. The Generalized Likelihood Ratio Test (GLRT)~\\cite{hero_statistical_2000} has the following form:\n\\begin{equation}\n\\label{eq:multi_sample_GLRT}\n\\begin{aligned}\n\\Lambda_{GLR} &= \\frac{\\max_{\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C\\in\\Theta_1}g(\\{\\bx_i\\}_{i=1}^n;\\{\\bmu_c,\\kappa_c,\\alpha_c\\}_{c=1}^C)}{\\max_{\\{\\bmu,\\kappa\\}\\in\\Theta_0}f(\\{\\bx_i\\}_{i=1}^n;\\{\\bmu,\\kappa\\})} \\\\\n&\\gtrless^{H_1}_{H_0} \\eta\n\\end{aligned}\n\\end{equation}\nwhere $\\Theta_0,\\Theta_1$ are the parameter spaces for the two hypotheses. The $f$ and $g$ functions for VMF and Watson distributions are defined in (\\ref{eq:mixture_density}), (\\ref{eq:Watson_mixture_density}) and (\\ref{eq:mixture_of_Ginv_VMF}), (\\ref{eq:mixture_of_Ginv_Watson}) respectively and the test statistic $\\Lambda_{GLR}$ can be calculated by the proposed EM algorithm. According to Wilks's theorem~\\cite{wilks_large-sample_1938} as $n$ approaches $\\infty$, the test statistic $2\\log{\\Lambda_{GLR}}$ will be asymptotically $\\chi^2$-distributed with degrees of freedom equal to $(p+1)(C-1)$, which is the difference in dimensionality of $\\Theta_0$ and $\\Theta_1$. Therefore, the threshold $\\eta$ in (\\ref{eq:multi_sample_GLRT}) can be determined by a given significance level $\\alpha$.\n\n\\section{Application to Crystallographic Orientation}\n\\label{sec:app_crystal_orientation_estimation}\nCrystal orientation and the grain distribution in polycrystalline materials determine the mechanical properties of the material, such as, stiffness, elasticity, and deformability. Locating the grain regions and estimating their orientation and dispersion play an essential role in detecting anomalies and vulnerable parts of materials.\n\nElectron backscatter diffraction (EBSD) microscopy acquires crystal orientation at multiple locations within a grain by capturing the Kikuchi diffraction patterns of the backscatter electrons ~\\cite{saruwatari_crystal_2007}. A Kikuchi pattern can be translated to crystal orientation through Hough Transformation analysis~\\cite{lassen_automated_1994} or Dictionary-Based indexing~\\cite{park_ebsd_2013}. The process of assigning mean orientation values to each grain is known as indexing. Crystal forms possess point symmetries, e.g. triclinic, tetragonal, or cubic, leading to a probability density of measured orientations that is invariant over an associated spherical symmetry group $\\mathcal{G}$. Therefore, when the type of material has known symmetries, e.g., cubic-type symmetry for nickel or gold, the $\\mathcal{G}$-invariant VMF and Watson models introduced in Section~\\ref{sec:spherical_symmetry_group} can be applied to estimate the mean orientation $\\bmu_g$ and the concentration $\\kappa_g$ associated with each grain. Furthermore, the clustering method along with the multi-sample hypothesis test in Section~\\ref{sec:clustering_spherical_symmetry_group} can be used to detect the underlying grains within a region.\n\n\\subsection{Simulation of Crystallographic Orientation}\n\\label{sec:simulation_orientations}\nTo simulate the crystallographic orientations, we first draw random samples from VMF and Watson distributions with $p=4$. The random variable $\\bx$ in a spherical distribution can be decomposed~\\cite{mardia_directional_1999}:\n\\begin{equation}\n\\label{eq:normal_tangent_decompose}\n\\bx=t\\bmu+\\sqrt{1-t^2}S_\\bmu(\\bx),\n\\end{equation}\nwhere $t=\\bmu^T\\bx$ and $S_\\bmu(\\bx)=(I_p-\\bmu\\bmu^T)\\bx\/\\|(I_p-\\bmu\\bmu^T)\\bx\\|$. Let $f(\\bx;\\bmu)$ be the p.d.f. of the distribution where $\\bmu$ is the mean direction. According to the normal-tangent decomposition property, for any rotationally symmetric distribution, $S_\\bmu(\\bx)$ is uniformly distributed on $S_{\\bmu^\\bot}^{p-2}$, the $(p-2)$-dimensional sphere normal to $\\bmu$, and the density of $t=\\bx^T\\bmu$ is given by:\n\\begin{equation}\n\\label{eq:tangent_density}\nt\\mapsto cf(t)(1-t^2)^{(p-3)\/2}.\n\\end{equation}\n\nFor VMF distribution, substituting (\\ref{eq:normal_tangent_decompose}) into (\\ref{eq:VMF_pdf}) and combining with (\\ref{eq:tangent_density}), we have the density of the tangent component $t$ as:\n\\begin{equation}\n\\label{eq:VMF_tangent_density}\n\\begin{split}\nf_v(t)&= C_v\\exp{\\{\\kappa t\\}}(1-t^2)^{(p-3)\/2} \\\\\nC_v&=\\left(\\frac{\\kappa}{2}\\right)^{(p\/2-1)}\\left(I_{p\/2-1}(\\kappa)\\Gamma\\left(\\frac{p-1}{2}\\right)\\Gamma\\left(\\frac{1}{2}\\right)\\right)^{-1}.\n\\end{split}\n\\end{equation}\n\nSimilarly, the density of the tangent component of Watson distribution is:\n\\begin{equation}\n\\label{eq:Watson_tangetn_density}\n\\begin{split}\nf_w(t)&= C_w\\exp{\\{\\kappa t^2\\}}(1-t^2)^{(p-3)\/2} \\\\\nC_w&=\\frac{\\Gamma(\\frac{p}{2})}{\\Gamma(\\frac{p-1}{2})\\Gamma(\\frac{1}{2})}\\frac{1}{\\bbM(\\frac{1}{2},\\frac{p}{2},\\kappa)}.\n\\end{split}\n\\end{equation}\nRandom samples from the density functions (\\ref{eq:VMF_tangent_density}) and (\\ref{eq:Watson_tangetn_density}) can be easily generated by rejection sampling. \n\nThe generated quaternions from VMF and Watson distributions are then mapped into the Fundamental Zone (FZ) with the symmetric group actions to simulate the wrap-around problem we observe in real data, i.e. observations are restricted to a single FZ. For cubic symmetry, the FZ in quaternion space is defined in the following set of equations: \\\\\n\\begin{minipage}{0.48\\linewidth}\n\\begin{equation}\n\\begin{cases}\n|q_2\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_3\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_4\/q_1|\\le\\sqrt{2}-1 \\\\\n|q_2\/q_1 + q_3\/q_1 + q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 - q_3\/q_1 + q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 + q_3\/q_1 - q_4\/q_1|\\le 1 \\\\\n|q_2\/q_1 - q_3\/q_1 - q_4\/q_1|\\le 1 \\nonumber\n\\end{cases}\n\\end{equation}\n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n\\begin{equation}\n\\label{eq:FZ_equations}\n\\begin{cases}\n|q_2\/q_1 - q_3\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 + q_3\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 - q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_2\/q_1 + q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_3\/q_1 - q_4\/q_1|\\le\\sqrt{2} \\\\\n|q_3\/q_1 + q_4\/q_1|\\le\\sqrt{2} \\\\\n\\end{cases}\n\\end{equation}\n\\end{minipage}\nwhere $q_i$ is the $i$-th component of quaternion $\\bq$.\n\n\\section{Experimental Results}\n\\label{sec:experiment}\n\\subsection{$\\mathcal G$-invariant EM-ML Parameter Estimation on Simulated Data}\nSets of $n$ i.i.d. samples were simulated from the VMF or Watson distributions using the method described in Sec.\\ref{sec:simulation_orientations} with given $\\bmu=\\bmu_o,\\kappa=\\kappa_o$ for the $m\\overline{3}m$ point symmetry group associated with the symmetries of cubic crystal lattice planes. The number of samples for each simulation was set to $n=1000$ and $\\kappa_o$ was swept from $1$ to $100$ while, for each simulation run, $\\bmu_o$ was selected uniformly at random. The experiment was repeated $100$ times and the average values of $\\hat{\\kappa}$ and the inner product $\\hat{\\bmu}^T \\bmu_o$ are shown in Fig.~\\ref{fig:mu_est} and \\ref{fig:kappa_est}. In the figures we compare performance for the following methods: (1) the naive ML estimator for the standard VMF or Watson model that does not account for the point group structure (labeled \"ML Estimator\"). (2) Mapping each of the samples $\\bx_i$ toward a reference direction $\\bx_{r}$ (randomly selected from $\\{\\bx_i\\}_{i=1}^n$), i.e. $\\bx_i\\mapsto\\bP_m\\bx_i$, where $\\bP_m=\\arg\\min_{\\bP\\in\\mathcal{G}} \\arccos{(\\bx_{r}^T\\bP\\bx)}$, to remove possible ambiguity. Then performing ML for the standard VMF or Watson distribution (labeled \"Modified ML\"). (3) Applying our proposed EM algorithm directly to the $n$ samples using the mixture of VMF distribution (\\ref{eq:VMF_Estep})-(\\ref{eq:VMF_Mstep_gamma}) (labeled \"EM-VMF\") (4) Applying our proposed EM algorithm to the mixture of Watson distribution (\\ref{eq:Watson_EM_Estep})-(\\ref{eq:Watson_Tp_func}) (labeled \"EM-Watson\").\n\n\nFigure \\ref{fig:mu_est} shows the inner product values $\\bmu_o^T\\hat{\\bmu}$. The proposed EM-VMF and EM-Watson estimators have similar performance in that they achieve perfect recovery of the mean orientation ($\\bmu_o^T\\hat{\\bmu}=1$) much faster than the other methods as the concentration parameter $\\kappa_o$ increases (lower dispersion of the samples about the mean) no matter whether the data is generated from VMF (Fig.~\\ref{fig:mu_est_VMFdata}) or Watson distribution (Fig.~\\ref{fig:mu_est_Watsondata}), indicating the robustness of the proposed approaches under model mismatch. Notice that when $\\kappa_o$ is small ($\\kappa_o<20$ for VMF data and $\\kappa_o<10$ for Watson data), none of the methods can accurately estimate the mean orientation. The reason is that when $\\kappa_o$ is small the samples become nearly uniformly distributed over the sphere. The threshold $\\kappa_o$ value at which performance starts to degrade depends on the choice of point symmetry group and the distribution used to simulate the data. In Fig.~\\ref{fig:kappa_est} it is seen that the biases of the proposed EM-VMF~\\cite{chen_parameter_2015} and EM-Watson $\\kappa$ estimators are significantly lower than that of the other methods compared. While the modified ML performs better than the naive ML estimator, its bias is significantly worse than the proposed EM-VMF and EM-Watson approaches. \n\n\\begin{figure}\n \\centering\n \\subfigure[VMF Simulated Data]{\n \\label{fig:mu_est_VMFdata}\n \\includegraphics[width=4.25cm]{figures\/Mu_est_VMF}}\n \\subfigure[Watson Simulated Data]{\n \\label{fig:mu_est_Watsondata}\n \\includegraphics[width=4.25cm]{figures\/Mu_est_Watson}}\n \\caption{Mean orientation estimator comparisons for $\\mathcal G$-invariant densities. Shown is the average inner product $\\bmu_o^T\\hat{\\bmu}$ of four estimators $\\hat{\\bmu}$ when $\\bmu_o$ is the true mean orientation as a function of the true concentration parameter $\\kappa_o$ for the data simulated from VMF (Fig.~\\ref{fig:mu_est_VMFdata}) and from Watson (Fig.~\\ref{fig:mu_est_Watsondata}) distribution. The naive estimator (\"ML Estimator\" in blue line) does not attain perfect estimation (inner product $=1$) for any $\\kappa_o$ since it does not account for the spherical symmetry group structure. The modified ML (green dashed line) achieves perfect estimation as $\\kappa_o$ becomes large. The proposed EM-ML methods (\"EM-VMF\", \"EM-Watson\") achieve perfect estimation much faster than the other methods even under model mismatch (EM-VMF for Watson simulated data and vice versa).}\n \\label{fig:mu_est}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\subfigure[VMF Simulated Data]{\n \\label{fig:kappa_est_VMFdata}\n \\includegraphics[width=4.25cm]{figures\/Kappa_est_VMF}}\n \\subfigure[Watson Simulated Data]{\n \\label{fig:kappa_est_Watsondata}\n \\includegraphics[width=4.25cm]{figures\/Kappa_est_Watson}}\n \\caption{Concentration parameter estimator bias as a function of the true concentration $\\kappa_o$ for data simulated from VMF (Fig.~\\ref{fig:kappa_est_VMFdata})\\cite{chen_parameter_2015} and from Watson (Fig.~\\ref{fig:kappa_est_Watsondata}) distributions. The bias of the naive ML (blue solid line) is large over the full range of $\\kappa_o$. The modified ML (green dashed line) estimates $\\kappa$ more accurately when $\\kappa_o$ is small. Our proposed EM-VMF and EM-Watson estimators (black dotted line and magenta dashed line) have lower bias than the other estimators.}\n \\label{fig:kappa_est}\n\\end{figure}\n\nFigure \\ref{fig:computation_time} shows the computation time of the estimation algorithms presented in Fig.~\\ref{fig:mu_est} and Fig.~\\ref{fig:kappa_est}. The computation time for all methods decreases as $\\kappa_o$ becomes larger. When $\\kappa_o$ is small ($\\kappa_o<20$ for VMF data and $\\kappa_o<10$ for Watson data), because the samples are almost uniformly distributed around the sphere, it is difficult for the EM algorithms to converge to the optimal solution and they therefore require maximum number of iterations to stop, forming the plateaus in Fig.~\\ref{fig:computation_time}. Notice that EM-Watson requires less time than EM-VMF even though it has more complicated E and M-steps. The reason is that EM-Watson uses only half of the symmetry operators, which corresponds to the size of the quotient group $\\mathcal{G}\/\\mathcal{I}$ as described in Section~\\ref{sec:ginv_Watson_dist}. By applying the hyperbolic sinusoidal simplification in Section~\\ref{sec:ginv_VMF_dist} (labeled \"EM-VMF-Hyper\"), we can further reduce the computation time by more than a factor of $2$ compared to the original EM-VMF. \n\n\\begin{figure}\n \\centering\n \\subfigure[VMF Simulated Data]{\n \\label{fig:computation_time_VMFdata}\n \\includegraphics[width=4cm]{figures\/ComputationTime_VMF}}\n \\subfigure[Watson Simulated Data]{\n \\label{fig:computation_time_Watsondata}\n \\includegraphics[width=4cm]{figures\/ComputationTime_Watson}}\n \\caption{Computation time for calculating the result in Fig.~\\ref{fig:mu_est} and Fig.~\\ref{fig:kappa_est}. EM-Watson (magenta dashed line) has less computation time than EM-VMF (black dotted line) because it uses only half of the symmetry operators. EM-VMF-Hyper (cyan circle line) which uses the hyperbolic sinusoidal simplification of EM-VMF reduces the computation time by more than a factor of $2$.}\n \\label{fig:computation_time} \n\\end{figure}\n\n\\subsection{$\\mathcal G$-invariant Clustering on Simulated Data}\nIn this section, we demonstrate the performance of our proposed EM approaches for clustering. Sets of $n$ i.i.d. samples were simulated from the VMF or Watson distributions with $\\kappa=\\kappa_o$ and one of two mean directions ($\\bmu_1,\\bmu_2$) to generate two clusters of samples. The spherical symmetry group is $m\\overline{3}m$ as before. The number of samples for each set was set to $n=1000$ and $\\kappa_o$ was swept from $1$ to $100$ while, for each set, $\\bmu_1,\\bmu_2$ was selected uniformly at random. The experiment was repeated $100$ times and the average values of the inner product $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ are shown in Fig.~\\ref{fig:mu_est_2clusters}. In the figure we compare performances of the following methods: (1) Cluster the samples by standard K-means algorithm with the distance defined by the arc-cosine of the inner product and then use the naive ML within each cluster to estimate the mean directions (labeled \"K-means\"). (2) \nCluster the samples by K-means with the distance defined as (\\ref{eq:sym_dist}) and then use the aforementioned modified ML estimator (labeled \"Modified K-means\"). (3) Apply our proposed multi-cluster EM-VMF algorithm to the $n$ samples directly (\\ref{eq:mVMF_mClusters_Estep})-(\\ref{eq:mVMF_mClusters_Mstep}) (labeled \"EM-VMF\") (4) Apply our multi-cluster EM-Watson algorithm to the $n$ samples directly (\\ref{eq:Watson_mClusters_Mstep_hatT})-(\\ref{eq:Watson_mClusters_Mstep_kappa}) (labeled \"EM-Watson\").\n\nFigure \\ref{fig:mu_est_2clusters} shows the average inner product values $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ from the mean direction estimation. The proposed EM-VMF and EM-Watson are able to correctly cluster the samples and achieve perfect recovery of the two mean orientations much faster than the other K-means approaches. Notice that the region where all the methods fail is larger than the single cluster case since multiple clusters increase the difficulty of parameter estimation. Again, no matter whether the samples are simulated from VMF or Watson distribution, our proposed approaches perform equally well under both cases.\n\nTo further test the ability to detect multiple clusters given a set of samples, we generate $1000$ sets of samples. Each set has $1000$ samples and is assigned randomly to label $0$ or $1$. If the set is labeled $0$, the samples are generated from a single distribution; If the set is labeled $1$, then the samples in the set are randomly generated from two distributions with different means. The GLRT is used with the four aforementioned clustering methods to test whether the samples in each set are uni-modal or multi-modal. The Receiver Operating Characteristic (ROC) curves of the test results are shown in Fig.~\\ref{fig:ROC}. The naive K-means with ML estimator which does not consider the symmetry group actions fails to distinguish whether the multiple modes are from actual multiple distributions or due to the wrap-around effect from the fundamental zone mapping. Therefore, this approach tends to over-estimate the goodness of fit of the $H_1$ model for true negative cases and under-estimate it for true positive cases, resulting in a result that is even worse than random guessing. The modified K-means performs better than K-means but worse than our proposed EM-VMF and EM-Watson algorithms. \n\n\\begin{figure}\n \\centering\n \\subfigure[VMF Simulated Data]{\n \\label{fig:mu_est_2clusters_VMFdata}\n \\includegraphics[width=4.25cm]{figures\/Mu_est_2clusters_VMF}}\n \\subfigure[Watson Simulated Data]{\n \\label{fig:mu_est_2clusters_Watsondata}\n \\includegraphics[width=4.25cm]{figures\/Mu_est_2clusters_Watson}}\n \\caption{Mean orientation estimator comparisons for samples generated from two different means. Shown is the average inner product $(\\hat{\\bmu}_1^T\\bmu_1+\\hat{\\bmu}_2^T\\bmu_2)\/2$ of four methods when $\\bmu_1,\\bmu_2$ are the true mean orientations as a function of the true concentration parameter $\\kappa_o$ for the data simulated from VMF (Fig.~\\ref{fig:mu_est_2clusters_VMFdata}) and from Watson (Fig.~\\ref{fig:mu_est_2clusters_Watsondata}) distributions. The K-means with naive estimator (\"K-means\" in blue line) does not attain perfect estimation for any $\\kappa_o$. A modified K-means with ML estimator (\"modified K-means\" in green dashed line) achieve perfect estimation as $\\kappa_o$ becomes large. The proposed EM-VMF and EM-Watson methods (\"EM-VMF\", \"EM-Watson\") achieves perfect estimation much faster than the other methods.}\n \\label{fig:mu_est_2clusters} \n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\subfigure[VMF Simulated Data]{\n \\label{fig:ROC_VMFdata}\n \\includegraphics[width=4cm]{figures\/Hypotest_VMF_kappa50}}\n \\subfigure[Watson Simulated Data]{\n \\label{fig:ROC_Watsondata}\n \\includegraphics[width=4cm]{figures\/Hypotest_Watson_kappa50}}\n \\caption{ROC curve for detecting bi-modally distributed samples. The samples are uni-modal or bi-modal distributed from VMF (Fig.~\\ref{fig:ROC_VMFdata}) or Watson (\\ref{fig:ROC_Watsondata}) distributions with $\\kappa_o=50$. The naive K-means with ML estimator cannot cluster the samples well and estimate the mean directions accurately, resulting in poor detection which is even worse than random guessing. The modified K-means (green dashed line) performs better than K-means but is still unsatisfactory. Our proposed EM-VMF (black dots) and EM-Watson (magenta dashed line) methods have very good performance in this detection task.}\n \\label{fig:ROC} \n\\end{figure}\n\n\\subsection{EM-ML orientation estimation for IN100 Nickel Sample}\nWe next illustrate the proposed EM-VMF and EM-Watson orientation estimators on a real IN100 sample acquired from US Air Force Research Laboratory (AFRL) \\cite{park_ebsd_2013}. The IN100 sample is a polycrystalline Ni superalloy which has cubic symmetry in the $m\\overline{3}m$ point symmetry group. EBSD orientation measurements were acquired on a $512\\times 384$ pixel grid, corresponding to spatial resolution of $297.7$ nm. The Kikuchi diffraction patterns were recorded on a $80\\times 60$ photosensitive detector for each of the pixels. \n\nFigure \\ref{fig:IN100} (a) shows a $200\\times 200$ sub-region of the full EBSD sample where the orientations are shown in the inverse pole figure (IPF) coloring obtained from the OEM EBSD imaging software and (b) is the back-scattered electron (BSE) image. Note that the OEM-estimated orientations in some grain regions of the IPF image are very inhomogeneous, having a mottled appearance, which is likely due to a fundamental zone wrap-around problem. As an alternative, we apply a combination of the proposed EM estimators (EM-VMF or EM-Watson) and the GLRT~(\\ref{eq:multi_sample_GLRT}) with $C=2$ and significance level $\\alpha=0.05$ to detect multi-modal distributions within each OEM-segmented region. Figure \\ref{fig:IN100} (c)(e) show the estimates of the mean orientations of the regions\/sub-regions, where the sub-regions surrounded by white boundaries indicate those that have been detected as deviating from the distribution of the majority of samples from the same region. The multi-modally distributed regions may be due to undetected grains, inaccurate segmentation, or noisy orientation observations. To distinguish the latter situations from the first in which the region really consists of two grains, the misalignment\/noise test introduced in~\\cite{chen_coercive_2015} can be used. Figures \\ref{fig:IN100} (d)(f) show the estimated concentration parameter $\\kappa$ for the regions\/sub-regions. Note that the estimated $\\kappa$ are large for most of the regions\/sub-regions because those regions which have multi-modally distributed samples are detected and their concentration parameters are estimated separately for each sub-region. \n\n\n\\begin{figure}[htb]\n \\centering\n \\subfigure[IPF from OEM]{\n \t\\includegraphics[width=3.2cm]{figures\/EA}}\n \\subfigure[BSE from OEM]{\n \t\\includegraphics[width=3.8cm]{figures\/BSE}}\n \n \n \\subfigure[EM-VMF $\\hat{\\bmu}$]{\n \t\\includegraphics[width=3.2cm]{figures\/Detected_Grains_VMF}}\t\n \\subfigure[EM-VMF $\\hat{\\kappa}$]{\n \t\\includegraphics[width=3.8cm]{figures\/Detected_Kappa_VMF}}\n \n \n \\subfigure[EM-Watson $\\hat{\\bmu}$]{\n \t\\includegraphics[width=3.2cm]{figures\/Detected_Grains_Watson}}\n \\subfigure[EM-Watson $\\hat{\\kappa}$]{\n \t\\includegraphics[width=3.8cm]{figures\/Detected_Kappa_Watson}}\n \\caption{A $200\\times 200$ sub-region of the IN100 sample. (a) is the IPF image for the Euler angles extracted from EBSD by OEM imaging software. IPF coloring in some grains is not homogeneous, likely due to symmetry ambiguity. (b) is the BSE image of the sample. (c)(e) show the estimates of the mean orientations of the regions\/sub-regions using a combination of the proposed EM estimators, EM-VMF and EM-Watson respectively, and the GLRT~(\\ref{eq:multi_sample_GLRT}) to detect multi-modal distributions within each OEM-segmented region. The sub-regions surrounded by white boundaries indicate those that have been detected as deviating from the distribution of the majority of samples from the same region. (d)(f) show the estimated concentration parameter $\\kappa$ for the regions\/sub-regions. Note that the estimated $\\kappa$ are large for most of the regions\/sub-regions because those regions which have multi-modally distributed samples are detected and their concentration parameters are estimated separately for each sub-region.}\n\\label{fig:IN100}\n\\end{figure}\n\\section{Conclusion}\n\\label{sec:conclusion}\nA hyperbolic $\\mathcal G$-invariant von Mises-Fisher distribution was shown to be equivalent to the distribution proposed in~\\cite{chen_parameter_2015}. The advantage of the hyperbolic form is parameter estimation can be performed with substantially fewer computations. A different group invariant orientation distribution was introduced, called the $\\mathcal{G}$-invariant Watson distribution, and an EM algorithm was presented that iteratively estimates its orientation and concentration parameters. We introduced multi-modal generalizations of these $\\mathcal G$-invariant distributions using mixture models and showed that these can be used to effectively cluster populations of orientations that have spherical symmetry group invariances. The mixture of VMF and Watson models were applied to the problem of estimation of mean grain orientation parameters in polycrystalline materials whose orientations lie in the $m\\overline{3}m$ point symmetry group. Application of the finite mixture representation to other types of groups would be worthwhile future work.\n\n\n\\section*{Acknowledgment}\nThe authors are grateful for inputs from Megna Shah, Mike Jackson and Mike Groeber. AOH would like to acknowledge financial support from USAF\/AFMC grant FA8650-9-D-5037\/04 and AFOSR grant FA9550-13-1-0043. MDG would like to acknowledge financial support from AFOSR MURI grant FA9550-12-1-0458.\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n$B$-meson decays are an excellent source of information about the CKM mechanism and allow us to test our\n understanding of the CP violation. In nonleptonic $B$\ndecays we must deal with final states interactions (FSI) as well, since they may modify the values of the extracted\n parameters. It is hard to take FSI into consideration properly since\n there are a lot of possible decay channels.\n\\\\\n\nDuring the recent years several authors have investigated various possible corrections\n due to FSI. Most of the analyses take into account the elastic and inelastic effects\n arising from intermediate states containing light quarks ($u$, $d$, $s$) [1-6] and apply\n symmetries of strong interactions (isospin, SU(3)) to reduce the number of parameters.\n Some authors argued that intermediate states containing charmed quarks ($c$) may also play an important role [7-13].\n\\\\\n\nIn the present paper we analyse $B$ decays into two light noncharmed pseudoscalar mesons.\n We consider FSI originating only from the intermediate states containing c quarks.\nIn sections 2 and 3 we introduce our parametrisation and relations between the amplitudes.\n Section 4 contains the description of the fit procedure and our results together with the CP\n asymmetry predictions. Finally, a short summary is given in section 5.\n\n\\section{Short-distance amplitudes}\n\n The decays of $B$ meson into two noncharmed pseudoscalar mesons are characterised by 10 $SU(3)_{f}$ invariant\n amplitudes corresponding to the specific quark-line diagrams. As in \\cite{ZL,ZL2} we use four\n dominant amplitudes: tree $T(T^{'})$, colour-suppressed $C(C^{'})$, penguin\n $P(P^{'})$ and singlet penguin $S^{'}$. Unprimed (primed) amplitudes denote\n strangeness conserving (violating) processes and are related to each other.\n Topological decompositions of decay amplitudes can be found in \\cite{ZL}.\n\\\\\n\n We use the Wolfenstein parameters: $\\lambda=0.222$,\n $A=0.832$, $\\bar{\\rho}=0.224$ and $\\bar{\\eta}=0.317$ \\cite{fl}.\n All relations bellow are calculated up to $O(\\lambda^4)$ unless explicitly written\n otherwise. Terms proportional to $\\lambda^4$ are kept on account of complex factor in $P^{'}$, which may interfere with FSI correction. We assume that all short-distance (SD) strong phases\n are negligible. For the tree amplitude we have\n\\begin{equation}\nT^{'} = \\frac{V_{us}}{V_{ud}} \\frac{f_{K}}{f_{\\pi}} T =0.278 T,\n\\end{equation}\nwhere $\\frac{f_{K}}{f_{\\pi}}$ is the SU(3) breaking factor. Both $T$ and $T^{'}$\namplitudes have a weak phase equal $\\gamma$.\nWe assume that SD penguin amplitudes are dominated by $t$ quark contribution.\nWhen terms of order $\\lambda^{4}$ are included, the strangeness violating penguin amplitude $P^{'}$ acquires a small weak\nphase. Thus, $P^{'}$ can be represented as a sum of two terms, the second one due to the $O(\\lambda^4)$ correction:\n\\begin{equation}\nP^{'} = \\frac{V_{ts}}{V_{td}} P = -(5.241+0.105 e^{i\\gamma} )|P|\n\\end{equation}\nPenguin amplitude $P$ has weak phase -$\\beta$. We used in our fits the value of $\\beta=24^{o}$ consistent with the world average.\nThe singlet penguin has the same phase as penguin $P'$:\n\\begin{equation}\nS^{'}=e^{i arg(P')}|S'|\n\\end{equation}\n\\\\\n\n\nFinally, we accept relations between the tree and colour-suppressed amplitudes:\n\\begin{equation}\nC=\\xi T,\n\\end{equation}\n\n\\begin{equation}\nC^{'}=(\\xi -(1+\\xi )\\delta_{EW} e^{-i\\gamma})T^{'}\n\\end{equation}\nwhere $\\xi=0.17$ and $\\delta_{EW}=0.65$. The last equation includes electroweak penguin\n$P_{EW}^{'}$. The EW penguin contribution $\\sim \\delta_{EW} e^{-i\\gamma}$ was calculated (see e.g. \\cite{Neu2,Neu:98})\nwithout $\\lambda^{4}$ corrections. This fact should not affect the fits much since $P_{EW}^{'} \\sim S^{'}$ \\cite{h} and the small correction\n in $S^{'}$ is\npractically invisible in the fits (the only changes we\nobserved were in the asymmetry for the $B^{+} \\to \\eta^{'} K^{+}$ decay channel).\n\n\n\n\n\\section{Long-distance charming penguins }\nIt was argued [7-13] that the intermediate states composed of charmed mesons ($D\\bar{D}$, etc.), generated from the\n$b \\rightarrow c \\bar{c} d(s) $ tree amplitudes $T_{c}^{(')}$, may lead via rescattering to amplitudes of \npenguin topology with an internal $c$ quark (the \"charming penguin\"). Our calculations are similar as in\nthe case of long-distance $u$-type penguins \\cite{ZL2}. Assuming SU(3) symmetry, we can redefine\npenguins:\n\\begin{equation}\nP^{(')} \\rightarrow P^{(')}+id_{c}T_{c}^{(')}\n\\end{equation}\nwhere $d_{c}$ is related to the size of the LD charming penguin and is a complex number in general.\nBecause we do not have information about $d_{c}$ (or $T_{c}^{(')}$), it is convenient to\nintroduce the following parametrisation:\n\\begin{equation}\nid_{c}T_{c}^{(')}=P^{(')}_{cLD}e^{i \\delta_{c}}\n\\end{equation}\nStrong phase $\\delta_{c}$ and size $P^{(')}_{cLD}$ of the charming penguin are additional free\nparameters\nin our fits. The weak phases are determined by the tree amplitudes $T^{(')}_{c}$ and are either $\\pi$ or 0.\nWe can eliminate $P^{'}_{cLD}$ using the relation\n\\begin{equation}\n\\frac{P^{'}_{cLD}}{P_{cLD}}=\\frac{T^{'}_{c}}{T_{c}}=\\frac{V_{cs}}{V_{cd}}=-4.388\n\\end{equation}\n\\\\\n\nShort-distance charming penguin $P_{c}^{'}$ has the same weak phase as $P_{cLD}^{'}$.\nIt can be included in a new redefined charming penguin\n\\begin{equation}\nP^{(')}_{cef}e^{i \\delta} = P^{(')}_{c}+P^{(')}_{cLD}e^{i \\delta_{c}}\n\\end{equation}\nwith new effective size and strong phase.\n\n\n\n\n\\section{Results of fits }\nWe minimise function f defined as:\n\\begin{equation}\nf =\\sum_i{\\frac{(B_i^{\\rm theor}-B_i^{\\rm exp})^2}{(\\Delta B_i^{\\rm\nexp})^2}}\n\\end{equation}\nwhere $B_i^{\\rm theor(exp)}$ denote theoretical (experimental) CP-averaged\n branching fractions\nand $\\Delta B_{i}^{\\rm exp}$ is an experimental error for i-th decay channel.\nThe sum is over all 16 decay channels as in \\cite{ZL2,chpZ}.\nExperimental branching ratios and their errors are listed in Tables 1 and 2.\nThe connection between the amplitudes and branching ratios was corrected in our calculations for the lifetime\ndifference between $B^{+}$ and $B^{0}$:\n\n\\begin{equation}\n\\frac{\\tau_{B^{+}}}{\\tau_{B^{0}}}=1.068\n\\end{equation}\n\\\\\nWe considered two sets of data. The first one was the same as in \\cite{ZL2}. The second one was used\nin \\cite{chpZ}. Data in Table 2 are more recent and differ from the previous ones in a couple of entries.\nWe performed fits in three general cases:\n\\begin{enumerate}\n\\item {without long-distance charming penguin contributions and with $|T|$, $|P|$, $|S'|$, $\\gamma$\ntreated as free parameters}\n\\item {with long-distance charming penguins described by real $P_{cef}$ as an additional parameter and\n$\\delta=0$, which is consistent with\n calculations done in \\cite{chpZ} but without any assumed connection between $P_{c}$ and $P_{t}$}\n\\item {with long-distance charming penguins described by two additional free parameters: $\\delta$, $P_{cef}$.}\n\\end{enumerate}\n\n\n\\begin{table}[h]\n\\caption{Fits to the first set of data (in units of $10^{-6}$)} \\label{tab:fit1}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nDecay channel & Exp &SD amplitudes &\\multicolumn{3}{|c|}{Charming penguin} \\\\\n\n\n & &only &(case 2) & \\multicolumn{2}{|c|}{(case3)}\\\\\n\n & & (case 1) &$\\delta=0^{o}$ &$\\gamma$ free &$\\gamma=64.5^o$ \\\\\n\n\n\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$ &$5.8\\pm 1.0$ &$5.01$ &5.65\t &$5.73$ &$5.85$\t\t \\\\\n$(B^+ \\to K ^+ \\bar{K}^0)$ &$0.0\\pm 2.0$ &$0.68$ &0.71\t &$2.10$ &$1.81$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta)$ &$2.9\\pm 1.1$ &$2.15$ &1.76\t &$2.47$ &$2.24$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta ')$ &$0.0\\pm 7.0$ &$1.07$ &0.88\t &$1.24$ &$1.12$\t\t \\\\\n\\hline\n$(B^0_d \\to \\pi ^+ \\pi ^- )$ &$4.7\\pm 0.5$ &$4.90$ &4.78\t &$4.76$ &$4.75$\t\t \\\\\n$(B^0_d \\to\\pi ^0 \\pi ^0)$ &$1.9\\pm 0.7$ &$0.62$ &0.73\t &$1.50$ &$1.36$\t\t \\\\\n$(B^0_d \\to K^+ K^-)$ &$0.0\\pm 0.6$ &$0.00$ &0 \t &$0.00$ &$0.00$\t\t \\\\ \t\t\t \n$(B^0_d \\to K^0 \\bar{K}^0)$ &$0.0\\pm 4.1$ &$0.62$ &0.66\t &$1.94$ &$1.67$\t\t \\\\ \t \t \n\n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$ &$18.1\\pm 1.7$ &$18.40$ &19.21\t &$18.67$ &$20.41$\t\t \\\\ \t \n$(B^+ \\to \\pi ^0 K ^+ )$ &$12.7\\pm 1.2$ &$13.11$ &13.10\t &$11.61$ &$10.63$\t\t \\\\ \t \n$(B^+ \\to\\eta K^+)$ &$4.1\\pm 1.1$ &$2.46 $ &2.30\t &$4.30 $ &$3.96$\t\t \\\\ \t \n$(B^+ \\to\\eta ' K^+)$ &$75\\pm 7.0$ &$73.00$ &73.37\t &$68.91$ &$69.69$\t\t \\\\ \t \n\\hline\n$(B^0_d \\to\\pi ^- K^+)$ &$18.5\\pm 1.0$ &$18.76$ &18.60\t &$18.38$ &$18.60$\t\t \\\\ \t \n$(B^0_d \\to\\pi ^0 K^0)$ &$10.2\\pm 1.2$ &$6.20$ &6.57\t &$7.76$ &$9.12$\t\t \\\\ \t \n$(B^0_d \\to\\eta K^0)$ &$0.0\\pm 9.3$ &$1.81$ &1.79\t &$3.19$ &$4.22$\t\t \\\\ \t \n$(B^0_d \\to\\eta ' K^0)$ &$56\\pm 9.0$ &$66.28$ &67.36\t &$62.35$ &$66.12$\t\t \\\\ \t \n\\hline\n\n$|T|$& &$2.60$ &2.76\t &$2.78$ &$2.81$ \\\\\n\n$|P|$& &$0.79$ &1.45\t &$2.59$ &$1.92$ \\\\\n\n$|S'|$& &$1.75$ &1.72\t &$2.46$ &$3.02$ \\\\\n\n$P_{cef}$& & &-0.77\t &$-2.81$ &$-2.32$ \\\\\n\n$\\gamma$& &$103^o$ &$94^o$\t &$110^o$ &$64.5^o$ \\\\\n\n$\\delta$& & &$0^o$\t &$\\pm18^o$ &$\\pm26^o$ \\\\\n\n$f_{m}$& &$15.36$ &14.79\t &$6.37$ &$9.39$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\\begin{table}[h]\n\\caption{Fits to the second set of data (in units of $10^{-6}$)} \\label{tab:fit2}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|l|c|c|c|c|}\n\\hline\nDecay channel & Exp &SD amplitudes &\\multicolumn{3}{|c|}{Charming penguin} \\\\\n\n\n & &only &(case 2) & \\multicolumn{2}{|c|}{(case3)}\\\\\n\n & & (case 1) &$\\delta=0^{o}$ &$\\gamma$ free &$\\gamma=64.5^o$ \\\\\n\n\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$ &$5.3\\pm 0.8$ &$4.27$ &5.32\t &$5.05$ &$5.40$\t\t \\\\\n$(B^+ \\to K ^+ \\bar{K}^0)$ &$0.0\\pm 2.4$ &$0.69$ &0.96\t &$2.55$ &$1.58$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta)$ &$4.2\\pm 0.9$ &$2.66$ &2.04\t &$3.04$ &$2.29$\t\t \\\\\n$(B^+ \\to\\pi ^+ \\eta ')$ &$0.0\\pm 4.5$ &$1.33$ &1.02\t &$1.52$ &$1.14$\t\t \\\\\n\\hline\t\t\t\t\t\t \t\t \t \n$(B^0_d \\to \\pi ^+ \\pi ^- )$ &$4.6\\pm 0.4$ &$5.09$ &4.76\t &$4.75$ &$4.70$\t\t \\\\\n$(B^0_d \\to\\pi ^0 \\pi ^0)$ &$1.9\\pm 0.5$ &$0.51$ &0.83\t &$1.65$ &$1.18$\t\t \\\\\n$(B^0_d \\to K^+ K^-)$ &$0.0\\pm 0.6$ &$0.00$ &0 \t &$0.00$ &$0.00$\t\t \\\\\n$(B^0_d \\to K^0 \\bar{K}^0)$ &$0.0\\pm 1.8$ &$0.64$ &0.89\t &$2.35$ &$1.46$\t\t \\\\\n\t\t\t\t\t\t \t\t \t \n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$ &$21.8\\pm 1.4$ &$19.10$ &22.11\t &$22.44$ &$21.57$ \t \\\\\n$(B^+ \\to \\pi ^0 K ^+ )$ &$12.8\\pm 1.1$ &$11.97$ &12.45\t &$10.92$ &$11.39$ \t \\\\\n$(B^+ \\to\\eta K^+)$ &$3.2\\pm 0.7$ &$2.03 $ &1.57\t &$2.71$ &$3.04$\t\t \\\\\n$(B^+ \\to\\eta ' K^+)$ &$77.6\\pm 4.6$ &$74.02$ &76.18\t &$75.27$ &$74.64$ \t \\\\\n\\hline\t\t\t\t\t\t \t\t \t \n$(B^0_d \\to\\pi ^- K^+)$ &$18.2\\pm 0.8$ &$17.57$ &18.20\t &$19.33$ &$19.01$ \t \\\\\n$(B^0_d \\to\\pi ^0 K^0)$ &$11.9\\pm 1.5$ &$6.86$ &8.03\t &$9.86$ &$9.14$\t\t \\\\\n$(B^0_d \\to\\eta K^0)$ &$0.0\\pm 4.6$ &$1.76$ &1.63\t &$3.85$ &$3.26$\t\t \\\\\n$(B^0_d \\to\\eta ' K^0)$ &$65.2\\pm 6.0$ &$68.66$ &72.32\t &$73.14$ &$70.76$ \t \\\\\n\\hline\n\n$|T|$& &$2.36$ &2.68\t &$2.61$ &$2.7$\t\t \\\\\n\t\t\t\t\t\t \t\t \t \n$|P|$& &$0.83$ &2.06\t &$2.63$ &$1.9$\t\t \\\\\n\t\t\t\t\t\t \t\t \t \n$|S'|$& &$1.77$ &1.69\t &$2.96$ &$2.61$\t\t \\\\\n\t\t\t\t\t\t \t\t \t \n$P_{cef}$& &\t &-1.45\t &$-3.05$ &$-2.07$ \t \\\\\n\t\t\t\t\t\t \t\t \t \n$\\gamma$& &$85^o$ &$68^o$\t &$22^o$ &$64.5^o$ \t \\\\\n$\\delta$& &\t &$0^o$\t &$\\pm19^o$ &$\\pm26^o$\t \\\\\n\n$f_{m}$& &$27.98$ &24.73\t &$14.97$ &$15.71$ \t \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[h]\n\n\\includegraphics[angle=-90,width=0.5\\textwidth]{cv.eps}\n\\includegraphics[angle=-90,width=0.5\\textwidth]{cv2.eps}\n\n\\caption{\\small Dependence of $f_{m}(\\gamma)$ on $\\gamma$ for the first (left) and second (right) set of data.\nSolid lines denote case without charming penguin, dashed lines - case with charming penguin and $\\delta =0$, \ndotted lines - case with charming penguin and $\\delta$ let free.}\n\n\\end{figure}\n\n\n\n\nResults of the fits are contained in Tables 1, 2. The branching fractions were calculated for the best\nfits and for the fit with fixed $\\gamma=64.5^{o}$. The minimums $f_{m}$ obtained by minimising $f$ of (10) are showed in the last rows of the tables.\nIn general, the fitted values of $\\gamma$ are far from the standard model prediction.\nTo find out what happens one should study the dependence of the fitted function on $\\gamma$.\nLet us denote by $f_{m}(\\gamma)$ the minimum values of $f$ obtained when keeping $\\gamma$ fixed.\nThe function $f_{m}(\\gamma)$ is obtained either by setting $P_{cef}$=0, or by assuming $\\delta=0$ while letting\n $P_{cef}$ free, or by letting both $P_{cef}$ and $\\delta$ free.\nFigure 1 shows $f_{m}(\\gamma)$ for the first (left) and second (right) set of data. The worst fits are those\nwithout charming penguins (solid lines). The minimal values were achieved for $\\gamma=103^o$ and\n$\\gamma=85^o$ respectively. For both fits with charming penguin and the strong phase $\\delta_{c}=0$ (dashed lines), the\nbest fit corresponds to $\\gamma$\nshifted down by $9^{o}$($17^{o}$) and a slightly lower value of $f_{m}$. In the third case shown (dotted\n lines) $\\delta$ was let free. For the first set of data, the $f_{m}(\\gamma)$ is fairly small over the whole\n region shown $(\\gamma \\in (0,120^{o}))$.\n For the second, more recent set of data this region is restricted to about $10^{o}-80^{o}$. Since the values of\n $f_{m}$ differ a little in the above-mentioned region we should rather think of an allowed range of $\\gamma$.\n\\\\\n\nThe values of fitted parameters $|P|$,$|S^{'}|$,$|P_{cef}|$ vary for different values of $\\gamma$.\nThe most stable are the ratio $\\frac{P_{cef}}{P}$ and the strong phase $\\delta$. $|\\frac{P_{cef}}{P}|$\nchanges from 1.1 to 1.3(1.2) only. The function $f_{m}(\\delta ,\\gamma)$ has a deep minimum around\n$ \\delta \\approx \\pm 20^{o}$ (Fig.2) for a wide range of fits with fixed $\\gamma$. Both positive and negative signs of $\\delta$ are allowed\nas the fitted function is\nsymmetric under $\\delta \\leftrightarrow -\\delta$. The fact that the ratio $|\\frac{P_{cef}}{P}|$ is close to unity is in agreement with the calculation in \\cite{buras}. On the other hand, for the\n best fits with $\\delta=0$ the ratio $|\\frac{P_{cef}}{P}|$ is about 0.53(0.7). This value for the second set of data is higher than that assumed in \\cite{chpZ}.\n\\begin{figure}[h]\n\\begin{center}\n\n\\includegraphics[width=0.9\\textwidth]{delta_23.eps}\n\n\\caption{\\small Dependence of $f_{m}(\\gamma ,\\delta)$ on $\\delta$ for selected\nvalues of $\\gamma$, dashed (solid) lines denote the first (second) set of data.}\n\\end{center}\n\n\\end{figure}\n\n\\begin{table}[h]\n\\caption{Asymmetries generated by charming penguin for $\\delta>0$ (for $\\delta<0$ asymmetries are of opposite\nsign)}\n\\begin{center}\n{\\footnotesize\n\\begin{tabular}{|l|c|c|c|c|c|}\n\\hline\nDecay channel & \\multicolumn{2}{|c|}{First set of data}&\\multicolumn{2}{|c|}{Second set of data} &Experiment\\\\\n & $\\gamma$ fitted&$\\gamma=64.5^o$ &$\\gamma$ fitted&$\\gamma=64.5^o$ & \\\\\n\\hline \\hline\n$(B^+ \\to \\pi ^+ \\pi ^0)$ &$0$ &0 &$0$ &0&$-0.07 \\pm 0.14$\\\\\t \t\t\t\t\t\t\t\t \n$(B^+ \\to K ^+ \\bar{K}^0)$ &-0.93&-0.90 &-0.90&-0.98&-\\\\ \t\t \t\t\t\t\t\t\t\t \n$(B^+ \\to\\pi ^+ \\eta)$ &0.48&0.87 &0.47&0.76&$-0.44 \\pm 0.18 \\pm 0.01$\\\\\t\t\t\t\t\t\t\t \n$(B^+ \\to\\pi ^+ \\eta ')$&0.48&0.87 &0.47&0.76&-\\\\\t\t\t \t\t\t\t\t\t\t\t \n\\hline\n$(B^+ \\to \\pi ^+ K ^0 )$&0.11&0.08 &0.04&0.07&$0.02 \\pm 0.06$\\\\\t \t\t\t\t\t\t\t\t \n$(B^+ \\to \\pi ^0 K ^+ )$ &-0.21&-0.28 &-0.09&-0.23&$0.00 \\pm 0.12$\\\\\t \t\t\t\t\t\t\t\t \n$(B^+ \\to\\eta K^+)$ &0&0 &0&0&$-0.52 \\pm 0.24 \\pm 0.01$\\\\ \t\t\t\t\t\t\t\t \n$(B^+ \\to\\eta ' K^+)$ &0.006&-0.004 &0.005&-0.004&$0.02 \\pm 0.042$\\\\ \t\t\t\t\t\t\t\t \n\\hline\n$(B^0_d \\to\\pi ^- K^+)$ & -0.19&-0.24 & -0.075&-0.21&$-0.09 \\pm0.03$\\\\ \t\t\t\t\t\t\t\t \n\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\nCharming penguins with a nonvanishing strong phase may be a source of direct CP asymmetries.\nThe predicted values were calculated for the same points as in Tables 1,2. The results are\ngiven in Table 3 together with the averages from Belle, BABAR and CLEO experiments \\cite{Rosner}.\nThe main features are large asymmetries in the $\\Delta S=0$ sector with relatively small asymmetries\nfor the $\\Delta S=1$ decays channels. We are not able to predict the absolute signs of the asymmetries\nsince we have two allowed signs of $\\delta$. The asymmetry for $(B^+ \\to \\pi ^+ K ^0 )$\nis a pure $\\lambda^{4}$ effect and shows a potential influence of this correction.\n\n\n\n\\section{Conclusions}\nOur results permit to draw the following conclusions:\n\\begin{enumerate}\n\\item{Even without the charming penguins the value of angle $\\gamma$ extracted from the fit depends on the details of data. More recent data prefer the value of $\\gamma$ \nmore in accordance with the expectations of the standard model. }\n\\item{If we admit the non-zero value of the charming penguin (with strong phase equal zero), the fitted values of $\\gamma$ may move toward the SM value by $10^{o}-15^{o}$.}\n\\item{Admitting strong phase of the charming penguin as a free parameter leads to a relatively flat function $f_{m}(\\gamma)$ i.e. it allows a wide range of $\\gamma$.\n This means that there is probably too much freedom in the fits. However, the fitted strong phase $\\delta$ is\n relatively stable and close to $\\pm 20^{o}$.}\n\\end{enumerate}\n\n\n{\\it Acknowledgements}. I would like to thank P. \\.Zenczykowski for helpful discussions and comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum states can be well described in terms of Wigner functions.\nThe states with Gaussian Wigner function have been of particular\ninterests in context of quantum information processing. Entanglement\nin such states in terms of quadratures are also well studied. On the\nother hand, the quantum states with non-Gaussian Wigner function are\nalso quite important. For example, a single-photon state, which\nfinds many applications in quantum information processing, shows\nnon-Gaussian behavior in phase-space. In a recent experiment, a\nnon-Gaussian state has been produced by homodyne detection technique\nfrom a single-mode squeezed state of light \\cite{wenger,kim}.\nFor certain non-Gaussian states, Wigner functions can take negative\nvalues. Such negativity refers to nonclassicality of these states.\nThese states are useful in entanglement distillation\n\\cite{eisert,cirac}, loophole-free tests of Bell's inequality\n\\cite{bell}, and quantum computing \\cite{sanders}. A specific class\nof such nonclassical states has been shown to be similar to the\nSchrodinger kitten state, in the sense that their Wigner functions\nshow negativity at the origin of phase space \\cite{science,polzik}.\nIt is well known that the Schrodinger cat states \\cite{knight},\nwhich are quantum superpositions of coherent states, are\nnon-classical in nature and are very important to study the interface\nof quantum and classical worlds. Superposition of coherent states\nwith low amplitudes creates Schrodinger kitten states. Most of the\nexperiments to prepare the Schrodinger cat states have been\nperformed in cavities or bound systems. Thus they are not much\nuseful in quantum information networks though they have the\nnon-Gaussian nature which is required in certain quantum\ncommunication protocols. In \\cite{science,polzik}, it has been shown\nhow to prepare an Schrodinger kitten state in an optical system, by\nsubtracting a single photon from a squeezed vacuum state. This\noptical kitten state would overcome the limitations of bound\nsystems. Repeated photon-subtractions can lead to conditional\ngeneration of arbitrary single-mode state \\cite{cerf}. We note that\nsimilar non-Gaussian states could be prepared by adding a single\nphoton to a squeezed vacuum state (see \\cite{tara,bellini} for\ndetails of photon-added coherent states, which are also non-Gaussian\nstates). These states are equivalent to single-photon subtracted\nsqueezed vacuum state and exhibit similar behavior in phase space.\nIt is worth to mention that non-Gaussian two-mode entangled states\ncan be prepared by subtracting a photon from a two-mode squeezed\nstate \\cite{sasaki,paris}.\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{sq_figs}Plots of Wigner functions of single-photon\nsubtracted squeezed states for (a) $r=0.31$ and (b) $r=0.8$ with\n$\\theta=0$.}\n\\end{center}\n\\end{figure*}\n\\begin{figure}\n\\caption{\\label{cond}Variation of $C$ in phase space for $r=0.31$\nand $\\theta=0$.}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\caption{\\label{ellipse}Contour plot for $C=$ constant in phase space for\n(a) $r=0.31$ and (b) $r=0.8$.}\n\\end{center}\n\\end{figure}\n\nIn this paper, we focus our study on the nonclassical properties and\ndecoherence of single-photon subtracted squeezed vacuum states which\nare optically produced single-mode non-Gaussian states. The\nstructure of the paper is as follows. In Sec. II, we introduce the\nphoton-subtracted squeezed states and discuss its nonclassical\nproperties in terms of the sub-Poissonian statistics and the\nnegativity of its Wigner function. We derive a compact expression of\nthe Wigner function and find the region in phase space where it\nbecomes negative. We show that there is an upper bound of the\nsqueezing parameter for this state to exhibit sub-Poissonian\nstatistics. In Sec. III, we study the effects of two different model\nof decoherence: photon-number decay and phase damping. In both\ncases, we derive analytical expressions for the time-evolution of\nthe state and its Wigner function. We discuss the loss of\nnonclassicality due to decoherence. We show through the study of\nevolution of the Wigner function how the state decays to vacuum as a\nresult of photon-number decay. We further show that phase damping\nleads to much slower decoherence than the photon-number decay.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics{fig5.eps}}\n\\end{center}\n\\caption{\\label{Qdecay}Variation of the $Q$-parameter\nwith time in presence of decoherence due to decay of\nphoton for squeezing parameters $r=0.31$ (red line) and $r=0.8$\n(green line).}\n\\end{figure}\n\n\\begin{figure}\n\\caption{\\label{negative_time}Variation of $P\/(a^2-4|c|^2)^{5\/2}$ in\nphase space for $r=0.31$, $\\theta=0$, and $\\kappa t=0.1$.}\n\\end{figure}\n\n\\section{Photon-subtracted squeezed states}\nAn unnormalized single-mode squeezed vacuum state is given by\n\\begin{equation}\n|\\psi\\rangle\\equiv\nS(\\zeta)|0\\rangle\\;,\\;S(\\zeta)=\\exp\\left[\\frac{\\zeta}{2} a^{\\dag\n2}-\\frac{\\zeta^*}{2} a^2\\right]\\;,\n\\end{equation}\nwhere $S(\\zeta)$ is the squeezing operator, $\\zeta=re^{i\\theta}$ is\nthe complex squeezing parameter, and $a$ is the annihilation\noperator. The Wigner function of this state is Gaussian and positive\nin phase space. When $p~(>0)$ number of photons are subtracted from\nsuch states, the state can be written as\n\\begin{equation}\n|\\psi\\rangle_p \\equiv a^p S(\\zeta)|0\\rangle \\equiv\na^p\\exp\\left[\\frac{\\xi}{2}a^{\\dag 2}\\right]|0\\rangle\\;,\n\\end{equation}\nwhere $\\xi=\\tanh(\\zeta)$. For odd $p=2m+1$, the normalized form of\nthis state can be written as\n\\begin{equation}\n\\label{odd}|\\psi\\rangle_p =\\frac{1}{N_o}\\sum_{s=0}^\\infty\n\\frac{(\\xi\/2)^{s+m+1}}{(s+m+1)!}\\frac{(2s+2m+2)!}{\\sqrt{2s+1}}|2s+1\\rangle\\;,\n\\end{equation}\nwhile for even $p=2m$, the state becomes\n\\begin{equation}\n\\label{even}|\\psi\\rangle_p=\\frac{1}{N_e}\\sum_{s=0}^\\infty\n\\frac{(\\xi\/2)^{s+m}}{(s+m)!}\\frac{(2s+2m)!}{\\sqrt{2s}}|2s\\rangle\\;,\n\\end{equation}\nwhere $N_o$ and $N_e$ are the normalization constants. In this\npaper, we focus on the case when a single photon is subtracted from\nthe squeezed vacuum, i.e., for $m=0$ in (\\ref{odd}).\n\\begin{figure}\n\\begin{center}\n\\caption{\\label{ellipse1}Plot of $16C'\/(1-e^{-2\\kappa t})$ in phase\nspace for (a) $r=0.31$ and (b) $r=0.8$ at times (i) $\\kappa t=0.05$,\n(ii) $\\kappa t=0.1$, (iii) $\\kappa t=0.3$, and (iv) $\\kappa t=0.5$.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{small_sq}Wigner function of single-photon\nsubtracted squeezed states for $\\theta=0$ and $r=0.31$ at (a)\n$\\kappa t=0.05$, (b) $\\kappa t=0.1$, (c) $\\kappa t=0.3$, and (d)\n$\\kappa t=0.5$.}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{large_sq}Wigner function of single-photon\nsqueezed states for $\\theta=0$ and $r=0.8$ at (a) $\\kappa t=0.1$,\n(b) $\\kappa t=0.3$, (c) $\\kappa t=0.5$, and (d) $\\kappa t=0.7$.}\n\\end{center}\n\\end{figure*}\n\\subsection{Negativity of the Wigner function}\nThe Wigner function of the squeezed vacuum state $|\\psi\\rangle$ is\ngiven by\n\\begin{equation}\n\\label{w_sq}W_{\\rm\nsq}(\\alpha,\\alpha^*)=\\frac{2}{\\pi}\\exp(-2|\\tilde{\\alpha}|^2)\\;,\n\\end{equation}\nwhere $\\tilde{\\alpha}=\\alpha\\cosh(r)-\\alpha^*e^{i\\theta}\\sinh(r)$.\nThe function is Gaussian in phase space, as shown in Fig.\n\\ref{vacuum}. We now calculate the Wigner function of the state\n$|\\psi\\rangle_p$. This state can be rewritten as\n\\begin{equation}\n\\label{rel}|\\psi\\rangle_p =a^p S(\\zeta)|0\\rangle=S(\\zeta)S^\\dag\n(\\zeta)a^pS(\\zeta)|0\\rangle\\;.\n\\end{equation}\nUsing the relation\n\\begin{equation}\nS^\\dag(\\zeta)a^pS(\\zeta)=[\\cosh(r)a+e^{i\\theta}\\sinh(r)a^{\\dag}]^p\\;,\n\\end{equation}\nwe get the following from (\\ref{rel})\n\\begin{equation}\n\\label{state}|\\psi\\rangle_1\\equiv S(\\zeta)|1\\rangle\\;,\n\\end{equation}\nfor $p=1$. For a density matrix $\\tilde{\\rho}(a,a^\\dag)$, we can\nwrite the following:\n\\begin{equation}\n\\label{relation}S(\\zeta)\\tilde{\\rho}(a,a^\\dag)S^\\dag\n(\\zeta)=\\tilde{\\rho}[S(\\zeta)aS^\\dag (\\zeta),S(\\zeta)a^\\dag\nS^\\dag(\\zeta)]\\;.\n\\end{equation}\nWe write the Wigner function of $\\tilde{\\rho}(a,a^\\dag)$ as\n$W_{\\tilde{\\rho}}(\\alpha,\\alpha^*)$. Using the identities\n\\begin{eqnarray}\nS(\\zeta)aS^\\dag (\\zeta) &=&a\\cosh(r)-a^\\dag e^{i\\theta}\\sinh(r)\\;,\\\\\nS(\\zeta)a^\\dag S^\\dag(\\zeta) &=& a^\\dag\n\\cosh(r)-ae^{-i\\theta}\\sinh(r)\\;,\n\\end{eqnarray}\nin (\\ref{relation}), we can thus write the Wigner function of the\ndensity matrix $\\rho(a,a^\\dag)=S(\\zeta)\\tilde{\\rho}(a,a^\\dag)S^\\dag\n(\\zeta)$ as\n\\begin{equation}\n\\label{formula}W_{\\rho}(\\alpha,\\alpha^*)=W_{\\tilde{\\rho}}(\\tilde{\\alpha},\\tilde{\\alpha}^*)\\;,\n\\end{equation}\nwhere we have used the linearity property of the Wigner function.\nFor the state (\\ref{state}), $\\tilde{\\rho}=|1\\rangle\\langle 1|$ and\nits Wigner function is given by\n\\begin{equation}\n\\label{wsq}W_{\\tilde{\\rho}}(\\alpha,\\alpha^*)=\\frac{2}{\\pi}(4|\\alpha|^2-1)e^{-2|\\alpha|^2}\\;.\n\\end{equation}\nThus, the Wigner function of the single-photon subtracted squeezed\nvacuum state becomes\n\\begin{equation}\n\\label{wig1}W_\\rho(\\alpha,\\alpha^*)=\\frac{2}{\\pi}(4|\\tilde{\\alpha}|^2-1)e^{-2|\\tilde{\\alpha}|^2}\\;,\n\\end{equation}\nwhere we have used (\\ref{formula}) and (\\ref{wsq}). Clearly, the\nWigner function (\\ref{wig1}) is non-Gaussian in phase-space. We show\nthe plot of this Wigner function in the phase-space in Figs.\n\\ref{sq_figs} for different squeezing parameters. As an evidence of\nnon-classicality of the state, squeezing in one of the quadratures\nis clear in the plots. Also there is some negative region of the\nWigner function in the phase-space which is another evidence of the\nnon-classicality of the state. The function becomes negative in\nphase space, when\n\\begin{equation}\n\\label{neg_cond}|\\tilde{\\alpha}|^2<\\frac{1}{4}\\;.\n\\end{equation}\nWe show in Fig. \\ref{cond} the variation of\n$C=|\\tilde{\\alpha}|^2-\\frac{1}{4}$ in phase space. The negative\nregion of $C$ corresponds to the negativity of the Wigner function.\nNote that $C=$ constant corresponds to ellipse in phase space, as\nshown in Fig. \\ref{ellipse}.\n\nNote that the photon-subtracted squeezed states are similar to\nSchrodinger kitten states \\cite{science,polzik} because Wigner functions\nexhibit the same characteristics in phase-space especially for the\nlarge values of the squeezing parameters. Moreover, in both cases,\nWigner function becomes negative in the center of phase space.\n\n\\subsection{Sub-Poissonian nature of the photon-subtracted state}\nThe nonclassicality of the state $|\\psi\\rangle_1$ can also be\nanalyzed by studying its sub-Poissonian character in terms of the\nMandel's $Q$-parameter \\cite{mandel} which is defined by\n\\begin{equation}\n\\label{Qpara}Q=\\frac{\\langle a^{\\dag 2}a^2\\rangle-\\langle a^\\dag\na\\rangle^2}{\\langle a^\\dag a\\rangle}\\;.\n\\end{equation}\nThe negativity of the $Q$-parameter refers to sub-Poissonian\nstatistics of the state. However in \\cite{tara_Q}, it has been shown\nthat a state can be nonclassical even if $Q$ is positive. A similar\nsituation occurs in the present case. For the state (\\ref{odd}), we\nfind\n\\begin{eqnarray}\n\\langle a^{\\dag\n2}a^2\\rangle &=&\\sum_{n=0}^\\infty n(n-1)\\rho_{n,n}=\\frac{3|\\xi|^4(3+2|\\xi|^2)}{N_o^2(1-|\\xi|^2)^{7\/2}}\\;,\\nonumber\\\\\n\\label{avg1}\\langle a^\\dag a\\rangle &=&\\sum_{n=0}^\\infty\nn\\rho_{n,n}=\\frac{|\\xi|^2(1+2|\\xi|^2)}{N_o^2(1-|\\xi|^2)^{5\/2}}\\;,\n\\end{eqnarray}\nwhere the normalization constant is given by\n\\begin{equation}\nN_o^2=\\frac{|\\xi|^2}{(1-|\\xi|^2)^{3\/2}}\\;.\n\\end{equation}\nFrom (\\ref{avg1}), we find that $Q$ becomes negative for\n$|\\xi|\\lesssim 0.43$, which is satisfied for $r\\lesssim 0.46$. We\nemphasize that the Wigner function has negative region for all\nvalues of $r$, and thus the photon-subtracted squeezed state is\nnonclassical for all $r$, though it does not exhibit sub-Poissonian\nphoton statistics above certain squeezing threshold.\n\n\\section{Models of decoherence}\nWe next consider how this state evolves under decoherence. The\ndecoherence of the single-mode state (\\ref{odd}) can be due to decay\nof photons to the reservoir or due to phase damping.\n\n\\subsection{Amplitude decay model}\nWhen the photons decay to reservoir, the corresponding Markovian\ndynamics of the state is well described by the following equation:\n\\begin{equation}\n\\frac{d}{dt}\\rho=-\\kappa(a^\\dag a\\rho-2a\\rho a^\\dag+\\rho a^\\dag a)\\;,\n\\end{equation}\nwhere $\\kappa$ is the rate of decay. The solution of this equation\ncan be written as\n\\begin{equation}\n\\label{sol}\\rho(t)=\\sum_{n,n'}\\rho_{n,n'}(t)|n\\rangle\\langle n'|\\;,\n\\end{equation}\nwhere the density matrix element $\\rho_{n,n'}(t)$ can be found by\nusing the Laplace transformation and the iteration methods\n\\cite{gsa_po}. To see this, let us start with the time-dependent\nequation for $\\rho_{n,n'}$:\n\\begin{equation}\n\\dot{\\rho}_{n,n'}=-\\kappa(n+n')\\rho_{n,n'}+2\\kappa\\sqrt{(n+1)(n'+1)}\\rho_{n+1,n'+1}\\;.\n\\end{equation}\nUsing the new subscripts $q=n-n'$ and $p=(n+n')\/2$, the above\nequation transforms into\n\\begin{equation}\n\\label{pq}\\dot{\\rho}_{p,q}=-2\\kappa p\\rho_{p,q}+2\\kappa\n\\sqrt{(p+1)^2-(q\/2)^2}\\rho_{p+1,q}\\;.\n\\end{equation}\nTaking Laplace transformation of (\\ref{pq}) and using the original\nsubscript $n$ and $n'$, we can write the time-dependent solution for\nthe density matrix elements as\n\\begin{eqnarray}\n\\rho_{n,n'}(t)&=&e^{-\\kappa\nt(n+n')}\\sum_{r=0}^\\infty\\sqrt{\\left(^{n+r}C_r\\right)\\left(^{n'+r}C_r\\right)}\\nonumber\\\\\n\\label{rhonn}&&\\times (1-e^{-2\\kappa t})^r\\rho_{n+r,n'+r}(t=0)\\;,\n\\end{eqnarray}\nwhere for the single-photon subtracted squeezed vacuum\n\\begin{eqnarray}\n\\rho_{n+r,n'+r}(t=0)&=&\\frac{1}{N_o^2}\\frac{(\\xi\/2)^{(n+r+1)\/2}(\\xi^*\/2)^{(n'+r+1)\/2}}{(\\frac{n+r+1}{2})!(\\frac{n'+r+1}{2})!}\\nonumber\\\\\n&&\\label{init}\\times\\frac{(n+r+1)!(n'+r+1)!}{\\sqrt{(n+r)!(n'+r)!}}\\;.\n\\end{eqnarray}\n\nWe next calculate the parameter $Q$ [Eq. (\\ref{Qpara})] for the\nstate (\\ref{sol}). We have found that\n\\begin{eqnarray}\n\\langle a^{\\dag 2}a^2\\rangle &=&\\sum_{n=0}^\\infty\nn(n-1)\\rho_{n,n}(t)\\;,\\nonumber\\\\\n\\label{avg2}\\langle a^\\dag a\\rangle &=&\\sum_{n=0}^\\infty n\\rho_{n,n}(t)\\;,\n\\end{eqnarray}\nwhere $\\rho_{n,n}(t)$ is given by (\\ref{rhonn}) and (\\ref{init}) for\n$n=n'$ in case of state $|\\psi\\rangle_1$. Using Eqs. (\\ref{avg2}), we\nplot $Q$ with time in Fig. \\ref{Qdecay}. It is easy to see that\nat long times ($\\kappa t\\rightarrow \\infty$), $Q$ vanishes. This is\nbecause at this limit, $\\rho_{n,n}(t)$ vanishes for all non-zero $n$\nand $\\rho_{0,0}(t\\rightarrow \\infty)=1$, i.e., the state decays to\nvacuum. Thus the averages (\\ref{avg2}) vanish and $Q$ also vanishes.\n\n\\subsubsection{Evolution of Wigner function}\nThe evolution of the Wigner function is governed by the following\nequation:\n\\begin{equation}\n\\label{wignereq}\\frac{\\partial W}{\\partial\nt}=\\kappa\\left[\\frac{\\partial}{\\partial\n\\alpha}\\alpha+\\frac{\\partial}{\\partial\n\\alpha^*}\\alpha^*+\\frac{\\partial^2}{\\partial\n\\alpha\\partial\\alpha^*}\\right]W(\\alpha,\\alpha^*)\\;.\n\\end{equation}\nThe solution can be written as\n\\begin{eqnarray}\n\\label{wignersol}W(\\alpha,\\alpha^*,t)&=&\\frac{2}{\\pi(1-e^{-2\\kappa\nt})}\\int\nd^2\\alpha_0W(\\alpha_0,\\alpha_0^*,0)\\nonumber\\\\\n&&\\exp\\left\\{-2\\frac{|\\alpha-\\alpha_0e^{-\\kappa\nt}|^2}{(1-e^{-2\\kappa t})}\\right\\}\\;,\n\\end{eqnarray}\nwhere $W(\\alpha_0,\\alpha_0^*,0)$ is the Wigner function of the\ninitial state. It is easy to verify this solution putting\n(\\ref{wignersol}) in (\\ref{wignereq}). The time-evolution of the\nWigner function of the squeezed vacuum state $|\\psi\\rangle$ can be\neasily calculated analytically using the following integral\nidentity \\cite{gsa_prdeq,puri}:\n\\begin{eqnarray}\n\\label{ident}&&\\int\nd^2\\alpha\\exp[-|\\alpha|^2]\\exp\\left(-\\frac{\\mu}{\\tau}\\alpha^2-\\frac{\\nu}{\\tau}\\alpha^{*2}-\\frac{z^*\\alpha}{\\sqrt{\\tau}}+\\frac{z\\alpha^*}{\\sqrt{\\tau}}\\right)\\nonumber\\\\\n&&=\\frac{\\pi\\tau}{\\sqrt{\\tau^2-4\\mu\\nu}}\\exp\\left(-\\frac{\\mu\nz^2+\\nu z^{*2}+\\tau|z|^2}{\\tau^2-4\\mu\\nu}\\right)\\;.\n\\end{eqnarray}\nUsing Eq. (\\ref{wsq}) and the above identity in Eq.\n(\\ref{wignersol}), we get the following:\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\frac{4}{\\pi(1-e^{-2\\kappa t})}\n\\frac{\\exp\\left[-\\frac{2|\\alpha|^2}{1-e^{-2\\kappa t}}\\right]}{\\sqrt{a^2-4|c|^2}}\\nonumber\\\\\n&&\\times\\exp\\left[\\frac{b^2c^*+b^{*2}c+a|b|^2}{a^2-4|c|^2}\\right]\\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na&=&2\\cosh(r)+2\\frac{e^{-2\\kappa t}}{1-e^{-2\\kappa t}}\\;,\\nonumber\\\\\nb&=&\\frac{2\\alpha^*e^{-\\kappa t}}{1-e^{-2\\kappa\nt}}\\;,\\;c=e^{-i\\theta}\\sinh(r)\\;.\n\\end{eqnarray}\nClearly the Wigner function of the squeezed vacuum state is Gaussian\nat all times.\n\nWe now calculate the time-dependence of the Wigner function of the\nstate $|\\psi\\rangle_1$. The initial Wigner function, as given by\n(\\ref{wig1}), can be rewritten as\n\\begin{equation}\n\\label{alter_W}W(\\alpha,\\alpha^*,0)=\\frac{2}{\\pi}D\\left.\\left[e^{-\\lambda|\\tilde{\\alpha}|^2}\\right]\\right|_{\\lambda=2}\\;,\\;D=-4\\frac{d}{d\\lambda}-1\\;.\n\\end{equation}\nUsing (\\ref{alter_W}) and (\\ref{wignersol}), we can find the\nfollowing expression for the Wigner function:\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\left(\\frac{2}{\\pi}\\right)^2\\frac{1}{1-e^{-2\\kappa\nt}}D\\int\nd^2\\alpha_0\\exp[-\\lambda|\\tilde{\\alpha}_0|^2]\\nonumber\\\\\n&&\\times \\left.\\exp\\left[-2\\frac{|\\alpha-\\alpha_0e^{-\\kappa\nt}|^2}{1-e^{-2\\kappa t}}\\right]\\right|_{\\lambda=2}\\;,\n\\end{eqnarray}\nwhere\n$\\tilde{\\alpha}_0=\\alpha_0\\cosh(r)-\\alpha_0^*e^{i\\theta}\\sinh(r)$.\nSimplifying the above expression using Eq. (\\ref{ident}), we get\n\\begin{eqnarray}\nW(\\alpha,\\alpha^*,t)&=&\\frac{32P}{\\pi(1-e^{-2\\kappa\nt})}\\frac{\\exp\\left[-\\frac{2|\\alpha|^2}{1-e^{-2\\kappa\nt}}\\right]}{(a^2-4|c|^2)^{5\/2}}\\nonumber\\\\\n&&\\label{timeW}\\times\\exp\\left[\\frac{b^2c^*+b^{*2}c+a|b|^2}{a^2-4|c|^2}\\right]\\;,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nP&=&(1-x^2)\\{\\sinh(2r)(e^{i\\theta}b^2+e^{-i\\theta}b^{*2})\\nonumber\\\\\n&&+2[(x+1)^2+4x\\sinh^2(r)]\\}\\nonumber\\\\\n\\label{P}&&+2\\{(1+x^2)\\cosh(2r)+2x\\}|b|^2\n\\end{eqnarray}\nand\n\\begin{equation}\nx=\\frac{e^{-2\\kappa t}}{1-e^{-2\\kappa t}}\\;.\n\\end{equation}\nClearly, the Wigner function is non-Gaussian due to the presence of\nthe polynomial $P$. This becomes negative when the polynomial $P$\nbecomes negative. In Fig. \\ref{negative_time}, we have plotted\n$C'(\\alpha,\\alpha^*,t)=P\/(a^2-4|c|^2)^{5\/2}$ in phase space to show\nthe negative region for Wigner function.\n\nNote that at the center of the phase space ($\\alpha= \\alpha^*=0$),\nthe Wigner function is maximally negative. At the center,\n\\begin{equation}\nC'(0,0,t)=\\frac{2(1-x^2)\\{(x+1)^2+4x\\sinh^2(r)\\}}{(a^2-4|c|^2)^{5\/2}}\\;,\n\\end{equation}\nwhich becomes negative when $(1-x^2)$ becomes negative. This leads\nto the following condition:\n\\begin{equation}\n\\kappa t<\\kappa t_0=\\frac{1}{2}\\ln(2)\\;,\n\\end{equation}\nwhich is independent of the squeezing parameter $r$. Thus the Wigner\nfunction has certain negative region for the time\n$tt_0$, the ellipse\n$16C'\/(1-e^{-2\\kappa t})=$ constant interchanges its minor and major\naxes. We show this behavior in Figs. \\ref{ellipse1} for different\nvalues of $r$. Note that at times much larger than decoherence\ntime-scale $1\/\\kappa$ (i.e., for $\\kappa t\\rightarrow \\infty$),\n$P\\rightarrow 2$ and thus becomes constant throughout the phase\nspace.\n\nUsing Eq. (\\ref{timeW}), we show the variation of Wigner function at\ndifferent time-scales in Figs. \\ref{small_sq}. It is easy to see how\nthe negative region of the Wigner function gradually diminishes. At\nlong times $\\kappa t\\rightarrow \\infty$, the Wigner function becomes\n\\begin{equation}\nW(\\alpha,\\alpha^*,\\infty)= \\frac{2}{\\pi}e^{-2|\\alpha|^2}\\;,\n\\end{equation}\nwhich corresponds to vacuum state. We have shown this in Fig.\n\\ref{vacuum}(b). This can also be understood from Eq. (\\ref{rhonn}).\nFor $\\kappa t\\rightarrow \\infty$, $\\rho_{0,0}$ approaches unity,\nwhereas all other density matrix elements vanish. This means that at\nlong times, the state decays to vacuum, as we have discussed\nearlier.\n\nWe next study the time-evolution of the Wigner function for the case\nof large squeezing, i.e., large values of $\\zeta$. In this case the\nsingle photon subtracted squeezed state becomes similar to a\nSchrodinger cat state. For large times, such an optical cat state\ndecays to vacuum. Thus the Wigner function becomes Gaussian, as\ndiscussed above. We show this evolution for large squeezing in Figs.\n\\ref{large_sq}.\n\n\\subsection{Effect of phase damping}\nWe now study the effect of phase-damping on the state\n$|\\psi\\rangle_1$. Such damping can be described by the following\nmaster equation:\n\\begin{equation}\n\\dot{\\rho}=-\\kappa_p(A^{\\dag}A\\rho-2A\\rho A^\\dag+\\rho A^\\dag A)\\;,\n\\end{equation}\nwhere $A=a^\\dag a$ is the number operator and $\\kappa_p$ is the\ncorresponding rate of decoherence. The solution of this equation can\nbe easily found as (\\ref{sol}) where\n\\begin{equation}\n\\label{sol_phase}\\rho_{n,n'}(t)=\\exp[-(n-n')^2\\kappa_p\nt]\\rho_{n,n'}(0)\\;.\n\\end{equation}\nIt is easy to see that only the diagonal elements $\\rho_{n,n}$ do\nnot decay due to dephasing. Thus at long times, we can write\n\\begin{equation}\n\\rho(t\\rightarrow \\infty)=\\sum_{n=0}^\\infty\n\\rho_{n,n}(0)|n\\rangle\\langle n|\\;,\n\\end{equation}\nwhich refers to a mixed state.\n\nUsing Eqs. (\\ref{avg2}) and (\\ref{sol_phase}), we next calculate the\nparameter $Q$. We find that the averages\n$\\langle a^{\\dag 2}a^2\\rangle$ and $\\langle a^\\dag a\\rangle$ do not\ndepend upon time, because in case of phase damping\n$\\rho_{n,n}(t)=\\rho_{n,n}(0)$. Thus $Q$ remains the same for all\ntimes.\n\\begin{figure*}\n\\begin{center}\n\\caption{\\label{wig_phase_fig}Wigner function in phase\nspace at long times in presence of phase damping for (a) $r=0.31$\nand (b) $r=0.8$.}\n\\end{center}\n\\end{figure*}\nHowever, the corresponding Wigner function has certain\ntime-dependence. We find that at long times, the Wigner function\nbecomes\n\\begin{equation}\n\\label{wigner_phase}W(\\alpha,\\alpha^*,\\infty)=\\sum_{n=0}^\\infty\n\\rho_{n,n}(0)W_{|n\\rangle\\langle n|}(\\alpha,\\alpha^*)\\;,\n\\end{equation}\nwhere $W_{|n\\rangle\\langle n|}(\\alpha,\\alpha^*)$ is the Wigner\nfunction of a Fock state $|n\\rangle$ as given by\n\\begin{equation}\nW_{|n\\rangle\\langle\nn|}(\\alpha,\\alpha^*)=(-1)^n\\frac{2}{\\pi}e^{-2|\\alpha|^2}L_n(4|\\alpha|^2)\\;.\n\\end{equation}\nThe function (\\ref{wigner_phase}) refers to a highly nonclassical\nstate. It is interesting to note that all the Fock states have\nindependent contributions to the Wigner function at long times,\nweighted by their initial population $\\rho_{n,n}(0)$. On the other\nhand, in case of decoherence due to photon-number decay, only the\nvacuum state survives. In Fig. \\ref{wig_phase_fig}, we plot the\nWigner function (\\ref{wigner_phase}) in phase space for different\nsqueezing. Note that the Wigner function has negative region at long\ntimes representing nonclassicality for all $r$, even if the state\ndoes not exhibit sub-Poissonian statistics for $r\\gtrsim 0.46$\n(because $Q$ is positive). In fact, if $Q$ is positive, it does not\nmean that the the state is classical. In such cases, we have to use\nother parameters to test the nonclassicality. Several parameters\nhave been introduced in this context \\cite{tara_Q,nonclass}. We can use\nhierarchy of these parameters which have been shown to be especially\nuseful in context of cat states. Here we illustrate the utility of one such\nparameter, e.g., the $A_3$ parameter as defined by \\cite{tara_Q}\n\\begin{equation}\n\\label{a3}A_3=\\frac{{\\rm det}[m^{(3)}]}{{\\rm det}[\\mu^{(3)}]-{\\rm\ndet}[m^{(3)}]}\\;,\n\\end{equation}\nwhere\n\\begin{equation}\nm^{(3)}=\\left(\\begin{array}{ccc} 1&m_1&m_2\\\\\nm_1&m_2&m_3\\\\\nm_2&m_3&m_4\n\\end{array}\\right)\\;,\\;\\mu^{(3)}=\\left(\\begin{array}{ccc} 1&\\mu_1&\\mu_2\\\\\n\\mu_1&\\mu_2&\\mu_3\\\\\n\\mu_2&\\mu_3&\\mu_4\n\\end{array}\\right)\\;,\n\\end{equation}\n$m_s=\\langle a^{\\dag s}a^s\\rangle$, $\\mu_s=\\langle (a^\\dag\na)^s\\rangle$ and det indicates determinant of the matrix. The state exhibits phase-insensitive nonclassical\nproperties if $A_3$ lies between 0 and -1 \\cite{tara_Q}. For the\nstate $|\\psi\\rangle_1$ we have found that $A_3$ remains negative for\n$|\\xi|\\lesssim 0.6$ which corresponds to $r\\lesssim 0.7$. Clearly\n$A_3$ is a stronger measure of nonclassicality than $Q$ because it\nleads to a larger upper bound of $r$ to exhibit nonclassicality.\nFurther, comparing the Wigner functions in Figs. \\ref{wig_phase_fig} with those at $t=0$ [see\nFigs. \\ref{sq_figs}], we find that the Wigner function varies very\nslowly with time for small squeezing. But for large squeezing, the\nvariation is faster. Although we can conclude that phase damping\nleads to much slower decoherence than amplitude damping.\n\n\\section{Conclusions}\nIn conclusion, we have studied how a class of non-Gaussian states\nevolves in presence of decoherence. We have considered a\nsingle-photon subtracted squeezed vacuum state, the Wigner function\nof which is similar to that of a Schrodinger kitten state. We have\nfound an upper bound for squeezing parameter for which this state\nexhibits sub-Poissonian photon statistics. However, the state\nremains nonclassical for all values of the squeezing parameter\nbecause the Wigner function becomes negative around central region\nin phase space. Next, we have studied how the state evolves in\npresence of two different kinds of decoherence, viz., amplitude\ndecay and phase damping. We have found analytical expressions for\nthe time-evolution of the state and the Wigner function in both\ncases. In case of amplitude decay, the Wigner function loses its\nnon-Gaussian nature and becomes Gaussian at long times,\ncorresponding to vacuum. On the other hand, phase damping leads to\nmuch slower decoherence than amplitude damping. The state remains\nnonclassical at long times.\n\n\\begin{acknowledgments}\nA.B. gratefully acknowledges the partial support from the Women in\nScience and Engineering program in University of Southern\nCalifornia, Los Angeles, USA. G.S.A. kindly acknowledges support\nfrom NSF grant no. CCF0524673.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThis paper continues the series on spectroscopic orbits of stars\nbelonging to hierarchical systems \\citep{paper1,paper2,paper3}. It is\nmotivated by the need to improve statistics of orbital elements in\nstellar hierarchies. Statistics will inform us on the processes of\ntheir formation and dynamical evolution, as outlined in the previous\npapers of this series. This work augments the collection of\nobservational data on stellar hierarchies assembled in the multiple\nstar catalog \\citep[MSC;][]{MSC}.\n\n\n\\begin{deluxetable*}{c c rr l cc rr r c }\n\\tabletypesize{\\scriptsize} \n\\tablecaption{Basic parameters of observed multiple systems\n\\label{tab:objects} } \n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{WDS} & \n\\colhead{Comp.} &\n\\colhead{HIP} & \n\\colhead{HD} & \n\\colhead{Spectral} & \n\\colhead{$V$} & \n\\colhead{$V-K$} & \n\\colhead{$\\mu^*_\\alpha$} & \n\\colhead{$\\mu_\\delta$} & \n\\colhead{RV} & \n\\colhead{$\\overline{\\omega}$\\tablenotemark{a}} \\\\\n\\colhead{(J2000)} & \n & & & \n\\colhead{Type} & \n\\colhead{(mag)} &\n\\colhead{(mag)} &\n\\multicolumn{2}{c}{ (mas yr$^{-1}$)} &\n\\colhead{(km s$^{-1}$)} &\n\\colhead{(mas)} \n}\n\\startdata\n04125$-$3609 &A & 19639 & 26758 & F3V & 7.12 & 1.12 & 61 & 24 & 35.40 & 7.94 \\\\\n &B & 19646 & 26773 & F2IV & 7.91 & 0.92 & 61 & 24 & 35.98 & 8.03 \\\\\n12059$-$4951 &A & \\ldots & 105080 & G3V & 9.13 & 1.43 & 25 &$-$15 & 50.19 & 9.99 \\\\\n &B & \\ldots & 105081 & G0V & 9.18 & 1.42 & 31 &$-$20 & 50.09 & 7.20 \\\\\n12283$-$6146 &A & 60845 & 108500 & G3V & 6.82 & 1.64 & 71 &$-$160 & 40.02 & 19.93 \\\\\n &D & \\ldots & \\ldots & \\ldots& 13.70 & 4.54& 73 &$-$169 & \\ldots & 19.91 \\\\\n12404$-$4924 &A & 61840 & 110143 & G0V & 7.60 & 2.00 & $-$28 & $-$112 & 6.76 & 18.59 \\\\ \n15275$-$1058 &A & 75663 & 137631 & G0 & 8.14 & 1.35&$-$65 &$-$35 & $-$56.0 v & 9.29 \\\\\n &B & \\ldots & \\ldots & G0 & 9.21 & 1.50&$-$61 &$-$35 & $-$56.82 & 7.69 \\\\\n15410$-$1449 &A & 76816 & 139864 & F8V & 9.47 & 1.62&$-$26 &$-$1 & $-$38.94 & 3.23 \\\\\n &B & \\ldots & \\ldots & \\ldots& 9.74 & 2.51&$-$25 &$-$2 & $-$50.9 v & 3.15 \\\\ \n15577$-$3915 &A & 78163 & 142728 & G3V & 9.04 & 1.54& 17 & 6 & 9.41 & 10.42 \\\\\n &B & \\ldots & \\ldots & \\ldots&10.30 & \\ldots& 31 & 4 & 6.78 v & 13.57 \\\\\n16005$-$3605 &A & 78416 & 143215 & G1V &8.65 & 1.32 &$-$26 &$-$41 & 1.60 & 9.33 \\\\\n &B & \\ldots & \\ldots & \\ldots&9.32 & 1.31 &$-$28 &$-$41 & 1.43 & 9.31 \\\\\n16253$-$4909 &AB& 80448 & 147633 & G2V & 7.5? &2.3? &$-$95 &$-$94 & $-$2.08 & 19.66 \\\\\n17199$-$1121 &A & 84789 & 156769 & F2 & 9.11 & 1.37& 6 & 13 & 5.62 & 5.33 \\\\\n &B & \\ldots & \\ldots & \\ldots& 9.89 & 1.37& 5 & 12 & 5.97 & 5.34 \n\\enddata\n\\tablenotetext{a}{Proper motions and parallaxes are \n from the {\\it Gaia} DR2 \\citep{Gaia,Gaia1}.}\n\\end{deluxetable*}\n\nThe systems studied here are presented in Table~\\ref{tab:objects}.\nOnly one of them (HIP 61840) is a simple binary belonging to the 67 pc\nsample of solar-type stars; others contain from three to five components and\nare also relatively close to the Sun. Their principal components are\nmain sequence stars with spectral types from F2V to G3V. The data in\nTable~\\ref{tab:objects} are collected from Simbad and {\\it Gaia} DR2 \\citep{Gaia},\nthe radial velocities (RVs) are determined here (variable RVs are\nmarked by 'v').\n\nThe structure of this paper is similar to the previous ones. The data\nand methods are briefly recalled in Section~\\ref{sec:obs}, where the\nnew orbital elements are also given. Then in Section~\\ref{sec:obj}\neach system is discussed individually. The paper closes with a short\nsummary in Section~\\ref{sec:sum}.\n\n\n\\section{Observations and data analysis}\n\\label{sec:obs}\n\n\n\\subsection{Spectroscopic observations}\n\nThe spectra used here were taken with the 1.5 m telescope sited at the\nCerro Tololo Inter-American Observatory (CTIO) in Chile and operated\nby the SMARTS Consortium.\\footnote{\n \\url{http:\/\/www.astro.yale.edu\/smarts\/}} The observing time was\nallocated through NOAO. Observations were made with the CHIRON\noptical echelle spectrograph \\citep{CHIRON} by the telescope operators\nin service mode. In two runs, 2017 August and 2018 March, the author\nalso observed in classical mode. All spectra are taken in the slicer\nmode with a resolution of $R=80,000$ and a signal to noise ratio of at\nleast 20. Thorium-Argon calibrations were recorded for each target.\n\nRadial velocities are determined from the cross-correlation\nfunction (CCF) of echelle orders with the binary mask based on the solar\nspectrum, as detailed in \\citep{paper1}. The RVs derived by this\nmethod should be on the absolute scale if the wavelength calibration\nis accurate. The CHIRON RVs were checked against standards from\n\\citep{Udry1998}, and a small offset of $+0.15$ km~s$^{-1}$ was found\nin \\cite{paper3}. \n\nThe CCF contains two dips in the case of double-lined systems studied\nhere. The dip width is related to the projected rotation speed $V\n\\sin i$, while its area depends on the spectral type, metallicity, and\nrelative flux. Table~\\ref{tab:dip} lists average parameters of the\nGaussian curves fitted to the CCF dips. It gives the number of\naveraged measurements $N$ (blended CCFs were not used), the dip\namplitude $a$, its dispersion $\\sigma$, the product $a \\sigma$\nproportional to the dip area (hence to the relative flux), and the\nprojected rotation velocity $V \\sin i$, estimated from $\\sigma$ by the\napproximate formula given in \\citep{paper1}. The last column\nindicates the presence or absence of the lithium 6708\\AA ~line in\nindividual components.\n\n\\begin{deluxetable*}{l l c cccc c} \n\\tabletypesize{\\scriptsize} \n\\tablecaption{CCF parameters\n\\label{tab:dip} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{Comp.} & \n\\colhead{$N$} & \n\\colhead{$a$} & \n\\colhead{$\\sigma$} & \n\\colhead{$a \\sigma$} & \n\\colhead{$V \\sin i$ } & \n\\colhead{Li}\n\\\\\n & & & &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{ 6708\\AA}\n}\n\\startdata\nHIP 19639 & Aa & 7 & 0.061 & 12.53 & 0.765 & 21.7 &N\\\\\nHIP 19639 & Ab & 7 & 0.033 & 5.90 & 0.193 & 8.7 &N\\\\\nHIP 19646 & B & 2 & 0.042 & 18.87 & 0.787 & 33: &N\\\\ \nHD 105080 & A & 3 & 0.389 & 3.68 & 1.429 & 2.5 &N\\\\ \nHD 105081 & Ba & 11 & 0.179 & 3.82 & 0.684 & 3.1 &N\\\\\nHD 105081 & Bb & 11 & 0.167 & 3.78 & 0.630 & 3.0 &N\\\\\nHIP 60845 & Aa & 2 & 0.183 & 3.83 & 0.702 & 3.2 &N\\\\\nHIP 60845 & Ab & 2 & 0.124 & 4.01 & 0.497 & 3.8 &N\\\\\nHIP 60845 & BC & 2 & 0.430 & 3.57 & 1.535 & 2.0 &N\\\\ \nHIP 61840 & Aa & 9 & 0.189 & 4.51 & 0.853 & 5.3 &Y\\\\\nHIP 61840 & Ab & 9 & 0.124 & 4.13 & 0.511 & 4.2 &Y\\\\\nHIP 75663 & A & 5 & 0.223 & 6.85 & 1.528 & 10.7 &Y\\\\\nHIP 75663 & Ba & 12 & 0.161 & 4.88 & 0.789 & 6.3 &Y\\\\ \nHIP 75663 & Bb & 12 & 0.155 & 4.87 & 0.754 & 6.3 &Y\\\\\nHIP 76816 & Aa & 6 & 0.110 & 8.14 & 0.899 & 13.3 &Y\\\\ \nHIP 76816 & Ab & 6 & 0.030 & 4.49 & 0.137 & 5.3 &Y\\\\\nHIP 76816 & B & 5 & 0.506 & 3.78 & 1.913 & 3.0 &Y\\\\\nHIP 78163 & Aa & 9 & 0.197 & 4.08 & 0.803 & 4.0 &Y\\\\\nHIP 78163 & Ab & 9 & 0.192 & 4.05 & 0.778 & 4.0 &Y\\\\\nHIP 78163 & B & 3 & 0.371 & 4.67 & 1.732 & 5.8 &N\\\\ \nHIP 78416 & Aa & 9 & 0.047 & 15.43 & 0.725 & 27: &Y\\\\\nHIP 78416 & Ab & 9 & 0.041 & 13.18 & 0.536 & 23: &Y\\\\\nHIP 78416 & B & 4 & 0.058 & 22.68 & 1.324 & 40: &Y\\\\ \nHIP 80448 & Aa & 2 & 0.101 & 12.30 & 1.247 & 21.3 &Y\\\\ \nHIP 80448 & Ab & 2 & 0.023 & 9.78 & 0.221 & 16.5 &Y\\\\\nHIP 80448 & B & 3 & 0.171 & 8.62 & 1.469 & 14.3 &Y\\\\ \nHIP 84789 & Aa & 6 & 0.043 & 12.41 & 0.536 & 21.5 &N\\\\\nHIP 84789 & Ab & 6 & 0.048 & 10.88 & 0.522 & 18.6 &N\\\\\nHIP 84789 & B & 2 & 0.036 & 27.49 & 0.996 & 49: & N\n\\enddata \n\\end{deluxetable*}\n\n\n\n\\subsection{Speckle interferometry}\n\nInformation on the resolved subsystems is retrieved from the\nWashington Double Star Catalog \\citep[WDS;][]{WDS}. It is complemented\nby recent speckle interferometry at the Southern Astrophysical\nResearch (SOAR) telescope. The latest publication \\citep{SAM18}\ncontains references to previous papers.\n\n\\subsection{Orbit calculation}\n\nAs in Paper 3 \\citep{paper3}, orbital elements and their errors are\ndetermined by the least-squares fits with weights inversely\nproportional to the adopted errors. The IDL code {\\tt\n orbit} \\citep{ORBIT}\\footnote{Codebase: \\url{http:\/\/www.ctio.noao.edu\/~atokovin\/orbit\/} and \n\\url{http:\/\/dx.doi.org\/10.5281\/zenodo.61119} }\nis used. It can fit spectroscopic, visual, or combined\nvisual\/spectroscopic orbits. Formal errors of orbital elements are\ndetermined from these fits. The elements of spectroscopic orbits are\ngiven in Table~\\ref{tab:sborb} in common notation.\n\n\n\n\\begin{deluxetable*}{l l cccc ccc c c} \n\\tabletypesize{\\scriptsize} \n\\tablecaption{Spectroscopic orbits\n\\label{tab:sborb} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{$P$} & \n\\colhead{$T$} & \n\\colhead{$e$} & \n\\colhead{$\\omega_{\\rm A}$ } & \n\\colhead{$K_1$} & \n\\colhead{$K_2$} & \n\\colhead{$\\gamma$} & \n\\colhead{rms$_{1,2}$} &\n\\colhead{$M_{1,2} \\sin^3 i$} \n\\\\\n& & \\colhead{(days)} &\n\\colhead{(+24,00000)} & &\n\\colhead{(degree)} & \n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{(km~s$^{-1}$)} &\n\\colhead{ (${\\cal M}_\\odot$) } \n}\n\\startdata\nHIP 19639 & Aa,Ab & 2.35254 & 58002.5732 & 0.0 & 0.0 & 39.474 & 49.194 & 35.400 & 0.36 & 0.094 \\\\\n && $\\pm$0.00005 & $\\pm$0.0019 & fixed & fixed & $\\pm$0.160 & $\\pm$0.298 & $\\pm$0.100 & 0.87 & 0.076 \\\\\nHD 105081 & Ba,Bb & 30.427 & 58214.038 & 0.418 & 82.1 & 46.856 & 47.493 & 50.099 & 0.07 & 1.00 \\\\\n && $\\pm$0.006 & $\\pm$0.023 & $\\pm$0.001 & $\\pm$0.3 & $\\pm$0.117 & $\\pm$0.118 & $\\pm$0.041 & 0.05 & 0.98 \\\\\nHIP 60845 & Aa,Ab & 6.3035 & 58195.6279 & 0.0 & 0.0 & 31.361 & 32.128 & 40.018 & 0.10 & 0.084 \\\\\n && $\\pm$0.0001 & $\\pm$0.0017 & fixed & fixed & $\\pm$0.063 & $\\pm$0.114 & $\\pm$0.036 & 0.27 & 0.082\\\\\nHIP 61840 & Aa,Ab& 9.6717 & 58194.383 & 0.007 & 357.0 & 56.665 & 60.744 & 6.745 & 0.05 & 0.84 \\\\\n && $\\pm$0.0008 & $\\pm$0.345 & $\\pm$0.001 & $\\pm$13.0 & $\\pm$0.525 & $\\pm$0.562 & $\\pm$0.048 & 0.05 & 0.78\\\\\nHIP 75663 & Ba,Bb & 22.8704 & 58204.4963 & 0.613 & 260.1 & 49.001 & 49.622 & $-$56.847 & 0.29 & 0.56 \\\\\n && $\\pm$0.0047 & $\\pm$0.020 & $\\pm$0.001 & $\\pm$0.3 & $\\pm$0.126 & $\\pm$0.134 & $\\pm$0.045 & 0.27 & 0.56\\\\\nHIP 76816 & Aa,Ab & 6.95176 & 58197.3528 & 0.0 & 0.0 & 45.790 & 62.448 & $-$39.145 & 0.29 & 0.53 \\\\\n && $\\pm$0.00002 & $\\pm$0.0063 & fixed & fixed & $\\pm$0.306 & $\\pm$0.570 & $\\pm$0.186 & 1.13 & 0.39 \\\\\nHIP 78163 & Aa,Ab & 21.8186 & 58202.091 & 0.577 & 301.0 & 45.429 & 45.671 & 9.411 & 0.04 & 0.47\\\\\n && $\\pm$0.0015 & $\\pm$0.014 & $\\pm$0.002 & $\\pm$0.3 & $\\pm$0.161 & $\\pm$0.186 & $\\pm$0.045 & 0.04 & 0.46\\\\\nHIP 78416 & Aa,Ab & 21.0802 & 58197.2344 & 0.708 & 99.1 & 64.636 & 71.737 & 1.636 & 0.16 & 1.03 \\\\\n && $\\pm$0.0030 & $\\pm$0.0072 & $\\pm$0.003 & $\\pm$0.3 & $\\pm$0.372 & $\\pm$0.509 & $\\pm$0.112 & 0.61 & 0.93\\\\\nHIP 80448 &Aa,Ab & 2.2699 & 58195.5948 & 0.0 & 0.0 & 73.124 & 108.452 & $-$1.848 & 0.64 & 0.84 \\\\\n && $\\pm$0.0002 & $\\pm$0.0033 & fixed & fixed & $\\pm$0.592 & $\\pm$0.698 & $\\pm$0.270 & 0.51 & 0.57\\\\\nHIP 84789 &Aa,Ab & 2.2758 & 58196.7968 & 0.0 & 0.0 & 78.044 & 79.004 & 5.624 & 0.40 & 0.46 \\\\\n && $\\pm$0.0001 & $\\pm$0.0020 & fixed & fixed & $\\pm$0.181 & $\\pm$0.182 & $\\pm$0.073 & 0.21 & 0.45\n\\enddata \n\\end{deluxetable*}\n\n\\begin{deluxetable*}{l l cccc ccc} \n\\tabletypesize{\\scriptsize} \n\\tablecaption{Visual orbits\n\\label{tab:vborb} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{$P$} & \n\\colhead{$T$} & \n\\colhead{$e$} & \n\\colhead{$a$} & \n\\colhead{$\\Omega_{\\rm A}$ } & \n\\colhead{$\\omega_{\\rm A}$ } & \n\\colhead{$i$ } \\\\\n& & \\colhead{(years)} &\n\\colhead{(years)} & &\n\\colhead{(arcsec)} & \n\\colhead{(degree)} & \n\\colhead{(degree)} & \n\\colhead{(degree)} \n}\n\\startdata\nHD 105080 & Aa,Ab & 91.6 & 1999.15 & 0.40 & 0.176 & 5.4 & 224.1 & 40.4 \\\\\nHIP 60845 & A,BC & 690 & 1826.7 & 0.20 & 2.485 & 104.4 & 156.6 & 141.4 \\\\\nHIP 60845 & B,C & 28.2 & 1990.2 & 0.166 & 0.221 & 162.8 & 77.3 & 156.1 \\\\\n & & $\\pm$0.2 & $\\pm$0.3 & $\\pm$0.009 & $\\pm$0.003 & $\\pm$6.5 & $\\pm$5.9 & $\\pm$2.5 \\\\\nHIP 80448 & A,B & 1950 & 1926.58 & 0.63 & 4.644 & 75.5 & 267.9 & 152.1 \\\\\nHIP 80448 & Ba,Bb & 20.0 & 2018.08 & 0.42 & 0.176 & 6.1 & 150.9 & 109.1 \\\\\n & & $\\pm$0.3 & $\\pm$0.24 & $\\pm$0.02 & $\\pm$0.003 & $\\pm$1.2 & $\\pm$4.9 & $\\pm$0.8 \n\\enddata \n\\end{deluxetable*}\n\n\n\nFor two multiple systems, the resolved measurements of inner and outer\npairs are represented by visual orbits. Simultaneous fitting of inner\nand outer orbits is done using the code {\\tt orbit3.pro} described by\n\\citet{TL2017}; the code is available in \\citep{ORBIT3}.\\footnote{Codebase: \\url{http:\/\/dx.doi.org\/10.5281\/zenodo.321854}}\nIt accounts for the wobble in the trajectory of the outer pair caused\nby the subsystem. The wobble amplitude is $f$ times smaller than the\ninner semimajor axis, where the wobble factor $f = q_{\\rm\n in}\/(1+q_{\\rm in})$ depends on the inner mass ratio $q_{\\rm in}$.\nThe elements of visual orbits are given in Table~\\ref{tab:vborb}. As\nouter orbits are poorly constrained, I do not list their errors. The\nouter orbits serve primarily to model the observed part of the\ntrajectory for the determination of $f$. In the figures illustrating\nthese orbits, the observed trajectories are plotted relative to the\nprimary component of each system, on the same scale.\n\nIndividual RVs of spectroscopic binaries and their residuals to the\norbits are presented in Table~\\ref{tab:rv}. The HIP or HD number and\nthe system identifier (components joined by comma) in the first two\ncolumns define the binary. Then follow the Julian date, the RV of the\nprimary component $V_1$, its adopted error $\\sigma_1$ (blended CCF\ndips are assigned large errors), and its residual (O$-$C)$_1$. The\nlast three columns give velocities, errors, and residuals of the\nsecondary component. Table~\\ref{tab:rvconst} contains RVs of other\ncomponents, both constant and variable. Finally,\nTable~\\ref{tab:speckle} lists position measurements used for the\ncalculation of visual orbits. It contains the HIP or HD number, system\nidentification, date of observation, position angle $\\theta$,\nseparation $\\rho$, adopted error $\\sigma_\\rho$ (errors in radial and\ntangential directions are considered to be equal), and the residuals\nto the orbits in $\\theta$ and $\\rho$. The last column indicates the\nmeasurement technique. Measurements of the outer systems are of two\nkinds: when the inner pair is unresolved (e.g. HIP 60845 A,BC), they\nrefer to the photo-center of the inner pair, while resolved\nmeasurements refer to the individual components (e.g. HIP 60845\nA,B). The orbit-fitting code accounts for reduced wobble amplitude of\nunresolved (photo-center) measurements compared to resolved ones.\n\n\n\n\\begin{deluxetable*}{r l c rrr rrr} \n\\tabletypesize{\\scriptsize} \n\\tablecaption{Radial velocities and residuals (fragment)\n\\label{tab:rv} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{Date} & \n\\colhead{$V_1$} & \n\\colhead{$\\sigma_1$} & \n\\colhead{(O$-$C)$_1$ } &\n\\colhead{$V_2$} & \n\\colhead{$\\sigma_2$} & \n\\colhead{(O$-$C)$_2$ } \\\\\n & & \n\\colhead{(JD $+$24,00000)} &\n\\multicolumn{3}{c}{(km s$^{-1}$)} &\n\\multicolumn{3}{c}{(km s$^{-1}$)} \n}\n\\startdata\n 19639 & Aa,Ab & 57985.8910 & 68.74 & 0.30 & 0.16 & $-$6.37 & 0.60 & $-$0.42 \\\\\n 19639 & Aa,Ab & 57986.8980 & 15.11 & 0.30 & 0.18 & 59.76 & 0.60 & $-$1.15 \\\\\n 19639 & Aa,Ab & 58052.6260 & 31.22 & 10.00 & 2.23 & \\ldots &\\ldots & \\ldots \\\\\n 19639 & Aa,Ab & 58053.6080 & 22.97 & 0.50 & 1.26 & 54.35 & 1.00 & 1.89 \n\\enddata \n\\end{deluxetable*}\n\n\\begin{deluxetable}{r l r r } \n\\tabletypesize{\\scriptsize} \n\\tablecaption{Radial velocities of other components\n\\label{tab:rvconst} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{Comp.} & \n\\colhead{Date} & \n\\colhead{RV} \\\\ \n & & \n\\colhead{(JD $+$24,00000)} &\n\\colhead {(km s$^{-1}$)} \n}\n\\startdata\n19646 & B & 57985.8932& 36.005 \\\\\n19646 & B & 58193.5386& 35.944 \\\\ \n105080 & A & 58193.7546& 50.182 \\\\ \n105080 & A & 58194.6308& 50.198 \\\\\n105080 & A & 58195.6489& 50.179 \\\\\n60845 & BC & 57985.4627& 42.462 \\\\\n60845 & BC & 58193.7521& 42.524 \\\\ \n60845 & BC & 58194.6447& 42.514 \\\\\n60845 & BC & 58195.6586& 42.523 \\\\\n60845 & BC & 58177.7617& 42.513 \\\\\n60845 & BC & 58232.5972& 42.540 \\\\\n60845 & BC & 58242.5361& 42.513 \\\\\n75663 & A & 57986.4885& $-$50.737 \\\\ \n75663 & A & 58177.8100& $-$56.446 \\\\\n75663 & A & 58193.8257& $-$57.181 \\\\\n75663 & A & 58195.8323& $-$57.305 \\\\\n76816 & B & 57986.4996& $-$50.912 \\\\\n76816 & B & 58193.8413& $-$50.916 \\\\ \n76816 & B & 58194.8267& $-$50.921 \\\\\n76816 & B & 58195.8509& $-$50.912 \\\\ \n78163 & B & 57986.5130& 6.116 \\\\ \n78163 & B & 58194.8533& 7.152 \\\\ \n78163 & B & 58195.7872& 7.082 \\\\ \n78416 & B & 57986.5221& 0.752 \\\\\n78416 & B & 58193.8595& 1.568 \\\\ \n78416 & B & 58194.8446& 1.816 \\\\ \n78416 & B & 58195.7774& 1.581 \\\\ \n80448 & B & 58193.8678& 7.509 \\\\\n80448 & B & 58195.8008& 7.752 \\\\\n80448 & B & 58194.8621& 7.679 \\\\\n80448 & B & 58228.8113& 7.515 \\\\ \n80448 & B & 58233.8463& 7.353 \\\\\n80448 & B & 58246.6693& 8.395 \\\\\n80448 & B & 58248.8529& 8.176 \\\\\n80448 & B & 58256.7395& 7.532 \\\\\n80448 & B & 58257.8020& 7.916 \\\\\n84789 & B & 57986.5346& 6.669 \\\\\n84789 & B & 58195.8671& 5.260 \n\\enddata \n\\end{deluxetable}\n\n\\begin{deluxetable*}{r l l rrr rr l} \n\\tabletypesize{\\scriptsize} \n\\tablecaption{Position measurements and residuals (fragment)\n\\label{tab:speckle} }\n\\tablewidth{0pt} \n\\tablehead{ \n\\colhead{HIP\/HD} & \n\\colhead{System} & \n\\colhead{Date} & \n\\colhead{$\\theta$} & \n\\colhead{$\\rho$} & \n\\colhead{$\\sigma_\\rho$} & \n\\colhead{(O$-$C)$_\\theta$ } & \n\\colhead{(O$-$C)$_\\rho$ } &\n\\colhead{Ref.\\tablenotemark{a}} \\\\\n & & \n\\colhead{(yr)} &\n\\colhead{(\\degr)} &\n\\colhead{(\\arcsec)} &\n\\colhead{(\\arcsec)} &\n\\colhead{(\\degr)} &\n\\colhead{(\\arcsec)} &\n}\n\\startdata\n 60845 & B,C & 1939.4600 & 357.8 & 0.3500 & 0.1500 & 4.2 & 0.1351 & M \\\\\n 60845 & B,C & 1956.3800 & 184.1 & 0.3100 & 0.0500 & 8.8 & 0.0956 & M \\\\\n 60845 & B,C & 2018.1639 & 92.6 & 0.1769 & 0.0050 & 1.9 & 0.0070 & S \\\\\n 60845 & A,BC & 1880.3800 & 270.5 & 2.4300 & 0.2500 & 2.2 & 0.3463 & M \\\\\n 60845 & A,BC & 1991.2500 & 201.1 & 2.0510 & 0.0100 & $-$1.2 & $-$0.0042 & H \\\\\n 60845 & A,B & 2018.1639 & 187.1 & 2.0965 & 0.0050 & 0.2 & $-$0.0018 & S\n\\enddata \n\\tablenotetext{a}{\nH: Hipparcos;\nS: speckle interferometry at SOAR;\ns: speckle interferometry at other telescopes;\nM: visual micrometer measures;\nG: Gaia DR2.\n}\n\\end{deluxetable*}\n\n\n\\section{Individual objects}\n\\label{sec:obj}\n\nFor each observed system, the corresponding Figure shows a typical\ndouble-lined CCF (the Julian date and components' designatios are marked\non the plot) together with the RV curve representing the orbit. In the\nRV curves, squares denote the primary component, triangles denote the\nsecondary component, while the full and dashed lines plot the orbit. \n\n\\subsection{HIP 19639 and 19646 (triple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig1.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 19639 Aa,Ab.\n\\label{fig:19639}\n}\n\\end{figure}\n\nThe 50\\arcsec ~pair of bright stars HIP~19639 and 19646 was identified\nas a visual binary by J.F.W.~Hershel in 1838. The {\\it Gaia} DR2\nastrometry leaves no doubt that this pair is physical: the components\nhave common proper motion (PM), distance, and RV. The outer orbital\nperiod is of the order of 240 kyr. \\citet{N04} found that the\ncomponent A is a double-lined binary; its 2.35-day circular orbit is\ndetermined here for the first time (Figure~\\ref{fig:19639}). Two\nspectra (JD 2458052 and 2458053) were taken with the NRES spectrograph,\nas described in \\citep{paper3}.\n\nThe CCF of the stronger component Aa is wide and asymmetric, while the\nCCF of Ab is narrower; their widths correspond to approximate\nprojected rotation velocities $ V \\sin i$ of 21.7 and 8.7 km~s$^{-1}$,\nrespectively, while the ratio of the CCF areas implies $\\Delta V_{\\rm\n Aa,Ab} = 1.50$ mag. Wide and shallow dips lead to large RV errors\nand large residuals to the orbit. The mass ratio in the inner pair is\n$q_{\\rm Aa,Ab} = 0.82$. The RV of the component B its close to the\ncenter-of-mass velocity of A. The component B also has a wide CCF\ncorresponding to $ V \\sin i$ of $\\sim$33 km~s$^{-1}$. \n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig2.eps}\n\\caption{Location of three components of the HIP~19639 system on the\n color-magnitude diagram (squares). The full line is a 2 Gyr\n isochrone for solar metallicity \\citep{Dotter2008}, where asterisks\n and numbers mark masses.\n\\label{fig:iso}\n}\n\\end{figure}\n\nComponents of the triple system HIP~19639 are placed on the\ncolor-magnitude diagram (CMD) in Figure~\\ref{fig:iso}, using the\ndistance modulus of 5.47 mag. The $V$ magnitudes of Aa and Ab are\nestimated from the combined magnitude of A and the spectroscopic\nmagnitude difference of 1.5 mag. The $V-K$ color of Ab, not measured\ndirectly, is assumed to place it on the main sequence. It is clear\nthat both Aa and B are located above the main sequence, near the\nturn-off. Their positions match reasonably well the 2 Gyr isochrone\nand correspond to the masses of 1.6 and 1.5 ${\\cal M}_\\odot$. The mass\nof Ab deduced from the isochrone is 1.28 ${\\cal M}_\\odot$, matching\nthe spectroscopic mass ratio, while the radii of Aa and Ab are 2.7 and\n1.1 $R_\\odot$. The orbital axis $a_1 + a_2 = 10.6 R_\\odot$ means that\nthe binary is detached. However, contact and mass transfer are\nimminent when Aa expands further.\n\nThe spectroscopic mass sum of the inner binary is only 0.18 ${\\cal\n M}_\\odot$. The mass sum estimated above, 2.88 ${\\cal M}_\\odot$,\nimplies an inclination $i_{\\rm Aa,Ab} = 23\\fdg4$, or $\\sin i_{\\rm\n Aa,Ab} = 0.40$, hence the synchronous rotation velocities of Aa and\nAb are 23.8 and 9.7 km~s$^{-1}$, respectively, in agreement with the\nmeasured CCF width. Summarizing, this is an interesting triple system\nwhere the inner close binary is caught at evolutionary phase preceding\nthe mass transfer.\n\n\n\\subsection{HD 105080 and 105081 (2+2 quadruple) }\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig3.eps}\n\\caption{CCF (left) and RV curve (right) of HD 105081 Ba,Bb.\n\\label{fig:105080}\n}\n\\end{figure}\n\n\nTwo nearly equal stars HD~105080 and 105081 form a 12\\farcs9 physical\nbinary, first measured by J.~Hershel in 1835. Each of these stars is\na close pair, making the whole system quadruple. The pair Aa,Ab is a\nknown visual binary RST~4958 with a small magnitude difference. Since\nits discovery in 1942 by R. A. Rossiter, it was observed only episodically.\nBy adding three speckle measures made at SOAR in 2017 and 2018, a\npreliminary orbit with $P = 91.6$ years can be fitted to the\nobservations (Table~\\ref{tab:vborb}). Double lines were reported for\nthis star in the literature, although there could be a confusion with\nthe double-lined component B. The RV of A is practically coincident\nwith the center-of-mass velocity of B. The {\\it Gaia} DR2 parallax of\nA has a large error of 1\\,mas, being affected by the Aa,Ab pair. I\nadopt the parallax of B as the distance to the system. Both\ncomponents are then located on the CMD very close to each other, above\nthe main sequence. This distance and the visual orbit of Aa,Ab\ncorrespond to the mass sum of 1.7 ${\\cal M}_\\odot$; however, the orbit\nis poorly constrained.\n\nThe component B (HD~105081), which is only slightly fainter than A in\nthe $V$ and $G$ bands, is revealed here to be a twin double-lined pair\nwith $P=30.4$ days and eccentricity $e=0.42$\n(Figure~\\ref{fig:105080}). The spectroscopic mass sum of\nBa and Bb, 1.98 ${\\cal M}_\\odot$, and the mass sum inferred from the\nabsolute magnitudes, 2.26 ${\\cal M}_\\odot$, imply inclination\n$i_{\\rm Ba,Bb}= 73\\degr$. The CCF dips of Ba and Bb are narrow and\ndeep, hence the residuals to the orbit are small, only\n0.07~km~s$^{-1}$ .\n\n\n\n\\subsection{HIP 60845 (quintuple)}\n\n\n\\begin{figure}\n\\plotone{fig4.eps}\n\\caption{Quintuple system HIP 60845 (WDS J12283$-$6146). The positions\n of three components A, BC, and D on the sky and their motions are\n illustrated in the upper panel. Periods and masses are indicated.\n The lower panel shows the observed motion of the subsystems A,BC and\n B,C and their orbits. In this plot, the coordinate origin coincides\n with the main star A, the wavy line shows the motion of the\n component B around A according to the orbit. Small crosses depict\n measurements of A,BC where the pair BC was unresolved, asterisks\n depict the resolved measurents of A,B. The orbit of B,C is plotted\n on the same scale around the coordinate origin by the dashed\n line and triangles. \n\\label{fig:HIP60845}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig5.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 60845 Aa,Ab. \n\\label{fig:60845}\n}\n\\end{figure}\n\n\nThis multiple system is located within 50\\,pc from the Sun and\ncontains at least five stars arranged in a rare 3-tier hierarchy\nillustrated in Figure~\\ref{fig:HIP60845}. The widest 7\\farcs9 pair\nRST~4499 AB,D is physical, based on its stable relative position and\nthe {\\it Gaia} DR2 parallaxes and PMs of the components. The fast\n(175 mas~yr$^{-1}$) PM facilitates discrimination between physical and\noptical companions, despite the high stellar density in this field and\nthe faintness of D ($V = 13.4$ mag). The period of AB,D estimated\nfrom its projected separation $\\rho_{\\rm AB,D} = 7\\farcs88$ is about 4\nkyr and corresponds to the chracteristic orbital velocity $\\mu^* =\n2 \\pi \\rho_{\\rm AB,D}\/P_{\\rm AB,D} = 13$ mas~yr$^{-1}$. The relative\nPM between A and D, measured by {\\it Gaia} and corrected for the\norbital motion of A, is 11 mas~yr$^{-1}$; it is directed almost\nexactly toward A (position angle 240\\degr), as indicated by the arrow\nin Figure~\\ref{fig:HIP60845}. If D moves on an eccentric orbit, it\nwill come close to A,BC in $\\sim$700 years, disrupting the system.\nAlternatively, the observed motion might correspond to a highly\ninclined and not very eccentric outer orbit, in which case the system\ncould be dynamically stable. If the pair AB,D is bound, the true\nseparation between A and D cannot exceed its projected separation by\nmore than $\\sim$2 times, given their relative speed of 11\nmas~yr$^{-1}$ and the total mass sum of 4.5 ${\\cal M}_\\odot$.\n\n\nThe visual binary A,BC (CPO~12), for which a crude orbit with\n$P=2520$\\,years and semimajor axis of 5\\farcs4 has been published by\n\\citet{USN2002}, occupies the intermediate hierarchical level. This\norbit is poorly constrained by the century-long observed arc. I\ncomputed an alternative orbit with $P_{\\rm A,BC} = 690$ years, with \nsmaller eccentricity and semimajor axis (Table~\\ref{tab:vborb}). This\norbit makes more sense, given the threat to dynamical stability posed\nby the outer component D. Even then, the period ratio $P_{\\rm AB,D}\/\nP_{\\rm A,BC} \\sim 5$ is comparable to the dynamical stability\nlimit. On the other hand, the nearly circular orbit of the inner pair\nB,C (RST~4499) with $P=28.2$\\,years is definitive. The visual orbits and\nthe estimated mass sums match the {\\it Gaia} DR2 parallax reasonably\nwell. Both orbits are retrograde and have small inclinations.\n\nThe visual primary star A was identified as a spectroscopic binary by\n\\citet{N04}. Now its 6.3-day double-lined orbit is determined\n(Figure~\\ref{fig:60845}). The eccentricity does not differ from zero\nsignificantly, hence the circular orbit is imposed. The masses of Aa\nand Ab are almost equal, as are their CCF dips. Given the small\nangular distance between A and BC, 2\\farcs06, the light is mixed in\nthe fiber, so the CCF often has 3 dips; the CCF shown in\nFigure~\\ref{fig:60845} is an exception recorded with good seeing and\ncareful guiding. The magnitude difference between Ab and Aa is 0.37\nmag, hence their individual $V$ magnitudes are 7.79 and 8.16 mag. By\ncomparing the mass sum of Aa and Ab estimated from their absolute\nmagnitudes, 2.2 ${\\cal M}_\\odot$, with the spectroscopic mass sum of\n0.167 ${\\cal M}_\\odot$, I find that the orbit of Aa,Ab has a small\ninclination of $i_{\\rm Aa,Ab} \\approx 25\\degr$. The synchronous\nrotation of the component Aa, of one solar radius, implies the\nprojected rotation of $V \\sin i = 3.3$ km~s$^{-1}$, close to the\nmeasured value. The three inner orbits could be close to coplanarity,\ngiven their small inclinations. \n\nThe CCF dip corresponding to the combined light of BC is narrow and\nhas a constant RV of 42.5 km~s$^{-1}$, close to the center-of-mass\nvelocity of A. Slow axial rotation, location of components on the\nmain sequence in the CMD, and non-detection of the lithium line\nsuggest that this quintuple system is relatively old.\n\n\n\\subsection{HIP 61840 (binary)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig6.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 61840 Aa,Ab.\n\\label{fig:61840}\n}\n\\end{figure}\n\nUnlike the rest of the objects in this paper, this star is a simple\nspectroscopic binary without additional components. It belongs to the\n67 pc sample of solar-type stars \\citep{FG67a} and is young, as\ninferred from the chromoshperic activity, X-ray flux, and the presence\nof lithium in its atmosphere. The double-lined nature was announced\nby \\citet{Wichman2003} and \\citet{N04}, but the orbital period was, so\nfar, unknown. The object has been observed by speckle interferometry\nat SOAR in 2011 and 2016 and found unresolved.\n\nThe orbit with $P=9.67$ days has a small, but significantly non-zero\neccentricity $e= 0.007 \\pm 0.001$ (Figure~\\ref{fig:61840}). The\nresiduals to the circular orbit are 0.3 km~s$^{-1}$, 6$\\times$\nlarger than to the eccentric orbit. The masses of Aa and Ab estimated\nfrom absolute magnitudes are 1.24 and 1.13 ${\\cal M}_\\odot$, the\nspectroscopic mass sum is 1.62 ${\\cal M}_\\odot$, hence the inclination\nis $i = 62\\degr$. The measured projected rotation speed matches the\nsynchronous speed.\n\n\\subsection{HIP 75663 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig7.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 75663 Ba,Bb.\n\\label{fig:75663}\n}\n\\end{figure}\n\n\nThis system is quadruple. The outer 9\\farcs4 pair was discovered in\n1825 by W.~Struve. Its estimated period is 17\\,kyr. \\citet{N04} noted\ndouble lines in the visual secondary B. Its orbital period is 22.9\ndays (Figure~\\ref{fig:75663}), with a large eccentricity of $e=0.61$\nand the mass ratio $q_{\\rm Ba,Bb} = 0.997$ (a twin). The areas of the\nCCF dips of Ba and Bb are equal to within 2\\%. The masses estimated\nfrom absolute magnitudes are 1.02 ${\\cal M}_\\odot$ each, leading to\nthe orbital inclination of $i_{\\rm Ba,Bb}=55\\degr$. The axial\nrotation of Ba and Bb is faster than synchronous, as expected for such\neccentric orbit.\n\n\nThe RV of the main component A is variable according to the CHIRON\ndata (range from $-$50.7 to $-$57.3 km~s$^{-1}$) and the literature.\n\\citet{N04} made two measurements averaging at $-58.2$ km~s$^{-1}$ and\nsuspected RV variability, while {\\it Gaia} measured $-55.4\n\\pm 2.$ km~s$^{-1}$. The photo-center motion of A caused by the subsystem\nAa,Ab could explain the discrepancy between the {\\it Gaia} DR2\nparallaxes of A and B {\\bf (9.29$\\pm$0.16 and 7.69$\\pm$0.07 mas,\n respectively). A similar discrepancy of parallaxes exists in the HD\n 105080\/81 system, where A is a visual binary. } The period of Aa,Ab is not known; presumably it\nis longer than a year. \\citet{Isaacson2010} classified this star as\nchromospherically active and found the RV jiter of\n4.2\\,m~s$^{-1}$. The location of the component A on the CMD indicates\nthat it is slightly evolved and matches approximately the 4-Gyr\nisochrone. Lithium is detectable in the spectra of A and B.\n\n\n\\subsection{HIP 76816 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig8.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 76816 Aa,Ab.\n\\label{fig:76816}\n}\n\\end{figure}\n\n\nThis is a quadruple system located at 309\\,pc from the Sun. The\n5\\farcs4 visual binary HWE~37 is known since 1876; its estimated\nperiod is 33\\,kyr. Double lines in the component A were discovered by\n\\citet{Desidera2006}. I used their measurement to refine the period of\nthe circular orbit of Aa,Ab with $P=6.95$ days, determined here\n(Figure~\\ref{fig:76816}). The eccentric orbit has similar residuals,\nhence the circular solution is retained. The components Aa and Ab are\nunequal in the amplitudes of their CCF dips (area ratio 0.152, or 2.0\nmag difference) and of the RV variation (mass ratio 0.735). The RV of\nthe visual component B is also variable with a long, still unknown\nperiod. I measured its RV at $-$50.9 km~s$^{-1}$ (constant), while\n{\\it Gaia} measured $-41.4$ km~s$^{-1}$ and \\citet{Desidera2006}\nmeasured $-37.5$ km~s$^{-1}$; these RVs differ from the center-of-mass\nvelocity of A, $-39.94$ km~s$^{-1}$.\n\nThe matching {\\it Gaia} DR2 parallaxes place both A and B above the\nmain sequence. The component B is more evolved: it is brighter than A\nin the $K$ band (unless its $K$ magnitude measured by 2MASS is \ndistorted by the proximity of A, as happens with other close\npairs). The mass sum of Aa and Ab, estimated crudely from the absolute\nmagnitudes, is almost 3 ${\\cal M}_\\odot$, leading to the\norbital inclination of 42\\fdg5. The stars\nAa and Ab apparently rotate synchronously with the orbit.\n\n\n\\subsection{HIP 78163 (2+2 quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig9.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 78163 Aa,Ab.\n\\label{fig:78163}\n}\n\\end{figure}\n\n\nThis multiple system is composed by the outer 7\\farcs9 pair WG~185\n(estimated period 8\\,kyr) and the inner subsystem Aa,Ab discovered by\n\\citet{N04}. For the latter, I determined here the orbit with $P=\n21.8$ days (Figure~\\ref{fig:78163}), $e=0.58$, and mass ratio $q_{\\rm\n Aa,Ab} = 0.996$ (a twin). The RV of the component B measured with\nCHIRON ranges from 6.1 to 7.1 km~s$^{-1}$ and differs from the\ncenter-of-mass RV of the component A, 9.4 km~s$^{-1}$. Considering\nalso the {\\it Gaia} RV(B)=16.3 km~s$^{-1}$, I infer that B is a\nsingle-lined binary, possibly with a long period and a small RV\namplitude. Its photo-center motion could explain the slight \n discrepancy between {\\it Gaia} parallaxes and PMs of A\nand B. Therefore, the parallax of A, 10.42\\,mas, is likely the correct\none. \n\nThe twin components Aa and Ab have masses of one solar each.\nComparting them to $M \\sin^3 i$, the inclination of 50\\degr ~is\nderived. The stars A and B are located on the main\nsequence. Interestingly, lithium is detectable in the spectra of Aa\nand Ab, but not in B, which is a similar solar-mass star.\n\n\n\n\\subsection{HIP 78416 (triple or quadruple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig10.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 78416 Aa,Ab.\n\\label{fig:78416}\n}\n\\end{figure}\n\n\nThe outer 6\\farcs5 pair HWE~81, known since 1876, has an estimated\nperiod of 10\\,kyr. \\citet{N04} detected RV variability of the\ncomponent A, later found to be a double-lined binary by\n\\citet{Desidera2006}. The orbital period is 21 days and the\neccentricity $e=0.766$ is unusually high for such a short period\n(Figure~\\ref{fig:78416}). The wide and shallow CCF dips imply fast\naxial rotation. For this reason, the RVs are not measured very\naccurately and the residuals to the orbit are large, 0.2 and 0.6\nkm~s$^{-1}$. By comparing the estimated masses of Aa and Ab, 1.15 and\n1.03 ${\\cal M}_\\odot$ respectively, with $M \\sin^3 i$, I estimate the\norbital inclination of 74\\degr.\n\nThe visual component B also has a fast axial rotation of $V \\sin i \\sim\n40$ km~s$^{-1}$, degrading the accuracy of its RV measurement. The\nRVs of B measured with CHIRON, by {\\it Gaia}, and by \\citet{Desidera2006}\n(1.4, $-$0.7, and 0.5 km~s$^{-1}$ respectively) are reasonably close\nto the center-of-mass RV of A, 1.7 km~s$^{-1}$. Therefore, B is\nlikely a single star. All three stars Aa, Ab, and B have comparable\nmasses and similar colors. The component A, being a close pair, is\nlocated on the CMD just above B, as expected.\n\nThe RVs of Aa and Ab measured by \\citet{Desidera2006}, 58.9 and $-1.8$\nkm~s$^{-1}$, correspond to the center-of-mass velocity of 30.2\nkm~s$^{-1}$ and do not fit the present orbit with $\\gamma = 1.7$\nkm~s$^{-1}$. This discrepancy suggests that Aa,Ab is orbited by\nanother close companion. Further monitoring is needed, however, to\nprove this hypothesis. \n\nAccording to \\citet{Rizzuto2011}, this system belongs to the Sco OB2\nassociation with a probability of 74\\%. Fast axial rotation and the\npresence of lithium indicate a young age.\n\n\n\n\\subsection{HIP 80448 (2+2 quadruple)}\n\n\nThis young multiple system is located within 50\\,pc from the Sun. It\ncontains four components in a small volume. The outer pair A,B\n(COO~197) has an uncertain visual orbit with a millenium-long period\n\\citep{Ary2015b}. Its secondary component was resolved in 2004 into a\n0\\farcs13 pair CVN~27 Ba,Bb by \\citet{Chauvin2010}, using adaptive\noptics. Independently, a subsystem TOK~50 Aa,Ab with similar\nseparation was discovered in 2009 by \\citet{TMH10} using speckle\ninterferometry. In fact, the same subsystem Ba,Bb was wrongly\nattributed to the primary component; its published measures at SOAR\nwith angle inverted by 180\\degr ~match the preliminary orbit with\n$P_{\\rm Ba,Bb}=20$ years. The pair TOK~50 Aa,Ab does not exist. The\ncomponent Bb is fainter than Ba by 3.5 mag in the $V$ band and by 1.1\nmag in the $K$ band; its lines are not detected in the combined\nspectrum of all stars.\n\n\n\n\\begin{figure}\n\\plotone{fig11.eps}\n\\caption{Visual orbits of HIP~80448 A,B and Ba,Bb (WDS J16253$-$4909,\n COO~197 and CVN~27).\n\\label{fig:COO197}\n}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig12.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 80448 Aa,Ab. \n\\label{fig:80448}\n}\n\\end{figure}\n\n\n\nFigure~\\ref{fig:COO197} shows the positions of resolved components on\nthe sky and the fitted orbits. Considering that the outer orbit is not\nconstrained by the data, I fixed its period to $P_{\\rm A,B} = 1950$\nyears and its axis to $a_{\\rm A,B} = 4\\farcs64$ to adjust the\ndynamical mass sum to its estimated value, 3.5 ${\\cal M}_\\odot$. The\norbit of Ba,Bb with $P_{\\rm Ba,Bb} = 20$ years yields the mass sum of\n1.8 ${\\cal M}_\\odot$, close to the estimated one. The ``wobble'' in\nthe positions of A,Ba caused by the subsystem is clearly seen. Its\nrelative amplitude $f = 0.40$ corresponds to the mass ratio $q_{\\rm\n Ba,Bb} = f\/(1-f) = 0.67$ that argees with the magnitude difference.\n\nThe spectrum of the main component A (in fact, blended light of Aa,\nAb, and Ba) shows stationary lines of Ba and double lines of Aa and Ab\nin rapid motion; the subsystem Aa,Ab was discovered with CHIRON\n\\citep{survey}. It is found here that the orbital period is\n$P_{\\rm Aa,Ab} = 2.3$ days and the orbit is circular\n(Figure~\\ref{fig:80448}). The mass ratio $q_{\\rm Aa,Ab} = 0.67$ is\nsimilar to the mass ratio $q_{\\rm Ba,Bb}$, while the ratio of dip\nareas corresponds to $\\Delta V_{\\rm Aa,Ab} = 1.9$ mag. Comparison of\nestimated and spectroscopic mass sums leads to the orbital inclination\n$i_{\\rm Aa,Ab} = 66\\degr$ or $i_{\\rm Aa,Ab} = 114\\degr$. It is not\ndissimilar to the inclination of other inner pair, $i_{\\rm Ba,Bb}\n= 109\\degr$, but it is difficult to believe that these two subsystems\nhave coplanar orbits, given the huge difference of their periods.\n\nThe components Aa and Ab rotate synchronously with the orbit. The\nprojected rotation of Ba, $V \\sin i = 14.3$ km~s$^{-1}$, is relatively\nfast, supporting the thesis that this system is young. The presence of\nlithium also suggests youth. The four components are located in the\nCMD at about 0.7 mag above the main sequence.\n\n\nThe pair Ba,Bb is presently near the periastron of its 20 year\norbit. I measured the RV(Ba) from 7.51 to 8.18 km~s$^{-1}$, quite\ndifferent from the center-of-mass velocity of A, $-2.08$ km~s$^{-1}$.\nThis positive difference and the positive trend actually match the orbit\nof Ba,Bb; I predict that RV(Ba) will soon start to decrease. The\norbits of A,B and Ba,Bb are not coplanar, although both are\nretrograde. \n\n\n\n\\subsection{HIP 84789 (triple)}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{fig13.eps}\n\\caption{CCF (left) and RV curve (right) of HIP 84789 Aa,Ab. Note that\n the secondary component Ab has a dip with larger amplitude.\n\\label{fig:84789}\n}\n\\end{figure}\n\n\nThis 5\\farcs6 visual binary STF~2148, discovered in 1832 by W.~Struve,\nhas an estimated period of 17\\,kyr. Double lines in its primary\ncomponent A were noted by \\citet{N04}. The orbital period of the\nsubsystem Aa,Ab determined here is $P_{\\rm Aa,Ab} = 2.3$ days; it is a\ntwin pair with $q_{\\rm Aa,Ab} = 0.988$ (Figure~\\ref{fig:84789}). The deeper CCF\ndip belongs to the less massive component Ab; the more massive star Aa\nrotates a little faster and has the dip area 3\\% larger than Ab, as\nwell as the smaller RV amplitude. The RVs of both components are\nmeasured with large errors owing to the wide and low-contrast CCF\ndips; the residuals to the orbits are also large. An attempt to fit\nthe orbit with non-zero eccentricity does not result in smaller\nresiduals, hence the orbit is circular.\n\nThe estimated masses of Aa and Ab are 1.30 ${\\cal M}_\\odot$ each,\nleading to the orbital inclination of $i_{\\rm Aa,Ab} =\n45\\degr$. Assuming the radii of 1.3 $R_\\odot$, the synchronous\nrotation velocity is $V \\sin i = 20.2$ km~s$^{-1}$. The width of the\ncorrelation dips matches this estimate and suggests that Aa rotates\nslightly faster and Ab slightly slower than synchronous.\n \nThe two components A and B are located on the CMD above each other\n(they have the same color) because A contains two equal stars; the\nmass of B is very similar to the masses of Aa and Ab, 1.3 solar. The\ncomponent B is single, as inferred from the equality of its RV to the\ncenter-of-mass velocity of A. However, it rotates much faster, at $V\n\\sin i \\sim 49$ km~s$^{-1}$. Very likely, the rotation of Aa and Ab has\nbeen slowed down by tidal synchronization with the orbit.\n\n\n\n\\section{Summary}\n\\label{sec:sum}\n\n\n\\begin{figure}\n\\plotone{fig14.eps}\n\\caption{Eccentricity vs. period for members of hierarchical systems\n studied here (large triangles) and for 467 spectroscopic binaries\n from the MSC with primary masses from 0.5 to 1.5 solar\n (crosses). The dashed line shows the locus of HIP~78416 Aa,Ab for\n evolution with constant angular momentum, $P(1 - e^2)^{3\/2} = {\\rm\n const}$.\n\\label{fig:pe}\n}\n\\end{figure}\n\nProbably by accident, the periods of 9 spectroscopic systems within\nhierarchical multiples are equally divided between three distinct\ngroups: (i) circular orbits with $P \\approx 2.3$ days, (ii)\nintermediate periods between 6 and 9 days, circular or nearly\ncircular, and (iii) eccentric orbits with periods from 21 to 30 days.\nFigure~\\ref{fig:pe} places these orbits on the period-eccentricity\nplot. The plus signs are 467 spectroscopic binaries with primary\nmasses from 0.5 to 1.5 ${\\cal M}_\\odot$ from the MSC \\citep{MSC}. When\nthe eccentric orbits of the group (iii) are tidally circularized,\ntheir periods will match those of group (ii), suggesting that these\nsubsystems could be formed by a common mechanism, such as Kozai-Lidov\ncycles with dynamical tides \\citep{Moe2018}. The periods of group (i)\nare substantially shorter, so their formation history could be\ndifferent.\n\nSix out of the 10 double-lined binaries studied here are twins with\nmass ratio $q > 0.95$, while the lowest measured mass ratio is\n0.67. If the mass ratios were uniformly distributed in the interval\n(0.7, 1.0), where double lines are detectable, the fraction of twins\nwould be only 0.15, whereas in fact it is 0.6. It is established that\ntwins correspond to a well-defined peak in the mass ratio distribution\nof solar-type spectroscopic binaries \\citep{twins}. They are believed\nto be formed when a low-mass binary accretes a major part of its mass.\nThe mass influx also creates conditions for formation of additional\ncomponents, building stellar hierarchies ``from inside out''. Thus,\ntwins are naturally produced as inner components of multiple systems\nin the process of mass assembly.\n\nThe goal of this study was to determine unknown periods of\nspectroscopic subsystems in several multiple stars. Although this goal\nis reached, I discovered RV variability of other visual components\n(HIP 75663A, 76816B, and 78163B), converting these triples into 2+2\nquadruples. The periods of new subsystems, presumably long, remain\nunknown so far.\n\n\n\n\n\n\\acknowledgements\n\nI thank the operator of the 1.5-m telescope R.~Hinohosa for executing\nobservations of this program and L.~Paredes for scheduling and\npipeline processing. Re-opening of CHIRON in 2017 was largely due to\nthe enthusiasm and energy of T.~Henry.\n\nThis work used the SIMBAD service operated by Centre des Donn\\'ees\nStellaires (Strasbourg, France), bibliographic references from the\nAstrophysics Data System maintained by SAO\/NASA, and the Washington\nDouble Star Catalog maintained at USNO.\n\nThis work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia}\nData Processing and Analysis Consortium (DPAC,\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC\nhas been provided by national institutions, in particular the institutions\nparticipating in the {\\it Gaia} Multilateral Agreement.\n\n\\facilities{CTIO:1.5m, SOAR}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}