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+{"text":"\\section{Introduction}\nThe notion of rectifying curve was introduced by Chen \\cite{BYC03} as a curve in the Euclidean space such that its position vector always lies in the rectifying plane, and then investigated some properties of such curves. For further properties of rectifying curves, the reader can be consulted \\cite{BYC05} and \\cite{BYC18}. Again Ilarslan and Nesovic \\cite{IN08} studied the rectifying curves in Minkowski space and obtained some of its characterization. \n\\par\nIn \\cite{CKI18} Camci et. al associated a frame different from Frenet frame to curves on a surface and deduced some characterization of its position vector. In \\cite{PRG18A} and \\cite{PRG18B} the present authors studied rectifying and osculating curves and obtained some conditions for the invariancy of such curves under isometry. Also the invariancy of the component of position vector of rectifying and osculating curves along the normal and tangent line to the surface are obtained under isometry of surfaces.\n\\par \nMotivating by the above studies of curves whose position vectors are confined in some plane, in this paper we have investigated curves on a smooth surface with position vector always lying in the tangent plane of the smooth surface. By using the Gauss equation we have deduced the component of the position vector along the tangent, normal and binormal vector in simple form. By considering isometry between two smooth surfaces it is proved that curves on smooth surface whose position vector lies in the tangent plane are invariant. It is also shown that the length of position vector, tangential component and geodesic curvature of such curves are invariant under isomerty.\n\\section{Preliminaries}\nThis section is concerned with some preliminary notions of rectifying curves, osculating curves, isometry of surfaces and geodesic curvature (for details see, \\cite{AP01}, \\cite{MPDC76 }) which will be needed for the remaining.\n\\par\nAt every point of an unit speed parametrized curve $\\gamma(s)$ with atleast fourth order continuous derivative, there is an orthonormal frame of three vectors, namely, tangent, normal and binormal vectors. Tangent, normal and binormal vectors are denoted by $\\vec{t}$, $\\vec{n}$ and $\\vec{b}$. They are related by the Serret-Frenet equation given as\n\\begin{eqnarray}\n\\nonumber\nt'(s)&=&\\kappa \\ n(s),\\\\\n\\nonumber\nn'(s) &=& -\\kappa \\ t(s) + \\tau \\ b(s),\\\\\n\\nonumber\nb'(s)&=&-\\tau \\ n(s),\n\\end{eqnarray}\nwhere $\\kappa$ and $\\tau$ are respectively the curvature and torsion of $\\gamma(s)$. Rectifying, osculating and normal plane is generated by $\\{\\vec{t},\\vec{b}\\}$, $\\{\\vec{t},\\vec{n}\\}$ and $\\{\\vec{n},\\vec{b}\\}$ respectively. Curves whose position vector contained in rectifying, osculating and normal plane are respectively called rectifying, osculating and normal curves. \n\\begin{defn}\nLet $S$ and $\\bar{S}$ be smooth surfaces immersed in $\\mathbb{R}^3$. Then a diffeomorphism $f:S\\rightarrow\\bar{S}$ is called an isometry if the length of any curve on $S$ is invariant under $f$.\n\\end{defn}\n\\begin{defn}\nSuppose $\\gamma(s)$ is any unit speed parametrized curve on a smooth surface $S$. Then the tangent vector $\\gamma'(s)$ and the normal $\\vec{N}$ to the surface are mutually orthogonal and also $\\gamma''(s)$ and $\\gamma'(s)$ are orthogonal. Hence $\\gamma''(s)$ is represented by the the linear combination of $\\vec{N}\\times\\gamma'(s)$ and $\\vec{N}$ as\n\\begin{equation*}\n\\gamma''(s)=\\kappa_g\\vec{N}\\times\\gamma'(s)+\\kappa_n\\vec{N}.\n\\end{equation*}\nThen $\\kappa_g$ and $\\kappa_n$ are respectively called the geodesic curvature and normal curvature of $\\gamma(s)$ on $S$ given by the following:\n\\begin{eqnarray}\n\\nonumber\n\\kappa_g&=&\\gamma''\\cdot(\\vec{N}\\times\\gamma'),\\\\\n\\nonumber\n\\kappa_n&=&\\gamma''\\cdot\\vec{N}.\n\\end{eqnarray}\n\\end{defn}\n\\section{Curves on a surface whose position vector lies in the tangent plane}\nLet $S$ be a smooth surface and $\\phi$ be a surface patch at any point $p\\in S$. Let $\\gamma(s)$ be an unit speed parametrized curve in $\\phi$ passing through $p\\in S$. Then the tangent space of $S$ at $p$ is generated by two linearly independent vectors $\\phi_u$ and $\\phi_v$, where $\\phi_u=\\frac{\\partial \\phi}{\\partial u}$ and $\\phi_v=\\frac{\\partial \\phi}{\\partial v}$. If $\\gamma(s)$ be a curve on $S$ whose position vector lies in $T_{\\gamma(s)}S$ then the equation of $\\gamma(s)$ is given by\n\\begin{equation}\\label{sc2}\n\\gamma(s)=\\lambda(s)\\phi_u+\\mu(s)\\phi_v,\n\\end{equation}\nwhere $\\lambda(s)$ and $\\mu(s)$ are two functions of $s$.\\\\\nDifferentiating equation $(\\ref{sc2})$ we get\n\\begin{equation*}\n\\gamma'(s)=\\lambda'\\phi_u+\\mu'\\phi_v+\\lambda (u'\\phi_{uu}+v'\\phi_{uv})+\\mu(u'\\phi_{uv}+v'\\phi_{vv}).\n\\end{equation*}\nThe Gauss Equation for the surface patch $\\phi$ of $S$ with normal vector $\\vec{N}$ is given by\n\n\\begin{equation}\\label{sc3}\n\\begin{cases}\n\\phi_{uu}&=\\Gamma_{11}^1\\phi_u+\\Gamma_{11}^2\\phi_v+L\\vec{N},\\\\\n\\phi_{uv}&=\\Gamma_{12}^1\\phi_u+\\Gamma_{12}^2\\phi_v+M\\vec{N},\\\\\n\\phi_{vv}&=\\Gamma_{22}^1\\phi_u+\\Gamma_{22}^2\\phi_v+N\\vec{N},\\\\\n\\end{cases}\n\\end{equation}\nwhere $\\{L, \\ M, \\ N\\}$ are the coefficients of the second fundamental form of $S$, and the Christoffel symbols $\\Gamma_{ij}^k$ are given by \n\\begin{eqnarray}\n\\nonumber\n\\Gamma_{11}^1=\\frac{E_uG-2F_uF+E_vF}{2(EG-F^2)},&& \\ \\Gamma_{11}^2=\\frac{2F_uE-E_vE-E_uF}{2(EG-F^2)},\\\\\n\\nonumber\n\\Gamma_{12}^1=\\frac{E_vG-G_uF}{2(EG-F^2)},&& \\ \\Gamma_{12}^2=\\frac{G_uE-E_vF}{2(EG-F^2)},\\\\\n\\nonumber\n\\Gamma_{11}^1=\\frac{2F_vG-G_uF-G_uG}{2(EG-F^2)},&& \\ \\Gamma_{11}^2=\\frac{G_vE-2F_vF+G_uF}{2(EG-F^2)},\n\\end{eqnarray}\nwhere $\\{E, \\ F, \\ G\\}$ are the coefficients of the second fundamental form of $S$.\nUsing equation $(\\ref{sc3})$ in equation $(\\ref{sc2})$ we get\n\\begin{eqnarray}\n\\nonumber\n\\vec{t}=\\gamma'(s)=&&\\{\\lambda'+u'\\lambda\\Gamma_{11}^1+(v'\\lambda+u'\\mu) \\Gamma_{12}^1+v'\\mu\\Gamma_{22}^1\\}\\phi_u+\\{\\mu'+u'\\lambda\\Gamma_{11}^2+(v'\\lambda+u'\\mu)\\Gamma_{12}^2\\\\\n\\nonumber\n&&+v'\\mu\\Gamma_{22}^2\\}\\phi_v+\\{u'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N\\}\\vec{N},\n\\end{eqnarray}\ni.e., $$\\vec{t}=A_1\\phi_u+A_2\\phi_V+A_3\\vec{N},$$ where $A_1$, $A_2$ and $A_3$ are respectively given by \n\n\\begin{equation}\\label{sc4}\n\\begin{cases}\nA_1=&\\lambda'+u'\\lambda\\Gamma_{11}^1+(v'\\lambda+u'\\mu) \\Gamma_{12}^1+v'\\mu\\Gamma_{22}^1,\\\\\nA_2=&\\mu'+u'\\lambda\\Gamma_{11}^2+(v'\\lambda+u'\\mu)\\Gamma_{12}^2+v'\\mu\\Gamma_{22}^2,\\\\\nA_3=& u'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N.\n\\end{cases}\n\\end{equation}\n\n\\par\nBut the tangent plane is generated by $\\phi_u$ and $\\phi_v$, hence\n\\begin{eqnarray}\n\\nonumber\nu'\\lambda L+v'\\lambda M+u'\\mu M+v'\\mu N=0,\\\\\n\\nonumber\\frac{\\lambda}{\\mu}=-\\frac{u'L+v'M}{u'M+v'N}.\n\\end{eqnarray}\nSo the tangent vector of $\\gamma(s)$ is given by\n\\begin{equation*}\n\\vec{t}=A_1\\phi_u+A_2\\phi_v.\n\\end{equation*}\nHere we consider that the curvature $\\kappa$ of $\\gamma$ is always positive. The normal vector $\\vec{n}$ is given by\n\\begin{eqnarray}\n\\nonumber\n\\vec{n}&=&\\frac{1}{\\kappa}[A_1'\\phi_u+A_2'\\phi_v+A_1(\\phi_{uu}u'+\\phi_{uv}v')+A_2(\\phi_{uv}u'+\\phi_{vv}v')],\\\\\n\\nonumber\n&=& \\frac{1}{\\kappa}\\{B_1\\phi_u+B_2\\phi_v+B_3\\vec{N}\\},\n\\end{eqnarray}\nwhere $B_1,\\ B_2$ and $ B_3$ are respectively given by\n\n\\begin{equation}\\label{sc5}\n\\begin{cases}\nB_1&= A_1'+u'A_1\\Gamma_{11}^1+(v'A_1+u'A_2)\\Gamma_{12}^1+v'\\Gamma_{22}^1A_2,\\\\\nB_2&= A_2'+u'A_1\\Gamma_{11}^2+(v'A_1+u'A_2)\\Gamma_{12}^2+v'\\Gamma_{22}^2A_2,\\\\\nB_3&= u'A_1L+(v'A_1+u'A_2)M+v'A_2N.\n\\end{cases}\n\\end{equation}\nNow the binormal vector $\\vec{b}$ of $\\gamma(s)$ is given by \n\\begin{equation}\\label{sc6}\n\\vec{b}=\\frac{1}{\\kappa}\\{ (A_1B_2-A_2B_1)\\vec{N}+A_1B_3(F\\phi_u-E\\phi_v)+A_2B_3(G\\phi_u-F\\phi_v)\\}.\n\\end{equation}\\\\\n\\begin{thm}\nLet $\\gamma(s)$ be an unit speed parametrized curve on $S$ whose position vector lies in the tangent plane $T_{\\gamma(s)}S$. Then the following statements hold:\n\\par\n$(i)$ The distance function $\\rho=\\|\\gamma\\|$ is given by $\\rho=\\lambda^2E+2\\lambda\\mu F+\\mu^2G$.\\\\\n\n$(ii)$ The tangential component of the position vector of the curve $\\gamma(s)$ is given by $$\\langle \\vec{t},\\gamma \\rangle=\\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G.$$ \n\n$(iii)$ The normal component of the position vector is given by $$\\langle \\vec{n},\\gamma \\rangle=\\frac{1}{\\kappa}(\\lambda B_1E+(\\lambda B_2+\\mu B_1)F+\\mu B_2G).$$\n\n$(iv)$ The component of the position vector along binormal vector is given by $$\\langle \\vec{b},\\gamma \\rangle = \\frac{1}{\\kappa}\\{A_1B_3\\mu( F^2-EG)+A_2B_3\\lambda( EG- F^2)\\},$$\nwhere $\\{A_1,\\ A_2,\\ A_3\\}$ and $\\{B_1,\\ B_2,\\ B_3\\}$ are described in equations $(\\ref{sc4})$ and $(\\ref{sc5})$ respectively.\\\\\n\\end{thm}\n\\begin{proof}\nLet $\\gamma(s)$ be an unit speed parametrized curve on $S$ whose position vector lies in the tangent plane $T_{\\gamma(s)}S$ and curvature $\\kappa>0$. Then $$\\gamma(s)=\\lambda\\phi_u+\\mu\\phi_v.$$ Therefore\n\\begin{eqnarray}\n\\nonumber\n\\rho=\\langle\\gamma,\\gamma\\rangle&=&\\langle\\lambda\\phi_u+\\mu\\phi_v,\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda^2\\langle\\phi_u,\\phi_u\\rangle+\\lambda\\mu\\langle\\phi_u,\\phi_v\\rangle+\\lambda\\mu\\langle\\phi_v,\\phi_u\\rangle+\\mu\\langle\\phi_v,\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda^2E+2\\lambda\\mu F+\\mu^2G.\n\\end{eqnarray}\nWhich proves $(i)$.\\par\nThe component of $\\gamma$ along the tangent vector is obtained as\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{t},\\gamma\\rangle&=&\\langle A_1\\phi_u+A_2\\phi_v,\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda A_1\\langle\\phi_u,\\phi_u\\rangle+\\lambda A_2\\langle\\phi_u,\\phi_v\\rangle+\\mu A_1\\langle\\phi_v,\\phi_u\\rangle+\\mu A_2\\langle\\phi_v,\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G,\n\\end{eqnarray}\nwhere $A_1,\\ A_2,\\ A_3$ are given in equation $(\\ref{sc4})$. This proves $(ii)$.\\par\nWe also find the component of $\\gamma$ along the normal vector, which is given by\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{n},\\gamma\\rangle&=&\\langle \\frac{1}{\\kappa}\\{B_1\\phi_u+B_2\\phi_v+B_3\\vec{N},\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=& \\frac{1}{\\kappa}\\{\\lambda B_1\\langle\\phi_u,\\phi_u\\rangle+\\lambda B_2\\langle\\phi_u,\\phi_v\\rangle+\\mu B_1\\langle\\phi_v,\\phi_u\\rangle+\\mu B_2\\langle\\phi_v,\\phi_v\\rangle\\},\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}(\\lambda B_1E+(\\lambda B_2+\\mu B_1)F+\\mu B_2G),\n\\end{eqnarray}\nwhere $B_1,\\ B_2,\\ B_3$ are given in equation $(\\ref{sc5})$. Hence statement $(iii)$ is proved.\\par\nAgain the component of $\\gamma$ along the binormal vector is given by\n\\begin{eqnarray}\n\\nonumber\n\\langle\\vec{b},\\gamma\\rangle&=&\\langle \\frac{1}{\\kappa}\\{(A_1B_2-A_2B_1)\\vec{N}+A_1B_3(F\\phi_u-E\\phi_v)+A_2B_3(G\\phi_u-F\\phi_v)\\},\\lambda\\phi_u+\\mu\\phi_v\\rangle,\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}\\{A_1B_3(F\\lambda\\langle\\phi_u,\\phi_u\\rangle+F\\mu\\langle\\phi_u,\\phi_v\\rangle-E\\lambda\\langle\\phi_u,\\phi_v\\rangle-E\\mu\\langle\\phi_v,\\phi_v\\rangle)+A_2B_3(G\\lambda\\langle\\phi_u,\\phi_u\\rangle\\\\\n\\nonumber\n&&-F\\lambda\\langle\\phi_v,\\phi_u\\rangle+G\\mu\\langle\\phi_u,\\phi_u\\rangle-F\\mu\\langle\\phi_v,\\phi_v\\rangle)\\},\\\\\n\\nonumber\n&=&\\frac{1}{\\kappa}\\{A_1B_3\\mu( F^2-EG)+A_2B_3\\lambda( EG- F^2)\\},\n\\end{eqnarray}\nwhere $A_1,\\ A_2,\\ A_3,\\ B_1,\\ B_2,\\ B_3$ are given in equation $(\\ref{sc4})$ and $(\\ref{sc5})$. Thus statement $(iv)$ is proved.\n\\end{proof}\nNow suppose that $\\bar{S}$ is an another surface isometric to $S$ and $f:S\\rightarrow\\bar{S}$ is the isometry. Then the curve $f\\circ\\gamma(s)$ in $\\bar{S}$ is expressed as \n\\begin{eqnarray}\\label{sc7}\n\\nonumber\n\\bar{\\gamma}(s)&=& f\\circ\\gamma(s)=f(\\lambda\\phi_u+\\mu\\phi_v),\\\\\n\\nonumber\n&=&\\lambda f_*(\\phi_u)+\\mu f_*(\\phi_v),\\\\\n&=&\\lambda(\\bar{\\phi}_u)+\\mu(\\bar{\\phi}_v),\n\\end{eqnarray}\nfor some smooth functions $\\lambda$ and $\\mu$. The position vector of $\\bar{\\gamma}$ lies in $T_{f(p)}\\bar{S}$. Hence the isometry $f$ transforms a curve on a surface with position vector in the tangent plane to the same and $\\lambda,\\ \\mu$ does not change under $f$. Hence\n\\begin{eqnarray}\n\\nonumber\n\\frac{u'L+v'M}{u'M+v'N}=\\frac{u'\\bar{L}+v'\\bar{M}}{u'\\bar{M}+v'\\bar{N}},\\\\\n\\nonumber\nu'^2(L\\bar{M}-\\bar{L}M)+v'^2(M\\bar{N}-\\bar{M}N)+u'v'(L\\bar{N}-\\bar{L}N)=0,\n\\end{eqnarray} \nwhere $\\{L, \\ M, \\ N\\}$ and $\\{\\bar{L}, \\ \\bar{M}, \\ \\bar{N}\\}$ are the coefficients of the second fundamental form of $S$ and $\\bar{S}$ respectively.\n\\begin{thm}\nLet $f:S\\rightarrow\\bar{S}$ be an isometry and the position vector of two curves $\\gamma(s)$ and $\\bar{\\gamma}(s)$ lies in $T_pS$ and $T_{f(p)\\bar{S}}$ respectively. Then the following statements hold:\n\\par\n$(i)$ The distance function $\\rho=\\|\\gamma\\|$ is invariant under the isometry. i.e., $\\gamma(s)=\\bar{\\gamma}(s)$.\n\n$(ii)$ The tangential component of the position vector of the curve $\\gamma(s)$ is invariant under the isometry $f$. i.e., $\\langle \\vec{t},\\gamma \\rangle=\\langle \\vec{\\bar{t}},\\bar{\\gamma} \\rangle$. \n\n$(iii)$ \nThe geodesic curvature of $\\gamma$ is invariant under the isometry $f$.\n\\end{thm}\n\\begin{proof}\nLet $f:S\\rightarrow\\bar{S}$ be an isometry. Since $\\phi(u,v)$ is a surface patch for the surface $S$ at $p$, hence $\\bar{\\phi}(u,v)=f\\circ\\phi(u,v)$ is also a surface patch for $\\bar{S}$   . Suppose $\\{E,F,G\\}$ and $\\{\\bar{E},\\bar{F},\\bar{G}\\}$ are the coefficients of first fundamental forms of $\\phi$ and $\\bar{\\phi}$ respectively. Then we have\n\\begin{equation}\\label{i1}\n\\bar{E}=E,\\quad \\bar{F}=F\\quad \\text{and}\\quad \\bar{G}=G.\n\\end{equation} \nDifferentiating first relation of equation $(\\ref{i1})$ we get\n\\begin{equation}\\label{i2}\n\\bar{E}_u=\\frac{\\partial}{\\partial u}(\\bar{\\phi}_u\\cdot\\bar{\\phi}_u)=\\frac{\\partial}{\\partial u}(\\phi_u\\cdot\\phi_u)=E_u.\n\\end{equation}\nSimilarly \n\\begin{equation}\\label{i3}\n\\begin{cases}\n\\bar{F}_u=F_u,\\qquad \\bar{G}_u=G_u,\\qquad \\bar{E}_v=E_v,\\qquad\\\\ \\bar{F}_v=F_v,\\qquad and \\qquad \\bar{G}_v=G_v.\n\\end{cases}\n\\end{equation}\\\\\nNow using $(\\ref{i1})$ we have  \n\\begin{eqnarray}\n\\nonumber\n\\rho&=\\lambda^2E+2\\lambda\\mu F+\\mu^2G\\\\\n\\nonumber\n&=\\lambda^2\\bar{E}+2\\lambda\\mu \\bar{F}+\\mu^2\\bar{G}\\\\\n\\nonumber\n&=\\bar{\\rho}.\n\\end{eqnarray} \nHence the statement $(i)$ is proved.\\par\nSince $\\bar{A_1}$ and $\\bar{A_2}$ are functions of $\\lambda$, $\\mu$ and Christoffel symbols, hence by virtue of the equations $(\\ref{i1}),\\ (\\ref{i2})$ and $(\\ref{i3})$ we see that $\\bar{A_1}$ and $\\bar{A_2}$ are invariant under the isometry $f$ . Thus\n\\begin{eqnarray}\n\\nonumber\n\\langle \\vec{\\bar{t}},\\bar{\\gamma} \\rangle&=&\\bar{\\lambda A_1}\\bar{E}+(\\lambda \\bar{A_2}+\\bar{\\mu} \\bar{A_1})\\bar{F}+\\bar{\\mu} \\bar{A_2}\\bar{G},\\\\\n\\nonumber\n&=& \\lambda A_1E+(\\lambda A_2+\\mu A_1)F+\\mu A_2G,\\\\\n\\nonumber\n&=& \\langle\\vec{t},\\gamma\\rangle.\n\\end{eqnarray}\nThis proves $(ii)$. \\par\nNow \n\\begin{eqnarray}\n\\nonumber\n\\kappa_g &=& \\gamma''\\cdot(\\vec{N}\\times\\gamma'),\\\\\n\\nonumber\n&=& \\gamma''\\{(\\phi_u\\times\\phi_v)\\times(A_1\\phi_u+A_2\\phi_v)\\},\\\\\n\\nonumber\n&=&\\gamma''\\cdot\\{A_1(E\\phi_v-F\\phi_u)+A_2(F\\phi_v-G\\phi_u)\\},\\\\\n\\nonumber\n&=&(B_1\\phi_u+B_2\\phi_v+B_3\\vec{N})\\cdot\\{A_1(E\\phi_v-F\\phi_u)+A_2(F\\phi_v-G\\phi_u)\\},\\\\\n\\nonumber\n&=&B_1A_1(EF-FE)+B_1A_2(F^2-EG)+B_2A_1(EG-F^2)+A_2B_2(FG-GF),\\\\\n\\nonumber\n&=&(B_1A_2-B_2A_1)(F^2-GE).\n\\end{eqnarray}\nSince $\\bar{\\gamma}$ is also a curve whose position vector lies in the tangent plane $T_{f(p)}\\bar{S}$ of $\\bar{S}$, hence $$\\bar{\\kappa}_g=(\\bar{B}_1\\bar{A}_2-\\bar{B}_2\\bar{A}_1)(\\bar{F}^2-\\bar{G}\\bar{E}).$$ \nSince $\\bar{B_1}$ and $\\bar{B_2}$ are the functions of $\\bar{A_1}$, $\\bar{A_2}$, $\\bar{A_1'}$, $\\bar{A_2'}$ and Christoffel symbols, so by using the equations $(\\ref{i1}),\\ (\\ref{i2})$ and $ (\\ref{i3})$ we can say that $\\bar{B_1}$ and $\\bar{B_2}$ are invariant under the isometry $f$. In view of $(\\ref{i1})$, the last equation yields $$\\bar{\\kappa}_g=\\kappa_g,$$\nwhich proves $(iii)$.\n\\end{proof}\n\n\\section{acknowledgment}\n The second author greatly acknowledges to The University Grants Commission, Government of India for the award of Junior Research Fellow.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nDespite the apparent cross-disciplinary interactions among scientific fields, a\nformal description of their evolution is lacking. Here we describe a novel\napproach to study the dynamics and evolution of scientific fields using a\nnetwork-based analysis. We build an {\\it idea} network consisting of American\nPhysical Society Physics and Astronomy Classification Scheme (PACS) numbers as\nnodes representing scientific concepts. Two PACS numbers are linked if there\nexist publications that reference them simultaneously. We locate scientific\nfields using a community finding algorithm, and describe\nthe time evolution of these fields over the course of 1985-2006. The communities we identify map\nto known scientific fields, and their age depends on their size and\nactivity. We expect our approach to quantifying the evolution of ideas to be\nrelevant for making predictions about the future of science and thus help to\nguide its development. \n\n\\section*{Introduction}\n\nCross-fertilization between different scientific fields has been recognized for\nits ability to encourage new developments and innovative thinking. For this\nreason, multidisciplinary approaches to research are becoming more popular.\nSome recent examples include applying physics techniques to the study of\nbiological phenomena \\cite{frauenfelder}, deriving an understanding of the\nnature of critical phenomena from renormalization techniques in particle\nphysics \\cite{wilson} drawing inferences about the early universe from findings\nin terrestrial superfluid experiments \\cite{zurek}, and using statistical\nphysics to analyze technological and social systems \\cite{dorogovtsev}. \n\nIn an effort to move beyond anecdotal evidence of the benefit of\ninterdisciplinary discourse for science, in this paper we study the dynamics of\ngroups, or ``communities\", of ideas using a statistical physics approach.  We\nattempt to quantify the evolution of ideas and subdisciplines within physics as\nthey emerge, interact, merge, stagnate, and desist. The quest for describing\nthe development of scientific fields is not new. There have been\nepidemiological \\cite{goffman, tabah} and network-based approaches (citation\nand collaboration networks)\n~\\cite{price,newmancoauth1,newmancoauth2,newmancoauth3,lehmann,borner,leydesdorff,bollen1,boyack} aiming\nto gain insight into the spread of scientific ideas. Recently the temporal\nevolution of several scientific disciplines have been modeled with a\ncoarse-grained approach \\cite{luis}. \n\nHere we build a scientific concept network consisting of American Physical\nSociety PACS numbers as nodes representing scientific concepts.   The American Institute of Physics (AIP) develops and maintains the PACS scheme as a service to the physics community in aiding the\nclassification of scientific literature and information retrieval.  Two PACS numbers are linked if there\nexist publications that reference them simultaneously. Our approach differs\nfrom previous methods in that it provides a direct, unsupervised description of\nscientific fields and uses techniques such as community finding and tracking\nfrom the field of network physics. This approach provides means to quantify how\nideas and movements in science appear and fade away.  Because this method makes\nit possible to measure the current and past state of the relationship between\nscientific concepts, it may also help to make predictions about the future of\nscience and thus inform efforts to guide its development. In this paper, we\nentertain some of the quantitative questions that this method permits;\nspecifically, we seek to answer questions about the relationship between size,\nlifetime, and activity of scientific fields.  \n\n\nVarious local to global topological measures have been introduced to unveil the\norganizational principles of complex networks\n\\cite{albertrev,newmanbook,caldarelli}. One such measure that allows the\ndiscovery of organizational principles of networks is community finding.  There\nhave been a number of methods to find the communities in networks which\ndescribe the inherent structure or functional units of a network\n\\cite{newmangirvan, palla1, clauset, gulbahce}. One of these is CFinder, a\nclique percolation method (CPM) introduced by Palla et al. \\cite{palla1}, which finds\noverlapping communities and is especially suitable for studying the evolution\nof scientific fields since scientific concepts are often shared among multiple\nfields. We use this CPM to track the evolution of physics.\n\n\\section*{Results}\n\n\\noindent{\\bf Building the Network} \n\n\nData were collected from the American Physical Society's (APS) {\\it Physical\nReview} database from 1977-2007. Journals included in the study are {\\it\nPhysical Review Letters}, {\\it Physical Review \\{A} through {\\it E\\}}, and {\\it\nPhysical Review Special Topics: Accelerators and Beams}. Papers in this\ndatabase contain a list of author-assigned PACS codes, where each PACS code\nrefers to a specific topic in physics.  PACS itself is hierarchical, which is\nevident in the structure of the codes with up to 5 levels of topic\nspecification.  For example the PACS code `64.60.aq' has 5 levels where the\nfirst digit `6' represents the first level (in this case `condensed matter'),\n`4' represents the second (e.g. `equations of state, phase equilibria, and\nphase transitions'), the third and fourth digits `60' together represent the\nthird level (e.g., `general studies of phase transitions') while the last two\ncharacters `aq' carry information pertaining to the fourth and fifth levels of\nspecification (e.g. `specific approaches applied to phase transitions' and\n`networks', respectively).  \n\n\nPACS codes are not static, rather,  the coding scheme is periodically updated with the addition and deletion of codes.  In order to (at least partially) account for this effect, the scientific concept network was constructed such that the nodes in the network represent individual PACS codes using the first four\ndigits of specification, where changes to scheme are less probable. This network and the related material is available on\nour website \\cite{datalink}.  In our network, an {\\it edge} occurs between two\nnodes if the two PACS codes they represent are cited in the same paper; one\npaper in the database often contributes many nodes and edges to the\nnetwork. Furthermore, edges are weighted by the number of papers that contain\nthat edge. We introduce two measures, node and edge cutoffs, to control for noise \nin the network (see Methods section).\n\nThe entire PACS network from 1977-2007 after both noise measurements were\napplied has 803 nodes and 23707 distinct edges.  The \\emph{degree} of a node is\nthe number of edges shared by the node.  The weighted cumulative degree\ndistribution follows a stretched exponential with the form,  $P(k)\\sim\\\n\\mbox{exp} [-(k\/842)^{0.53}]$ as shown in Fig. \\ref{DegreeFig}A. The\ndistribution has a similar form in the unweighted case.  The dynamic\nclassification scheme of the American Physical Society, implemented by the\naddition, splitting and removal of codes, may be preventing the formation of\nlarge hubs, thus keeping the specification of the codes more useful. The\nstretched exponential distribution may be the result of a sublinear-linear\nattachment type growth \\cite{krapivsky}.\n\n\nThe PACS network also exhibits a weak but apparent hierarchical structure\nmeasured by the dependence of the \\emph{clustering coefficient} on (unweighted)\ndegree.  For a node $i$, the clustering coefficient is given as $C_i =\n2n_i\/k_i(k_i-1)$, where $n_i$ is the number of edges that link the neighbors of\nnode $i$, and $k_i$ is the degree of the node. The clustering coefficient for a\nnode is the ratio of the number of triangles through node $i$ over the possible\nnumber of triangles that could pass through node $i$ \\cite{barabasi}.  A purely\nhierarchical network will have a $\\langle C \\rangle$ that scales as a power of\n$k$, $\\langle C \\rangle \\sim k^{-1}$, while a random network will have a\nclustering coefficient that is constant with $k$ \\cite{barabasi}.  For this\nnetwork,  $\\langle C(k) \\rangle \\sim k^{-0.29}$ , shown in Fig. \n\\ref{DegreeFig}B. This dependence is not surprising given the hierarchical\nstructure of the classification scheme.\n\n\\noindent{\\bf Defining Communities in Physics} \n\n\nPapers published between 1985 and 2006 were used to study the community\nevolution of the network; 1985 appears to be the first year when all journals present ({\\it Physical Review E} began publication in 1993)\nconsistently used the PACS data scheme, and 2007 was thrown out to exclude\nincomplete data from the analysis. The journal {\\it Physical Review Special\nTopics: Accelerators and Beams} was not included because of an irregular\npublishing schedule. After the noise measures were carried out, the edge\nweights were no longer used, and the network became an unweighted network with\nrespect to the community evolution analysis. The data were organized into 44\ntime bins, with each bin representing a 0.5 year time period. Once a paper (and\nthe edges and nodes it contains) appears in the analysis, it is assigned a\nlifetime, $l$, of 0 or 2.5 years. This assignment is an attempt to more\nrealistically capture the nature of scientific dissemination, as well as the\ndelay in time from publication to assimilation by the field. The analysis of\ncommunity evolution begins at the time bin subsequent to the lapse of the\nassigned lifetime.  Thus the first time bin, $t=0$, for a paper lifetime of\n$l=2.5$ refers to the latter half of 1987 since we start the analysis in 1985.  \n\nIn order to study the evolution of different fields in physics, one must first\nfind these fields in our network. We hypothesize that scientific fields are\nrepresented by communities in our PACS network. These communities are found\nusing the CFinder algorithm, which is based on a clique percolation\nmethod\\cite{palla1}.  Figs. \\ref{comdiag} and \\ref{comdiaga}  present\nexamples of the community structure extracted utilizing CFinder.  \n\nFor each community, the code (using only the first two digits) that encompasses\nthe largest fraction of nodes in the community was found. Its name, specified by\nthe PACS scheme, is then used to label the community.  If a community has\nmultiple codes which compose the same largest fraction of nodes in that\ncommunity, then the community is assigned multiple labels. As shown in\nFigs.~\\ref{comdiag} and ~\\ref{comdiaga}, we observe that the analysis captures \nexpected scientific connections among fields in physics.  For example, in 1997, particle physics is linked to both general relativity and astrophysics.  It is also worthwhile to note the emergence of biophysics as a community in the 2005 analysis.  \\\\ \n\n\\noindent{\\bf Community Evolution and Dynamics} \n\n\nIn order to track the evolution of scientific fields, after identifying\ncommunities at each individual time interval, it is necessary to match the\ncommunities between adjacent time steps. We implemented a community evolution\nalgorithm developed by Palla et al.~\\cite{palla2} to match the communities\nbetween time bins (see Methods section).\n\nTo gain a better understanding of the dynamics of evolving communities, we\ndefined two properties of each community: size and activity. A value for each of\nthese measures can be assigned to every community for each individual time bin.\nThe size $s$ of a community is the number of nodes contained within that\ncommunity at time $t$. Size can be interpreted as a measure of a community's\nbreadth: communities with a small size encompass only a few distinct ideas,\nwhile large communities encompass many distinct ideas.  (The cumulative size distribution was calculated for different times and is displayed in Fig. S1.)\n\n\nThe activity $\\alpha$ of a community is defined as the number of papers that\ncontain at least one node from that community at time $t$. As one expects,\nthere is a strong correlation between size and activity (see Fig. S2).\n\nNext, we study the relationship between the age or lifetime of a community\nversus its size and activity. The age of a community at time $t$ is simply the\nnumber of time bins the community has been present in the evolution analysis:\n$\\tau=t-t_0+1$, where $t_0$ is the time bin in which the community was born. In\norder to study the dependence of age on size, in each time bin, the current age\n$\\tau$ and size $s$ are recorded. Using all communities from all time\nintervals, the median age is calculated for communities with the same size as\nshown in Fig.  \\ref{fig2}A.  There is a trend of $\\tau$ increasing with\nsize $s$.  Thus, it would appear that older communities tend to contain more\nnodes, and that longer lived fields tend to encompass many distinct ideas.   Values for both the Pearson correlation coefficient, $p$, and the Spearman's rank correlation coefficient, $\\rho$, were calculated between $\\tau$ and $s$ using the raw, unbinned data.  $\\rho=1-6\\sum_i\\frac{d_i^2}{N(N^2-1)}$ where $N$ is the number of data points and $d_i$ is the difference in the statistical rank of the corresponding values for each data point.  For $l=2.5$, the Pearson correlation coefficient was $p=0.4772$ while the  Spearman's $\\rho$ was calculated to be $ \\rho=0.5913$.\n\n\nIn order to measure the dependence of age on activity, the current age $\\tau$\nis recorded along with the current activity $\\alpha$ of every community in each\ntime step. Because of the wide range of possible values for activity and noise\nin the data, the values of $\\alpha$  are sorted into 100 equally sized bins.\nThe median age is calculated for all communities within the same activity\ninterval. There is a trend of $\\tau$ increasing with activity as shown in\nFig.  \\ref{fig2}B which can be partially understood by the strong correlation\nbetween size and activity. Further we note an apparent phase transition in\nactivity; as shown in Fig.  \\ref{fig2}B after some critical value, communities\ntend to be longer lived.   This transition also appears for $l=0$ (see Fig. S3).  Lifetime as a function of size, $\\tau(s)$, for $l=0$ is shown in Fig. S4. Again, the Pearson correlation coefficient and the Spearman's rank correlation coefficient were calculated for $l=2.5$ using the raw, unbinned data between $\\tau$ and $\\alpha$, with $p=0.3283$ and $\\rho=0.3764$. \n\n\n\\section*{Discussion}\n\n\nIn this paper, we have developed an approach that enables the quantitative\nstudy of the evolution of physics fields, specifically by following the\ndynamical connections between various ideas within physics. From our\ninvestigation, we have shown that long lived communities tend to be larger, and\nare associated with a higher number of papers. \n\nOur approach opens up an interesting possibility of being able to predict\ncommunity dynamics and impact from the current network structure. Furthermore,\nthis method can be easily adapted to other scientific fields using different\ndatabases. One such is the INSPEC database which has comprehensive coverage of\nresearch activity in computer science and engineering in addition to physics,\nand has an expert-assigned classification scheme rather than author-based\nassignments. \n\n\n\\section*{Materials and Methods}\n\\subsection*{Noise Measures }\nA node cutoff is introduced such that in a given time interval a node must\nappear at least twice to be included in the network. This measure eliminates\nmany of the typographical errors occurring in the database. The edge cutoff,\nhowever, takes into account the random expectation of two PACS codes\nco-occurring in the same paper. For this cutoff, the weight of an edge between\nnodes $i$ and $j$, $W_{ij}$, which is the number of papers that both codes $i$\nand $j$ appear in, is compared to the weight expected at random, $E_{ij}=n_i\nn_j\/N$, where $n_i$ and $n_j$ are the number of papers containing nodes $i$ and\n$j$ respectively, and $N$ is the total number of papers present in the time\ninterval.   If $W_{ij}\/E_{ij} > 1.2$, then the appearance of the edge is\nsignificant compared to random appearance, and we include it in the network.\\\\ \n\\subsection*{CFinder} \nThe CFinder algorithm is described in detail in Ref.\n\\cite{palla1}.  A community is defined as a union of all $k$-cliques (complete\nsubgraphs of size $k$) that can be reached from each other through a series of\nadjacent $k$-cliques (where adjacency means sharing $k - 1$ nodes)\n\\cite{palla1}.  \\\\ \n\\subsection*{Picking a $k$ value}\nFor this study, $k=9$ was principally used (for l=2.5) because it appears to produce a\nlarge number of communities while discouraging the formation of giant\ncommunities. Further, by keeping $k$ constant, we keep the resolution constant\nfor the entire analysis. Picking an appropriate $k$ value for the analysis is\ndone by considering two properties: the number of communities present, and the\npresence of overly large communities \\cite{palla1}. It is desirable to have a\nlarge number of communities, so as to increase the statistical quality of\nmeasurements made on the network. Fig.  S6 plots the number of\npresent communities for each time step for  $k=8, 9$, and $10$, for $l=2.5$. As\ndemonstrated, the number of communities found using the choice of $k=10$ tends\nto be less than the other parameter choices, making it less favorable in terms\nof improving statistical quality.\n\nA $k$ value must also be large enough to avoid the introduction of overly large\ncommunities that obscure the actual community structure of the network\n\\cite{palla1}. To quantify this property, we use the quantity $r$ which is the\nratio of the size of the largest community to the second largest community for a\ngiven time bin.  Thus while some distribution in the sizes of communities is\nnecessary, $r$ should not be overly large.   Fig.  S7 plots the measure\n$r$ against all time bins for $l=2.5$.  For $k=8$, the values of $r$ tends to be larger than (signifying giant communities) than those calculated from the other two parameter values, making it an unfavorable parameter choice. \n \n\\subsection*{Community Matching}\nThe community matching algorithm is described in detail in Ref. \\cite{palla2}.\nIn this analysis, an appropriate $k$-value is used rather than a constant\nedge-weight cutoff.  A running stationarity measure is described in Appendix S1 and Figure S5.  The merger of two communities is described in Appendix S1 and Figures S8 and S9.\n\\section*{Acknowledgments} \nThe authors would like to thank the American Physical Society and the American\nInstitute of Physics for the use of their data. The authors would like to\nacknowledge Gergely Palla, Sune Lehmann, Albert-L\\'aszl\\'o Barab\\'asi, Tam\\'as\nVicsek, Aric Hagberg, Hristo Djidjev, Luis Bettencourt, and Michael Ham for\nuseful discussions. \n\n\n\\section{Community Dynamics}\nOnce community structure is established, a variety of different measurements are performed on the dynamics of the evolving communities.\n\nThe cumulative community size distribution appears long tailed over one decade, which is robust as a function of $t$ (years), as shown in Fig. S\\ref{cumulsize}.\n\nFig. S\\ref{svsa} plots for all time intervals the size of every\ncommunity against its activity $\\alpha$ for paper lifetime $l=2.5$.  There appears to be a positive correlation between the two measures, and this trend is observed for $l=0$ (not shown).\n\n\nThe dependence of age on activity was measured and Fig.~S\\ref{tauaval0} shows the results for the $\\tau$\nvs. $\\alpha$ measurements for $k=7$ with a paper lifetime of $l= 0$ years, where $\\alpha$ values were binned because of the wide\nrange in $\\alpha$, as well as to reduce noise.  In both cases  (the one presented here and the one presented in the letter) there is a trend of age increasing as activity increases (though less apparently for $l=0$) and, as expected given the correlation between $\\alpha$ and $s$, one sees a similar relationship between age and size, as shown in Fig. S\\ref{tausval0}.  Thus, older communities also tend to encompass more publications, a result that\nagrees with naive expectation. Further, we note an apparent  phase transition in\n both paper lifetime cases (more apparent for l=2.5 than for $l=0$); after some critical $\\alpha$, communities tend to be\nlonger lived. \n\n\n\nFurther understanding of the community dynamics can be gained by studying the\nvolatility of the evolving communities, a measure of how much communities tend\nto change between subsequent time steps.  To see this, we define an \\emph{age dependent}\nrunning stationarity $\\xi(\\tau)$ based on community correlation and stationarity\npresented by Palla et al.~\\cite{palla2}. The correlation $C(t,t')$ between two\nstates of the same community $A(t)$ at times $t$ and $t'$ is \n\\begin{equation*} \nC(t,t')=\\left|\\frac{A(t) \\cap A(t')}{A(t) \\cup A(t')}\\right|.\n\\end{equation*} \n\n\\noindent Then the running stationarity, $\\xi(\\tau)$, of that community is the\naverage correlation between subsequent time steps up to age $\\tau$, \n\\begin{equation*}\n\\xi(\\tau)=\\frac{1}{\\tau-1}\\sum_{t'=t_0}^{t_0+\\tau-2}C(t',t'+1). \n\\end{equation*}\n\nThe running stationarity, $\\xi(t)$, is plotted against lifetime for every\ncommunity with $\\tau>1$ along with its current age at time $t$, for all $t$,\nfor $l=0$ in Fig. S\\ref{run_statl0}. This result is qualitatively similar to results obtained using\nrandomized correlations.  For larger values of $l$, the distribution shifts to\nlarger values of $\\xi(t)$.\n\n\\section{Picking a $k$ value}\n\nThroughout the paper, $k=9$ is principally used (for l=2.5) because it appears to produce a\nlarge number of communities while discouraging the formation of giant\ncommunities. Further, by keeping $k$ constant, we keep the resolution constant\nfor the entire analysis. Picking an appropriate $k$ value for the analysis is\ndone by considering two properties: the number of communities present, and the\npresence of overly large communities \\cite{palla1}. It is desirable to have a\nlarge number of communities, so as to increase the statistical quality of\nmeasurements made on the network. Fig. S\\ref{commnum} plots the number of\npresent communities for each time step for  $k=8, 9$, and $10$, for $l=2.5$. As\ndemonstrated, the number of communities found using the choice of $k=10$ tends\nto be less than the other parameter choices, making it less favorable in terms\nof improving statistical quality.\n\nA $k$ value must also be large enough to avoid the introduction of overly large\ncommunities that obscure the actual community structure of the network\n\\cite{palla1}. To quantify this property, we use the quantity $r$ which is the\nratio of the size of the largest community to the second largest community for a\ngiven time bin.  Thus while some distribution in the sizes of communities is\nnecessary, $r$ should not be overly large.   Fig. S\\ref{rfig} plots the measure\n$r$ against all time bins for $l=2.5$.  For $k=8$, the values of $r$ tend to become larger (signifying giant communities) than those calculated\nfrom the other two parameter values, making it an unfavorable parameter choice. \\\\\n\n\\section{Merging of communities}\nLastly, we present an example of a merger between two communities.  Tracking\na nuclear physics community, Fig. S\\ref {ctrack} shows the size of\nthat community as a function of time for $k=9$ and a community of similar nodes with $k=10$, using $l=2.5$.\n\nWith $k=9$, it appears that this particle physics community abruptly dies at\n$t=4$ years.  Increasing the cohesiveness of communities by increasing to $k=10$\ndemonstrates that a community composed of similar nodes continues to propagate\npast this time of apparent death. Thus, it seems that the nuclear physics community is still present in the network, but has become absorbed by another\ncommunity.  \n\nFig. S\\ref{merge} plots a community at $t=4$ years with the nodes from\nthe nuclear physics community displayed in green.  We can assign a label\nto this community in the usual manner using the nodes present just before the\napparent death of the nuclear physics community. Doing so, the absorbing\ncommunity is comprised of the `physics of elementary particles and fields: specific reactions and phenomenology' in the time bin prior to its absorption of the particle physics community.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nTime reversal and inversion are two of the most important symmetries for the formation of Cooper pairs in a superconductor. Interesting new features of superconductivity arise when one or both of them are missing. One intriguing phase in the absence of time reversal symmetry is the so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state - a spatially modulated superconducting condensate induced by the spin-splitting of the Fermi surface due to high magnetic fields.~\\cite{PhysRev.135.A550,SovPhysJETP.20.762} This phase has been discussed in many contexts, for condensed matter systems~\\cite{Nature.425.51,Bianchi_FFLO} and ultra-cold atomic gases,~\\cite{Liao} as well as in nuclear physics.~\\cite{RevModPhys.76.263} The other interesting class is represented by superconductors without inversion center in the crystal lattice. These so-called non-centrosymmetric superconductors are also characterized by spin-splitting of the Fermi surface, in this case due to anti-symmetric spin-orbit coupling, and form Cooper pairs of mixed parity.~\\cite{Springer} Interestingly, both types of situations have been recently discussed in the context of Ce-based heavy Fermion superconductors. A phase observed in CeCoIn$_5$ at low-temperatures and rather high magnetic fields is considered a possible candidate for an FFLO state.~\\cite{Nature.425.51,Bianchi_FFLO,JPSJ.76.051005,kumagai2011} Most striking properties of unusually high upper critical fields have been reported for CeIrSi$_3$ and CeRhSi$_3$ which both have a non-centrosymmetric crystal lattice.~\\cite{Springer} In our study we would like to consider a combined effect of spin-splitting through magnetic fields and spin-orbit coupling in an artificially structured heavy Fermion superconductor based on CeCoIn$_5$. \n\n\nMizukami and his coworkers succeeded recently in fabricating superlattices of \nCeCoIn$_5$ and YbCoIn$_5$, the former a heavy Fermion and the latter an ordinary metal.~\\cite{Nat.Phys.7.849} Surprisingly this system is superconducting, if the number of CeCoIn$_5$ layers within a unit cell exceeds two. In view of the fact that the upper critical field remains amazingly high despite the low transition temperature,~\\cite{Nat.Phys.7.849} the superconductivity in the superlattice is \nrobust against paramagnetic depairing effect. \nIt was proposed that the superstructure of this material would have locally \nnon-centrosymmetric features while having global inversion centers and may \nlead to unusual properties of the superconducting phase.~\\cite{Maruyama-Sigrist-Yanase} \nActually the multilayered structure would induce a spatially dependent Rashba-type of \nspin-orbit coupling~\\cite{JPSJ.79.084701} which would influence the spin structure of \nthe electronic bands and suppress the paramagnetic depairing effect.~\\cite{Maruyama-Sigrist-Yanase} \nRecent measurement of the angular variation of the upper critical field found evidence \nfor such a behavior.~\\cite{Goh}\nIn this study, we investigate an unusual form of the superconducting order parameter induced by \nthe modulated Rashba spin-orbit coupling in the magnetic field. \nAs it will turn out to involve a modulation on the length scale of lattice constant, \nthe phase may be viewed as a pair-density wave (PDW) state. \nHere we will restrict to models of bi- and tri-layer systems as depicted schematically in Fig.~\\ref{fig:fig1}.\n\\begin{figure}[htbp]\n  \\includegraphics[width=85mm]{fig1.eps}\n  \\caption{(Color online) Schematic figures of (a) bi- and (b) tri-layer system.\n            Thick bars describe the two dimensional conducting planes \n            which are coupled with each other through the inter-layer coupling $t_{\\perp}$. \n            The structures of modulated Rashba spin-orbit coupling and \n            superconducting order parameters in BCS and PDW states are  \n            shown in the right hand side of each figure. See the text for details. \n    \\label{fig:fig1}}\n\\end{figure} \n\n\\section{FORMULATION}\nFor our discussion we analyze the following model of a multilayered system having the inter-layer hopping and Rashba spin-orbit coupling, \n\\begin{eqnarray}\nH&=&\\sum_{{\\bm k},s,m}\\xi({\\bm k})c^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm}\n~+t_\\perp \\sum_{{\\bm k},s,\\langle m,m'\\rangle} c^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm'} \\nonumber \\\\\n&&+\\sum_{{\\bm k},{\\bm k}',m}V({\\bm k},{\\bm k}')c^\\dagger_{{\\bm k}\\uparrow m}c^\\dagger_{-{\\bm k}\\downarrow m}\nc_{-{\\bm k}'\\downarrow m}c_{{\\bm k}'\\uparrow m} \\nonumber\\\\\n&&+\\sum_{{\\bm k},s,s',m}\\alpha_m {\\bm g}({\\bm k})\\cdot {\\bm \\sigma}_{ss'}c^\\dagger_{{\\bm k}sm}c_{{\\bm k}s'm} \n\\nonumber \\\\\n&&-\\mu_{\\rm B}H\\sum_{{\\bm k},s,m}sc^\\dagger_{{\\bm k}sm}c_{{\\bm k}sm}, \n\\label{eq:eq1}\n\\end{eqnarray}\nwhere \n$m$ is the index of layers. Note that we restrict here to one set of superconducting layers and ignore the \nnormal metal part of the superlattice, assuming that the coupling\nbetween heavy Fermion and normal metal part is small.\nThis is a reasonable approximation for the superlattice of CeCoIn$_5$, \nsince the large mismatch in the Fermi velocities between CeCoIn$_5$ and YbCoIn$_5$ at the interface suppresses the proximity effect.~\\cite{She-Balatsky}\nFor the band structure we assume a simple nearest-neighbor hopping tight binding form \n$\\xi({\\bm k}) = -2t (\\cos k_x + \\cos k_y) -\\mu$ on a square lattice with the chemical potential $\\mu\/t = 2$, \nand a small inter-layer hopping $t_\\perp\/t =0.1 $. The symbol $\\langle m,m'\\rangle$ denotes the summation over nearest-neighbor layers. It is important to note that the following results do not qualitatively depend on the details of the band structure. We also use an $s$-wave superconducting state by assuming $V({\\bm k},{\\bm k}')=-V$ (onsite attractive interaction) for simplicity. Although CeCoIn$_5$ is believed to be a $d$-wave superconductor and would have a small odd-parity component admixed with the local non-centrosymmetricity, the features found below \nare independent of the pairing symmetry as long as spin-singlet pairing is predominant.\nIn fact, we have confirmed that the model for $d$-wave superconductivity gives \nthe qualitatively same results (see Appendix). \nHence, for the sake of numerical accuracy, we restrict ourselves to the simpler $s$-wave state. \nOur choice $V\/t=1.7$ gives rise to the critical temperature \n$k_{\\rm B}T_{{\\rm c}0}\/t = 0.0255$ \nfor both bi-layer and tri-layer systems with $\\alpha=0$, ignoring phase fluctuations. \n\nFor our purpose it is sufficient to choose a $g$-vector with Rashba spin-orbit structure, \n${\\bm g}({\\bm k})=(-\\sin k_y,\\sin k_x,0)$ without going into microscopic details of the \ncompound.~\\cite{JPSJ.77.124711}\nThe coupling constants $\\alpha_m$ are layer dependent and antisymmetric with respect to \nreflection at the center of the multilayer structure. In this way the existence of a global inversion center is guaranteed. For example, $(\\alpha_1,\\alpha_2)=(\\alpha,-\\alpha)$ for bi-layers, \nand $(\\alpha_1,\\alpha_2,\\alpha_3)=(\\alpha,0,-\\alpha)$ for tri-layers (see Fig.~\\ref{fig:fig1}). \nParamagnetic limiting and also eventually the PDW phase enter through the Zeeman coupling term, i.e. the last term in Eq.~(\\ref{eq:eq1}).\n\nIn our discussion we restrict to magnetic fields along {\\it c}-axis \nand analyze the model on the basis of the Bogoliubov-de Gennes equation. \nWhile we include the layer dependence of order parameter $\\Delta_m = \n- \\sum_{{\\bm k}} V \\langle c_{-{\\bm k}\\downarrow m}c_{{\\bm k}\\uparrow m} \\rangle$, \nwe ignore the spatial modulation in the plane, such as the vortex state. This simplification \nis not crucial on a qualitative level, if we assume short coherence lengths and a large Ginzburg-Landau parameter\n$ \\kappa $, the ratio of coherence length and London penetration depth.  \nAlthough phase fluctuations may play an important role for the critical temperature, \nwe focus on the low-temperature and high-magnetic field phase, and ignore this aspect.\n\n\\section{SUPERCONDUCTING PHASES}\n\\subsection{Bi-layer system}\nFirst, we address the results for the bi-layer system. \nFigure~\\ref{fig:fig2} shows the $T$-$H$ phase diagram for several values of $\\alpha\/t_\\perp$.\n\\begin{figure*}[htbp]\n  \\begin{tabular}{cc}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2a.eps}\n    \\end{minipage}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2b.eps}\n    \\end{minipage}\n  \\end{tabular}\n  \\begin{tabular}{cc}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2c.eps}\n    \\end{minipage}\n    \\begin{minipage}{0.5\\hsize}\n      \\includegraphics[width=70mm]{fig2d.eps}\n    \\end{minipage}\n  \\end{tabular}\n  \\caption{$T$-$H$ phase diagram of bi-layer systems \nfor (a) $\\alpha\/t_\\perp=0$, (b) $\\alpha\/t_\\perp=1$, (c) $\\alpha\/t_\\perp=2$, \nand (d) $\\alpha\/t_\\perp=3$. \nThe solid and dashed lines show the second- and first-order phase transition lines, \nrespectively.\nThe ``BCS'' and ``PDW'' are explained in the text. \nThe temperature $T$ and magnetic field $\\mu_{\\rm B}H$ in Fig.~\\ref{fig:fig2} are \nnormalized by the critical temperature $T_{\\rm c0}$ at $\\alpha\/t_\\perp=0$. \n    \\label{fig:fig2}}\n\\end{figure*}\nFor $\\alpha\/t_\\perp=0$ [Fig.~\\ref{fig:fig2}(a)], we obtain the conventional phase diagram of a\nsuperconductor subject to paramagnetic limiting.~\\cite{JPSJ.76.051005} In this case  \nthe order parameter is the same in both layers having the same phase,\n$(\\Delta_1,\\Delta_2)=(\\Delta,\\Delta)$. This state we call Bardeen-Cooper-Schrieffer (BCS) state\nis stabilized basically by the inter-layer ``Josephson'' coupling. \nThe superconducting phase transition turns first order at low temperatures, \nbecause of the paramagnetic depairing effect. The in-plane FFLO phase with \nan inhomogeneous order parameter in the two-dimensional plane is supposed to\nappear around the first order transition line in Fig.~\\ref{fig:fig2}(a). However, in our\napproach we ignore the possibility of in-plane modulations as mentioned above. \n\n\nWe now turn to the role of the layer-dependent Rashba spin-orbit coupling. \nInterestingly, the order parameter configuration with a sign change, $(\\Delta_1,\\Delta_2)=(\\Delta,-\\Delta)$, \nbecomes more stable at high magnetic fields for $\\alpha\/t_\\perp \\agt  1$. This state is called \nthe pair-density wave (PDW) state in Figs.~\\ref{fig:fig2}(b-d),\nsince the order parameter modulates on length scales of the crystal lattice constant. \nThe PDW state was proposed for the $Q$-phase in the bulk \nCeCoIn$_5$,~\\cite{PhysRevLett.102.207004,aperis2010} \nbut has not been established in microscopic models so far.  \nWith increasing the spin-orbit coupling $ \\alpha $ the PDW state is stabilized in the large parameter range. \nThus, we here find that the PDW state may be realized in multilayer systems having \n``weak inter-layer coupling'' and ``moderate spin-orbit coupling''. \nWe would like to stress that the PDW state is stabilized in the purely superconducting multilayers, \nwhile a similar inhomogeneous superconducting state can be induced by the proximity effect \nin the ferromagnet-superconductor junction.~\\cite{Buzdin}  \n\n\n\nThe stability of the PDW state can be understood on the basis of the spin susceptibility \nand the band structure investigated in Ref.~\\onlinecite{Maruyama-Sigrist-Yanase}. \nIn the absence of spin-orbit coupling the bi-layer system forms two bands, \nthe bonding band and anti-bonding band. Cooper pairs in the PDW state \nwould form by inter-band pairing, in contrast to the BCS state which is an intra-band pairing state.\nObviously, the PDW state would not be stable under these circumstances.  \nOn the other hand, the band structure is described by the two spin-split bands \non each layer for large $\\alpha\/t_\\perp$. In this case, both BCS and PDW states occur as intra-band Cooper pairing states. \nAlthough the zero-field transition temperature of the PDW state\nis lower than the onset of superconductivity (BCS state), \na magnetic field along the $c$-axis suppresses the BCS state \nmore strongly through paramagnetic depairing. Indeed, the PDW state\nhas a larger spin susceptibility than the BCS state.~\\cite{Maruyama-Sigrist-Yanase} \nTherefore, the PDW state is more robust against paramagnetic limiting than the BCS state,\nas shown in Figs.~\\ref{fig:fig2}(b-d). \n\n\n\\subsection{Tri-layer system}\n\\begin{figure}[htbp]\n  \\includegraphics[width=70mm]{fig3.eps}\n  \\caption{$T$-$H$ phase diagram of tri-layer systems for $\\alpha\/t_\\perp=3$.\n    The dash-dotted line shows the crossover field determined by the criterion \n    $\\Delta_{\\rm in} = \\Delta_{\\rm out}$. \n    \\label{fig:fig3}}\n\\end{figure}\nNew aspects can be found for the tri-layer system.\nThe order parameter has now three components, such that we distinguish\nthe BCS state as $(\\Delta_1,\\Delta_2,\\Delta_3)=(\\Delta_{\\rm out},\\Delta_{\\rm in},\\Delta_{\\rm out})$ \nand the PDW state as\n$(\\Delta_1,\\Delta_2,\\Delta_3)=(\\Delta_{\\rm out},0,-\\Delta_{\\rm out})$. \nFor $\\alpha\/t_\\perp\\alt 2$, the phase diagram is qualitatively the same \nas shown in Fig.~\\ref{fig:fig2} for bi-layer systems.  \nThe range of the PDW state is slightly reduced because of the \ncomplete suppression of order parameter in the inner layer. \nIncreasing spin-orbit coupling to $\\alpha\/t_\\perp=3$ yields an intriguing modification of the \nphase diagram (see Fig.~\\ref{fig:fig3}). An further first-order phase transition line\nappears within the BCS state, which is absent for bi-layer systems. \n\n\nIn order to characterize the first order phase transition in the BCS state, \nwe show the magnetic field dependence of the order parameter in Fig.~\\ref{fig:fig4}.\n\\begin{figure}[htbp]\n \\includegraphics[width=70mm]{fig4a.eps}\n \\includegraphics[width=70mm]{fig4b.eps}\n \\caption{(Color online) Magnetic field dependence of the order parameters \n in the inner layer $\\Delta_{\\rm in}$ (solid lines) and outer layers $\\Delta_{\\rm out}$ (dashed lines) \n for $\\alpha\/t_\\perp=3$. \n (a) $T\/T_{{\\rm c}0}=0.04$ and (b) $T\/T_{{\\rm c}0}=0.28$. \n \\label{fig:fig4}}\n\\end{figure}\nAt $H=0$, the order parameter of the inner layer $\\Delta_{\\rm in}$ is slightly larger than \nthat of the outer layers $\\Delta_{\\rm out}$. Upon increasing the magnetic field for $T\/T_{{\\rm c}0}=0.04$ at $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0} = 2.04$ an abrupt drop of $\\Delta_{\\rm in}$ occurs, while $\\Delta_{\\rm out}$ is only weakly affected [see Fig.~\\ref{fig:fig4}(a)]. Indeed, with the layer-dependent spin-orbit coupling the paramagnetic limiting behavior is different for the inner and outer layers. The spin-susceptibility in the superconducting phase is more strongly diminished for the inner than the outer layer when $\\alpha\/t_\\perp > 1$, as the Rashba spin-orbit coupling vanishes by symmetry for the inner layer.~\\cite{Maruyama-Sigrist-Yanase} \nThe high-field BCS state is stabilized by the more field resistant order parameter component $\\Delta_{\\rm out}$ of the outer layer. Note that the inter-layer hopping is responsible for the fact that the order parameter components are not behaving independently.  \nAs the temperature is increased, the abrupt first order phase transition in the BCS state \nturns into a crossover at the critical end point $T\/T_{\\rm c0} \\sim 0.12$. In this regime\nthe order parameter of the inner layer decreases continuously, as shown in Fig.~\\ref{fig:fig4}(b). \n\n\\begin{figure}[htbp]\n \\includegraphics[width=70mm]{fig5a.eps}\n \\includegraphics[width=70mm]{fig5b.eps}\n \\includegraphics[width=70mm]{fig5c.eps}\n \\caption{(Color online) DOS in the inner layer $\\rho_{\\rm in}(\\omega)$ (solid lines) \n and outer layers $\\rho_{\\rm out}(\\omega)$ (dashed lines) at $T\/T_{{\\rm c}0}=0.04$.  \n (a) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=0$ (low field BCS state), \n (b) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=2.25$ (high field BCS state), \n and (c) $\\mu_{\\rm B}H\/k_{\\rm B}T_{{\\rm c}0}=3$ (PDW state), respectively.\n The other parameters are the same as Fig.~\\ref{fig:fig3}.\n \\label{fig:fig5}}\n\\end{figure}\nThe different phases also influence the quasiparticle spectrum which we will discuss here looking at\nthe density of states (DOS) layer-resolved for the tri-layer system, as depicted in Fig.~\\ref{fig:fig5} (solid line for inner and dashed line for outer layer). \nThe DOS of the outer layer always shows a gap at the Fermi energy. \nIn the absence of a magnetic field the DOS of both the inner and outer layers are \nessentially identical [see Fig.~\\ref{fig:fig5}(a)]. Differences appear for finite fields. \nIn Fig.~\\ref{fig:fig5}(b) we show the DOS in the high-field BCS state, \nwhich is different for the inner and outer layers. \nThe quasiparticle spectrum of the inner layer is gapless, although the order parameter $ \\Delta_{\\rm in} $ does not disappear. Surprisingly, a small gap on the inner layer opens up again when we enter the PDW state as revealed by Fig.~\\ref{fig:fig5}(c), although the order parameter $ \\Delta_{\\rm in} $ vanishes in this case. The origin of this somewhat counter-intuitive behavior lies in the inter-layer hopping and the energy shifts due to the magnetic field leading to the pair breaking. The inner layer is obviously affected by paramagnetic limiting in the high-field BCS state yielding subgap quasiparticle states. On the other hand, in the PDW state the paramagnetic pair breaking effects almost disappear.~\\cite{Maruyama-Sigrist-Yanase}\nIn this case the intrinsic order parameter of the inner layer vanishes and a small gap appears in \nthe quasiparticle excitations because the quasiparticles are extended over all layers \ndue to the inter-layer hopping. \n\n\n\\section{SUMARRY AND DISCUSSION}\nAs mentioned earlier, possible realizations of the situation discussed here could \nbe found in artificially grown superlattices of ${\\rm CeCoIn_5}$~\\cite{Nat.Phys.7.849} \nas well as in multilayer high-$T_{\\rm c}$ cuprates.~\\cite{Mukuda,Shimizu} \nThese multilayer systems are quasi-two-dimensional superconductors and likely \nnear the Pauli limit as they are rather robust against orbital depairing due to the very short\ncoherence lengths.~\\cite{Nat.Phys.7.849,Obrien}\nConsequently, they satisfy the condition for the phenomenology analyzed in our study. \nIndeed, the reduced paramagnetic limiting due to the Rashba spin-orbit coupling of \nthe outer layers has been confirmed in recent experiments for superlattices ${\\rm CeCoIn_5}$.~\\cite{Goh}\nThe experiment is approaching to the FFLO state for the field along {\\it ab}-plane.~\\cite{Goh} \nOur study shows that the PDW state may be stabilized in the magnetic field along {\\it c}-axis. \nWhile this lies in the measurable range for the ``low-$T_{\\rm c}$'' system CeCoIn$_5$, \nfields revealing the high-field BCS or PDW state are likely beyond 100 T in high-$T_{\\rm c}$ cuprates.    \n\nFinally, we would like to comment on several points of our results. \nFirst, we restricted to pairing interaction in the spin singlet channel. Spin-singlet pairing is essential \nfor the observation of the phase diagram including different BCS states and the PDW state.\nIf spin triplet pairing compatible with the Rashba-spin-orbit coupling would be dominant, \nparamagnetic limiting would not play a role at all, since the spin susceptibility would keep the\nfull value of the normal-state Pauli spin susceptibility for magnetic fields along the \n$c$-axis.~\\cite{Maruyama-Sigrist-Yanase}\nTherefore, no change of phase would be induced by a magnetic field. \nFor the intermediate situation, if spin singlet and spin triplet pairing are \nof comparable strength an intriguing mixing of the two channels is possible leading\nto a complex phase structure involving spontaneous time reversal symmetry breaking. This\nsituation will be discussed elsewhere. \nNote, however, that the pairing interactions in the spin triplet channel are expected to be weak \nin the bulk\/superlattice of ${\\rm CeCoIn_5}$ and high-$T_{\\rm c}$ cuprates. \nEven then, weak interactions in the spin triplet channel gives rise to the uniform \nspin triplet Cooper pairs in the PDW state. \n\n\n\nSecond, we focused on the Pauli-limited superconductor with large Maki parameter, \nneglecting orbital depairing effects, in particular, the aspect of a mixed phase including\nflux lines. Although orbital depairing may play quantitatively important roles for the field along {\\it c}-axis, \nthe PDW state survives moderate orbital depairing effects (moderate Maki parameters) because the spatial modulation of vortex state \nalong the {\\it ab}-plane is not affecting the $\\pi$-phase shift along {\\it c}-axis profoundly. \nThe situation is quite different for the helical superconducting state in Rashba-type \nnon-centrosymmetric superconductors, which is obscured even by the weak orbital \ndepairing effect.~\\cite{PhysRevB.78.220508} \n\n\nThird, the PDW state is stabilized not only in the bi- and tri-layers but also in systems with\nmore layers. In such systems the layers towards the center are subject to paramagnetic limiting\nand it is expected that within the BCS state we encounter a progressive demolishing of superconductivity \nin these layers upon increasing field. Eventually, at high enough magnetic field the superconducting\nsystem behaves like the bi-layer system in the PDW state. Due to the fact that these outer layers\nare well separated the condition for the appearance of the PDW state is less stringent than in the real bi-layer system. These views have been confirmed by us numerically for the 4-layer system.\n\nIn conclusion, we have studied the superconducting state in multilayer systems \nincluding layer dependent antisymmetric spin-orbit coupling due to the coordination of the layers. \nWe found that in a magnetic field along the $c$-axis spin-orbit coupling influences the \nphase of the order parameters of different layers. We find a first order phase transition separating two \nstates different by symmetry: the high-field BCS state which is even under reflection at the center of the multilayer system and the PDW state which is odd under reflection. The latter phase \nallows for a considerable enhancement of the upper critical field and generates the uniform order \nparameter for the spin triplet superconductivity. \nThis phenomenology \nrelies on the fact that orbital depairing is weak, i.e. the coherence length of the superconductor is very short, and that the dominant pairing is in a spin singlet channel. This may apply to the artificially grown multilayer version of CeCoIn$_5$ and some multilayered high-$ T_{\\rm c} $ cuprates. \n\n\\begin{acknowledgements}\nThe authors are grateful to D.F. Agterberg, M. H. Fischer, S. K. Goh, J. Goryo, D. Maruyama, Y. Matsuda, \nT. Shibauchi, and H. Shishido for fruitful discussions. \nThis work was supported by KAKENHI (Grant No. 24740230, 23102709, 21102506 and 20740187). \nWe also are grateful for financial support of the \nSwiss Nationalfonds, the NCCR MaNEP and the Pauli Center for Theoretical Studies of ETH Zurich. \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION} \n \nHierarchical clustering scenario, including the cold dark matter\n(CDM) model, may be the most established one for \nreconstructing various observational properties in the \ncosmological structure from the galaxies to the clusters of \ngalaxies.\nPress $ \\& $ Schechter (1974) firstly proposed an analytical formalism\nwhich derives the number density of bound virialized objects of the \nmass at any given epoch, with the assumption that the primordial \ndensity fluctuations is random Gaussian field. The mass function\npredicted by the PS theory shows reasonably\nthe good agreement with N-body simulations even if it has more low\nmass objects (e.g. Lacey $\\&$ Cole 1994).   \nTo reconstruct the observational properties in  \ntheoretical galaxy formation scenario,  \nthere are also approaches which study the history of the mass growth for \nbound objects and the characteristic times (e.g. Lacey $ \\& $ Silk 1991; \nKauffmann, White  $ \\& $ Guiderdoni 1993; Cole et al. 1994).\nMost of them were based on the extended PS formalism, which was\nproposed by Bower (1991) and Bond et al. (1991).  It\ncan derive the number density of objects of a certain mass at a \ngiven time subject to a larger object at a later time.  Using the \nformalism, Lacey $ \\& $ Cole (1993; LC) calculated the ``merger'' rate.  \n\nThe PS formalism, however, has a limitation for \ndescribing the history of the mass growth about the individual objects.  \nThe ``merging process'' described with the PS approach in LC, cannot be \ninterpreted as the same meaning of the merger in astronomical sense, \nin which the objects lose their identity.  \nIn the mass growth history for the astronomical objects, \nthe continuous accretion onto a bound object  \nwithout the loss of identity has different meanings from \nthe mass accumulation with the loss of the identity in the major merger.  \nThe formalism based with the\nPS approach cannot imply the distinction between ``tiny'' and ``notable'' \ncaptures.  For solving this problem,\nManrique $ \\& $ Salvador-Sole (1995, 1996) proposed a \nformalism named ``ConflUent System of Peak trajectories'' (CUSP)\nformalism as an extension of adaptive windowing\nby Appel and Jones (1990).  They can be categorized into\na type of ``peak'' theory (Doroshkevch 1970; Adler 1981;\nBardeen et al. 1986, hereafter BBKS), \nwhich can count the number of  \nthe peaks of the density related to \nthe collapsing threshold, applying a\nlow-pass filter of the bound object scale to the fluctuation field.  \nThe CUSP formalism, unfortunately, needs the iterative \ncalculation to estimate the destruction\nand the reformation rates with mergers. This point \nbecomes a disadvantage \nwhen we try to apply this scheme to the semi-analytical\nstudies for the galaxy formation, including\nthe mass accumulation history of bound \nobjects with Monte Carlo method as shown in LC.  \n\nIn order to give analytical description \nfor the destruction and the reformation rate in the merger,\nwe developed a new approach called \n{\\it the skeleton tree formalism}, using the\ntopological characteristics in the smoothing of the random field\nwith Gaussian filter.\nWith the appearance of {\\it\nsloping saddles} in the landscape of the smoothed field,\nwe can pick up the merging events, \nand distinguish the merger and the accretion.\nThe topological feature in the landscape \nwith the sloping saddles can be \nextremely reduced into {\\it skeleton trees}.\n\nIn this paper, we will focus on the concept and the\nbasic description of {\\it the skeleton tree formalism}.  The outline of the\npaper is as follows. In Section 2, \nin order to distinguish the\naccretion and the merging in our\ncontext, we sketch the topological characteristics in the landscape\nof the smoothed field with critical points; the peaks and the sloping saddles,\ndefine the merger events with appearance \nof the sloping saddles,  and reconstruct the growth history of the objects\nwith {\\it skeleton tree} picture.\nIn Sections 3, 4, and 5, we formulate the\nmathematical description of the constraints, the\nprobability distribution functions, and the scale functions\nfor the critical points.   In Section 6, we\nshows the results of the evolution rates with the\naccretion and the merging obtained from {\\it the skeleton tree formalism}\nwith its consistency.\nFinally, we present our conclusion in Section 7.  We have relegated\nthe details of the derivations to five appendices.  \n\n\\section{Identification of Accretion and Merging}\n\n\\subsection{Hierarchical Evolution from Fluctuations and Filtering Process} \n\nWe will express the density fluctuation field \nas the functions of the comoving \nspatial coordinate $ \\mbox{\\bf r} $ and $ \\mbox{\\bf k} $ ; \n\\begin{eqnarray} \n \\delta(\\mbox{\\bf r})= \\int d^3\\mbox{\\bf k} \ne^{i {\\mbox{\\bf  k r}} } \\delta ({\\bf k}) \\; .\n\\end{eqnarray} \nInterested collapse objects of a comoving scale $ R $ \ncan be identified as peaks greater than a threshold, whose \nfields are smoothed with a low-pass filter of the resolution scale $ R $.  \nThe fluctuation, smoothed with the selection function\n$ S(\\mbox{\\bf r};R) $,  \ncan be expressed as \n\\begin{equation} \nF(\\mbox{\\bf r};R) = \\frac{\\int d^3\\mbox{\\bf  r}_0 \nS(\\mbox{\\bf r}_0;R) \n\\delta (\\mbox{\\bf  r}- \\mbox{\\bf  r}_0)} \n{ \\int d^3\\mbox{\\bf  r}_0 S(\\mbox{\\bf  r}_0;R) } \\; . \n\\end{equation} \nThis Fourier transform is represented with the window function which \nis the Fourier transforms of the selection function; \n\\begin{equation}\nF(\\mbox{\\bf k};R) = \\delta({\\mbox{\\bf k}}) \nW(\\mbox{\\bf k};R) \\; .  \n\\end{equation}\nIn our interest cases, the window function works as a low-pass filter.\n\nWe shall restrict ourselves to isotropic homogeneous Gaussian random\nfields with zero mean as descriptions of the initial fluctuations.  \nFor the field, the power\nspectrum is then only a function of $ k = \\vert \\mbox{\\bf k} \\vert $;\n$\\vert \\delta_k \\vert^2 = \\vert \\delta ({\\bf k}) \\vert$.  \nThe fluctuation spectrum filtered with the scale $ R$ is \n\\begin{equation} \nP(k;R)=\\vert \\delta_k \\vert^2 W(k;R)^2 \\; .\n\\end{equation} \nIn this paper, \nwe take a normalized isotropic Gaussian filter:  \n\\begin{equation}\nW(\\mbox{\\bf k};R) = W(k;R) \\equiv e^{ -k^2 R^2\/2 } \\; . \n\\end{equation}\nIn the next subsection, we will\ndiscuss the reason related to its unique property for the\nsmoothing.  \n\nIn the linear theory of gravitational instability\nfor the structure formation, \nthe amplitude of the field in the overdensity area \nfirstly grows in proportion to $ D(t) $, \nwhere $ D(t) $ is the linear growth factor.\nAccording to BBKS, a \nbound object collapses from the area of a comoving scale $ R $\nwhen the density of a peak in the \nfluctuation smoothed over the resolution scale\n$ R $ exceeds above a fixed threshold\n$ \\delta_{c;0} $.  \nInstead of viewing the peaks to be growing in density \namplitude relative to the fixed threshold $ \\delta_{c;0} $,  \nwe can interpret that the threshold level $ \\delta_c $ is decreasing \nas $ \\delta_{c;0} D(t_{0}) D(t)^{-1} $ with fixing \nthe initial fluctuation field $ F(\\mbox{\\bf r};R) $, where \n$\\delta_{c;0}$ was determined from the threshold at the\npresent time $t_0$.  \nIn this paper, we take Einstein-de Sitter\nmodel: $ \\Omega_0 =1, \\lambda_0 =0 $, in which the relative threshold level\n$ \\delta_c \/ \\delta_{c;0}= D(t_{0}) D(t)^{-1} = (1+z) $, where $ z $\nis the redshift at the time $ t $.\n\nFor standard initial fluctuations like CDM models, in general, \nthe rms of the smoothed field $ < F ({\\bf r}; R) ^2 > $ is  decreasing\nas increasing $ R $.   For such a fluctuation,\nas decreasing the collapse threshold with time evolution, \nwe can pick the collapse objects in the larger scale, which\ngives a reasonable sketch of hierarchical clustering picture.\nWe will relate the filtering process and \nthe hierarchical clustering description in the next.  \n\n\n\\subsection{Landscape of the Fluctuation Field in Position \nand Resolution Space}  \n\nConsider the random field in one dimensional (1-D) positional \ncoordinate of $ x $ is smoothed with a low-pass filter of\na resolution scale $ R $.  This field\nis reproduced as a landscape which extends in two dimensional (2-D)\nextended space of $ (x, R) $.\nA smoothing peak with a low-pass filter make a ridge  \nwhich is running along the direction of increasing\n$ R $ in the landscape.\nThe threshold \nlevel of the collapse can be interpreted as an ocean surface which makes a \n``shoreline'' and some ``lakes'' and ``islands'' in the landscape.\nA ``cape'' on the shoreline of $ \\delta_c $, \ncan be identified as a bound object at $ \\delta_c $.   \nThen,  we can count the number of the bound objects of the scale $ R $ \nto pick up the capes at the $ R $ in the landscape.  \nAs the level of the ocean surface of $ \\delta_c $ \nis decreasing with the evolution of the universe, \nthe shoreline moves to the offing.  It means that the bound objects \ngrow their scale continuously.  \n\nHowever, if ``islands'' or ``lakes'' appear in the landscape,\nthey confuse the identification of ``capes'' as bound objects.\nAs an example, let consider \nan island in the offing of a cape on a shoreline. In this case, \nwe have also another offing side cape of the island; there are two \ncapes which can be counted as the bound objects of ``clouds''.  \nIn these situations, the cloud of the island cape  \ncontains the smaller cloud of the cape on the shore.   \nThese problems appear  \nwhen we take general filters except the Gaussian filter.  \nBond et al. (1991) had shown the filter dependency of \nthe landscape ( see their Figure 1 and 2 ), in which they \ntake sharp $k$-space filter and Gaussian filter.\nIn the case of the sharp $k$-space filter, \nthe ridges cannot decrease monotonically as \nthe resolution scale $ R $ increases.  \nOn the other hand,  in the case of the Gaussian filter, the ridge decreases \nmonotonically as the resolution scale $ R $ increases.  \nThey represent about this feature as there is  no ``lake'' of finite \nextent, and the ocean shoreline have no bays.  Another \nproperty of the Gaussian filtering is that the variance is also monotonically \ndecreasing as increasing $ R $.  In general, this picture is also valid in 3-D\nrandom field.  With the help of this feature,\nwe can distinguish the accretion and the merge in the landscape\nproduced from initial fluctuation fields.  \n\n\n\\subsection{Definition of Monotonic Accretion and Merging in the Landscape} \n\nIn order to understand the monotonic evolution of the field smoothed \nwith the Gaussian filter of the\nresolution scale $ R $, we rewrite the derivative of the field as  \n\\begin{eqnarray} \n\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} = \nR \\nabla^2 F(\\mbox{\\bf r}; R) \\; .  \n\\end{eqnarray} \nThis is identical to a diffusion equation of the variables \n($R^2$, $\\mbox{\\bf r}$).   For all critical points of $ \\nabla^2 F <0 $ \n($ \\nabla^2 F > 0 $) like peaks (holes), \nit guarantees the monotonically decreasing (increasing) as the scale \nincreasing as \n\\begin{eqnarray} \n\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} < 0  \\; ,    \n(\\frac{\\partial F(\\mbox{\\bf r}; R)}{\\partial R} > 0)  \\; .   \n\\end{eqnarray} \nThis monotonicity of the peak smoothing also \nguarantees that the peak runs continuously on the ridge to \nthe shore cape without the island in the landscape.   \nIt can be reasonable that the smoothing and the merging of peaks\nare defined as the continuous accretion growth and the\nmerging event of bound objects.   \nIf we have islands in the landscape with other filters \nof Gaussian, however, we cannot distinguish the accretion and the \nmerging with the confusion for the scale identification of\nthe related bound objects.  Fortunately,\nwe can exclude this problem\nas long as using the Gaussian filter which guarantees the absence of\nisland in the landscape as shown above.  \nThis is the reason that we take the Gaussian filter in this paper.  \n\nA ridge in the landscape, then, \nrepresents continuous accretion growth of a bound object.\nOn the other hand, some ridges terminate on the slope of\nneighboring ridges.  \nThe vanishing point of the ridge on the \nslope of the neighboring ridge can be defined as a type of critical \npoints.  We call it as {\\it sloping saddle} since\nit is a saddle point on the slope of the neighbor peak.\nThe sloping saddle can represent the \nreasonable feature that a bound \nobject loses the identity loss in the merger, associated with\na tree structure in which the branches of the ridges \nare nested at the junctions of the sloping saddles.\nThen, we can translate the topology in the landscape to the tree structure\nnamed with {\\it skeleton tree}, which consists of\nthe accretion branches and the merger junction picked up with the\nsloping saddles.  \n\n\\subsection{Skeleton Tree Picture}\n\n\\begin{figure}\n\\begin{center}\n\\psbox[scale=0.45]{fig1.eps}\n\\end{center}\n\\caption{ The schematic presentation for the abstraction steps \nfrom the smoothing fluctuation field to the {\\it skeleton tree} picture.\n(a)  Schematic representation of the landscape.  \nThe symbols of $\\odot$ and $\\otimes$ means the peak and the sloping\nsaddle, respectively.  The peak trajectory with filter smoothing\nmakes the ridge.  A ridge terminates at a sloping\nsaddle at $ (R_c , \\delta_c) $ which is associated with a \nridge of the neighboring peak\nthrough the resolution scale interval from $R_c$\nto $R_l$.  (b)  The critical points and the ridges are picked up as\nthe abstract tree of the field around a sloping saddle.\n(c) The tree is presented by the reordering in $ ( R,\\delta ) $ space.\nThe hatched area means the merger.  \n(d) The graphical presentation of the smoothing peaks with\nthe merger and the accretion in {\\it the skeleton tree formalism}.  } \n\\end{figure}\n\nWe will describe the abstraction from the landscape to the tree structure \nas presented in Fig. 1 schematically.\nIn Fig. 1 (a), the local structure of the landscape is represented \nwith contour lines, and the different classes of interest \ncritical points are marked;\nthe peak as $\\odot$ and the sloping saddle as $ \\otimes $, and   \nthe ridges are represented as the dashed lines.  \nIn this example, the sloping saddle appears on a resolution scale \n$ R_c $ at a threshold $ \\delta_c $.\nA ridge terminates at the sloping saddle\n$ \\otimes(R_c, \\delta_c) $.  It means that a bound \nobject of the scale $ R_c $ loses its identity at\nthe threshold $ \\delta_c $.  There is a\nridge neighboring with the sloping saddle, on which the peak has \nthe density of $ \\delta_s $ at the same resolution scale \nof $ \\otimes(R_c, \\delta_c) $.  The peak on the neighboring ridge is\nreforming around the sloping saddle. Then, \nwe can consider that the reformation starts from $ (R_c,\n\\delta_s) $ of a peak neighboring with the sloping saddle.  \nThe merger with the destruction and the reformation  \nstarts from $ \\delta_s $ and ends at $ \\delta_l $ with the scale\nrange from $ R_s $ to $ R_l $ in the case of Fig. 1 (a).  \n\nAs shown in Fig. 1 (b) as an abstraction of Fig. 1 (a), then, \nwe can introduce a tree structure associated with  \nthe sloping saddle $ \\otimes(R_c, \\delta_c) $ and three peaks;  \nthe neighboring peak $ \\odot(R_c,\\delta_s) $ of $ \\otimes(R_c,\n\\delta_c) $ in the same resolution $ R_c$,  \nthe progenitor peak $ \\odot(R_s,\\delta_s) $ on the same ridge of $ \\otimes(R_c,\n\\delta_c) $ at $ \\delta_s $ of the density of\n$ \\odot(R_c,\\delta_s) $, and the reformed peak \n$ \\odot(R_l,\\delta_l) $ on the same ridge of the neighboring\n$ \\odot(R_c,\\delta_s) $.  \n\nFor describing the history of the hierarchical clustering with the merging, \nit is convenient to reorder the tree structure along the \nthreshold level instead with the filtering scale.  Because the \nthreshold level is identical to the time as monotonically mapped\nwith $ \\delta_c=\\delta_{c;0} D(t)^{-1} $.  \nFig. 1 (c) schematically shows this reconstructed feature  \nfrom the Fig. 1 (b).  \nThe continuous accretion growth \nis represented as the dashed lines along the ridges, while \nthe merger is represented as the hatched area in Fig. 1 (c).\nIn this case, the merger occurs during the interval \nbetween $ \\delta_l $ and $\\delta_s $.\nEven if we can follow the neighboring peak on the\nsame ridge over $ \\delta_c $ in the landscape, however, we have two peaks\nat a threshold during the merger after the reordering with the\nthreshold, as shown in the hatched area in Fig 1. (c).\nIt means that the reforming peak on the ridge\nneighboring with the sloping saddle also \nloses its identity with the merger.  Then,\nthe neighboring ridge is divided into two part of a disappearing peak\n$ > \\delta_c $ and a reforming peak $ < \\delta_c $.  \n\nThe merger feature is simplified as a joint, which connects three\nlines of two disappearing peaks  \nand the reforming peak at a point of\nthe sloping saddle as shown in Fig. 1 (d), where the lines  \nare the abstraction of the ridges as the continuous accretion growth.\nIn the Figure, the line processes of the accretion branches\nare connected with the point process of the merger\nat the $ (R_c,\\delta_c) $,\nwhich makes a {\\it skeleton tree}.\n\nIn the derivation of the {\\it skeleton tree},\nwe have neglected the detail features of three peaks around $ \\otimes(R_c,\n\\delta_c) $ except counting the number of the associated lines,\nand approximated as \nthe destruction and the reformation occur at $ (\\delta_c,R_c) $ \ninstantaneously.  \nThis is reasonable since no other parameter sets\nexcept $ (\\delta_c,R_c) $ can be defined without any extra\nparameter for modeling the merger event.  The consistency of this\nsimple picture can be proved with the result in the section 6.  \n\nWe can practically reproduce accretion and merging with \nthe script of {\\it skeleton tree} as shown in in Fig 1(d);\nthe arrowed line is related to \nthe mass growth with continuous accretion, and the joint of hatching\ntriangle is a merger event with the destruction and the reformation\nwith the mark $ \\otimes $ of the sloping saddle.  \nWe note that the branching number at the merging \npoints is always two in the progenitor side.\nThis is natural since the merger rate \nof three or more multi peaks are negligible to that of double peaks.  \n\n\\section{The Constraints of Critical Points}\n\n\\subsection{Line and Point Processes in the Skeleton Tree}\n\nAs shown in the previous section,\n{\\it the skeleton tree formalism} is described by a set of the line and  \npoint processes of the smoothing peak along the ridge and the sloping saddle\nin the extended space $ ({\\bf r}, R) $.  The line process of the\nsmoothing peaks in $ ({\\bf r}, R) $ is equal to the point process in the \noriginal spatial coordinate $ {\\bf r} $.  \n\nTheir density fields of these point processes \nare described as sums of $ \\delta $ functions :  \n\\begin{eqnarray} \nn_{pk} ({\\bf r};R) & = & \n\\sum_i \\delta^{(3)}(\\mbox{\\bf r}-\\mbox{\\bf r}_{pk,i}) \\;,\n\\\\\n{\\bf n}_{ss} ({\\bf r}, R) & = & \\sum_i\n\\delta^{(3)}(\\mbox{\\bf r}-\\mbox{\\bf r}_{ss,i})\n\\delta ( R - R_{ss,i})  \\; ,\n\\end{eqnarray}\nwhere the subscripts of $ pk$ and $ss$ means peaks and sloping\nsaddles, $ R_{ss} $ is the resolution scale of the\nsloping saddles.  \nSince $ n_{pk} d^{3}{\\bf r} $ and $ {\\bf n}_{ss} d^{3}{\\bf r} d R $\nare the numbers in 3-D infinite volume of $ d^3 {\\bf r} $ and \n4-D infinite volume of $ d^{3}{\\bf r} d R $,  \nwe call the former and the later\nas the spatial density and the instantaneous spatial density,\nrespectively.   \n\nWe can express the point processes entirely in terms of the field and\nits derivatives with the spatial coordinate $ {\\bf r} $\nand the resolution scale $R$.  \nIn the neighborhood around a critical point, with its constraint of\n$ \\nabla F( {\\bf r})\\vert_{cr}  = 0 $\n, we can expand the field in a Taylor series:   \n\\begin{eqnarray}  \nF(\\mbox{\\bf r}) & \\simeq & \nF(\\mbox{\\bf r}_{cr})  +\\frac{1}{2!}\n\\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\Delta {\\bf r}\n\\nonumber \\\\\n& & \\makebox[0.5cm]{} + \\frac{1}{3!}\n\\nabla \\otimes \\nabla \\otimes \\nabla\nF \\vert_{cr} \\Delta {\\bf r} \\Delta {\\bf r}\n\\Delta {\\bf r} \\; ,\n\\end{eqnarray}\nand its derivatives can be also expanded as \n\\begin{eqnarray} \n\\nabla F (\\mbox{\\bf r})\n& \\simeq & \\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\; , \\\\\n\\nabla \\otimes \\nabla F (\\mbox{\\bf r})\n& \\simeq & \\nabla \\otimes \\nabla F (\\mbox{\\bf r}) \\vert_{cr} +\n\\nabla \\otimes \\nabla \\otimes \\nabla F \\vert_{cr} \\Delta {\\bf r} \\;  , \n\\end{eqnarray}\nwhere \n$ \\Delta {\\bf r} = {\\bf r} - {\\bf r}_{cr} \\; $\nand the suffix $ cr $ means the value at the critical point.  \n\nThe critical points can be divided into non-degenerate one and \ndegenerate one.  The extrema like a peak and hole    \ncan be categorized into non-degenerate one.       \nProvided the condition of the non-degenerate extrema \n$ {\\mbox{det} (\\nabla \\otimes \\nabla F) \\vert_{cr}  \\neq 0} \\; $,\nEq. (11) can be rewritten to \n\\begin{eqnarray}\n{\\bf r} - {\\bf r}_{cr} = ( \\nabla \\otimes \\nabla F \\vert_{cr} )^{-1}  \n\\nabla F (\\mbox{\\bf r}) \\; . \n\\end{eqnarray}\nUsing the second derivatives of the field,  \nthe number density of the extrema \ncan be represented as  \n\\begin{eqnarray} \nn_{ex}& = & \n  \\delta^{(3)}( \n  (\\nabla \\otimes \\nabla F)^{-1} \\nabla F  ) \\nonumber \\\\\n & = &\n\\vert  \\mbox{det}\n(\\nabla \\otimes \\nabla F) \\vert \\delta^{(3)}(\\nabla F) \\; . \n\\end{eqnarray} \nIn order to describe the point process for the extrema,\nthus, it is enough to take the terms to the \norder of the second derivatives.\n\nIn the degenerate case of $ \\nabla \\otimes \\nabla F \\vert_{cr}  = 0 \\; $,\nhowever, we cannot describe the displacement vector\nonly with the first and second derivatives of the field as\nthe non-degenerate case of Eq. (13).  \nThe sloping saddle is a kind of the degenerate \ncritical points.  \n\nUnder the transformation of \nthe principal axis in the spatial coordinate, the part of six\ncomponents \nrelated to the second derivatives $ \\nabla \\otimes \\nabla F $ \nin the covariance matrix becomes diagonal.  It means that \nthe second derivatives have three eigenvalues as \n\\begin{eqnarray} \n(F_{11},F_{22},F_{33})= - \\sigma_2 (\\lambda_1,\\lambda_2,\\lambda_3) \\; , \n F_{\\alpha \\beta}=0 \\; (\\beta \\neq \\alpha) \\; , \n\\end{eqnarray}\nwhere $ \\sigma_2 $ is the rms of $ \\nabla \\otimes \\nabla F $\n( see the definition in Appendix A ).   \nAll the eigenvalues are not null in the non-degenerate cases.\nOn the other hand, the degenerate critical points\nhas one null eigenvalue at least as $ \\lambda_3 =0 $, where \nwe assumed $ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 $ for convenience.\nThis is the reason for the break of the non-degenerate condition as  \n$ \\mbox{det} \\nabla \\otimes \\nabla F \\vert_{cr}  = 0 \\; $ \nin the degenerate case.  In general,\na sloping saddle has a neighboring peak at the degenerate direction.  \nIn this degenerate direction, we cannot take the inversion of\nEq. (11) as $ x_3 - x_{3,cr} = F_{33}\\vert_{cr}^{-1} F_3 $ since $\nF_{33} \\vert_{cr} = 0\n$.\n\nIn order to describe the point process of the sloping saddles,  \nwe use the expansions\nof $ F_{33} $ at the degenerate direction and $ \\nabla^{(2)} F $\nfor the rest 2-D non-degenerate components \nin a couple of Eq. (12) and Eq. (11) as \n\\begin{eqnarray}\nF_{33} & \\simeq & \\sum^3_{\\beta=1} F_{33\\beta}  \n\\Delta x_\\beta \\simeq F_{333} \\Delta x_3 \\; ,  \\\\\n\\nabla^{(2)} F\n& \\simeq & \\nabla^{(2)} \\otimes \\nabla^{(2)} F \\Delta {\\bf r}^{(2)}\n\\; ,\n\\end{eqnarray}\nwhere the suffice of $ (2) $ means the 2-D non-degenerate space.  \nThe set of the expansions derives\n\\begin{eqnarray}\n{\\bf r}^{(2)} - {\\bf r}_{cr}^{(2)} & = & \n( \\nabla^{(2)} \\otimes \\nabla^{(2)} F \\vert_{cr} )^{-1}  \n\\nabla^{(2)} F (\\mbox{\\bf r}) \\; ,  \\\\\nx_3 -x_{3,ss} & = & F_{333}^{-1} F_{33} \\; .\n\\end{eqnarray} \n\nFurthermore, we should remember that the sloping saddle is defined\nin the extended space with the resolution scale $ R $.  The gradient\n$\\nabla F (\\mbox{\\bf r}) $ can be expanded with the derivative of $R$; \n\\begin{eqnarray} \n\\nabla F (\\mbox{\\bf r})\n\\simeq \\frac{d \\nabla F  }{d R} \\vert_{ss} (R-R_{ss})\n= \\nabla^2 \\nabla F \\vert_{ss}\\: R (R-R_{ss}) \\; . \n\\end{eqnarray}\nIn the degenerate direction, this gives \n\\begin{eqnarray}\n R-R_{ss} = \\left( \\sum_{\\alpha=1}^3 F_{3 \\alpha \\alpha} R\n\\right)^{-1} F_3 \\; .\n\\end{eqnarray} \n\nWe obtain the constraint for the instantaneous spatial density\nof the sloping saddles as \n\\begin{eqnarray}\n{\\bf n}_{ss} & = & \\sum_i\n\\delta^{(3)}(\\mbox{\\bf r}- \\mbox{\\bf r}_{ss,i}) \\delta ( R - R_{ss,i}) \n\\nonumber \\\\\n& = &\n\\vert  \\mbox{det}\n( \\nabla^{(2)} \\otimes \\nabla^{(2)} F) \n\\vert \\delta^{(2)}(\\nabla^{(2)} F) \\nonumber \\\\\n&  & \\makebox[1cm]{} \\times \n\\vert F_{333} \\vert \\delta (F_{33})  \\; \n{ \\vert {\\sum_{\\alpha=1}^3 F_{3 \\alpha \\alpha} R} \\vert}  \n\\delta (F_3)  \\; .\n\\end{eqnarray}\n\n\\subsection{Constraints of Peaks and Sloping Saddles} \n\nUnder the transformation to the diagonal principal axis,  \nthe peaks require the condition of \n$ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 \\geq 0 $  \nwhich should be added to the above condition of the extrema: Eq. (14).  \nWith the additional condition of the peaks,  \nthe constraint for the spatial distributions of the peaks is \n\\begin{eqnarray} \nC(\\mbox{\\bf F}^{(10)} \\vert \\mbox{\\rm peaks} ) \n& = &\\sigma_2^3 \n\\vert \\lambda_1 \\lambda_2 \\lambda_3 \\vert \\prod_{i=1}^3  \n\\delta(F_{\\alpha}) \\prod_{\\alpha=1}^3 \\theta(\\lambda_{i}) \\; , \n\\end{eqnarray}\nwhere $ \\theta(\\lambda_{i}) $ is the Heaviside function.   \n\nThe sloping saddles requires the additional condition of \n$ \\lambda_{i}>0 \\; (i=1,2) $ as excluding the merger of the\nhole.  With the constraint of Eq. (22),\nthe constraint for the instantaneous spatial distribution of\nthe sloping saddle can be described as \n\\begin{eqnarray} \n& & C( \\mbox{\\bf F}^{(20)} \\vert \\; \\mbox{\\rm s.saddles} ) = \\sigma_2^2 \n\\sigma_3^2 R  \n \\vert \\lambda_1 \\lambda_2 w_3 (w_1+w_2+w_3) \\vert \\nonumber \\\\\n& & \\makebox[1cm]{} \\times \n\\delta(\\lambda_3) \\prod_{\\alpha=1}^3 \n\\delta(F_{\\alpha}) \\prod_{i=1}^2 \\theta(\\lambda_{i}) \\; , \n\\end{eqnarray}  \nwhere we use $ \\sigma_3 w_{\\alpha} = F_{3 \\alpha \\alpha} $.\n\n\\section{Probability Distribution Function} \n   \nThe joint probability distribution \nof n-dimensional random variables with multivariate Gaussian can \nbe described as \n\\begin{eqnarray} \nP( \\mbox{\\bf  F}^{(n)} ) d \\mbox{\\bf  F}^{(n)} = \\frac{\n\\exp [ \\frac{1}{2} \\mbox{\\bf  F}^{(n)\\:T} \\mbox{\\bf C}^{(n \\times n) -1} \n\\mbox{\\bf  F}^{(n)} ] }\n{ (2\\pi)^n \\vert det \\mbox{\\bf C}^{(n)} \\vert^{1\/2} } \nd \\mbox{\\bf  F}^{(n)} \\; . \n\\end{eqnarray} \nThe convariance matrix $ \\mbox{\\bf C}^{(n \\times n)} $ \nis defined as the expectation \nvalue of the direct product of the vector $ \\mbox{\\bf F}^{(n)} $ : \n\\begin{eqnarray} \nC^{(n \\times n)}_{\\alpha \\beta}= < F^{(n)}_{\\alpha} F^{(n)}_{\\beta} > \\; ,  \n\\end{eqnarray} \nwhere the suffix $ (n)$ of the $ \\mbox{\\bf F}^{(n)} $ \nis the dimension of the \nparameter space.\n\nIn general, \nthe conditional probability of the event A\nwith the constraint event B is given by the Bayes formula\n$ P( \\mbox{A}\\vert\n\\mbox{B}) = P( \\mbox{A}, \\mbox{B}) \/ P ( \\mbox{B}) $. \nThe joint probability, then, can be expanded with the conditional\nprobabilities. \nIf the parameters in the vector are all Gaussian distributed, we can\ndirectly obtain the covariance matrix of the conditional probability.\nThere is a general theorem which is extremely useful when\nwe calculate the above joint probability from the\nconditional probabilities.    \nFor the count of the peaks, we need only the 10 dimension parameters \n$ \\mbox{\\bf F}^{(10)} = (F, \\nabla F, \\nabla \\otimes \\nabla F ) $.\nFor the count of the sloping saddles, \nwe should extend the parameter space to the 20 dimension \n$ \\mbox{\\bf F}^{(20)} = (F, \\nabla F, \\nabla \\otimes \\nabla F, \n\\nabla \\otimes \\nabla \\otimes \\nabla F) $.\nThe 20 dimension covariance matrix can be found with the explicit forms of\nthe parameters in Appendix A.  With the help of the theorem \n(see Appendix B1),  the divided conditional probabilities\nare described for the case of the peaks and the sloping saddles\nin Appendix B2 and B3, respectively.  \n\nFor a type of critical points\nin a n-dimensional parameter space described with constraints, in general, \nthe probability weighted density is\n\\begin{eqnarray}\nn_{cr}({\\mbox{\\bf F}}^{(n)}) d \\mbox{\\bf  F}^{(n)}\n= P({\\mbox{\\bf F}}^{(n)}) C({\\mbox{\\bf F}}^{(n)} \\mid \n\\mbox{cr. points} ) d \\mbox{\\bf  F}^{(n)} \\; .\n\\end{eqnarray} \nThen, \nthe ensemble averaged density of the type of critical points \nrestricted with m-dimension parameters, can be obtained from \nthe integration over (n-m) dimension parameters; \n\\begin{eqnarray} \n< n_{cr}({\\mbox{\\bf F}}^{(m)}) > d \\mbox{\\bf  F}^{(m)} \n = \\int d {\\mbox{\\bf F}}^{(n-m)} \nn_{cr}({\\mbox{\\bf F}}^{(n)})\nd \\mbox{\\bf  F}^{(m)}\n\\; . \n\\end{eqnarray}\nThe ensemble averaged density gives the scale function we interest.  \n   \n\\section{Scale Functions of Critical Points} \n\nWe will derive the differential number densities of all the peaks,\nthe nesting peaks, the non-nesting peaks,  and   \nthe instantaneous differential number density of the sloping\nsaddles,    \nwhich are presented as the scale functions and\nthe instantaneous scale function of \n$ N_{pk}(R,\\delta) dR$, $ N_{nest}(R,\\delta) dR$,\n$ N_{pk}(R,\\delta) dR$, and $ {\\bf N}_{ss}(R,\\delta) dR d \\delta $,\nrespectively.\n\nThe scale functions for all the peaks, the non-nesting,\nand nesting peaks are briefly\ngiven in the appendix C1, C2, and C3,  according to BBKS and \nthe CUSP formalism.  \nWe had checked that the contribution of the nesting peaks \nis negligible as shown in Fig. 2 .  Thus,\nwe treat the scale function of peaks as that of the non-nesting peaks\nhereafter. \n\nIn the first subsection, then, \nwe sketch the derivations of the scale functions of peaks.   \nIn the next subsection, \nwe present the new calculation for the instantaneous scale function\nfor the sloping saddles.  \n\n\\subsection{Scale Function of Peaks}  \n\n\\begin{figure*}\n\\psbox[scale=0.9]{fig2.eps}\n\\caption{ The scale functions are presented\nfor various fluctuation models of the initial density fluctuation.\nThe solid and dotted lines represent the scale function of the\nall peaks and the nesting peaks.  The dashed line is that of the\nPS formalism with the same filter.\n}\n\\end{figure*}\n\nThe scale function of $ N_{pk}(R,\\delta) dR$  \ncan be calculated from the ensemble-averaged density of\nthe peak $ {\\cal N}_{pk} \n(\\nu; R) d \\nu = < n_{pk} (\\nu ;R) > d \\nu $,\nwhere $ \\nu = \\delta\/ \\sigma_0$,\naccording to Manrique $\\&$ Salvador-Sole (1995). \nThe scale function means the differential number density\nper infinitesimal range of $ R $ at a fixed $ \\nu $, not from that \nper infinitesimal range of $ \\nu $ at a fixed $ R$.   \nThe forward one and the last one are denoted \nby a roman capital and a caligraphic capital which is the same\nas the notation of BBKS, respectively.   \nThe spatial density for the peaks of the non-degenerate \ncritical points with \na certain filtering scale can be \ncalculated by the same way of BBKS.  \n\nIn order to obtain  \nthe density of peaks with up-crossing a certain $ \\delta_c $ \nin the range of the filtering scale between $ R - \\Delta R$ and $ R $, \nwe should pick up the critical points which are equal \nto or smaller than $ \\delta_c $ \nat the filtering scale of $ R $ and becomes larger than $ \\delta_c $ at the \nsmaller filtering scale of $ R - \\Delta R $.  It means that \nwe should count the peaks of \n$ \\delta_c + (d F_{pk}\/ dR) \\Delta R < F_{pk} < \\delta_c $ \non scale $ R $ .  As shown in the discussion of the Gaussian filtering, \nthe condition of the counting is expressed as \n\\begin{eqnarray} \n\\delta_c + \\nabla^2 F  R \\Delta R < F < \\delta_c  \\;  .  \n\\end{eqnarray} \n\nAs obtained for the peaks in BBKS, in general,   \nwe can obtain the spatial number density of the peaks \nin infinite ranges $ d \\nu dx $ of the density contrast $ \\nu $ \nand the second derivative of it with a certain filtering scale $ R $;  \n$ {\\cal N}_{pk} (\\nu,x ;R) d \\nu dx $.   We can \ntransform this number density to the number satisfying the condition of \nthe our counting;  \n\\begin{eqnarray} \nN_{pk}(R,\\delta_c) & = & \\lim_{\\Delta R \\rightarrow 0}\n \\frac{1}{\\Delta R} \\int^{\\infty}_{0} dx \\int^{\\nu_{c}}_{\\nu_{c,b}} \nd \\nu {\\cal N}_{pk}(\\nu,x ;R) \\nonumber \\\\ \n& = &  {\\cal N}_{pk}(\\nu_c ;R) <x>_{pk}\n\\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) \nR \\;, \\\\ \n\\nu_{c,b} & = &\\nu_c \n-x \\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) R \\Delta R  \\; ,\n\\end{eqnarray}   \nwhere $ < x >_{pk} $ is the mean value of $ x $ defined \nfrom the distribution function as \n\\begin{eqnarray} \n< x >_{pk} = \\frac{ H_{pk}(\\gamma, \\gamma \\nu_c ) }\n{ G_{pk}(\\gamma, \\gamma \\nu_c) } \\; , \n\\end{eqnarray}\nIn practical treatment, we can use the averaged mapping relation\n\\begin{eqnarray}\n d \\nu_c \\simeq\n<x>_{pk} \\left( \\frac{\\sigma_2(R)}{\\sigma_0(R)} \\right) \nR d R \\; .\n\\end{eqnarray} \n\nFig. 2 illustrates $ N_{pk}(R,\\delta_c) R R_0^2$ and $ N^{nest} (R,\n\\delta_c) R R_0^2$ versus $ R\/R_0 $  with the present\ncollapsing threshold condition $ \\delta_c =1.68 $ in the power-law\nfluctuations of $ n= 0, -1 $, and $ -2 $.  The former and the latter\nare represented with the solid  and dotted lines, respectively.    \nThe critical scale of $\nR_{0} $, was determined by $ \\delta_c\/ \\sigma_0 (R_0) = 1 $\nwith $ \\delta_c =1.68 $.  \nThe number of the nesting peaks is\nnegligible compared with that of all peaks. Hereafter, then, we will \ntake $ N (R,\\delta_c) = N_{pk}(R,\\delta_c) $.  \nThe PS scale functions, obtained from the same filtered random fields, \nare also presented with the dashed lines.\nWe also present the two scale functions of peaks for\nstandard CDM (SCDM) model with the normalization \nof 8 Mpc with the bias factor $ b= 1$.  \nEven if \nthe two functions in the small scale are different from each other between  \nthe peak theory and the PS formalism as pointed out by Appel $\\&$\nJones (1991),  they deviate relatively little over the cosmological interest\ninterval ( 2 decades in the filter scale, 6 decades in the mass scale\naround $ R_0 $).\n\nFig. 2 shows that\nthe scale functions of the peaks $ N_{pk}(R,\\delta_c) R R_0^2$ \nbecome proportional to $ R^{-3}$ asymptotically in the small scale.\nAs shown in BBKS, the cumulative number is a useful quantity which can \nbe evaluated analytically: \n\\begin{eqnarray}\nn_{pk}( \\nu_c=- \\infty ) = \\int^{\\infty}_{\\nu_c=-\\infty} {\\cal N}_{pk}\nd \\nu =\n\\frac{29-6\\sqrt{6}}{ 5^{3\/2} 2 (2 \\pi)^2 }\nR_{\\ast}^{-3} \\; ,\n\\end{eqnarray} \nAs $ R \\rightarrow 0 $, the density contrast $ \\nu_c \\rightarrow\n0 $ and the peaks of the low contrast $ \\nu_c $ dominates\nin the cumulative number.  Then,\n$\nN_{pk}(R,\\delta_c) R R_0^2 \\propto\nn_{pk}(\\nu_c \\rightarrow 0; R_{\\ast} ) \\propto\nn_{pk}(\\nu_c=- \\infty ; R_{\\ast} ) $ in the small scale.   \nSince $ R_{\\ast}$ is proportional to\nthe resolution scale $ R $,  the asymptotic feature seen in Fig. 2 \nis not unexpected thing.   \n\n\\subsection{Instantaneous Scale Function of Sloping Saddles}\n\nThe instantaneous scale function of sloping saddles can be\ndirectly calculated as its ensemble-averaged density:\n\\begin{eqnarray} \n\\mbox{\\bf N}_{ss}(R,\\nu_c) d R d \\nu_c & = & < {\\bf n}_{ss} (\\nu_c;R)\n> d R d \\nu_c \\; .\n\\end{eqnarray}\nThe brief calculation is described in the appendix C.  \n\nWe should note that the scale function is defined as the \ndensity in the infinite interval $ d R d \\nu_c$.  \nThe scale function of the peaks \n$ N_{pk}(R,\\delta_c) dR $ introduced before is the differential density  \nof peaks per infinitesimal ranges of the resolution scale $ R $ at a \nfixed density threshold $ \\delta_c $. On the other hand,\nthe instantaneous scale function of the sloping saddles\n$ \\mbox{\\bf N}_{ss}(R,\\delta_c) $\nis the differential density per infinitesimal ranges \nof $ R $ and $ \\delta $.  Their difference come from the fact that\nthe point processes of peaks and sloping saddles can be defined\nin the 3-D spatial coordinate and the 4-D space extended with $ R $,\nrespectively.  Then, the instantaneous scale function is responsible\nfor the time evolution properties of the scale function\nwith the merging process as seen in the {\\it skeleton tree};\nthe line of the smoothing peak connected with other lines at the\njunction of the sloping saddle.  In the next section,\nusing {\\it the skeleton tree formalism},  \nwe will present the evolution of the scale functions with\nthe distinction between the merger and the accretion according to\n{\\it the skeleton tree formalism}.   \n\n\\section{Evolution with Merger and Accretion}\n\n\\subsection{Instantaneous Scale Functions for the Disappearing and\nReforming Peaks}\n\nAs discussed in the sketch of the {\\it skeleton tree} picture with Fig. 1,\nthe reformation of a peak can be only identified with the destruction\nof two peaks around a sloping saddle.  \nIn the above section, \nwe can obtain the instantaneous number density of\nthe merger of a scale $ R $ at $ \\delta_c $ in the \ninfinite interval of $ d \\delta_c $.\nThus, with changing between the variables $ \\delta_c $ and $ \\nu_c $,\nthe instantaneous scale functions of the destruction and the\nreformation for the peaks of the scale $ R $ at $ \\delta_c $\nare described as \n\\begin{eqnarray} \n\\mbox{\\bf N}^d (R,\\delta_c) d R \\; d \\delta_c  & \\simeq & \n2 \\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; , \\\\\n\\mbox{\\bf N}^f (R,\\delta_c) d R \\; d \\delta_c  & \\simeq & \n\\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; , \n\\end{eqnarray} \nwhere the superscripts of $f$ and $d$ mean the reformation and the\ndestruction in the merger, respectively.  \n\nWith the formula,\nthe net change of the differential number density of the peaks in $ d R $ \nduring the interval of $ d \\delta_c $ is simplified to  \n\\begin{eqnarray}\nS (R,\\delta_c) d R \\; d \\delta_c\n& = &  \n\\mbox{\\bf N}^f (R,\\delta_c) d R \\; d \\delta_c - \n\\mbox{\\bf N}^d (R,\\delta_c) d R \\; d \\delta_c  \\nonumber \\\\\n& = & \n- \\mbox{\\bf N}_{ss}(R,\\delta_c) d R \\; d \\nu_c \\; .\n\\end{eqnarray}\nIt means the net destruction of the peaks is the same as the\nappearance of the sloping saddles.\nThis is reasonable since the formation of the peaks\ndoes not occur in exact sense and all peaks only disappear at the\nsloping saddle in the original filtering process. \nFrom the concept of the {\\it skeleton tree},\nthe peak neighboring with a sloping saddle\nis interpreted to lose its identity with the\nmerger.  We considered that \nthe ridge of the neighboring peak \nis divided into two part of a disappearing peak\n$ > \\delta_c $ and a reforming peak $ < \\delta_c $,\neven if the ridge of the neighboring peak continues over $ \\delta_c $. Thus,\nthe reforming number of the peaks can compensate\nfor the destruction of them, and the appearing number of the sloping saddles\nlefts as the net disappearing number of the peaks.\nThe consistency of this formula can be checked with the conservation\nequation for the scale function of the peaks, as shown\nin the next subsection.   \n\n\\subsection{Instantaneous Rates for Accretion, Destruction and Reformation}  \n\n\\begin{figure*}\n\\begin{center}\n\\psbox[scale=0.9]{fig3.eps}\n\\end{center}\n\\caption{ The rates for the evolution processes at the present with\n$ \\delta_c = 1.68 $. \nThe solid, dotted, short dashed, long dashed,\nand long dash-dotted \nlines mean the reformation rate $ r^f \\vert dt \/d \\delta_c \\vert $,\nthe destruction rate $ r^d \\vert dt \/d \\delta_c \\vert $,\nthe relative accretion rate $ r^a \\vert dt \/d \\delta_c \\vert $,\nthe relative accretion rate $ r^a \\vert dt \/d \\delta_c \\vert $\nduring the merger, and  \nthe shift rate $ r^s \\vert dt \/d \\delta_c \\vert $, respectively.  \nThe evolution rate of the scale function \n$ \\vert  \\partial \\ln N_{pk} (R,t)\/ \\partial t\ndt \/d \\delta_c \\vert $ and\nthe right hand side of the conservation equation (51) is presented\nby short dash-dotted line and crosses. The absolute\nvalues are presented for the two last rates.  \n}\n\\end{figure*}\n\nWe will consider the instantaneous scale growth rate for a peak of \n$ R $ as the continuous scale growth of the accretion.\nThe accretion growth rate can be described \nwith the scaled Laplacian $ x $ as \n\\begin{eqnarray} \n{\\dot R}_{acc}(R, x, \\delta_c ) d t \n& \\equiv & \\frac{1}{x \\sigma_2 R} d \\delta_c \\; .\n\\end{eqnarray}\nWe should note that the growth rate \nfor a peak depends on its particular value $ x $.\nAccording to Manrique $\\&$ Salvador-Sole (1996),\nthe mean growth rate for the objects of the scale $ R $ can be expressed as \n\\begin{eqnarray} \n{\\dot R}_{acc}(R, t) d t \n& \\equiv &\n< \\frac{1}{x \\sigma_2 R} > d \\delta_c \\simeq \\frac{1}{<x>_{pk}\\sigma_2 R}\nd \\delta_c \\; ,  \n\\end{eqnarray} \nwhere $ < \\; > $\nmeans the average of the function with $ x $ and we used the relation \n\\begin{eqnarray} \n< \\; x^{-1} \\;  > & \\simeq &\n\\frac{ \\int_0^{\\infty} dx \\; x^{-1} \\; N_{pk} (R, \\delta_c, x) dR } \n{ \\int_0^{\\infty} dx N_{pk} (R, \\delta_c, x) dR } \\nonumber \\\\\n& = & \\frac{ \\int_0^{\\infty} dx {\\cal N}_{pk}(\\nu_c, x; R)\n\\frac{\\sigma_2}{\\sigma_0} d R } \n{ \\int_0^{\\infty} dx \\; x \\; {\\cal N}_{pk} (\\nu_c, x, R)\n\\frac{\\sigma_2}{\\sigma_0} d R } \\nonumber \\\\\n& = & \\frac{G_{pk} (\\gamma, \\gamma \\nu_c)}\n{ H_{pk} (\\gamma, \\gamma \\nu_c)} = <x>_{pk}^{-1}\\; . \n\\end{eqnarray}\nFrom this mean scale growth rate, we can define \nthe relative growth rate with the accretion as\n\\begin{eqnarray}\nr^a (R,t) dt & \\equiv &\\frac{ \\partial {\\dot R}_{acc} (R,t')}{ \\partial R}\n\\; .\n\\end{eqnarray}\n\nWith the similar consideration applied for the merging objects,\nwe can introduce the mean scale\ngrowth rate and relative growth rate contributed\nwith continues accretion during the merger phase;   \n\\begin{eqnarray} \n{\\dot R}_{acc,m}(R, t) d t \n& \\simeq & \\frac{1}{<x>_{ss}\\sigma_2 R}\nd \\delta_c \\; ,  \\\\ \nr^a_m (R,t) dt & \\equiv & \\frac{ \\partial {\\dot R}_{acc,m} (R,t')}\n{ \\partial R} \\; ,   \n\\end{eqnarray}\nwhere\n$\n<x>_{ss} = H_{ss} (\\gamma, \\kappa, \\gamma \\nu_c) \/\nG_{ss} (\\gamma, \\kappa, \\gamma \\nu_c) \\;$ . \n\n\nThe instantaneous destruction and reformation rates can be defined directly\nfrom their instantaneous scale functions as \n\\begin{eqnarray}\nr^d (R,t) \\; dt & = & \\frac{ {\\bf N}^d (R, \\delta) d R } { N (R,t) d \nR} \\; d \\delta \\; \\\\\nr^f (R,t) \\; dt & = & \\frac{ {\\bf N}^f (R, \\delta) d R } { N (R,t) d \nR} \\; d \\delta \\; . \n\\end{eqnarray}  \nThus, the conservation equation for the scale function $ N (R,t) $ \nis given as \n\\begin{eqnarray}\n\\frac{ \\partial N (R,t) }{\\partial t} +\n\\frac{ \\partial  ( {\\dot R}_{acc} N (R,t) ) }{ \\partial R}  =  S(R,t) \\; ,\n\\end{eqnarray}\nwhere $ S(R,t) $ can be given as\nthe net source term with the destruction and reformation\nrates of $ r^d (R,t) $ and $ r^f (R,t) $ ;   \n\\begin{eqnarray} \nS(R,t) = [ r^f (R,t) - r^d (R,t) ] N (R,t) \\; . \n\\end{eqnarray}\nThe conservation equation can be rewritten to\n\\begin{eqnarray} \n\\frac{ d \\ln N (R,t) }{d t} = r^f (R,t) - r^d (R,t) - r^a (R,t)\\; , \n\\end{eqnarray}\nwhich can be rewritten to\n\\begin{eqnarray} \n\\frac{ \\partial \\ln N (R,t) }{\\partial t} = r^s(R,t)  + \nr^f (R,t) - r^d (R,t) - r^a (R,t)\\; , \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nr^s(R,t) = -{\\dot R}_{acc} \\frac{\\partial \\ln N (R,t)}{\\partial R} \\; ,\n\\end{eqnarray}\nis the rate of shift in the scale space with the\naccretion for the number density distribution.\n\n\nThe inverses of these rates of $ r^{a,f,d,s}(R,t) $ give the time\nscales of the individual processes.      \nIn Fig. 3, the rates for the reformation,\nthe destruction, the relative growth with accretion, the relative\ngrowth with continuous accretion for merging objects\nand the shift in the scale with the accretion \nat the present are presented with solid, dotted, short\ndashed, long dashed, and long dash-dotted lines.  The left hand side \nand the right hand side of the conservation equation (52) are also\npresented with short dash-dotted line and crosses,  the former\n$ \\partial \\ln N(R,t) \\partial t $ can be calculated with the partial\ntime derivative of $  N_{pk}(R,t) $ and the latter\nis calculated from the set of the\nindividual rates as  $r^s(R,t)  +  r^f (R,t) - r^d\n(R,t) - r^a (R,t)$.   \n\nWe can see \nthat the partial time derivative of $ \\partial \\ln N(R,t) \/ \\partial\nt $ can be reproduced well from  \nthe right hand side\nof the conservation equation from the individual calculations of \n$r^s(R,t), r^f (R,t), r^d (R,t),$  and $ r^a (R,t)$ as shown in Fig. 3.\nEspecially,  around the scale of $ R \\simeq R_0 $,\nboth of them shows the same feature, in which the values\nshapely switch from negative to\npositive.  Then, we can verify the consistency of our formalism with\nthe equality between the left hand side and the right hand side\nof the conservation equation.  \n\nThe rates of the reformation and the destruction have a maximum\naround $ R \\simeq R_0 $ and rapidly decrease for the larger\nscale.  On the other hand, \nthe accretion dominates the merger processes of the reformation and\nthe destruction and the rate becomes almost constant \nfor the large objects ( $ R >> R_0 $ ).\nIt is consistent with the picture that \nthe large object are growing in the cosmic time scale.\nThese mean that the larger\nobjects of $ R > R_0 $ grows with the accretion in the linear regime\nof the gravitational instability,\nwhile the smaller objects are accumulated into\nthe larger objects with the merging in the non-linear gravitational\ngrowth regime under the critical scale of $ R_0 $, which is related to \nthe present threshold $ \\delta_c $.  In the theory of\ngravitational instability,  the smaller than $ R_0$ can become in\nnon-linear growth.  \nWe should remark that the merger process is the most efficient\naround the critical scale $R_0$ even if the merger still dominates\nthe accretion in the small scale.  \nThese properties of the dominating merger around $ R_0 $\nare already suggested from the previous\nN-body calculations (e.g. Navaro, Frenk, $\\&$ White 1997 ).  \n\n\\begin{figure}\n\\begin{center}\n\\psbox[scale=0.45]{fig4.eps}\n\\end{center}\n\\caption{ The evolution of growth rates \nat fixed masses for the SCDM model in an Einstein-de Sitter universe.  \nThe solid and dashed lines present the merger and the accretion\ngrowth rate $ r^f \\cdot t $ and\n$ r^a \\cdot t $.\nIn the figure, the highest curves at the right is for $ M\/M_0 = \n10^{-2} $, and successive curves are for $ M\/M_0 = \n10^{-1.5}, 10^{-1}, 10^{-0.5}, 1, 10^{0.5}$ and $ 10^{1}$.   }\n\\end{figure}\n\nFrom the view point of the mass growth history for a object of the\nscale $ R $,  the results in Fig. 3 should indicate that  \nthe dominant growth process switches\nfrom the accretion to the merger accumulation around\nthe threshold of $ \\delta\n= \\sigma_0 (R) $ and\nthe merger accumulation rate becomes maximum and decreases rapidly.  \nWe can see directly this feature in Fig. 4, which represents the\nevolution of the growth rate with the reformation and the accretion\nat fixed scales.\nFrom these results, in general, the growth of a halo \nfirstly starts with the accretion process, \nsecondly switches to the merging and is suppressed soon after the\nmerger dominates.  This feature also was remarked from the\nsimulations.\n\nWe should remark that the results in Fig. 4\nare not the mean growth histories for the individual objects \nsince the fixed scales cannot be directly related to the final \nscales of the objects at the present due to the successive destruction\nand the reformation cooperated with the accretion.\nIn order to reconstruct the mean\ngrowth history for a individual objects, we should extend\nthe basic of {\\it the skeleton tree formalism} including the\nbackground effects.  This extension of {\\it the skeleton tree\nformalism } will be described in the\nfollowing papers.  \n\n\\section{Conclusion}\n\nWe have derived an analytical expression for the statistics and the\nevolution of the cosmic bound objects \nwith the reformation, the destruction and the accretion. It is \napplicable to any hierarchical clustering models in which structure\ngrows via gravitational instability.  {\\it The skeleton formalism} \nis derived as a natural expansion of the peaks theory of BBKS.\nIn the landscape reproduced from the random field with the Gaussian filter, \nwe have followed the smoothing of the peaks as\nthe accretion growth of the objects, and \nhave picked up the ``sloping saddles'' as the merging\nevents with the destruction and the reformation of the objects.\nThe line and point processes of the peaks and sloping saddles in the\nlandscape produce the tree structure.  Then, we call our scheme as\n{\\it the skeleton tree formalism}.   \nWith {\\it the skeleton tree formalism}, we can estimate the rates\nof the reformation, destruction, and the accretion\nin any hierarchical clustering models.\nThe set of these rates \ncan reproduce the evolution of the scale function of the\nobjects with the conservation equation.  This reproduction of\nthe evolving scale function verifies the self-consistency of\n{\\it the skeleton tree formalism}.\nWith the rate calculation of the individual processes, we can find \nthe merger processes are efficient around the critical scale of $ R_0\n$ determined as $ \\delta_c = \\sigma_0(R_0) $.\nThe dominant growth process of the\nobjects switches from the accretion to the merger accumulation  \naround the critical threshold related to the scale.\nIt is important to reproduce the\nmass growth history with distinguish between the accretion and the\nmerger when we try to reproduce the cosmic structure and the galaxy formation\nin the hierarchical scenario.  \n\n\n\\section*{acknowledgments}\nThe author is grateful to H.J. Mo and S.D.M. White  \nfor warm hospitality and the discussions \nwhen H. H. visited in MPA in Garching.  This works was partially \nsupported by a Grant-in-Aid from Japanese Ministry of Education, \nScience and Culture.   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{intro}}\nA fundamental problem in cognition concerns the identification of the principles guiding human decision-making. Identifying the mechanisms of decision-making would indeed have manifold implications, from psychology to economics, finance, politics, philosophy, and computer science. In this regard, the predominant theoretical paradigm rests on a classical conception of logic and probability theory. According to this paradigm, people take decisions by following the rules of Boole's logic, while the probabilistic aspects of these decisions can be formalized by Kolmogorov's probability theory \\cite{k1933}. However, increasing experimental evidence on conceptual categorization, probability judgments and behavioral economics confirms that this classical conception is fundamentally problematical, in the sense that the cognitive models based on these mathematical structures are not capable of capturing how people concretely take decisions in situations of uncertainty.\n\nIn the last decade, an alternative scientific paradigm has caught on which applies a different modeling scheme. The research that uses the mathematical formalism of quantum theory to model situations and processes in cognitive science is becoming more and more accepted in the scientific community, having attracted the interest of renowned scientists, funding institutions, media and popular science. And, quantum models of cognition showed to be more effective than traditional modeling schemes to describe situations like the `Guppy effect', the `combination problem', the `prisoner's dilemma', the `conjunction and disjunction fallacies', `similarity judgments', the `disjunction effect', `violations of the Sure-Thing principle', `Allais', `Ellsberg' and `Machina paradoxes' (see, e.g., \\cite{ga2002,ag2005a,ag2005b,a2009a,pb2009,k2010,bpft2011,bb2012,abgs2013,ags2013,hk2013,pb2013,wbap2013,ast2014}).\n\nThere is a general acceptance that the use of the term `quantum' is not directly related to physics, neither this research in `quantum cognition' aims to unveil the microscopic processes occurring in the human brain. The term `quantum' rather refers to the mathematical structures that are applied to cognitive domains. The scientific community engaged in this research does not instead have a shared opinion on how and why these quantum mathematical structures should be employed in human cognition. Different hypotheses have been put forward in this respect. Our research team in Brussels has been working in this domain since early nineties, providing pioneering and substantial contributions to its growth, and we think it is important  to expose the epistemological foundations of the quantum theoretical approach to cognition we developed in these years. This is the main aim of the present paper. \n\nOur approach was inspired by a two decade research on the mathematical and conceptual foundations of quantum physics, quantum probability and the fundamental differences between classical and quantum structures \\cite{a82,a86,p1989,a99a}. We followed an axiomatic and operational-realistic approach to quantum physics, in which we investigated how the mathematical formalism of quantum theory in Hilbert space can be derived from more intuitive and physically justified axioms, directly connected with empirical situations and facts. This led us to elaborate a `State Context Property' (SCoP) formalism, according to which any physical entity is expressed in terms of the operationally well defined notions of `state', `context' and `property', and functional relations between these notions. If suitable axioms are imposed to such a SCoP structure, then one obtains a mathematical representation\nthat is isomorphic to a Hilbert space over complex numbers (see, e.g., \\cite{bc81}).\n\nLet us shortly explain the \n`operational-realistic' connotation characterizing\nour approach, because doing so we can easily point out its specific strength, and the reason why it introduces an essentially new element to the domain of psychology. `Operational' stands for the fact that all fundamental elements in the formalism  are directly linked to the measurement settings and operations that are performed in the laboratory of experimentation. `Realistic' means that we introduce \nin an operational way\nthe notion of `state of an entity', considering such a `state' as representing an aspect of the\nreality of the considered entity at a specific moment or during a specific time-span. Historically, the notion of `state of a physical entity' was the `easy' part of the physical theories that were the predecessors of quantum theory, and it was the birth of quantum theory that forced physicists to take also \nseriously\nthe role of measurement and hence the value of an operational approach. The reason is that `the reality of a physical entity' was considered to be a simple and straightforward notion in classical physics and hence the\n`different modes of reality of  a\nsame physical entity' were described by its `different states'. That measurements would intrinsically play a role, also in the description of the reality of a physical entity, only became clear in quantum physics for the case of micro-physical entities.  \n\nIn psychology, things historically evolved in a different way. Here, one is in fact confronted with what we call `conceptual entities', such as `concepts' or `conceptual combinations', and more generally with `any cognitive situation which is presented to the different participants in a psychology experiment. Due to their nature, conceptual entities and cognitive situations are `much less real than physical entities', which makes the notion of `state of a conceptual  entity' a highly non-obvious one in psychology. And, as far as we know, the notion of state is never explicitly introduced in psychology, although it appears implicitly within the reasoning that is made about experiments, their setups and results.\nPossibly, the notion of `preparation of the experiment' will be used for what we call `the state of the considered conceptual entity' in our approach. Often, however, the notion of state is also associated with the `belief system' of the participant in the experiment. In our approach we keep both notions of\n`state' and `measurement' on equal footing, whether our description concerns a physical entity or a conceptual entity. In this way, we can make optimal use of the characteristic methodological strengths of each one of the notions.  It is in doing so that we observed that there is an impressive analogy between the operational-realistic description of a physical entity and the operational-realistic description of a conceptual entity, in particular for what concerns the measurement process and the effects of context on the state of the entity.  As a matter of fact, one can give a SCoP description of a conceptual entity and its dynamics \\cite{ga2002,ag2005a,ag2005b}. This justifies the investigation of quantum theory as a unified, coherent and general framework to model conceptual entities, as quantum theory is a natural candidate to model context effects and context-induced state transformations. Hence, the quantum theoretical models that we worked out for specific cognitive situations strictly derive from such investigation of quantum theory as a scientific paradigm for human cognition. In this respect, we think that each predictive success of quantum modeling can be considered as a confirmation of such general validity. It is however important to observe that, recently, potential deviations from Hilbert space modeling were discovered in two cognitive situations, namely, `question order effects' \\cite{pnas} and `response replicability' \\cite{plosone}. According to some authors, question order effects can be represented by sequential quantum measurements of incompatible properties \\cite{bb2012,pb2013,pnas}. However, such a representation seems to be problematical, as it cannot reproduce the pattern observed in response replicability \\cite{plosone}, nor it can exactly fit experimental data \\cite{bketal2015, asdb2015a}. We put forward an alternative solution for these effects within a `hidden measurement formalism' elaborated by ourselves (see, e.g., \\cite{a86, asdb2014,asdb2015b,asdb2015c,a1998, a1999} and references therein), which goes beyond the Hilbert space formulation of quantum theory (probabilities), though it remains compatible with our operational-realistic description of conceptual entities \\cite{asdb2015a,asdb2015d}.\n\nFor the sake of completeness, we summarize the content of this paper in the following.\n\nIn Section~\\ref{foundations}, we present the epistemological foundations of the quantum theoretical approach to human cognition we developed in Brussels. We operationally describe a conceptual entity in terms of concrete experiments that are performed in psychological laboratories. Specifically,  the conceptual entity is the reality of the situation which every participant in an experiment is confronted with, and the different states of this conceptual entity are the different modes of reality of this experimental situation. There are contexts influencing the reality of this experimental situation, and the relevant ones of these contexts are elements of the SCoP structure, the theory of our approach, and their influence on the experimental situation is described as a change of state of the conceptual entity under consideration. There are also properties of this experimental situation, the relevant ones being elements of the SCoP structure, and they can be actual or potential, their `amount of actuality' (i.e. their `degree of availability in being actualized') being described by a probability measure.  The operational analogies between physical and conceptual entities suggest to represent the latter by means of the mathematical formalism of quantum theory in Hilbert space. Hence, we assume, in our research, the validity of quantum theory as a scientific paradigm for human cognition. On the basis of this assumption, we provide a unified presentation in Section \\ref{Hilbertmodeling} of the results obtained within a quantum theoretical modeling in knowledge representation, decision theory under uncertainty and behavioral economics. We emphasize that our research allowed us to identify new unexpected deviations from classical structures \\cite{asv2015,PhilTransA2015}, as well as new genuine quantum structures in conceptual combinations \\cite{as2014,asSpinWind2015,IQSA1}, which could not have been identified \nat the same fundamental level as it was possible in our approach\nif we would have adopted the \nmore traditional perspective only inquiring into the observed deviations from classical probabilistic structures. In Section~\\ref{challenges}, we analyze question order effects and response replicability and explain why a quantum theoretical modeling in Hilbert space of these situations is problematical. Finally, we present in Section~\\ref{beyond} a novel solution we recently elaborated for these cognitive situations \\cite{asdb2015a,asdb2015d}. The solution predicts a violation of the Hilbert space formalism, more specifically, the Born rule for probabilities is put at stake. We however emphasize that this solution remains  compatible with the general operational and realistic description of cognitive entities and their dynamics  given in Section~\\ref{foundations}.  In Section~\\ref{final}, we conclude our article by offering a few additional remarks, further emphasizing the coherence and advantage of our theoretical approach.  We stress, to conclude this section, that the deviation above from Hilbert space modeling should not be considered as an indication that we should better come back to more traditional classical approaches. On the contrary, we believe that new mathematical structures, more general than both pure classical and pure quantum structures, will be \nneeded in the modeling of cognitive processes.\n\n\n\n\\section{An operational-realistic foundation of cognitive psychology\\label{foundations}}\nMany quantum physicists agree that the phenomenology of microscopic particles is intriguing, but what is equally curious is the quantum mathematics that captures the mysterious quantum phenomena. Since the early days of quantum theory, indeed, scholars have been amazed by the the success of the mathematical formalism of quantum theory, as it was not clear at all how it had come about. This has inspired a long-standing research on the foundations of the Hilbert space formalism of quantum theory from physically justified axioms, resting on well defined empirical notions, more directly connected with the operations that are usually performed in a laboratory. Such an operational justification would make the formalism of quantum theory more firmly founded.\n\nOne of the well-known approaches to the foundations of quantum physics and quantum probability is the `Geneva-Brussels approach', initiated by Jauch \\cite{j68} and Piron \\cite{p76}, and further developed by our Brussels research team (see, e.g., \\cite{a82,a99a}). This research produced a formal approach, called `State Context Property' (SCoP) formalism, where any physical entity can be expressed in terms of the basic notions of `state', `context' and `property', which arise as a consequence of concrete physical operations on macroscopic apparatuses, such as preparation and registration devices, performed in spatio-temporal domains, such as physical laboratories. Measurements, state transformations, outcomes of measurements, and probabilities can then be expressed in terms of these more fundamental notions. If suitable axioms are imposed on the mathematical structures underlying the SCoP formalism, then the Hilbert space structure of quantum theory emerges as a unique mathematical representation, up to isomorphisms \\cite{bc81}. \n\nThere are still difficulties connected with the interpretation of some of these axioms and their physical justification, in particular for what concerns compound physical entities \\cite{a82}. But, \nbut this research line was a source of inspiration for the operational approaches applying the quantum formalism outside the microscopic domain of quantum physics \\cite{aa1995,aabg2000}. In particular, \nas we already mentioned in the Introduction,\na very similar realistic and operational representation of conceptual entities can be given for the cognitive domain, in the sense that the SCoP formalism can again be employed to formalize the more abstract conceptual entities in terms of states, contexts, properties, measurements and probabilities of outcomes \\cite{ga2002,ag2005a,ag2005b}.\n\nLet us first consider the empirical phenomenology of cognitive psychology. Like in physics, where laboratories define precise spatio-temporal \ndomains, we can introduce `psychological laboratories' where cognitive experiments are performed. These experiments are performed on \nsituations that are specifically `prepared' for the experiments, including experimental devices, and, for example, structured questionnaires, human participants that interact with the questionnaires in written answers, or each other, e.g., an interviewer and an interviewed. Whenever empirical data are collected from the responses of several participants, a statistics of the obtained outcomes arises. Starting from these empirical facts, we identify in our approach entities, states, contexts, measurements, outcomes and probabilities of outcomes, as follows. \n\nThe complex of experimental procedures conceived by the experimenter, the experimental design and setting and the cognitive effect that one wants to analyze, define a conceptual entity $A$, and are usually associated with a preparation procedure of a state of $A$. Hence, like in physics, the preparation procedure sets the \ninitial\nstate $p_A$ of the conceptual entity $A$ under study. Let us consider, for example, a questionnaire where a participant is asked to rank on a 7-point scale the membership of a list of items with respect to the concepts {\\it Fruits}, {\\it Vegetables} and their conjunction {\\it Fruits and Vegetables}. The questionnaire defines the states $p_{Fruits}$, $p_{Vegetables}$ and $p_{Fruits \\ and \\ Vegetables}$ of the conceptual entities {\\it Fruits}, {\\it Vegetables} and {\\it Fruits and Vegetables}, respectively. It is true that cognitive situations exist where the preparation procedure of the state of a conceptual entity is hardly controllable. Notwithstanding this, the state of the conceptual entity, defined by means of such a preparation procedure, is a `state of affairs'. It indeed expresses a `reality of the conceptual entity', in the sense that, once prepared in a given state, such condition is independent of any measurement procedure, and can be confronted with the different participants in an experiment, leading to outcome data and their statistics, exactly like in physics.\n\nA context $e$ is an element that can provoke a change of state of the conceptual entity. For example, the concept {\\it Juicy} can function as a context for the conceptual entity {\\it Fruits} leading to {\\it Juicy Fruits}, which can then be considered as a state of the conceptual entity {\\it Fruits}. A special context is the one introduced by the measurement itself. Indeed,  when the cognitive experiment starts, an interaction of a cognitive nature occurs between the conceptual entity $A$ under study and a participant in the experiment, in which the state $p_{A}$ of the conceptual entity $A$ generally changes, being transformed to  another state $p$. Also this cognitive interaction is formalized by means of a context $e$. For example, if the participant is asked to choose among a list of items, say, {\\it Olive}, {\\it Almond}, {\\it Apple}, etc., the most typical one with respect to {\\it Fruits}, and the answer is {\\it Apple}, then the\ninitial\nstate $p_{Fruits}$ of the conceptual entity {\\it Fruits} changes to $p_{Apple}$, i.e. the state describing the situation `the fruit is an apple',  as a consequence of the contextual interaction with the participant.\n\nThe change of the state of a conceptual entity due to a context may be either `deterministic', hence in principle predictable under the assumption that the state before the context acts is known, or `intrinsically probabilistic', in the sense that only the probability $\\mu(p,e,p_{A})$ that the  state $p_{A}$ of $A$ changes to the state $p$ is given. In the example above on typicality estimations, the typicality of the item {\\it Apple} for the concept {\\it Fruits} is formalized by means of the transition probability $\\mu(p_{Apple},e, p_{Fruits})$, where the context $e$ is the context of the typicality measurement.\n\nLike in physics, an important role is played by experiments with only two outcomes, the so-called `yes-no experiments'. Suppose that in an opinion poll a participant is asked to answer the question: ``Is Gore honest and trustworthy?''. Only two answers are possible: `yes' and `no'. Suppose that, for a given participant, the answer is `yes'. Then, the state  $p_{Honesty}$ of the conceptual entity {\\it Honesty and Trustworthiness} (which we will  \ndenote by {\\it Honesty}, for the sake of simplicity) changes to a new state \n$p_{Gy}$, which is the state describing the situation `Gore is honest'.\nHence, we \ncan\ndistinguish a class of  yes-no measurements on conceptual entities, as we do in physics. \n\nThe third step is the mathematical representation. We have seen that the Hilbert space formalism of quantum theory is general enough to capture an operational description of any entity in the micro-physical domain. Then, the strong analogies between the realistic and operational descriptions of physical and conceptual entities, in particular for what concerns the measurement process, suggest us to apply the same Hilbert space formalism when representing cognitive situations. Hence, each conceptual entity $A$ is associated with a Hilbert space ${\\mathcal H}$, and the state $p_A$ of $A$ is represented by a unit vector $|A\\rangle \\in \\mathcal H$. A yes-no measurement is represented by a spectral family $\\{M, \\mathbbmss{1}-M \\}$, where $M$ denotes an orthogonal projection operator over the Hilbert space $\\mathcal H$, and $\\mathbbmss{1}$ denotes the identity operator over $\\mathcal H$. The probability that the `yes' outcome is obtained in such a yes-no measurement when the conceptual entity $A$ is in the state represented by $|A\\rangle$ is \nthen\ngiven by the Born rule $\\mu(A)=\\langle A|M|A \\rangle$. For example, $M$ may represent an item $x$ that can be chosen in relation to a given concept $A$, so that its membership weight is given by $\\mu(A)$. \n\nThe Born rule obviously applies to measurement with more than two outcomes too.  For example, a typicality measurement involving a list of  $n$ different items $x_1$, \\ldots, $x_n$ with respect to a concept $A$ can be represented as a spectral measure $\\{M_1, \\ldots, M_n \\}$, where $\\sum_{k=1}^n M_k = \\mathbbmss{1}$ and $M_kM_l = \\delta_{kl}M_k$, such that the typicality $\\mu_{k}(A)$ of the item $x_k$ with respect to the concept $A$ is again given by the Born rule $\\mu_{k}(A)=\\langle A|M_k|A\\rangle$.\n\nAn interesting aspect concerns the final state of a conceptual entity $A$ after a human judgment. As above, we can assume the existence of a nonempty class of cognitive measurements that are ideal first kind measurements in the standard quantum sense, i.e. that satisfy the `L\\\"{u}ders postulate'. For example, if the typicality measurement of a list of items $x_1$, \\ldots, $x_n$ with respect to a concept $A$ gave the outcome $x_k$, then the final state of the conceptual entity after the measurement is represented by the unit vector $|A_k\\rangle=\\frac{M_{k}|A\\rangle}{\\sqrt{\\langle A |M_{k}|A\\rangle}}$. This means that the weights $\\mu_{k}(A)$ given by the Born rule can actually be interpreted as transition probabilities $\\mu(p_k,e,p_{A})$, where $e$ is the context producing the transitions from the initial state $p_{A}$ of the conceptual entity $A$, represented by the unit vector $|A\\rangle$, to one of the $n$ possible outcome states $p_k$, represented by the unit vectors $|A_k\\rangle$.\n \nSo, how can a Hilbert space model be actually constructed for a cognitive situation? To answer this question let us consider  again a conceptual entity $A$, in the state $p_{A}$, a cognitive measurement on $A$ described by means of  a context $e$, and suppose that the measurement has $n$ distinct outcomes, $x_1$, $x_2$, \\ldots, $x_n$. A quantum theoretical model for this situation can be constructed as follows. We associate $A$ with a\n$n$-dimensional complex Hilbert space $\\cal H$, and then consider an orthonormal base $\\{ |e_1\\rangle, |e_2\\rangle, \\ldots, |e_n\\rangle \\}$ in $\\cal H$ (since ${\\cal H}$ is isomorphic to the Hilbert space ${\\mathbb C}^{n}$, the orthonormal base of $\\cal H$ can be the canonical base of ${\\mathbb C}^{n}$). Next, we represent the cognitive measurement described by $e$ by means of the spectral family $\\{M_1, M_2, \\ldots, M_n \\}$, where $M_k=|e_k\\rangle\\langle e_k|$, $k=1,2,\\ldots,n$. Finally, the probability that the measurement $e$ on the conceptual entity $A$ in the state $p_{A}$\ngives the outcome $x_k$ is given by $\\mu_{k}(A)=\\langle A|M_k|A\\rangle=\\langle A|e_k\\rangle\\langle e_k|A\\rangle=|\\langle e_k|A\\rangle|^2$.\n\nWhat about the interpretation of the Hilbert space formalism above? Two major points should now be reminded, namely: \n\n(i) the states of conceptual entities describe the `modes of being' of these conceptual entities;\n\n(ii) in a cognitive experiment, a participant \nacts as a (measurement) context for the conceptual entity, changing its state.\n\nThis means that, as we mentioned already, the  state $p_{A}$ of the conceptual entity $A$ is represented in the Hilbert space formalism by the unit vector $|A\\rangle$, the possible outcomes $x_k$ of the experiment by the base vectors $|e_k\\rangle$, and the action of a participant (or the overall action of the ensemble of participants) as the state transformation $|A\\rangle \\to |e_k\\rangle$ induced by the orthogonal projection operator $M_k=|e_k\\rangle\\langle e_k|$, if the outcome $x_k$ is obtained, so that the probability of occurrence of $x_k$ can also be written as $\\mu_{k}(A)=\\mu(|e_k\\rangle,e,|A\\rangle)$, where $e$ is the measurement context associated with the spectral family $\\{M_1, M_2, \\ldots, M_n \\}$.\n\nIt follows from (i) and (ii) that a state, hence a unit vector in the Hilbert space representation of states, does not describe the subjective \nbeliefs \nof a person, or collection of persons, about a conceptual entity. Such subjective  beliefs are rather incorporated in the cognitive interaction between the cognitive situation and the human participants deciding on that cognitive situation. In this respect, our operational quantum approach to human cognition  is also a realistic one, and thus it departs from other approaches that apply the mathematical formalism of quantum theory to model cognitive processes \\cite{k2010,bb2012,hk2013,pb2013,pnas,plosone}. Of course, one could say that the difference between interpreting the quantum state as a `state of belief' of a participant in the experiment, or as a `state of a conceptual entity', i.e. a `state of the situation which the participant is confronted with during an experiment', is only a question of philosophical interpretation, but comes to the same when it concerns the methodological development of the approach. Although this is definitely partly true, we do not fully agree with it. Interpretation and methodology are never completely separated. A certain interpretation, hence giving rise to a specific view on the matter, will give rise to other ideas of how to further develop the approach, how to elaborate the method, etc., than another interpretation, with another view, will do. We believe that an operational-realistic approach, being balanced between attention for idealist as well as realist philosophical \ninterpretations,\ncarries in this sense a particular strength, \nprecisely\ndue to this balance. A good example of this is how we were inspired to use the superposition principle of quantum theory in our modeling of concepts as conceptual entities. We represented the combination of two concepts by a state that is the linear superposition of the states describing the component concepts. This way of representing combined conceptual entities captures the nature of emergence, exactly like in physics. It would not be obvious to put forward \nthis\ndescription when state of beliefs are the focus of what can be predicted.\n\nWe stress a third point that is important, in our opinion. For most situations, we interpret the effect of the cognitive context on a conceptual entity in a decision-making process as an `actualization of pure potentiality'. Like in quantum physics, the (measurement) context does not reveal pre-existing properties of the entity but, rather, it makes actual properties that were only potential in the initial state of the entity (unless the initial state is already an eigenstate of the measurement in question, like in physics) \\cite{ga2002,ag2005a,ag2005b}.\n\nIt follows from the previous discussion that our research investigates the validity of quantum theory as a general, unitary and coherent theory for human cognition. Our quantum theoretical models, elaborated for specific cognitive situations and data, derive from quantum theory as a consequence of the assumptions about this general validity. As such, these models are subject to the technical and epistemological constraints of quantum theory. \nIn other terms,\nour quantum modeling rests on a `theory based approach', and should be distinguished from an `ad hoc modeling based approach', only devised to fit data. In this respect, one should be suspicious of models in which free parameters are added on an `ad hoc' basis to fit the data more closely in specific experimental situations. In our opinion, the fact that our `theory derived model' reproduces different sets of experimental data constitutes in itself a convincing argument to support its advantage over traditional modeling approaches and to extend its use to more complex cognitive situations (in that respect, see also our final remarks in Section~\\ref{final}).\n\nWe present in Section \\ref{Hilbertmodeling} the results obtained in our quantum theoretical approach in the light of the epistemological perspective of this section.\n\n\n\n\n\\section{On the modeling effectiveness of Hilbert space\\label{Hilbertmodeling}}\nThe quantum approach to cognition described  in Section \\ref{foundations} produced concrete models in Hilbert space, which faithfully matched different sets of experimental data collected to reveal `decision-making errors' and `probability judgment errors'. This allowed us to identify genuine quantum structures in the cognitive realm. We present a reconstruction of the attained results in the following.\n\nThe first set of results concerns knowledge representation and conceptual categorization and combination. James Hampton collected data on how people rate membership of items with respect to pairs of concepts and their combinations, conjunction \\cite{h1988a}, disjunction \\cite{h1988b} and negation \\cite{h1997}. By using the data in \\cite{h1988b}, we reconstructed the typicality estimations of 24 items with respect to the concepts {\\it Fruits} and {\\it Vegetables} and their disjunction {\\it Fruits or Vegetables}. We showed that the concepts {\\it Fruits} and {\\it Vegetables} interfere when they combine to form {\\it Fruits or Vegetables}, and the state of the latter can be represented by the linear superposition of the states of the former.  This behavior is analogous to that of quantum particles interfering in the double-slit experiment when both slits are open. The data are faithfully represented in a 25-dimensional Hilbert space over complex numbers \\cite{abgs2013,ags2013}.\n\nIn the data collected on the membership estimations of items with respect to pairs $(A,B)$ of concepts and their conjunction `$A$ and $B$' and disjunction `$A$ or $B$', Hampton found systematic violations of the rules of classical (fuzzy set) logic and probability theory. For example, the membership weight of the item {\\it Mint} with respect to the conjunction {\\it Food and Plant} is higher than the membership weight of {\\it Mint} with respect to both {\\it Food} and {\\it Plant} (`overextension'). Similarly, the membership weight of the item {\\it Ashtray} with respect to the disjunction {\\it Home Furnishing or Furniture} is lower than the membership weight of {\\it Ashtray} with respect to both {\\it Home Furnishing} and {\\it Furniture} (`underextension'). We showed that overextension and underextension are natural expressions of `conceptual emergence' \\cite{a2009a,ags2013}. Namely, whenever a person estimates the membership of an item  $x$ with respect to the pair $(A,B)$ of concepts and their combination $C(A,B)$, two processes act in the person's mind. The first process is guided by `emergence', that is, the person estimates the membership of  $x$ with respect to the new emergent concept $C(A,B)$. The second process is guided by `logic', that is, the person separately estimates the membership of $x$ with respect to $A$ and $B$ and applies \na probabilistic logical calculus to estimate the membership of $x$ with respect to $C(A,B)$  \\cite{IQSA2}. More important, the new concept $C(A,B)$ emerges from the concepts $A$ and $B$, exactly as the linear superposition of two quantum states emerges from the component states. A two-sector Fock space faithfully models Hampton's data, and was later successfully applied to the modeling of more complex situations involving concept combinations (see, e.g., \\cite{IQSA2,s2015}).\n\nIt is interesting to note that the size of deviation of classical probabilistic rules due to overextension and underextension generally depends on the item  $x$ \nand the specific combination $C(A,B)$ of the concepts $A$ and $B$ that are investigated. However, we recently performed a more general experiment in which we asked the participants to rank the membership of items with respect to the concepts $A$, $B$, their negations `not $A$', `not $B$', and the conjunctions `$A$ and $B$', `$A$ and not $B$', `not $A$ and $B$', and `not $A$ and not $B$'.  We surprisingly found that the size of deviation from classicality in this experiment does not depend on either the item or the pair of concepts or the specific combination, but shows to be a numerical constant. Even more surprisingly, our two-sector Fock space model correctly predicts the value of this constant, capturing in this way a deep non-classical mechanism connected in a fundamental way with the mechanism of conceptual formation itself rather than only specifically with the mechanism of conceptual combination \\cite{asv2015,PhilTransA2015}.\n\nDifferent concepts entangle when they combine, where `entanglement' is meant in the standard quantum sense. We proved this feature of concepts in two experiments. In the first experiment, we asked the participants to choose the best example for the conceptual combination {\\it The Animal Acts} in a list of four examples, e.g., {\\it The Horse Growls}, {\\it The Bear Whinnies}, {\\it The Horse Whinnies} and {\\it The Bear Growls}. By suitably combining exemplars of {\\it Animal} and exemplars of {\\it Acts}, we performed four joint measurements on the combination {\\it The Animal Acts}. The expectation values violated the `Clauser-Horne-Shimony-Holt' version of Bell inequalities \\cite{b1964,chsh1969}. The violation was such that, not only the state of {\\it The Animal Acts} was entangled, but also the four joint measurements were entangled, in the sense that they could not be represented in the Hilbert space ${\\mathbb C}^{4}$ as the (tensor) product of a measurement performed on the concept {\\it Animal} and a measurement performed on the concept {\\it Acts} \\cite{as2014}. In the second experiment, performed on the conceptual combination {\\it Two Different Wind Directions}, we confirmed the presence of quantum entanglement, but we were also able to prove that the empirical violation of the marginal law in this type of experiments is due to a bias of the participants in picking wind directions. If this bias is removed, which is what we did in an ensuing experiment on {\\it Two Different Space Directions}, one can show that people pick amongst different space directions exactly as coincidence spin measurement apparatuses pick amongst different spin directions of a compound system in the singlet spin state. In other words, entanglement in concepts can be proved from only the statistics of the correlations of joint measurements on combined concepts, exactly as in quantum physics \\cite{asSpinWind2015}.\n\nSince concepts exhibit genuine quantum features when they combine pairwise, it is reasonable to expect that these features should be reflected in the statistical behavior of the combination of several identical concepts. Indeed, we detected quantum-type indistinguishability in an experiment on the combination of identical concepts, such as the combination {\\it Eleven Animals}. More specifically, we found significant evidence of deviation from the predictions of classical statistical theories, i.e. `Maxwell-Boltzmann distribution'. This deviation has \nclear analogies with the deviation of quantum mechanical from classical mechanical statistics, due to indistinguishability of microscopic quantum particles, that is, we found convincing evidence of the presence of `Bose-Einstein distribution'. In the experiment, indeed, people do \nnot seem to distinguish two identical concepts in the combination of $N$ identical concepts, which is more evident in more abstract than in more concrete concepts, as expected \\cite{IQSA1}. \n\nThe second set of results concern `decision-making errors under uncertainty'. In the `disjunction effect' people prefer action $x$ over action $y$ if they know that an event $A$ occurs, and also if they know that $A$ does not occur, but they prefer $y$ over $x$ if they do not know whether $A$ occurs or not. The disjunction effect violates a fundamental principle of rational decision theory, Savage's `Sure-Thing principle' and, more generally, the total probability rule of classical probability \\cite{s1954}. This preference of sure over unsure choices violating the Sure-Thing principle was experimentally detected in the `two-stage gamble' and in the `Hawaii problem' \\cite{ts1992}. In the experiment on a gamble that can be played twice, the majority of participants prefer to bet again when they know they won in the first gamble, and also when they know they lost in the first gamble, but they generally prefer not to play when they do not know whether they won or lost. In the Hawaii problem, most students decide to buy the vacation package when they know they passed the exam, and also when they know they did not pass the exam, but they generally decide not to buy the vacation package when they do not know whether they passed or not passed the exam. We recently showed that, in both experimental situations, this `uncertainty aversion' can be explained as an effect of underextension of the conceptual entities $A$ and `not $A$' with respect to the conceptual disjunction `$A$ or not $A$', where the latter describes the situation of not knowing which event, $A$ or `not $A$', will occur. The concepts $A$ and `not $A$' interfere in the disjunction `$A$ or not $A$', which determines its underextension. A Hilbert space model in ${\\mathbb C}^{3}$ allowed us to reproduce the data in both experiments on the disjunction effect \\cite{s2015}.\n\nEllsberg's thought experiments, much before the disjunction effect, revealed that the Sure-Thing principle is violated in concrete decision-making under uncertainty, as people generally prefer known over unknown probabilities, instead of maximizing their expected utilities. In the famous `Ellsberg three-color example', an urn contains 30 red balls and 60 balls that are either yellow or black, in unknown proportion. One ball will be drawn at random from the urn. The participant is firstly asked to choose between betting on `red' and betting on `black'. Then, the same participant is asked to choose between betting on `red or yellow' and betting on `black or yellow'. In each case, the `right' choice will be awarded with $\\$100$. As the events `betting on red' and `betting on black or yellow' are associated with known probabilities, while their counterparts are not, the participants will prefer betting on the former than betting on the latter, thus revealing what Ellsberg called `ambiguity aversion', and violating the Sure-Thing principle \\cite{e1961}. This pattern of choice has been confirmed by several experiments in the last thirty years \\cite{ms2014}. Recently, Machina identified in a couple of thought experiments, the `50\/51 example' and the `reflection example', a similar mechanism guiding human preferences in specific ambiguous situations, namely, `information symmetry' \\cite{m2009,blhp2011}, which was experimentally confirmed in \\cite{lhp2010}. In our quantum theoretical approach, ambiguity aversion and information symmetry are two possible cognitive contexts influencing human preferences in uncertainty situations and changing the states of the `Ellsberg and Machina conceptual entities', respectively. Hence, an ambiguity aversion context will change the state of the Ellsberg conceptual entity in such a way that `betting on red' and `betting on black or yellow' are finally preferred. In other terms, the novel element of this approach is that the initial state of the conceptual entity, in its Hilbert space representation, can also change because of the pondering of the participants in relation to certain choices, before being collapsed into a given outcome.  This opens the way to a generalization of rational decision theory with quantum, rather than classical, probabilities \\cite{asJMP2015}.\n\nThe results above provide a strong confirmation of \nthe quantum theoretical approach presented in Section \\ref{foundations}, and we expect that further evidence will be given in this direction in the years to come. In the next section we instead intend to analyze some situations where deviations from Hilbert space modeling of human cognition apparently occur.\nWe will see in Section~\\ref{beyond} that these deviations are however compatible with the general  operational-realistic framework  portrayed in Section~\\ref{foundations}.\n\n\n\n\\section{Deviating from Hilbert space\\label{challenges}} \nAs mentioned in Section~\\ref{foundations}, if suitable axioms are imposed on the SCoP formalism, the Hilbert space structure of quantum theory can be shown to emerge uniquely, up to isomorphisms \\cite{bc81}. However, we also know that certain experimental situations can violate some of these axioms. This is the case for instance when we consider entities formed by experimentally separated sub-entities, a situation that cannot be described by the standard quantum formalism \\cite{a82,a1984}. Similarly, one may expect that the structural shortcomings of the standard quantum formalism can also manifest in the ambit of psychological measurements, in the form of data that cannot be exactly modeled (or jointly modeled) by means of the specific Hilbert space geometry and the associated Born rule. The purpose of this section is to describe two paradigmatic examples of situations of this kind: `question order effects' and `response replicability'. In the following section, we then show how the quantum formalism can be naturally completed to also faithfully model these data, in a way that remains consistent with our operational-realistic approach. \n\nLet us first remark that the mere situation of having to deal with a set of data for which we don't have yet a faithful Hilbert space model should not make one necessarily search for an alternative more general quantum-like mathematical structure as a modeling environment. Indeed, it is very well possible that the adequate Hilbert space model has not yet been found. Recently, however, a specific situation was identified and analysed indicating that the standard quantum formalism in Hilbert space would not be able to be used to model it \\cite{plosone}. This situation combines two phenomena: `question order effects' and `response replicability'. We start by explaining `question order effects' and how the cognitive situation in which they appear can be represented in Hilbert space.\n\nFor this we come back to the yes-no experiment of Section~\\ref{foundations}, where participants are asked: ``Is Gore honest and trustworthy?''. This experiment gives rise to a two-outcome measurement performed on the conceptual entity {\\it Honesty} in the initial state $p_H$, represented by the unit vector $|H\\rangle\\in {\\mathcal H}$, where ${\\mathcal H}$ is a two-dimensional Hilbert space if we assume the measurement to be non-degenerate, or more generally a $n$-dimensional Hilbert space if we also admit the possibility of sub-measurements. Denoting $\\{M_G, \\bar{M}_G = \\mathbbmss{1}-M_G\\}$ the spectral family associated with this measurement, the probability of the `yes' outcome (i.e. to answer `yes' to the question about Gore's honesty and trustworthiness) is then given by the Born rule $\\mu_{Gy}(H)=\\langle H|M_G|H\\rangle$, and of course $\\mu_{Gn}(H)=\\langle H|\\bar{M}_G|H\\rangle = 1-\\mu_{Gy}(H)$ is the probability for the `no' outcome. We then consider a second measurement performed on the conceptual entity {\\it Honesty}, but this time associated with the question: ``Is Clinton honest and trustworthy?''. We denote $\\{M_C, \\bar{M}_C = \\mathbbmss{1}-M_C\\}$ the spectral family associated with this second measurement, so that the probabilities for the `yes' and `no' outcomes are again given by $\\mu_{Cy}(H)=\\langle H|M_C|H\\rangle$ and $\\mu_{Cn}(H)=\\langle H|\\bar{M}_C|H\\rangle$, respectively. \n\nStarting from these two measurements, it is possible to conceive sequential measurements, corresponding to situations where the respondents are subject to the Gore and Clinton questions in a succession, one after the other, in different orders. Statistical data about `Clinton\/Gore' sequential measurements were reported in a seminal article on question order effects \\cite{m2002} and further analyzed in \\cite{bb2012,wb2013}. More precisely, after fixing a rounding error in \\cite{wb2013}, we have the following sequential (or conditional) probabilities \\cite{asdb2015a}: \n\\begin{eqnarray}\n&\\mu_{CyGy}(H)=0.4899 \\quad \\mu_{CyGn}(H)= 0.0447 \\quad \\mu_{CnGy}(H)=0.1767 \\quad \\mu_{CnGn}(H)= 0.2887 \\label{Prob-Clinton-Gore1}\\\\\n&\\mu_{\\rm GyCy}(H)=0.5625 \\quad \\mu_{GyCn}(H) = 0.1991 \\quad \\mu_{GnCy}(H) = 0.0255 \\quad \\mu_{GnCn}(H) = 0.2129\n\\label{Prob-Clinton-Gore2}\n\\end{eqnarray}\nwhere (\\ref{Prob-Clinton-Gore1}) corresponds to the sequence where first the Clinton and then the Gore measurements are performed, whereas (\\ref{Prob-Clinton-Gore2}) corresponds to the reversed order sequence for the measurements. Considering that the probabilities in each of the four columns above are sensibly different, these data describe typical `question order effects'. \n\nQuantum theory is equipped with a very natural tool to model question order effects: `incompatible measurements', as expressed by the fact that two self-adjoint operators, and the associated spectral families, in general do not commute. More precisely, the Hilbert space expression for the probability that, say, we obtain the answer $CyGn$ when we perform first the Clinton measurement and then the Gore one, is \\cite{bb2012,wb2013}: $\\mu_{CyGn}(H)= \\langle H|M_C\\bar{M}_GM_C|H\\rangle$. Similarly, the probability to obtain the outcome $GnCy$, for the sequential measurement in reversed order, is: $\\mu_{GnCy}(H)= \\langle H|\\bar{M}_GM_C\\bar{M}_G|H\\rangle$. Since we have the operatorial identity $\\bar{M}_GM_C\\bar{M}_G-M_C\\bar{M}_GM_C=(M_G-M_C)[M_G,M_C]$, the difference $\\mu_{GnCy}(H)-\\mu_{CyGn}(H)$ will generally be non-zero if $[M_G,M_C]\\neq 0$, i.e. if the spectral families associated with the two measurements do not commute. In the following we will analyse whether non-compatibility within a standard quantum approach can cope in a satisfying way with these question order effects, and show that a simple `yes' to this question is not possible. Indeed, a deep problem already comes to the surface in relation to the phenomenon of `response replicability'.\n\nConsider again the Gore\/Clinton measurements: if a respondent says `yes' to the Gore question, then is asked the Clinton question, then again is asked the Gore question, the answer given to the latter will almost certainly be a `yes', independently of the answer given to the intermediary Clinton question. This phenomenon is called `response replicability'. If in addition to question order effects also response replicability is jointly modeled in  Hilbert space quantum mechanics, a contradiction can be detected, as shown in \\cite{plosone}. Let us indicate what are the elements that produce this contradiction. In standard quantum mechanics only if a state is an eigenstate of the considered measurement the outcome `yes' will be certain in advance. Also, measurements that can transform an arbitrary initial state into an eigenstate are ideal measurements called of the first kind.  According to response replicability, outcomes that once have been obtained for a measurement will have to become certain in advance if this same measurement is performed a second time. This means that the associated measurements should be ideal and of the first kind. For the case of the Gore\/Clinton measurements, and the situation of response replicability mentioned above, this means that the Gore measurement should be ideal and of the first kind. But  one can also consider the situation \nwhere first the Clinton measurement is performed, then the Gore measurement and afterwards the Clinton measurement again. A similar analysis leads then to the Clinton measurement needing to be ideal and of the first kind. This means however that after more than three measurements that alternate between Clinton and Gore, the state needs to have become an eigenstate of both measurements. As a consequence, both measurements can be shown to be represented by commuting operators. The proof of the contradiction between `response replicability' and 'non-commutativity' worked out in \\cite{plosone} is formal and also more general than the intuitive reasoning presented above -- for example, the contradiction is also proven when measurements are represented by positive-operator valued measures instead of projection valued measures, which is what we have considered here -- and hence indicates that the non-commutativity of the self-adjoint operators needed to account for the question order effects cannot be realised together with the `ideal and first kind' properties\nneeded to account for the response replicability within a standard quantum Hilbert space setting.      \n\nAlthough refined experiments would be needed to reveal the possible reasons for response replicability, it is worth to put forward some intuitive ideas, as we have been developing a quantum-like but more general than Hilbert space formalism within our Brussels approach to quantum cognition \\cite{asdb2014,asdb2015b,asdb2015c}, and we believe that we can cope with the above contradiction within this more general quantum-like setting in a very natural way. It seems to be a plausible hypothesis that response replicability is, at least partly, due to a multiplicity of effects, that however take place during the experiment itself, such as desire of coherence, learning, fear of being judged when changing opinion, etc. And a crucial aspect for both  question order effects and response replicability appearing\nin the Gore\/Clinton situation is that the sequential measurements need to be carried out with the same participant, who has to be tested again and again. This is different than\nthe situation in quantum physics, where order effects appear for non-commuting observables also when sequential measurements are performed with different apparatuses. \nHence, both question order effects and response replicability seem to be the consequence of `changes taking place in the way each subject responds probabilistically to the situation -- described by the state of the conceptual entity in our approach -- he or she is confronted with during a measurement'. Since the structure of the probabilistic response to a specific state is fixed in quantum mechanics, being determined by the Born rule, it is clear that such a change of the probabilistic response to a given measurement, when it is repeated in a sequence of measurements, cannot be accounted for by the standard quantum formalism. And it is exactly such structure of the probabilistic response to a same measurement with respect to a given state that can be varied in the generalized quantum-like theory that we have been developing \\cite{asdb2014,asdb2015b,asdb2015c}. This is the reason that, when we became aware of the contradiction identified in \\cite{plosone}, we were tempted to investigate whether in our generalized quantum-like theory the contradiction would vanish, and response replicability would be jointly\nmodelizable with question order effects. And indeed, we could obtain a positive result with respect to this issue \\cite{asdb2015a}, which we will now sketch in the next section. \n\n\n\n\n\\section{Beyond-quantum models\\label{beyond}}\nWe presented in Section \\ref{challenges} two paradigmatic situations in human cognition that cannot be modeled together using the standard quantum formalism. We want now to explain how the latter can be naturally extended to also deal with these situations, still remaining in the ambit of a unitary and coherent framework for cognitive processes. \n\nFor this, we introduce a formalism where the probabilistic response with respect to a specific experimental situation, i.e. a state of the conceptual entity under consideration, can vary, and hence can be different than the one compatible with the Born rule of standard quantum theory. This formalism, called \nthe `extended Bloch representation' of quantum mechanics \\cite{asdb2014}, \nexploits\nin its most recent formulation the fact that the states of a quantum entity (described as ray-states or density matrix-states) can be uniquely mapped into a convex portion of a generalized unit Bloch sphere, in which also measurements can be represented in a natural way, by means of appropriate simplexes having the eigenstates as vertex vectors. A measurement can then be described as a process during which an abstract point particle (representing the initial state of the quantum entity) enters into contact with the measurement simplex, which then, as if it was an elastic and disintegrable hyper-membrane, can collapse to one of its vertex points (representing the outcomes states) or to a point of one of its sub-simplexes (in case the measurement would be degenerate). \n \nWe do not enter here into the details of this remarkable process, and refer the reader to the detailed descriptions  in \\cite{asdb2015a,asdb2014,asdb2015b,asdb2015c}. For our present purposes, it \nwill be\nsufficient to observe that a measurement simplex, considered as an abstract membrane that can collapse as a result of some uncontrollable environmental fluctuations, can precisely model that aspect of a measurement that in the quantum jargon is called `wave function collapse'. More precisely, when the abstract point particle enters into contact with the `potentiality region' represented by such membrane, it creates some `tension lines' partitioning the latter into different subregions, one for each possible outcome. The collapse of the membrane towards one of the vertex points then depends on which subregion disintegrates first, so that the different outcome probabilities can be expressed as the relative Lebesgue measures of these subregions (the larger a subregion, the higher the associated probability). In other terms, this membrane's mechanism, with the tension lines generated by the abstract point particle, is a mathematical representation of a sort of `weighted symmetry breaking' process. Now, thanks to the remarkable geometry of simplexes, it can be proven that if the membrane is chosen to be uniform, thus having the same probability of disintegrating in any of its points (describing the different possible measurement-interactions), the collapse probabilities are exactly given by the Born rule. In other terms, the latter can be derived, and explained, as being the result of a process of actualization of potential hidden-measurement interactions, so that the extended Bloch representation constitutes a possible solution to the measurement problem.\n\nThus, when the membrane is uniform, the `way of choosing' an outcome is precisely the `Born way'. However, a uniform membrane is a very special situation, and it is natural to also consider membranes whose points do not all\nhave the same probability of disintegrating, i.e. membranes whose disintegrative processes are described by non-uniform probability densities $\\rho$, which we simply call $\\rho$-membranes. Non-uniform $\\rho$-membranes can produce outcome probabilities different from the standard quantum ones and give rise to probability models different from the Hilbertian one (even though the state space is a generalized Bloch sphere derived from the Hilbert space geometry\\footnote{More general state spaces can also be considered, in what has been called the `general tension-reduction' (GTR) model \\cite{asdb2015b,asdb2015c,asdb2015d}.}). But this is exactly what one needs in order to account, in a unified framework, for the situation we encounter when combining the phenomena of `response replicability' and `question order effects', as previously described and analysed in \\cite{plosone}.\n\nWe thus see that it is possible to naturally complete the quantum formalism to obtain a finer grained description of psychological experiments in which the probabilistic response of a measurement with respect to a state can be different to the one described by the Born rule. Additionally, our generalized quantum-like theory also explains why, despite the fact that individual measurements are possibly associated with different non-Born probabilities, the Born rule nevertheless appears to be a very good approximation to describe numerous experimental situations. This is related to the notion of `universal measurement', firstly introduced by one of us in \\cite{a1998} and further analyzed in \\cite{asdb2014,asdb2015b,asdb2015c,asdb2014b}. In a nutshell, a universal measurement is a measurement whose probabilities are obtained by averaging over the probabilities of all possible quantum-like measurements sharing a same set of outcomes, in a same state space. In other terms, a universal measurement corresponds to an average over all possible non-uniform $\\rho$-membranes, associated with a given measurement simplex. Following a strategy similar to that used in the definition of the `Wiener measure', it is then possible to show that if the state space is Hilbertian (more precisely, a convex set of states inscribed in a generalized Bloch sphere, inherited from a Hilbert space), then the probabilities of a universal measurement are precisely those predicted by the Born rule. \n\nIn \\cite{asdb2015a} we could show that the joint situation of question order effects and response replicability for the data collected with respect to the Gore\/Clinton measurements, and others, is modelizable\nwithin our generalized quantum theory by introducing non-Born type measurements. However, we  were also able to provide a better modeling of the question order effects data as such. Indeed, using standard Born-probability quantum theory it was only possible to model approximately these data in earlier studies \\cite{wb2013}. This is due to the existence of a general algebraic equality about sequential measurements in standard quantum mechanics which is the following \\cite{asdb2015a,wb2013,Niestegge2008}:\n\\begin{equation}\nQ\\equiv M_GM_CM_G - M_CM_GM_C + \\bar{M}_G\\bar{M}_C\\bar{M}_G - \\bar{M}_C\\bar{M}_G\\bar{M}_C=0\n\\label{QQ-equality}\n\\end{equation}\nwhere $\\{M_G, \\bar{M}_G = \\mathbbmss{1}-M_G\\}$ and $\\{M_C, \\bar{M}_C = \\mathbbmss{1}-M_C\\}$ are the spectral families associated with the Hilbert model of the Gore and Clinton measurements introduced in Section \\ref{challenges}.\nTaking the average $q=\\langle H|Q|H\\rangle$, one thus obtains, more specifically: \n\\begin{equation}\nq\\equiv\\mu_{GyCy}(H) - \\mu_{CyGy}(H) + \\mu_{GnCn}(H) - \\mu_{CnGn}(H) =0.\n\\label{QQ-equality2}\n\\end{equation}\nThis equality has been called the `QQ-equality', and can be used as a test for the quantumness of the probability model, but only in the sense that a quantum model, necessarily, has to obey it, although the fact that it does so is not a guarantee that the model will be Hilbertian. Inserting the experimental values (\\ref{Prob-Clinton-Gore1})-(\\ref{Prob-Clinton-Gore2}) into (\\ref{QQ-equality2}), one finds $q=0.0032\\neq 0$. \nThis value is small (being only $0.32\\%$ of the maximum value $q$ can take, which is 1), which is the reason that approximate modeling can be obtained within standard quantum mechanics \\cite{wb2013}. Note however that (\\ref{QQ-equality}) does not depend on the dimension of the Hilbert space considered, which means that even in higher dimensional Hilbert spaces, if degenerate measurements \nare\nconsidered, an exact modeling \nwould\nstill be impossible to obtain. We have reasons to believe that also question order effects, with the QQ-equality standing in the way of an exact modeling of the data, contain an indication for the need to turn to a more general quantum-like theory, such as the one we used to cope with the joint phenomenon of question order effects and response replicability. We \npresent some arguments in this regard in the following of this section.\n\nFirst,  \nwe\nnote that in case one would choose a two-dimensional Hilbert space, which is the natural choice when dealing with two-outcome measurements, additional equalities can be written that are this time strongly violated by the data. As an example, consider the quantity \\cite{asdb2015a}:\n\\begin{eqnarray}\nq'&\\equiv& \\mu_{CyGn}(H)\\mu_{CnGn}(H) - \\mu_{CnGy}(H)\\mu_{CyGy}(H)\\\\\n&=& \\langle H|M_C\\bar{M}_GM_C|H\\rangle \\langle H|\\bar{M}_C\\bar{M}_G\\bar{M}_C|H\\rangle - \\langle H|\\bar{M}_CM_G\\bar{M}_C|H\\rangle \\langle H|M_CM_GM_C|H\\rangle\n\\label{QQ-equality3}\n\\end{eqnarray}\nIf the Hilbert space is two-dimensional, one can write $M_G=|G\\rangle\\langle G|$, $\\bar{M}_G=|\\bar{G}\\rangle\\langle \\bar{G}|$, as well as $M_C=|C\\rangle\\langle C|$, $\\bar{M}_C=|\\bar{C}\\rangle\\langle \\bar{C}|$. Replacing these expressions into (\\ref{QQ-equality3}) one finds, after some easy algebra, that $q'=0$. However, inserting the experimental values (\\ref{Prob-Clinton-Gore1})-(\\ref{Prob-Clinton-Gore2}) into (\\ref{QQ-equality3}), one finds $q'=-0.073\\neq 0$, which not only \nis not zero, but also $29.2\\%$ of the maximum value that $q'$ can take (which is $0.25$).\n\nSecond, let us repeat our intuitive reasoning as to why measurements in the situation of response replicability carry non-Bornian probabilities. Due to the local \ncontexts\nof the collection of sequential measurements, Gore, Clinton, and then Gore again, the third measurement internally changes into a non-Bornian one, and more specifically a deterministic one for the considered state, since response replicability means that for all subsequent Gore measurements the same outcome is assured. It might well be the case, although an intuitive argument would be more complex \nto give\nin this case, that also for the situation of question order effects, precisely because they only appear if a same human mind is sequentially interrogated, non-Bornian probabilities would be required. An even stronger hypothesis, which we plan to investigate in the future, is that most individual human minds, and perhaps even all, would carry in general non-Bornian probabilities, so that the success of standard quantum mechanics and Bornian probabilities would be mainly an effect of averaging over a sufficiently large set of different human minds, which effectively is what happens in a standard psychological experiment. If this last hypothesis is true, the violation of the Born rule for question order effects and  response replicability would be quite natural, since the same human mind is needed to provoke these effects. Indeed, our analysis in \\cite{asdb2015b,asdb2015c} shows that standard quantum probabilities in the modeling of human cognition can be explained by considering that in numerous experimental situations the average over the different participants will be quite close to that of a universal measurement, which as we observed is exactly given by the Born rule. In other terms, even if the probability model of an individual psychological measurement could be non-Hilbertian, it will generally admit a first order approximation, and when the states of the conceptual entity under investigation can be described by means of a Hilbert space structure, this first order approximation will precisely correspond to the quantum mechanical Born rule. \n\nIf the above considerations provide an interesting piece of explanation as to why the Born rule is generally successful also beyond the micro-physical domain, at the same time it also contains a plausible reason of why it will possibly be not successful in all experimental situations, i.e. when the average is either not large enough, or when the experiment is so conceived that it doesn't apply as such. This could be the typical situation of question order effects and response replicability, since in this case we do not consider an average over single measurements, but over sequential (conditional) measurements. And this could be an explanation of why Hilbertian symmetries like those described above can be easily violated and that it will not be possible, by means of the Born rule, to always obtain an exact fit of the data \\cite{asdb2015a,asdb2015d}.\n\nAdditionally, as we said, it allowed us to precisely fit the data by using the extended Bloch representation,\nand more specifically \nsimple one-dimensional locally uniform membranes inscribed in a 3-dimensional Bloch sphere that \ncan\ndisintegrate (i.e. break) only inside a connected internal region \\cite{asdb2015a}. Thanks to this modeling, we could also understand that the reason the Clinton\/Gore and similar data appear to almost obey the QQ-equality (\\ref{QQ-equality2}) is quite different from the reason the equality is obeyed by pure quantum probabilities. Indeed, in a pure quantum model two specific contributions to the $q$-value (\\ref{QQ-equality2}), called the `relative indeterminism' and `relative asymmetry' contributions, are necessarily both identically zero, whereas we \ncould\nshow, using our extended model, that for the data (\\ref{Prob-Clinton-Gore2}),\nand similar data,  \nthese two contributions are both very different from zero, but happen to almost cancel each other, thus explaining why the $q=0$ equality is almost obeyed, although the probabilities are manifestly non-Bornian \\cite{asdb2015a}.\n\n\n\n\\section{Final considerations\\label{final}}\nIn this article we explained the essence of the operational-realistic approach to cognition developed in Brussels, which in turn originated from the foundational approach to physics elaborated initially in Geneva and then in Brussels (in what has become known as the `Geneva-Brussels school'). Our emphasis was that this approach is sufficiently general, and fundamental, to provide a unitary framework that can be used to coherently describe, and realistically interpret, not only standard quantum theory, but also its natural extensions, like the extended Bloch model and the GTR-model. In this final section we offer some additional comments on our approach to cognition, taking into consideration the confusion that sometimes exists between `ad hoc (phenomenological) models' and `theoretical (first principle) models', as well as the critique that a Hilbertian model (and a fortiori its possible extensions) is suspicious because it allows `too many free parameters' to obtain an exact fit (and not just an approximate fit) for all the experimental data. \n\nIn that respect, it is worth emphasizing that the principal focus of our `theory of human cognition' is not to model as precisely as possible the data gathered in psychological measurements. A faithful modeling of the data is of course an essential part of it, but our aim is actually more ambitious. In putting forward our methodology, consisting in looking at instances of decision-making as resulting from an interaction of a decision-maker with a conceptual entity, we look first of all for a theory truly describing `the reality of the cognitive realm to which a conceptual entity belongs', and additionally also `how human minds can interact with the latter so that decision-making can occur'.\n\nIn this sense, each time we have put forward a model for some specific experimental data, it has always been our preoccupation to also make sure that (i) the model was extracted following the logic that governs our theory of human cognition, and (ii) that whatever other experiments would be performed by a human mind interacting with that same cognitive-conceptual entity under consideration, also the data of these hypothetical additional experiments could have been modeled exactly in the same way. Clearly, this requirement -- that `all possible experiments and data' have to be modeled in an equivalent way -- poses severe constraints to our approach, and it is not a priori evident that this would always be possible. However, we are convinced that the fundamental idea underlying our methodology, namely that of looking upon a decision as an interaction of a human mind with a conceptual entity in a specific state (with such state being independent of the human minds possibly interacting with it), equips the theory of exactly those degrees of freedom that are needed to model `all possible data from all possible experiments'.\n\nAs we already explained in the foregoing, in all this we have been guided by how physical theories deal with data coming from the physical domain. They indeed satisfy this criterion and are able to model all data from all possible experiments that can be executed on a given physical entity. What we have called `conceptual entity' is what in physics corresponds to the notion of `physical entity'. Now, in our approach we might be classified as adhering to an idealistic philosophy, i.e. believing that the conceptual entities ``really exist,'' and are not mere creations of our human culture. Our answer to this objection is the following: to profit of the strength of the approach it is not mandatory to take a philosophical stance in the above mentioned way, in the sense that we are not obliged to attribute more existence to what we call a conceptual entity than that attributed, for example, to `human culture' in its entirety. The importance of the approach lies in considering such a conceptual entity as independently existing from any interaction with a human mind, and describe the continuously existing interactions with human minds as processes of the `change of state of the conceptual entity', and whenever applicable also as processes of the `change of context'. And again, let us emphasize that this `hidden-interaction' methodology is inspired by its relevance to physical theories. Our working hypothesis is that in this way it will be possible to advantageously model, and better understand, all of human cognition experimental situations.\n\nHaving said this, we observe that the interpretation of the quantum formalism that is commonly used in cognitive domains is a subjectivist one, very similar to that interpretation of quantum theory known as `quantum Bayesianism', or `QBism' \\cite{FuchsEtal2014}. In a sense, this interpretation is the polar opposite of our realistic (non-subjectivistic) operational approach. Indeed, QBism originates from a strong critique \\cite{fms2014} of the famous Einstein-Podolsky-Rosen reality criterion \\cite{epr1935}, whereas at the foundation of the Geneva-Brussels approach there is the idea of taking such criterion not only extremely seriously, but also of using it more thoroughly, as a powerful demarcating tool separating `actually existing properties' from `properties that are only available to be brought into actual existence', and therefore exist in a potential sense \\cite{sdb2011}. In other terms,  a quantum state is not considered in QBism as a description of the actual properties of a physical entity, but of the beliefs of the experimenter about it. Similarly, for the majority of authors in quantum cognition, a quantum state is a description of the state of belief of a participant, and not of the actual state of the conceptual entity that interacts with the participants. In ultimate analysis, this difference of perspectives is about taking a clear position regarding the key notion of `certainty': is certainty (probability $1$ assignments) just telling us something about the very firm belief of a subject, or also about some objective properties of the world (be it physical or cultural)? In the same way, are probabilities only shared personal beliefs, based on habit, or also elements of reality (considering that in principle their values can be predicted with certainty)? Although we certainly agree that it is not necessary to take a final stance on these issues to advantageously exploit the quantum mathematics in the modeling of many experimental situations, both in physics and cognition, we also think that the explicative power of a pure subjectivist view rapidly diminishes when we have to address the most remarkable properties of the physical and conceptual entities, like non-locality (non-spatiality) and the non-compositional way with which they can combine. \n\nIt is important to emphasize that the subjectivist view is also a consequence of the absence, in the standard quantum formalism, of a meaningful description of what goes on `behind the scenes' during a measurement. On the other hand, the hidden-measurement paradigm, as implemented in the extended Bloch representation \\cite{asdb2014}, or even more generally in the GTR-model \\cite{asdb2015b,asdb2015c,asdb2015d}, offers a credible description of the dynamics of a measurement process, in terms of a process of actualization of potential interactions, thus explaining a possible origin of the quantum indeterminism. This certainly allows understanding the so-called `collapse of the state vector' as an objective process, either produced by a macroscopic apparatus in a physics laboratory, or by a mind-brain apparatus in a psychological laboratory. As we tried to motivate in the second part of this article, this completed version of the quantum formalism also allowed us to describe those aspects of a psychological measurements -- the possible different ways participants can choose an outcome -- that would be impossible to model by remaining within the narrow confines not only of the standard formalism, but also of a strict subjectivistic interpretation of it. \n\nTo conclude, a final remark is in order. Quantum cognition is undoubtedly a fascinating field of investigation also for physicists, as it offers the opportunity to take a new look at certain aspects of the quantum formalism and use them to possibly make discoveries also in the physical domain. We already mentioned the example of `entangled measurements', that were necessary to exactly model certain correlations. Entangled (non-separable) measurements are usually not considered in the physics of Bell inequalities, while they are widely explored in quantum cryptography, teleportation and information. However, it is very possible that this stronger form of entanglement will prove to be useful for the interpretation of certain non-locality tests and the explanation of `anomalies' that were identified in EPR-Bell experiments \\cite{as2014}. Also, for what concerns the notion of `universal measurement', which is quite natural in psychological measurements, since data are obtained from a collection of different minds, could it be that `universal averages' also happen in the physical domain? In other terms, could it be that a single measurement apparatus is actually more like `a collection of different minds' than `a single Born-like mind'? Considering that the origin of the observed deviations from the Born rule, in situations of sequential measurements, can be understood as the ineffectiveness of the averaging process in producing the Born prescription, is it possible to imagine, in the physics laboratory, similar experimental situations where these deviations would be equally observed, thus confirming that the hypothesis of `hidden measurement-interactions' would be a pertinent one also beyond the psychological domain? Whatever the verdict will be, we certainly live in a very stimulating time for foundational research; a time where the conceptual tools that once helped us building a deeper understanding of the `microscopic layer' of our physical reality are now proving to be instrumental for understanding our human `mental layer'; but also a time where all this is also coming back to physics, not only in the form of possible new experimental findings, but also of possible new and deeper understandings \\cite{aerts2009,aerts2010a,aerts2010b,aerts2013,aerts2014}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}