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+{"text":"\\section{Introduction}\\label{sec1}\nTwo-dimensional Fermions and Boson-Fermion correspondence are well-known in mathematical physics. Meanwhile Young diagrams and symmetric functions are of interest to many researchers and have many applications in mathematics including combinatorics and representation theory of the symmetric and general linear group. There are many relations between them.\n\nThe Kakomtsev-Petviashvili (KP) hierarchy\\cite{DKJM} is one of the most important integrable hierarchies and it arises in many different fields of mathematics and physics such as enumerative algebraic geometry, topological field and string theory. Schur functions have close relations with the tau functions of KP hierarchy. Schur functions in variables $x_1,x_2,\\cdots,x_n$ are well known to give the characters of finite dimensional irreducible representations of the general linear groups $GL(n)$\\cite{Mac,FH}. From \\cite{MJD,NR,Jing}, Schur functions can be realized from vector operators and these vertex operators correspond to free Fermions acting on Bosonic Fock space. It turns out that the Boson-Fermion correspondence and the Jacobi-Trudi formula are the same thing, which tells us that Schur functions are solutions of differential equations in KP hierarchy, and the linear combinations of Schur functions with coefficients satisfied some relations (p\\\"uker relations) are also tau functions of KP hierarchy.\n\nThe $\\pi$-type symmetric functions are upgraded from Schur functions in the same setting. The linear basis of $\\pi$-type symmetric functions provides the structure of the universal character ring of group $H_\\pi$ (subgroup of $GL(n)$)\\cite{BJK,HW,Litt}. Like Schur functions, $\\pi$-type symmetric functions can also be realized from vertex operators which are constructed in \\cite{FJK}. Then free Fermions can be constructed and there exists for sure an integrable system. In this paper, we will construct this integrable system, and find that the $\\pi$-type symmetric functions are the solutions of differential equations in this integrable system.\n\nThe paper is organized as follows. In section \\ref{sect2}, we recall the $\\pi$-type symmetric functions and vertex operators associated with them. In section \\ref{sect3}, we recall Schur functions and KP hierarchy. In section \\ref{sect4}, we define $\\pi$-type Fermions and construct $\\pi$-type Boson-Fermion correspondence from which we can calculate $\\pi$-type symmetric functions. In section \\ref{sect5}, we construct the $\\pi$-type KP hierarchy and analyze its tau functions.\n\\section{$\\pi$-type symmetric function and vertex operator}\\label{sect2}\nWe begin this section with some notational preliminaries\\cite{FJK}. Let $\\Lambda({\\bf x})$ be the ring of symmetric functions of a countably infinite alphabet of variables ${\\bf x}=\\{x_1,x_2,\\cdots\\}$. The power sum symmetric functions $p_n({\\bf x})$ are\n\\[\np_n({\\bf x})=\\sum_{k}x_k^n.\n\\]\nThe operators $p_n$ and $n\\partial_{p_n}$ ($\\partial_{p_n}:=\\partial\/\\partial_{p_n}$) give a representation of the infinite dimensional Heisenberg algebra generated by $a_n, n\\in\\Z, n\\neq 0$ with the relation\n\\begin{equation}\\label{hei}\n[a_n,a_m]=n{\\delta}_{n+m,0}.\n\\end{equation}\nThe vertex operator are defined with the help of Heisenberg algebras\n\\begin{eqnarray}\n&&M(z,{\\bf x})=\\prod_k\\frac{1}{1-zx_k}=\\exp(\\sum_{n=1}^\\infty\\frac{p_n}{n}z^n)\\\\\n&&L(z,{\\bf x})=\\prod_k{(1-zx_k)}=\\exp(-\\sum_{n=1}^\\infty\\frac{p_n}{n}z^n)\n\\end{eqnarray}\nwhere $L(z,{\\bf x})=M(z,{\\bf x})^{-1}$. In the special case $z=1$, we set $M({\\bf x})=M(1,{\\bf x})$ and $L({\\bf x})=L(1,{\\bf x})$. When ${\\bf x}$ is to be understood, we often write $L(z,{\\bf x})$ and $M(z,{\\bf x})$ by $L(z)$ and $M(z)$ respectively for short.\n\nFor a Young diagram $\\lambda$, let $\\mathcal{T}^\\lambda$ denote the set of semistandard tableaux $T$ of shape $\\lambda$ with entries from $\\{1,2,\\cdots,n\\}$, and let ${\\bf x}^T=x_1^{\\#1}x_2^{\\#2}\\cdots x_n^{\\#n}$ where $\\#k$ is the number of entries $k$ in $T$, then the Schur function\n\\[S_\\lambda({\\bf x})=\\sum_{T\\in\\mathcal{T}^\\lambda}{\\bf x}^T\n\\]\nSchur functions are an orthonormal basis of the ring $\\Lambda({\\bf x})$. The operation of symmetric function skew is defined by duality as\n\\[\n\\langle g^\\perp f|h\\rangle=\\langle f|g\\cdot f\\rangle\n\\]\nGiven two Schur functions $S_\\lambda$ and $S_\\mu$, the skew Schur function $S_\\mu^\\perp S_\\lambda=S_{\\lambda\/\\mu}$.\n\nThe plethysm is defined as follows\\cite{Mac}. Let $f({\\bf x})=\\sum_i y_i$. Consider these monomials as elements of a new countably infinite alphabet ${\\bf y}=\\{y_1,y_2,\\cdots\\}$. Then for any Schur function $S_\\lambda$, the plethysm of $f$ by $S_\\lambda$, $S_\\lambda[f]({\\bf x}):=S_\\lambda ({\\bf y})$ is the symmetric function of the composite alphabet. For any Young diagram $\\pi$,\n\\begin{eqnarray}\n&&M_\\pi(z,{\\bf x})=\\prod_{T\\in \\mathcal {T}^\\pi}\\frac{1}{1-z{\\bf x}^T}=\\sum_{r\\geq 0}z^rS_{(r)}[S_\\pi]({\\bf x})\\label{ml1}\\\\\n&&L_\\pi(z,{\\bf x})=\\prod_{T\\in \\mathcal {T}^\\pi}{(1-z{\\bf x}^T)}=\\sum_{r\\geq 0}(-1)^rz^rS_{(1^r)}[S_\\pi]({\\bf x})\n\\end{eqnarray}\nfor arbitrary $\\pi$, we have\n\\[\nS_\\lambda^\\pi({\\bf x})=L^\\perp_\\pi({\\bf x})S_\\lambda\n\\]\nthe symmetric functions of this type correspond to the branching rule from a module of the general linear group to (generically indecomposable) module of the $H_\\pi$ subgroup.\n\nDefine\n\\begin{equation}\nV_\\pi(z)=M(z)L^\\perp(z^{-1})\\prod_{k>0}L^\\perp_{\\pi\/(k)}(z^k),\n\\end{equation}\nthen we have\\begin{equation}\nS_\\lambda^\\pi=[{\\bf z}^\\lambda]V_\\pi(z_1)V_\\pi(z_2)\\cdots V_\\pi(z_k)\\cdot 1\n\\end{equation}\nwhere $[{\\bf z}^\\lambda]$ means selecting the coefficient of $z_1^{\\lambda_1}z_2^{\\lambda_2}\\cdots z_k^{\\lambda_k}$. In order to obtain the complete set of exchange relations between the $\\pi$-type vertex operators it is necessary to introduce suitably constructed dual vertex operators $V^*_\\pi(z)$.\n\\begin{thm}(theorem 1 in \\cite{FJK})\nFor each partition $\\pi$ and any $z$ let\n\\begin{eqnarray}\nV_\\pi(z)&:=&M(z)L^\\perp(z^{-1})\\prod_{k>0}L^\\perp_{\\pi\/(k)}(z^k),\\\\\nV_\\pi^*(z)&:=&L(z)M^\\perp(z^{-1})\\prod_{k\\geq 0}M^\\perp_{\\pi\/(1^{2k+1})}(z^{2k+1})\\prod_{k>0}L^\\perp_{\\pi\/(k)}(z^{2k}),\n\\end{eqnarray}\nwhere it is to be understood that all the Schur functions in $M(w),L(w),M^\\perp(w)$ and $L^\\perp(w)$, for any $w$, depend on the same sequence of variables $(x_1,x_2,\\cdots)$ whose specification, for the sake of simplicity, has been suppressed.\n\nFurthermore, let the associated full vertex operators $X^\\pi(z)$ and $X^{*\\pi}(z)$, constructed by adjoining zero mode contributions, be defined by\n\\begin{eqnarray}\nX^\\pi(z)&=&V_\\pi(z)e^Kz^{H_0}:=\\sum_{n\\in\\Z+1\/2}X_j^\\pi z^{-j-1\/2+H_0}\\\\\nX^{*\\pi}(z)&=&V_\\pi^*(z)e^{-K}z^{-H_0}:=\\sum_{n\\in\\Z+1\/2}X_j^{*\\pi} z^{-j-1\/2-H_0}\n\\end{eqnarray}\nthen we have the modes $X^\\pi(z)$ and $X^{*\\pi}(z)$ fulfil the free Fermion anticommutation relations of  a complex Clifford algebra:\n\\[\n\\{X_i^\\pi, X_j^\\pi\\}=0,\\ \\{X_i^{*\\pi}, X_j^{*\\pi}\\}=0,\\ \\{X_i^\\pi, X_j^{*\\pi}\\}=\\delta_{i+j,0}.\n\\]\nwhere $\\{\\cdot,\\cdot\\}$ signifies an anticommutator.\n\\end{thm}\n\\section{Schur function, vertex operator and KP hierarchy}\\label{sect3}\nLet $\\C[{\\bf x}]=\\C[x_1,x_2,\\cdots]$ be the polynomial ring of infinitely many variables.\nAlthough the number of variables is infinite, each polynomial itself is a finite sum of monomials, so involves only finitely many of the variables.\nBosons are operators  $\\{a_n\\}_{n\\in{\\Z},n\\neq 0}$ satisfying relations (\\ref{hei}).\nThe representation of Bosons on $\\C[{\\bf x}]$ is $a_n=\\frac{\\partial}{\\partial x_n}, a_{-n}=nx_n$ for $n>0$. Denote $\\frac{\\partial}{\\partial x_n}$ by $\\partial_n$. Define\n\\begin{equation}\\label{pqn}\n\\text{exp}(\\sum_{m\\geq 1}x_m k^m)=\\sum_{n\\geq 0} P_{(n)} k^n,\\quad \\text{exp}(\\sum_{m\\geq 1}\\frac{\\partial_m}{m} k^m)=\\sum_{n\\geq 0} Q_{(n)} k^n\n\\end{equation}\nwhen $i<0$, we set $P_{(i)}=0,\\ Q_{(i)}=0$. For any $m$, $P_{(m)}$ is a polynomial of variables $x_1,x_2,\\cdots, x_m$, $P_{(m)}=P_{(m)}(x_1,x_2,\\cdots,x_m)$.\nIn fact,\nReplacing\n$\nx_n$ with the power sum symmetric function $p_n$,\nwe get\n$\n P_{(n)} =S_{(n)}\n$,\nwhere $S_{(n)}$ is the Schur polynomial of Young diagram $(n)$. Let ${\\bf x}=(x_1,x_2,\\cdots)$ and $x_n=p_n(\\tilde { x}_1,\\tilde x_2,\\cdots)$. Therefore,\nfor any Young diagram $\\lambda=(\\lambda_1,\\lambda_2,\\cdots,\\lambda_l)$,\n\\begin{equation}\\label{ps}\nP_{\\lambda}({\\bf x})=S_{\\lambda}(\\tilde {\\bf x})=\\text{det}(h_{\\lambda_i-i+j}(\\tilde{\\bf x}))_{1\\leq i,j\\leq l}\n\\end{equation}\nwhere $h_{n}(\\tilde{\\bf x})$ is the $n$th complete symmetric function, i.e.,\n\\[\nh_{n}(\\tilde{\\bf x})=\\sum_{i_1\\leq\\cdots\\leq i_n}\\tilde x_{i_1}\\cdots\\tilde x_{i_n}.\n\\]\n\nIn the following,\nwe do not distinguish Young diagram $\\lambda$, $P_\\lambda$ and $S_\\lambda$. The actions of $P_\\lambda$ and $Q_\\lambda$ on Young diagram $\\mu$ are defined to be\\cite{WWY2,W}\n\\begin{equation}\nP_\\lambda\\cdot\\mu:=\\lambda\\cdot\\mu,\\ \\ \\ Q_\\lambda\\cdot\\mu:={\\mu\/\\lambda}.\n\\end{equation}\nwhere the multiplication $\\lambda\\cdot\\mu$ satisfies the Littlewood-Richardson rule.\n\nIntroduce the vertex operators\n\\begin{eqnarray}\nV^{\\pm}(k)=\\sum_{n\\in\\Z}V_n^{\\pm}k^n=\\exp(\\pm\\sum_{n=1}^\\infty x_nk^n)\\exp(\\mp\\sum_{n=1}^\\infty \\frac{\\partial_n}{n}k^{-n}).\n\\end{eqnarray}\nThe operator $X_n^+$ is a raising operators of the Schur function, i.e.,\n\\begin{equation}\nS_\\lambda({\\bf \\tilde x})=P_\\lambda({\\bf x})=V_{\\lambda_1}^+V_{\\lambda_2}^+\\cdots V_{\\lambda_l}^+\\cdot1\n\\end{equation}\nfor a partition $\\lambda=(\\lambda_1,\\lambda_2,\\cdots,\\lambda_l)$.\n\nBy Boson-Fermion correspondence, there are three vector spaces which are isomorphic to each other, the polynomial ring $\\C[{\\bf x}]=\\C[x_1,x_2,\\cdots]$ of infinitely many variables ${\\bf x}=(x_1,x_2,\\cdots)$ which is called the Bosonic Fock space, the charge zero part of the Fermionic Fock space $\\mathcal{F}$ which is the vector space based by the set of Maya diagrams, and the vector space $Y$ based by the set of Young diagrams. Therefore a Maya diagram $|u\\rangle$ can be written as $$|u\\rangle=|\\lambda, n\\rangle=|P_\\lambda, n\\rangle$$\nwhere $n$ is the charge of $|u\\rangle$. In special case, if the charge $n=0$, we also write the Maya diagram  $|u\\rangle$ as $|\\lambda\\rangle$.\n\nLet $f(z,{\\bf x})$ be a function in space\n$\\C[z,z^{-1},x_1,x_2,\\cdots].$\nDefine operators\n\\begin{equation}\ne^K f(z,{\\bf x}):=zf(z,{\\bf x}),\\ \\ k^{H_0}f(z,{\\bf x}):=f(kz,{\\bf x}).\n\\end{equation}\nDefine the generating functions\\cite{MJD}\n\\begin{eqnarray}\nX(k)&=&\\sum_{j\\in \\Z+\\frac{1}{2}}X_{j}k^{-j-\\frac{1}{2}}=V^{+}(k)e^K k^{H_0},\\label{xx1}\\\\\nX^*(k)&=&\\sum_{j\\in \\Z+\\frac{1}{2}}X^*_{j}k^{-j-\\frac{1}{2}}=V^{-}(k)e^K k^{H_0}.\\label{xx2}\n\\end{eqnarray}\nIt can be checked that\\begin{equation}\n\\{X_i,X_j\\}=0,\n\\{X_i^*, X_j^*\\}=0,\n\\{X_i,X_j^*\\}=\\delta_{i+j,0}.\n\\end{equation}\n\n\\begin{dfn}\nFor an unknown function $\\tau=\\tau({\\bf x})$,  the bilinear equation\n\\begin{equation}\n\\sum_{j\\in{\\Z+\\frac{1}{2}}}X^*_j\\tau\\otimes X_{-j}\\tau=0\n\\end{equation}\nis called the KP hierarchy.\n\\end{dfn}\n\\section{$\\pi$-type Boson-Fermion correspondence}\\label{sect4}\nWe begin this section by recalling the definition of Maya diagram. Let an increasing sequence of half-integers\\cite{MJD}\n\\[\n{\\bf u}=\\{u_j\\}_{j\\geq 1},\\ \\text{with}\\ u_1<u_2<u_3<\\cdots,\n\\]\nsatisfy $u_{j+1}=u_j+1$ for all sufficiently large $j$.  Putting a black stone on the position $u_j$ for all $j$ and a white stone on every other half-integer position, we get a Maya diagram and denote it by $|{\\bf u}\\rangle$. Specially, the Maya diagram $|1\/2,3\/2,5\/2,\\cdots\\rangle$ is denoted by $|\\text{vac}\\rangle$.\n\nFermions $\\psi_j, \\psi^*_j,\\ j\\in\\frac{1}{2}+\\Z$ are operators satisfying \\[\n\\{\\psi_i,\\psi_j\\}=0,\n\\{\\psi_i^*, \\psi_j^*\\}=0,\n\\{\\psi_i,\\psi_j^*\\}=\\delta_{i+j,0}.\n\\]\nThe actions of Fermions $\\psi_j, \\psi^*_j$ on Maya diagrams are determined by\n\\begin{eqnarray}\n\\psi_j|{\\bf u}\\rangle &=&  \\begin{cases}\n (-1)^{i-1}| \\cdots,u_{i-1},u_{i+1},\\cdots\\rangle & \\text{ if } u_i=-j \\ \\text{for some} \\ i, \\\\\n  0& \\text{ otherwise},\n\\end{cases}  \\\\\n\\psi^*_j |{\\bf u}\\rangle &=&  \\begin{cases}\n (-1)^{i}| \\cdots,u_{i},j,u_{i+1},\\cdots\\rangle & \\text{ if } u_i<j<u_{i+1} \\ \\text{for some} \\ i, \\\\\n  0& \\text{ otherwise}.\n\\end{cases}\n\\end{eqnarray}\nThe generating functions of Fermions are\n\\[\n\\psi(k)=\\sum_{j\\in\\Z+1\/2}\\psi_jk^{-j-1\/2},\\ \\psi^*(k)=\\sum_{j\\in\\Z+1\/2}\\psi^*_jk^{-j-1\/2}.\n\\]\nThe normal order is defined as usual. Let\n\\begin{equation}\\label{hhx}\nH_n=\\sum_{j\\in\\Z+1\/2}:\\psi_{-j}\\psi_{j+n}^*: \\ \\ \\text{and}\\ \\ \\ H(x)=\\sum_{n=1}^\\infty x_nH_n.\\end{equation}\nFor Maya diagrams $|{\\bf u}\\rangle$ and $|{\\bf v}\\rangle$, the pair $\\langle{\\bf v}|{\\bf u}\\rangle$ is defined by the formula\n\\[\n\\langle{\\bf v}|{\\bf u}\\rangle=\\delta_{v_1+u_1,0}\\delta_{v_2+u_2,0}\\cdots.\n\\]\n\nThe Boson-Fermion correspondence is the correspondence $$\\Phi:\\mathcal{F}\\rightarrow \\C[z,z^{-1},x_1,x_2,x_3,\\cdots]$$\ngiven by\\[\n|u\\rangle\\mapsto\\Phi(|u\\rangle):=\\sum_{l\\in\\Z}\\langle l|\\exp(H(x))|u\\rangle\n\\]\nwhich is an isomorphism of vector spaces, where $|l\\rangle$ is the Maya diagram obtained from $|\\text{vac}\\rangle$ by sliding the diagram bodily $l$ steps to the right (that is, $-l$ steps to the left if $l<0$), and the operator $H(x)$ is defined in (\\ref{hhx}).\n\nUnder Boson-Fermion correspondence, the Fermions $\\psi_j, \\psi^*_j$, respectively, correspond to the operators $X_j, X_j^*$ defined in equations (\\ref{xx1}) and (\\ref{xx2}).\n\nLet $\\lambda$ be a Young diagram, and $\\lambda'$ be its conjugate. The Frobenius notation $\\lambda=(n_1,\\cdots,n_l|m_1,\\cdots,m_l)$ describes the Young diagram $\\lambda$ by $n_i=\\lambda_i-i$, $m_i=\\lambda'_i-i$, where $l$ is the number of the boxes in the NW-SE diagonal line of $\\lambda$.\n\nThe Boson-Fermion correspondence tells us that the basis vector\n\\[\n\\psi_{n_1}\\cdots\\psi_{n_l}\\psi^*_{m_1}\\cdots\\psi^*_{m_l}|\\text{vac}\\rangle \\ \\  \\text{for} \\ n_1<\\cdots<n_l<0\\ \\text{and}\\ m_1<\\cdots<m_l<0\n\\]\nof Fermionic Fock space of charge zero goes over into the Schur function $S_\\lambda$ multiplied by $(-1)^{\\sum_{i=1}^l(m_i+{1}\/{2})+l(l-1)\/2}$, where $\\lambda=(-n_1-1\/2,\\cdots,-n_l-1\/2|-m_1-1\/2,\\cdots,-m_l-1\/2)$. Under the correspondence between Young diagrams and Maya diagrams, we can also write $S_\\lambda$ as\n\\begin{equation}\\label{slam}\nS_\\lambda({\\bf x})=\\langle \\text{vac}|e^{H(x)}|\\lambda \\rangle\n\\end{equation}\n\nIn the following, we will define $\\pi$-type Boson-Fermion correspondence, from which we will get $\\pi$-type symmetric functions.\nDefine\n\\begin{equation}\nL^\\perp (z)=\\exp(-\\sum_{n\\geq 1}\\frac{H_n}{n}z^n)\n\\end{equation}\nwhich corresponds to $L^\\perp (z)$ defined in Section 2 by Boson-Fermion correspondence, that is why we use the same notation.\n\nAccording to the definition of operators $P_\\lambda({\\bf x})$ in equations (\\ref{pqn}) and (\\ref{ps}), The operator $P_\\lambda(H_n)$ is defined by replacing $x_i$ in $P_\\lambda({\\bf x})$ with $\\frac{1}{i}H_{in}$,  which is a plethysm in fact. For example, $P_{(2)}({\\bf x})=\\frac{1}{2}x_1^2+x_2$, then $P_{(2)}(H_n)=\\frac{1}{2}H_{n}^2+\\frac{1}{2}H_{2n}$.\n\nDefine \\begin{equation}\nL^\\perp_\\pi(z)=\\exp(-\\sum_{n=1}^\\infty\\frac{z^n}{n}P_\\pi(H_n))\n\\end{equation}\nwhere $\\pi$ is a Young diagram. In special case, if $\\pi=\\emptyset $, the operator $L^\\perp_\\pi(z)=1$; if $\\pi=(0)$, the operator $L^\\perp_\\pi(z)=1-z$ and if $\\pi=(1)$, the operator $L^\\perp_\\pi(z)=L^\\perp(z)$. When $z=1$, we denote $L^\\perp_\\pi(z)$ by $L^\\perp_\\pi$.\n\n\\begin{dfn}\nDefine $\\pi$-type Fermions by\n\\begin{eqnarray}\n\\psi_j^\\pi&:=&L^\\perp_\\pi\\psi_j(L^\\perp_\\pi)^{-1}\\\\\n\\psi_j^{*\\pi}&:=&L^\\perp_\\pi\\psi_j^*(L^\\perp_\\pi)^{-1}\n\\end{eqnarray}\n\\end{dfn}\nIt is easy to check that the $\\pi$-type Fermions satisfy\n\\[\n\\{\\psi_i^\\pi,\\psi_j^\\pi\\}=0,\n\\{\\psi_i^{*\\pi}, \\psi_j^{*\\pi}\\}=0,\n\\{\\psi_i^\\pi,\\psi_j^{*\\pi}\\}=\\delta_{i+j,0}.\n\\]\n\\begin{prp}\nUnder Boson-Fermion correspondence, $\\pi$-type Fermions $\\psi_j^\\pi,\n\\psi_j^{*\\pi}$ correspond to $X^\\pi_j, X_j^{*\\pi}$ defined in Theorem 2.1 respectively. Therefore, the conclusion in Theorem 2.1 holds naturally.\n\\end{prp}\n\\begin{proof}\nUnder Boson-Fermion correspondence, the operator $H_n, n>0$ corresponds to $\\partial_n$, then $L^\\perp_\\pi(z)$ corresponds to\n\\[\n\\exp(-\\sum_{n=1}^\\infty\\frac{z^n}{n}P_\\pi(\\partial_n))\n\\]\nwhich is the same as $L^\\perp_\\pi(z)$ defined in Section 2. In appendix A of \\cite{FJK}, they have proved that\n\\begin{eqnarray*}\nL^\\perp_\\pi(z)M(w)&=&M(w)\\prod_{k\\geq 0}L^\\perp_{\\pi\/(k)}(zw^K),\\\\\nL^\\perp_\\pi(z)L(w)&=&L(w)\\prod_{k\\geq 0}L^\\perp_{\\pi\/(1^{2k})}(zw^{2k})M^\\perp_{\\pi\/(1^{2k+1})}(zw^{2k+1})\n\\end{eqnarray*}\nwhere $$M(z)=\\exp(\\sum_{n=1}^\\infty {x_n}z^n)$$ appeared in $X(z)$ and $L(z)=(M(z))^{-1}$ appeared in $X^*(z)$, and we know that Fermions $\\psi_j$ and $\\psi_j^*$ correspond to $X_j$ and $X^*_j$ under Boson-Fermion correspondence, respectively.\nThen we obtain the conclusion.\n\\end{proof}\n\nIn the following, we will generalize the Boson-Fermion correspondence to $\\pi$-type, from which we can calculate $\\pi$-type symmetric functions. It turns out that the classical Boson-Fermion correspondence is the special case $\\pi=\\emptyset$ of the $\\pi$-type Boson-Fermion correspondence.\n\n\\begin{dfn}\nLet $\\mathcal{F}$ denote the Fermionic Fock space based by the set of Maya diagrams, define\n\\begin{equation}\n\\Phi_\\pi: \\mathcal{F}\\rightarrow\\C[z,z^{-1},x_1,x_2,\\cdots]\n\\end{equation}\nby \\begin{equation}\n\\Phi_\\pi(|u\\rangle)=\\sum_{l\\in\\Z}z^l\\langle l|e^{H(x)}L^\\perp_\\pi|u\\rangle\n\\end{equation}\nwhere $|u\\rangle$ is a Maya diagram.\n\\end{dfn}\n\n\\begin{prp}\nThe correspondence $\\Phi_\\pi:\\mathcal{F}\\rightarrow\\C[z, z^{-1},x_1,x_2,\\cdots]$ defined above is an isomorphism of vector spaces.\n\\end{prp}\n\nUnder the correspondence between Maya diagrams and Young diagrams, we know that the charge zero Maya diagram  $|u\\rangle$ corresponds to a Young diagram denoted by $\\lambda$, and we denote $|u\\rangle$ by $|\\lambda\\rangle$, then we get\n\n\\begin{prp}\nFor a Maya diagram $|\\lambda\\rangle$ which corresponds to the Young diagram $\\lambda$, the $\\pi$-type symmetric functions $S_\\lambda^\\pi({\\bf x})$ can be obtained from\n\\begin{equation}\nS_\\lambda^\\pi({\\bf x})=\\langle \\text{vac}|e^{H(x)}L^\\perp_\\pi|\\lambda\\rangle\n\\end{equation}\n\\end{prp}\n\nFrom the relations between $\\psi_j^\\pi,\\psi_j^{*\\pi}$ and $\\psi_j,\\psi_j^*$, we have\n\\begin{prp} If Young diagram $\\lambda=(-n_1-1\/2,\\cdots,-n_l-1\/2|-m_1-1\/2,\\cdots,-m_l-1\/2)$ in the Frobenius notation, then $S_\\lambda^\\pi({\\bf x})$ can be obtained from\n\\[\n\\langle \\text{vac}|e^{H(x)}\\psi^\\pi_{n_1}\\cdots\\psi^\\pi_{n_l}\\psi^{*\\pi}_{m_1}\\cdots\\psi^{*\\pi}_{m_l}|\\text{vac}\\rangle \\ \\  \\text{for} \\ n_1<\\cdots<n_l<0\\ \\text{and}\\ m_1<\\cdots<m_l<0\n\\]\n by multiplying $(-1)^{\\sum_{i=1}^l(m_i+{1}\/{2})+l(l-1)\/2}$.\n\\end{prp}\nTake an example, we will calculate $S_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}^{(2)}$ in two ways. We will need the actions of $H_n$ on Maya diagrams.\nFrom the actions of Fermions on Maya diagrams, we get\nthe action of $H_1$ on Maya diagram, that is $H_1$ sending a Maya diagram $|{\\bf{u}}\\rangle$ to\nthe sum over all Maya diagrams who can be obtained from $|{\\bf{u}}\\rangle$ by moving a black stone  to the right.\nDefine\n\\begin{equation}\\label{hh}\n\\text{exp}(\\sum_{m\\geq 1}\\frac{H_m}{m} k^m)=\\sum_{n\\geq 0} Q_{(n)} k^n,\\ \\ \\text{exp}(\\sum_{m\\geq 1}\\frac{H_{-m}}{m} k^m)=\\sum_{n\\geq 0} P_{(n)} k^n\n\\end{equation}\nThe action of $Q_{(m)}$ on Maya diagram is that $Q_{(m)}$ sending a Maya diagram $|{\\bf{u}}\\rangle$ to\nthe sum over all Maya diagrams who can be obtained from $|{\\bf{u}}\\rangle$ by moving black stones $n$ times to the right and no one black stone is moved twice. and $Q_{(1^m)}$ sends a Maya diagram $|{\\bf{u}}\\rangle$ to\nthe sum over all Maya diagrams who can be obtained from $|{\\bf{u}}\\rangle$ by moving black stones $n$ times to the right and no two adjacent black stones move at the same time\\cite{Wang,W}. The actions of $P_\\lambda$ on Maya diagram is similar to that of $Q_\\lambda$ on Maya diagram but shifting the black stones to the left. Then\nthe action of $H_m$ on a Maya diagram can be obtained from the actions of $P_n, Q_n$ on this Maya diagram, and we get that the action of $H_m$ on Maya diagram is moving black stones $|m|$ times, to the right if $m>0$ and to the left if $m<0$. Then we can calculate  $S_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}^{(2)}$.\nThe first way, since\n\\begin{eqnarray*}\n\\psi^{(2)}_{-3\/2}&=&\\psi_{-3\/2}-\\psi_{1\/2}-\\psi_{-1\/2}Q+\\psi_{3\/2}Q+\\cdots\\\\\n\\psi^{*(2)}_{-1\/2}&=&\\psi^*_{-1\/2}+\\psi^*_{1\/2}Q+\\psi^*_{3\/2}Q_{2}+\\cdots\n\\end{eqnarray*}\nthen\n\\begin{eqnarray*}\nS_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}^{(2)} &=&\\langle\\text{vac}|e^{H(x)}\\psi_{-3\/2}^{(2)}\\psi^{*(2)}_{-1\/2}|\\text{vac}\\rangle\\\\\n&=&\\langle\\text{vac}|e^{H(x)}(\\psi_{-3\/2}-\\psi_{1\/2})\\psi^*_{-1\/2}|\\text{vac}\\rangle\\\\\n&=& S_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}-S_0=\\frac{1}{2}x_1^2+x_2-1.\n\\end{eqnarray*}\nand in the second way, we know that for Maya diagram\n\\[\n\\begin{tikzpicture}\n\\draw (-3.4,-0.1) node{$|\\gamma\\rangle=$};\n\\fill (0,0) circle(0.1);\n\\draw (0,-0.5) node{$\\frac{1}{2}$};\n\\draw (0.5,0) circle(0.1);\n\\draw (0.5,-0.5) node{$\\frac{3}{2}$};\n\\fill (1,0) circle(0.1);\n\\draw (1,-0.5) node{$\\frac{5}{2}$};\n\\fill (1.5,0) circle(0.1);\n\\draw (1.5,-0.5) node{$\\frac{7}{2}$};\n\\fill (2,0) circle(0.1);\n\\draw (2,-0.5) node{$\\frac{9}{2}$};\n\\draw (2.5,0) node{$\\cdots$};\n\\fill (-0.5,0) circle(0.1);\n\\draw (-0.6,-0.5) node{$-\\frac{1}{2}$};\n\\draw (-1,0) circle(0.1);\n\\draw (-1.1,-0.5) node{$-\\frac{3}{2}$};\n\\draw (-1.5,0) circle(0.1);\n\\draw (-1.6,-0.5) node{$-\\frac{5}{2}$};\n\\draw (-2,0) circle(0.1);\n\\draw (-2.1,-0.5) node{$-\\frac{7}{2}$};\n\\draw (-2.5,0) node{$\\cdots$};\n\n\n\\end{tikzpicture}\n\\]\nwe have that $H_m\\gamma=0$ when $m>2$. Then\n\\begin{eqnarray*}\nS_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}^{(2)}&=&\\langle\\text{vac}|e^{H(x)}e^{-\\sum_{n=1}^\\infty\\frac{1}{2n}(H_n^2+H_{2n})}\\psi_{-3\/2}\\psi_{-1\/2}|\\text{vac}\\rangle\\\\\n&=&\\langle\\text{vac}|e^{H(x)}(1-\\frac{1}{2}(H_1^2+H_2))|\\gamma\\rangle\\\\\n&=&\\langle\\text{vac}|e^{H(x)}(1-Q_2)|\\gamma\\rangle\\\\\n&=& S_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}-S_0=\\frac{1}{2}x_1^2+x_2-1.\n\\end{eqnarray*}\n\\section{$\\pi$-type KP hierarchy}\\label{sect5}\nIn this section, we will define the $\\pi$-type KP hierarchy and discuss its tau functions.\n\\begin{dfn}\nFor an unknown charge zero state $|u\\rangle$ in $\\mathcal{F}$, the bilinear equation\n\\begin{equation}\\label{pikp0}\n\\sum_{j\\in\\Z+\\frac{1}{2}}\\psi^{*\\pi}_j|u\\rangle\\otimes\\psi^{\\pi}_{-j}|u\\rangle=0\n\\end{equation}\nis called the $\\pi$-type KP hierarchy.\n\\end{dfn}\nUnder Boson-Fermion correspondence, this definition can be written into\n\\begin{dfn}\nFor an unknown function $\\tau=\\tau({\\bf x})$, the bilinear equation\n\\begin{equation}\\label{pikp}\n\\sum_{j\\in\\Z+\\frac{1}{2}}X^{*\\pi}_j\\tau\\otimes X^{\\pi}_{-j}\\tau=0\n\\end{equation}\nis called the $\\pi$-type KP hierarchy.\n\n\\end{dfn}\n\nWe write $V_\\pi(z)=\\sum_{n\\in \\Z}V^\\pi_nz^n$ and $V_\\pi^*(z)=\\sum_{n\\in\\Z}V^{*\\pi}_nz^n$. It can be check that $V^\\pi_n$ and $V^{*\\pi}_m$ satisfy\n$V^\\pi_nV^\\pi_m+V^\\pi_{m-1}V^\\pi_{n+1}=0$, $V^{*\\pi}_nV^{*\\pi}_m+V^{*\\pi}_{m-1}V^{*\\pi}_{n+1}=0$ and $V^\\pi_nV^{*\\pi}_m+V^{*\\pi}_{m+1}V^\\pi_{n-1}=\\delta_{n+m,0}$. From the relations between $V^\\pi_n, V^{*\\pi}_n$ and $X^\\pi_n, X^{*\\pi}_n$, the equation (\\ref{pikp}) can be rewritten into\n\\begin{equation}\\label{pikp1}\n\\sum_{n+m=-1}V^{*\\pi}_n\\tau\\otimes V^{\\pi}_m\\tau=0\n\\end{equation}\n\nSuppose $\\tau=\\tau({\\bf x})$ is a solution of $\\pi$-type KP hierarchy (\\ref{pikp1}), then $V^\\pi(\\alpha)\\tau$ solves (\\ref{pikp1}) again with an arbitrary constant $\\alpha\\in\\C^\\times$.\n\nIn the following, we will discuss the differential equations in the $\\pi$-type KP hierarchy and their solutions. From the relations between Fermions and $\\pi$-type Fermions, the equation (\\ref{pikp0}) can be rewritten into\n\\[\n\\sum_{j\\in\\Z+\\frac{1}{2}} L^\\perp_\\pi\\psi_j^*(L^\\perp_\\pi)^{-1}\\tau\\otimes L^\\perp_\\pi\\psi_{-j}(L^\\perp_\\pi)^{-1}\\tau=0\n\\]\nMultiplied by $(L^\\perp_\\pi)^{-1}\\otimes (L^\\perp_\\pi)^{-1}$, the equation above turns into\n\\begin{equation}\\label{pikp2}\n\\sum_{j\\in\\Z+\\frac{1}{2}}\\psi^*_j M^\\perp_\\pi\\tau\\otimes \\psi_{-j}M^\\perp_\\pi\\tau=0\n\\end{equation}\nFrom this, we can obtain\n\\begin{prp}\\label{relation}\n If $\\tau$ is a solution of the $\\pi$-type KP hierarchy, then $M^\\perp_\\pi\\tau$ is a solution of KP hierarchy. If $\\tau$ is a solution of KP hierarchy, then $L^\\perp_\\pi\\tau$ is a solution of $\\pi$-type KP hierarchy.\n \\end{prp}\n\n From (\\ref{ml1}), we have\n \\[\n M^\\perp_\\pi=\\sum_{n\\geq 0}(S_{(n)}[S_\\pi]({\\bf x}))^\\perp\n \\]\n From the definition of plethysm, the relation $S_{(n)}[S_\\pi]=\\sum_{\\lambda}a_{(n)\\pi}^\\lambda S_\\lambda$ holds for $|\\lambda|=n\\cdot|\\pi|$, and it has been proved that $a_{(n)\\pi}^\\lambda$ are nonnegative integers\\cite{Mac}. Then\n \\[\n M^\\perp_\\pi=\\sum_{n\\geq 0}\\sum_\\lambda a_{(n)\\pi}^\\lambda (S_\\lambda)^\\perp=\\sum_{n\\geq 0}\\sum_\\lambda a_{(n)\\pi}^\\lambda Q_\\lambda\n \\]\n The action of $P_\\lambda$ on Maya diagram is clearly known\\cite{Wang}. Let $\\lambda=(\\lambda_1,\\lambda_2,\\cdots,\\lambda_k)$ be a Young diagram. The action of $P_\\lambda$ on Maya diagram $|{\\bf{u}}\\rangle$ includes $k$ steps corresponding to $k$ in $\\lambda$. The first step is $P_{(\\lambda_1)}$ acting on $|{\\bf{u}}\\rangle$ which we have introduced above, and the position, where the black stone is moved, is labeled by $1$; The second step is $P_{(\\lambda_2)}$ acting on all the Maya diagrams obtained from $P_{(\\lambda_1)}\\cdot|{\\bf{u}}\\rangle$ and the position, where the black stone is moved, is labeled by $2$; Continuing until the $k$th step, the operator $P_{(\\lambda_k)}$ acts on all the Maya diagrams obtained from $P_{(\\lambda_{k-1})}\\cdots P_{(\\lambda_2)}P_{(\\lambda_1)}\\cdot|{\\bf{u}}\\rangle$, and the position, where the black stone is moved, is labeled by $k$. We define $P_\\lambda$ sending Maya diagram $|{\\bf{u}}\\rangle$ to the sum over all Maya diagrams obtained from $k$ steps above and satisfied the following situation: from right to left, one looks at the first $l$ entries in the list (for any $l$ between $1$ and $\\lambda_1+\\lambda_2+\\cdots+\\lambda_k$), each integer $p$ between $1$ and $k-1$ occurs at least as many times as the next integer $p+1$.\n\n Choose $\\tau$ in equation (\\ref{pikp2}) in the form of linear combination of all charge zero Maya diagrams $\\tau=\\sum_{|u\\rangle}c(|u\\rangle)|u\\rangle$, where the coefficient $c(|u\\rangle)\\in\\C$. Let $|\\mu\\rangle$ be a Maya diagram of charge $1$ and $|\\nu\\rangle$ of charge $-1$, the coefficient of $|\\mu\\rangle\\otimes |\\nu\\rangle$ in equation (\\ref{pikp2}) is zero. From this, we will get many differential equations whose solutions include $\\pi$-type symmetric functions.\n\\begin{prp} In $\\pi$-type KP hierarchy, the tau function $\\tau$ is a solution if and only if the coefficients $c(|u\\rangle)$ in $\\tau=\\sum_{|u\\rangle}c(|u\\rangle)|u\\rangle$ satisfy Pl\\\"uker relations, i.e.,\nthe following equation holds for any $|\\mu\\rangle$ and $|\\nu\\rangle$ whose charges are $1$ and $-1$ respectively\n \\begin{equation}\\label{pieq1}\n\\sum_{j}(-1)^j\\sum_{(n),\\lambda}\\sum_{(m),\\lambda'}a_{(n)\\pi}^\\lambda a_{(m)\\pi}^{\\lambda'}c(P_\\lambda|\\mu-\\mu_j\\rangle)c(P_{\\lambda'}|\\nu+\\mu_j\\rangle)=0\n\\end{equation}\nhere, Maya diagrams are signed Maya diagrams whose definition can be found in \\cite{MJD}.\n\\end{prp}\n\n When $\\pi=\\emptyset$, the equation (\\ref{pieq1}) turns into\n \\begin{equation}\\label{piempt}\n\\sum_{j}(-1)^jc(|\\mu-\\mu_j\\rangle)c(|\\nu+\\mu_j\\rangle)=0\n\\end{equation}\nwhich is the Pl\\\"uker relations of KP hierarchy (the equation (10.3)  in \\cite{MJD}).\n\n When $\\pi=(1)=\\begin{tikzpicture}\n\\draw [step=0.2](0,0) grid(0.2,0.2);\n\\end{tikzpicture}$, the equation (\\ref{pieq1}) turns into\n \\begin{equation}\n\\sum_{j}(-1)^j\\sum_{n,m\\geq 0}c(P_n|\\mu-\\mu_j\\rangle)c(P_m|\\nu+\\mu_j\\rangle)=0\n\\end{equation}\nwhich in fact is the same as (\\ref{piempt}).\n\nWhen $\\pi=(2)=\\begin{tikzpicture}\n\\draw [step=0.2](0,0) grid(0.4,0.2);\n\\end{tikzpicture}$, since $a_{(n)(2)}^\\lambda =1$ if and only if $\\lambda$ is an even partition ($\\lambda_i$ are even numbers) of $2n$, otherwise $a_{(n)(2)}^\\lambda =0$, then the equation (\\ref{pieq1}) turns into\n\\begin{equation}\\label{pi22}\n\\sum_{j}(-1)^j\\sum_{\\lambda,\\lambda' \\text{even}}c(P_\\lambda|\\mu-\\mu_j\\rangle)c(P_{\\lambda'}|\\nu+\\mu_j\\rangle)=0.\n\\end{equation}\n In special case, let \\begin{equation}\n\\begin{tikzpicture}\n\\draw (-3.4,-0.1) node{$\\mu=$};\n\\fill (0,0) circle(0.1);\n\\draw (0,-0.5) node{$\\frac{1}{2}$};\n\\fill (0.5,0) circle(0.1);\n\\draw (0.5,-0.5) node{$\\frac{3}{2}$};\n\\fill (1,0) circle(0.1);\n\\draw (1,-0.5) node{$\\frac{5}{2}$};\n\\fill (1.5,0) circle(0.1);\n\\draw (1.5,-0.5) node{$\\frac{7}{2}$};\n\\fill (2,0) circle(0.1);\n\\draw (2,-0.5) node{$\\frac{9}{2}$};\n\\draw (2.5,0) node{$\\cdots$};\n\\fill (-0.5,0) circle(0.1);\n\\draw (-0.6,-0.5) node{$-\\frac{1}{2}$};\n\\draw (-1,0) circle(0.1);\n\\draw (-1.1,-0.5) node{$-\\frac{3}{2}$};\n\\draw (-1.5,0) circle(0.1);\n\\draw (-1.6,-0.5) node{$-\\frac{5}{2}$};\n\\draw (-2,0) circle(0.1);\n\\draw (-2.1,-0.5) node{$-\\frac{7}{2}$};\n\\draw (-2.5,0) node{$\\cdots$};\n\n\n\\end{tikzpicture}\n\\end{equation}\nand\\begin{equation}\n\\begin{tikzpicture}\n\\draw (-3.4,-0.1) node{$\\nu=$};\n\\draw (0,0) circle(0.1);\n\\draw (0,-0.5) node{$\\frac{1}{2}$};\n\\draw (0.5,0) circle(0.1);\n\\draw (0.5,-0.5) node{$\\frac{3}{2}$};\n\\fill (1,0) circle(0.1);\n\\draw (1,-0.5) node{$\\frac{5}{2}$};\n\\fill (1.5,0) circle(0.1);\n\\draw (1.5,-0.5) node{$\\frac{7}{2}$};\n\\fill (2,0) circle(0.1);\n\\draw (2,-0.5) node{$\\frac{9}{2}$};\n\\draw (2.5,0) node{$\\cdots$};\n\\draw (-0.5,0) circle(0.1);\n\\draw (-0.6,-0.5) node{$-\\frac{1}{2}$};\n\\fill (-1,0) circle(0.1);\n\\draw (-1.1,-0.5) node{$-\\frac{3}{2}$};\n\\draw (-1.5,0) circle(0.1);\n\\draw (-1.6,-0.5) node{$-\\frac{5}{2}$};\n\\draw (-2,0) circle(0.1);\n\\draw (-2.1,-0.5) node{$-\\frac{7}{2}$};\n\\draw (-2.5,0) node{$\\cdots$};\n\n\n\\end{tikzpicture}\n\\end{equation}\nthe equation (\\ref{pi22}) equals\n\\begin{equation}\n\\sum_{\\lambda,\\lambda' \\text{even}}(c(\\lambda)c(\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.3);\n\\end{tikzpicture})-c(\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.15);\n\\end{tikzpicture})c(\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\draw [step=0.15](0,-0.15) grid(0.15,0);\n\\end{tikzpicture})+c(\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture})c(\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.3);\n\\end{tikzpicture}))=0\n\\end{equation}\nwhere $\\lambda\\cdot\\mu=\\sum_{\\nu}N_{\\lambda\\mu}^\\nu\\nu,\\ N_{\\lambda\\mu}^\\nu\\in\\Z_{\\geq 0}$ satisfies the Littlewood-Richardson rule. Replacing $c(\\lambda)$ by $Q_\\lambda\\tau$, we get the differential equation\n\\begin{equation}\n\\sum_{\\lambda,\\lambda' \\text{even}}(Q_{\\lambda}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.3);\n\\end{tikzpicture}}\\tau-Q_{\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\draw [step=0.15](0,-0.15) grid(0.15,0);\n\\end{tikzpicture}}\\tau+Q_{\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.3);\n\\end{tikzpicture}}\\tau)=0\n\\end{equation}\nWe can similarly write the differential equations in $\\pi$-type KP hierarchy, i.e., replacing $c(\\lambda)$ by $Q_\\lambda\\tau$ in (\\ref{pieq1}), therefore,\n\\begin{prp}\nThe differential equations in the $\\pi$-type KP hierarchy are\n \\begin{equation}\n\\sum_{j}(-1)^j\\sum_{(n),\\lambda}\\sum_{(m),\\lambda'}a_{(n)\\pi}^\\lambda a_{(m)\\pi}^{\\lambda'}Q_{P_\\lambda|\\mu-\\mu_j\\rangle}\\tau\\cdot Q_{P_{\\lambda'}|\\nu+\\mu_j\\rangle}\\tau=0\n\\end{equation}\nwhere $|\\mu\\rangle$ and $|\\nu\\rangle$  are two Maya diagrams whose charges are $1$ and $-1$ respectively. Choose $|\\mu\\rangle$ and $|\\nu\\rangle$ as before, we get\n \\begin{equation}\\label{pieq11}\n\\sum_{(n),\\lambda}\\sum_{(m),\\lambda'}a_{(n)\\pi}^\\lambda a_{(m)\\pi}^{\\lambda'}(Q_{\\lambda}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.3);\n\\end{tikzpicture}}\\tau-Q_{\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\draw [step=0.15](0,-0.15) grid(0.15,0);\n\\end{tikzpicture}}\\tau+Q_{\\lambda\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\lambda'\\cdot\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.3);\n\\end{tikzpicture}}\\tau)=0\n\\end{equation}\n\\end{prp}\nThe solutions of these equations is known from the discussions before. When $\\pi=\\emptyset$, the equation (\\ref{pieq11}) turns into\n\\[\n\\tau\\cdot Q_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.3);\n\\end{tikzpicture}}\\tau-Q_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\draw [step=0.15](0,-0.15) grid(0.15,0);\n\\end{tikzpicture}}\\tau+Q_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.3,0.15);\n\\end{tikzpicture}}\\tau\\cdot Q_{\\begin{tikzpicture}\n\\draw [step=0.15](0,0) grid(0.15,0.3);\n\\end{tikzpicture}}\\tau=0\n\\]\nwhich is the KP equation\n\\[\n\\frac{3}{4}\\frac{\\partial^2u}{\\partial x_2^2}=\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial u}{\\partial x_3}-\\frac{3}{2}u\\frac{\\partial u}{\\partial x}-\\frac{1}{4}\\frac{\\partial^3u}{\\partial x^3}\\right).\n\\]\n\nThen we have the following remark.\n\\begin{rmk}\nThe $\\pi$-type KP hierarchy is quite different from other types of classical KP systems from the point of different types of Lie algbras (like BKP, CKP and so on \\cite{DKJM,MJD}).\nThe relation between the tau functions of BKP, CKP and so on and the tau function of the classical KP hierarchy is more complicated than the relation between the tau function of the $\\pi$-type KP hierarchy and the tau function of the classical KP hierarchy as mentioned in the Proposition \\ref{relation}.\n\\end{rmk}\n\n\n\n\n\\section*{Acknowledgements}\nThe authors gratefully acknowledge the support of Professors Ke Wu, Zi-Feng Yang, Shi-Kun Wang.\nChuanzhong Li is supported by the National Natural Science Foundation\nof China under Grant No. 11571192 and K. C. Wong Magna Fund in Ningbo University.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\nThe determination of\nthe proton form-factors in the time-like region is planned at PANDA (FAIR) in\nthe process of annihilation of proton and antiproton to an electron-positron pair,\nfor values of antiproton momentum up to 15 GeV\/c \\cite{epja}. Radiative corrections\n(RC) due to the emission of real and virtual photons do affect the measurement of\nthe experimentally observable quantities, in particular of the differential cross section.\nThe individual determination of the electric and magnetic proton form factors\nrequires the precise knowledge of the angular distribution of the final lepton,\nin shape and in absolute value. The motivation of this paper to calculate radiative\ncorrections to this process and to provide the formulae for the differential cross\nsections with the adequate accuracy to be used in the experiment.\n\nA lot of attention was paid to the cross-process $e^+e^-\\to \\mu^+\\mu^-(\\gamma)$ years ago\n(see Ref. \\cite{Ku77,Skrzypek:1992vk,Be81}), where the total cross section as well as different distributions were\nconsidered. The purpose of this work is to update and generalize the results obtained in Born approximation and in the lowest order of perturbation theory for the process of radiative annihilation of proton and antiproton to a lepton pair at high energies.\n\nAmong the first papers devoted to the annihilation of proton pair to a lepton pair one should  refer to Ref. \\cite{Zichichi:1962ni}, where the possibility to\nmeasure the nucleon form factors in the time-like region of momentum transfer was discussed.\nIn the paper \\cite{BR65} the case of polarized particles was considered. More recently, single and double spin polarization observables were calculated in frame of different models and applied to the world data \\cite{ETG05} in space-like as well as in time-like region. In Ref. \\cite{Ga06} the expression of the cross section and the polarization observables in terms of form factors was extended to the two photon exchange mechanism, in a model independent formalism.\n\nThe matrix element of the annihilation process\n\\be\np(p_+)+\\bar{p}(p_-) \\to e^+(q_+)+e^-(q_-)\n\\label{eq:reac}\n\\ee\nin Born approximation has the form:\n\\ba\nM_B=\\frac{4\\pi\\alpha}{q^2}\\bar{v}(p_-)\\Gamma_{\\mu} u(p_+)\\bar{u}(q_-)\\gamma_{\\mu} v(q_+),\n\\ea\nwith $q=p_++p_-$, $q^2=s$, $p_\\pm^2=M^2$, $q_\\pm^2=m^2$, $M(m)$ is the proton(electron) mass, and\n\\ba\n\\Gamma_{\\mu}(q)=F_1^{str}(s)\\gamma_{\\mu}+\\frac{1}{4M}[\\gamma_{\\mu},\\hat q]F_2^{str}(s),\n\\ea\nwhere  $F_{1,2}^{str}(s)$ - are the Dirac and Pauli form factors of proton.\n\nFor large-angle scattering, the terms of the order of $m^2\/M^2$ compared to the ones of order of unity can be neglected ($1+\\mathcal O(m^2\/M^2))$.\n\nThe matrix element squared summed over the spin states is:\n\\ba\n\\sum_{spin}|M|^2=16 \\cdot (\\frac{4\\pi\\alpha}{s})^2 \\cdot \\frac{s^2}{4}\\{|G_{M}(s)|^2(1+\\cos^2\\theta)+(1-\\beta^2)\n|G_E(s)|^2\\sin^2\\theta\\}\n\\ea\nwhere the magnetic and electric form factors of proton are\n\\ba\nG_E(s)=F_1^{str}(s) + \\eta F_2^{str}(s), \\quad G_M(s)=F_1^{str}(s)+F_2^{str}(s),\n\\quad \\eta=\\frac{s}{4M^2}, \\quad \\beta^2=1-\\frac{4M^2}{s}.\n\\ea\n$M$ is the proton mass, $\\beta$ is the proton velocity.\n\nLet us introduce the kinematics invariants\n\\ba\nt&=&(p_- - q_-)^2=-\\frac{s}{4}(1+\\beta^2-2\\beta \\cos\\theta),\\quad\nu=(p_+ - q_-)^2=-\\frac{s}{4}(1+\\beta^2+2\\beta \\cos\\theta)\n\\nn\\\\\ns&=&\\frac{4M^2}{1-\\beta^2}, \\quad\ns+t+u=2M^2,\n\\ea\nwhere $\\theta$ -is the scattering angle between the direction of motion of the antiproton and the electron in the center of mass system (cms).\n\nThe differential cross section in Born approximation has the form \\cite{Zichichi:1962ni}:\n\\ba\n\\frac{d\\sigma_B}{dO_-}=\\frac{\\alpha^2}{4s\\beta}\\biggl\\{|G_M(s)|^2(1+\\cos^2\\theta)+(1-\\beta^2)\n|G_E(s)|^2\\sin^2\\theta\\  \\biggr\\}\n\\ea\nThe total cross section is\n\\ba\n\\sigma_{tot}=\\frac{4\\pi\\alpha^2}{3\\beta s}\\biggl[|G_M(s)|^2+\\frac{1}{2}|G_E(s)|^2 (1-\\beta^2)\n\\biggr].\n\\ea\n\n\\section{Virtual and soft photon emission}\n\nBelow we will suppose the proton to be point-like particle $G_E(s)=G_M(s)=1$, and RC to the proton and electron vertex can be written in the form:\n\\begin{gather}\nF_{1}(s)=1+\\frac{\\alpha}{\\pi}F_{1}^{(2)}(s)+...\\,,\\quad\nF_{2}(s)=\\frac{\\alpha}{\\pi}F_{2}^{(2)}(s)+...,\\quad\nF_e(s)=1+\\frac{\\alpha}{\\pi}F_e^{(2)}(s)+...\n\\end{gather}\nMoreover we will consider the case of non-relativistic proton and antiproton,\n$\\beta \\sim 1$.\n\nThe Born matrix with one-loop corrections can be written in the form:\n\\ba\nM_{BV}=\\frac{1}{1-\\Pi(s)}\\biggl\\{M_{1B}\\left [1+\\frac{\\alpha}{\\pi}F_{e}^{(2)}(s)\n+\\frac{\\alpha}{\\pi}F_1^{(2)}(s)\\right ]+\\frac{\\alpha}{\\pi}M_{2B}F_2^{(2)}(s)\\biggr\\}\n-\\left (\\frac{\\alpha}{\\pi}\\right )^2M_{box},\n\\ea\nwith\n\\ba\nM_{1B}&=&\\frac{4\\pi\\alpha}{s}\\bar{v}(p_-)\\gamma_{\\mu} u(p_+)\\bar{u}(q_-)\\gamma_{\\mu} v(q_+), \\nn \\\\\nM_{2B}&=&\\frac{4\\pi\\alpha}{s}\\frac{1}{4M}\\bar{v}(p_-)(\\gamma_\\mu \\hat{q}-\\hat{q}\\gamma_\\mu)u(p_+)\n\\bar{u}(q_-)\\gamma_\\nu v(q_+).\n\\ea\n\nThe real parts of Dirac $F_1^{(2)}(s)$ and Pauli $F_2^{(2)}(s)$ proton form-factors (of QED origin) can be written as:\n\\cite{Akhiezer,Be81}:\n\\ba\nRe F_1^{(2)}(s)&=&\\left (\\ln\\frac{M}{\\lambda}-1\\right )\\left (1-\\frac{1+\\beta^2}{2\\beta}L_\\beta\\right)\n\\nn \\\\\n&+&\\frac{1+\\beta^2}{2\\beta}\\left [\\frac{1}{3}\\pi^2 +\\Li\\left (\\frac{1-\\beta}{1+\\beta}\\right)-\n\\frac{1}{4}L^2_\\beta -L_\\beta\\ln\\left (\\frac{2\\beta}{1+\\beta}\\right)\\right]-\n\\frac{1}{4\\beta}L_\\beta,\\nn \\\\\nRe F_2^{(2)}(s)&=&-\\frac{1-\\beta^2}{4\\beta}L_\\beta,\\quad\nL_\\beta=\\ln\\frac{1+\\beta}{1-\\beta}.\n\\ea\nOnly the Dirac form factor is relevant for the lepton:\n\\ba\nRe F_e^{(2)}(s)=\\left(\\ln\\frac{m}{\\lambda}-1\n\\right )(1-L_e)-\\frac{1}{4}L_e-\\frac{1}{4}L_e^2+2\\xi_2,\\quad L_e=\\ln\\frac{s}{m^2}.\n\\ea\nThe polarization of vacuum operator $\\Pi(s)=\\Pi_e(s)+\\Pi_{\\mu}(s)+\\Pi_{\\tau}(s)+\\Pi_{hadr}(s)$ has a standard form:\n\\ba\nRe \\Pi_e(s)&=&\\frac{\\alpha}{3\\pi}\\left(L_e-\\frac{5}{3}\\right), \\quad\nIm \\Pi_e(s)=-\\frac{\\alpha}{3}, \\nn \\\\\nRe \\Pi_{\\mu}(s)&=&-\\frac{\\alpha}{\\pi}\\left[\\frac{8}{9}-\\frac{\\beta_{\\mu}^2}{3}-\n\\beta_{\\mu}\\left( \\frac{1}{2}-\\frac{1}{6}\\beta_{\\mu}^2\\right )L_{\\mu}\\right ],\n~\nIm \\Pi_{\\mu}(s)=-\\frac{\\alpha}{\\pi}\\cdot \\frac{\\pi}{2}\\left(1-\\frac{1}{3}\\beta_{\\mu}^2\\right ), \\nn \\\\\nL_{\\mu}&=&\\ln\\frac{1+\\beta_{\\mu}}{1-\\beta_{\\mu}}, \\quad \\beta_{\\mu}=\\sqrt{1-\\frac{4M_{\\mu}^2}{s}}.\n\\ea\nThe contribution of hadron states to the vacuum polarization is \\cite{Eidelman}\n\\ba\n\\Pi_{hadr}(s)=-\\frac{s}{2\\pi^2\\alpha}\\int\\limits_{4M_\\pi^2}^\\infty\nds'\\frac{\\sigma_{e^+e^-\\to hadr}(s')}{s'-s-i0}.\n\\ea\nIn particular for a charged pion pair as hadron state we have\n\\ba\n\\sigma_{e^+e^-\\to hadr}(s)=\\frac{\\pi \\alpha^2}{3s}\\beta_\\pi^3|F_\\pi(s)|^2,\n\\quad\n\\beta_\\pi=\\sqrt{1-\\frac{4M_\\pi^2}{s}}.\n\\ea\nwith $F_\\pi(s)$ is pion form factor. Setting $F_\\pi(s)=1$ we have\n$$\n\\Pi_{\\pi^+\\pi^-}(s)=\\frac{2\\alpha}{\\pi}\\left [\\frac{1}{12}\\ln\\frac{1+\\beta_{\\pi}}{1-\\beta_{\\pi}}-\\frac{2}{3}-2\\beta^2_{\\pi}-\ni\\frac{\\beta^3_{\\pi}}{12}\\right ].\n$$\nThe contribution of the box type Feynman diagrams describing the two virtual photons annihilation mechanism is\n\\ba\nM_{box}=\\int\\frac{d^4k}{i\\pi^2}\\frac{1}{(k^2-\\lambda^2)((q-k)^2-\\lambda^2)}\n\\bar{v}(p_-)\\gamma_{\\nu}\\frac{\\hat {p}_+ -\\hat {k}+M}{(p_+ -k)^2-M^2} \\gamma_{\\mu}\nu(p_+)\n \\nn \\\\\n\\times \\bar {u}(q_-)\n\\biggl[\\gamma_{\\mu}\\frac{\\hat {q}_- -\\hat {k}+m}{(q_- -k)^2-m^2}\\gamma_{\\nu} +\n\\gamma_{\\nu}\\frac{-\\hat {q}_+ +\\hat {k}+m}{(q_+-k)^2-m^2}\\gamma_{\\mu}\\biggr]v(q_+).\n\\ea\nThe interference of Born and box-type amplitudes contribution to the odd part of differential cross section is:\n\\ba\n\\label{eq::18}\n\\frac{d\\sigma_{odd}}{d O_-}=\\frac{-\\alpha^3}{2\\pi s^2 \\beta}I(t,u,s), \\quad\nI(t,u,s)=(1-P(t,u))\\int\\frac{d^4k}{i\\pi^2} \\cdot \\frac{S_e S_p}{(0)(q)(p)(m)},\n\\ea\nwhere $P(t,u)$ is the exchange operator $P(t,u)f(t,u)=f(u,t)$,\nand\n\\ba\n(0)&=&k^2-\\lambda^2,~ (q)=(q-k)^2-\\lambda^2, ~(p)=k^2-2p_+k+i0,~(m)=k^2-2q_-k+i0,  \\nn \\\\\nS_e&=&\\frac{1}{4}Tr\\hat{q}_+\\gamma_\\lambda\\hat{q}_-\\gamma_{\\mu}(\\hat{q}_--\\hat{k})\\gamma_\\nu, \\nn \\\\\nS_p&=&\\frac{1}{4}Tr(\\hat{p}_++M)\\gamma_\\lambda(\\hat{p}_--M)\\gamma_\\nu(\\hat{p}_+-\\hat{k}+M)\\gamma_{\\mu}.\n\\label{eq::denom}\n\\ea\n\nUsing the set of one-loop integrals listed in Appendix B we obtain\n\\ba\nI(t,u,s)&=&(u-t)\\biggl[\\left (\\frac{2M^2}{\\beta^2}+t+u \\right )I_{0qp}-\\frac{\\pi^2}{6}+\\frac{1}{2}L_\\beta^2-\\frac{1}{\\beta^2}L_\\beta \\biggr]\n\\nn\\\\\n&+&(2t+s)\\biggl[\\frac{1}{2}L_{ts}^2-\\Li\\left (\\frac{-t}{M^2-t}\\right)\\biggr]\n-(2u+s)\\biggl[\\frac{1}{2}L_{us}^2-\\Li(\\frac{-u}{M^2-u})\\biggr]\n \\nn \\\\\n&+&(ut-M^2(s+M^2))\\biggl[\\frac{1}{t}L_{ts}-\\frac{1}{u}L_{us}+\\frac{u-t}{ut}L_s\\biggr]\n \\nn \\\\\n&+&\\frac{2}{s}\\left [ t^2+u^2-4M^2(t+u)+6M^4\\right]L_{tu}(L_{M\\lambda}+L_s),\n\\label{eq:20}\n\\ea\nwith\n\\begin{gather}\nL_s=\\ln\\frac{s}{M^2}, \\quad L_{tu}=\\ln\\frac{M^2-t}{M^2-u},\n\\quad L_{ts}=\\ln\\frac{M^2-t}{s}, \\nn \\\\\nL_\\beta=\\ln\\frac{1+\\beta}{1-\\beta}, \\quad L_{us}=\\ln\\frac{M^2-u}{s}, \\quad L_{M\\lambda}=\\ln\\frac{M^2}{\\lambda^2},\n\\end{gather}\nand\n\\ba\nI_{0qp}=\\frac{1}{s\\beta}\\left[L_sL_\\beta-\\frac{1}{2}L_\\beta^2-\\frac{\\pi^2}{6}+\n2\\Li\\left(\\frac{1+\\beta}{2}\\right)-2\\Li\\left(\\frac{1-\\beta}{2}\\right)-\n2\\Li\\left(-\\frac{1-\\beta}{1+\\beta}\\right)\\right].\n\\ea\nFor large values of the kinematic invariants $s\\sim-t\\sim-u\\gg M^2$ our result is in agreement with the result previously obtained by I. Kriplovich \\cite{Kr73}:\n\\ba\nI(t,u,s)&=&\\frac{2}{s}(t^2+u^2)L_{tu}[L_{M\\lambda}+L_s]+\\frac{1}{2}(t-u)L^2_{tu}+\n(t-u)L_{ts}L_{us}+uL_{ts}-tL_{us}\\nn\\\\\n&=&s(1+c^2)\\left\\{\\ln\\frac{1-c}{1+c}\\ln\\frac{s}{\\lambda^2}+\n\\frac{1}{2(1+c^2)} \\left [c\\left (\\ln^2\\frac{1+c}{2}+\\ln^2\\frac{1-c}{2}\\right )\n\\right . \\right .\\nn\\\\\n&&\\left .\\left .-(1+c)\\ln\\frac{1-c}{2}+(1-c)\\ln\\frac{1+c}{2}\\right ]\\right\\}.\n\\ea\nIt is useful to note that the coefficient of $L_{M\\lambda}$, see Eq. (\\ref{eq:20})\n$$\\frac{2}{s}(t^2+u^2-4M^2(t+u)+6M^4)=s(2-\\beta^2\\sin^2\\theta),$$\nis proportional to the Born matrix element squared.\n\nThe differential cross section, including Born amplitudes and virtual corrections, becomes:\n\\ba\n\\frac{d\\sigma_{BV}}{dO_-}=\\frac{\\alpha^2}{4 s \\beta}\\biggl\\{\n\\biggl|\\frac{1}{1-\\Pi}\\biggr|^2 \\biggl[(2-\\beta^2 \\sin^2\\theta)\\left [1+\\frac{2\\alpha}{\\pi}(F_e^{(2)}(s)+F_1^{(2)}(s))\\right ]+\\nn \\\\\n\\frac{4\\alpha}{\\pi} Re F_2^{(2)}(s)\\biggr]-\\frac{2\\alpha}{\\pi s}I(t,u,s)\\!\\!\\biggr\\}.\n\\ea\n\nThe soft real photons emission can be taken into account in the standard way:\n\\ba\n\\label{eq::25}\n\\frac{d\\sigma^{soft}}{d\\sigma_{B}}=-\\frac{4\\pi \\alpha}{(2\\pi)^3}\\int \\frac{d^3k}{2\\omega}\\left(\\frac{p_+}{p_+ k}-\n\\frac{p_-}{p_- k}+\\frac{q_-}{q_- k}-\\frac{q_+}{q_+ k} \\right)^2 =\n\\frac{d\\sigma^{soft}_{even}}{d\\sigma_B}+\\frac{d\\sigma^{soft}_{odd}}{d\\sigma_B},\n\\ea\nwhere the photon energy in CMS is constrained by: $\\omega<\\Delta E$,\n$\\Delta E\\ll E=\\sqrt{s}\/2$. We obtain for the even part:\n\\ba\n\\frac{d\\sigma^{soft}_{even}}{d\\sigma_B}&=&\\frac{\\alpha}{\\pi}\n\\left \\{ -2\n\\left [\\ln\\left (\\frac{2\\Delta E}{\\lambda}\\right )-\n\\frac{1}{2\\beta}L_{\\beta}\\right ]-2\\ln\\left (\\frac{\\Delta E \\cdot m}\n{\\lambda E}\\right )\\right .\n\\nn \\\\\n&+&2 \\left . \\frac{1+\\beta^2}{2\\beta}\\left[\\ln\\left(\\frac{2 \\Delta E}{\\lambda}\\right)L_{\\beta}-\n\\frac{1}{4}L_{\\beta}^2+\n\\Phi(\\beta)\\right]+2\\left[\\ln(\\frac{2\\Delta E}{\\lambda})L_e -\\frac{1}{4}L_e^2-\\frac{\\pi^2}{6}\\right]\\right \\},\n\\nn\n\\ea\nwith $\\Phi(\\beta)$ :\n\\ba\n\\Phi(\\beta)&=&\\frac{\\pi^2}{12}+L_\\beta\\ln\\frac{1+\\beta}{2\\beta}\n+\\ln\\frac{2}{1+\\beta}\\ln(1-\\beta)+\\frac{1}{2}\\ln^2(1+\\beta)-\\frac{1}{2}\\ln^22\n\\nn\\\\\n&-&\\Li\\left(\\beta\\right)+\\Li\\left(-\\beta\\right)-\n\\Li\\left(\\frac{1-\\beta}{2}\\right), \\quad \\Phi(1)=-\\frac{\\pi^2}{6}.\n\\label{eq:eqA4}\n\\ea\nFor the case $s\\sim-t\\sim-u\\gg M^2$ we have \\cite{Ku85}\n\\be\n\\frac{d\\sigma_{BSV}}{d\\sigma_B}=\\frac{1}{|1-\\Pi|^2}D^2_{p\\Delta} D^2_{e\\Delta}\n\\left [1+\\frac{\\alpha}{\\pi}(K^p+K^e)_{SV}\\right ]\n-4\\frac{\\alpha}{\\pi}\\ln\\frac{\\Delta E}{E}\\ln\\frac{1-c}{1+c}+\\frac{\\alpha}{\\pi}\nK^{odd}_{SV}\n\\ee\nwith\n\\ba\nD_{p\\Delta}&=&1+\\frac{\\alpha}{2\\pi}(L_s-1)\\biggl(2\\ln\\frac{\\Delta E}{E}+\\frac{3}{2}\\biggr), \\nn \\\\\nD_{e\\Delta}&=&1+\\frac{\\alpha}{2\\pi}(L_e-1)\\biggl(2\\ln\\frac{\\Delta E}{E}+\\frac{3}{2}\\biggr), \\nn \\\\\nK_{SV}^p&=&K_{SV}^e=\\frac{\\pi^2}{3}-\\frac{1}{2},\n \\nn \\\\\nK_{SV}^{odd}&=&\\frac{4}{(1+\\cos^2\\theta)}\n\\left[ \\cos\\theta\\left(\\ln^2\\sin\\frac{\\theta}{2}+\\ln^2\\cos\\frac{\\theta}{2}\\right )\n+\\sin^2\\frac{\\theta}{2}\\ln\\cos\\frac{\\theta}{2}-\\cos^2\\frac{\\theta}{2}\n\\ln\\sin\\frac{\\theta}{2}\\right] \\nn \\\\\n&&\n-4\\ln^2\\sin\\frac{\\theta}{2}+4\\ln^2\\cos\\frac{\\theta}{2}\n+2\\Li\\left(\\sin^2\\frac{\\theta}{2}\\right)-2\\Li\\left(\\cos^2\\frac{\\theta}{2}\\right).\n\\ea\nLet us note that $K_{SV}^{odd}$ has a different sign than for the annihilation process $ e^++e^-\\to \\mu^++\\mu^-$.\n\n\\section{Hard photon emission}\n\nLet now consider hard  photon  emission contribution:\n\\be\n\\bar p(p_-)+p(p_+)\\to e^+(q_+)+e^-(q_-)+\\gamma(k)\n\\ee\nThe differential cross section is the sum of contributions of\nemission from leptons and hadrons \\cite{Ku77}:\n\\be\nd\\sigma=\\displaystyle\\frac{\\alpha^3}{2\\pi s}R\\displaystyle\\frac{dx_-dx dc dz}\n{\\sqrt{D(c,a_-,z)}}; \\,\\,D(c,a_-,z)=(z_1-z)(z-z_2), \\nn \\\\\n~R=\\displaystyle\\frac{s}{16(4\\pi\\alpha)^3}\\sum_{spin}|M|^2=R_{even}+R_{odd},\n\\ee\nwith\n\\be\nx_{\\pm}=\\displaystyle\\frac{E_{\\pm}}{E},~x=\\displaystyle\\frac{\\omega}{E},\\nn \\\\\n~z=\\cos(\\vec{k},\\vec{p}_-),~c=\\cos(\\vec{q}_-,\\vec{p}_-), \\nn \\\\\n~a_-=\\cos(\\vec{k},\\vec{q}_-),~c_+=\\cos(\\vec{q}_+,\\vec{p}_-),\n\\ee\nand\n\\be\n~z_{1,2}=a_-c\\pm \\sqrt{(1-a_-^2)(1-c^2)},~ a_-=1-\\frac{2(1-x_+)}{xx_-}.\n\\ee\nThe energy-momentum conservation gives the relations\n\\ba\nx_++x_-+x=2, \\,\\,\\, x_+c_++x_-c+x z=0.\n\\ea\nThe quantity $R$ has been previously calculated in \\cite{Ku77,Be81}. For $s\\gg M^2$ we have:\n\\ba\nR_{even}&=&-\\frac{M^2s}{2s'^2}\\left [\\displaystyle\\frac{t^2+u^2}{(p_-k)^2}+ \\displaystyle\\frac{t'^2+u'^2}{(p_+k)^2}\\right ]\n-\\displaystyle\\frac{m^2}{2s}\\left [\\displaystyle\\frac{t^2+u'^2}{(q_-k)^2}+ \\displaystyle\\frac{t'^2+u^2}{(q_+k)^2}\\right ]\\nn\\\\\n&&\n+\\displaystyle\\frac{t^2+t'^2+u^2+u'^2}{4s'}\\left (\\displaystyle\\frac{s}{p_+k p_-k}+ \\displaystyle\\frac{s'}{q_+k q_-k}\\right );\\nn\\\\\nR_{odd}&=&\\frac{t^2+t'^2+u^2+u'^2}{4s'}\\left (\\displaystyle\\frac{t}{p_+k\\cdot q_+k}+\n\\displaystyle\\frac{t'}{p_-k\\cdot q_-k}-\n\\displaystyle\\frac{u'}{p_-k\\cdot q_+k}-\n\\displaystyle\\frac{u}{p_+k\\cdot q_-k}\n\\right ),\\nn\n\\ea\nwith\n\\ba\nt&=&-2p_+q_+,~u=-2p_+q_-,~s=2p_+p_-,~\nt'=-2p_-q_-,~u'=-2p_-q_+,~s'=2q_+q_-,\\nn\\\\\n&&t+t'+u+u'+s+s'=0.\n\\nn\n\\ea\n\nPerforming the simplifications and omitting the contributions of terms of order\n\\ba\nO\\left (\\frac{M^2}{s}\\right )\n\\ea\ncompared to the ones which lead to contributions of the order of unity,\nwe obtain\n\\ba\nR_{even}&=&R_M+R_p+R_q, \\nn \\\\\nR_M&=&-(1+P_c)\\frac{2M^2}{s\\bar{x}^2}\\frac{x_-^2}{x^2}\\frac{\\sigma_+}{(1-\\beta z)^2}-\n\\frac{m^2}{s}(1+c^2)\\biggl[\\frac{1}{\\bar{x}_+^2}+\\frac{1}{\\bar{x}_-^2}\\biggr]; \\nn \\\\\nR_p&=&2(1+P_c)\\frac{X}{\\bar{x}x^2(1-\\beta z)};\\,\\,\\,\nR_q=\\frac{X}{\\bar{x}_+\\bar{x}_-}, \\qquad\n\\ea\nwhere $\\sigma_+=\\bar{x}^2(1-c)^2+(1+c)^2$, $\\bar{x}=1-x,\\bar{x}_\\pm=1-x_\\pm$ and\nthe exchange operator $P_c$ is defined as $P_cf(c)=f(-c)$, (see Eq. (\\ref{eq::18}), $P_c=P(t,u)$).\nThe quantity\n\\ba\nX=\\frac{t^2+t'^2+u^2+u'^2}{s^2}\n\\ea\ncan be written in two (equivalent) forms. In the case when it is convoluted\nwith the \"singular\" factor $1\/(1-\\beta z)$, keeping in mind that one has to further\nintegrate on $x_-$ and $z$ (see details in Appendix A) it has the form:\n\\ba\nX&=&A_1x_-^2+A_2(1-z)^2+A_3x_-(1-z)(c-t)+ \\nn \\\\\n&&A_4x_-^2(c-t)^2+A_5x_-(1-z)+A_6x_-^2(c-t),\n\\ea\nwith $x_-=2\\bar{x}\/r, r=2-x(1-t)$. Here and further we use the notation $t=a_-=1-2\\bar{x}_+\/(xx_-)$. The coefficients $A_i$ are\n\\ba\nA_1&=&\\frac{1+\\bar{x}^2}{4\\bar{x}^2}\\sigma_+~; A_2=\\frac{1}{2}x^2;~A_3=\\frac{x^3}{2\\bar{x}}; \\nn \\\\\nA_4&=&\\frac{x^2(1+\\bar{x}^2)}{4\\bar{x}^2};~ A_5=\\frac{x}{2\\bar{x}}[\\bar{x}^2(1-c)-(1+c)];~\nA_6=-\\frac{x}{2\\bar{x}^2}[\\bar{x}^3(1-c)+(1+c)].\n\\label{eq:eqai}\n\\ea\n\nFor \"nonsingular\" terms entering in $R_q$ we have (following from the definition):\n\\ba\nX&=&\\frac{1}{2}[x_+^2(1+c_+^2)+x_-^2(1+c^2)]= \\nn \\\\\n&&\\frac{1}{2}[(2-x_--x)^2+(x_-c+zx)^2+x_-^2(1+c^2)].\n\\ea\nThe contribution which is odd under the action of $P_c$ has the form (in agreement with Ref. \\cite{Ku77})\n\\ba\nR_{odd}&=&\\frac{1}{\\bar{x}}\\biggl[\\frac{2}{x^2(1-\\beta^2z^2)}[xx_+(c_+-z)+xx_-(z-c)]+\n\\frac{\\bar{x}}{\\bar{x}_+\\bar{x}_-}[x_-\\bar{x}_-c-x_+\\bar{x}_+c_+]+2(x_-c-x_+c_+) \\nn \\\\\n&& +\\frac{x_+(1+c_+)}{2x\\bar{x}_-(1+\\beta z)}[2x_+x_-(1-c c_+)+xx_-(1-z c)-x_+\\bar{x}_+(1-c_+)-xx_-(1-c)]-\\nn \\\\\n&&\\frac{x_+(1-c_+)}{2x\\bar{x}_-(1-\\beta z)}[2x_+x_-(1-c c_+)+xx_-(1-z c)-x_+\\bar{x}_+(1+c_+)-xx_-(1+c)]+ \\nn \\\\\n&&\\frac{x_-(1-c)}{2x\\bar{x}_+(1-\\beta z)}[2x_+x_-(1-c c_+)+xx_+(1-z c_+)-x_+\\bar{x}_+(1+c_+)-xx_-(1+c)]- \\nn \\\\\n&&\\frac{x_-(1+c)}{2x\\bar{x}_+(1+\\beta z)}[2x_+x_-(1-c c_+)+xx_+(1-z c_+)-x_+\\bar{x}_+(1-c_+)-xx_-(1-c)]\\biggr]. \\qquad\n\\ea\nLet now focus on the distribution $d\\sigma\/(d x d c)$.\n\nThe integration on the two other variables ($x_-,z$) (see Appendix \\ref{AppendixA}) leads to\n\\ba\n\\int\\frac{dx_- dz}{\\pi \\sqrt{D}}R_M=-\\frac{8\\bar{x}}{x}(1+P_c)\n\\frac{\\sigma_+}{A^4}-\\frac{2\\bar{x}}{x}(1+c^2), \\quad\nA=2-x(1-c).\n\\ea\nUsing the relations\n\\ba\n\\frac{1}{\\bar{x}_+\\bar{x}_-}=\\frac{1}{x}\\left[\\frac{1}{\\bar{x}_+}+\\frac{1}{\\bar{x}_-}\\right],\n\\quad\nx_+c_++x_-c+z x=0,\n\\ea\nand performing the integration over $z$, the expression for $R_q$ can be written as\n\\ba\nR_q=\\frac{1+c^2}{2x} \\left [(1+\\bar{x}^2)\\left (\\frac{1}{\\bar{x}_+}+\\frac{1}{\\bar{x}_-}\\right )+\\frac{2x\\bar{x}}{x_-^2}-2x\\right].\n\\ea\nThe integration on $x_-$ leads to\n\\ba\n\\int\\frac{dx_- dz}{\\pi \\sqrt{D}}R_q=\\frac{1+\\bar{x}^2}{x}(1+c^2)\\ln\\frac{s\\bar x}{m^2}\n\\ea\nand\n\\ba\n\\int\\frac{dx_- dz}{\\pi \\sqrt{D}}R_p&=&(1+P_c)\\frac{4}{x}\\biggl[4\\bar{x}^2A_1J_1+A_2J_2+ \\nn \\\\\n&&2\\bar{x}A_3J_3+4\\bar{x}^2A_4J_4+2\\bar{x}A_5J_5+4\\bar{x}^2A_6J_6 \\biggr],\n\\ea\nwith $A_i$ given above and $J_i$ listed in Appendix A. Note that\nonly quantity $J_1$ contains the logarithmically-enhanced contribution\n\\ba\nJ_1=\\frac{1}{A^4}\\ln\\frac{s}{M^2}+\\Delta J_1. \\nn\n\\ea\nMoreover, the coefficients $A_2-A_6$ as well as the quantity\n$\\Delta J_1$ are proportional to $x$ at small $x$.\n\nPerforming the integration over $z$ we obtain\n\\ba\n\\frac{d\\sigma_{odd}^{hard}}{d x d c}&=&\\frac{\\alpha^3}{s}(1-P_c)\\int\\limits_{\\bar{x}-\\frac{m^2x}{s\\bar{x}}}^{1-\\frac{m^2x}{s\\bar{x}}}\n\\frac{d x_-}{x\\bar{x}}\n\\left\\{c\\left (2x\\bar{x}_+-\\frac{x}{\\bar{x}_+}+\\frac{x\\bar{x}^2}{\\bar{x}_-}\\right ) \\right.\n\\nn\\\\\n&+&\\frac{1}{\\sqrt{R}}\\left [\\frac{1+c}{\\bar{x}_+}(c^2+c x+1-x+x^2)\n-\\frac{1-c}{\\bar{x}_-}(c^2\\bar{x}^2+c x\\bar{x}+1-x+x^2)\\right .\n\\nn \\\\\n&-& x A(1+c^2)+x^2c(1-c)^2-2x(c^3-c^2+c+1)+2(1-c^2)\\Big] \\Big \\},\n\\ea\nwith $R=(c-t)^2+(1-\\beta^2)(1-c^2)$.\n\nPerforming the integration over $x_-$ we obtain\n\\ba\n\\frac{d\\sigma_{odd}}{d x d c}&=&\\frac{\\alpha^3}{2s}(1-P_c)\\frac{1}{A}\n\\left\\{\\left [c^2+x c+1-x+x^2][\\frac{4}{x}\\ln\\frac{1+c}{2}+\n\\frac{1+c}{\\bar{x}}\\ln\\frac{s\\bar{x}(1+c)^2}{m^2A^2}\\right ] \\right.\\nn \\\\\n&&+\\left [-\\frac{4}{x}\\ln\\frac{\\bar{x}(1-c)}{2}+(1-c)\\ln\\frac{s\\bar{x}^3(1-c)^2}{m^2A^2}\\right ]\\left[c^2\\bar{x}^2+c x\\bar{x}+1-x+x^2\\right ]+ \\nn \\\\\n&&2c(2c+x(1-c))\\ln\\frac{4s\\bar{x}}{m^2A^2}+2(1+c^2)\\left[\\frac{6\\bar{x}}{A}-(2-x)-\\frac{A}{4\\bar{x}}(1+\\bar{x}^2)\\right] +\\nn \\\\\n&&\\left .\\left(-\\frac{4}{A}+\\frac{2-x}{\\bar{x}}\\right)\\left[x^2c(1-c)^2-2x(c^3-c^2+c+1)+2(1-c^2)\\right]\\right\\}.\n\\ea\nPerforming the integration on the photon energy fraction $\\Delta=\\frac{\\Delta E}{E}<x< 1-\\frac{4m^2}{s}$\nwe obtain (see Appendix A)\n\\ba\n\\frac{d\\sigma_{odd}^{hard}}{dc}&=&\\frac{\\alpha^3}{2s}\\biggl\\{4(1+c^2)\\ln\\frac{1+c}{1-c}\\ln\\frac{1}{\\Delta}+\nc\\left[ -6-\\frac{\\pi^2}{3}-\\ln^2\\left(\\frac{1-c}{1+c}\\right)\\right]+2(1-3c^2)\\ln\\frac{1+c}{1-c}+ \\nn \\\\\n&&\n\\frac{6}{1-c}\\left(-1+2\\frac{\\ln\\frac{2}{1+c}}{1-c}\\right)-\\frac{6}{1+c}\n\\left(-1+2\\frac{\\ln\\frac{2}{1-c}}{1+c}\\right)+ \\nn \\\\\n&&\n2(1+c^2)\\biggl[\\int\\limits_{\\frac{1-c}{2}}^{\\frac{1+c}{2}}\n\\frac{dx}{x}\\ln(1-x)+\\frac{1}{2}\\ln^2\\left (\\frac{1-c}{2}\\right )-\\frac{1}{2}\\ln^2\\left (\\frac{1+c}{2}\\right )\\biggr]\\biggr\\}.\n\\label{eq::46}\n\\ea\n\nThe odd part of differential distribution does not have logarithmic enhancement.\n\nThe logarithmically - enhanced contribution from hard\nphoton emission to the even part of differential cross section has the form:\n\\ba\n\\frac{d\\sigma^{hard}_{even}}{dx dc} =\n\\frac{\\alpha}{2\\pi}\\cdot \\biggl\\{\\biggl[\\frac{d\\sigma_B(\\bar{x} p_-, p_+)}{d c}+\n\\frac{d\\sigma_B(p_-,\\bar{x} p_+)}{d c}\\biggr] \\cdot\n\\frac{1+\\bar{x}^2}{x}\\biggl(\\ln\\frac{s}{ M^2}-1\\biggr) +\\nn \\\\\n2\\frac{d\\sigma_B(p_-,p_+)}{d c}\\biggl[\\frac{1+\\bar{x}^2}{x}\\ln\\left (\\frac{s}{m^2}-1\\right )\\biggr]+\n\\frac{\\pi\\alpha^2}{s}K(x,c)\\biggr\\},\n\\ea\nwith\n\\ba\nK(x,c)&=&(1+c^2)\\left (x+\\frac{1+\\bar{x}^2}{x}\\ln\\bar{x}\\right )+(1+P_c)\\left [\\frac{4x\\sigma_+}{A^4}+\\frac{4}{x}Q(x,c)\\right ], \\nn \\\\\nQ(x,c)&=&(1+\\bar{x}^2)\\Delta J_1\\sigma_++A_2J_2+2\\bar{x}A_3J_3+4\\bar{x}^2A_4J_4+2\\bar{x}A_5J_5+4\\bar{x}^2A_6J_6.\n\\ea\n\n\\section{Discussion}\n\nExtracting the logarithmically - enhanced terms in virtual and soft photon (27) and hard photon (47)\nemission and keeping in mind the factorization theorem \\cite{Ku85} we can express the differential cross section for the process\n\\be\n\\bar p(p_-)+p(p_+)\\to e_+(y_+)+e_-(y_-)+(\\gamma(k))\n\\ee\nin the form\n\\ba\n\\displaystyle\\frac{d\\sigma}{dc dy_+ dy_-}&=&\n\\int dx_+dx_-{\\cal D}(x_+,L_s){\\cal D}(x_-,L_s)\n\\displaystyle\\frac{d\\sigma_B(x_-p_-,x_+p_+,z_+,z_-)}{d c} \\nn \\\\\n&&\\frac{1}{|\\Pi(sx_+x_-)|^2}\\left (1+\\displaystyle\\frac{\\alpha}{\\pi} K\\right )\n\\displaystyle\\frac{1}{z_+z_-}{\\cal D}\\left (\\displaystyle\\frac{ y_+}{z_+},L_e\\right )\n{\\cal D}\\left (\\displaystyle\\frac{y_-}{z_-},L_e\\right )+ \\nn \\\\\n&&+\\left( \\displaystyle\\frac{d\\sigma}{d c}\\right )^{odd},\n\\ea\nwhere $y_{\\pm}$ is the energy fraction of the final lepton. The non-singlet structure function ${\\cal D}$ of the leptons is \\cite{Ku85,Skrzypek:1992vk}\n\\ba\n{\\cal D}(x,L)&=& \\delta(x-1)+ \\frac{b}{4} P^{(1)}(x)+\\frac{1}{2!}(\\frac{b}{4})^2 P^{(2)}(x)+...,\\nn\\\\\nP^{(1)}(x) &=& \\left(\\frac{1+x^2}{1-x}\\right)_+,\n\\qquad\nP^{(2)}(x) =P \\otimes P(x),\n\\ea\nor equivalently\n\\ba\n{\\cal D}(x,L)=\\frac{1}{2}b(1-x)^{\\frac{b}{2}-1}\\left (1+\\frac{3 b}{8}\\right )-\\frac{1}{4}b(1+x)+O(b^2),\n\\quad\nb=\\frac{2\\alpha}{\\pi}(L-1).\n\\ea\n\nThe shifted Born cross section for the hard subprocess is:\n\\be\n\\frac{d\\sigma_B(x_-p_-,x_+p_+,z_+,z_-)}{d c}=\\displaystyle\\frac{d\\sigma_B(x_-,x_+ )}{d c}=\n\\displaystyle\\frac{4\\pi\\alpha^2}{s}\n\\displaystyle\\frac{x_-^2(1-c)^2+x_+^2(1+c)^2}{[x_-(1-c)+x_+(1+c)]^4},\n\\ee\nand the energy fraction of the leptons in the hard subprocess is:\n\\be\nz_-=\\displaystyle\\frac{x_-^2(1-c)+x_+^2(1+c)}{x_-(1-c)+x_+(1+c)};~\nz_+=\\displaystyle\\frac{2x_- x_+}{x_-(1-c)+x_+(1+c)};~c=\\cos\\widehat{\\vec p_-\\cdot\\vec q_-}.\n\\ee\nThe explicit form of $(d\\sigma\/d c)^{odd}$ is the sum of soft, virtual and hard photon\nemission contributions ( see (\\ref{eq::18},\\ref{eq::25},\\ref{eq::46}):\n\\ba\n\\frac{d \\sigma^{odd}}{d c}& =& \\frac{\\alpha^3}{2 s}F(c), \\nn \\\\\nF(c)&=&c\\left(-6-\\frac{\\pi^2}{3}+2 \\ln\\frac{2}{1+c}\n\\ln\\frac{2}{1-c}+\\ln\\frac{4}{1-c^2}\\right )+3(1-2c^2)\\ln\\frac{1+c}{1-c}+ \\nn \\\\\n&&\\frac{6}{1-c}\\left(-1+\\frac{2}{1-c}\\ln\\frac{2}{1+c}\\right )\n-\\frac{6}{1+c}\\left(-1+\\frac{2}{1+c}\\ln\\frac{2}{1-c}\\right )+ \\nn \\\\\n&&4(1+c^2)\\left[Li_2\\left (\\frac{1-c}{2}\\right )-Li_2\\left(\\frac{1+c}{2}\\right)\\right ].\n\\ea\nThe relevant charge-asymmetry\n\\ba\n\\label{eq::56}\nA(c)=\\frac{d\\sigma(c)-d\\sigma(-c)}{d\\sigma(c)+d\\sigma(-c)}=\\frac{\\alpha}{\\pi}\\frac{F(c)}{1+c^2},\n\\ea\nturns out to be rather large to be measured in experiment: $|A|\\sim 4-5\\%$. It is presented\nin Fig. \\ref{Fig1}. We note the numerical agreement with the calculation of Ref. \\cite{Ku77}.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{Fig1.eps}\n\\caption{Asymmetry  A(c) $\\cdot \\frac{\\pi}{\\alpha}$ ( see(\\ref{eq::56}))\nas a function of $c, \\,\\,\\,c=\\cos\\widehat{\\vec p_{-} \\vec q_{-}}$.}\n\\label{Fig1}\n\\end{figure}\n\nThe non - leading terms (which do not contain the \"large logarithms\" $L_e, L$) are contained in the factor\n$(1+\\frac{\\alpha}{\\pi}K)$ with $K=K_{SV}+K_{hard}(x_+,x_-,c), \\,\\, K_{SV}=\\frac{2}{3}\\pi^2-1$ and\n\\ba\nK_{hard}(x_-,x_+,c)&=&\\frac{[x_-(1-c)+x_+(1+c)]^4}{8[(x_-^2(1-c)^2+x_+^2(1+c)^2]}F(x_-,x_+,c); \\nn \\\\\nF(x_-,x_+,c)&=&\\frac{1}{2}(1+c^2)[\\bar{x}_-+\\frac{1+x_-^2}{\\bar{x}_-}\\ln x_-+\\bar{x}_++\\frac{1+x_+^2}{\\bar{x}_+}\\ln x_+] \\nn \\\\\n&&+\\frac{4r_-\\bar{x}_-}{A_-^4}+\\frac{4}{\\bar{x}_-}Q(\\bar{x}_-,c)+\\frac{4r_+\\bar{x}_+}{A_+^4}+\n\\frac{4}{\\bar{x}_+}Q(\\bar{x}_+,-c),\n\\ea\nwith\n\\ba\nr_-=x_-^2(1-c)^2+(1+c)^2,\\qquad A_-=1+x_-+c(1-x_-); \\nn \\\\\nr_+=(1-c)^2+x_+^2(1+c)^2,\\qquad A_+=1+x_+-c(1-x_+).\n\\ea\nThe quantity\n\\ba\n\\label{eq::59}\nK_{hard}\\left (\\frac{1}{2},\\frac{1}{2},c\\right )=\\frac{1}{8(1+c^2)}F\\left (\\frac{1}{2},\\frac{1}{2},c\\right )\n\\ea\nis presented in Fig.\\ref{Fig2}.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{Fig2.eps}\n\\caption{K-factor for even part of cross-section  $K_{hard}(c)$ (see (\\ref{eq::59}))\nas a function  of $c, \\,\\,\\,c=\\cos\\widehat{\\vec p_{-} \\vec q_{-}}$.}\n\\label{Fig2}\n\\end{figure}\n\nIn the application of the present calculation to the kinematical range covered by PANDA ($s=15$ GeV$^2$) we can neglect the corrections related\nwith neutral weak currents (annihilation through neutral $Z$-boson). Indeed, the relevant\ncorrection $(1+\\delta_W)$ is very close to unity $\\delta_W\\sim a_v s\/M_Z^2<10^{-3}$.\n\nThe distributions on the energy fractions of the muons (Dalitz-plot distribution),\nthe inclusive distribution and the total cross-section\nof the process $e^+e^-\\to \\mu^+\\mu^-\\gamma$ was considered\nin the papers \\cite{Ku77},\\cite{Be81}. The relevant results for the process\nconsidered here\ncan be obtained by a simple replacement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\n\\noindent\nWebsites are networks of interlinked pages, and a good website makes it easy and intuitive for users to navigate its content.\nOne way of making navigation intuitive and content discoverable is to provide carefully placed hyperlinks.\nConsider for instance Wikipedia, the free encyclopedia that anyone can edit. The links that connect articles are essential for presenting concepts in their appropriate context, for letting users find the information they are looking for, and for providing an element of serendipity by giving users the chance to explore new topics they happen to come across without intentionally searching for them.\n\n\nUnfortunately, maintaining website navigability can be difficult and cumbersome, especially if the site is large and changes frequently \\cite{devanbu1999chime,Nentwich:2002:XCC}. For example, about 7,000 new pages are created on Wikipedia every day~\\cite{wikipedia-stats}.\nAs a result, there is an urgent need to keep the site's link structure up to date, which in return requires much manual effort and does not scale as the website grows, even with thousands of volunteer editors as in the case of Wikipedia. Neither is the problem limited to Wikipedia; on the contrary, it may be even more pressing on other sites, which are often maintained by a single webmaster. Thus, it is important to provide automatic methods to help maintain and improve website hyperlink structure.\n\n\n\nConsidering the importance of links, it is not surprising that the problems of finding missing links and predicting which links will form in the future have been studied previously \\cite{liben2007link,milne+witten2008_link,noraset2014adding,west-et-al2009a}.\nPrior methods typically cast improving website hyperlink structure simply as an instance of a link prediction problem in a given network;\n\\textit{i.e.}, they extrapolate information from existing links in order to predict the missing ones.\nThese approaches are often based on the intuition that two pages should be connected if they have many neighbors in common, a notion that may be quantified by the plain number of common neighbors, the Jaccard coefficient of sets of neighbors, and other variations \\cite{adamic03,liben2007link}.\n\nHowever, simply considering the existence of a link on a website is not all that matters. Networks in general, and hyperlink graphs in particular, often have traffic flowing over their links,\nand\na link is of little use if it is never traversed. For example, in the English Wikipedia, of all the 800,000 links added to the site in February 2015, the majority (66\\%) were not clicked even a single time in March 2015, and among the rest, most links were clicked only very rarely. For instance, only 1\\% of links added in February were used more than 100 times in all of March (\\Figref{fig:new_link_usage_ccdf}).\nConsequently, even if we could perfectly mimic Wikipedia editors (who would {\\em a priori} seem to provide a reasonable gold standard for links to add), the suggested links would likely have little impact on website navigation.\n\nFurthermore, missing\\hyp link prediction systems can suggest a large number of links to be inserted into a website. But\nbefore links can be incorporated, they typically need to be examined by a human, for several reasons.\nFirst, not all sites provide an interface that lets content be changed programmatically.\nSecond, even when such an interface is available, determining where in the source page to place a suggested link may be hard, since it requires finding, or introducing, a phrase that may serve as anchor text~\\cite{milne+witten2008_link}.\nAnd third, even when identifying anchor text automatically is possible, human verification and approval might still be required in order to ensure that the recommended link is indeed correct and appropriate.\nSince the addition of links by humans is a slow and expensive process, presenting the complete ranking of all suggested links to editors is unlikely to be useful.\nInstead, a practical system should choose a well composed subset of candidate links of manageable size.\n\n\n\\begin{figure}\n \\centering\n\t\\includegraphics{FIG\/new_link_usage_ccdf.pdf}\n\n\t\\caption{Complementary cumulative distribution function of the number of clicks received in March 2015 by links introduced in the English Wikipedia in February 2015.\n\t}\n\t\\vspace{-3mm}\n \\label{fig:new_link_usage_ccdf}\n\\end{figure}\n\n\nThus, what is needed is an automatic approach to improving website hyperlink structure that accounts for website navigation patterns and suggests a limited number of highly relevant links based on the way they will impact the website's navigability. \n\n\\xhdr{Present work}\nHere we develop an automatic approach for suggesting missing but potentially highly used hyperlinks for a given website. \nOur approach is language- and website\\hyp independent and is based on using access logs as collected by practically all web servers.\nSuch logs record users' natural browsing behavior, and we show that server logs contain strong signals not only about which existing links are presently used, but also about how heavily currently nonexistent links would be used if they were to be added to the site.\nWe build models of human browsing behavior, anchored in empirical evidence from log data, that allow us to predict the usage of any potentially introduced hyperlink $(s,t)$ in terms of its {\\em clickthrough\\ rate,} \\textit{i.e.}, the probability of being chosen by users visiting page $s$.\nThe models are based on the following intuition: Consider a page $s$ not linked to another page $t$.\nA large fraction of users visiting $s$ and later on, via an indirect path, $t$, suggests that they might have taken a direct shortcut from $s$ to $t$ had the shortcut existed. Even though one could hope to detect such events by parsing the logs, our approach is even more general. One of our models is able to estimate the fraction of people who would take a shortcut $(s,t)$ only based on the pairwise page transition matrix. That is, we are able to estimate navigation probabilities over longer paths by combining information about individual link transitions.\n\nThe second important aspect of our solution is that we only suggest a small number of the top $K$ most useful links, where $K$ is a `link budget' constraint.\nConsider a na\\\"ive approach that first ranks all candidate links by their probability of being used and then simply returns the top $K$.\nAn extreme outcome would be for all $K$ links to be placed in a single page $s$. \nThis is problematic because it seems {\\em a priori} more desirable to connect a large number of pages reasonably well with other pages than to connect a single page extremely well.\nThis intuition is confirmed by the empirical observation that the expected number of clicks users take from a given page $s$ is limited and not very sensitive to the addition of new links out of $s$. Thus we conclude that inserting yet another out-link to $s$ after some good ones have already been added to it achieves less benefit than adding an out-link to some other page $s'$ first. This diminishing\\hyp returns property, besides accurately mirroring human behavior,\nleads to an optimization procedure that does not over\\hyp optimize on single source pages but instead spreads good links across many pages.\n\nWe demonstrate and evaluate our approach on two very different websites: the full English Wikipedia and Simtk, a small community of biomedical researchers.\nFor both websites we utilize complete web server logs collected over a period of several months and demonstrate that our models of human browsing behavior are accurate and lead to the discovery of highly relevant and useful hyperlinks.\nHaving full access to the logs of Wikipedia, one of the highest\\hyp traffic websites in the world, is particularly beneficial, as it allows us to perform a unique analysis of human web\\hyp browsing behavior.\nOur results on Simtk\\ show that our method is general and applies well beyond Wikipedia.\n\nThe following are the main contributions of this paper:\n\\begin{itemize}\n \\itemsep -2pt\\topsep-10pt\\partopsep-10pt \n\\item We introduce the link placement problem under budget constraints, and design objective functions anchored in empirically observed human behavior and exposing an intuitive diminishing\\hyp returns property. We also provide an efficient algorithm for optimizing these objectives (\\Secref{sec:link placement}).\n\\item We identify signals in the logs that can indicate the usage of a future link before it is introduced, and we operationalize these signals in simple but effective methods for finding promising new links and estimating their future clickthrough\\ rates (\\Secref{sec:estimating clickthrough rates}).\n\\item We characterize the usage and impact of newly added links by analyzing human navigation traces mined from Wikipedia's server logs (\\Secref{sec:effects}).\n\\end{itemize}\n\n\\section{The link placement problem}\n\\label{sec:link placement}\n\n\\noindent\nFirst we introduce the {\\em link placement problem under budget constraints}. In this context, the budget refers to a maximum allowable number of links that may be suggested by an algorithm.\nAs mentioned in the introduction, realistic systems frequently require a human in the loop\nfor verifying that the links proposed by the algorithm are indeed of value\nand for inserting them.\nAs we need to avoid overwhelming the human editors with more suggestions than they can respond to, we need to carefully select our suggestions.\n\nFormally, the task is to select a set $A$ of suggested links of a given maximum size $K$, where the quality of the chosen set $A$ is determined by an {\\em objective function} $f$:\n\\begin{eqnarray}\n\\text{maximize} && f(A) \\\\\n\\text{subject to} && |A| \\leq K.\n\\end{eqnarray}\nIn principle, any set function $f : 2^V \\rightarrow \\mathbb{R}$\nmay be plugged in (where $V$ is the set of pages on the site, and $2^V$ is the set of all subsets of $V$),\nbut the practical usefulness of our system relies vitally on a sensible definition of $f$.\nIn particular, $f$ should be based on a reasonable model of human browsing behavior and should score link sets for which the model predicts many clicks above link sets for which it predicts fewer clicks.\n\nIn the following we introduce two simplified models of human browsing behavior and propose three objective functions $f$ based on these models.\nThen, we show that two of these objectives have the useful property of diminishing returns and discuss the implications thereof.\nFinally, we show that all three objectives can be maximized exactly by a simple and efficient greedy algorithm.\n\n\\subsection{Browsing models and objective functions}\n\\label{sec:objective functions}\n\n\\noindent\nWe first define notation, then formulate two web\\hyp browsing models, and finally design three objective functions based on the models.\n\n\\xhdr{Notation}\nWe model hyperlink networks as directed graphs $G=(V,E)$, with {\\em pages} as nodes and {\\em hyperlinks} as edges.\nWe refer to the endpoints of a link $(s,t)$ as {\\em source} and {\\em target,} respectively.\nUnlinked pairs of nodes are considered potential {\\em link candidates}.\nWe denote the overall number of views of a page $s$ by $\\Count{s}$. When users are in $s$, they have a choice to either stop or follow out-links of $s$. To simplify notation, we model stopping as a link click to a special sink page $\\varnothing \\notin E$ that is taken to be linked from every other page.\nFurther, $\\Count{st}$ counts the transitions from $s$ to $t$ via direct clicks of the link $(s,t)$.\nFinally, a central quantity in this research is the {\\em clickthrough\\ rate} $p_{st}$ of a link $(s,t)$.\nIt measures the fraction of times users click to page $t$, given that they are currently on page $s$:\n\\begin{equation}\np_{st} = \\Count{st} \/ \\Count{s}.\n\\end{equation}\nThe {\\em stopping probability} is given by $p_{s\\varnothing}$. Note that, since in reality users may click several links from the same source page, $p_{st}$ is generally not a distribution over $t$ given $s$.\n\n\\vspace{1.7mm}\nNext we define two models of human web\\hyp browsing behavior.\n\n\\xhdr{Multi-tab browsing model}\nThis model captures a scenario where the user may follow several links from the same page $s$, \\textit{e.g.}, by opening them in multiple browser tabs.\nWe make the simplifying assumption that, given $s$, the user decides for each link $(s,t)$ independently (with probability $p_{st}$) if he wants to follow it.\n\n\\xhdr{Single-tab browsing model}\nThis is a Markov chain model where the user is assumed to follow exactly one link from each visited page $s$.\nThis corresponds to a user who never opens more than a single tab, as might be common on mobile devices such as phones.\nThe probability of choosing the link $(s,t)$ is proportional to $p_{st}$.\n\n\\vspace{1.7mm}\nBased on these browsing models, we now define objective functions that model the value provided by the newly inserted links $A$. Note that the objectives reason about the clickthrough\\ rates $p_{st}$ of links $A$ that do not exist yet, so in practice $p_{st}$ is not directly available but must be estimated from data. We address this task in \\Secref{sec:estimating clickthrough rates}.\n\n\\xhdr{Link-centric multi-tab objective}\nTo capture the value of the newly inserted links $A$, this first objective assumes the multi-tab browsing model.\nThe objective computes the expected number of new-link clicks from page $s$ as $\\sum_{t : (s,t) \\in A} p_{st}$ and aggregates this quantity across all pages $s$:\n\\begin{equation}\nf_1(A) = \\sum_s w_s \\sum_{t : (s,t) \\in A} p_{st}.\n\\label{eqn:f_1}\n\\end{equation}\nIf we choose $w_s = \\Count{s}$ then $f_1(A)$ measures the expected number of clicks received by all new links $A$ jointly when $s$ is weighted according to its empirical page-view count.\nIn practice, however, page-view counts tend to follow heavy\\hyp tailed distributions such as power laws~\\cite{adar08revisitation}, so a few top pages would attract a disproportional amount of weight.\nWe mitigate this problem by using $w_s = \\log\\Count{s}$, \\textit{i.e.}, by weighting $s$ by the order of magnitude of its raw count.\n\n\n\\xhdr{Page-centric multi-tab objective}\nOur second objective also assumes the multi-tab browsing model.\nHowever, instead of measuring the total expected number of clicks on all new links $A$ (as done by $f_1$),\nthis objective measures the expected number of source pages on which at least one new link is clicked\n(hence, we term this objective {\\em page\\hyp centric,} whereas $f_1$ is link\\hyp centric):\n\\begin{equation}\nf_2(A) = \\sum_s w_s \\left(1 - \\prod_{t : (s,t) \\in A}  1 - p_{st} \\right).\n\\label{eqn:f_2}\n\\end{equation}\nHere, the product specifies the probability of no new link being clicked from $s$, and one minus that quantity yields the probability of clicking at least one new link from $s$.\nAs before, we use $w_s = \\log\\Count{s}$.\n\n\\xhdr{Single-tab objective}\nUnlike $f_1$ and $f_2$, our third objective is based on the single-tab browsing model.\nIt captures the number of page views upon which the one chosen link is one of the new links $A$:\n\\begin{equation}\nf_3(A) = \\sum_s w_s \\frac{\\sum_{t : (s,t) \\in A} p_{st}}{\\sum_{t : (s,t) \\in A} p_{st} + \\sum_{t : (s,t) \\in E} p_{st}}.\n\\label{eqn:f_3}\n\\end{equation}\nThe purpose of the denominator is to renormalize the independent probabilities of all links---old and new---to sum to 1, so $p_{st}$ becomes a distribution over $t$ given $s$.\nAgain, we use $w_s = \\log\\Count{s}$.\n\n\n\\subsection{Diminishing returns}\n\\label{sec:diminishing returns}\n\n\\noindent\nNext, we note that all of the above objective functions are monotone, and that $f_2$ and $f_3$ have a useful diminishing\\hyp returns property.\n\nMonotonicity means that adding one more new link to the solution $A$ will never decrease the objective value. It holds for all three objectives\n(we omit the proof for brevity):\n\\begin{equation}\nf(A \\cup \\{(s,t)\\}) - f(A) \\geq 0.\n\\end{equation}\nThe difference $f(A \\cup \\{(s,t)\\}) - f(A)$ is called the {\\em marginal gain} of $(s,t)$ with respect to $A$.\n\nThe next observation is that, under the link\\hyp centric multi-tab objective $f_1$, marginal gains never change: regardless of which links are already present in the network, we will always have\n\\begin{equation}\nf_1(A \\cup \\{(s,t)\\}) - f_1(A) = w_s p_{st}.\n\\label{eqn:f1_marg_gain}\n\\end{equation}\nThis means that $f_1$ assumes that the solution quality can be increased indefinitely by adding more and more links to the same source page $s$.\nHowever, we will see later (\\Secref{sec:competition between links}) that this may not be the case in practice: adding a large number of links to the same source page $s$ does not automatically have a large effect on the total number of clicks made from $s$.\n\nThis discrepancy with reality is mitigated by the more complex objectives $f_2$ and $f_3$.\nIn the page\\hyp centric multi-tab objective $f_2$, further increasing the probability of at least one new link being taken becomes ever more difficult as more links are added.\nThis way, $f_2$ discourages adding too many links to the same source page.\n\nThe same is true for the single\\hyp tab objective $f_3$:\nsince the click probabilities of all links from $s$---old and new---are required to sum to 1 here, it becomes ever harder for a link to obtain a fixed portion of the total probability mass of 1 as the page is becoming increasingly crowded with strong competitors.\n\n\nMore precisely, all three objective functions $f$ have the \\textit{submodularity} property (proof omitted for brevity):\n\\begin{equation}\nf(B \\cup \\{(s,t)\\}) - f(B) \\leq f(A \\cup \\{(s,t)\\}) - f(A),\n\\hspace{2mm} \\text{for $A \\subseteq B,$}\n\\label{eqn:diminishing returns}\n\\end{equation}\nbut while it holds with equality for $f_1$ (\\Eqnref{eqn:f1_marg_gain}), this is not true for $f_2$ and $f_3$. When marginal gains decrease as the solution grows, we speak of {\\em diminishing returns.}\n\n\nTo see the practical benefits of diminishing returns, consider two source pages $s$ and $u$, with $s$ being vastly more popular than $u$, \\textit{i.e.}, $w_s \\gg w_u$.\nWithout diminishing returns (under objective $f_1$), the marginal gains of candidates from $s$ will nearly always dominate those of candidates from $u$ (\\textit{i.e.}, $w_s p_{st} \\gg w_u p_{uv}$ for most $(s,t)$ and $(u,v)$), so $u$ would not even be considered before $s$ has been fully saturated with links.\nWith diminishing returns, on the contrary, $s$ would gradually become a less preferred source for new links.\nIn other words, there is a trade-off between the popularity of $s$ on the one hand (if $s$ is frequently visited then links that are added to it get more exposure and are therefore more frequently used), and its `crowdedness' with good links, on the other.\n\nAs a side note we observe that, when measuring `crowdedness', $f_2$ only considers the newly added links $A$, whereas $f_3$ also takes the pre\\hyp existing links $E$ into account.\nIn our evaluation (\\Secref{sec:global evaluation}), we will see that the latter amplifies the effect of diminishing returns.\n\n\n\n\\subsection{Algorithm for maximizing the objectives}\n\\label{sec:optimization}\n\n\\noindent\nWe now show how to efficiently maximize the three objective functions.\nWe first state two key observations that hold for all three objectives and then describe an algorithm that builds on these observations to find a globally optimal solution:\n\n\\begin{enumerate\n   \\itemsep -2pt\\topsep-10pt\\partopsep-10pt \n  \\item For a single source page $s$, the optimal solution is given by the top $K$ pairs $(s,t)$ with respect to $p_{st}$.\n  \\item Sources are `independent': the contribution of links from $s$ to the objective does not change when adding links to sources other than $s$. This follows from \\Eqnref{eqn:f_1}--\\ref{eqn:f_3}, by noting that only links from $s$ appear in the expressions inside the outer sums.\n\\end{enumerate}\n\n\n\\begin{algorithm}[t]\n\\caption{Greedy marginal-gain link placement}\n\\begin{algorithmic}[1]\n\\label{alg:greedy}\n\\INPUT Hyperlink graph $G=(V,E)$; source-page weights $w_s$;\\\\clickthrough\\ rates $p_{st}$; budget $K$; objective $f$\n\\OUTPUT Set $A$ of links that maximizes $f(A)$ subject to $|A| \\leq K$\n\\STATE $Q \\leftarrow$ new priority queue\n\\FOR{$s \\in V$}\n  \\STATE $\\Sigma_E \\leftarrow \\sum_{(s,t) \\in E}p_{st}$ \\hspace{1mm} {\\em \/\/ sum of $p_{st}$ values of old links from $s$}\n  \\STATE $\\Sigma \\leftarrow 0$ \\hspace{3mm} {\\em \/\/ sum of new $p_{st}$ values seen so far for $s$}\n  \\STATE $\\Pi \\leftarrow 1$ \\hspace{3mm} {\\em \/\/ product of new $(1-p_{st})$ values seen so far for $s$}\n  \\STATE $C \\leftarrow 0$ \\hspace{3mm} {\\em \/\/ current objective value for $s$}\n  \\FOR{$t \\in V$ in decreasing order of $p_{st}$}\n    \\STATE $\\Sigma \\leftarrow \\Sigma + p_{st}$\n    \\STATE $\\Pi \\leftarrow \\Pi \\cdot (1 - p_{st})$\n    \\STATE \\textbf{if} $f = f_1$ \\textbf{then} $C' \\leftarrow w_s \\cdot \\Sigma$\n    \\STATE \\textbf{else if} $f = f_2$ \\textbf{then} $C' \\leftarrow w_s \\cdot (1-\\Pi)$\n    \\STATE \\textbf{else if} $f = f_3$ \\textbf{then} $C' \\leftarrow w_s \\cdot \\Sigma \/ (\\Sigma+\\Sigma_E)$\n    \\STATE Insert $(s,t)$ into $Q$ with the marginal gain $C'-C$ as value\n    \\STATE $C \\leftarrow C'$\n  \\ENDFOR\n\\ENDFOR\n\\STATE $A \\leftarrow$ top $K$ elements of $Q$\n\\end{algorithmic}\n\\end{algorithm}\n\nThese observations imply that the following simple, greedy algorithm always produces an optimal solution%\n\\footnote{\nFor maximizing general submodular functions (\\Eqnref{eqn:diminishing returns}), a greedy algorithm gives a\n$(1-1\/e)$\\hyp\napproximation \\cite{nemhauser1978analysis}.\nBut in our case, observation~2 makes greedy optimal:\nour problem is equivalent to finding a maximum\\hyp weight basis of a uniform matroid of rank $K$ with marginal gains as weights; the greedy algorithm is guaranteed to find an optimal solution in this setting \\cite{edmonds1971matroids}.\n}\n(pseudocode is listed in Algorithm~\\ref{alg:greedy}).\nThe algorithm processes the data source by source.\nFor a given source $s$, we first obtain the optimal solution for $s$ alone by sorting all $(s,t)$ with respect to $p_{st}$ (line 7; \\textit{cf.}\\ observation 1).\nNext, we iterate over the sorted list and compute the marginal gains of all candidates $(s,t)$ from $s$ (lines 7--15).\nAs marginal gains are computed, they are stored in a global priority queue (line 13); once computed, they need not be updated any more (\\textit{cf.}\\ observation 2).\nFinally, we return the top $K$ from the priority queue (line 17).\n\n\\section{\\hspace{-1mm}Estimating clickthrough rates}\n\\label{sec:estimating clickthrough rates}\n\n\\noindent\nIn our above exposition of the link placement problem, we assumed we are given a clickthrough\\ rate $p_{st}$ for each link candidate $(s,t)$.\nHowever, in practice these values are undefined before the link is introduced.\nTherefore, we need to estimate what the clickthrough\\ rate for each nonexistent link would be in the hypothetical case that the link were to be inserted into the site.\n\nHere we propose four ways in which historical web server logs can be used for estimating $p_{st}$ for a nonexistent link $(s,t)$. Later (\\Secref{sec:pst evaluation}) we evaluate the resulting predictors empirically.\n\n\\xhdr{Method 1: Search proportion}\n\\label{sec:search}\nWhat we require are indicators of users' need to transition from $s$ to $t$ before the direct link $(s,t)$ exists.\nMany websites provide a search box on every page, and thus one way of reaching $t$ from $s$ without taking a direct click is to `teleport' into $t$ by using the search box. Therefore, if many users search for $t$ from $s$ we may interpret this as signaling the need to go from $s$ to $t$, a need that would also be met by a direct link.\nBased on this intuition, we estimate $p_{st}$ as the {\\em search proportion} of $(s,t)$, defined as the number of times $t$ was reached from $s$ via search, divided by the total number $\\Count{s}$ of visits to $s$.\n\n\\xhdr{Method 2: Path proportion}\n\\label{sec:indirect navigation}\nBesides search, another way of reaching $t$ from $s$ without a direct link is by navigating on an indirect path.\nThus, we may estimate $p_{st}$ as the {\\em path proportion} of $(s,t)$, defined as $\\PathCount{st}\/\\Count{s}$, where $\\PathCount{st}$ is the observed number of navigation paths from $s$ to $t$.\nIn other words, path proportion measures the average number of paths to $t$, conditioned on first seeing $s$.\n\n\\xhdr{Method 3: Path-and-search proportion}\nLast, we may also combine the above two metrics.\nWe may interpret search queries as a special type of page view, and the paths from $s$ to $t$ via search may then be considered indirect paths, too. Summing search proportion and path proportion, then, gives rise to yet another predictor of $p_{st}$. We refer to this joint measure as the {\\em path-and-search proportion}.\n\n\\xhdr{Method 4: Random-walk model}\n\\label{sec:random walks}\nThe measures introduced above require access to rather granular data:\nin order to mine keyword searches and indirect navigation paths, one needs access to complete server logs.\nProcessing large log datasets may, however, be computationally difficult.\nFurther, complete logs often contain personally identifiable information, and privacy concerns may thus make it difficult for researchers and analysts to obtain unrestricted log access.\nFor these reasons, it is desirable to develop clickthrough\\ rate estimation methods that manage to make reasonable predictions based on more restricted data.\nFor instance, although the Wikipedia log data used in this paper is not publicly available, a powerful dataset derived from it, the matrix of pairwise transition counts between all English Wikipedia articles, was recently published by the Wikimedia Foundation \\cite{wulczyn15}.\nWe now describe an algorithm for estimating path proportion{}s solely based on the {\\em pairwise} transition counts for those links that already exist.\n\n\\begin{algorithm}[t]\n\\caption{Power iteration for estimating path proportion{}s from pairwise transitions probabilities}\n\\begin{algorithmic}[1]\n\\label{alg:power-iter}\n\\INPUT Target node $t$, pairwise transition matrix $P$ with $P_{st}=p_{st}$\n\\OUTPUT Vector $Q_t$ of estimated path proportion{}s for all source pages when the target page is $t$\n\\STATE $Q_t \\leftarrow (0, \\dots, 0)^T$\n\\STATE $Q_{tt} \\leftarrow 1$\n\\STATE $Q_t' \\leftarrow (0, \\dots, 0)^T$ \\hspace{1.5mm} {\\em \/\/ used for storing the previous value of $Q_t$}\n\\WHILE{$\\|Q_t - Q_t'\\| > \\varepsilon$}\n  \\STATE $Q_t' \\leftarrow Q_t$\n  \\STATE $Q_t \\leftarrow P Q_t$ \\hspace{6mm} {\\em \/\/ recursive case of \\Eqnref{eqn:Q_st}}\n  \\STATE $Q_{tt} \\leftarrow 1$ \\hspace{9mm} {\\em \/\/ base case of \\Eqnref{eqn:Q_st}}\n\\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\nAs defined above, the path proportion{} measures the expected number of paths to $t$ on a navigation trace starting from $s$;\nwe denote this quantity as $Q_{st}$ here.\nFor a random walker navigating the network according to the empirically measured transitions probabilities, the following recursive equation holds:\n\\begin{eqnarray}\nQ_{st} = \\begin{cases}\n        1 & \\text{if $s=t$,}\\\\\n        \\sum_{u} p_{su} Q_{ut} & \\text{otherwise.}\n        \\end{cases}\n\\label{eqn:Q_st}\n\\end{eqnarray}\nThe base case states that the expected number of paths to $t$ from $t$ itself is 1 (\\textit{i.e.}, the random walker is assumed to terminate a path as soon as $t$ is reached).\nThe recursive case defines that, if the random walker has not reached $t$ yet, he might do so later on, when continuing the walk according to the empirical clickthrough\\ rates.\n\n\\xhdr{Solving the random\\hyp walk equation}\nThe random\\hyp walk equation (\\Eqnref{eqn:Q_st}) can be solved by power iteration.\nPseudocode is listed as Algorithm~\\ref{alg:power-iter}.\nLet $Q_t$ be the vector containing as its elements the estimates $Q_{st}$ for all pages $s$ and the fixed target $t$.\nInitially, $Q_t$ has entries of zero for all pages with the exception of $Q_{tt}=1$, as per the base case of \\Eqnref{eqn:Q_st} (lines~1--2).\nWe then keep multiplying $Q_t$ with the matrix of pairwise transition probabilities, as per the recursive case of \\Eqnref{eqn:Q_st} (line~6), resetting $Q_{tt}$ to 1 after every step (line~7).\nSince this reset step depends on the target $t$, the algorithm needs to be run separately for each $t$ we are interested in.\n\nThe algorithm is guaranteed to converge to a fixed point under the condition that $\\sum_{s} p_{st} < 1$ for all $t$ (proof omitted for conciseness), which is empirically the case in our data.\n\n\n\\section{Datasets and log processing}\n\\label{sec:data}\n\n\\noindent\nIn this section we describe the structure and processing of web server logs. We also discuss the properties of the two websites we work with, Wikipedia and Simtk.\n\n\\subsection{From logs to trees}\n\n\\xhdr{Log format}\nWeb server log files contain one entry per HTTP request, specifying {\\em inter alia:}\ntime\\-stamp, requested URL, referer URL, HTTP response status, user agent information, client IP address, and proxy server information via the HTTP X\\hyp Forwarded\\hyp For header.\nSince users are not uniquely identifiable by IP addresses (\\textit{e.g.}, several clients might reside behind the same proxy server, whose IP address would then be logged), we create an approximate user ID by computing an MD5 digest of the concatenation of IP address, proxy information, and user agent.\nCommon bots and crawlers are discarded on the basis of the user agent string.\n\n\\xhdr{From logs to trees}\n\\label{sec:from log to trees}\nOur goal is to analyze the traces users take on the hyperlink network of a given website.\nHowever, the logs do not represent these traces explicitly, so we first need to reconstruct them from the raw sequence of page requests.\n\nWe start by grouping requests by user ID and sorting them by time.\nIf a page $t$ was requested by clicking a link on another page $s$, the URL of $s$ is recorded in the referer field of the request for $t$.\nThe idea, then, is to reassemble the original traces from the sequence of page requests by joining requests on the URL and referer fields while preserving the temporal order.\nSince several clicks can be made from the same page, \\textit{e.g.}, by opening multiple browser tabs, the navigation traces thus extracted are generally trees.\n\nWhile straightforward in principle, this method is unfortunately unable to reconstruct the original trees perfectly.\nAmbiguities arise when the page listed in the referer field of $t$ was visited several times previously by the same user.\nIn this situation, linking $t$ to all its potential predecessors results in a directed acyclic graph (DAG) rather than a tree.\nTransforming such a DAG into a tree requires a heuristic approach.\nWe proceed by attaching a request for $t$ with referer $s$ to the {\\em most recent} request for $s$ by the same user.\nIf the referer field is empty or contains a URL from an external website, the request for $t$ becomes the root of a new tree.\n\nNot only is this greedy strategy intuitive, since it seems more likely that the user continued from the most recent event, rather than resumed an older one;\nit also comes with a global optimality guarantee.\nMore precisely, we observe that in a DAG $G$ with edge weights $w$, attaching each internal node $v$ to its minimum\\hyp weight parent $\\argmin_u w_{uv}$\nyields a minimum spanning tree of $G$. \nWe follow this approach with time differences as edge weights. Hence, the trees produced by our heuristic are the best possible trees under the objective of minimizing the sum (or mean) of all time differences spanned by the referer edges.\n\n\\xhdr{Mining search events}\nWhen counting keyword searches, we consider both site\\hyp internal search (\\textit{e.g.}, from the Wikipedia search box) and site\\hyp external search from general engines such as Goo\\-gle, Yahoo, and Bing.\nMining internal search is straightforward, since all search actions are usually fully represented in the logs.\nAn external search from $s$ for $t$ is defined to have occurred if $t$ has a search engine as referer, if $s$ was the temporally closest previous page view by the user, and if $s$ occurred at most 5 minutes before $t$.%\n\n\n\n\\subsection{Wikipedia data}\n\n\\xhdr{Link graph}\nThe Wikipedia link graph is defined by nodes representing articles in the main namespace, and edges representing the hyperlinks used in the bodies of articles.\nFurthermore, we use the English Wikipedia's publicly available full revision history (spanning all undeleted revisions from 2001 through April 3, 2015) to determine when links were added or removed.\n\n\\xhdr{Server logs}\nWe have access to Wikimedia's full server logs, containing all HTTP requests to Wikimedia projects. We consider only requests made to the desktop version of the English Wikipedia.\nThe log files we analyze comprise 3 months of data, from January through March 2015.\nFor each month we extracted around 3 billion navigation trees.\n\\Figref{fig:tree_metrics} summarizes structural characteristics of trees by plotting the complementary cumulative distribution functions (CCDF) of the tree size (number of page views) and of the average degree (number of children) of non-leaf page views per tree.\nWe observe that trees tend to be small:\n77\\% of trees consist of a single page view;\n13\\%, of two page views;\nand 10\\%, of three or more page views.\nAverage degrees also follow a heavy\\hyp tailed distribution. Most trees are linear chains of all degree-one nodes, but page views with larger numbers of children are still quite frequent;\n\\textit{e.g.}, 6\\% (44 million per month) of all trees with at least two page views have an average degree of 3 or more.\n\n\\begin{figure}[t]\n    \\centering\n    \\hspace{-3mm}\n    \\subfigure[Tree properties]{\n        \\includegraphics{FIG\/tree_properties.pdf}\n        \\label{fig:tree_metrics}\n    }\n    \\hspace{-3mm}\n    \\subfigure[New-link usage]{\n    \\includegraphics{FIG\/new_link_pst_ccdf_BY-DEGREE.pdf}\n     \\label{fig:new_link_pst_ccdf}\n     }\n\\vspace{-3mm}\n\\caption{\nWikipedia dataset statistics. (a)~CCDF of size and average degree of navigation trees. (b)~CCDF of clickthrough\\ rate for various source\\hyp page out-degrees. (Log--log scales.)\n}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\\subsection{Simtk{} data}\n\n\\noindent\nOur second log dataset stems from Simtk.org, a website where biomedical researchers share code and information about their projects. We analyze logs from June 2013 through February 2015.\n\nContrasting Simtk{} and Wikipedia, we note that the Simtk{} dataset is orders of magnitude smaller than the Wikipedia dataset, in terms of both pages and page views, at hundreds, rather than millions, of pages, and tens of millions, rather than tens of billions, of page views. Simtk{}'s hyperlink structure is also significantly less dense than Wikipedia's. On the one hand, this means that there is much improvement to be made by a method such as ours; on the other hand, it also means that the navigation traces mined from the logs are less rich than for Wikipedia. Therefore, instead of extracting trees, we extract {\\em sessions,} defined here as sequences of page views with idle times of no more than one hour \\cite{halfaker2015user} between consecutive events.\nFurther, to be consistent, we compute path proportion{}s (\\Secref{sec:estimating clickthrough rates}) by tallying up how often $s$ occurred before $t$ in the same session, rather than on a path in the same tree.\n\n\\section{\\hspace{-3mm}Evaluation: Effects of new links}\n\\label{sec:effects}\n\n\\noindent\nThe goal of this section is to investigate the effects of adding new links and assess some of the assumptions we made when formulating the link placement problem (\\Secref{sec:link placement}).\nIn particular, we answer the following questions:\nFirst, how much click volume do new links receive?\nSecond, are several new links placed in the same source page independent of each other, or do they compete for clicks?\n\nTo answer these questions, we consider the around 800,000 links $(s,t)$ added to the English Wikipedia in February 2015 and study all traces involving $s$ in the months of January and March 2015.\n\n\n\n\\subsection{Usage of new links}\n\\label{sec:usage of new links}\n\n\\noindent\nIn the introduction we have already alluded to the fact that new links tend to be rarely used, to the extent that 66\\% of the links added in February were not used even a single time in March, and only 1\\% were used more than 100 times (\\Figref{fig:new_link_usage_ccdf}).\n\n\\begin{figure}[t]\n    \\centering\n    \\hspace{-3mm}\n    \\subfigure[]{\n    \\includegraphics{FIG\/pst_vs_src_outdeg.pdf}\n \\label{fig:pst_vs_src_outdeg}\n    }\n    \\hspace{-3mm}\n    \\subfigure[]{\n    \\includegraphics{FIG\/pst_vs_rel_link_position.pdf}\n \\label{fig:pst_vs_rel_link_position}\n     }\n\\vspace{-3mm}\n\\caption{\nClickthrough\\ rate as function of (a)~source-page out-degree and (b)~relative position of link in wiki markup of source page (0 is top; 1 is bottom). Lower bin boundaries on $x$-axis.\n}\n\\vspace{-3mm}\n\\end{figure}\n\n\nWhile \\Figref{fig:new_link_usage_ccdf} plots absolute numbers, \\Figref{fig:new_link_pst_ccdf} shows the complementary cumulative distribution function (CCDF) of the clickthrough\\ rate $p_{st}$, stratified by the out-degree of the source page $s$.\nWe observe that, depending on source-page out-degree, between 3\\% and 13\\% of new links achieve over 1\\% clickthrough, with higher values for lower\\hyp degree source pages.\nThis finding is confirmed by \\Figref{fig:pst_vs_src_outdeg}, which plots the clickthrough\\ rate $p_{st}$ against source-page out-degree and shows a strong negative trend.\n\n\nFinally, \\Figref{fig:pst_vs_rel_link_position} confirms the fact that the popularity of a link is also correlated with its position in the source page \\cite{lamprecht2015quo}, with links appearing close to the top of the page achieving a clickthrough\\ rate about 1.6 times as high as that of links appearing in the center, and about 3.7 times as high as that of links appearing at the bottom.\n\n\n\n\\subsection{Competition between links}\n\\label{sec:competition between links}\n\n\\noindent\nTraces are generally trees, not chains (\\Secref{sec:from log to trees}).\nWhile in chains, $p_{st}$ would form a distribution over $t$ given $s$, \\textit{i.e.}, $\\sum_t p_{st} = 1$, this is not necessary in trees, where several next pages may be opened from the same page $s$, such that, in the most extreme case (when all $p_{st}=1$), $\\sum_t p_{st}$ may equal the out-degree of $s$.\n\nIn the multi-tab browsing model with its independence assumption\n(\\Secref{sec:objective functions}), we would see no competition between links; the larger the out-degree of $s$, the larger the expected number of clicks from $s$ for fixed $p_{st}$.\nIn its pure form, this model seems unlikely to be true, since it would imply a reader scanning the entire page, evaluating every link option separately, and choosing to click it with its probability $p_{st}$.\nIn a less extreme form, however, it is well conceivable that adding many good links to a page $s$ might significantly increase the number of links a given user chooses to follow from $s$.\n\nIn \\Secref{sec:usage of new links} we already saw that links from pages of higher out-degree tend to have lower individual clickthrough\\ rates, which may serve as a first sign of competition, or interaction, between links.\nAs our link placement method is allowed to propose only a limited number of links, the question of interaction between links is of crucial importance. Next we investigate this question more carefully.\n\nFirst we define the {\\em navigational degree} $\\Deg{s}$ of $s$ to be the total number of transitions out of $s$, divided by the total number of times $s$ was seen as an internal node of a tree, \\textit{i.e.}, without stopping there:\n\\begin{equation}\n    \\Deg{s} = \\frac{\\sum_{t\\neq\\varnothing} \\Count{st}}{\\Count{s} - \\Count{s\\varnothing}}.\n\\end{equation}\nIn other words, the navigational degree of $s$ is simply the average number of transitions users make out of $s$, given that they do not stop in $s$.  \nWe also define the {\\em structural degree,} or {\\em out-degree,} of $s$ to be the number of pages $s$ links to.\n\nNext, we examine the relation of the structural degree of $s$ with\n(1)~the probability of stopping at $s$ and\n(2)~the navigational degree of $s$,\nacross a large and diverse set of pages $s$.\n\n\\Figref{fig:structdeg_vs_pstop}, which has been computed from the transition counts for 300,000 articles in January 2015, shows that stopping is less likely on pages of larger structural degree, with a median stopping probability of 89\\% for pages with less than 10 out-links, and 76\\% for pages with at least 288 out-links.\nAdditionally, given that users do not stop in $s$, they make more clicks on average when $s$ offers more links to follow (median 1.00 for less than 10 out-links \\textit{vs.}\\ 1.38 for 288 or more out-links; \\Figref{fig:structdeg_vs_navdeg}).\n\nThese effects could be explained by two hypotheses.\nIt is possible that\n(i)~simply adding more links to a page also makes it more likely that more links are taken;\nor (ii)~structural degree may be correlated with latent factors such as `interestingness' or `topical complexity':\na more complex topic $s$ will have more connections (\\textit{i.e.}, links) to other topics that might be relevant for understanding $s$;\nthis would lead to more clicks from $s$ to those topics, but not simply because more link options are present but because of inherent properties of the topic of $s$.\n\nTo decide which hypothesis is true, we need to control for these inherent properties of $s$.\nWe do so by tracking the same $s$ through time and observing whether changes in structural degree are correlated with changes in navigational degree for fixed $s$ as well.\nIn particular, we consider two snapshots of Wikipedia, one from January 1, 2015, and the other from March 1, 2015.\nWe take these snapshots as the approximate states of Wikipedia in the months of January and March, respectively.\nThen, we compute structural and navigational degrees, as well as stopping probabilities, based exclusively on the links present in these snapshots,\nobtaining two separate values for each quantity, one for January and one for March.\nFor a fixed page $s$, the difference between the March and January values now captures the effect of adding or removing links from $s$ on the stopping probability and the navigational degree of $s$.\n\n\\Figref{fig:structdeg-delta_vs_pstop-delta} and \\ref{fig:structdeg-delta_vs_navdeg1-delta} show that this effect is minuscule, thus lending support to hypothesis ii from above: as structural degree grows or shrinks, both stopping probability (\\Figref{fig:structdeg-delta_vs_pstop-delta}) and navigational degree (\\Figref{fig:structdeg-delta_vs_navdeg1-delta}) vary only slightly, even for drastic changes in link structure;\n\\textit{e.g.}, when 100 links or more are added, the median relative increase in navigational degree is still only 0.3\\%, and when 100 links are deleted, the median relative decrease is only 0.1\\%.\n\n\nThis is not to say that adding links has no effect at all, but this effect stems from extreme, rather than typical, values, as indicated by the upward (downward) shift of interquartile ranges (\\Figref{fig:structdeg-delta_vs_navdeg1-delta}) as links are added (removed).\n\nIn a nutshell, simply adding more links does not increase the overall number of clicks taken from a page. Instead, links compete with each other for user attention.\nThis observation has important implications for user modeling and hence for budget\\hyp constrained link placement algorithms:\none should avoid spending too much of one's budget on the same source page, since this will not increase click volume indefinitely;\nrather, one should spread high\\hyp clickthrough\\ links across many different source pages.\n\nThese findings justify, {\\em post hoc,} the diminishing\\hyp returns properties of the objective functions $f_2$ and $f_3$ (\\Secref{sec:diminishing returns}).\n\n\n\\begin{figure}[t]\n    \\centering\n    \\hspace{-3mm}\n    \\subfigure[]{\n        \\includegraphics{FIG\/structdeg_vs_pstop.pdf}\n        \\label{fig:structdeg_vs_pstop}\n    }\n    \\hspace{-3mm}\n    \\subfigure[]{\n        \\includegraphics{FIG\/structdeg_vs_navdeg.pdf}\n        \\label{fig:structdeg_vs_navdeg}\n    }\n\n    \\hspace{-3mm}\n    \\subfigure[]{\n        \\includegraphics{FIG\/structdeg-delta_vs_pstop-delta.pdf}\n        \\label{fig:structdeg-delta_vs_pstop-delta}\n    }\n    \\hspace{-3mm}\n    \\subfigure[]{\n        \\includegraphics{FIG\/structdeg-delta_vs_navdeg1-delta.pdf}\n        \\label{fig:structdeg-delta_vs_navdeg1-delta}\n    }\n\\vspace{-5mm}\n\\caption{\nAcross different source pages, structural degree is (a) negatively correlated with stopping probability, and (b) positively correlated with navigational degree.\nWhen fixing the source page, however, structural degree has only a small effect on (c) stopping probability and (d) navigational degree. The $y$-axes in (c) and (d) are relative with respect to the values from January.\nLower bin boundaries on $x$-axes.\nBoxes show quartiles; whiskers show the full range without outliers.\n}\n    \\label{fig:impact_of_struct_deg}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation: Link placement}\n\\label{sec:evaluation}\n\n\\noindent\nNext we demonstrate the universality of our approach by evaluating it on two very different websites, Wikipedia and Simtk.\n\n\\subsection{Wikipedia}\n\\label{sec:wikipedia evaluation}\n\n\\noindent\nThe analysis of results for Wikipedia is divided into three parts. First, we show that the estimation methods from \\Secref{sec:estimating clickthrough rates} are suited for predicting the clickthrough\\ rates of new links.\nSince the evaluation set of added links is known only after the fact, a practically useful system must have the additional ability to identify such links on its own before they are introduced.\nTherefore, we then check whether a large predicted $p_{st}$ value indeed also means that the link is worthy of addition.\nLast, we investigate the behavior of our algorithm for link placement under budget constraints.\n\nWe also developed a graphical user interface that makes it easy to add missing links to Wikipedia:\nusers are shown our suggestions and accept or decline them with the simple click of a button \\cite{metapage}.\n\n\\subsubsection{Estimating clickthrough\\ rates}\n\\label{sec:pst evaluation}\n\n\\noindent\nAs described in \\Secref{sec:effects}, we have identified 800,000 links that were added to the English Wikipedia in February 2015. Here we evaluate how well we can predict the clickthrough\\ rates $p_{st}$ of these links in March 2015 from log data of January 2015, \\textit{i.e.}, before the link was added. In particular, we compare five different methods (\\Secref{sec:estimating clickthrough rates}):\n\\begin{enumerate}\n \\itemsep -2pt\\topsep-10pt\\partopsep-10pt \n\\item search proportion,\n\\item path proportion,\n\\item the combined path-and-search proportion,\n\\item random walks, and\n\\item a mean baseline.\n\\end{enumerate}\nThe mean baseline makes the same prediction for all candidates originating in the same source page $s$, namely the average clickthrough\\ rate of all out-links of $s$ that already exist.\n\n\\begin{table}[tb]\n  \\centering\n  {\\small\n\\begin{tabular}{rrrr}\n  \\hline\n & Mean absolute err. & Pearson corr. & Spearman corr. \\\\ \n  \\hline\n  Path prop. & \\textbf{0.0057} ($\\pm$0.0003) & 0.58 ($\\pm$0.12) & \\textbf{0.64} ($\\pm$0.01) \\\\ \n  Search prop. & 0.0070 ($\\pm$0.0004) & 0.49 ($\\pm$0.22) & 0.17 ($\\pm$0.02) \\\\ \n  P\\&S prop. & \\textbf{0.0057} ($\\pm$0.0003) & \\textbf{0.61} ($\\pm$0.13) & \\textbf{0.64} ($\\pm$0.01) \\\\ \n  Rand. walks & 0.0060 ($\\pm$0.0004) & 0.53 ($\\pm$0.13) & 0.59 ($\\pm$0.02) \\\\ \n  Mean baseln. & 0.0072 ($\\pm$0.0004) & 0.20 ($\\pm$0.06) & 0.34 ($\\pm$0.02) \\\\ \n   \\hline\n\\end{tabular}\n}\n\\caption{Comparison of clickthrough\\ rate estimation methods on Wikipedia, with bootstrapped 95\\% confidence intervals.\n}\n  \\label{tbl:pst_evaluation}\n\\vspace{-3mm}\n\\end{table}\n\n\nTable~\\ref{tbl:pst_evaluation} evaluates the results using three different metrics: mean absolute error, Pearson correlation, and Spearman rank correlation. The table shows that across all metrics path proportion and path-and-search proportion perform best.\nFurther, it is encouraging to see that the random\\hyp walk--based predictor, which only requires the pairwise transition matrix, is not lagging behind by much.\nAlso, recall from \\Figref{fig:pst_vs_src_outdeg} that $p_{st}$ is to a large part determined by source\\hyp page out-degree.\nThis is why the mean baseline performs quite well (mean absolute error of 0.72\\%, \\textit{vs.}\\ 0.57\\% achieved by the best\\hyp performing method) even though it predicts the same value for all links from the same page (but see the next section for when the baseline fails).\nFor a graphical perspective on the relation between predicted and ground-truth values, we refer to \\Figref{fig:pindir_vs_pst} and \\ref{fig:psearch_vs_pst}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\subfigure[]{\n  \\includegraphics{FIG\/pindir_vs_pst.pdf}\n  \\label{fig:pindir_vs_pst}\n  }\n  \\hspace{-1mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/psearch_vs_pst.pdf}\n  \\label{fig:psearch_vs_pst}\n  }\n  \\hspace{-1.5mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/prec_at_k.pdf}\n  \\label{fig:prec_at_k}\n  }\n  \\hspace{-1.5mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/global_opt\/jaccard_coefficient.pdf}\n  \\label{fig:jaccard}\n  }\n\n  \\subfigure[]{\n  \\includegraphics{FIG\/global_opt\/prior_cum_clickthrough.pdf}\n  \\label{fig:prior_cum_clickthrough}\n  }\n  \\hspace{-1.5mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/global_opt\/num_targets_per_source.pdf}\n  \\label{fig:num_targets_per_source}\n  }\n  \\hspace{-1.5mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/global_opt\/cumulative_click_volume.pdf}\n  \\label{fig:cumulative_click_volume}\n  }\n  \\hspace{-1.5mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/global_opt\/avg_click_volume.pdf}\n  \\label{fig:avg_click_volume}\n  }\n \n \n\\vspace{-3mm}\n\\caption{\nWikipedia results.\n(a)~Path proportion \\textit{vs.}\\ clickthrough\\ rate.\n(b)~Search proportion \\textit{vs.}\\ clickthrough\\ rate (log--log; black lines correspond to gray dots kernel\\hyp smoothed on linear scales).\n(c)~Precision at $k$ on link prediction task (\\Secref{sec:predicting link addition}; logarithmic $x$-axis); path-and-search proportion nearly identical to path proportion{} and thus not shown.\n(d--h)~Results of budget\\hyp constrained link placement for objectives of \\Eqnref{eqn:f_1}--\\ref{eqn:f_3} (\\Secref{sec:global evaluation});\n`prior cumulative clickthrough' refers to second term in denominator of \\Eqnref{eqn:f_3}.\n}\n\\vspace{-3mm}\n\\end{figure*}\n\n\n\n\\subsubsection{Predicting link addition}\n\\label{sec:predicting link addition}\n\n\\noindent\nOur models for clickthrough\\ prediction reason about the hypothetical scenario that a link $(s,t)$ is added to the site.\nIn the previous subsection we showed that the models manage to predict the future usage of a link fairly well for the subset of links that were actually added.\nBut, as stated above, this set is not known when the predictors are deployed in practice.\n\nIntuitively, a large predicted clickthrough\\ rate $p_{st}$ should also imply that $(s,t)$ is a useful link and should thus be added.\nHere we test whether this is actually the case by evaluating whether our high\\hyp ranking predictions using the January data correspond to links that were added to Wikipedia in February.\n\nFor this task, we need a ground\\hyp truth set of positive and negative examples.\nWe require a non\\hyp negligible number of positive examples for which our methods can potentially predict a high $p_{st}$ value.\nSince our methods are based mainly on path and search counts, we therefore consider all links added in February with at least 10 indirect paths or searches in January; call this set $L$.\nSince in reality most page pairs are never linked, we also need to add a large number of negative examples to the test set.\nWe do so by including all out-links of the sources, and in-links of the targets, appearing in $L$.\nUsing this approach, we obtain a set of about 38 million link candidates, of which 9,000 are positive examples.\nThe aforementioned threshold criterion is met by 7,000 positive, and 104,000 negative, examples.\nTherefore, while there is a fair number of positive examples with many indirect paths and searches, they are vastly outnumbered by negative examples with that same property; this ensures that retrieving the positive examples remains challenging for our methods\n(\\textit{e.g.}, returning all links meeting the threshold criterion in random order yields a precision of only 6\\%).\n\nGiven this labeled dataset, we evaluate the precision at $k$ of the candidate ranking induced by the predicted $p_{st}$ values.\nThe results, plotted in \\Figref{fig:prec_at_k}, indicate that our methods are well suited for predicting link addition.\nPrecision stays high for a large set of top predictions:\nof the top 10,000 predictions induced by the path\\hyp proportion and random\\hyp walk measures, about 25\\% are positive examples; among the top 2,000 predictions, as many as 50\\% are positive.\nAs before, random walks perform nearly as well as the more data\\hyp intensive path proportion, and better than search proportion.\n\nNote that the mean baseline, which performed surprisingly well on the clickthrough\\ rate prediction task (\\Secref{sec:pst evaluation}), does not appear in the plot.\nThe reason is that it fails entirely on this task, producing only two relevant links among its top 10,000 predictions.\nWe conclude that it is easy to give a good $p_{st}$ estimate when it is known which links will be added.\nHowever, predicting whether a link should be added in the first place is much harder.\n\n\n\n\\subsubsection{Link placement under budget constraints}\n\\label{sec:global evaluation}\n\n\\noindent\nThe above evaluations concerned the quality of predicted $p_{st}$ values.\nNow we rely on those values being accurate and combine them in our budget\\hyp constrained link placement algorithm (\\Secref{sec:optimization}).\nWe use the $p_{st}$ values from the path\\hyp proportion method.\n\nFirst, we consider how similar the solutions produced by the different objective functions are.\nFor this purpose, \\Figref{fig:jaccard}) plots the Jaccard coefficients between the solutions of $f_1$ and $f_2$ (orange) and between the solutions of $f_1$ and $f_3$ (blue) as functions of the solution size $K$.\nWe observe that $f_1$ and $f_2$ produce nearly identical solutions.\nThe reason is that clickthrough\\ rates $p_{st}$ are generally very small, which entails that \\Eqnref{eqn:f_1} and \\ref{eqn:f_2} become very similar (formally, this can be shown by a Taylor expansion around 0).\nObjective $f_3$, on the contrary, yields rather different solutions;\nit\ntends to favor source pages with fewer pre\\hyp existing high\\hyp clickthrough\\ links (due to the second term in the denominator of \\Eqnref{eqn:f_3}; \\Figref{fig:prior_cum_clickthrough}) and attempts to spread links more evenly over all source pages (\\Figref{fig:num_targets_per_source}).\nIn other words, $f_3$ offers a stronger diminishing\\hyp returns effect: the marginal value of more links decays faster when some good links have already been added to the same source page.\n\nNext, we aim to assess the impact of our suggestions on real Wikipedia users.\nRecall that we suggest links based on server logs from January 2015.\nWe quantify the impact of our suggestions in terms of the total number of clicks they received in March 2015 (contributed by links added by editors after January).\n\\Figref{fig:cumulative_click_volume}, which plots the total March click volume as a function of the solution size $K$, shows that (according to $f_3$) the top 1,000 links received 95,000 clicks in total in March 2015, and the top 10,000 received 336,000 clicks.\nRecalling that two-thirds of all links added by humans receive no click at all (\\Figref{fig:new_link_usage_ccdf}), this clearly demonstrates the importance of the links we suggest.\n\\Figref{fig:avg_click_volume} displays the average number of clicks per suggested link as a function of $K$.\nThe decreasing shape of the curves implies that higher\\hyp ranked suggestions received more clicks, as desired.\nFinally, comparing the three objectives, we observe that $f_3$ fares best on the Wikipedia dataset:\nits suggestions attract most clicks (\\Figref{fig:cumulative_click_volume} and \\ref{fig:avg_click_volume}), while also being spread more evenly across source pages (\\Figref{fig:num_targets_per_source}).\n\nThese results show that the links we suggest fill a real user need.\nOur system recommends useful links before they would normally be added by editors, and it recommends additional links that we may assume would also be clicked frequently if they were added.\n\nTable~\\ref{tbl:examples} illustrates our results by listing the top link suggestions.\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{c|c}\n\n\\hspace{-5mm}\n\\subfigure[Page-centric multi-tab objective $f_2$]{\n{\\small\n\\begin{tabular}{lll}\n& \\textbf{Source} & \\textbf{Target} \\\\\n* & ITunes Originals -- Red Hot  &  Road Trippin' Through Time \\\\\n  & \\hspace{2mm} Chili Peppers & \\\\\n* & Internat.\\ Handball Federation &  2015 World Men's Handball Champ. \\\\\n  & Baby, It's OK! & Olivia (singer) \\\\\n  & Payback (2014) & Royal Rumble (2015) \\\\\n* & Nordend & Category:Mountains of the Alps \\\\\n* & Gran Hotel & Grand Hotel (TV series) \\\\\n* & Edmund Ironside & Edward the Elder \\\\\n* & Jacob Aaron Estes &  The Details (film) \\\\\n  & Confed.\\ of African Football &  2015 Africa Cup of Nations \\\\\n  & Live Rare Remix Box & Road Trippin' Through Time\n\\end{tabular}\n}\n}\n&\n\\subfigure[Single-tab objective $f_3$]{\n{\\small\n\\begin{tabular}{lll}\n& \\textbf{Source} & \\textbf{Target} \\\\\n* & ITunes Originals -- Red Hot  &  Road Trippin' Through Time \\\\\n  & \\hspace{2mm} Chili Peppers & \\\\\n* & Gran Hotel & Grand Hotel (TV series) \\\\\n* & Tithe: A Modern Faerie Tale & Ironside: A Modern Faery's Tale \\\\\n* & Nordend & Category:Mountains of the Alps \\\\\n* & Jacob Aaron Estes &  The Details (film) \\\\\n& Blame It on the Night &  Blame (Calvin Harris song) \\\\\n* & Internat.\\ Handball Federation &  2015 World Men's Handball Champ. \\\\\n* & The Choice (novel) & Benjamin Walker (actor) \\\\\n  & Payback (2014) & Royal Rumble (2015) \\\\\n* & Just Dave Van Ronk & No Dirty Names\n\\end{tabular}\n}\n}\n\n\\end{tabular}\n\\vspace{-3mm}\n\\caption{\nTop 10 link suggestions of Algorithm~\\ref{alg:greedy} using objectives $f_2$ and $f_3$.\nClickthrough\\ rates $p_{st}$ estimated via path proportion{} (\\Secref{sec:estimating clickthrough rates}).\nObjective $f_1$ (not shown) yields same result as $f_2$ but includes an additional link for source page \\textit{Baby, It's OK!}, demonstrating effect of diminishing returns on $f_2$.\nAsterisks mark links added by editors after prediction time, independently of our predictions.\n}\n\\vspace{-3mm}\n\\label{tbl:examples}\n\\end{table*}\n\n\n\n\n\\subsection{Simtk}\n\\label{sec:simtk evaluation}\n\n\\noindent\nIt is important to show that our method is general, as it relies on no features specific to Wikipedia.\nTherefore, we conclude our evaluation with a brief discussion of the results obtained on the second, smaller dataset from Simtk.\n\nFor evaluating our approach, we need to have a ground-truth set of new links alongside their addition dates.\nUnlike for Wikipedia, no complete revision history is available for Simtk{}, but we can nonetheless assemble a ground truth by exploiting a specific event in the site's history:\nafter 6 months of log data, a sidebar with links to related pages was added to all pages.\nThese links were determined by a recommender system once and did not change for 6 months.\nOur task is to predict these links' clickthrough\\ rates $p_{st}$ for the 6 months after, based on log data from the 6 months before.\n\n\n\n\\begin{table}[tb]\n  \\centering\n{\\small\n\\begin{tabular}{rrrr}\n  \\hline\n & Mean absolute err. & Pearson corr. & Spearman corr. \\\\ \n  \\hline\nPath prop. & 0.020 ($\\pm$0.003) & \\textbf{0.41} ($\\pm$0.10) & \\textbf{0.50} ($\\pm$0.09) \\\\ \n  Mean baseln. & \\textbf{0.013} ($\\pm$0.003) & $-$0.01 ($\\pm$0.11) & $-$0.27 ($\\pm$0.10) \\\\ \n   \\hline\n\\end{tabular}\n}\n\\vspace{-3mm}\n\\caption{Performance of path\\hyp proportion clickthrough\\ estimation on Simtk, with bootstrapped 95\\% confidence intervals.}\n  \\label{tbl:pst_evaluation_simtk}\n\\end{table}\n\nThe results on the $p_{st}$ prediction task are summarized in Table~\\ref{tbl:pst_evaluation_simtk};\nwe compare path proportion{} to the mean baseline (\\textit{cf.}~\\Secref{sec:pst evaluation}).\nAt first glance, when considering only mean absolute error, it might seem as though the former were outperformed by the latter.\nBut upon closer inspection, we realize that this is simply because the baseline predicts very small values throughout, which tend to be close to the ground-truth values, but cannot discriminate between good and bad candidates, as signified by the negative correlation coefficients.\nPath proportion, on the contrary, does reasonably well, at Pearson (Spearman) correlation coefficients of 0.41 (0.50).\nThis correlation can be observed graphically in \\Figref{fig:pindir_vs_pst_simtk}.\n\nLast, we analyze the solutions of our budget\\hyp constrained link placement algorithm.\nOur first observation is that the two multi\\hyp tab objectives $f_1$ and $f_2$ differ much more on Simtk\\ than on Wikipedia\n(with Jaccard coefficients between 40\\% and 60\\%, compared to over 90\\%),\nwhich shows that the page\\hyp centric multi\\hyp tab objective $f_2$ is not redundant but adds a distinct option to the repertoire.\n\n\\begin{figure}[t]\n  \\centering\n  \\hspace{-3mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/simtk\/pindir_vs_pst.pdf}\n  \\label{fig:pindir_vs_pst_simtk}\n  }\n \n \n \n \n \n  \\hspace{-3mm}\n  \\subfigure[]{\n  \\includegraphics{FIG\/simtk\/num_targets_per_source.pdf}\n  \\label{fig:num_targets_per_source_simtk}\n  }\n\\vspace{-5mm}\n\\caption{\nSimtk results.\n(a)~Path proportion \\textit{vs.}\\ clickthrough\\ rate (black line corresponds to gray dots kernel\\hyp smoothed on linear scales).\n(b)~Solution concentration of budget\\hyp constrained link placement; dotted vertical line: number of distinct pages.\n}\n \\label{fig:results_simtk}\n\\vspace{-1mm}\n\\end{figure}\n\nWe also observe that the value of the single\\hyp tab objective $f_3$ saturates after inserting around as many links as there are pages (750).\nThe reason is that only very few links were available and clicked before the related\\hyp link sidebar was added, so the second term in the denominator of \\Eqnref{eqn:f_3} is typically much smaller than the first term and the numerator; with the first link added to $s$, the ratio in \\Eqnref{eqn:f_3} becomes close to 1, and the contribution of $s$ to $f_3$ close to its possible maximum $w_s$.\nThereafter, the objective experiences extreme diminishing returns for $s$.\n\\Figref{fig:num_targets_per_source_simtk} confirms that $f_3$ starts by inserting one link into each of the around 750 pages.\n\nWe conclude that\nwith few pre\\hyp existing links, the page\\hyp centric multi-tab objective $f_2$ might be better suited than the single-tab objective $f_3$, as it provides a more balanced trade-off between\nthe number of links per source page\nand predicted clickthrough\\ rates.\n\n\\section{Discussion and related work}\n\\label{sec:discussion}\n\n\\noindent\nThere is a rich literature on the task of link prediction and recommendation, so rather than trying to be exhaustive, we refer the reader to the excellent survey by Liben\\hyp Nowell and Kleinberg~\\cite{liben2007link} and to some prior methods for the specific case of Wikipedia \\cite{adafre+derijke2005,milne+witten2008_link,noraset2014adding,west-et-al2009a}, as it is also our main evaluation dataset.\nThe important point to make is that most prior methods base their predictions on the static structure of the network, whereas we focus on how users interact with this static structure by browsing and searching on it.\n\nAmong the few link recommendation methods that take usage data into account are those by Grecu \\cite{grecu2014navigability} and West \\textit{et al.}\\ \\cite{west2015mining}.\nThey leverage data collected through a human\\hyp computation game in which users navigate from a given start to a given target page on Wikipedia, thereby producing traces that may be used to recommend new `shortcut' links from pages along the path to the target page.\nAlthough these approaches are simple and work well, they are limited by the requirement that the user's navigation target be explicitly known.\nEliciting this information is practically difficult and, worse, may even be fundamentally impossible: often users do not have any target in mind, \\textit{e.g.}, during exploratory search \\cite{marchionini2006exploratory} or when browsing the Web for fun or distraction.\nOur approach alleviates this problem, as it does not rely on a known navigation target, but rather works on passively collected server logs.\n\nWeb server logs capture user needs and behavior in real time, so another clear advantage of our log\\hyp based approach is the timeliness of the results it produces.\nThis is illustrated by \\Tabref{tbl:examples}, where most suggestions refer to current events, films, music, and books.\n\nOur random\\hyp walk model (\\Secref{sec:random walks}) builds on a long tradition of Markovian web navigation models \\cite{davison2004learning,downey07models,singer2015hyptrails}.\nAmong other things, Markov chains have been used to predict users' next clicks \\cite{davison2004learning,Sarukkai2000377} as well as their ultimate navigation targets \\cite{west+leskovec2012www}.\nOur model might be improved by using a higher\\hyp order Markov model, based on work by Chierichetti \\textit{et al.}\\ \\cite{chierichetti2012web}, who found that human browsing is around 10\\% more predictable when considering longer contexts.\n\nFurther literature on human navigation includes information foraging \\cite{chi01scent,olston03stenttrails,pirolli2007information,west+leskovec2012www}, decentralized network search \\cite{helic2013models,kleinberg2000,trattner2012exploring}, and the analysis of click trails from search result pages \\cite{bilenko08trails,downey2008understanding,white10scenic}.\n\nIn the present work we assume that frequent indirect paths between two pages indicate that a direct link would be useful.\nWhile this is confirmed by the data (\\Figref{fig:pindir_vs_pst}), research has shown that users sometimes deliberately choose indirect routes \\cite{teevan04teleport}, which may offer additional value \\cite{white10scenic}.\nDetecting automatically in the logs when this is the case would be an interesting research problem.\n\nOur method is universal in the sense that it uses only data logged by any commodity web server software, without requiring access to other data, such as the content of pages in the network.\nWe demonstrate this generality by applying our method on two rather different websites without modification.\nWhile this is a strength of our approach, it would nonetheless be interesting to explore if the accuracy of our link clickthrough\\ model could be improved by using machine learning to combine our current estimators, based on path counts, search counts, and random walks, with domain\\hyp specific elements such as the textual or hypertextual content of pages.\n\nA further route forward would be to extend our method to other types of networks that people navigate.\nFor instance,\ncitation networks, knowledge graphs, and some online social networks could all be amenable to our approach.\nFinally, it would be worthwhile to explore how our methods could be extended to not only identify links inside a given website but to also links between different sites.\n\nIn summary, our paper makes contributions to the rich line of work on improving the connectivity of the Web. We hope that future work will draw on our insights to increase the usability of websites as well as the Web as a whole.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n\\xhdr{Acknowledgments}\nThis research has been supported in part by NSF\nIIS-1016909,             \nIIS-1149837,      \nIIS-1159679,             \nCNS-1010921,             \nNIH R01GM107340,\nBoeing,                   \nFacebook,\nVolkswagen,                \nYahoo,\nSDSI,\nand Wikimedia Foundation.\nRobert West acknowledges support by a Stanford Graduate Fellowship.\n}\n\n\\vspace{-2mm}\n\\bibliographystyle{abbrv}\n\\vspace{0mm}\n{\\small\n\\input{bibliography.bbl}\n}\n\n\\hide{\n\n\\section{Properties of objective functions}\n\\label{app:objectives}\n\\subsection{Monotonicity}\n\\begin{equation}\nf_2(A \\cup (u,v)) - f_2(A) =  w_s \\left(1 - \\prod_{(s,t) \\in A}  1 - p_{st} \\right) p_{uv}\n\\end{equation}\n\\begin{equation}\nf_3(A) = \\sum_s w_s \\frac{\\sum_{(s,t) \\in A} p_{st}}{\\sum_{(s,t) \\in A} p_{st} + \\sum_{(s,t) \\in E} p_{st}}.\n\\end{equation}\nLet $\\alpha = \\sum_{(s,t) \\in A} p_{st}$ and $\\beta = \\sum_{(s,t) \\in A} p_{st} + \\sum_{(s,t) \\in E} p_{st}$. Note that $\\alpha < \\beta$\n\\begin{equation}\nf_3(A \\cup \\{(s,t)\\}) - f_3(A) = w_s \\left(\\frac{ \\alpha + p_{st}}{\\beta + p_{st}} - \\frac{\\alpha}{\\beta} \\right) > 0\n\\end{equation}\n\\subsection{Top k optimality}\nLet $A_k$ be a $k$ sized subset of outlinks with the top-$k$ $p_{st}$ values for a given page as $s$. Let $p^A_1, p^A_2 \\ldots p^A_k$ be the $p_{st}$ values in decreasing order\\\\\nLet there be another $k$ sized subset $B_k$ which has a higher score but does not contain the top-k outlinks. Let $p^B_1, p^B_2 \\ldots p^B_k$ be the $p_{st}$ values in decreasing order.\n\nConsider $f_2$,\n\\begin{equation}\nf_2(A_k) - f_2(B_k) =  w_s \\left( \\prod_{i \\in 1 \\ldots k}  (1 - p^B_i)  - \\prod_{i \\in 1 \\ldots k}  (1 - p^A_i) \\right)\n\\end{equation}\nBut we know from the top-k optimility that\n\\[\np^A_i \\ge p^B_i \\implies (1 - p^B_i) \\ge (1 - p^A_i) \\text{   } \\forall i \\in 1 \\ldots k\n\\]\nThus\n\\[\nf_2(A_k) - f_2(B_k) \\ge 0\n\\]\nwith equality occuring at $p^A_i = p^B_i \\text{   }\\forall i \\in 1 \\ldots k$\nThus the top-$k$ values of $p_{st}$ for a source, yeild the optimal solution of size $k$\n\nNow consider $f_3$,\nLet $\\alpha^A = \\sum_{(s,t) \\in A_k} p_{st}$ and $\\beta^A = \\sum_{(s,t) \\in A_k} p_{st} + \\sum_{(s,t) \\in E} p_{st}$. Note that $\\alpha^A < \\beta^A$\nNow $\\alpha^B = \\sum_{(s,t) \\in B_k} p_{st} = \\alpha^A - c$, where $c$ is a constant.\nSimilarly $\\beta^B = \\alpha^B - c$\n\\begin{equation}\nf_3(A_k) - f_3(B_k) = w_s \\left(\\frac{\\alpha^A}{\\beta^A} - \\frac{\\alpha^B - c}{\\beta^B - c} \\right)  \\ge 0\n\\end{equation}\nWith equality obtained only under the condition that $p^A_i = p^B_i \\text{   }\\forall i \\in 1 \\ldots k$\n\n\n\n\ndiminishing returns (technical term: submodularity)\n\n}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUsing  classical submanifold techniques, a lot of results on the geometry  \nof totally umbilical   submanifolds (and other special hypersurfaces) in ambient manifolds of special geometries were obtained \\cite{BCO, Chen1, chen1, chen2, chen3, chen4, ok, to, tsu, yam}. As one example, O. Kowalski \\cite{Kow} used the Codazzi-Mainardi equation to prove  the following  elementary and well-known result:\n\\begin{thm} \\label{bfact}Every totally umbilical connected hypersurface of an Einstein manifold of dimension greater or equal to 3 is of constant mean curvature.\n\\end{thm}\n\nExamples of ambient Riemannian Einstein manifolds $(\\widetilde M^{m+1}, \\widetilde g)$ of dimension \n $m+1 \\geq 3$ are Riemannian ${\\mathop{\\rm Spin}}$ manifolds carrying an $\\alpha$-Killing spinor ($\\alpha\\in \\mathbb C$), i.e., a spinor field $\\psi$ \nsatisfying the equation \n\\begin{equation}\\label{def:Killing} \\widetilde \\nabla_X \\psi = \\alpha X\\cdot \\psi,\\end{equation}\nfor any  vector $X$ tangent to $\\widetilde M$, where $\\widetilde \\nabla$ denotes the spinorial Levi-Civita connection on the spinor bundle and $``\\cdot\"$ the Clifford multiplication, compare Section~\\ref{prelim}.\\medskip    \n\n\nFor ${\\mathop{\\rm Spin}}$ manifolds it is known that the Killing constant $\\alpha$ has to be zero (parallel spinor), a nonzero real constant (real Killing spinor) or a nonzero purely imaginary constant (imaginary Killing spinor) \\cite{BHMM}. When $\\alpha$ is real, such spinors characterize the limiting case in the Friedrich's and Hijazi's inequalities which provide a lower bound for the eigenvalues of the Dirac operator involving the infimum of the scalar curvature or the first eigenvalue of the Yamabe operator \\cite{frfr, hij84, hij86}. Moreover, the existence of $\\alpha$-Killing spinors leads to restrictions on the geometry and topology of the manifold. In fact besides being Einstein (and even Ricci-flat when $\\alpha =0$), $\\widetilde M$ is automatically compact if $\\alpha$ is real and noncompact if $\\alpha$ is purely imaginary.  Complete simply connected ${\\mathop{\\rm Spin}}$ manifolds with real, parallel or imaginary Killing  spinors  have been classified by Wang \\cite{wang}, B{\\\"a}r \\cite{bar1} and Baum  \\cite{baum1, baum2, baum3} and the existence is glued to the holonomy of the manifold. This classification gives, in some dimensions, other examples than the most symmetric ones as Euclidean space, the sphere  or the hyperbolic space. These examples are relevant to physicists in general relativity where the Dirac operator plays a central role. \\medskip\n \nTechniques from ${\\mathop{\\rm Spin}}$ geometry  have been successfully used to produce striking advances in extrinsic geometry (see e.g. the study of CMC or minimal surfaces in homogeneous 3-spaces which arise in Thurston's classification of 3-dimensional geometries and Alexandrov-type theorems as in \\cite{HM13, HS14, HMZ01, HMR03, Bar98}). It is remarkable that, in many extrinsic results, ${\\mathop{\\rm Spin}}$ geometrical tools - in particular special\/natural spinor fields and  the Dirac operator - have played a central role and  inspired further research directions.\\medskip\n\nWhen shifting from the from the classical ${\\mathop{\\rm Spin}}$ geometry to $\\mathrm{Spin}^c$ geometry, the situation is more general and many obstacles appear since the $\\mathrm{Spin}^c$ structure will not only depend on the geometry of the manifold but also on the connection (and hence the curvature) of the auxiliary line bundle associated with the fixed $\\mathrm{Spin}^c$ structure. From a physical point of view, spinors model fermions while $\\mathrm{Spin}^c$-spinors can be interpreted as fermions coupled to an electromagnetic field. Transferring the idea to use spinorial methods in the study of submanifolds to the $\\mathrm{Spin}^c$ world, allows us to cover more ambient geometric structures (CR structures, K\\\"ahler and Sasaki structures). Indeed, O. Hijazi, S. Montiel and F. Urbano constructed on K\\\"ahler-Einstein  manifolds  with  positive  scalar  curvature \\cite{HMU}, $\\mathrm{Spin}^c$\nstructures  carrying K\\\"ahlerian  Killing  spinors.  The  restriction  of  these  spinors  to  minimal  Lagrangian\nsubmanifolds provides topological and geometric restrictions on these submanifolds (see \\cite{RN12, Na11} for other applications of the use of $\\mathrm{Spin}^c$ geometry in  extrinsic geometry). Equation \\eqref{def:Killing} on $\\mathrm{Spin}^c$ manifolds has been studied  by A. Moroianu \\cite{moroi}  when $\\alpha$ is real and by the authors \\cite{grosse-nakad} when $\\alpha$ is purely imaginary. In fact, a complete simply connected manifold has a parallel $\\mathrm{Spin}^c$ spinor if and only if it is isometric to the Riemannian product between a simply connected K\\\"ahler manifold (with its canonical or anti-canonical $\\mathrm{Spin}^c$ structure) and a simply connected ${\\mathop{\\rm Spin}}$ manifold carrying a parallel spinor. The only simply connected $\\mathrm{Spin}^c$ manifolds admitting real non-parallel Killing spinors other than the ${\\mathop{\\rm Spin}}$ manifolds are the non-Einstein Sasakian manifolds\nendowed with their canonical or anti-canonical $\\mathrm{Spin}^c$ structure. Beside that, complete $\\mathrm{Spin}^c$ manifolds\nwith imaginary Killing spinors are isometric to a special warped product of a $\\mathrm{Spin}^c$ manifold with a parallel spinor with  $\\mathbb R$. These classification results, stated above, show that, in contrast to the ${\\mathop{\\rm Spin}}$ case,  $\\mathrm{Spin}^c$ manifolds carrying an $\\alpha$-Killing spinor are not in general Einstein manifolds. For this reason, we start by extending  Theorem~\\ref{bfact} to ambient Riemannian $\\mathrm{Spin}^c$ manifolds carrying an $\\alpha$-Killing spinor. In fact, our main result is:\n\\begin{thm}\\label{Exto}\nEvery connected totally umbilical  hypersurface $M^m$  of a Riemannian $\\mathrm{Spin}^c$ manifold $\\widetilde M^{m+1}$ with $m+1\\geq 5$  carrying a real (including parallel) Killing spinor or of dimension $m+1\\geq 3$ carrying an imaginary Killing spinor is of constant mean curvature. \\end{thm}\nHere, we recall that if $\\widetilde{M}$ is already ${\\mathop{\\rm Spin}}$, Theorem~\\ref{Exto} is just a special case of Theorem~\\ref{bfact} because in this case $\\widetilde M$ is Einstein. Using the classification of Killing spinors on $\\mathrm{Spin}^c$ manifolds cited above we thus in particular obtained:\n\n\\begin{corollary}\n Every connected totally umbilical  hypersurface $M^m$  of a K\u00e4hler or a Sasakian manifold has constant mean curvature.\n\\end{corollary}\n\nFor K\u00e4hler manifolds the last statement is known from \\cite[Thm. 4.2]{Chen1} and we merely give a spinorial proof here. The counterpart for Sasakian manifolds was not known before to our best knowledge.\\medskip\n\nWe will show by counterexamples that  Theorem~\\ref{Exto} is sharp in the sense that it fails if the ambient $\\mathrm{Spin}^c$ manifold is of dimension  $3$ or $4$  carrying a parallel or  real Killing spinor.  The proof of Theorem~\\ref{Exto} relies on two  families of differential forms naturally associated to the spinor  obtained by the restriction of the $\\alpha$-Killing spinor to the hypersurface $M$. These differential forms and their exterior derivatives  involve the mean curvature $H$ of the isometric immersion and hence allow to deduce that $H$ is constant. Dependent on whether $\\alpha$ is real or imaginary, the proof of Theorem~\\ref{Exto} differs in these cases and is carried out separately (see Section~\\ref{Mmain}). \\medskip\n \nAs further applications of Theorem~\\ref{Exto}, we give some no-existence results of extrinsic hyperspheres in some special complete  ${\\mathop{\\rm Spin}}$ manifolds.\n\\begin{thm}\\label{THMp1}\nThere are no extrinsic hyperspheres in \n\\begin{enumerate}[(i)]\n\\item complete manifolds with holonomy $G_2$ and Spin$(7)$.\n \\item complete simply connected 3-Sasakian manifold of dimension $4m+3$  which is not of constant curvature\n\\item complete simply connected Sasakian Einstein manifold of dimension $4m+3$, $m \\geq 2$ which is not 3-Sasakian \n\\item compact Sasakian-Einstein manifold of dimension $2m+1$, $m \\geq 2$ which are not  locally symmetric.\n\\item homogeneous warped product  $\\widetilde M = N \\times_f  \\mathbb R$ where $f(t) = e^{4 \\mu t}$ ($\\mu \\in \\mathbb R^*$) and $N$ a complete Riemannian ${\\mathop{\\rm Spin}}$ manifold with a parallel spinor and of non-constant sectional curvature.\n\\end{enumerate}\n\\end{thm}\n\n Note that (i) was already obtained in \\cite{JMS} and we give here just a spinorial proof. Moreover, Theorem~\\ref{THMp1} is a particular case of the more general Theorem~\\ref{pa-ap}. In fact, we prove that there are no extrinsic hyperspheres in Riemannian ${\\mathop{\\rm Spin}}$ manifolds of non-constant sectional curvature and carrying an $\\alpha$-Killing spinor field. \n\n \n\n\\section{Preliminaries}\\label{prelim}\n\nIn this section, we briefly review \nsome basic facts about $\\mathrm{Spin}^c$ structures on oriented Riemannian manifolds and their hypersurfaces \\cite{friedrich, LM, BHMM, Bar98, Na11, bfg}.\n\n\\subsection{Hypersurfaces and induced \\texorpdfstring{$\\mathrm{Spin}^c$}{Spinc} structures}\\hfill \\medskip\n\n\\textbf{Spin$^c$ structures on manifolds:} Let $(\\widetilde M^{m+1}, \\widetilde g)$ be a Riemannian $\\mathrm{Spin}^c$ manifold of dimension $m+1 \\geq 3$ without\nboundary. On such a manifold, we have  a Hermitian complex vector bundle $\\Sigma \\widetilde M$ endowed with a natural scalar product $\\<., .\\>$ and with a connection $\\widetilde \\nabla $ which parallelizes the metric. We denote by $\\Re\\<.,.\\>$ the real part of the scalar product $\\<., .\\>$. This complex vector bundle, called the $\\mathrm{Spin}^c$ bundle, is endowed with a Clifford multiplication denoted by $``\\cdot\"$, $\\cdot\\colon T\\widetilde M \\rightarrow \\mathrm{End}_{\\mathbb C} (\\Sigma \\widetilde M)$, such that at every point $x \\in \\widetilde M$, defines an irreducible representation of the corresponding Clifford algebra. Hence, the complex rank of $\\Sigma \\widetilde M$ is $2^{[\\frac {m+1}{2}]}$. The Clifford multiplication can be extended to exterior products of the tangent bundle and to differential forms, such that $(v_1\\wedge \\ldots \\wedge v_k)\\cdot \\varphi\\mathrel{:=} v_1\\cdot \\ldots \\cdot v_k\\cdot \\varphi$ if the $v_i$'s are mutually orthogonal and such that $v^\\sharp\\cdot \\psi\\mathrel{:=}  v\\cdot \\psi$ for all vector fields $v_i,v$ and spinors $\\psi$ and where $.^\\sharp$ denotes the isomorphism $TM\\to T^*M$ induced by the metric.\\medskip\n\nGiven a $\\mathrm{Spin}^c$ structure on $(\\widetilde M^{m+1}, g)$, one can prove that the determinant line bundle $\\mathrm{det} (\\Sigma \\widetilde M)$ has a root of index $2^{[\\frac{m+1}{2}]-1}$. We denote\nby $\\widetilde L$ this root line bundle over $\\widetilde M$ and it is called the auxiliary line bundle associated with the $\\mathrm{Spin}^c$ structure. Locally, a ${\\mathop{\\rm Spin}}$ structure always exists. We denote by $\\Sigma' \\widetilde M$ the (possibly globally non-existent)\nspinor bundle. Moreover, the square root of the auxiliary line bundle $\\widetilde L$\nalways exists locally. But, $\\Sigma\\widetilde M = \\Sigma' \\widetilde M \\otimes {\\widetilde L}^{\\frac 12}$ exists globally.  This essentially means that, while\nthe spinor bundle and ${\\widetilde L}^{\\frac 12}$\nmay not exist globally, their tensor product (the $\\mathrm{Spin}^c$  bundle) is\ndefined globally. Thus, the connection $\\widetilde \\nabla$ on $\\Sigma \\widetilde M$ is the twisted connection of the one on the\nspinor bundle (coming from the Levi-Civita connection) and a fixed connection on $\\widetilde L$. \\medskip \n\nWe may now define the Dirac operator $\\widetilde D$ acting on the space of smooth\nsections of $\\Sigma\\widetilde M$  by the composition of the metric connection and\nthe Clifford multiplication. In local coordinates this reads as\n\\begin{align*}\\widetilde D =\\sum_{j=1}^{m+1} e_j \\cdot \\widetilde \\nabla_{e_j},\n \\end{align*}\nwhere $\\{e_1,\\ldots,e_{m+1}\\}$ is a local oriented orthonormal tangent frame. It is a first order elliptic operator, formally self-adjoint with respect to the $L^2$-scalar product and satisfies the Schr\\\"odinger-Lichnerowicz formula \n\\begin{align}\\label{eq:Lich}\n{\\widetilde D}^2=\\widetilde\\nabla^*\\widetilde \\nabla+\\frac{1}{4}\\widetilde {\\mathrm{scal}}+\\frac{\\i}{2}\\widetilde\\Omega\\cdot,\n\\end{align}\nwhere $\\widetilde\\nabla^*$ is the adjoint of $\\widetilde\\nabla$\nwith respect to the $L^2$-scalar product, $\\widetilde {\\mathrm{scal}}$ is the scalar curvature of $\\widetilde M$, $\\i\\widetilde \\Omega$ is the curvature of\nthe auxiliary line bundle $\\widetilde L$ associated with the fixed connection ($\\widetilde \\Omega$ is a real $2$-form on $\\widetilde M$) and $\\widetilde \\Omega \\cdot$\nis the extension of the Clifford multiplication to differential forms. For any $X \\in \\Gamma(T\\widetilde M)$ and any spinor field $\\psi \\in \\Gamma (\\Sigma \\widetilde M)$ , the Ricci identity is given by\n\\begin{align}\\label{Ricci-identity}\n\\sum_{k=1}^{m+1} e_k \\!\\cdot\\! \\widetilde{\\mathcal{R}}(e_k,X) \\psi=\n\\frac 12 \\widetilde {\\mathop{\\rm Ric}}(X) \\!\\cdot\\! \\psi-\\frac{\\i}{2} (X\\lrcorner\\widetilde\\Omega)\\cdot\\psi,\n\\end{align}\nwhere $\\widetilde{{\\mathop{\\rm Ric}}}$ is the Ricci curvature of $(\\widetilde M^{m+1}, g)$ and $\\widetilde {\\mathcal{R}}$ is the curvature tensor of the spinorial connection $\\widetilde\\nabla$. \\medskip\n\nWhen $m$ is even, the complex volume form $\\widetilde\\omega_{\\mathbb C} \\mathrel{:=} \\i^{[\\frac{m+2}{2}]} e_1 \\cdot\\ldots \\cdot e_{m+1}$ acts on $\\Sigma \\widetilde M$ as the identity, i.e., $\\widetilde\\omega_\\mathbb C \\!\\cdot\\!\\psi = \\psi$ for any spinor $\\psi \\in \\Gamma(\\Sigma \\widetilde M)$. Besides, if $m$ is odd, we have $\\widetilde\\omega_\\mathbb C^2 =1$. We denote by $\\Sigma^\\pm \\widetilde M$ the eigenbundles corresponding to the eigenvalues $\\pm 1$, hence $\\Sigma\\widetilde  M = \\Sigma^+ \\widetilde M \\oplus \\Sigma^- \\widetilde M$ and a  spinor field $\\psi$ can be written as $\\psi = \\psi^+ + \\psi^-$. The conjugate $\\overline \\psi$ of $\\psi$ is defined  by $\\overline \\psi = \\psi^+ - \\psi^-$.\\medskip\n\n As pointed out, the Clifford multiplication can be extended to differential forms and one sees that\n\\begin{eqnarray*}\n\\<\\delta \\cdot \\psi, \\psi\\>= (-1)^{\\frac{k(k+1)}{2}}\\  \\overline{\\<\\delta\\cdot\\psi, \\psi\\>}\n\\end{eqnarray*}\nfor any $k$-form $\\delta$ and a spinor field $\\psi\\in \\Gamma (\\Sigma \\widetilde M)$. \nThis directly implies that for  mutually orthogonal vector fields $v_{1},\\ldots, v_k$ we have\n\\begin{align}\\label{cli-diff}\n\\<v_1\\cdot v_2\\cdot \\ldots \\cdot v_k \\cdot \\psi, \\psi\\>\\in \\left\\{\\begin{matrix}\n                                                                   \\mathbb R & \\text{for }k\\equiv 0,3\\ \\text{ mod  } 4\\\\\n                                                                   \\i \\mathbb R & \\text{for }k\\equiv 1,2\\ \\text{ mod  } 4.                                                                                                                                     \\end{matrix}\n \\right.\n\\end{align}\n\n\\textbf{Spin$^c$ structures on  hypersurfaces:} The following can be e.g. found in \\cite{r2}.  Any $\\mathrm{Spin}^c$ structure on  $ (\\widetilde M^{m+1}, g)$ induces a $\\mathrm{Spin}^c$ structure on an oriented hypersurface $(M^m, g)$ of dimension $m\\geq 2$, and we have \n$$ \\Sigma M\\simeq \\left\\{\n\\begin{array}{l}\n\\Sigma {\\widetilde M}_{|_M} \\ \\ \\ \\ \\ \\ \\text{\\ \\ \\ if\\ $m$ is even,} \\\\\\\\\n \\Sigma^+ {\\widetilde M}_{|_M}   \\ \\text{\\ \\ \\ \\ \\ \\ if\\ $m$ is odd.}\n\\end{array}\n\\right.\n$$\nFurthermore Clifford multiplication by a vector field $X$, tangent to $M$, is given by \n\\begin{align*}\nX\\!\\cdot\\!_M\\phi = (X\\!\\cdot\\!\\nu\\!\\cdot\\! \\psi)_{|_M},\n\\end{align*}\nwhere $\\psi \\in  \\Gamma(\\Sigma \\widetilde M)$ (or $\\psi \\in \\Gamma(\\Sigma^+ \\widetilde M)$ if $m$ is odd),\n$\\phi$ is the restriction of $\\psi$ to $M$, ``$\\!\\cdot\\!_M$'' the Clifford multiplication  on $M$ and $\\nu$ is the unit normal\nvector field of $M$ in $\\widetilde M$. Also, when $m$ is odd, we obtain $\\Sigma M \\simeq \\Sigma^- \\widetilde M \\vert_M$. With this identification, the Clifford multiplication is given by\n$X\\!\\cdot\\!_M\\phi = -(X\\!\\cdot\\!\\nu\\!\\cdot\\! \\psi)_{|_M}$. In particular, we have $\\Sigma \\widetilde M \\simeq \\Sigma M \\oplus \\Sigma M$.\\medskip\n\nMoreover, the corresponding auxiliary line bundle $L$ on $M$ is the restriction to $M$ of $\\widetilde L$ and the curvature $2$-form $\\i\\Omega $ on $L$ is given by  $\\i\\Omega = \\i \\widetilde \\Omega\\vert_{M}$.  For every\n$\\psi \\in \\Gamma(\\Sigma \\widetilde M)$ ($\\psi \\in \\Gamma(\\Sigma^+ \\widetilde M)$ if $m$ is odd), the real 2-forms\n$\\Omega$ and $\\widetilde \\Omega$ are related by\n\\begin{align*}\n(\\widetilde\\Omega \\!\\cdot\\!\\psi)_{|_M} = \\Omega\\!\\cdot\\!_M\\phi -\n(\\nu\\lrcorner\\widetilde\\Omega)\\!\\cdot\\!_M\\phi.\n\\end{align*}\nWe denote by $\\nabla$ the $\\mathrm{Spin}^c$  connection on $\\Sigma M$. Then, for all $X\\in \\Gamma(TM)$, we have the $\\mathrm{Spin}^c$ Gauss formula:\n\\begin{equation}\\label{eq_spincgauss}\n(\\widetilde\\nabla_X\\psi)_{|_M} =  \\nabla_X \\phi + \\frac 12 \\mathrm{II} X \\!\\cdot\\!_M\\phi,\n\\end{equation}\nwhere $\\mathrm{II}$ denotes the Weingarten map of the hypersurface. Denoting by $D$  the Dirac operator on $M$ and  by the same symbol any spinor and its restriction to $M$, we have\n\\begin{equation*}\nD \\phi = \\frac{m}{2}H\\phi -\\nu\\!\\cdot\\! \\widetilde D\\phi-\\widetilde \\nabla_{\\nu}\\phi,\n\\end{equation*}\nwhere $H = \\frac 1m \\mathrm{tr\\,}(\\mathrm{II})$ denotes the mean curvature and $D^M = D$ if $m$ is even and $D^M= D\\oplus(-D)$ if $m$ is odd.\n\n\n\n\\section{Totally umbilical  hypersurfaces of \\texorpdfstring{$\\mathrm{Spin}^c$}{Spinc} manifolds carrying an \\texorpdfstring{$\\alpha$}{alpha}-Killing spinor}\nLet $(\\widetilde{M}^{m+1}, \\widetilde g)$ be a  Riemannian $\\mathrm{Spin}^c$ manifold with an $\\alpha$-Killing spinor $\\psi$ of Killing constant $\\alpha\\in \\mC$. It is known that for $m\\geq 1$, the Killing constant $\\alpha$ has to be purely real or purely imaginary \\cite[Theorem 1.1] {grosse-nakad}. Moreover, if $\\alpha$ is real, then $\\psi$ has constant norm since, for any $X \\in \\Gamma(T\\widetilde M)$, we have\n\\begin{align}\\label{eq:realconst} X(|\\psi|^2)=2\\Re \\< \\widetilde \\nabla_X\\psi, \\psi\\>=2\\alpha \\Re \\< X\\cdot \\psi, \\psi\\>=0.\\end{align}\nHence, real Killing spinors have no zeros. When $\\alpha$ is purely imaginary, the function $\\vert \\psi\\vert$ is a non-constant and\nnowhere vanishing function \\cite{baum1, grosse-nakad}. In this case, the set of zeros of $\\psi$ is discrete \\cite{grosse-nakad, rad, Lich1, KR}. \nUsing the definition~\\eqref{def:Killing} of an $\\alpha$-Killing spinor $\\psi$, we have \n\\begin{align*}\\widetilde{D}\\psi = \\sum_{j=1}^{m+1} e_j\\!\\cdot\\! \\widetilde{\\nabla}_{e_j} \\psi= (m+1)\\alpha \\psi\\quad \\text{and}\\quad\n              \\widetilde{D}^2\\psi= (m+1)^2\\alpha^2 \\psi.\n                                                       \\end{align*}\nThen, the Schr\\\"odinger-Lichnerowicz formula \\eqref{eq:Lich} on $\\widetilde{M}$ gives \n\\begin{align*}\n m(m+1)\\alpha^2\\psi= \\frac{\\widetilde{\\text{scal}}}{4}\\psi+\\frac{\\i}{2} \\widetilde{\\Omega} \\!\\cdot\\! \\psi.\n\\end{align*}\n\n\\emph{From now on we assume that $(M, g)$ is an oriented totally umbilical  hypersurface of $(\\widetilde M, \\widetilde g)$.} Totally umbilical  means $\\mathrm{II}X=HX$ for all $X\\in \\Gamma(TM)$. Note that this implies $(\\nabla_Y\\mathrm{II})(X)=dH(Y)X$ for all $X,Y \\in \\Gamma(TM)$.\\medskip \n\nWe choose the local  orthonormal frame $e_i$ on $\\widetilde{M}$ such that \n $\\{e_1 , \\ldots, e_m\\}$ is a local  orthonormal frame of $M$, $\\nabla e_i =0$ and that $e_{m+1}=\\nu$ a unit normal vector to $M$. The Ricci identity \\eqref{Ricci-identity} on $\\widetilde M$ for $X = \\nu$ applied to the $\\alpha$-Killing spinor $\\psi$ reads\n\\begin{align}\\label{ri-equation}  \\frac{\\widetilde{{\\mathop{\\rm Ric}}}(\\nu,\\nu)}{2}\\nu\\!\\cdot\\!\\psi+ \\frac 12  \\sum_{j=1}^m \\widetilde {\\mathrm{Ric}}(\\nu, e_j) e_j \\!\\cdot\\! \\psi - \\frac{\\i}{2} (\\nu\\lrcorner \\widetilde \\Omega)\\!\\cdot\\! \\psi = 2m \\alpha^2 \\nu \\!\\cdot\\! \\psi.\\end{align}\nwhere we also used the calculation $ \\widetilde{R}(e_k,\\nu)\\psi=2\\alpha^2 e_k\\cdot \\nu\\cdot \\psi$.\nNow, the Codazzi-Mainardi equation \\cite[Prop. 33]{ON}  gives that \n\\begin{align*}\n\\widetilde g(\\widetilde R (X, Y)U, \\nu)&= g(\\nabla_X \\mathrm{II})(Y), U) - g((\\nabla_Y \\mathrm{II})(X), U)  \\\\\n&= dH(X) \\<Y, U\\> - dH(Y) \\<X, U\\\n\\end{align*}\nfor all $X, Y, U \\in \\Gamma(TM)$. Hence,  \n$$\\widetilde {\\mathrm{Ric}} (X, \\nu) =\\sum_{l =1}^m \\widetilde g(\\widetilde R (X, e_l) e_l, \\nu)=\n(m-1) dH(X).$$\nReplacing this in Equation \\eqref{ri-equation} and taking then the Clifford multiplication by $\\nu$, we obtain for $\\phi=\\psi|_M$ that \\begin{align} \\label{riccianddH}\n-\\frac 12 \\widetilde {\\mathop{\\rm Ric}}(\\nu, \\nu) \\varphi - \\frac {(m-1)}{2} \\d H\\!\\cdot\\!_M \\varphi   +\\frac{\\i}{2} (\\nu\\lrcorner \\widetilde \\Omega)\\!\\cdot\\!_M \\varphi = -2m \\alpha^2 \\varphi.\n\\end{align}\n\\begin{lemma}\nAssume that $(M, g)$ is a totally umbilical  oriented hypersurface of $(\\widetilde M, \\widetilde g)$ carrying an $\\alpha$-Killing spinor $\\psi$. Then, for $\\phi=\\psi|_M$  we have\n\\begin{enumerate}\n\\item{{\\bf The Schr\\\"odinger-Lichnerowicz formula on $M$:}}\n\\begin{align}\\label{eq:LichM}\n m(m-1)\\left( \\alpha^2+\\frac{ H^2}{4}\\right)\\phi +\\frac{m-1}{2} dH\\!\\cdot\\!_M \\phi = \\frac{\\text{scal}}{4} \\phi +\\frac{\\i}{2} \\Omega^M\\!\\cdot\\!_M\\phi. \n\\end{align}\n\\item{{\\bf The Ricci identity on $M$:}}\n\\begin{align}\\nonumber\n\\frac 12 \\big(\\mathrm{Ric} (X) - \\i (X \\lrcorner\\Omega^M)\\big)\\!\\cdot\\!_M \\varphi =& -\\frac 12 dH\\!\\cdot\\!_M X\\!\\cdot\\!_M \\varphi - \\frac m2 dH(X) \\varphi \\\\ & + \\frac{(m-1)}{2} H^2 X\\!\\cdot\\!_M\\varphi +2(m-1) \\alpha^2 X\\!\\cdot\\!_M \\varphi. \\label{Ricci-hyper}\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nUsing Equation \\eqref{eq_spincgauss} we have:\n\\begin{align}\n \\nabla_{e_i}\\phi&=\\widetilde{\\nabla}_{e_i}\\phi- \\frac{H}{2}e_i\\!\\cdot\\!_M \\phi= \\alpha {e_i}\\!\\cdot\\! \\phi-\\frac{H}{2}e_i\\!\\cdot\\!_M \\phi\\label{eq_spincgaussKill}\n \\end{align}\n and \n \\begin{align}\n \\nabla_{e_j}\\nabla_{e_i}\\phi&=\\nabla_{e_j}\\!\\left( \\alpha {e_i}\\!\\cdot\\! \\phi-\\frac{H}{2}e_i\\!\\cdot\\!_M \\phi\\right)\n\\! =\\! \\alpha \\nabla_{e_j} ({e_i}\\!\\cdot\\! \\phi)-\\frac{dH(e_j)}{2}e_i\\!\\cdot\\!_M \\phi-\\frac{H}{2} e_i\\!\\cdot\\!_M \\nabla_{e_j}\\phi\\nonumber\\\\\n &=\\alpha \\widetilde{\\nabla}_{e_j} ({e_i}\\!\\cdot\\! \\phi)-\\frac{\\alpha H}{2} e_j\\!\\cdot\\! \\nu\\!\\cdot\\! e_i\\!\\cdot\\! \\phi-\\frac{dH(e_j)}{2}e_i\\!\\cdot\\!_M \\phi \\nonumber \\\\\n & \\quad -\\frac{H}{2} e_i\\!\\cdot\\! \\nu \\!\\cdot\\! \\left(\\alpha {e_j}\\!\\cdot\\! \\phi-\\frac{H}{2}e_j\\!\\cdot\\! \\nu\\!\\cdot\\! \\phi\\right)\\nonumber\\\\\n &=\\alpha^2 e_i\\!\\cdot\\! e_j \\!\\cdot\\!  \\phi + \\alpha H \\delta_{ij} \\nu\\!\\cdot\\! \\phi-\\frac{\\alpha H}{2} e_j\\!\\cdot\\! \\nu\\!\\cdot\\! e_i\\!\\cdot\\! \\phi-\\frac{dH(e_j)}{2}e_i\\!\\cdot\\!_M \\phi\\nonumber\\\\ \n &\\quad -\\frac{H}{2} e_i\\!\\cdot\\! \\nu \\!\\cdot\\! \\left(\\alpha {e_j}\\!\\cdot\\! \\phi-\\frac{H}{2}e_j\\!\\cdot\\! \\nu\\!\\cdot\\! \\phi\\right)\\nonumber\\\\\n &=\\left( \\alpha^2 +\\frac{H^2}{4}\\right) e_i\\!\\cdot\\! e_j \\!\\cdot\\!  \\phi -\\frac{dH(e_j)}{2}e_i\\!\\cdot\\!_M \\phi.\\label{eq:nn}\n\\end{align}\nNow, we calculate \n\\begin{align*}\n-\\nabla^*\\nabla \\phi =&\\sum_{i=1}^m \\nabla_{e_i} \\nabla_{e_i}\\phi\n    = -m \\left( \\alpha^2 +\\frac{H^2}{4}\\right)   \\phi -\\frac{1}{2}d H \\!\\cdot\\!_M \\phi, \n\\end{align*}\nand for the Dirac operator on the hypersurface we obtain \n\\begin{align}\n D\\phi&=  \\sum_{i=1}^m e_i\\!\\cdot\\!_M\\!\\left( \\alpha {e_i}\\!\\cdot\\! \\phi-\\frac{H}{2}e_i\\!\\cdot\\!_M \\phi \\right) = m \\alpha \\nu\\!\\cdot\\! \\phi+m\\frac{H}{2} \\phi. \\nonumber \n \\end{align}\n Hence, we have \n \\begin{align}\n D^2\\phi &= \\sum_{i=1}^m  e_i\\!\\cdot\\!_M \\nabla_{e_i}\\!\\left( m \\alpha \\nu\\!\\cdot\\! \\phi+m\\frac{H}{2} \\phi\\right) \\nonumber \\\\\n &= m \\alpha\\sum_{i=1}^m e_i\\!\\cdot\\!_M {\\nabla}_{e_i} \\!\\left( \\nu\\!\\cdot\\! \\phi\\right) +\\frac{m}{2} d H\\!\\cdot\\!_M\\phi +\\frac{mH}{2}D\\phi \\nonumber\\\\\n &=m^2\\alpha^2\\varphi+\\frac{m}{2}\\d H\\!\\cdot\\!_M \\varphi+\\frac{m^2}{4}H^2\\phi.\\nonumber\n\\end{align}\nThen, the Schr\\\"odinger-Lichnerowicz formula \\eqref{eq:Lich} on $M$ implies\n\\begin{align*}\n m(m-1)\\left( \\alpha^2+\\frac{ H^2}{4}\\right)\\phi +\\frac{m-1}{2} dH\\!\\cdot\\!_M \\phi = \\frac{\\text{scal}}{4} \\phi +\\frac{\\i}{2} \\Omega^M\\!\\cdot\\!_M\\phi, \n\\end{align*}\nwhich proves the first identity of the Lemma. Furthermore, using again Equation \\eqref{eq:nn}, we obtain for the $\\mathrm{Spin}^c$ curvature tensor $\\mathcal{R}$ on $M$ and for $i\\neq j$:\n\\begin{align*}\n\\mathcal{R}_{e_j, e_i} \\varphi =&  \\nabla_{e_j}\\nabla_{e_i}\\varphi - \\nabla_{e_i}\\nabla_{e_i}\\varphi\\\\\n =&2\\left( \\alpha^2 +\\frac{H^2}{4}\\right) e_i\\!\\cdot\\! e_j \\!\\cdot\\!  \\phi -\\frac{dH(e_j)}{2}e_i\\!\\cdot\\!_M \\phi  +\\frac{dH(e_i)}{2}e_j\\!\\cdot\\!_M \\phi.\n\\end{align*}\nHence this implies that\n\\begin{align*}\n\\sum_{j=1}^m e_j\\!\\cdot\\!_M \\mathcal{R}_{e_j, e_i} \\varphi \n =&2(m-1)\\left( \\alpha^2 +\\frac{H^2}{4}\\right) e_i\\!\\cdot\\!_M \\!\\cdot\\!  \\phi -\\frac{dH}{2}\\!\\cdot\\!_Me_i\\!\\cdot\\!_M \\phi  -\\frac{m}{2}dH(e_i) \\phi.\n\\end{align*}\nThe last identity together with the Ricci identity \\eqref{Ricci-identity} on $M$ can be written as:\n\\begin{align*}\n\\frac 12 \\big(\\mathrm{Ric} (X) - \\i (X \\lrcorner\\Omega^M)\\big)\\!\\cdot\\!_M \\varphi =& -\\frac 12 dH\\!\\cdot\\!_M X\\!\\cdot\\!_M \\varphi - \\frac m2 dH(X) \\varphi \\\\ & + \\frac{(m-1)}{2} H^2 X\\!\\cdot\\!_M\\varphi +2(m-1) \\alpha^2 X\\!\\cdot\\!_M \\varphi. \\qedhere\n\\end{align*}\\end{proof}\n\n \\section{Differential forms on the totally umbilical  hypersurface \\texorpdfstring{$M$}{M} build from the  \\texorpdfstring{$\\alpha$}{alpha}-Killing spinor} \nIn this section, we consider again that $(M, g)$ is a totally  umbilical  oriented hypersurface of $(\\widetilde M, \\widetilde g)$ carrying an $\\alpha$-Killing spinor $\\psi$. \n \n\\begin{lemma}\\label{lem_xi}  Let $\\xi$ be the vector field on $M$ defined by $g(X, \\xi)= -\\i \\< X\\!\\cdot\\!_M \\phi, \\phi\\>$.  Then on $M$ we have  \\begin{align}\\label{eq:xi} \\xi\\lrcorner \\Omega^M =& (m-1) |\\varphi|^2 dH,\\\\\n\\label{eq:dH}\ndH(\\xi ) =& 0.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} We recall that the Killing constant $\\alpha$ for $m\\geq 2$ is either purely real or purely imaginary. Thus, for $m\\geq 1$ we have $(m-1)\\alpha^2\\in \\mathbb R$. Then the real part of the scalar product with $\\varphi$ of the Ricci identity \\eqref{Ricci-hyper} together with \\eqref{cli-diff} gives\n$$-\\frac{\\i}{2} (X\\lrcorner \\Omega^M \\!\\cdot\\!_M \\varphi, \\varphi) =\\left(\\frac{1}{2}dH(X) - \\frac m2 dH(X)\\right) |\\varphi|^2 = \\frac{(1-m)}{2} dH(X) |\\varphi|^2.$$\nSince $(\\xi\\lrcorner \\Omega^M)(X)=-(X\\lrcorner \\Omega^M)(\\xi)= -g(X\\lrcorner \\Omega^M, \\xi)$, we obtain \\eqref{eq:xi} and hence \\eqref{eq:dH}.\n\\end{proof}\n\nNext we define differential forms on $M$ depending on whether $\\alpha$ is real or imaginary. The first of these forms has been introduced in \\cite{herzlich-moroianu}.\n\n\\begin{lemma}\nOn the totally umbilical  hypersurface $M$ of $\\widetilde M$, we define differential $p$-forms $\\omega_p$ by  \n\\begin{align*}\n\\omega_p(e_1, e_2, \\ldots, e_p) \\mathrel{:=} \\<(e_1\\wedge e_2\\wedge\\ldots\\wedge e_p)\\!\\cdot\\!_M\\varphi, \\varphi\\>. \\end{align*} If the Killing constant  $\\alpha$ is \\emph{real}, we have for all $p\\geq 1$,\n\\begin{align}\\label{d-omega}\n& d\\omega_p = \\tfrac{1}{2} H (1-(-1)^p) \\omega_{p+1},\\\\\n\\label{dH-real}\n& dH \\wedge \\omega_{2p} =0.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof} Using $[e_i,e_j]=0$, Equations \\eqref{eq_spincgaussKill} and \\eqref{cli-diff}, we have \n\\begin{align*}\n (p+1)& d\\omega_p (e_1, e_2, \\ldots, e_p, e_{p+1}) \\\\ \n =&\n  \\sum_{j=1}^{p+1} (-1)^{j-1} e_j\\big(\\omega_p(e_1, e_2, \\ldots, \\hat e_j, \\ldots, e_p, e_{p+1})\\big)  \\\\ \n=& \\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M  \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M  e_{p+1}\\!\\cdot\\!_M \\left( \\alpha e_j \\!\\cdot\\! \\varphi - \\tfrac{1}{2} H e_j\\!\\cdot\\!_M \\varphi\\right), \\varphi\\> \\\\ \n& + \\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M  \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M  e_{p+1}\\!\\cdot\\!_M \\varphi, \\alpha e_j \\!\\cdot\\! \\varphi - \\tfrac{1}{2} H e_j\\!\\cdot\\!_M \\varphi\\> \\\\\n=& \\alpha \\sum_{j=1}^{p+1} (-1)^{j-1} \\left(\\<e_1\\!\\cdot\\! \\nu\\!\\cdot\\!  \\ldots \\!\\cdot\\! \\hat e_j \\!\\cdot\\!\\hat\\nu  \\ldots \\!\\cdot\\!  e_{p+1}\\!\\cdot\\! \\nu  \\!\\cdot\\! e_j \\!\\cdot\\!\\varphi, \\varphi\\> \\right.  \\\\ \n& \\left.+ \n\\<e_1\\!\\cdot\\! \\nu\\!\\cdot\\!  \\ldots \\!\\cdot\\! \\hat e_j \\!\\cdot\\!\\hat\\nu  \\ldots \\!\\cdot\\!  e_{p+1}\\!\\cdot\\! \\nu  \\!\\cdot\\!\\varphi, e_j\\!\\cdot\\! \\varphi\\>\\right)\\\\ \n& \n-\\tfrac{1}{2} H  \\sum_{j=1}^{p+1} (-1)^{j-1} \\left(\\<e_1\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M     e_{p+1}\\!\\cdot\\!_M  e_j \\!\\cdot\\!_M \\varphi, \\varphi\\>\\right. \\\\ &\n\\left. +\\<e_1\\!\\cdot\\!_M  \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M     e_{p+1}\\!\\cdot\\!_M  \\varphi, e_j\\!\\cdot\\!_M \\varphi\\>\\right) \\\\ \n=&\n-(p+1)\\tfrac{1}{2} H \\Big((-1)^p -1\\Big) w_{p+1} (e_1, e_2, \\ldots, e_p, e_{p+1}).\n\\end{align*}\nThis proves \\eqref{d-omega}. In particular,  we obtained for all $k\\geq 1$\n\\begin{equation*}\n \\left\\{\n\\begin{array}{rcl}\nd\\omega_{2k}&=&0,\\\\\nd\\omega_{2k-1}&=&H \\omega_{2k}.\n\\end{array}\\right.\n\\end{equation*}\nDifferentiating the last equality  we obtain $dH \\wedge \\omega_{2k}  =0$ for any $k \\geq 1$.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\nOn the totally umbilical oriented hypersurface $M$ of $\\widetilde M$, we define differential $p$-forms $\\eta_p$ by  \n\\begin{align*}\\eta_p(e_1, e_2, \\ldots, e_p) \\mathrel{:=} \\<(e_1\\wedge e_2\\wedge \\ldots\\wedge e_p)\\!\\cdot\\!_M \\varphi, \\nu \\!\\cdot\\!\\varphi\\>.\\end{align*}\nIf the Killing constant $\\alpha$ is in $\\i\\mathbb R\\setminus \\{0\\}$, we have  for $p\\geq 1$,\n\\begin{align}\n& d\\eta_p = -\\tfrac{1}{2} H (1 +(-1)^p)\\eta_{p+1},\\nonumber\\\\\n& dH \\wedge \\eta_{2p-1} = 0,  \\label{dH-img}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWith an analog calculation as in the last lemma and using $\\alpha\\in \\i \\mathbb R$  we obtain\n\\begin{align*}\n(p+1) & d\\eta_p (e_1, e_2, \\ldots, e_p, e_{p+1}) \\\\ \n=&\\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M  \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M e_{p+1}\\!\\cdot\\!_M \\big( \\alpha e_j \\!\\cdot\\! \\varphi - \\tfrac{1}{2} H e_j\\!\\cdot\\!_M \\varphi\\big), \\nu\\!\\cdot\\!\\varphi\\> \\\\ & +\\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M  e_{p+1}\\!\\cdot\\!_M \\varphi, \\nu\\!\\cdot\\! \\widetilde \\nabla_{e_j}  \\varphi - \\tfrac{1}{2} H e_j\\!\\cdot\\!_M\\nu\\!\\cdot\\! \\varphi\\> \\\\\n=& \n -\\tfrac{1}{2} H  \\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M  \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M   e_{p+1}\\!\\cdot\\!_M  e_j \\!\\cdot\\!_M \\varphi, \\nu\\!\\cdot\\! \\varphi\\> \\\\ & \n -\\tfrac{1}{2} H  \\sum_{j=1}^{p+1} (-1)^{j-1} \\<e_1\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M\\hat e_j\\!\\cdot\\!_M \\ldots \\!\\cdot\\!_M  e_{p+1}\\!\\cdot\\!_M  \\varphi, e_j\\!\\cdot\\!_M\\nu\\!\\cdot\\! \\varphi\\> \\\\\n=&-(p+1)\\tfrac{1}{2} H \\Big((-1)^p +1\\Big) \\eta_{p+1} (e_1, e_2, \\ldots, e_p, e_{p+1}),\n\\end{align*}\nand, thus, for all $p\\geq 1$\n\\begin{equation*}\n \\left\\{\n\\begin{array}{rl}\nd\\eta_{2p-1}=&0,\\\\\nd\\eta_{2p}=&-H \\eta_{2p-1}.\n\\end{array}\\right.\n\\end{equation*}\nDifferentiating  the last equality then again gives  $dH \\wedge \\eta_{2p-1}  =0$ for any $p \\geq 1$. \n\\end{proof}\n\n\\section{Proof of the main result: Theorem ~\\ref{Exto}}\\label{Mmain}\nThe goal of this section is to prove Theorem~\\ref{Exto}. If $\\widetilde{M}$ is spin, $\\Omega^M=0$ and the statement follows directly from \\eqref{eq:dH}. For the general $\\mathrm{Spin}^c$ case we split the proof into the two cases:  \n\n{\\it Case 1}: The $\\alpha$-Killing spinor $\\psi$ is  a real Killing spinor ($\\alpha \\in \\mathbb R$)\n\n{\\it Case 2}: The $\\alpha$-Killing spinor $\\psi$ is an  imaginary Killing spinor ($\\alpha \\in i \\mathbb R \\setminus\\{0\\}$).\\medskip\n\nFirst we note, that the hypersurfaces in Section~\\ref{Mmain} is not assumed to be orientable. But since all our calculations are local, we at least have locally always an induced $\\mathrm{Spin}^c$ structure as in Section~\\ref{prelim} and can use all the spinorial formula from above.\n\n\\begin{proof}[\\underline{Proof of Theorem~\\ref{Exto} for Case 1}]\nWe prove this by contradiction. In fact, assume that $dH$ is not identically zero. Then, there is a point $x\\in M$ and a neighborhood $U$ of $x$ where $\\text{grad}_g H$ is nonzero. Hence, we find a local orthonormal frame $(e_1,\\ldots, e_{m-1}, Z= \\frac{\\text{grad}_g H}{|\\text{grad}_g H|})$ of $TU$. Then,  \n$(e_1,\\ldots, e_{m-1}, Z, \\nu)$ is a local orthonormal frame of $\\widetilde M$ on $U$. Note that then $dH\\cdot_M = \\text{grad}_g \\cdot_M$.\\medskip \n\n \n First we prove the claim for $m>4$:  From Equation (\\ref{dH-real}), it is clear that for $2k \\leq m-1$ and for each subset $i_1, \\ldots, i_{2k}$ of $\\{1, \\ldots, m-1\\}$, we have \n$$\\omega_{2k} (e_{i_1}, \\ldots, e_{i_{2k}}) = 0.$$\nThus the spinors in  $\\{\\varphi\\}\\cup \\{ e_{i_1}\\!\\cdot\\!_M e_{i_2} \\!\\cdot\\!_M \\varphi\\}_{i_1<i_2}\\cup ..\\cup \\{{e_{i_1}\\!\\cdot\\!_M\\ldots\\cdot_Me_{i_{2l}}\\!\\cdot\\!_M} \\varphi\\}_{i_1<\\ldots <i_{2l}}$, where the $i_j\\in \\{1, \\ldots, m-1\\}$, $l= [\\frac{m-1}{2}]$ and $\\phi=\\psi|_x\\in \\Sigma_x\\widetilde{M}$, are mutually orthogonal. Hence they span a complex vector subspace of $\\Sigma_x \\widetilde{M}$ of complex dimension \n$$\\binom{m-1}{0} + \\binom{m-1}{2} + \\ldots +\\binom{m-1}{2l}=2^{m-2}.$$\nSince $\\mathrm{dim} (\\Sigma_x M) = 2^{[\\frac m2]}$, we obtain\n$ 2^{[\\frac m2]} \\geq 2^{m-2}$.\nIt follows that $[\\frac m2] \\geq m-2$, so $m \\leq 4$, which is a contradiction and finishes the proof for $m>4$.\\medskip\n\nLet now $m=4$. In dimension $4$  the spinor bundle splits into positive and negative spinors $\\Sigma \\widetilde{M}|_M\\cong \\Sigma M\\cong \\Sigma^+M\\oplus \\Sigma^-M$, both $\\Sigma^{\\pm}M$ have $\\mathbb C^ 2$-fibers, and we have $\\phi\\mathrel{:=} \\psi|_M=\\phi^++\\phi^-$ with $\\phi_{\\pm}\\in \\Gamma(\\Sigma^\\pm M)$. Moreover,  $e_i\\cdot_M\\colon \\Gamma(\\Sigma^\\pm M)\\to \\Gamma (\\Sigma^\\mp M)$ and   $\\overline {\\varphi}=- e_1\\!\\cdot\\!_M e_2\\!\\cdot\\!_M e_3\\!\\cdot\\!_M Z\\!\\cdot\\!_M \\varphi$.\\medskip \n\nUsing Equation \\eqref{dH-real} we have\n\\begin{align*}\n 0=&(dH\\wedge \\omega_2)(Z, e_2,e_3)= dH(Z) \\omega_2(e_2,e_3)= dH(Z) \\< e_2\\cdot_M e_3\\cdot_M \\varphi, {\\varphi}\\>\\\\ =& \\< Z\\cdot_M e_2 \\cdot_M e_3\\cdot_M \\varphi , dH\\cdot_M\\varphi\\>\\\\\n = &- \\<e_1\\cdot_Me_1\\cdot_M  e_2 \\cdot_M e_3\\cdot_M Z\\cdot_M \\varphi , dH\\cdot_M\\varphi\\>\n =\\< \\overline{\\varphi}, dH\\cdot_Me_1\\cdot_M\\varphi\\>\n\\end{align*}\nand analogously $\\< dH\\cdot_Me_i\\cdot_M\\varphi, \\overline{\\varphi}\\>=0$ for $i=1,2,3$.\\medskip \n\nLet $\\xi$ be as defined in Lemma~\\ref{lem_xi}. Then \\eqref{eq:dH} implies that $\\xi$ is in the span of $\\{e_1,e_2,e_3\\}$. Taking the Clifford multiplication with $dH\\!\\cdot\\!_M$ in the Ricci identity  \\eqref{Ricci-hyper} for $X = \\xi$ and then the imaginary part of the scalar multiplication with $\\overline\\varphi$, we obtain\n$$0=\\<\\xi\\lrcorner \\Omega^M,Z\\>\\<\\varphi,\\overline{\\varphi}\\>.$$\nTogether with \\eqref{eq:xi} this implies  $\\vert dH\\vert^2 |\\varphi|^2 \\<\\varphi,\\overline{\\varphi}\\>=0$. Since $\\varphi\\neq 0$ for a real Killing spinor and $dH|_U \\neq 0$ by assumption, we obtain $\\<\\varphi,\\overline{\\varphi}\\>=0$.  Let $X \\in \\Gamma(TM)$ with $|X|=1$. Using $\\nu\\cdot \\colon \\Gamma(\\Sigma^\\pm M)\\to \\Gamma(\\Sigma^\\pm M)$, see \\cite[p. 31]{ginoux-these},  we calculate $\\widetilde\\nabla_X \\overline \\varphi = -\\alpha X\\!\\cdot\\!\\overline \\varphi $. Differentiating  $\\<\\varphi,\\overline{\\varphi}\\>=0$ and using Equation \\eqref{eq_spincgauss} we then obtain\n$$\\alpha \\<X\\cdot \\varphi, \\overline \\varphi\\> - \\tfrac{1}{2} H \\<X\\cdot_M\\varphi, \\overline \\varphi\\> + \\tfrac{1}{2} H \\<\\varphi, X\\cdot_M\\overline\\varphi\\> - \\alpha \\<\\varphi, X\\cdot \\overline \\varphi\\> =0.$$\nHence, we have \n\\begin{align}\n2\\alpha  \\<X\\cdot \\varphi, \\overline \\varphi\\> =& H  \\<X\\cdot_M\\varphi, \\overline \\varphi\\>.\\label{derivative2}\n\\end{align}\nLet also $e_4\\mathrel{:=} Z$. We calculate using $\\nabla_{e_j} e_i=0$ (and hence $\\widetilde{\\nabla}_{e_j} e_i = H \\delta_{ij} \\nu$) that \n\\begin{align}\n e_j\\<e_i\\cdot \\phi, \\overline\\phi\\>&= \\< \\widetilde \\nabla_{e_j} (e_i\\cdot \\phi),\\overline{\\phi}\\> +\\<e_i\\cdot \\phi, \\widetilde\\nabla_{e_j} \\overline{\\phi}\\>\\nonumber\\\\\n &= \\< H\\delta_{ij} \\nu\\cdot \\phi +  e_i\\cdot \\widetilde{\\nabla}_{e_j} \\phi,\\overline{\\phi}\\> +\\<e_i\\cdot \\phi, -\\alpha e_j\\cdot \\overline{\\phi}\\>\\nonumber\\\\ \n &= \\< H\\delta_{ij} \\nu\\cdot \\phi, \\overline{\\phi}\\>+ \\< \\alpha e_i \\cdot e_j\\cdot \\phi,\\overline{\\phi}\\> -\\<e_i\\cdot \\phi, \\alpha e_j\\cdot \\overline\\phi\\>\\nonumber\\\\ \n &= H\\delta_{ij} \\<  \\nu\\cdot \\phi, \\overline{\\phi}\\>\\label{eq:aux1}\n\\end{align}\nand\n\\begin{align}\n e_j&\\<e_i\\cdot_M \\phi, \\overline\\phi\\>= \\< e_i\\cdot_M \\nabla_{e_j} \\phi,\\overline{\\phi}\\> +\\<e_i\\cdot_M \\phi, \\nabla_{e_j} \\overline{\\phi}\\>\\nonumber\\\\\n &= \\< \\alpha e_i\\cdot_M \\cdot e_j\\cdot \\phi\\!-\\!\\tfrac{1}{2} H e_i\\cdot_M e_j\\cdot_M \\phi,\\overline{\\phi}\\> \\!+\\!\\<e_i\\cdot_M \\phi, -\\alpha e_j\\cdot \\overline\\phi-\\tfrac{1}{2} H e_j\\cdot_M \\overline{\\phi}\\>\\nonumber\\\\ \n &=-H \\< e_i \\cdot_M e_j\\cdot_M \\phi, \\overline{\\phi}\\>.\\label{eq:aux2}\n\\end{align}\nBy \\eqref{cli-diff} and $\\overline{\\phi}=-e_1\\!\\cdot\\!_M e_2\\!\\cdot\\!_Me_3\\!\\cdot\\!_MZ\\!\\cdot\\!_M \\phi$,  the left hand sides of both of the equations \\eqref{eq:aux1} and \\eqref{eq:aux2} are real. On the other hand the right hand side of \\eqref{eq:aux1} is imaginary and the one of \\eqref{eq:aux2} is imaginary for $i\\neq j$ and $0$ for $i=j$. Hence, all sides have to be zero. Using this when differentiating \\eqref{derivative2} for $X=e_i$ in direction of $Z$, we obtain \n\\[ Z(H) \\<e_i\\cdot_M \\phi, \\overline{\\phi}\\>=0 \\text{ for all }i=1,\\ldots, 4.\\]\nHence, $\\<e_i\\cdot_M \\phi, \\overline{\\phi}\\>=0$ and thus\n\\begin{equation}\\label{eq:re4} \\Re\\<e_i\\cdot_M \\phi_+, \\phi_-\\>=0  \\text{ for all }i=1,\\ldots, 4.\\end{equation}\nWe note that in dimension $4$ every non-zero element $\\psi\\in \\Sigma_+M|_y$ for $y\\in U$ gives rise to a real basis $e_i\\cdot_M \\psi$ of $\\Sigma_+M|_y$ with respect to the scalar product $(.,.)\\mathrel{:=} \\Re \\< .,.\\>$. \nHence, Equation \\eqref{eq:re4} implies that at each point $y\\in U$, either $\\varphi_+ = 0$ or $\\varphi_-$ is perpendicular to the four dimensional real vector space $\\Sigma_+M|_y$ w.r.t this real scalar product, i.e, $\\varphi_- = 0$.\\medskip\n\nSince $|\\varphi|^2=|\\varphi_+|^2+|\\varphi_-|^2$ is of constant norm  by \\eqref{eq:realconst}, we obtain that $\\varphi_+ = 0$ or $\\varphi_- =0$ on all of $U$. Assume that $\\varphi_- = 0$ on $U$ (the other case is analogous), then \n $$0=\\nabla_X \\varphi_- = \\alpha X\\!\\cdot\\! \\varphi_+ + \\tfrac{1}{2} H X\\!\\cdot\\!_M \\varphi_+.$$ \nThe real part of the scalar product of the last identity with $X\\!\\cdot\\!_M \\varphi_+$ gives\n$$ \\tfrac{1}{2} H |X|^2 \\vert\\varphi_+\\vert^2 = 0.$$\nSince $\\varphi$ is  non-zero, $\\varphi_+$ has no zeros on $U$ and we get that $H=0$ on $U$.  Thus, $dH = 0$ on $U$ which gives the contradiction.\n\\end{proof}\n\n\\begin{proof}[\\underline{Proof of Theorem~\\ref{Exto} for Case 2}]  Assume that $dH$ is not identically zero. Then, there is a point $x\\in M$ and a neighborhood $U$ of $x$ where $\\text{grad}_g H$ is nonzero. Hence, we find a local orthonormal frame $(e_1,\\ldots, e_{m-1}, Z= \\frac{\\text{grad}_g H}{|\\text{grad}_g H|})$ of $TU$. Then, we have with $(e_1,\\ldots, e_{m-1}, Z, \\nu)$ again a local orthonormal frame of $\\widetilde M$ on $U$. \\medskip \n\nOn all of $U$ we have by Equation \\eqref{dH-img} that $d H \\wedge \\eta_1 = 0$. Then with $dH(e_i)=0$ we obtain\n$$0 = d H \\wedge \\eta_1 \\left(\\frac{\\text{grad}_g H}{|\\text{grad}_g H|^2}, e_i\\right) = \\eta_1 (e_i) = -\\<e_i \\!\\cdot\\!\\varphi, \\varphi\\>$$ for all $1\\leq i\\leq m-1$ which will used in following without any further comment. \\medskip\n\n\\underline {We consider three different subcases}: First assume that $\\<dH\\cdot \\varphi, \\varphi\\>=\\<\\nu \\cdot \\varphi, \\varphi\\>=0$ on $U$. Note that for all $X\\in \\Gamma (TM)$ the vector $V$, defined on $\\widetilde{M}$ by $\\widetilde{g}(V,X) \\mathrel{:=} \\i \\< X\\!\\cdot\\! \\varphi, \\varphi\\>$ , vanishes on $U$, see \\cite{rad, baum1, baum2, baum3, grosse-nakad}. From \\cite{baum1, rad} we have $\\widetilde{\\nabla}_X V=2\\alpha |\\varphi|^2 X$ for all $X\\in \\Gamma(T\\widetilde{M})$. Since $V\\equiv 0$, this implies that $\\varphi\\equiv 0$ on $U$.\nThis gives a contradiction in the first case.\\medskip \n\nSecond let $\\<dH\\cdot \\varphi, \\varphi\\>=0$ and let $\\<\\nu \\cdot \\varphi, \\varphi\\>$ be nonzero on a possibly smaller $U$. In particular, we can make $U$ small enough such that $\\phi$ has no zeros on $U$. Then, \nthe   imaginary part  of the  scalar product  of Equation~\\eqref{riccianddH} with $\\nu\\!\\cdot\\!\\varphi$  gives\n\\begin{align*}\\widetilde{\\text{Ric}}(\\nu, \\nu) = 4m\\alpha^2.\\end{align*}\nReinserting into Equation \\eqref{riccianddH} gives\n\\begin{align}\\label{contr}\n\\frac{m-1}{2} dH\\!\\cdot\\!_M\\varphi =& \\frac{\\i}{2} (\\nu\\lrcorner\\widetilde \\Omega)\\!\\cdot\\!_M \\phi,\n\\end{align}\nand hence \n\\begin{align*}\n\\frac{m-1}{2} \\<dH\\!\\cdot\\!_M\\varphi, Z\\!\\cdot\\!_M \\phi\\> =& \\frac{\\i}{2} \\left(\\sum_{i=1}^{m-1} \\widetilde \\Omega(\\nu, e_i)\\< e_i\\!\\cdot\\!_M \\varphi, Z\\!\\cdot\\!_M \\phi\\> + \\widetilde \\Omega(\\nu, Z)|Z\\!\\cdot\\!_M \\varphi|^2\\right). \n\\end{align*}\nTaking the imaginary part of the last equality implies $\\widetilde \\Omega(\\nu, Z)|\\phi|^2=0$.  Since $\\phi$ has no zeros, we obtain  $\\widetilde \\Omega (Z, \\nu) =0$. The real part of the scalar product of Equation \\eqref{eq:LichM} with $e_i\\cdot \\phi$ then gives\n\\begin{align*} \\frac{m-1}{2}&\\< dH\\cdot \\nu\\cdot \\phi, e_i\\cdot \\phi\\>\\\\ = &-\\frac{1}{2}\\text{Im} \\left( \\sum_{j<k} \\Omega^M(e_j,e_k)\\< e_j\\!\\cdot\\! e_k\\!\\cdot\\! \\phi, e_i\\!\\cdot\\! \\phi\\> + \\sum_{j} \\Omega^M(e_j,Z)\\< e_j\\!\\cdot\\! Z \\!\\cdot\\! \\phi, e_i\\!\\cdot\\! \\phi\\> \\right) =0, \\nonumber\n\\end{align*}\nwhere the last equality uses Equation \\eqref{cli-diff} and $\\< Z\\cdot \\phi,\\phi\\>=\\<e_i\\cdot \\phi,\\phi\\>=0$.\nThus, taking the scalar product of Equation \\eqref{contr} with $e_i\\cdot \\phi$ implies \n\\begin{align*}\n0= \\sum_{j=1}^{m-1} \\widetilde \\Omega(\\nu, e_j)\\< e_j\\!\\cdot\\!_M \\varphi, e_i\\cdot  \\phi\\>. \n\\end{align*}\nBy taking the imaginary part we obtain $\\widetilde \\Omega (\\nu, e_i)=0$ and hence $\\nu\\lrcorner \\widetilde{\\Omega}=0$. Reinserting in Equation \\eqref{contr} implies $dH=0$ which gives the contradiction in the second case.\\medskip \n\nThe third case covers the remaining possibility that $\\< dH\\cdot \\varphi, \\varphi\\>$ is nonzero at a point in $U$. The next calculations will be carried out at this point. Taking the real part of the scalar product of Equation \\eqref{eq:LichM} with $e_i\\cdot \\phi$ gives\n\\begin{align}\\label{eq:iHnu0} \\frac{m-1}{2}&\\< dH\\cdot \\nu\\cdot \\phi, e_i\\cdot \\phi\\> = \\frac{\\i}{2} {\\Omega^M} (e_i, Z)\\< Z\\cdot \\phi, \\phi\\>.\n\\end{align}\nOn the other hand taking the scalar product of the Ricci identity \\eqref{Ricci-hyper}  for $X=e_i$ with $\\nu\\cdot\\varphi$  gives \n \\begin{align*}\n - \\frac{1}{2}& {\\mathop{\\rm Ric}}\\left(e_i,Z\\right) \\<Z\\cdot \\varphi, \\varphi\\> + \\frac{\\i}{2} \\Omega^M\\left(e_i, Z\\right) \\< Z\\cdot \\varphi, \\varphi\\>\n= -\\frac 12 \\< dH \\cdot \\nu \\cdot \\varphi,  e_i \\cdot\\varphi\\>.\n\\end{align*}\n\nThe imaginary part of the last identity gives ${\\mathop{\\rm Ric}}\\left(e_i,Z\\right)=0$. Reinserting this into the above equation and using Equation \\eqref{eq:iHnu0}  implies $\\< dH \\cdot \\nu \\cdot \\varphi,  e_i \\cdot\\varphi\\>=\\Omega^M\\left(e_i, Z\\right)=0$.\\medskip\n\nTaking again the scalar product of the Ricci identity \\eqref{Ricci-hyper} but this time for $X=Z$ then \ngives \n \\begin{align*}\n  \\frac{1}{2} &{\\mathop{\\rm Ric}}\\left(Z ,Z\\right)Z \\cdot_M \\varphi  = \\frac{1-m}{2} |dH| \\varphi + \\left(\\frac 12 (m-1)H^2 - 2(m-1) \\alpha^2\\right) Z\\cdot_M\\varphi.\n\\end{align*}\nThe real part of the scalar product with of the last identity with $\\varphi$ gives that $\\frac{1-m}{2} | dH| \\vert \\varphi\\vert^2= 0$. But  $\\varphi$ has no zeros on $U$ and $m>1$, so $dH=0$ which gives the desired contradiction for the remaining case.\n\\end{proof}\n\nThe following example shows that the dimension constraint  for Case 1 in Theorem~\\ref{Exto} is necessary. \n\n\\begin{example}There exist totally umbilical connected  hypersurfaces with non-constant mean curvature in Riemannian $\\mathrm{Spin}^c$ manifolds of dimension $3$ or $4$ and  carrying parallel or real Killing spinors:\n\n\\textit{\\underline{Dimension $3$:}} The product of the canonical $\\mathrm{Spin}^c$ structure on $\\mathbb S^2$ with the ${\\mathop{\\rm Spin}}$ structure on $\\mathbb R$ defines a $\\mathrm{Spin}^c$ structure on the manifold $\\widetilde M = \\mathbb S^2 \\times \\mathbb R$ \\cite {moroi}. This $\\mathrm{Spin}^c$ structure carries a parallel spinor \\cite{moroi}. Totally umbilical  hypersurfaces (which are not totally geodesic) of $\\mathbb S^2 \\times \\mathbb R$ have been classified in \\cite{Souam1}. Moreover, they are not of constant mean curvature \\cite[ Remark 10]{Souam1}.  We point out that  $\\widetilde M = \\mathbb S^2 \\times \\mathbb R$ is ${\\mathop{\\rm Spin}}$ but does not carry a real or parallel Killing ${\\mathop{\\rm Spin}}$ spinor.\n\n\\textit{\\underline{Dimension $4$:}} The $\\mathrm{Spin}^c$ manifold $\\widetilde M = \\mathbb S^2 \\times \\mathbb H^2$  carries a parallel spinor for the product of the canonical $\\mathrm{Spin}^c$ structure on $\\mathbb S^2$ with the canonical $\\mathrm{Spin}^c$ structure on $\\mathbb H^2$. In \\cite{danielthesis}, the author classified totally umbilical  hypersurfaces of $\\mathbb S^2 \\times \\mathbb H^2$  (see \\cite[Theorem 4.5.3]{danielthesis}) and showed that these hypersurfaces are not of constant mean curvature in general. We also point out that  $\\widetilde M = \\mathbb S^2 \\times \\mathbb H^2$ is ${\\mathop{\\rm Spin}}$ but does not carry a real or parallel Killing ${\\mathop{\\rm Spin}}$ spinor. \n\\end{example}\n\n\\section{Extrinsic hyperspheres in Riemannian \\texorpdfstring{${\\mathop{\\rm Spin}}$}{Spin} manifolds}\n\nIn this section, we give some additional information if the ambient manifold carrying a Killing spinor is already spin. As a first corollary, we get:\n\\begin{cor}\\label{cor-E}\nLet $M^m \\hookrightarrow \\widetilde M^{m+1}$ be a totally umbilical   isometric immersion. Assume that $\\widetilde M$ is a ${\\mathop{\\rm Spin}}$ manifold with a Killing spinor $\\psi$ of Killing constant $\\alpha$ (could be zero, real or purely imaginary). Then, $M$ is Einstein with scalar curvature ${\\mathop{\\rm scal}} = m(m-1)(H^2 + 4 \\alpha^2)$.  \n\\end{cor}\n\\begin{proof}[Proof of Corollary~\\ref{cor-E}]\nFrom Thm \\ref{Exto}, $H$ is constant. By the Ricci identity \\eqref{Ricci-hyper}, we have $$\\frac 12 \\mathrm{Ric}(e_j) \\!\\cdot\\!_M \\phi = 2 (m-1) \\left[\\alpha^2 + \\frac {H^2}{4}\\right]e_j \\!\\cdot\\!_M \\phi.$$  This means that $M$ is Einstein with constant scalar curvature ${\\mathop{\\rm scal}} = m(m-1)(H^2 + 4 \\alpha^2)$.  \n\\end{proof}\nFor later use, we recall here \\textbf{Koiso's Theorem.} \n\\begin{thm}\\cite[Thm. B]{koiso} \\label{koi}Let $M$ be a totally umbilical  Einstein hypersurface in a complete Einstein manifold $(\\widetilde M, \\widetilde g)$. Then the only possible cases are: \n\\begin{enumerate}\n\\item $g$ has  positive Ricci curvature. Then $g$ and $\\widetilde g$ have constant sectional curvature. \n\\item $\\widetilde g$ has negative Ricci curvature. If $\\widetilde M$ is compact or homogeneous, then $g$ and $\\widetilde g$ have constant sectional curvature.\n\\item $g$ and $\\widetilde g$ have zero Ricci curvature. If $\\widetilde M$ is simply connected, then $\\widetilde M = (\\overline M, \\overline g) \\times \\mathbb R$ where $\\overline M$ is totally geodesic hypersurface in $\\widetilde M$ which contains $M.$ \n\\end{enumerate}\n\\end{thm}\n\nAn important special case of totally umbilical  hypersurfaces with constant mean curvature are totally geodesic hypersurfaces (when the mean curvature $H$ is  zero). The other cases are called extrinsic hypersphere (when the mean curvature is a nonzero constant).\\medskip \n\nFrom Theorem~\\ref{koi} and Corollary \\ref{cor-E}, we deduce the following result:\n\\begin{thm}\\label{pa-ap}\nLet $\\widetilde M$ be a complete Riemannian ${\\mathop{\\rm Spin}}$ manifolds of non-constant sectional curvature  that carry an $\\alpha$-Killing spinor. \nIf $\\alpha \\in \\i\\mathbb R\\setminus\\{0\\}$, we assume moreover that $\\widetilde M$ is homogeneous. Then, there are no extrinsic hyperspheres in $\\widetilde M$.\n \\end{thm}\n\\begin{proof} Assume that $M$ is an extrinsic hypersphere ($H\\neq 0$) in a Riemannian ${\\mathop{\\rm Spin}}$ manifold with an $\\alpha$-Killing spinor. By Corollary \\ref{cor-E}, $M$ is  Einstein with scalar curvature $m(m-1)(H^2 + 4 \\alpha^2)$.  If  $H^2 + 4 \\alpha^2 > 0$, the Ricci curvature of $M$ is positive.  If  $H^2 + 4 \\alpha^2\\leq 0$, then $\\alpha\\in \\i\\mathbb R\\setminus\\{0\\}$ and hence  the Ricci curvature of $\\widetilde{M}$ is negative  and hence, in both cases, we have by Koiso's theorem \\ref{koi} that $\\widetilde g$ is of constant sectional curvature, which is a contradiction. \n\\end{proof}\nTheorem~\\ref{THMp1} is a particular case of  Theorem~\\ref{pa-ap} as is easily seen as follows: All the manifolds appearing in this Theorem are  ${\\mathop{\\rm Spin}}$, complete, with $\\alpha$-Killing spinor and  of non-constant sectional curvature (see \\cite{boyer} and \\cite[Prop~3.1]{Gou}).\\medskip \n\nOne can add further examples for Theorem~\\ref{pa-ap}, such as $6$-dimensional nearly K\\\"ahler manifolds which are not K\\\"ahler and of non-constant sectional curvature and  $7$-dimensional weak $G_2$ manifolds of non-constant sectional curvature.\\medskip \n\nThe completeness assumptions in Theorem~\\ref{pa-ap} is necessary not only because we want to use Koiso's theorem but also because otherwise, every\nmanifold is an extrinsic hypersphere in its (non-complete) metric cone. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Definitions}\n\\section{Notation}\nWe will use standard set-theoretic notation following e.g. \\cite{Jech}.  \n$\\mathbb{R}} %{{\\Bbb R}$ will denote the real line. For a set $X$, $P(X)$ denotes the power set of $X$ and $|X|$ denotes the cardinality of $X$. If $\\kappa$ is a cardinal number then\n$$\\begin{array}{r@{\\; =\\; }l}\n  [X]^\\kappa & \\{A\\subseteq X:\\ |A|=\\kappa\\}\\\\  \n  {[X]^{<\\kappa}} & \\{A\\subseteq X:\\ |A|<\\kappa\\}\\\\  \n  {[X]^{\\le\\kappa}} & \\{A\\subseteq X:\\ |A|\\le\\kappa\\}\\\\\n  \\end{array}\n$$\n\nLet $X$ be any uncountable Polish space with $\\mathcal{I}$ an arbitrary $\\sigma$-ideal on $\\mathcal{P}(X)$ and let us recall the cardinal coefficients of $\\mathcal{I}$\n\\begin{itemize}\n\\item $\\non(\\mathcal{I})=\\min \\{ |F|:\\; F\\subseteq X\\land F\\notin \\mathcal{I}\\},$\n\\item $\\add(\\mathcal{I})=\\min \\{ |\\mathcal{A}|:\\; \\mathcal{A}\\subseteq \\mathcal{I}\\land \\bigcup\\mathcal{A}\\notin \\mathcal{I} \\},$\n\\item $\\cof(\\mathcal{I})=\\min \\{ |\\mathcal{B}|:\\; \\mathcal{B}\\subseteq \\mathcal{I}\\land (\\forall A\\in \\mathcal{I})(\\exists B\\in\\mathcal{B}) A\\subseteq B\\},$\n\\item $\\cov(\\mathcal{I})=\\min \\{ |\\mathcal{A}|:\\; \\mathcal{A}\\subseteq \\mathcal{I}\\land \\bigcup\\mathcal{A} = X\\},$\n\\item for a fixed family of perfect subsets $\\mathcal{P} \\subseteq Perf(X)$ let\\\\ $\\cov_h(\\mathcal{I})=\\min \\{ |\\mathcal{A}|:\\; \\mathcal{A}\\subseteq \\mathcal{I}\\land (\\exists P\\in \\mathcal{P})\\; P\\subseteq \\bigcup\\mathcal{A}\\}$.\n\\end{itemize}\nLet us recall the definition of a bounding number.\n$$ \n \\mathfrak{b} = \\min \\{| \\mathcal{B} |:\\ \\mathcal{B} \\subseteq \\omega^\\omega \\land (\\forall x \\in \\omega^\\omega) (\\exists y \\in \\mathcal{B}) \\; \\neg (s \\le^* x) \\} \n $$ \n\nIn \\cite{Marczewski} Marczewski introduced the notion of $s$-measurability and the $s_0$-ideal. Recalling these definitions we have:\n\\begin{definition}[Marczewski ideal $s_0$] Let $X$ be any fixed uncountable Polish space. Then we say that $A\\in \\mathcal{P}(X)$ is in $s_0$ iff\n$$\n(\\forall P\\in Perf(X))(\\exists Q\\in Perf(X))\\; Q\\subseteq P\\land Q\\cap A=\\emptyset.\n$$\n\\end{definition}\nNotice that for this ideal we have $\\cov(s_0)=\\cov_h(s_0)$ and this cardinal is the same for all uncountable Polish spaces. \nTo see this use the fact that in any uncountable Polish space there is a disjoint maximal antichain $\\mathcal{A}$ (of cardinality $\\mathfrak{c}$) consisting of Cantor perfect sets. \nFrom this it follows that $B\\in s_0$ if and only if $(\\forall A\\in\\mathcal{A})\\; B\\cap A\\in s_0$.\n\n\\begin{definition}[$s$-measurable set] Let $X$ be any fixed uncountable Polish space. Then we say that $A\\in \\mathcal{P}(X)$ is {\\bf $s$-measurable} iff\n$$\n(\\forall P\\in \\Perf(X))(\\exists Q\\in \\Perf(X))\\; Q\\subseteq P\\land (Q\\subseteq A\\lor Q\\cap A=\\emptyset).\n$$\nMoreover, a set $A\\in \\mathcal{P}(X)$ is a {\\bf Bernstein set} if\n$$\n(\\forall P\\in Perf(X))\\; P\\cap A\\ne\\emptyset\\land P\\cap A^c\\ne \\emptyset,\n$$\n(where $A^c$ denotes complement of the set A in space $X$).\n\\end{definition}\n\n\\begin{definition} Let $X$ be any uncountable Polish space and let us consider a cardinal $\\kappa$. We say that the family \n$\\mathcal{A}\\subseteq P(X)$ is {\\bf $\\kappa$-point family} iff $|\\{ A\\in\\mathcal{A}:\\ x\\in A\\}|<\\kappa$ for all $x\\in X$. \n\\end{definition}\nWe say that $\\mathcal{A}$ is point-finite family if $\\mathcal{A}$ is $\\omega$-point family and  $\\mathcal{A}$ is countable-point family if $\\mathcal{A}$ is $\\omega_1$-point family.\n\nWe say that $\\sigma$-ideal $\\mathcal{I}$ of subsets of some Polish space $X$ has Borel base if for any set $A\\in \\mathcal{I}$ there is a Borel set $B\\in Bor(X)\\cap\\mathcal{I}$ such that $A\\subseteq B$. \nClassical examples of ideals possesing Borel base on the real line are \n\\begin{itemize}\n \\item the $\\sigma$-ideal $\\mathcal{I}=[\\mathbb{R}]^{\\le\\omega}$ of all countable subsets,\n \\item the $\\sigma$-ideal $\\mathcal{M}$ of meager subsets, \n \\item the $\\sigma$-ideal $\\mathcal{N}$ of null subsets with respect to Lebesgue measure.\n\\end{itemize}\n\nFor fixed $\\sigma$-ideal $\\mathcal{I}$ with Borel base we say that a subset $A\\subseteq X$ of Polish space $X$ is measurable with respect to $\\mathcal{I}$ iff $A$ belongs to $\\sigma$-algebra $Bor[\\mathcal{I}]$ generated by Borel subsets of $X$ and $\\sigma$-ideal $\\mathcal{I}$. \n\nIn the first part of this paper we consider subsets connected to $\\sigma$-ideal without Borel base generated by trees. We are interested in measurability connected to Laver trees and Miller trees and it's interplay with m.a.d. families.\n\nIn the second part we investigate subsets connected to $\\sigma$-ideals with Borel base.\nWe discuss the difference between measurability and complete nonmeasurability of unions of small sets.\n\n\\section{m.a.d. families and their $s$, $l$ and $m$-measurability}\n\nFor every tree $T\\subseteq \\omega^{<\\omega}$ let $[T]$ be the set of all branches of $T$ which is defined as follows:\n$$\n[T]=\\{ x\\in \\omega^\\omega:\\; (\\forall n\\in\\omega)\\; x\\upharpoonright n\\in T\\}.\n$$\n\nWe say that a tree $T\\subseteq \\omega^{<\\omega}$ is called a {\\bf Laver tree} iff there is a node $s\\in T$ such that, \nfor every node $t\\in T$ if $s\\subseteq t$ then $t$ is infinitely spliting i.e. $\\{ n\\in\\omega:\\; s^\\frown n\\in T\\}$ is infinite.\n\nThe set of all Laver trees is denoted by the $\\Laver$. Moreover, recalling th definition of the ideal $l_0$, we have\n\\begin{definition}[ideal $l_0$] We say that $A\\in \\mathcal{P}(\\omega^\\omega)$ is in $l_0$ iff\n$$\n(\\forall T\\in \\Laver)(\\exists Q\\in \\Laver)\\; Q\\subseteq T\\land [Q]\\cap A=\\emptyset.\n$$\n\\end{definition}\n\\begin{definition}[$l$-measurable set] We say that $A\\in \\mathcal{P}(\\omega^\\omega)$ is {\\bf $l$-measurable} iff for every Laver tree $T\\in \\Laver$ there is a Laver tree $S\\in\\Laver$ such that\n$$\n(S\\subseteq T\\land [S]\\subseteq A)\\lor (S\\subseteq T\\land [S]\\cap A=\\emptyset).\n$$\n\\end{definition}\n\nWe say that a tree $T\\subseteq \\omega^{<\\omega}$ is called a {\\bf Miller tree} iff there is a node $s\\in T$ such that, \nfor every node $t\\in T$ if $s\\subseteq t$ then there is $t'$ such that $t\\subseteq t'$ and $t'$ is infinitely spliting.\n\nThe set of all Miller trees is denoted by the $\\Miller$. Moreover, recalling th definition of the ideal $m_0$, we have\n\\begin{definition}[ideal $m_0$] We say that $A\\in \\mathcal{P}(\\omega^\\omega)$ is in $m_0$ iff\n$$\n(\\forall T\\in \\Miller)(\\exists Q\\in \\Miller)\\; Q\\subseteq T\\land [Q]\\cap A=\\emptyset.\n$$\n\\end{definition}\n\\begin{definition}[$m$-measurable set] We say that $A\\in \\mathcal{P}(\\omega^\\omega)$ is {\\bf $m$-measurable} iff for every Miller tree $T\\in \\Miller$ there is a Miller tree $S\\in\\Miller$ such that\n$$\n(S\\subseteq T\\land [S]\\subseteq A)\\lor (S\\subseteq T\\land [S]\\cap A=\\emptyset).\n$$\n\\end{definition}\n\n\nIt is well known by Judah, Miller, Shelah see \\cite{JMS} and {Repick{\\'y} } see \\cite{Rep} that $add(s_0)\\le cov(s_0)\\le cof(\\mathfrak{c})\\le non(s_0)=\\mathfrak{c}<cof(s_0)\\le 2^\\mathfrak{c}$.\nCleary $\\omega_1\\le \\add(l_0)\\le \\cov(l_0)\\le \\mathfrak{c}$ holds. \nMoreover, in \\cite{Goldstern}, Goldstern, {Repick{\\'y} }\\!\\!, Shelah and Spinas showed that it is relatively consistent with $\\ZFC$ that $\\add(l_0)<\\cov(l_0)$.\n\n\nUsing the natural isomorphism of the perfect set $P=[T]$, where $T$ is Laver tree, with $\\omega^\\omega$, we have $cov_h(l_0)=cov(l_0)$.\n\n\n\n\n\n\nLet us recall the definition of almost disjoint family. Any family of sets $\\mathcal{A}\\subseteq [\\omega]^\\omega$ is an {\\bf { a.d.\\,}--family} on $\\omega$ if\n$$\n(\\forall a,b\\in\\mathcal{A})\\; a\\ne b\\longrightarrow a\\cap b\\in [\\omega]^{<\\omega}.\n$$\nTwo reals $f,g\\in\\omega^\\omega$ in Baire space are {\\bf eventually different { e.d.\\,}} iff $f\\cap g$ is a finite subset of $\\omega\\times\\omega$. \nLet us observe that an { e.d.\\,} family $\\mathcal{A}\\subseteq\\omega^\\omega$ is an { a.d.\\,} family on $\\omega\\times\\omega$. \nFor this reason we will identify the notions of eventually different family and almost disjoint { a.d.\\,} family. \nMaximal almost disjoint (or eventually different) families with respect to inclusion are called {\\bf { m.a.d.\\,} families}.\n\nEvery { a.d.\\,} family is a meager subset of the Cantor space and every ${ e.d.\\,}$ family is a meager subset of the Baire space. \nIt is natural to ask whether the existence { m.a.d.\\,} families that are either $s$-measurable or $s$-nonmeasurable can be proven in $\\ZFC$ alone. \nOne can find a consistent example of a { m.a.d.\\,} family $\\mathcal{A}$ of cardinality smaller than $\\mathfrak{c}$ (see \\cite{Kunen}, for example). \nIn a this case $\\mathcal{A}\\in s_0$ since we have $\\non(s_0)=\\mathfrak{c}$. Furthermore it is well known that there exists a perfect { a.d.\\,} family and therefore not all { m.a.d.\\,} families are in $s_0$.\n\n\n\\begin{theorem}\\label{nonmeasurable_mad} There exists a $s$-nonmeasurable  m.a.d. family in Baire space.\n\\end{theorem}\n\\begin{proof} Fix $T\\subseteq \\omega^{}<\\omega$ a perfect tree such that $[T]$ is a.d. in $\\omega^\\omega$. Let us enumerate $Perf(T)=\\{ T_\\alpha: \\alpha<\\mathfrak{c}\\}$ a family of all perfect subsets of $T$. \nBy transfinite reccursion let us define\n$$\n\\{ (a_\\alpha,d_\\alpha,x_\\alpha)\\in [T]^2\\times \\omega^\\omega:\\alpha<\\mathfrak{c}\\}\n$$\nsuch that for any $\\alpha<\\mathfrak{c}$ we have:\n\\begin{enumerate}\n \\item $a_\\alpha,d_\\alpha\\in T_\\alpha$,\n \\item $\\{ a_\\xi:\\xi<\\alpha\\}\\cap \\{ d_\\xi:\\xi<\\alpha\\}=\\emptyset$,\n \\item $\\{ a_\\xi:\\xi<\\alpha\\}\\cup \\{ x_\\xi:\\xi<\\alpha\\}$ is a.d.,\n \\item $\\forall^\\infty n\\; x_\\alpha(n)=d_\\alpha(n)$ but $x_\\alpha\\ne d_\\alpha$.\n\\end{enumerate}\nNow assume that we are in $\\alpha$-th step construction and we have required sequence\n$$\n\\{ (a_\\xi,d_\\xi,x_\\xi)\\in [T]^2\\times \\omega^\\omega:\\xi<\\alpha\\}\n$$\nwhich have size at most $\\omega|\\alpha|<\\mathfrak{c}$ then we can choose in $[T_\\alpha]$ (of size $\\mathfrak{c}$) $a_\\alpha,d_\\alpha\\in [T_\\alpha]$ which fulfills the first condition. \nThen choose any $x_\\alpha\\in \\omega^\\omega$ different than $d_\\alpha$ but $(\\forall^\\infty n) d_\\alpha(n)=x_\\alpha(n)$ then $x\\in\\omega^\\omega\\setminus [T]$ and \n$$\n\\{ a_\\xi:\\xi<\\alpha\\}\\cup \\{ x_\\xi:\\xi<\\alpha\\}\n$$\nforms an a.d. family in $\\omega^\\omega$. Then $\\alpha$-th step construction is completed. By transfinite induction theorem we have required sequence of the length $\\mathfrak{c}$. \nNow set $A_0 = \\{ a_\\alpha:\\alpha < \\mathfrak{c}\\}\\cup \\{ x_\\alpha:\\alpha<\\mathfrak{c}\\}$ and let us extend it to any maximal a.d. family $A$. It is easy to chect that $A$ is required $s$-nonmeasurable m.a.d. family in the Baire space $\\omega^\\omega$.\n\\end{proof}\n\nThe next theorem generalizes result obtained in \\cite{R1}.\n\\begin{theorem}\\label{d_l_mad} There exists a { m.a.d.\\,} family of functions \n$\\mathcal{A}\\subseteq \\omega^\\omega$ such that $\\mathcal{A}$ is not $s,l,m$-measurable at the same time, and there is an dominating subfamily $\\mathcal{A}'\\in [\\mathcal{A}]^{\\le \\mathfrak{d}}$ in Baire space $\\omega^\\omega$.\n\\end{theorem}\n\\begin{proof} Now by assumption there is a dominating family $\\mathcal{D}_0\\subseteq \\omega^\\omega$ of size $\\mathfrak{d}$. Then we show the existence an a.d. dominating family $\\mathcal{D}$ of the same size. \nTo do let $\\mathcal{P} = \\{ A_m\\in [\\omega]^\\omega:\\; m\\in\\omega \\}$ be a partition of $\\omega$ ont infinite subsets. Now let us construct a tree as follows: \n$T_{-1}=\\{ \\emptyset \\}$, next $T_0=\\{ (0,n): n\\in\\omega\\}$. \nNow assume that we have defined $T_n$ for a fixed $n\\in\\omega$ and let us enumerate $T_n=\\{ s_k: k\\in\\omega\\}$ then for any $m\\in\\omega$ let us set $A_m={ k_i\\in\\omega: k\\in\\omega}$ as an increasing sequence and define \n$T_{n+1,m} = \\{ s_m\\cup \\{(n+1, k_i)\\}:i\\in\\omega\\}$ and then let $T_{n+1}=\\bigcup_{m\\in\\omega} T_{n+1,m}$ and finally $T=\\bigcup_{n\\in\\omega\\cup\\{ -1\\}} T_n$.  \nIt is easy to observe that $[T]$ forms a a.d. family of reals in $\\omega^\\omega$. \n\nNow let us define an embedding $F:\\mathcal{D}_0\\to [T]$ as follows: pick an arbitrary element $d\\in\\mathcal{D}_0$ which is an union $\\bigcup\\{ d\\upharpoonright n: n\\in\\omega\\}$ \nthen assign to $d\\upharpoonright 0=\\emptyset\\in T_{-1}$ and to $d\\upharpoonright 1$ $t_0=d\\upharpoonright 1=\\{ (0,d(0))\\}$. Now let us assume that we have assigned for a fixed \n$d\\upharpoonright n$  $t_n\\in T_n$ for $n\\in\\omega$. Then there is unique $m\\in\\omega$ such that $t_n\\in T_{n,m}$ but $A_m=\\{ k_i:i\\in\\omega\\}$ is represented by the increasing sequence \n$(k_i)_{i\\in\\omega}\\in\\omega^\\omega$ then $d\\upharpoonright n+1$ is assigned to $t_{n+1} = t_n\\cup \\{ (n+1,w)\\}$ where $w=k_{d(n+1)}$ which is a greater than $d(n+1)$ of course. \nFrom the construction we see that $t_{n+1}\\in T_{n+1}$ and for any $n\\in\\omega$ $t_n\\subseteq t_{n+1}$. Now let $f(d)=\\bigcup\\{ t_n\\in T_n: n\\in\\omega:\\}\\in [T]$. \nIt easy to see that this construction ensure that $f$ is one to one mapping and for any $d\\in \\mathcal{D}_0$ $d\\le f(d)$. Now let $\\mathcal{D}=\\{ 4\\cdot f(d): d\\in\\mathcal{D}_0\\}\\subseteq (4\\mathbb{N})^\\omega$ \nwhich forms a dominating family in $\\omega^\\omega$ of size equal to $\\mathfrak{d}=|\\mathcal{D}_0|$.\n\nNow let us choose a.d. trees $S\\subseteq (4\\mathbb{N}+1)^{<\\omega}$, $M\\subseteq (4\\mathbb{N}+2)^{<\\omega}$ and $L\\subseteq (4\\mathbb{N}+3)^{<\\omega}$ where $S$ is a perfect tree, $M$ is Miller \nand last $L$ is a Laver tree.\n\nLet us enumerate $Perf(S)=\\{ T_\\alpha: \\alpha<\\mathfrak{c}\\}$ a family of all perfect subsets of $S$ and analogously $Miller(M)=\\{ M_\\alpha:\\alpha<\\mathfrak{c}\\}$, $Laver(L)=\\{ L_\\alpha:\\alpha<\\mathfrak{c}\\}$. \nBy transfinite reccursion let us define\n$$\n\\{ w_\\alpha \\in [S]^2\\times \\omega^\\omega\\times [M]^2\\times \\omega^\\omega\\times [L]^2\\times \\omega^\\omega:\\alpha<\\mathfrak{c}\\}\n$$\nwhere $w_\\alpha=(a^s_\\xi,d^s_\\xi,x^s_\\xi,a^s_\\xi,d^s_\\xi,x^s_\\xi,a^s_\\xi,d^s_\\xi,x^s_\\xi,)$ for any $\\alpha<\\mathfrak{c}$, and such that for any $\\alpha<\\mathfrak{c}$ we have:\n\\begin{enumerate}\n \\item $a^s_\\alpha,d^s_\\alpha\\in S_\\alpha$,\n \\item $\\{ a^s_\\xi:\\xi<\\alpha\\}\\cap \\{ d^s_\\xi:\\xi<\\alpha\\}=\\emptyset$,\n \\item $\\{ a^s_\\xi:\\xi<\\alpha\\}\\cup \\{ x^s_\\xi:\\xi<\\alpha\\}$ is a.d.,\n \\item $\\forall^\\infty n\\; x^s_\\alpha(n)=d^s_\\alpha(n)$ but $x^s_\\alpha\\ne d^s_\\alpha$.\n \n \\item $a^m_\\alpha,d^m_\\alpha\\in M_\\alpha$,\n \\item $\\{ a^m_\\xi:\\xi<\\alpha\\}\\cap \\{ d^m_\\xi:\\xi<\\alpha\\}=\\emptyset$,\n \\item $\\{ a^m_\\xi:\\xi<\\alpha\\}\\cup \\{ x^m_\\xi:\\xi<\\alpha\\}$ is a.d.,\n \\item $\\forall^\\infty n\\; x^m_\\alpha(n)=d^m_\\alpha(n)$ but $x^m_\\alpha\\ne d^m_\\alpha$.\n\n \\item $a^l_\\alpha,d^l_\\alpha\\in L_\\alpha$,\n \\item $\\{ a^l_\\xi:\\xi<\\alpha\\}\\cap \\{ d^l_\\xi:\\xi<\\alpha\\}=\\emptyset$,\n \\item $\\{ a^l_\\xi:\\xi<\\alpha\\}\\cup \\{ x^l_\\xi:\\xi<\\alpha\\}$ is a.d.,\n \\item $\\forall^\\infty n\\; x^l_\\alpha(n)=d^l_\\alpha(n)$ but $x^l_\\alpha\\ne d^l_\\alpha$.\n\\end{enumerate}\nNow assume that we are in $\\alpha$-th step construction and we have required sequence\n$$\n\\{ w_\\alpha :\\; \\xi<\\alpha\\}\n$$\nwhich have size at most $\\omega\\cdot |\\alpha|<\\mathfrak{c}$. In case of perfect part we can choose in $[S_\\alpha]$ (of size $\\mathfrak{c}$) $a^s_\\alpha,d^s_\\alpha\\in [S_\\alpha]$ which fulfills the first condition. \nThen choose any $x^s_\\alpha\\in \\omega^\\omega$ different than $d^s_\\alpha$ but $(\\forall^\\infty n) d_\\alpha(n)=x_\\alpha(n)$ then $x^s_\\alpha\\in\\omega^\\omega\\setminus [S]$ and \n$$\n\\{ a_\\xi:\\xi \\le \\alpha\\}\\cup \\{ x_\\xi:\\xi \\le \\alpha\\}\n$$\nforms an a.d. family in $\\omega^\\omega$. In the same way we can choose rest points of our tuple for Miller and Laver trees. Then $\\alpha$-th step construction is completed. \nBy transfinite induction theorem we have required sequence of the length $\\mathfrak{c}$. Now set \n$$\nA_s = \\mathcal{D}\\cup \\{ a^s_\\alpha:\\alpha < \\mathfrak{c}\\}\\cup \\{ x^s_\\alpha:\\alpha<\\mathfrak{c}\\},\n$$\n$$\nA_m = \\mathcal{D}\\cup \\{ a^m_\\alpha:\\alpha < \\mathfrak{c}\\}\\cup \\{ x^m_\\alpha:\\alpha<\\mathfrak{c}\\}\n$$\nand\n$$\nA_l = \\mathcal{D}\\cup \\{ a^l_\\alpha:\\alpha < \\mathfrak{c}\\}\\cup \\{ x^l_\\alpha:\\alpha<\\mathfrak{c}\\}\n$$\nand let us extend the family $\\mathcal{A}=\\mathcal{D}\\cup\\mathcal{A}_s\\cup\\mathcal{A}_m\\cup\\mathcal{A}_l$ to any maximal a.d. family $A$. It is easy to check that $A$ is required $s,m$ \nand $l$-nonmeasurable m.a.d. family in the Baire space $\\omega^\\omega$ with a dominating subfamily of size $\\mathcal{D}$, what completes this proof.\n\\end{proof}\n\n\nNow, let us give some examples of subsets of Baire space which are nonmeasurable in one sense and measurable in other one at the same time. Similar results were obtained in \\cite{Brendle}.\n\\begin{theorem} There are subsets $A,B,C$ of the Baire space such that\n\\begin{itemize}\n \\item $A$ is $l$-measurable and not $s$-measurable,\n \\item $B$ is $m$-measurable  but not $s$-measurable,\n \\item $C$ is $l$-measurable but not $m$-measurable.\n\\end{itemize}\nMoreover, if $\\mathfrak{b}=\\mathfrak{c}$ then \n\\begin{itemize}\n \\item there is a not $l$-measurable set which is $s$-measurable\n \\item there is a not $m$-measurable set which is $s$-measurable.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof} To show the first part, let $T$ be $a.d.$-disjoint Laver tree (defined as in the proof of Theorem \\ref{d_l_mad}). Now let us define a perfect subtree $S$ of $T$ such that all levels of $S$ consist the first two numbers of $T$, i.e. $\\tau\\in S$ if $\\tau\\in T$ and for any $i\\in dom(\\tau)$ $\\tau(i)\\in\\{m,n\\}$ where $m=\\min\\{ s(i):s\\in T\\}$ and $n=\\min\\{ s(i): s\\in T\\land s(i)\\ne m\\}$. Now choose any $X\\subset [S]$ which is not $s$-measurable and then let $A=([T]\\setminus [S])\\cup X$. Now let us observe that $[S]\\in l_0$ and then $X\\in l_0$ what give the assertion.\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   \nThe similar argument shews the second clause.\n\nTo see next, let $T$ be as above and let $M\\subseteq T$ be a Miller tree defined as follows: on the odd levels of $T$ each node of $M$ does not split but in evan levles of $T$ the all nodes of $M$ uses half part of the level of $T$ i.e. if $s\\in M$ and $i\\in dom(s)$ is even then $s(i)\\in \\{a_{2k}:k\\in\\omega\\}$ where $\\{ a_k:k\\in\\omega\\}$ enumeration of \n$$\n\\{ m\\in\\omega: (\\exists \\tau\\in T)\\; \\tau=s\\upharpoonright i\\land \\tau^\\frown m\\in T\\}.\n$$\nFirstly we show that $[M]\\in l_0$, let us consider any Laver subtree $S\\subseteq T$ with a stem $s\\in S$ then let us find a node $\\tau\\in S$ which extend $s$ and $|\\tau|$ is odd but every node $\\tau'\\in M$ with $|\\tau'|$ is odd has no splitting one then we can find infinite set of splitting i.e. \n$$\nW_\\tau=\\{ n\\in\\omega:\\; \\tau^\\frown \\in S\\setminus M\\}\n$$\nand then we can a Laver subtree $S'$ of $S$ such that $[S']\\cap [M]$ is empty and then we have conclusion. Now let us consider any not $m$-measurable subset $Y\\subset [M]$ then $B=([T]\\setminus [M])\\cup Y$ is as we want.\n\nTo prove the next sentence let us enumerate all Laver subtrees of $T$ $T_\\alpha:\\alpha<\\mathfrak{c}\\}$ and all perfect subtrees $\\{ S_\\alpha<\\mathfrak{c}]|$ of $T$. Then let us define a transfinite sequence:\n$$\n(a_\\xi,d_\\xi,P_\\xi):\\xi<\\mathfrak{c}\n$$\nwith the following conditions, for any $\\xi<\\mathfrak{c}$\n\\begin{enumerate}\n \\item $a_\\xi,d_\\xi\\in [T_\\xi]$\n \\item for any $\\alpha<\\mathfrak{c}$ $\\{ a_\\eta:\\eta<\\xi\\}\\cap \\{ d_\\eta:\\eta<\\xi\\}$,\n \\item $P_\\xi\\subseteq S_\\xi$ and $P_\\xi$ is binary tree,\n \\item for any $\\eta<\\xi$ $P_\\eta\\cap \\{ a_\\beta:\\beta<\\xi\\}=\\emptyset$.\n\\end{enumerate}\nNow in $\\alpha$-th step construction let us consider $T_\\alpha\\subseteq T$ and $S_\\alpha$. Then let us consider a perfect set $P$ contained in $[S_\\alpha]$ generated by a binary tree $P_\\alpha$ such that $P\\cap \\{ a_\\xi:\\xi<\\alpha\\}=\\emptyset$ which is possible because any perfect can be partitioned onto $\\mathfrak{c}$ many perfect sets. Then choose $a_\\alpha\\in [T_\\alpha]\\setminus (\\bigcup_{\\xi<\\alpha} P_\\xi\\cup \\{ d_\\xi:\\xi<\\alpha\\})$ what is guaranteed by the fact that $\\mathfrak{b}=\\mathfrak{c}$ and $a_\\alpha$ is in unbounded in family $\\{ f_\\xi:\\xi<\\alpha\\}$ where\n$$\nf_\\xi(n)= \\max\\{ s(n):s\\in P_\\xi\\land n\\in dom(s)\\}\n$$\nThe chossing $d_\\alpha\\in [T_\\alpha]\\setminus \\{ a_\\xi:\\xi\\le\\alpha\\}$ finishes this step of construction.\n\nFinally the $\\{ a_\\xi:\\xi<\\mathfrak{c}\\}$ witness the required set.\n\nThe proof of the last sentence is similar to previous one when we replace the family of binary $\\{ P_\\xi<\\mathfrak{c}\\}$ trees by Miller trees such that every odd node has finite splitting.\n\\end{proof}\n\n\\section{Nonmeasurable and completely nonmeasurable unions}\nIt is known that for any $\\sigma$-ideal\n$\\mathcal{I}$ with a Borel base  if  $\\mathcal{A}$ is point-finite and its union is not in $\\mathcal{I}$ then  there exists a subfamily $\\mathcal{A}'$ of $\\mathcal{A}$ such that the union of its sets is not in the $\\sigma$-algebra generated by the Borel sets and $\\mathcal{I}$ (see \\cite{Buk}, \\cite{BCGR}).\n\nIt is known that within ZFC it is not possible to replace the as-\nsumption that the family $\\mathcal{A}$ is point-finite even by the one saying that\n$\\mathcal{A}$ is point-countable (see \\cite{Fre}).\n\nIn various cases it is possible to obtain more than nonmeasurability\nof the union of a subfamily of $\\mathcal{A}$. Namely, the intersection of this union\nwith any measurable set that is not in $\\mathcal{I}$\nis nonmeasurable (recall, the measurability is understood here in the sense of belonging to the $\\sigma$-algebra generated by the family of Borel sets and $\\mathcal{I}$). Such strong\nconclusion can be obtained for the ideal of first Baire category sets\nunder the assumption that $\\mathcal{A}$ is a partition, but without assuming anything about the regularity of the elements of $\\mathcal{A}$ (see \\cite{5P}).\n\nSome related topics are presented in \\cite[chapter 14]{Kha}. Namely, the problem of the existance of the family of first category sets whose union do not posses Baire property is discussed for general topological spaces of second Baire category.\n\nIn \\cite{FT}, the problem concerning null sets is discussed. It is shown that for every partition of unit inerval into Lebesgue null sets and for every $\\varepsilon>0$ we can find a subfamily such that its union has inner measure smaller than $\\varepsilon$ and outher measure grater than $1-\\varepsilon.$  \n\nIn paper \\cite{Z} it was shown how to obtain complete nonmeasurability of\nthe union of a subfamily of $\\mathcal{A}$ assuming that $\\mathcal{A}$ is point-finite family.\nHowever, the result requires some  set-theoretic assumptions. Namely, we need to assume that\nthere is no quasi-measurable cardinal smaller than $2^\\omega.$ (Recall that\n$\\kappa$ is quasi-measurable if there exists a $\\kappa$-additive ideal\n$\\mathcal{I}$ of subsets of $\\kappa$ such that the Boolean algebra $P(\\kappa)\/\\mathcal{I}$ satisfies countable chain condition.) By the Ulam theorem (see \\cite{Jech}) every quasi-measurable\ncardinal is weakly inaccessible, so it is a large cardinal. \n\nThe above result was strenghtened in paper \\cite{RZ1} where it was shown that it is enough to assume that  there is no quasi-measurable cardinal not greater than $2^\\omega.$\n\nThe problem concerning finding completely $\\mathcal{I}$-nonmeasurable sets were also discussed in papers \\cite{R}, \\cite{RZ2}. In those results the starting families fulfills some additional conditions.   \n\nThe aim of this section is to  discus the following problem. Let $\\mathcal{I}$ be a $\\sigma$-ideal of subsets of $\\mathbb{R}} %{{\\Bbb R}$.\nAssume that  $\\mathcal{P}\\subseteq\\mathcal{I}$. Is it possible that\nfor all $\\mathcal{A}\\subseteq\\mathcal{P}$\n$$\n\\bigcup\\mathcal{A} \\text{ is } \\mathcal{I}\\text{-nonmeasurable }\n$$\n$$\n\\Downarrow\n$$\n$$\n\\bigcup\\mathcal{A} \\text{ is completely} \\mathcal{I}\\text{-nonmeasurable}?\n$$\nWe will consider situations when $\\mathcal{P}$ is a partition of $\\mathbb{R}} %{{\\Bbb R}$, $\\mathcal{P}$ is point-finite family and $\\mathcal{P}$ is point-countable family. \n\nThrought this section $\\mathcal{I}$ will denote a $\\sigma$-ideal of subsets of $\\mathbb{R}} %{{\\Bbb R}$ satisfying the following conditions\n\\begin{enumerate}\n \\item  $\\mathcal{I} $ contain singletons, i.e. $[\\mathbb{R}} %{{\\Bbb R}]^\\omega\\subseteq\\mathcal{I},$\n \\item $\\mathcal{I}$ has Borel base, i.e. $(\\forall I \\in\\mathcal{I})(\\exists B \\in \\mbox{\\rm Borel} \\cap \\mathcal{I})(I\\subseteq B),$\n \\item $\\mathcal{I}$ is {\\bf translation invariant}, i.e. \n $$(\\forall I \\in\\mathcal{I})(\\forall x \\in\\mathbb{R}} %{{\\Bbb R})(x + I = \\{x + i :\\ i\\in I \\} \\in\\mathcal{I}).$$\n\\end{enumerate}\n\n\\begin{definition}\n Let $A\\subseteq\\mathbb{R}} %{{\\Bbb R}$. We say that \n \\begin{enumerate}\n  \\item $A$ is {\\bf $\\mathcal{I}$-nonmeasurable} if $A$ does not belong to the $\\sigma$-algebra generated by Borel sets and $\\sigma$-ideal $\\mathcal{I};$\n  \\item $A$ is {\\bf completely $\\mathcal{I}$-nonmeasurable} if $A\\cap B$ is $\\mathcal{I}$-nonmeasurable for every Borel set $B$ which does not belong to $\\mathcal{I}.$\n \\end{enumerate}\n\\end{definition}\n\nLet us remark that the folowing conditions are all equivalent:\n\\begin{enumerate}\n \\item $A$ is completely $\\mathcal{I}$-nonmeasurable,\n \\item $A\\cap B$ and $A\\cap (\\mathbb{R}} %{{\\Bbb R}\\setminus B)$ does not belong to $\\mathcal{I}$  for every Borel set $B$ such that $B, \\mathbb{R}} %{{\\Bbb R}\\setminus B\\notin\\mathcal{I},$\n \\item $A$ intersects every Borel set which does not belong to $\\mathcal{I}$ and does not contain any of such sets.\n\\end{enumerate}\n \nLet us notice that if $\\mathcal{I}$ is the ideal of countable sets then $A$ is completely $\\mathcal{I}$-nonmeasurable if and only if $A$ is a Bernstein set. That is why completely $\\mathcal{I}$-nonmeasurable sets are sometimes called $\\mathcal{I}$-Bernstein sets.\n\nIf $\\mathcal{I}$ is the ideal of Lebesgue null sets then $A$ is completely $\\mathcal{I}$-nonmeasurable if and only if its inner measure is zero and the inner measure of its complement is also zero.\n\nWe divide results into three groups: the first - ideals with Steinhaus property, the second - families consisting of finite sets and the third - families consisting of countable sets.\n\\subsection{Ideals with weaker Smital property}\nIn this subsection we will consider $\\sigma$-ideals possesing weaker Smital property. This notion was introduced in \\cite{lodz} and was invastigated in \\cite{MZ}.\nLet us recall the definition.\n\\begin{definition}\nWe say that $\\mathcal{I} $ has {\\bf weaker Smital property} if there exists a countable dense set $D$ such that\n$$\n (\\forall A\\in\\mbox{\\rm Borel}\\setminus\\mathcal{I})((A+D)^c\\in\\mathcal{I}).\n$$\nWe say that $D$ witnesses that $\\mathcal{I}$ has the weaker Smital property.\n\\end{definition}\n\n\nLet us notice that the weaker Smital property is implied by Smital property and Smital property is implied by Steinhaus property.\n\nLet us remark that the ideal $\\mathcal{N}$ of null subsets of $\\mathbb{R}} %{{\\Bbb R}$ and the ideal $\\mathcal{M}$ of meager subsets of $\\mathbb{R}} %{{\\Bbb R}$ have Steinhaus property.\nMore natural examples of ideals possesing weaker Smital property in euclidean spaces can be found in \\cite{lodz}.\n\nFrom the other hand, the ideal $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ of countable subsets of reals does not have weaker Smital property.\n\n\\begin{theorem}\nAssume $\\mathcal{I}$ has weaker Smital property. Then there exists a partition\n$\\mathcal{P}\\subseteq\\mathcal{I}$ of $\\mathbb{R}} %{{\\Bbb R}$ such that for every $\\mathcal{A}\\subseteq\\mathcal{P}$\n$$\n\\bigcup\\mathcal{A} \\text{ is } \\mathcal{I}\\text{-nonmeasurable }\n$$ \n$$\n\\Downarrow\n$$\n$$\n\\bigcup\\mathcal{A} \\text{ is completely } \\mathcal{I}\\text{-nonmeasurable.}\n$$\n\\end{theorem}\n\\begin{proof}\nLet $D$ be a set witnessing that $\\mathcal{I} $ has weaker Smital propoerty. We can assume that $D$ is a subgroup of $(\\mathbb{R},+)$\nFor $x, y \\in\\mathbb{R}} %{{\\Bbb R}$ let $x \\sim y \\leftrightarrow x-y\\in D.$ Set\n$$\n\\mathcal{P} = \\mathbb{R}} %{{\\Bbb R}\/\\sim = \\{x_\\alpha + D :\\ \\alpha\\in 2^\\omega \\}.\n$$\nTake $\\mathcal{A}\\subseteq\\mathcal{P}$ such that $\\bigcup\\mathcal{A}$ is  $\\mathcal{I}$-nonmeasurable.\nAssume that $\\bigcup\\mathcal{A}$ is not completely $\\mathcal{I}$-nonmeasurable.\nThen $\\bigcup\\mathcal{A}\\notin\\mathcal{I}$ and $\\bigcup(\\mathcal{P}\\setminus\\mathcal{A})\\notin\\mathcal{I}$ and at least one of this sets\ncontains $\\mathcal{I}$-possitive Borel set. Without loss of generality we can assume that $\\bigcup\\mathcal{A}$ contains $\\mathcal{I}$-possitive Borel set.\nBy weaker Smital property of $\\mathcal{I}$, the set $(\\bigcup\\mathcal{A} +D)^c$ belongs to $\\mathcal{I}$. Notice that \n$$\n\\bigcup\\mathcal{A} +D =\\bigcup\\mathcal{A}\\text{ and }\\bigcup\\mathcal{A}\\cap\\bigcup(\\mathcal{P}\\setminus\\mathcal{A})=\\emptyset\n$$\nContradiction.\n\\end{proof}\n\n\\subsection{Finite sets}\nIn this subsection we will deal with families consisting of finite sets.\n\\begin{theorem}\nLet $\\mathcal{P} \\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{<\\omega}$ be a partition of $\\mathbb{R}} %{{\\Bbb R}.$ Then\n\\begin{enumerate}\n \\item there is $\\mathcal{A}_0\\subseteq \\mathcal{P}$ such that $\\bigcup\\mathcal{A}_0$ is completely $\\mathcal{I}$-nonmeasurable;\n \\item there is $\\mathcal{A}_1\\subseteq \\mathcal{P}$ such that $\\bigcup\\mathcal{A}_1$ is $\\mathcal{I}$-nonmeasurable but is not\ncompletely $\\mathcal{I}$-nonmeasurable.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nFamily $\\mathcal{A}_0$ can be constructed in the standard way following construction of Bernstein set.\n\nTo prove the second part let us enumerate \n$$\n\\mathcal{P} = \\{Y_\\alpha : \\alpha\\in2^\\omega \\},\n$$\n$$\nY_\\alpha = \\{y_0^\\alpha , y_1^\\alpha , \\ldots , y_n^\\alpha \\},\\quad y_0^\\alpha<y_1^\\alpha< \\ldots  <y_n^\\alpha.\n$$\n\nDefine $X_k = \\{y_k^\\alpha : \\alpha\\in 2^\\omega \\}.$ Witout lost of generality $X_0\\notin\\mathcal{I}.$\nWe can find $r\\in\\mathbb{R}} %{{\\Bbb R}$ such that $X_0 \\cap (-\\infty, r ) \\notin\\mathcal{I}$ and\n$X_0 \\cap (r , +\\infty) \\notin\\mathcal{I}.$\n\nSet $\\mathcal{P}^+ = \\{Y_\\alpha :\\ y_0^\\alpha > r \\} \\subseteq\\mathcal{P}.$\nWe have that $\\bigcup\\mathcal{P}^+ \\subseteq (r , +\\infty)$ and $\\bigcup\\mathcal{P}\\notin\\mathcal{I}.$\nFind $\\mathcal{A}_1 \\subseteq\\mathcal{P}^+$ such that $\\bigcup\\mathcal{A}_1$ is $\\mathcal{I}$-nonmeasurable. \n$\\bigcup\\mathcal{A}_1$ is not\ncompletely $\\mathcal{I}$-nonmeasurable.\n\\end{proof}\n\n\\subsection{Countable sets}\nIn this subsection we will deal with families consisting of countable sets.\n\\begin{theorem} \nAssume that $\\mathcal{P}\\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ is a point-countable cover of $\\mathbb{R}} %{{\\Bbb R}.$ Then we can find\n$\\mathcal{A}\\subseteq\\mathcal{P}$ such that $\\bigcup\\mathcal{A}$ is completely $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable.\n\\end{theorem}\n\\begin{proof}\nWe will slightly modify the standard construction of Bernstein set. Let \n$$\n\\{Q_\\alpha:\\ \\alpha<2^\\omega\\}\n$$\nbe enumeration of all nonempty perfect subsets of $\\mathbb{R}} %{{\\Bbb R}$. By transfinite induction on $\\alpha<2^\\omega$ we will construct \n$$\nA_\\alpha\\in\\mathcal{P},\\quad x_\\alpha\\in Q_\\alpha\n$$\nsatisfying the following conditions\n\\begin{enumerate}\n \\item $A_\\alpha\\cap Q_\\alpha\\neq\\emptyset,$\n \\item $A_\\alpha\\cap\\{x_\\beta: \\beta<\\alpha\\}=\\emptyset,$\n \\item $x_\\alpha\\notin\\bigcup_{\\beta\\le\\alpha}A_\\beta.$\n\\end{enumerate}\nThe construction can be made because at $\\alpha$-step\n$$\nQ_\\alpha\\setminus\\left(\\bigcup\\{A\\in\\mathcal{P}:\\ \\exists\\beta<\\alpha\\ x_\\beta\\in A\\}\\cup\\bigcup_{\\beta<\\alpha}A_\\beta\\right)\\neq\\emptyset.\n$$\nSo, we can find $A_\\alpha$ and $x_\\alpha$ fulfilling our requirements.\n\nAt the end, we get $\\mathcal{A}=\\{A_\\alpha:\\ \\alpha<2^\\omega\\}$ such that $\\bigcup\\mathcal{A}\\cap Q_\\alpha\\neq\\emptyset$ and $\\bigcup\\mathcal{A}\\cap\\{x_\\alpha:\\ \\alpha< 2^\\omega\\}=\\emptyset$, what shows that $\\bigcup\\mathcal{A}$ is completely $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable. \n\\end{proof}\n\n\\begin{theorem}[$\\neg CH$]\\label{nch}\nAssume that $\\mathcal{P}\\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ is a partition of $\\mathbb{R}} %{{\\Bbb R}.$ Then we can find\n$\\mathcal{A}\\subseteq\\mathcal{P}$ such that $\\bigcup\\mathcal{A}$ is $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable but is not completely\n$[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable.\n\\end{theorem}\n\\begin{proof}\nTake $\\mathcal{A}\\subseteq\\mathcal{P}$ such that $|\\mathcal{A}| = \\omega_1.$\n$|\\bigcup\\mathcal{A}| = \\omega_1 < 2^\\omega.$ So, $\\bigcup\\mathcal{A}$ is $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable.\nFix $\\{Q_\\alpha : \\alpha\\in 2^\\omega \\}$ a family of pairwise disjoint perfect sets.\nThere exists $\\alpha$ such that $Q_\\alpha \\cap \\bigcup\\mathcal{A} = \\emptyset.$ \nSo, $\\bigcup\\mathcal{A}$ is not completely\n$[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable.\n\\end{proof}\n\n\\begin{theorem}[$CH$]\\label{ch}\nThere is $\\mathcal{P}\\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ a partition of $\\mathbb{R}} %{{\\Bbb R}$ such that for any $\\mathcal{A}\\subseteq\\mathcal{P}$\n$$ \\bigcup\\mathcal{A} \\text{ is } [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega} \\text{-nonmeasurable }\n$$\n$$\n\\Downarrow\n$$\n$$\n\\bigcup\\mathcal{A} \\text{ is completely } [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega} \\text{-nonmeasurable. }$$\n\\end{theorem}\n\\begin{proof}\nLet $\\{Q_\\alpha : \\alpha\\in\\omega_1\\}$ be an enumeration of all perfect subsets of $\\mathbb{R}} %{{\\Bbb R}.$\nWe can construct a partition $\\mathcal{P} = \\{X_\\alpha : \\alpha\\in\\omega_1 \\}\\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ \nin such a way that\n$X_\\alpha \\cap Q_\\beta \\neq \\emptyset$ for every $\\beta <\\alpha.$\nNow, take $\\mathcal{A} \\subseteq\\mathcal{P}$ such that $|\\mathcal{A}| = |\\mathcal{P}\\setminus\\mathcal{A} | = \\omega_1.$ Then\n$\\bigcup\\mathcal{A} \\cap Q_\\alpha \\neq\\emptyset$ and\n$\\bigcup(\\mathcal{P} \\setminus \\mathcal{A}) \\cap Q_\\alpha \\neq \\emptyset$ for every $\\alpha < \\omega_1.$\nSo, $\\bigcup\\mathcal{A}$ is completely $[\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$-nonmeasurable.\n\\end{proof}\n\nAs a consequence of Theorem \\ref{nch} and Theorem \\ref{ch} we get the following characterisation of Continuum Hypothesis.\n\\begin{cor}\nThe following statements are equivalent:\n\\begin{enumerate}\n \\item CH,\n \\item there is $\\mathcal{P}\\subseteq [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}$ a partition of $\\mathbb{R}} %{{\\Bbb R}$ such that for any $\\mathcal{A}\\subseteq\\mathcal{P}$\n$$\n\\bigcup\\mathcal{A} \\text{ is } [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}\\text{-nonmeasurable }\n$$\n$$\n\\Updownarrow\n$$\n$$\n\\bigcup\\mathcal{A} \\text{ is completely } [\\mathbb{R}} %{{\\Bbb R}]^{\\le\\omega}\\text{-nonmeasurable.}\n$$\n\\end{enumerate}\n\\end{cor}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}