diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkhpu" "b/data_all_eng_slimpj/shuffled/split2/finalzzkhpu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkhpu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nMultiplier ideals are an important tool in higher-dimensional algebraic geometry, and one can view them as measuring singularities. \nThey are defined as follows:\nlet $X$ be a smooth complex variety and ${\\mathfrak{a}} \\subseteq {\\mathcal{O}} _X$ be an ideal sheaf of $X$. \nSuppose that $\\pi:\\widetilde{X} \\to X$ is a log resolution of ${\\mathfrak{a}}$, that is, $\\pi$ is a proper birational morphism, \n$\\widetilde{X}$ is smooth and $\\pi^{-1}V({\\mathfrak{a}})=F$ is a divisor with simple normal crossing support. \nIf $K_{\\widetilde{X}\/X}$ is the relative canonical divisor of $\\pi$, then the multiplier ideal of ${\\mathfrak{a}}$ with exponent $c \\in \\R_{\\ge 0}$ is \n\\[\n\\J({\\mathfrak{a}}^c)=\\J(c \\cdot {\\mathfrak{a}})=\\pi_*{\\mathcal{O}}_{\\widetilde{X}}(K_{\\widetilde{X}\/X}-\\lfloor cF \\rfloor) \\subseteq {\\mathcal{O}}_X.\n\\]\nA positive rational number $c$ is called a {\\it jumping coefficient} if $\\J({\\mathfrak{a}}^{c-\\varepsilon})\\neq \\J({\\mathfrak{a}}^{c})$ for all $\\varepsilon>0$, \nand the minimal jumping coefficient is called the {\\it log-canonical threshold} of ${\\mathfrak{a}}$ and denoted by $\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}})$. \nSince multiplier ideals are defined via log resolutions, it is difficult to compute them in general \n(when the ideal ${\\mathfrak{a}}$ is a monomial ideal or a principal ideal generated by a non-degenerate polynomial, \nthere is a combinatorial description of the multiplier ideals $\\J({\\mathfrak{a}}^c)$. See \\cite{H} and \\cite{H2}). \nIn this paper, we will give algorithms for computing multiplier ideals using the theory of $D$-modules. \n\n\nThe Bernstein-Sato polynomial (or $b$-function) is one of the main objects in the theory of $D$-modules. \nIt has turned out that jumping coefficients are deeply related to Bernstein-Sato polynomials.\nFor a given polynomial $f \\in {\\mathbb{C}}[x_1, \\dots, x_n]$, \nthe Bernstein-Sato polynomial $b_f(s)$ of $f$ is the monic polynomial in one variable $b(s) \\in {\\mathbb{C}}[s]$ of minimal degree \nhaving the property that there exists a linear differential operator $P(x,s)$ such that $b(s) f^s =P(x,s)f^{s+1}$. \nKoll\\'ar \\cite{Ko} proved that the log canonical threshold of $f$ is the minimal root of $b_f(-s)$. \nFurthermore, Ein--Lazarsfeld--Smith--Varolin \\cite{ELSV} extended Koll\\'ar's result to higher jumping coefficients: \nthey proved that all jumping coefficients in the interval $(0,1]$ are roots of $b_f(-s)$. \nRecently Budur--Musta\\c{t}\\v{a}--Saito introduced the notion of Bernstein-Sato polynomials of arbitrary ideal sheaves \nusing the theory of $V$-filtrations of Kashiwara \\cite{Ka} and Malgrange \\cite{M}. \nThey then gave a criterion for membership of multiplier ideals in terms of their Bernstein-Sato polynomials, \nand proved that all jumping coefficients of ${\\mathfrak{a}}\\subset {\\mathcal{O}}_X$ in the interval $(\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}}),\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}})+1]$ \nare roots of the Bernstein-Sato polynomial of ${\\mathfrak{a}}$ up to sign. \n\nIt is difficult to compute Bernstein-Sato polynomials in general, \nbut Oaku \\cite{O1}, \\cite{O2}, \\cite{O3} gave algorithms for computing Bernstein-Sato polynomials $b_f(s)$ \nusing Gr\\\"obner bases in Weyl algebras \n(algorithms for computing Gr\\\"obner bases in Weyl algebras are implemented in some computer systems, \nsuch as Kan\/Sm1 \\cite{T} and Risa\/Asir \\cite{N}). \nIn this paper, we give algorithms for computing Budur--Musta\\c{t}\\v{a}--Saito's Bernstein-Sato polynomials \n(Theorems \\ref{Algorithm for global b-functions 1}, \\ref{Algorithm for global b-functions 2} and \\ref{Algorithm for local b-functions}). \nOur algorithms are natural generalizations of Oaku's algorithm. \n\nThe other ingredient of this paper is algorithms for computing multiplier ideals. \nThe algorithm for computing generalized Bernstein-Sato polynomials enables us to solve the membership problem for multiplier ideals, \nbut does not give a system of generators of multiplier ideals. \nWe modify the definition of Budur--Musta\\c{t}\\v{a}--Saito's Bernstein-Sato polynomials \nto determine a system of generators of the multiplier ideals of a given ideal (Definition \\ref{new Bernstein-Sato polynomial}). \nThen we obtain algorithms for computing our Bernstein-Sato polynomials\nand algorithms for computing multiplier ideals (Theorem \\ref{algorithm for computing multiplier ideals 1}, Theorem \\ref{algorithm for computing multiplier ideals 2}). \nOur algorithms are based on the theory of Gr\\\"obner bases in Weyl algebras \n(see \\cite{OT} and \\cite{SST} for a review of Gr\\\"obner bases in Weyl algebras and their applications). \nWe conclude the paper by presenting several examples computed by our algorithms. \\\\\n\\begin{acknowledgments}\\upshape\nThe author thanks Shunsuke Takagi for useful comments and discussions. \nHe also would like to thank Naoyuki Shinohara and Kazuhiro Yokoyama for warm encouragement and support. \n\\end{acknowledgments}\n\\section{Preliminaries}\n\\subsection{Gr\\\"obner bases in Weyl algebras}\nWe denote by ${\\mathbb{C}}$ the complex number field. \nWhen we use a computer algebra system, we may work with a computable field $\\Q(z_1,\\dots,z_l)\\subset {\\mathbb{C}}$ \nwhich is sufficient to express objects that appear in the computations. \nLet $X$ be the affine space ${\\mathbb{C}}^{d}$ with the coordinate system ${\\boldsymbol{x}}= (x_1,\\dots,x_d)$, \nand ${\\mathbb{C}}[{\\boldsymbol{x}}]={\\mathbb{C}}[x_1,\\dots,x_d]$ a polynomial ring over ${\\mathbb{C}}$ which is the coordinate ring of $X$. \nWe denote by ${\\partial_{\\boldsymbol{x}}} = (\\partial_{x_1} ,\\dots, \\partial_{x_d})$ the partial differential operators where \n$\\partial_{x_i} = \\frac{\\partial}{\\partial{x_i}}$. \nWe set \n\\[\nD_X={\\mathbb{C}}\\langle {\\boldsymbol{x}},{\\partial_{\\boldsymbol{x}}}\\rangle={\\mathbb{C}}\\langle x_1,\\dots,x_d,\\partial_{x_1},\\dots,\\partial_{x_d}\\rangle, \n\\]\nthe rings of differential operators of $X$, and call it the {\\it Weyl algebra} (in $d$ variables). \nThis ring is non-commutative ${\\mathbb{C}}$-algebra with the commutation rules \n\\[\nx_ix_j=x_jx_i,~~\\partial_{x_i}\\partial_{x_j}=\\partial_{x_j}\\partial_{x_i},\n~~\\partial_{x_i}x_j=x_j\\partial_{x_i}~~\\mathrm{for}~~i\\neq j,~~\\mathrm{and}~\\partial_{x_i}x_i=x_i\\partial_{x_i}+1.\n\\]\nWe write $\\langle P_1,\\dots,P_r\\rangle$ for the left ideal of $D_X$ generated by $P_1,\\dots,P_r\\in D_X$. \nWe use the notation \n${\\boldsymbol{x}}^{\\boldsymbol{\\mu}}=\\prod_{i=1}^d x_i^{{\\mu}_{i}}$, \nand \n${\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}}=\\prod_{i=1}^d\\partial_{x_i}^{{\\nu}_{i}}$ \nfor $\\boldsymbol{\\mu} = ({\\mu}_{1}, \\dots , {\\mu}_{d})$, $\\boldsymbol{\\nu}=({\\nu}_{1},\\dots,{\\nu}_{d})\\in{\\mathbb{Z}}_{\\ge 0}^d$. \nWe denote by $|\\boldsymbol{\\mu}| := {\\mu}_{1} + \\dots+ {\\mu}_{d}$ the side of $\\mu$. \nWe call a real vector $({\\boldsymbol{v}},{\\boldsymbol{w}})=(v_{1},\\dots,v_{d},w_{1},\\dots,w_{d})\\in \\R^{d}\\times\\R^{d}$ a {\\it weight vector} if \n\\[\nv_i+w_i\\ge 0 \\mathrm{~for~}i=1,2,\\dots,d.\n\\]\nWe define the ascending filtration $\\cdots \\subset F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_0{D_X}\\subset F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_1{D_X}\\subset \\cdots$ on $D_X$ \nwith respect to the weight vector $({\\boldsymbol{v}},{\\boldsymbol{w}})$ by \n\\[\nF^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}=\\{\\sum_{{\\boldsymbol{v}}\\cdot \\boldsymbol{\\mu}+{\\boldsymbol{w}}\\cdot \\boldsymbol{\\nu}\\le m} \na_{\\boldsymbol{\\mu\\nu}}{\\boldsymbol{x}}^{\\boldsymbol{\\mu}}{\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}}\\mid \na_{\\boldsymbol{\\mu\\nu}}\\in {\\mathbb{C}}\\}\\subset D_X \n\\]\nwhere ${\\boldsymbol{v}}\\cdot \\boldsymbol{\\mu}=\\sum v_i\\mu_i$ is the usual inner product of ${\\boldsymbol{v}}$ and $\\boldsymbol{\\mu}$. \nThen we have \n\\[\nF^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_{m_1}{D_X}\\cdot F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_{m_2}{D_X}\\subset F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_{m_1+m_2}{D_X}\n\\] \nfor all $m_1$, $m_2\\in {\\mathbb{Z}}$ by the conditions $v_i+w_i\\ge 0$ and the commutation rules of $D_X$. \nIn particular, $F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_0{D_X}$ is a sub-ring of $D_X$, and $F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}$'s are $F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_0{D_X}$-submodules of $D_X$. \nWe can define the associated graded ring of $D_X$ with respect to the filtration \n\\[\n\\mathop{\\mathrm{Gr}}\\nolimits^{({\\boldsymbol{v}},{\\boldsymbol{w}})}{D_X}=\\bigoplus_{m\\in {\\mathbb{Z}}} F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}\/ F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_{m-1}{D_X}. \n\\]\n\\begin{Definition}\nThe {\\it order} of $P\\in D_X$ is defined by \n\\[\n\\mathop{\\mathrm{ord}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(P)=\\min\\{m\\mid P\\in F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}\\}. \n\\]\nFor a non-zero $P\\in D_X$ with $\\mathop{\\mathrm{ord}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(P)=m$, the {\\it initial form} $\\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(P)$ of $P$ is the image of $P$ in $\\mathop{\\mathrm{Gr}}\\nolimits_m^{({\\boldsymbol{v}},{\\boldsymbol{w}})}{D_X}:=F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}\/ F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_{m-1}{D_X}$. \nFor a left ideal $I\\subset D_X$, the {\\it initial ideal} $\\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(I)$ of $I$ is the left ideal of \n$\\mathop{\\mathrm{Gr}}\\nolimits^{({\\boldsymbol{v}},{\\boldsymbol{w}})}{D_X}$ generated by all initial forms of elements in $I$. \nA finite subset $G$ of $D_X$ is called {\\it Gr\\\"obner basis} of $I$ with respect to $({\\boldsymbol{v}},{\\boldsymbol{w}})$ if $I$ is generated by $G$ and \n$\\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(I)$ is generated by initial forms of elements in $G$. \n\\end{Definition}\nIt is known that there is an algorithm for computing Gr\\\"obner bases (\\cite{SST} Algorithm 1.2.6). \nWe can compute the restriction of ideals to sub-algebras using Gr\\\"obner bases as in the commutative case. \n\\begin{Lemma}\\label{elimination of variables}\nLet $Z$ be a subsystem of $({\\boldsymbol{x}},\\partial_{\\boldsymbol{x}})$, and ${\\mathbb{C}}\\langle Z \\rangle$ a sub-algebra of $D_X$ generated by $Z$ over ${\\mathbb{C}}$. \nLet $({\\boldsymbol{v}},{\\boldsymbol{w}})$ be a weight vector such that $v_i>0$ (resp. $w_j>0$) if $x_i$ (resp. $\\partial_{x_j}$) is not a member of $Z$, \nand $v_i=0$ (resp. $w_j=0$) otherwise. \nLet $I$ be a left ideal of $D_X$ and $G$ a Gr\\\"obner basis of with respect to $({\\boldsymbol{v}},{\\boldsymbol{w}})$, \nthen $G\\cap {\\mathbb{C}}\\langle Z\\rangle$ is a system of generators of the left ideal $I\\cap {\\mathbb{C}}\\langle Z\\rangle$. \n\\end{Lemma}\nWe can also compute the intersection of ideals using elimination of variables as in the commutative case. \n\\begin{Lemma}\\label{intersection}\nLet $I$ and $J$ be left ideals of $D_X$. Then \n\\[\nI\\cap J=D_X[u](u I+(1-u)J)\\cap D_X. \n\\]\n\\end{Lemma}\n\\begin{proof}\nIf $P\\in I\\cap J$, then $P=uP+(1-u)P\\in D_X[u](u I+(1-u)J)\\cap D_X$. \nLet $P \\in D_X[u](u I+(1-u)J)\\cap D_X$. By substituting $1$ and $0$ to $u$, we have $P\\in I$ and $P\\in J$. \n\\end{proof}\nNote that substituting some $p\\in D_X$ to the variable $u$ makes sense only when $p$ is in the center of $D_X$, that is, $p\\in{\\mathbb{C}}$. \nIn this case, the left ideal of $D_X[u]$ generated by $u-p$ is a two-side ideal. \n\nFrom now on, we assume that the weight vector $({\\boldsymbol{v}},{\\boldsymbol{w}})$ satisfies \n\\[\nv_i+w_i=0 \\mathrm{~for~}i=1,2,\\dots,d. \n\\]\nThen $D_X$ has a structure of a graded algebra: \nWe set \n\\[\n[D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m:=\\{\\sum_{{\\boldsymbol{v}}\\cdot \\boldsymbol{\\mu}+{\\boldsymbol{w}}\\cdot \\boldsymbol{\\nu}= m} \na_{\\boldsymbol{\\mu\\nu}}{\\boldsymbol{x}}^{\\boldsymbol{\\mu}}{\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}}\\mid a_{\\boldsymbol{\\mu\\nu}}\\in {\\mathbb{C}}\\}\\subset D_X. \n\\]\nThen $F^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m{D_X}=\\bigoplus_{k\\le m}[D_X]_k^{({\\boldsymbol{v}},{\\boldsymbol{w}})}$ and $\\mathop{\\mathrm{Gr}}\\nolimits_m^{({\\boldsymbol{v}},{\\boldsymbol{w}})}{D_X}\\cong [D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m$ \nsince the commutation rules of $D_X$ are homogeneous of weight $0$. \nHence $D_X$ is a graded algebra $D_X=\\bigoplus_{m\\in {\\mathbb{Z}}} [D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m$ and isomorphic to $\\mathop{\\mathrm{Gr}}\\nolimits^{({\\boldsymbol{v}},{\\boldsymbol{w}})}{D_X}$. \nIn particular $[D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_0$ is a sub-ring of $D_X$. We call an element in \n$[D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m$ a {\\it homogeneous element} of degree $m$. \nA left ideal $J$ of $D_X$ is called a {\\it homogeneous ideal} if $J$ is generated by homogeneous elements. \n\n\\begin{Definition}\nFor $P=\\sum P_m\\in D_X$ with $P_m\\in [D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m$ and $m_0=\\mathop{\\mathrm{ord}}\\nolimits_{({\\boldsymbol{v}},{\\boldsymbol{w}})}(P)$, \nwe define the homogenization of $P$ with homogenizing variable $u_1$ to be \n\\[\nP^h=\\sum P_mu_1^{m_0-m}\\in D_X[u_1]. \n\\]\nwhere For a left ideal $J$ of $D_X$, we define the homogenization of $J$ to be the left ideal of $D_X[u_1]$ \n\\[\nJ^h=\\langle P^h\\mid P\\in J\\rangle. \n\\]\n\\end{Definition}\n\\begin{Definition}\nFor a left ideal $J$ of $D_X$, we set \n\\[\nJ^*=J^h\\cap D_X=\\bigoplus_{m\\in {\\mathbb{Z}}} (J\\cap [D_X]^{({\\boldsymbol{v}},{\\boldsymbol{w}})}_m), \n\\]\nthe ideal of $D_X$ generated by all homogeneous elements in $J$. \n\\end{Definition}\n\\begin{Lemma}\\label{homogeneous part}\nLet $J=\\langle P_1,\\dots,P_r\\rangle$ be a left ideal of $D_X$. Then \n\\[\nJ^*=D_X[u_1,u_2]\\langle P_1^h,\\dots,P_r^h,u_1u_2-1\\rangle\\cap D_X. \n\\]\n\\end{Lemma}\n\\begin{proof}\nIt is easy to see that \n\\[\nJ^h=(D_X[u_1,u_1^{-1}]\\langle P_1^h,\\dots,P_r^h\\rangle) \\cap D_X[u_1]=\\langle P_1^h,\\dots,P_r^h,u_1u_2-1\\rangle\\cap D_X[u_1]. \n\\]\nSince $J^*=J^h\\cap D_X$, we obtain the assertion. \n\\end{proof}\n\\subsection{Bernstein-Sato polynomials}\nBudur--Musta\\c{t}\\u{a}--Saito introduced generalized Bernstein-Sato polynomials (or $b$-function) of arbitrary varieties in \\cite{BMS} \nand proved relations between generalized Bernstein-Sato polynomials and multiplier ideals \nusing the theory of the $V$-filtration of Kashiwara and Malgralge. \n\nLet $X$ be the affine space ${\\mathbb{C}}^{n}$ with the coordinate ring ${\\mathbb{C}}[{\\boldsymbol{x}}]={\\mathbb{C}}[x_1,\\dots,x_n]$, and fix \nan ideal ${\\mathfrak{a}}$ of ${\\mathbb{C}}[{\\boldsymbol{x}}]$ with a system of generators ${\\boldsymbol{f}} =(f_1,\\dots,f_r)$. \nLet $Y=X\\times {\\mathbb{C}}^r$ be the affine space ${\\mathbb{C}}^{n+r}$ with the coordinate system $({\\boldsymbol{x}},{\\boldsymbol{t}}) = (x_1,\\dots, x_n,t_1,\\dots, t_r)$. \nThen $X\\times\\{0\\}=V(t_1,\\dots,t_r)\\cong X$ is a linear subspace of $Y$ with the defining ideal \n$I_{X\\times\\{0\\}}=\\langle t_1,\\dots,t_r\\rangle$. \nWe denote the rings of differential operators of $X$ and $Y$ by \n\\begin{eqnarray*}\nD_X&=&{\\mathbb{C}}\\langle {\\boldsymbol{x}}, {\\partial_{\\boldsymbol{x}}}\\rangle={\\mathbb{C}}\\langle x_1,\\dots,x_n,\\partial_{x_1},\\dots,\\partial_{x_n}\\rangle, \\\\\nD_Y&=&{\\mathbb{C}}\\langle {\\boldsymbol{x}},{\\boldsymbol{t}},{\\partial_{\\boldsymbol{x}}},{\\partial_{\\boldsymbol{t}}}\\rangle={\\mathbb{C}}\\langle x_1,\\dots, x_n, t_1,\\dots, t_r,\\partial_{x_1},\\dots,\\partial_{x_n},\n\\partial_{t_1},\\dots,\\partial_{t_r}\\rangle. \n\\end{eqnarray*}\nWe use the notation \n${\\boldsymbol{x}}^{\\boldsymbol{\\mu}_1}=\\prod_{i=1}^n x_i^{{\\mu}_{1i}}$, \n${\\boldsymbol{t}}^{\\boldsymbol{\\mu}_2}=\\prod_{j=1}^r t_j^{{\\mu}_{2i}}$, \n${\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}_1}=\\prod_{i=1}^n \\partial_{x_i}^{{\\nu}_{1i}}$, \nand ${\\partial_t}^{\\boldsymbol{\\nu}_2}=\\prod_{j=1}^r\\partial_{t_j}^{{\\nu}_{2j}}$ \nfor $\\boldsymbol{\\mu}_1 = ({\\mu}_{11}, \\dots , {\\mu}_{1n})$, $\\boldsymbol{\\nu}_1=({\\nu}_{11},\\dots,{\\nu}_{1n})\\in{\\mathbb{Z}}_{\\ge 0}^n$ \nand $\\boldsymbol{\\mu}_2 = ({\\mu}_{21},\\dots , {\\mu}_{2r})$, $\\boldsymbol{\\nu}_2=({\\nu}_{21},\\dots,{\\nu}_{2r}) \\in {\\mathbb{Z}}_{\\ge 0}^r$. \nThe ${\\mathbb{C}}[{\\boldsymbol{x}}]$-module $N_{{\\boldsymbol{f}}}:={\\mathbb{C}}[{\\boldsymbol{x}}][\\prod_i f_i^{-1},s_1,\\dots,s_r]\\prod_i f_i^{s_i}$, \nwhere $s_i$'s are independent variables and $\\prod_i f_i^{s_i}$ is a symbol, has a $D_X$-module structure as follows: \nThe action of ${\\mathbb{C}}[{\\boldsymbol{x}}]$ on $N_{{\\boldsymbol{f}}}$ is given by the canonical one, and the action of $\\partial_{x_i}$ are given by\n\\[\n\\partial_{x_j} (h\\prod_i f_i^{s_i})=\\biggl(\\partial_{x_j}(h)+h\\sum_{k=1}^{r} s_j\\frac{\\partial_{x_j}(f_k)}{f_k}\\biggr)\\prod_i f_i^{s_i}.\n\\] \nfor $h\\in {\\mathbb{C}}[{\\boldsymbol{x}}][\\prod_i f_i^{-1},s_1,\\dots,s_r]$. \nThis action is defined formally, but it has an obvious meaning when some integers are substituted for $s_i$'s. \nWe define $D_X$-linear actions $t_j$ and $\\partial_{t_j}$ on $N_{{\\boldsymbol{f}}}$ by \n\\[\nt_j(h(x,s_1,\\dots,s_r)\\prod_i f_i^{s_i})=h(x,s_1,\\dots,s_j+1,\\dots,s_r)f_j\\prod_i f_i^{s_i}\n\\] \nand \n\\[\n\\partial_{t_j}(h(x,s_1,\\dots,s_r)\\prod_i f_i^{s_i})=-s_jh(x,s_1,\\dots,s_j-1,\\dots,s_r)f_j^{-1}\\prod_i f_i^{s_i}. \n\\]\nfor $h(x,s_1,\\dots,s_r)\\in {\\mathbb{C}}[{\\boldsymbol{x}}][\\prod_i f_i^{-1},s_1,\\dots,s_r]$. \nThen it follows that $N_{{\\boldsymbol{f}}}$ is a $D_Y$-module because the actions defined above respect the commutation rules of $D_Y$. \nNote that $-\\partial_{t_i}{t_i} \\prod_i f_i^{s_i}=s_i\\prod_i f_i^{s_i}$ for all $i$. \n\\begin{Definition}[\\cite{BMS}]\\label{def of b-functions}\nLet $\\sigma =-(\\sum_i \\partial_{t_i}{t_i})$, and let $s$ be a new variable. \nThen the global generalized Bernstein-Sato polynomial $b_{{\\mathfrak{a}},g}(s)\\in {\\mathbb{C}} [s]$ of ${\\mathfrak{a}}=\\langle f_1,\\dots,f_r\\rangle$ and \n$g\\in {\\mathbb{C}}[{\\boldsymbol{x}}]$ is defined to be the monic polynomial of minimal degree satisfying \n\\begin{equation}\\label{exp1}\nb(\\sigma)g\\prod_i f_i^{s_i}=\\sum_{j=1}^{r}P_jgf_j\\prod_i f_i^{s_i}\n\\end{equation}\nfor some $P_j\\in D_X\\langle -\\partial_{t_j}t_k \\mid 1\\le j,k \\le r\\rangle$. \nWe define $b_{\\mathfrak{a}}(s)=b_{{\\mathfrak{a}},1}(s)$. \n\nFor a prime ideal ${\\mathfrak{p}}$ of ${\\mathbb{C}}[{\\boldsymbol{x}}]$, we define the local generalized Bernstein-Sato polynomial $b^{\\mathfrak{p}}_{{\\mathfrak{a}},g}(s)$ at ${\\mathfrak{p}}$ \nto be the monic polynomial of minimal degree satisfying \n\\begin{equation*}\nb(\\sigma)gh\\prod_i f_i^{s_i}=\\sum_{j=1}^{r}P_jgf_j\\prod_i f_i^{s_i}\n\\end{equation*}\nfor some $P_j\\in D_X\\langle -\\partial_{t_j}t_k \\mid 1\\le j,k \\le r\\rangle$ and $h \\not\\in {\\mathfrak{p}}$. \nWe define $b^{\\mathfrak{p}}_{\\mathfrak{a}}(s)=b^{\\mathfrak{p}}_{{\\mathfrak{a}},1}(s)$. \n\\end{Definition}\nNote that ${\\mathbb{C}}[x]_{\\mathfrak{p}} \\otimes_{{\\mathbb{C}}[x]} D_X$ is the ring of differential operators of $\\mathop{\\mathrm{Spec}}\\nolimits {\\mathbb{C}}[{\\boldsymbol{x}}]_{\\mathfrak{p}}$. \nIt is proved in \\cite{BMS} that generalized Bernstein-Sato polynomials are well-defined, \nthat is, they do not depend on the choice of generators of ${\\mathfrak{a}}$, and all their roots are negative rational numbers. \nThese facts follow from the theory of $V$-filtrations of Kashiwara \\cite{Ka} and Malgrange \\cite{M}. \nWhen ${\\mathfrak{a}}$ is a principal ideal generated by $f$, then $b_{\\mathfrak{a}}(s)$ coincides with the classical Bernstein-Sato polynomial $b_f(s)$ of $f$. \n\\subsection{V-filtrations}\nWe will briefly recall the definition and some basic properties of $V$-filtrations. \nSee \\cite{M}, \\cite{Ka}, \\cite{Sab} and \\cite{BMS} for details. \n\nWe fix the weight vector $({\\boldsymbol{w}},-{\\boldsymbol{w}})\\in {\\mathbb{Z}}^{2(n+r)}$, ${\\boldsymbol{w}}=((0,\\dots,0),(1,\\dots,1))\\in {\\mathbb{Z}}^{n}\\times{\\mathbb{Z}}^{r}$, that is, \nwe assign the weight $1$ to $\\partial_{t_j}$, $-1$ to $t_j$, and $0$ to $x_i$ and $\\partial_{x_i}$. \nThen \n\\[\nF_{m}^{({\\boldsymbol{w}},-{\\boldsymbol{w}})} D_Y=\\{\\sum_{-|\\boldsymbol{\\mu}_2|+|\\boldsymbol{\\nu}_2|\\le m} \na_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}{\\boldsymbol{x}}^{\\boldsymbol{\\mu}_1}{\\boldsymbol{t}}^{\\boldsymbol{\\mu}_2}{\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}_1}{\\partial_{\\boldsymbol{t}}}^{\\boldsymbol{\\nu}_2}\n\\mid a_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}\\in {\\mathbb{C}}\\}\n\\]\nIn this paper, we call the decreasing filtration $V^{m}D_Y:=F_{-m}^{({\\boldsymbol{w}},-{\\boldsymbol{w}})} D_Y$ on $D_Y$ \nthe {\\it V-filtration} of $D_Y$ along $X\\times\\{0\\}$ \n(some author call the increasing filtration $F^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}$ the $V$-filtration). \nNote that \n\\begin{eqnarray*}\nV^m{D_Y}&=&\\{\\sum_{|\\boldsymbol{\\mu}_2|-|\\boldsymbol{\\nu}_2|\\ge m} \na_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}{\\boldsymbol{x}}^{\\boldsymbol{\\mu}_1}{\\boldsymbol{t}}^{\\boldsymbol{\\mu}_2}{\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}_1}{\\partial_{\\boldsymbol{t}}}^{\\boldsymbol{\\nu}_2}\\mid \na_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}\\in {\\mathbb{C}}\\}\\\\\n&=&\\{P \\in D_Y \\mid P(I_{X\\times\\{0\\}} )^j \\subset (I_{X\\times\\{0\\}} )^{j+m} ~\\mathrm{for~~any}~ j \\ge 0\\},\n\\end{eqnarray*}\nwith the convention $I_{X\\times\\{0\\}}^{j}={\\mathbb{C}}[{\\boldsymbol{x}},{\\boldsymbol{t}}]$ for all $j\\le 0$. \n\\begin{Definition}\\label{Vfiltration}\nThe $V$-filtration along $X\\times\\{0\\}$ on a finitely generated left $D_Y$-module $M$ is an exhaustive decreasing filtration \n$\\{V^{{\\alpha}}M\\}_{\\alpha\\in \\Q}$ indexed by $\\Q$, such that: \n\\\\ {\\rm(i)} $V^{{\\alpha}}M$ are finitely generated $V^0D_Y$-submodules of $M$. \n\\\\ {\\rm(ii)} \n$\\{V^{{\\alpha}}M\\}_{\\alpha}$ is left-continuous and discrete, that is, \n$V^{\\alpha} M=\\bigcap_{{\\alpha}'<{\\alpha}}V^{{\\alpha}'}M$, and every interval contains only finitely many ${\\alpha}\\in \\Q$ \nwith $\\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha} M\\neq 0$. \nHere $\\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha} M := V^{{\\alpha}} M\/(\\bigcup_{{\\alpha}'>{\\alpha}} V^{{\\alpha}'}M)$. \n\\\\ {\\rm(iii)} \n$(V^iD_Y)(V ^{{\\alpha}}M) \\subset V^{{\\alpha}+i}M$ for any $i \\in {\\mathbb{Z}}$, ${{\\alpha}} \\in \\Q$.\n\\\\ {\\rm(iv)} \n$(V^iD_Y)(V ^{{\\alpha}}M) = V^{{\\alpha}+i}M$ for any $i > 0$ if ${{\\alpha}}\\gg 0$.\n\\\\ {\\rm(v)} \nthe action of $\\sigma+{\\alpha}$ is nilpotent on $\\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha} M$. \n\\end{Definition}\n\\begin{Remark}\\label{b from V}\n(i) The filtration $V$ is unique if it exists (\\cite{Ka}), \nand $D_Y$-submodule $D_Y \\prod_i f_i^{s_i}\\cong D_Y\/\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i}$ of $N_{{\\boldsymbol{f}}}$ has such a $V$-filtration (see \\cite{BMS}). \\\\\n(ii) For ${\\alpha}\\neq z\\in {\\mathbb{C}}$, the action of $\\sigma+z$ on $\\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha} M$ is invertible. \nHence, if $\\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha} M\\neq 0$, $u \\not\\in V^{{\\alpha} +\\varepsilon}M$ and $b(\\sigma)u \\in V^{{\\alpha} +\\varepsilon}M$ \nfor some $b(s)\\in {\\mathbb{C}}[s]$ and all sufficiently small $\\varepsilon >0$, then $s+{\\alpha}$ is a factor of $b(s)$. \n\\end{Remark}\nLet $\\iota : X \\rightarrow Y$ be the graph embedding $x \\mapsto (x, f_1(x),\\dots, f_r(x))$ of ${\\boldsymbol{f}} =(f_1,\\dots,f_r)$, \nand $M_{{\\boldsymbol{f}}}=\\iota_+{\\mathbb{C}}[{\\boldsymbol{x}}]$, where $\\iota_+$ denotes the direct image for left $D$-modules. \nThere is a natural isomorphism $M_{{\\boldsymbol{f}}} \\cong {\\mathbb{C}}[{\\boldsymbol{x}}]\\otimes_{{\\mathbb{C}}}{\\mathbb{C}}[\\partial_{t_1},\\dots, \\partial_{t_r}]$ (see \\cite{Bo}), \nand the action of ${\\mathbb{C}}[{\\boldsymbol{x}}]$ and $\\partial_{t_1}, \\dots, \\partial_{t_r}$ on $M_{{\\boldsymbol{f}}}$ is given by the canonical one, \nand the action of a vector field $\\xi$ on $X$ and $t_j$ are given by\n\\begin{eqnarray*}\n\\xi (g \\otimes {\\partial_t}^{\\nu} ) &=& \\xi g \\otimes {\\partial_t}^{\\nu} -\\sum_j (\\xi f_j)g\\otimes \\partial_{t_j} {\\partial_t}^{{\\nu}},\\\\\nt_j(g\\otimes {\\partial_t}^{\\nu}) &=& f_jg \\otimes {\\partial_t}^{\\nu} - {\\nu}_j g \\otimes {\\partial_t}^{{\\nu}-1_j}.\n\\end{eqnarray*}\nwhere $1_j$ is the element of ${\\mathbb{Z}}^r$ whose $i$-th component is $1$ if $i=j$ and $0$ otherwise. \n\\begin{Definition}[\\cite{BMS}]\nLet M be a $D_Y$-module with $V$-filtration. \nFor $u\\in M$, the Bernstein-Sato polynomial $b_u(s)$ of $u$ is the monic minimal polynomial\nof the action of $\\sigma$ on $V^0D_Yu\/V^1D_Yu$. \n\\end{Definition}\nBy the properties of $V$-filtration in Definition \\ref{Vfiltration}, \nthe induced filtration $V$ on $(V^0D_Y)u\/(V^1D_Y)u$ is finite (see \\cite{BMS} Section 2.1) \nThis guarantees the existence of $b_{u}(s)$. \nIf $u\\in V^{\\alpha} M$, than $V^0D_Yu \\subset V^{\\alpha} M$ and $V^1D_Yu \\subset V^{{\\alpha}+1} M$. \nHence, if we set ${\\alpha}_0=\\mathop{\\mathrm{max}}\\nolimits\\{{\\alpha} \\mid u \\in V^{\\alpha} M\\}$, \nthen $u\\not\\in V^{{\\alpha}_0+\\varepsilon}M$ and $b_u(\\sigma)u\\in V^1D_Yu \\subset V^{{\\alpha}_0+1} M\\subset V^{{\\alpha}_0+\\varepsilon}M$ \nfor sufficiently small $\\varepsilon >0$. Hence \n\\[\n\\mathop{\\mathrm{max}}\\nolimits\\{{\\alpha} \\mid u \\in V^{\\alpha} M\\} = \\min\\{{\\alpha} \\mid \\mathop{\\mathrm{Gr}}\\nolimits_V^{\\alpha}((V^0D_Y)u) \\neq 0\\} \n \\min\\{{\\alpha} \\mid b_u(-{\\alpha}) = 0\\}. \n\\]\nTherefore we conclude the next proposition. \n\\begin{Proposition}[\\cite{Sab}]\nLet M be a $D_Y$-module with $V$-filtration. \nThen \n\\[\nV^{\\alpha} M = \\{ u \\in M \\mid {\\alpha}\\le {\\alpha}' \\mbox{~~if~~} b_u(-{\\alpha}') = 0 \\}.\n\\]\n\\end{Proposition}\nSince we have a canonical injection $M_{{\\boldsymbol{f}}} \\rightarrow N_{{\\boldsymbol{f}}}={\\mathbb{C}}[{\\boldsymbol{x}}][\\prod_i f_i^{-1},s_1,\\dots,s_r]\\prod_i f_i^{s_i}$ \nthat sends $g\\otimes {\\partial_t}^{\\nu}$ to $g{\\partial_t}^{\\nu}\\prod_i f_i^{s_i}$, \nthe generalized Bernstein-Sato polynomial $b_{{\\mathfrak{a}},g}(s)$ coincides with $b_u(s)$ where $u=g\\otimes 1 \\in M_{{\\boldsymbol{f}}}$ \n(see Observation \\ref{obs} in the next section). \n\\subsection{Multiplier ideals}\nWe will recall the relations between generalized Bernstein-Sato polynomials and multiplier ideals following \\cite{BMS}. \nThe reader is referred to \\cite{L} for general properties of multiplier ideals. \nFor a positive rational number $c$, the multiplier ideal $\\J({\\mathfrak{a}}^c)$ is defined via a log resolution of ${\\mathfrak{a}}$. \nLet $\\pi: \\tilde{X}\\to X=\\mathop{\\mathrm{Spec}}\\nolimits {\\mathbb{C}}[{\\boldsymbol{x}}]$ be a log resolution of ${\\mathfrak{a}}$, \nnamely, $\\pi$ is a proper birational morphism, $\\tilde{X}$ is smooth, \nand there exists an effective divisor $F$ on $\\tilde{X}$ such that ${\\mathfrak{a}}{\\mathcal{O}}_{\\tilde{X}}={\\mathcal{O}}_{\\tilde{X}}(-F)$ \nand the union of the support of $F$ and the exceptional divisor of $\\pi$ has simple normal crossings. \nFor a given real number $c\\ge 0$, the multiplier ideal $\\J({\\mathfrak{a}}^c)$ associated to $c$ is defined to be the ideal \n\\[\n\\J({\\mathfrak{a}}^c)=H^0(\\tilde{X},{\\mathcal{O}}_{\\tilde{X}}( K_{\\tilde{X}\/X}-\\lfloor cF \\rfloor))\n\\]\nwhere $K_{\\tilde{X}\/X}=K_{\\tilde{X}}-\\pi^*K_{X}$ is the relative canonical divisor of $\\pi$. \nThis definition is independent of the choice of a log resolution $\\pi:\\tilde{X}\\to X$. \nThe reader is referred to \\cite{L} for general properties of multiplier ideals. \nBy the definition, if $c0$. \nThe multiplier ideals give a decreasing filtration on ${\\mathcal{O}}_X$, \nand there are rational numbers $0 =c_0< c_1 < c_2 < \\cdots$ such that \n$\\J ({\\mathfrak{a}}^{c_j}) = \\J({\\mathfrak{a}}^c) \\neq \\J({\\mathfrak{a}}^{c_{j+1}})$ for $c_j \\leq c< c_{j+1}$. \nThese $c_j$ for $j > 0$ are called the {\\it jumping coefficients}, \nand the minimal jumping coefficient $c_1$ is called the {\\it log-canonical threshold} of ${\\mathfrak{a}}$ and denoted by $\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}})$. \nBy the definition, it follows that multiplier ideals are integrally closed, \nand the multiplier ideal associated to the log canonical threshold is radical. \nIt is known that $\\J({\\mathfrak{a}}^{c})={\\mathfrak{a}}\\J({\\mathfrak{a}}^{c-1})$ for $c\\ge \\lambda({\\mathfrak{a}})$ where $\\lambda({\\mathfrak{a}})$ is the analytic spread of ${\\mathfrak{a}}$. \nRecall that analytic spread of ${\\mathfrak{a}}$ is the minimal number of elements needed to generate ${\\mathfrak{a}}$ up to integral closure, \nand thus $\\mu({\\mathfrak{a}})\\ge\\lambda({\\mathfrak{a}})$ where $\\mu({\\mathfrak{a}})$ is the minimal number of generators of ${\\mathfrak{a}}$. \nIn particular, if ${\\mathfrak{a}}$ is a principal ideal generated by $f$, then $\\J(f^c)=f\\J(f^{c-1})$ for $c\\ge 1$. \n\nBudur--Musta\\c{t}\\u{a}--Saito proved that the $V$-filtration on ${\\mathbb{C}}[x]$ is essentially equivalent to the filtration by multiplier ideals \nusing the theory of mixed Hodge modules (\\cite{Sa1}, \\cite{Sa2}), \nand gave a description of multiplier ideals in terms of generalized Bernstein-Sato polynomials. \n\\begin{Theorem}[\\cite{BMS}]\\label{multiplier ideals and b-functions}\nWe denote by $V$ the filtration on ${\\mathbb{C}}[{\\boldsymbol{x}}]\\cong {\\mathbb{C}}[{\\boldsymbol{x}}]\\otimes 1$ induced by the $V$-filtration on $\\iota_+ {\\mathbb{C}}[{\\boldsymbol{x}}]$. \nThen $\\J({\\mathfrak{a}}^c) = V^{c+\\varepsilon}{\\mathbb{C}}[{\\boldsymbol{x}}]$ and $V^{\\alpha} {\\mathbb{C}}[{\\boldsymbol{x}}] = \\J ({\\mathfrak{a}}^{{\\alpha}-\\varepsilon})$ \nfor any ${\\alpha}\\in \\Q$ and $0<\\varepsilon \\ll 1$. \nTherefore the following hold: \\\\\n{\\rm (i)} For a given rational number $c\\ge 0$ and a prime ideal ${\\mathfrak{p}}\\subset {\\mathbb{C}}[x]$, \n\\begin{eqnarray*}\n\\J({\\mathfrak{a}}^c) &=& \\{g\\in {\\mathbb{C}}[{\\boldsymbol{x}}] \\mid c < c' \\mbox{~~if~~} b_{{\\mathfrak{a}},g}(-c') = 0\\}, \\\\\n\\J({\\mathfrak{a}}^c)_{\\mathfrak{p}} \\cap {\\mathbb{C}}[{\\boldsymbol{x}}] &=& \\{g\\in {\\mathbb{C}}[{\\boldsymbol{x}}] \\mid c < c' \\mbox{~~if~~} b^{\\mathfrak{p}}_{{\\mathfrak{a}},g}(-c') = 0\\}. \n\\end{eqnarray*}\nIn particular, the log canonical threshold $\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}})$ of ${\\mathfrak{a}}=\\langle f_1,\\dots,f_r\\rangle$ is the minimal root of $b_{\\mathfrak{a}}(-s)$. \\\\\n{\\rm (ii)} All jumping coefficients of ${\\mathfrak{a}}$ in $[\\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}}), \\mathop{\\mathrm{lct}}\\nolimits({\\mathfrak{a}}) + 1)$ are roots of $b_{\\mathfrak{a}}(-s)$. \n\\end{Theorem}\nTherefore an algorithm for computing generalized Bernstein-Sato polynomials induces an algorithm for \nsolving membership problem for multiplier ideals, and in particular, an algorithm for computing log canonical thresholds. \n\\section{Algorithms for computing generalized Bernstein-Sato polynomials}\nIn this section, we obtain an algorithm for computing generalized Bernstein-Sato polynomials of arbitrary ideals. \nThe algorithms for computing classical Bernstein-Sato polynomials are given by Oaku (see \\cite{O1}, \\cite{O2}, \\cite{O3}). \nWe will generalize Oaku's algorithm to arbitrary $n$ and $g$. \n\nLet ${\\mathbb{C}}[{\\boldsymbol{x}}]={\\mathbb{C}}[x_1,\\dots,x_n]$ be a polynomial ring over ${\\mathbb{C}}$, \nand ${\\mathfrak{a}}$ an ideal with a system of generators ${\\boldsymbol{f}} =(f_1,\\dots,f_r)$, \nand we fix the weight vector $({\\boldsymbol{w}},-{\\boldsymbol{w}})\\in {\\mathbb{Z}}^{n+r}\\times{\\mathbb{Z}}^{n+r}$, ${\\boldsymbol{w}}=((0,\\dots,0),(1,\\dots,1))\\in {\\mathbb{Z}}^{n}\\times{\\mathbb{Z}}^{r}$. \n\\begin{Observation}\\label{obs}\nWe rewrite the definition of generalized Bernstein-Sato polynomials in several ways. \n\nRecall that $b_{{\\mathfrak{a}},g}(s)$ is the monic polynomial of minimal degree satisfying \n\\[\nb(\\sigma)g\\prod_i f_i^{s_i}=\\sum_{j=1}^{r}P_jgf_j\\prod_i f_i^{s_i}\n\\]\nfor some $P_j\\in D_X\\langle -\\partial_{t_j}t_k \\mid 1\\le j,k \\le r\\rangle$ (Definition \\ref{def of b-functions} (\\ref{exp1}))\nSince \n\\begin{eqnarray*}\n[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0&=&\\{\\sum_{|\\boldsymbol{\\nu}_2|-|\\boldsymbol{\\mu}_2|=0} \na_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}{\\boldsymbol{x}}^{\\boldsymbol{\\mu}_1}{\\boldsymbol{t}}^{\\boldsymbol{\\mu}_2}{\\partial_{\\boldsymbol{x}}}^{\\boldsymbol{\\nu}_1}{\\partial_{\\boldsymbol{t}}}^{\\boldsymbol{\\nu}_2}\n\\mid a_{\\boldsymbol{\\mu}_1 \\boldsymbol{\\mu}_2 \\boldsymbol{\\nu}_1 \\boldsymbol{\\nu}_2}\\in {\\mathbb{C}}\\}\\\\\n&=& D_X\\langle -\\partial_{t_j}t_k \\mid 1\\le j,k \\le r\\rangle, \n\\end{eqnarray*} \nand $\\sigma =-\\sum \\partial_{t_i}t_i$ is a homogeneous element of degree $0$,\nthe condition (\\ref{exp1}) is equivalent to saying that \n\\begin{eqnarray*}\n&&(b(\\sigma)g-\\sum_{j=1}^{r}P_jgf_j)\\prod_i f_i^{s_i}=0 \\mbox{\\quad for~~} {}^\\exists P_j\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0 \\\\\n&\\Longleftrightarrow&\nb(\\sigma)g-\\sum_{j=1}^{r}P_jgf_j\\in \\mathop{\\mathrm{Ann}}\\nolimits_{[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0}\\prod_i f_i^{s_i} \\mbox{\\quad for~~} {}^\\exists P_j\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0 \n\\nonumber\\\\\n&\\Longleftrightarrow&\nb(\\sigma)g\\in \\mathop{\\mathrm{Ann}}\\nolimits_{[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0}\\prod_i f_i^{s_i}+[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0 g{\\mathfrak{a}}. \n\\end{eqnarray*}\nSince $(I+J)\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0=I\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0+J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$ for homogeneous ideals $I$ and $J$, \nwe have \n\\begin{eqnarray*}\n&&\\mathop{\\mathrm{Ann}}\\nolimits_{[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0}\\prod_i f_i^{s_i}+[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0 g{\\mathfrak{a}}\\\\\n=&& (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0+(D_Y g{\\mathfrak{a}})\\cap[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0 \\\\\n=&&((\\mathop{\\mathrm{Ann}}\\nolimits_{[D_Y]}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}})\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0. \\\\\n\\end{eqnarray*}\nHence the condition (\\ref{exp1}) is equivalent to saying that \n\\begin{equation}\\label{exp2}\nb(\\sigma)g\\in (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}}. \n\\end{equation}\nSince $t_j\\prod_i f_i^{s_i}=f_j\\prod_i f_i^{s_i}$, the condition (\\ref{exp1}) is also equivalent to saying that \n\\begin{eqnarray}\n\\label{exp3}&&b(\\sigma)g\\prod_i f_i^{s_i}\\in (V^{1}{D_Y})g\\prod_i f_i^{s_i}=(F^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_{-1}{D_Y})g\\prod_i f_i^{s_i} \\\\\n\\label{exp4}\\Longleftrightarrow && b(\\sigma) \\in \\mathop{\\mathrm{in}}\\nolimits_{(-w,w)}(\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y} g\\prod_i f_i^{s_i}) \\\\\n\\label{exp5}\\Longleftrightarrow && b(\\sigma)g \\in \\mathop{\\mathrm{in}}\\nolimits_{(-w,w)}((\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y} \\prod_i f_i^{s_i})\\cap D_Yg). \n\\end{eqnarray}\nBy the expression (\\ref{exp3}), the generalized Bernstein-Sato polynomial $b_{{\\mathfrak{a}},g}(s)$ coincides with $b_u(s)$ where $u=g\\otimes 1 \\in M_{{\\boldsymbol{f}}}$. \n\\end{Observation}\nBy (\\ref{exp4}), the polynomial $b_{{\\mathfrak{a}},g}(-s-r)$ coincides with the $b$-function for $\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}g\\prod_i f_i^{s_i}$ \nwith the weight vector $({\\boldsymbol{w}},-{\\boldsymbol{w}})$ in \\cite{SST}, p.194. \nIn the case $g=1$, one can compute $b_{\\mathfrak{a}}(s)$ using loc. cit., p.196, Algorithm 5.1.6 by the next lemma. \n\\begin{Lemma}\\label{annfs}\n\\[\n\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y} \\prod_i f_i^{s_i}=\\langle t_i-f_i\\mid 1\\le i \\le r\\rangle + \\langle \\partial_{x_j}\n+\\sum_{i=1}^{r} \\partial_{x_j}(f_i)\\partial_{t_i}\\mid 1\\le j \\le n\\rangle. \n\\]\n\\end{Lemma}\n\\begin{proof}\nOne can prove the assertion similarly to the case $n=1$. See \\cite{SST} Lemma 5.3.3. \n\\end{proof}\nBy this lemma and Lemma \\ref{homogeneous part}, the homogeneous left ideals \n\\[\n(\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}},\\mbox{~~and~~} \\mathop{\\mathrm{in}}\\nolimits_{(-w,w)}((\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y} \\prod_i f_i^{s_i})\\cap D_Yg),\n\\] \nin (\\ref{exp2}) and (\\ref{exp5}) are computable. \nTherefore we can calculate generalized Bernstein-Sato polynomials if we obtain an algorithm for computing the ideal \n$\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[{\\boldsymbol{x}},s]\\mid b({\\boldsymbol{x}},\\sigma)\\in J \\}\\cong J \\cap {\\mathbb{C}}[x,\\sigma]$ for a given homogeneous ideal $J\\subset D_Y$. \nOne can compute this in the same way as \\cite{SST} Algorithm 5.1.6. \nThe algorithm calculates $J'=J\\cap {\\mathbb{C}}[x,\\sigma_1,\\dots,\\sigma_r]$ first where $\\sigma_i=-\\partial_{t_i}t_i$, \nthen computes $J' \\cap {\\mathbb{C}}[x,\\sigma]$. This algorithm requires $2r$ new variables. \nWe will give an algorithm for computing $J \\cap {\\mathbb{C}}[x,\\sigma]$ without computing $J'$. \n\\begin{Lemma}\\label{substitution}\nLet $J$ be a homogeneous left ideal of $D_Y$. The following hold: \\\\\n{\\rm(i)} $\\sigma $ is in the center of $[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$.\\\\\n{\\rm(ii)} $D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0=J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$.\\\\\n{\\rm(iii)} $\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[{\\boldsymbol{x}},s]\\mid b({\\boldsymbol{x}},\\sigma)\\in J \\}=D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap {\\mathbb{C}}[{\\boldsymbol{x}},s]$. \n\\end{Lemma}\n\\begin{proof}\n(i) \nSince the ring $[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$ is generated by $\\partial_{t_j}{t_k}$, $1\\le j,k\\le r$, over $D_X$, \nand $\\sigma $ commutes with any element of $D_X$, \nit is enough to show that $\\sigma (\\partial_{t_j}{t_k})=(\\partial_{t_j}{t_k})\\sigma$ for all $1\\le j,k\\le r$. \n\nIn the case $j\\neq k$, we obtain \n\\begin{eqnarray*}\n(\\partial_{t_j}{t_k})(\\partial_{t_j}{t_j})&=&\\partial_{t_j}^2t_jt_k,\\quad \\quad ~~~\n(\\partial_{t_j}{t_j})(\\partial_{t_j}{t_k}) = \\partial_{t_j}^2t_jt_k-\\partial_{t_j}t_k,\\\\\n(\\partial_{t_j}{t_k})(\\partial_{t_k}{t_k})&=&\\partial_{t_j}\\partial_{t_k}t_k^2-\\partial_{t_j}t_k,~~\n(\\partial_{t_k}{t_k})(\\partial_{t_j}{t_k}) = \\partial_{t_j}\\partial_{t_k}t_k^2,\n\\end{eqnarray*}\nthus $(\\partial_{t_j}{t_k})(\\partial_{t_j}{t_j}+\\partial_{t_k}{t_k})=(\\partial_{t_j}{t_j}+\\partial_{t_k}{t_k})(\\partial_{t_j}{t_k})$. \nHence \n\\begin{eqnarray*}\n(\\partial_{t_j}{t_k})\\sigma\n&=&-(\\partial_{t_j}{t_k})(\\partial_{t_j}{t_j}+\\partial_{t_k}{t_k}+\\sum_{\\ell\\neq j,k}\\partial_{t_j\\ell}{t_\\ell})\\\\\n&=&-(\\partial_{t_j}{t_j}+\\partial_{t_k}{t_k})(\\partial_{t_j}{t_k})-(\\sum_{\\ell\\neq j,k}\\partial_{t_j\\ell}{t_\\ell})(\\partial_{t_j}{t_k})\n=\\sigma(\\partial_{t_j}{t_k}).\n\\end{eqnarray*}\nIn the case $j=k$, it is obvious that $(\\partial_{t_j}{t_j})\\sigma=\\sigma(\\partial_{t_j}{t_j})$. \nTherefore $\\sigma$ is in the center of $[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$. \\\\\n(ii) The inclusion $D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0\\supset J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$ is trivial. \nWe will show the converse inclusion. Let \n\\[\nh=\\sum P_{\\ell}s^{\\ell}+Q(s)(s-\\sigma)\\in D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0\n\\]\nwhere $P_{\\ell}\\in J$ and $Q(s) \\in D_Y[s]$. \nTaking the degree zero part, we may assume $P_{\\ell}\\in J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$ and $Q[s]\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0[s]$. \nAs $\\sum P_{\\ell}s^{\\ell}-\\sum P_{\\ell}\\sigma^{\\ell}\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0[s](s-\\sigma)$, \nthere exists $Q^{'}(s)\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0[s]$ such that \n$h=\\sum P_{\\ell}\\sigma^{\\ell}+Q^{'}(s)(s-\\sigma)$. \nSince $h\\in D_Y$, we have $Q^{'}(s)=0$. Therefore $h=\\sum P_{\\ell}\\sigma^{\\ell}=\\sum \\sigma^{\\ell}P_{\\ell}\\in J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$. \\\\\n(iii) Let $b({\\boldsymbol{x}},\\sigma)\\in J$.\nSince $b({\\boldsymbol{x}},s)-b({\\boldsymbol{x}},\\sigma)\\in \\langle s-\\sigma \\rangle$, we have \n\\[\nb({\\boldsymbol{x}},s)=b({\\boldsymbol{x}},\\sigma)+(b({\\boldsymbol{x}},s)-b({\\boldsymbol{x}},\\sigma))\\in D_Y[s](J+\\langle s-\\sigma \\rangle). \n\\] \nConversely, if $b({\\boldsymbol{x}},s)\\in D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap {\\mathbb{C}}[{\\boldsymbol{x}},s]$, then \n\\[\nb({\\boldsymbol{x}},\\sigma)=b({\\boldsymbol{x}},s)-(b({\\boldsymbol{x}},s)-b({\\boldsymbol{x}},\\sigma))\\in D_Y[s](J+\\langle s-\\sigma \\rangle). \n\\]\nSince $b({\\boldsymbol{x}},\\sigma)\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$, and by (ii) , we conclude \n\\[\nb({\\boldsymbol{x}},\\sigma)\\in D_Y[s](J+\\langle s-\\sigma \\rangle)\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0=J\\cap [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0\\subset J. \n\\]\n\\end{proof}\n\\begin{Theorem}[Algorithm for global generalized Bernstein-Sato polynomials 1]\\label{Algorithm for global b-functions 1}\nLet \n\\[\nI_{{\\boldsymbol{f}}}=\\langle t_iu_1-f_i\\mid 1\\le i \\le r\\rangle + \\langle u_1\\partial_{x_j}\n+\\sum_{i=1}^{r} \\partial_{x_j}(f_i)\\partial_{t_i}\\mid 1\\le j \\le d\\rangle+\\langle u_1u_2-1\\rangle\n\\]\nbe a left ideal of $D_Y[u_1,u_2]$. Then compute the following ideals; \\\\\n1. $I_{{\\boldsymbol{f}},1}=I_{{\\boldsymbol{f}}}\\cap D_Y$, \\\\\n2. $I_{({\\boldsymbol{f}};g),2}=D_Y[s](I_{{\\boldsymbol{f}},1}+g{\\mathfrak{a}}+\\langle s-\\sigma \\rangle)\\ \\cap {\\mathbb{C}}[{\\boldsymbol{x}},s]$, \\\\\n3. $I_{({\\boldsymbol{f}};g),3}=I_{({\\boldsymbol{f}};g),2}:g=(I_{({\\boldsymbol{f}};g),2}\\cap \\langle g \\rangle)\\cdot g^{-1}$, \\\\\n4. $I_{({\\boldsymbol{f}};g),4}=I_{({\\boldsymbol{f}};g),3}\\cap {\\mathbb{C}}[s]$. \\\\\nThen $b_{{\\mathfrak{a}},g}(s)$ is the generator of $I_{({\\boldsymbol{f}};g),4}$. \n\\end{Theorem}\n\\begin{proof}\nBy Lemma \\ref{annfs} and Lemma \\ref{homogeneous part}, $I_{{\\boldsymbol{f}},1}= (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*$. \nAs $I_{{\\boldsymbol{f}},1}+D_Y g{\\mathfrak{a}}$ is a homogeneous ideal, we have \n\\[\nI_{({\\boldsymbol{f}};g),2}=\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[{\\boldsymbol{x}},s]\\mid b({\\boldsymbol{x}},\\sigma)\\in (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}}\\}\n\\]\nby Lemma \\ref{substitution}. Since $b_{{\\mathfrak{a}},g}(s)$ is the minimal generator of the ideal \n\\[\n\\{b(s)\\in {\\mathbb{C}}[s]\\mid b(\\sigma)g({\\boldsymbol{x}})\\in (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}}\\}, \n\\]\nit follows that $I_{({\\boldsymbol{f}};g),4}=(I_{({\\boldsymbol{f}};g),2}:g)\\cap {\\mathbb{C}}[s]=\\langle b_{{\\mathfrak{a}},g}(s)\\rangle$. \n\\end{proof}\n\\begin{Theorem}[Algorithm for global generalized Bernstein-Sato polynomials 2]\\label{Algorithm for global b-functions 2}\nLet \n\\[\n\\tilde{I}_{{\\boldsymbol{f}}}=\\langle t_i-f_i\\mid 1\\le i \\le r\\rangle + \\langle \\partial_{x_j}\n+\\sum_{i=1}^{r} \\partial_{x_j}(f_i)\\partial_{t_i}\\mid 1\\le j \\le d\\rangle\\subset D_Y, \n\\]\nand compute the following ideals; \\\\\n0. $\\tilde{I}_{({\\boldsymbol{f}};g),0}=\\tilde{I}_{{\\boldsymbol{f}}}\\cap D_Yg=D_Y[u](u\\tilde{I}_{{\\boldsymbol{f}}}+(1-u)g)\\cap D_Y$, \\\\\n1. $\\tilde{I}_{({\\boldsymbol{f}};g),1}=\\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{w}},-{\\boldsymbol{w}})}(\\tilde{I}_{({\\boldsymbol{f}};g),0})$, \\\\\n2. $\\tilde{I}_{({\\boldsymbol{f}};g),2}=(\\tilde{I}_{({\\boldsymbol{f}};g),1}+\\langle s-\\sigma \\rangle)\\ \\cap {\\mathbb{C}}[{\\boldsymbol{x}},s]$, \\\\\n3. $\\tilde{I}_{({\\boldsymbol{f}};g),3}=\\tilde{I}_{({\\boldsymbol{f}};g),2}:g=\\tilde{I}_{({\\boldsymbol{f}};g),2}\\cdot g^{-1}$, \\\\\n4. $\\tilde{I}_{({\\boldsymbol{f}};g),4}=\\tilde{I}_{({\\boldsymbol{f}};g),3}\\cap {\\mathbb{C}}[s]$. \\\\\nThen $b_{{\\mathfrak{a}},g}(s)$ is the generator of $\\tilde{I}_{({\\boldsymbol{f}};g),4}$. \n\\end{Theorem}\n\\begin{proof}\nBy Lemma \\ref{annfs} and Lemma \\ref{intersection}, $\\tilde{I}_{({\\boldsymbol{f}};g),0}= (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})\\cap D_Yg$. \nAs $\\tilde{I}_{({\\boldsymbol{f}};g),1}$ is a homogeneous ideal, we have \n\\[\n\\tilde{I}_{({\\boldsymbol{f}};g),2}=\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[{\\boldsymbol{x}},s]\\mid b({\\boldsymbol{x}},\\sigma)\\in \\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{w}},-{\\boldsymbol{w}})}((\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})\\cap D_Y g) \\}\n\\]\nby Lemma \\ref{substitution}. \nSince $\\tilde{I}_{({\\boldsymbol{f}};g),0}\\subset D_Yg$, we have $\\tilde{I}_{({\\boldsymbol{f}};g),1}\\subset \\mathop{\\mathrm{in}}\\nolimits_{({\\boldsymbol{w}},-{\\boldsymbol{w}})}(D_Yg)=D_Yg$. \nHence $\\tilde{I}_{({\\boldsymbol{f}};g),2}\\subset (D_Yg +\\langle s-\\sigma \\rangle)\\ \\cap {\\mathbb{C}}[{\\boldsymbol{x}},s]= {\\mathbb{C}}[{\\boldsymbol{x}},s]g$, \nand $\\tilde{I}_{({\\boldsymbol{f}};g),2}:g=\\tilde{I}_{({\\boldsymbol{f}};g),2}\\cdot g^{-1}$. \nSince $b_{{\\mathfrak{a}},g}(s)$ is the minimal generator of the ideal \n\\[\n\\{b(s)\\in {\\mathbb{C}}[s]\\mid b(\\sigma)g({\\boldsymbol{x}})\\in (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}}\\}, \n\\]\nit follows that $\\tilde{I}_{({\\boldsymbol{f}};g),4}=(\\tilde{I}_{({\\boldsymbol{f}};g),2}:g)\\cap {\\mathbb{C}}[s]=\\langle b_{{\\mathfrak{a}},g}(s)\\rangle$. \n\\end{proof}\n\\begin{Remark}\\label{same algorithm}\nNote that \n\\[\nI_{({\\boldsymbol{f}};g),3}=\\tilde{I}_{({\\boldsymbol{f}};g),3}=\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[s]\\mid b({\\boldsymbol{x}},\\sigma)g({\\boldsymbol{x}})\\prod_i f_i^{s_i}\\in (V^1D_Y) g(x)\\prod_i f_i^{s_i}\\}, \n\\]\nand in the case $g=1$, it follows that \n\\[\nI_{({\\boldsymbol{f}};1),2}=\\tilde{I}_{({\\boldsymbol{f}};1),2}=\\{b({\\boldsymbol{x}},s)\\in {\\mathbb{C}}[{\\boldsymbol{x}},s]\\mid b({\\boldsymbol{x}},\\sigma)\\prod_i f_i^{s_i} \\in (V^1D_Y) \\prod_i f_i^{s_i}\\}. \n\\]\n\\end{Remark}\nWe can compute local generalized Bernstein-Sato polynomials similarly to the classical case using primary decompositions \n(see \\cite{O1}, \\cite{O2}, and \\cite{O3}). \n\\begin{Theorem}[Algorithm for local generalized Bernstein-Sato polynomials]\\label{Algorithm for local b-functions}\nLet $I_{({\\boldsymbol{f}};g),3}=\\tilde{I}_{({\\boldsymbol{f}};g),3}$ be the ideals in Theorem \\ref{Algorithm for global b-functions 1} \nand Theorem \\ref{Algorithm for global b-functions 2} \nwith primary decompositions $I_{({\\boldsymbol{f}};g),3}=\\cap_{i=1}^{\\ell} {\\mathfrak{q}}_i$. \nThen $b^{\\mathfrak{p}}_{{\\mathfrak{a}},g}(s)$ is the generator of the ideal\n\\[\n\\bigcap_{{\\mathfrak{q}}_i \\cap {\\mathbb{C}}[{\\boldsymbol{x}}] \\subset {\\mathfrak{p}}}{\\mathfrak{q}}_i\\cap {\\mathbb{C}}[s].\n\\]\n\\end{Theorem}\n\\begin{proof}\nWe set $R={\\mathbb{C}}[{\\boldsymbol{x}}]$. \nBy the definition, $b^{\\mathfrak{p}}_{{\\mathfrak{a}},g}(s)$ is the generator of the ideal \n\\[\n\\{b(s)\\in {\\mathbb{C}}[s]\\mid b(\\sigma)g({\\boldsymbol{x}})h({\\boldsymbol{x}})\\in (\\mathop{\\mathrm{Ann}}\\nolimits_{D_Y}\\prod_i f_i^{s_i})^*+D_Y g{\\mathfrak{a}} \\mbox{\\quad for~~} {}^\\exists h\\not\\in {\\mathfrak{p}} \\}. \n\\]\nThis ideal equals to \n\\begin{eqnarray*}\n&&\\{b(s)\\in {\\mathbb{C}}[s]\\mid b(s)g({\\boldsymbol{x}})h({\\boldsymbol{x}})\\in I_{({\\boldsymbol{f}};g),2} \\mbox{\\quad for~~} {}^\\exists h\\not\\in {\\mathfrak{p}} \\}\\\\\n&=&\\{b(s)\\in {\\mathbb{C}}[s]\\mid b(s)h({\\boldsymbol{x}})\\in I_{({\\boldsymbol{f}};g),3} \\mbox{\\quad for~~} {}^\\exists h\\not\\in {\\mathfrak{p}} \\}\\\\\n&=&(I_{({\\boldsymbol{f}};g),3}\\otimes_{R} R_{\\mathfrak{p}} )\\cap {\\mathbb{C}}[s]. \n\\end{eqnarray*} \nSince $I_{({\\boldsymbol{f}};g),3}\\otimes_{R} R_{\\mathfrak{p}} =\\bigcap_{{\\mathfrak{q}}_i \\cap R\\subset {\\mathfrak{p}}}{\\mathfrak{q}}_i\\otimes_{R} R_{\\mathfrak{p}}$, \nand ${\\mathfrak{q}}_i\\otimes_{R} R_{\\mathfrak{p}}=R_{\\mathfrak{p}}[s]$ if and only if ${\\mathfrak{q}}_i \\cap R \\subset {\\mathfrak{p}}$, \nit follows that $b^{\\mathfrak{p}}_{{\\mathfrak{a}},g}(s)$ is the generator of the ideal \n$(\\bigcap_{{\\mathfrak{q}}_i \\cap R \\subset {\\mathfrak{p}}}{\\mathfrak{q}}_i)\\cap {\\mathbb{C}}[s]$. \n\\end{proof}\nThe algorithm for computing generalized Bernstein-Sato polynomials enables us to solve the membership problem for multiplier ideals, \nbut does not give a system of generators of multiplier ideals. \nWe have to compute $b_{{\\mathfrak{a}},g}(s)$ for infinitely many $g$ to obtain a system of generators. \n\\section{Algorithms for computing multiplier ideals}\nThe purpose of this section is to obtain algorithms for computing the system of generators of multiplier ideals. \nTo do this, we modify the definition of Budur--Musta\\c{t}\\v{a}--Saito's Bernstein-Sato polynomial. \n\nAs in the previous section, let ${\\mathbb{C}}[{\\boldsymbol{x}}]={\\mathbb{C}}[x_1,\\dots,x_n]$ be a polynomial ring over ${\\mathbb{C}}$, \nand ${\\mathfrak{a}}$ an ideal with a system of generators ${\\boldsymbol{f}} =(f_1,\\dots,f_r)$, \nand fix the weight vector $({\\boldsymbol{w}},-{\\boldsymbol{w}})\\in {\\mathbb{Z}}^{n+r}\\times{\\mathbb{Z}}^{n+r}$, ${\\boldsymbol{w}}=((0,\\dots,0),(1,\\dots,1))\\in {\\mathbb{Z}}^{n}\\times{\\mathbb{Z}}^{r}$. \nWe set $\\delta=1\\otimes 1\\in M_{{\\boldsymbol{f}}}=\\iota_+ {\\mathbb{C}}[{\\boldsymbol{x}}]$ and $\\overline{M}_{{\\boldsymbol{f}}}^{(m)}=(V^0D_Y) \\delta\/(V^m D_Y) \\delta$. \nThe induced filtration $V$ on the $\\overline{M}_{{\\boldsymbol{f}}}^{(m)}$ is finite by the definition of the $V$-filtration \n(Definition \\ref{Vfiltration}) as in the case $m=1$. \nFor $g\\in {\\mathbb{C}}[{\\boldsymbol{x}}]$, we denote by $\\overline{g\\otimes 1}$ the image of $g\\otimes 1=g\\delta$ in $\\overline{M}_{{\\boldsymbol{f}}}^{(m)}$. \n\\begin{Definition}\\label{new Bernstein-Sato polynomial}\nWe define $b_{{\\mathfrak{a}},g}^{(m)}(s)$ to be the monic minimal polynomial of the action of $\\sigma$ on \n$(V^0D_Y)\\overline{g\\otimes 1}\\subset \\overline{M}_{{\\boldsymbol{f}}}^{(m)}$. \nWe define $b^{(m)}_{{\\mathfrak{a}}}=b^{(m)}_{{{\\mathfrak{a}},1}}$. \n\\end{Definition}\nThe existence of $b_{{\\mathfrak{a}},g}^{(m)}(s)$ follows from the finiteness of the filtration $V$ on $\\overline{M}_{{\\boldsymbol{f}}}^{(m)}$, \nand the rationality of its roots follows from the rationality of the $V$-filtration. \nNote that $b_{{\\mathfrak{a}},g}^{(m)}(s)=1$ if and only if $g\\otimes 1 \\in (V^m D_Y) \\delta$.\n\\begin{Observation}\\label{on new b-function}\nSince the ring $V^0D_Y$ is generated by ${\\boldsymbol{t}}=t_1,\\dots,t_r$ over $[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$, \n$\\sigma$ is in the center of $[D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0$, and $t_i\\cdot\\overline{g\\otimes 1}=f_i\\cdot\\overline{g\\otimes 1}$, \nit follows that $b_{{\\mathfrak{a}},g}^{(m)}(s)$ is the monic polynomial $b(s)$ of minimal degree satisfying $b(\\sigma)\\overline{g\\otimes 1}=0$. \nThis is equivalent to saying \n\\[\nb(s)g \\prod_i f_i^{s_i}\\in V^mD_Y\\prod_i f_i^{s_i}. \n\\]\nSince $V^mD_Y$ is generated by all monomials in ${\\boldsymbol{t}}=t_1,\\dots,t_r$ of degree $m$ as $V^0D_Y$-module, \nand $t_j\\prod_i f_i^{s_i}=f_j\\prod_i f_i^{s_i}$, \nour Bernstein-Sato polynomial $b_{{\\mathfrak{a}},g}^{(m)}(s)$ is the monic polynomial $b(s)$ of minimal degree satisfying \n\\begin{equation}\\label{exp6}\nb(s)g \\prod_i f_i^{s_i}= \\sum_j P_j h_j \\prod_i f_i^{s_i} \\mbox{\\quad for~~} {}^\\exists P_j\\in [D_Y]^{({\\boldsymbol{w}},-{\\boldsymbol{w}})}_0, {}^\\exists h_j\\in {\\mathfrak{a}}^m.\n\\end{equation}\nHence if $g,h\\in{\\mathbb{C}}[{\\boldsymbol{x}}]$ are polynomials such that $g$ divides $h$, then $b_{{\\mathfrak{a}},h}^{(m)}(s)$ is a factor of $b_{{\\mathfrak{a}},g}^{(m)}(s)$. \nIn particular, $b_{{\\mathfrak{a}},g}^{(m)}(s)$ is a factor of $b_{{\\mathfrak{a}}}^{(m)}(s)$ for all $g\\in{\\mathbb{C}}[{\\boldsymbol{x}}]$. \n\\end{Observation}\nWe obtain a description of multiplier ideals in terms of our Bernstein-Sato polynomials similarly to \nTheorem \\ref{multiplier ideals and b-functions}. \n\\begin{Theorem}\\label{multiplier ideals and b-functions 2}\n{\\rm (i)} For a given rational number $c$ v$_{\\rm e}$, then we are in the super-escape regime, so that gravitational-focusing \ncan be neglected. Conversely, in the sub-escape and super-Hill regime v$_{\\rm H} <$ $\\sigma$ $<$ v$_{\\rm e}$ \ngravitational-focusing is significant. We do not consider the sub-Hill regime (i.e. $\\sigma <$ v$_{\\rm H}$), \nsince the Hill velocity is always very small in galactic nuclei. \n\nPutting this all together, the collision rate for a Keplerian nuclear disk can be written: \n\\begin{equation}\n\\label{eqn:gamma1}\n\\Gamma \\approx \\Big( \\frac{{\\Sigma}{\\Omega}R^2}{M} \\Big)\\Big(1 + \\Big( \\frac{v_{\\rm e}}{\\sigma} \\Big)^2 \\Big),\n\\end{equation} \nwhere \n\\begin{equation}\n\\label{eqn:omega}\n\\Omega = \\Big( \\frac{GM_{\\rm BH}}{r^3} \\Big)^{1\/2},\n\\end{equation}\nand the second term in Equation~\\ref{eqn:gamma1} smoothly transitions the collision rate into and out of the regime \nwhere gravitational-focusing becomes important. The stellar surface mass density is:\n\\begin{equation}\n\\label{eqn:surfdens}\n\\Sigma = \\Big( \\frac{M_{\\rm d}}{\\pi{a^2}} \\Big),\n\\end{equation}\nwhere M$_{\\rm d}$ is the total stellar mass of the disk, a is its scale length and h is its scale height. \nEquation~\\ref{eqn:gamma1} gives the mean rate at which a given star or object orbiting at a distance r \nfrom the central SMBH collides with other objects. \n\nNext, we consider an NSC for which the dominant stellar distribution is (approximately) spherical, pressure-supported \nand the stellar motions are predominantly random. Here, \nwe assume that the stellar orbits are distributed isotropically and the velocity distribution is isothermal (i.e., Maxwellian with \ndispersion $\\sigma_{\\rm 0}$) \\citep[e.g.][]{merritt13}. \n\nSimilar to Equation~\\ref{eqn:gamma1}, an order-of-magnitude estimate for the mean rate $\\Gamma$ at which a given star or object \nundergoes direct collisions with other objects is:\n\\begin{equation}\n\\label{eqn:gamma9}\n\\Gamma \\approx \\Big( \\frac{{\\rho}{\\sigma}R^2}{M} \\Big)\\Big(1 + \\Big(\\frac{v_{\\rm e}}{\\sigma}\\Big)^2\\Big),\n\\end{equation} \nwhere, as before, the second term smoothly transitions the collision rate into and out of the regime where gravitational-focusing \nbecomes important. In Equation~\\ref{eqn:gamma9}, we use the local velocity dispersion $\\sigma$, calculated as:\n\\begin{equation}\n\\label{eqn:sigloc}\n\\sigma(r)^2 = \\sigma_{\\rm 0}^2 + \\frac{GM_{\\rm BH}}{r}.\n\\end{equation} \nThis serves to modify the velocity dispersion very close to the central SMBH, where the stellar orbits begin transitioning into \nthe Keplerian regime.\n\nFinally, combining Equations~\\ref{eqn:gamma1} and~\\ref{eqn:gamma9}, we obtain a \ngeneral formula for the rate of collisions in a nuclear star cluster with or without a central SMBH:\n\\begin{equation}\n\\label{eqn:gamma10}\n\\Gamma \\approx ({\\rho}\\sigma + {\\Sigma}\\Omega)\\Big( \\frac{R^2}{M} \\Big)\\Big(1 + \\Big( \\frac{v_{\\rm e}}{\\sigma} \\Big)^2 \\Big),\n\\end{equation}\nAs we will show in Section~\\ref{app}, Equation~\\ref{eqn:gamma10} reasonably describes the collision rates in every \ngalactic nucleus in our sample, provided we set \\textit{either} $\\rho =$ 0 or $\\Sigma =$ 0. This is because, within our chosen \nouter limit of integration (i.e., 2 pc), the dominant NSC \nstructure is either well described by one of these two extremes, namely an (approximately) spherical pressure-supported \nsystem (i.e., the MW, M32 and M33) or a Keplerian disk (i.e., M31).\n\n\n\\subsection{Binary mergers due to Kozai-Lidov oscillations} \\label{kozai}\n\nIn this section, we derive order-of-magnitude estimates for the mean rate of binary mergers mediated \nby Kozai-Lidov oscillations with an SMBH acting as the outer triple companion, as a function \nof the host nuclear environment and distance from the central SMBH. \n\nThe characteristic time-scale for eccentricity oscillations to occur due to the Kozai-Lidov mechanism is \\citep{holman97,antonini16}:\n\\begin{equation}\n\\label{eqn:taukl}\n\\tau_{\\rm KL} \\approx P_{\\rm b}\\Big( \\frac{M_{\\rm b}}{M_{\\rm BH}} \\Big)\\Big( \\frac{r}{a_{\\rm b}} \\Big)^3(1-e_{\\rm BH}^2)^{3\/2}, \n\\end{equation} \nwhere M$_{\\rm b} =$ 2M is the binary mass, r is the distance from the SMBH, e$_{\\rm BH}$ is orbital eccentricity of the \nbinary's orbit about the central SMBH, a$_{\\rm b}$ is the binary orbital separation and P$_{\\rm b}$ is the binary \norbital period. The rate at which binaries merge due to Kozai-Lidov oscillations within a given distance from the SMBH \nis then:\n\\begin{equation}\n\\label{eqn:gammakl1}\n\\Gamma_{\\rm KL}(r) \\sim \\frac{{\\pi}f_{\\rm b}f_{\\rm i}{\\Sigma}r^2}{2M\\tau_{\\rm KL}},\n\\end{equation} \nfor a nuclear disk, and \n\\begin{equation}\n\\label{eqn:gammakl2}\n\\Gamma_{\\rm KL}(r) \\sim \\frac{2{\\pi}f_{\\rm b}f_{\\rm i}{\\rho}r^3}{3M\\tau_{\\rm KL}},\n\\end{equation} \nfor a spherical pressure-supported nucleus. Here, f$_{\\rm b}$ is the fraction \nof objects that are binaries, and \nf$_{\\rm i}$ is the fraction of binaries with their inner and outer orbital planes aligned such that Kozai \ncycles operate. For simplicity, we assume f$_{\\rm i} =$ 1. Thus, the calculated rates \nare strict upper limits, since they assume that every binary is aligned relative to the SMBH such that Kozai oscillations \nwill occur, which is certainly not the case. We will return to this issue in Section~\\ref{discussion}.\n\nImportantly, Equation~\\ref{eqn:taukl}, and hence Equations~\\ref{eqn:gammakl1} and~\\ref{eqn:gammakl2}, \nare only valid at distances from the SMBH smaller than \\citep{prodan15}:\n\\begin{equation}\n\\begin{gathered}\n\\label{eqn:rsc}\nr_{\\rm SC} \\approx 0.02\\Big( \\frac{a_{\\rm b}}{{\\rm AU}} \\Big)^{4\/3}\\Big( \\frac{M_{\\rm b}}{2 {\\rm M_{\\rm \\odot}}} \\Big)^{-2\/3} \\\\\n\\Big( \\frac{M_{\\rm BH}}{4 \\times 10^6 {\\rm M_{\\rm \\odot}}} \\Big)^{1\/3}\\Big( \\frac{1- e_{\\rm b}^2}{1 - e_{\\rm BH}^2} \\Big)^{1\/2} {\\rm pc},\n\\end{gathered}\n\\end{equation}\nwhere e$_{\\rm b}$ is the orbital eccentricity of the binary star. This is due to general relativistic, or \nSchwarzschild apsidal, precision in the binary star \\citep{holman97,blaes02,holman06}. \n\n\n\n\\section{The effect of mass loss on the photometric appearance of red giant branch stars} \\label{ehbform}\n\nIn this section, we quantify the effects of collisions and, specifically, the subsequent tidal stripping on \nthe photometric appearance of \nRGB stars. Our primary concern is the stellar evolution, so we do not specify the nature of the impactor. \nInstead, we look at how discrete mass loss events affect the subsequent time evolution of the \nRGB luminosity and radius. We note here that our calculated collision rates are sufficiently high that \nthe underlying RGB populations could be significantly affected, and these stars dominate the light \ndistribution in an old stellar population. Due to the high extinction toward the Galactic centre, for \nexample, these are the only (``old'') stars that can be observed.\n\nStars of initial mass\\footnote{In this section, unless otherwise stated, mass, luminosity and radius are \nexpressed in solar units, and our chosen unit of time is $10^6~$yr} $0.3 \\lap M\/M_{\\odot} \\lap 10$ become RGBs \nstars. Exhaustion of hydrogen in the core is followed by burning of hydrogen to helium in a thin shell; the \nluminosity at the beginning of this evolutionary stage, i.e. at the base of the giant branch (BGB), is \napproximated by \\citep[e.g.][]{1989ApJ...347..998E}:\n\\begin{equation}\n\\label{eqn:LBGB}\nL_{\\rm BGB} \\approx \\frac{2.15M^2+0.22M^5}{1+1.4\\times10^{-2}M^2+5\\times10^{-6}M^4},\n\\end{equation}\nwhere M is the mass of the star. The mass of the helium core gradually increases, and the star's \nradius and luminosity shoot up as the star climbs the giant branch. \nIn stellar evolution models, the radius and luminosity during the giant phase are determined \nalmost uniquely by the mass, $M_{\\rm c}$, of the (helium) core.\nApproximate relations for $0.17 \\lap M_{\\rm c}\/M_{\\odot} \\lap 1.4$ are \\citep{2006epbm.book.....E}:\n\\begin{eqnarray}\n\\label{eqn:LRRG}\nL_{\\rm GB} & \\approx & \\frac{10^{5.3}M_{\\rm c}^6}{1 + 10^{0.4}M_{\\rm c}^4 + 10^{0.5}M_{\\rm c}^5}, \\\\\nR _{\\rm GB} & \\approx & M^{-0.3}(L_{\\rm GB}^{0.4}+0.383L_{\\rm GB}^{0.76}).\n\\end{eqnarray}\nThe dominant energy source during these evolutionary phases\nis the CNO cycle, and so the rate of energy production is tied to the rate of increase of the core mass by\n\\begin{equation}\n\\label{eqn:Lpp}\nL_{\\rm GB} \\approx 1.44 \\times 10^{5} \\times \\frac{dM_{\\rm c}}{dt}.\n\\end{equation}\nGiven an initial core mass, Equations~\\ref{eqn:LRRG}-\\ref{eqn:Lpp} can be solved for the dependence of \n$M_{\\rm c}$, $L_{\\rm GB}$ and $R_{\\rm GB}$ on time.\nThe surface temperature is given by the Stefan-Boltzmann law (however, at late times, the \nmaximum radius is limited by mass loss). \nThe initial core mass -- that is, the mass of the hydrogen-exhausted\nhelium core at the time the main sequence turnoff is reached -- is given by \\citep{2006epbm.book.....E}:\n\\begin{equation}\nM_{\\rm c} \\approx \\frac{0.11 M^{1.2} + 0.0022M^2 + 9.6 \\times 10^{-5} M^4}\n{1 + 0.0018M^2 + 1.75\\times 10^{-4} M^3}.\n\\end{equation}\nFor instance, for $M = (1,2,3)$ M$_{\\odot}$, the initial core mass is\n$M_{\\rm c}\/M_{\\odot} = (0.11,0.26,0.43)$ and the initial core mass ratio is\n$M_{\\rm c}\/M_{\\odot} = (0.11,0.13,0.14)$.\n\nFor stars with masses $0.3 \\lap M\/M_{\\odot} \\lap 0.5$ the evolution on the RGB is interrupted when \n$M_{\\rm c}$ has grown to approximately $M$. Low mass stars, with $0.5 \\lap M\/M_{\\odot} \\lap 2$, \ndevelop degenerate He cores on the RGB and $3\\alpha$ reactions initiate a violent nuclear\nrunaway, called the helium-flash. Intermediate mass stars, with $2 \\lesssim M\/M_{\\odot} \\lesssim 10$,\ndevelop non-degenerate He cores, which growth only slightly on the RGB before the He burning begins. \nThe luminosity at the tip of the giant branch (TGB) is roughly $L_{\\rm TGB} \\approx L_{\\rm BGB} + 2 \\times 10^3$.\n\nMass loss can significantly alter the evolution of an RGB star. For instance, it can \nprevent the star from igniting He, and therefore terminate the star's evolution at an \nearlier stage. In order to study the response of RGB stars to episodes of intense mass loss, \nwe used the stellar evolution code~STARS \\citep{eggleton71,pols+95}. \nWe compute evolution tracks for stars with various ZAMS masses, applying the \\citet{dejager+88} \nprescription for the mass loss rate due to stellar winds. \nAt some point during the RGB phase, we switch to a constant mass loss regime at the rate \n$10^{-5}M_\\odot~\\rm yr^{-1}$. During this time we turn off changes in composition due to nuclear burning, \nwhile retaining energy production. When the stellar mass is reduced to a certain value, $M_{\\rm f}$, we\nhalt the rapid mass loss regime, and continue with \"normal\" stellar evolution by turning the \nde~Jager mass loss rate prescription back on, which in turn allows for composition changes to \nre-initiate. This numerical procedure allows us to study the evolution of stars undergoing \nan episode of mass loss on a time scale much shorter than their nuclear time scale, as is the case for \nan RGB star that is tidally stripped of its envelope.\n\nFigure~\\ref{fig:fig1} displays stellar evolution tracks on the RGB, after different amounts of envelope \nstripping. Stars will not reach the TGB if the total stellar mass is smaller (due to stellar winds) than \nthe He core mass at the TGB of the progenitor star. In this case, the star will instead climb the giant branch \nuntil the mass of its hydrogen exhausted He core is essentially equal to its total stellar mass, at which point it \nwill turn off the giant branch to the blue at the same luminosity as other stars of the same total mass. The \nstar will fail to ignite He and will subsequently cool off to become a white dwarf \nwith an He core. If the mass left in the star is instead large enough that its He core mass \ncan reach that of the parent star at the moment of He ignition, then the remnant \nwill also ignite He at roughly the same luminosity as its (unperturbed) parent star. In this scenario, the star \neventually cools off to become a white dwarf with a CO core. \nFigure~\\ref{fig:fig1} shows that, for stars of total mass $0.8 \\lesssim M\/M_{\\odot} \\lesssim 1.5$, \nto significantly alter the evolution and make the TGB fainter requires $M_{\\rm f} \\lesssim 0.7 M_{\\odot}$.\nFor $M \\gtrsim 2 M_{\\odot}$, we require instead $M_{\\rm f} \\approx 0.3 M_{\\odot}$ to be removed or, \nequivalently, more than $90\\%$ of the star's envelope \\citep[e.g.][]{pradam09}.\n\nWe find that the overall stars' evolution is relatively insensitive to the luminosity at which the mass loss \nepisode occurs. Following mass removal, a star will nearly return to its initial luminosity on the RGB, \nbut will move onto the point on the Hayashi track corresponding to its new mass. This is independent \nof the luminosity at which tidal stripping occurs. While the luminosity and evolutionary timescales \nremain virtually unchanged, the new radius of a stripped RGB star will increase (or decrease; see below) \nsignificantly as it reaches a new state of thermal equilibrium.\n\nFor $M_{\\rm c}\/M \\gap(\\lap) 0.2$, the star responds to mass loss by becoming smaller (larger).\nIn Figure~\\ref{fig:fig4}, the dashed line corresponds to an He core mass equal to 0.2M, \napproximately the mass ratio below which RGB stars respond to tidal stripping by expanding. Stars with \nmasses $M \\gtrsim 2 M_{\\odot}$ never rise above this critical mass ratio and will always expand on\nlosing mass. Lower mass stars, on the other hand, have $0.1 \\lap M_{\\rm c}\/M \\lap 0.15$ when they \nfirst leave the main sequence, and will expand upon losing mass. Farther up the giant branch, \nthe core mass increases relative to the total stellar mass, and these stars should shrink upon losing mass.\n\nFigure~\\ref{fig:fig2} shows the mass of the hydrogen exhausted core, $M_{\\rm c}$, at the TGB as a function \nof the ZAMS mass for both unperturbed and stripped stars. For unperturbed stars, there is a deep minimum \naround $M = 2 M_{\\odot}$ which corresponds approximately to the transition between stars with \ndegenerate, $M \\lesssim 2 M_{\\odot}$, and non-degenerate, $M \\gtrsim 2 M_{\\odot}$, He cores at the beginning \nof the giant branch \\citep{HS55}. Comparing the He core mass at the TGB with that at the time corresponding to the \nmain sequence turnoff (dot-dashed line in Figure~\\ref{fig:fig2}), we see that for $M\\gtrsim 2 M_{\\odot}$ the majority \nof the He core mass needed for ignition of He-burning is already in place before the star reaches the giant branch. \nA star with $M \\gtrsim 2M_{\\odot}$ that undergoes a strong mass loss episode on the RGB will still experience a \ncore-He burning phase and reach nearly the same luminosity as its parent star at the TGB. Therefore, for \n$M \\gtrsim 2 M_{\\odot}$, mass loss will have little effect on the luminosity corresponding to the TGB, unless most of \nthe envelope is suddenly removed near the BGB. With that said, however, mass loss during the RGB phase \ncould still make the tip of the asymptotic giant branch (AGB) non-negligibly fainter (for example, see the lower \npanels in Figure~\\ref{fig:fig1}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.73\\linewidth,angle=0]{fig1.eps}\n\\end{center}\n\\caption[Evolution tracks for stars undergoing an episode of mass loss on the giant branch]{Evolution tracks for stars with \nZAMS masses of $M = 1 M_{\\odot}$ and $3 M_{\\odot}$ undergoing an \nepisode of mass loss on the giant branch. The total stellar mass, $M_{\\rm f}$, remaining after some fraction of \nthe RGB envelope was (instantaneously) expelled is shown in blue. The star's evolution during the mass \nremoval phase is shown by the red line, with the blue lines depicting the subsequent evolution. \nStellar evolution tracks for unperturbed stars are shown via the black lines. For comparison, for each model, \nwe also show via the dashed orange lines the Eggleton stellar evolution tracks for unperturbed stars with \n$M = M_{\\rm f}$. The star symbols indicate the point on the H-R diagram where stars ignite He. The \ndot-dashed lines are K=12, 15 contours obtained using giant colours and bolometric corrections from \n\\citet{johnson66}. \nStars with $K >15$ are too faint to be resolved at the MW Galactic centre. For this plot, we assume \na distance of $8~\\rm kpc$ and an extinction of $A_{\\rm K}=3$, suitable to the line of sight extending from the \nSun to the Galactic Centre.\n\\label{fig:fig1}}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.8\\linewidth,angle=-90.]{fig2.eps}\n\\end{center}\n\\caption[The He core mass at the TGB as a function of the ZAMS mass.]{The He-core mass at the TGB is shown as a \nfunction of the ZAMS mass. The filled circles show the \nresults for unperturbed single stars. The star symbols show the hydrogen-exhausted He core mass for stars \nundergoing an episode of rapid mass loss on the giant branch. The symbols are larger for stars that ignite He. The \nindicated masses show the final stellar mass that remains after mass removal. The dashed line delineates the boundary \nbetween canonical stars (filled circles) that respond to mass loss by becoming bigger ($M_{\\rm c}\/M_{\\odot} < 0.2$) or \nsmaller ($M_{\\rm c}\/M_{\\odot} > 0.2$). The dot-dashed line shows the mass of the hydrogen-exhausted He core at the time \ncorresponding to the main sequence turnoff. All units are assumed to be solar.\n\\label{fig:fig2}}\n\\end{figure}\n\n\\section{Application to Local Group Galactic Nuclei} \\label{app}\n\nIn this section, we review the available observations for the central nuclear regions of our four \nLocal Group galaxies, namely the MW, M31, M32 and M33, in order to characterize their physical \nproperties for input to the rate estimates presented in Section~\\ref{rates}. We further present the \ncalculated collision rates for each nucleus as a function of distance from the cluster centre, using the \nobserved NSC properties.\n\nOur main results are shown in Figures~\\ref{fig:fig3} and~\\ref{fig:fig4}, which show the collision rates (both for a \nparticular object and the total integrated rates) for the MW (top left insets), M31 (top right insets), M32 (bottom left insets) and \nM33 (bottom right insets), as a function \nof distance from the cluster centre or SMBH. The solid black, dotted red, dashed blue, long-dashed green, dotted black and \ndot-dashed cyan lines show, respectively, the \nrates of MS+MS, MS+RGB, MS+WD, WD+RGB, BH+RGB and 1+2 collisions. \n\nFigure~\\ref{fig:fig3} shows the individual rates for a \n\\textit{specific} object to experience direct collisions with other objects of a given type, and must hence be multiplied by an \nappropriate fraction. \n\n We multiply the rates of MS+MS, MS+RGB, MS+WD, WD+RGB, BH+RGB and 1+2 collisions by factors of (1-f$_{\\rm b}$)f$_{\\rm MS}$, \n(1-f$_{\\rm b}$)f$_{\\rm MS}$, (1-f$_{\\rm b})$f$_{\\rm MS}$, (1-f$_{\\rm b})$f$_{\\rm RGB}$, (1-f$_{\\rm b})$f$_{\\rm RGB}$ and \n(1-f$_{\\rm b}$), respectively, where f$_{\\rm b}$ is the fraction of objects that \nare binaries and f$_{\\rm MS}$ is the fraction of single stars on the main-sequence. We require \n1 $=$ f$_{\\rm MS}$ + f$_{\\rm RGB}$ + f$_{\\rm WD}$ + f$_{\\rm NS}$ + f$_{\\rm BH}$, where f$_{\\rm RGB}$, f$_{\\rm WD}$, f$_{\\rm NS}$ and \nf$_{\\rm BH}$ are the fractions of single RGBs, WDs, NSs and BHs, respectively. We set f$_{\\rm b} =$ 0.01, f$_{\\rm MS} =$ 0.89, \nf$_{\\rm WD} =$ 0.10 and f$_{\\rm RGB} =$ 0.01 \\citep[e.g.][]{leigh07,leigh09,maeder09,leigh11b}. These fractions are \nonly approximate, but are representative of what we find upon integrating over a Kroupa initial mass function \\citep{kroupa95,kroupa95b}, \nover the appropriate mass ranges for an old stellar population. For the fractions of NSs and BHs, we\nset f$_{\\rm NS} =$ 0.01 and f$_{\\rm BH} =$ 0.001 \\citep[e.g.][]{alexander05,hopman06}. \n\nTo estimate the binary fraction for \neach NSC, f$_{\\rm b}$, we first calculate the expected average hard-soft boundary in the NSC for a typical binary.\nWe then calculate the fraction of the log-normal input period distribution, taken from \\citet{raghavan10} for solar-type binaries\nin the field, below this hard-soft boundary, $f_{\\rm P}$. We assume that, if the full period distribution could be occupied, this would\nyield a binary fraction of 50\\%. The (hard) binary fraction is then given as the product f$_{\\rm b} =$ 0.5f$_{\\rm P}$.\nThe resulting binary fractions are lower in clusters with hard-soft boundaries at smaller orbital periods (i.e. clusters with larger\nvelocity dispersions), as is consistent with observed star clusters \\citep[e.g.,][]{sollima08,milone12,leigh15}. This gives\nf$_{\\rm b} =$ 0.010, 0.002, 0.020 and 0.060 for, respectively, the MW, M31, M32 and M33. Finally, for the mean binary orbital\nseparation, we adopt the cluster hard-soft boundary.\n\nFigure~\\ref{fig:fig4} shows the total rates for \\textit{any} pair of objects to undergo a collision. These total rates are calculated \nby multiplying each collision rate in Figure~\\ref{fig:fig3} by the number of objects (MS, RGB or WD) in a given radial bin. We adopt a \nbin size of 0.01 pc, such that the y-axis corresponds to the number of collision products in each 0.01 pc \nradial interval, over a 100 Myr time interval. We multiply each rate by the calculated \nnumber of objects of the given type within a given radial bin, found by multiplying the mean density in each \nbin by its volume. An additional factor is included to correct for the fraction of objects of the desired type. That is, \nwe multiply the numbers of MS+MS, MS+RGB, MS+WD, WD+RGB, MS+NS, BH+RGB and 1+2 collisions by (1-f$_{\\rm b}$)f$_{\\rm MS}$, \n(1-f$_{\\rm b})$f$_{\\rm RGB}$, (1-f$_{\\rm b})$f$_{\\rm WD}$, (1-f$_{\\rm b})$f$_{\\rm WD}$, (1-f$_{\\rm b})$f$_{\\rm NS}$, \n(1-f$_{\\rm b})$f$_{\\rm BH}$ and f$_{\\rm b}$, respectively. We then calculate the total number of collision\/merger products \nfor each mechanism by integrating numerically over the 2-D and 3-D density profiles. That is, we sum over all radial bins to \nestimate the total number of collision products expected in a 100 Myr time interval, from 0.1 pc (i.e., the current \nlocation of the inner blue stars) to 2 pc (i.e., a rough estimate for the relevant size of the \nsurrounding old population) from the cluster centre. The results are shown in Table~\\ref{table:one} for \nall NSCs in our sample.\n\nIn making Figures~\\ref{fig:fig3} and~\\ref{fig:fig4}, we assume a mean single star \nmass and radius of 0.3 M$_{\\odot}$ and 0.3 R$_{\\odot}$, respectively, which are suitable for an old stellar \npopulation.\\footnote{We do not integrate over a present-day stellar mass function in calculating the collision rates, due \nmainly to uncertainties in the cluster age and initial stellar mass function for the NSCs in our sample. For example, \nsome authors have argued for a top-heavy initial mass function in the Galactic Centre \\citep[e.g.][]{paumard06}.} \nAll RGB stars, WDs, NSs and BHs are assumed to have masses of, respectively, 1 M$_{\\odot}$, \n1 M$_{\\odot}$, 2 M$_{\\odot}$ and 10 M$_{\\odot}$. All RGB stars are assumed to have radii\nof 10 R$_{\\odot}$, and all binaries have orbital separations equal to the hard-soft boundary. Hence, gravitational focusing\nis typically negligible for all collisions involving RGB stars (except if BHs are involved) and binaries, but is always significant for MS+MS, \nMS+WD, MS+NS and BH+RGB collisions. We caution that the \nrates shown in Figures~\\ref{fig:fig3} and~\\ref{fig:fig4} are order-of-magnitude estimates, \nand are highly sensitive to our assumptions for the input parameters. For example, an increase in the mean \nbinary orbital separation by a factor of only 2 translates into an \nincrease in both the corresponding rates and numbers by a factor of 8. Thus, binary destruction via 1+2 \ncollisions could be (or have been in the past) much more efficient than Figures~\\ref{fig:fig3} and~\\ref{fig:fig4} would \nsuggest, since we assume only very compact binaries. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig3.eps}\n\\end{center}\n\\caption[Individual rates of MS+MS, MS+RGB, MS+WD, WD+RGB, BH+RGB and 1+2 collisions for the MW, M31, M32 and M33]{The MS+MS (black solid lines), \nMS+RGB (red dotted lines), MS+WD (green long-dashed lines), WD+RGB (cyan dot-dashed lines), BH+RGB (dotted black lines) and 1+2 (blue dashed \nlines) collision rates are shown as a function of distance from the cluster centre (or SMBH) for the MW (top left panel), M31 (top right panel), \nM32 (bottom left panel) and M33 (bottom right panel). These are the rates for a \\textit{particular} object to undergo a collision. \nFor comparison, we also show our results for the MW \nassuming the density profile given by Equations 1 and 4 in \\citet{merritt10}, via an additional \ninset in the top left panel. The assumptions used as \ninput to each rate equation in Section~\\ref{rates} are discussed in Section~\\ref{app}. \n\\label{fig:fig3}}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig4.eps}\n\\end{center}\n\\caption[Total Rates of MS+MS, MS+RGB, MS+WD, WD+RGB, BH+RGB and 1+2 collisions for the MW, M31, M32 and M33]{The total rates of \nMS+MS (black solid lines), MS+RGB (red dotted lines), MS+WD (green long-dashed lines), WD+RGB (cyan dot-dashed lines), \nBH+RGB (dotted black lines) and 1+2 (blue dashed lines) collisions are shown as a function of distance from the \ncluster centre (or SMBH) for the MW (top left panel), M31 (top right panel), M32 (bottom left panel) and M33 (bottom right panel). These \nare the total rates for \\textit{any} pair of objects to undergo a collision. For \ncomparison, we also show our results assuming the density profile given by Equations 1 and 4 in \\citet{merritt10}, via an additional \ninset in the top left panel. In making this figure, we adopt a \nbin size of 0.01 pc, such that the y-axis corresponds to the number of collision products in each 0.01 pc radial interval, over a 100 Myr \ntime period. The assumptions used as input to each collision rate are the same as in Figure~\\ref{fig:fig3}.\n\\label{fig:fig4}}\n\\end{figure}\n\n\\begin{table*}\n\\caption{Total number of collision products expected in a 100 Myr time interval}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\nGalaxy & MS+MS & MS+RGB & MS+WD & WD+RGB & MS+NS & BH+RGB & 1+2 \\\\\n\\hline\nMilky Way & 7.1 $\\times$ 10$^1$ & 1.8 $\\times$ 10$^2$ & 2.5 $\\times$ 10$^1$ & 2.0 $\\times$ 10$^1$ & 5.0 $\\times$ 10$^0$ & 1.0 $\\times$ 10$^0$ & 6.5 $\\times$ 10$^1$ \\\\\nM31 & 3.2 $\\times$ 10$^4$ & 9.3 $\\times$ 10$^4$ & 1.1 $\\times$ 10$^4$ & 1.1 $\\times$ 10$^4$ & 2.2 $\\times$ 10$^3$ & 4.6 $\\times$ 10$^2$ & 1.5 $\\times$ 10$^3$ \\\\\nM32 & 4.3 $\\times$ 10$^3$ & 5.5 $\\times$ 10$^4$ & 1.4 $\\times$ 10$^3$ & 2.0 $\\times$ 10$^3$ & 2.6 $\\times$ 10$^2$ & 6.2 $\\times$ 10$^1$ & 7.9 $\\times$ 10$^3$ \\\\\nM33 & 1.3 $\\times$ 10$^4$ & 5.0 $\\times$ 10$^3$ & 4.9 $\\times$ 10$^3$ & 1.9 $\\times$ 10$^3$ & 9.7 $\\times$ 10$^2$ & 1.8 $\\times$ 10$^2$ & 4.3 $\\times$ 10$^4$ \\\\\n\\hline\n\\end{tabular}\n\\label{table:one}\n\\end{table*}\n\n\n\\subsection{The Milky Way} \\label{MW}\n\nAt the heart of the Milky Way (MW), there resides an NSC of mass \n$\\sim$ 9 $\\times$ 10$^6$ M$_{\\odot}$ ($<$ 100 arcsec) hosting a super-massive black hole \nof mass $\\sim$ 4 $\\times$ 10$^6$ M$_{\\odot}$ \\citep[e.g.][]{genzel96,chatzopoulos15}.\nWe assume a pressure-supported nucleus with an isothermal velocity dispersion \n$\\sigma_{\\rm 0} =$ 100 km\/s \\citep{merritt13}. Using Equation~\\ref{eqn:rinf}, this yields \nan influence radius for the central SMBH of $\\sim$ 1.7 pc. \n\nThe MW NSC is thought to have a core \\citep[e.g.][]{merritt10}, hence we adopt the stellar density profile of \\citet{stone15}:\n\\begin{equation}\n\\label{eqn:rho}\n\\rho(r) = \\frac{\\rho_{\\rm 0}}{(1 + r^2\/r_{\\rm c}^2)(1 + r^2\/r_{\\rm h}^2)},\n\\end{equation}\nwhere r is the distance from the origin, r$_{\\rm c}$ is the core radius and r$_{\\rm h}$ is the half-mass radius. Equation~\\ref{eqn:rho} \nhas been chosen to produce a flat core inside a roughly isothermal cluster.\\footnote{These assumptions are consistent \nwith the observational constraints available for the MW NSC \\citep[e.g.][]{merritt10}. However, we note that, formally, \na certain amount of anisotropy is needed for there to exist a core (in a Keplerian potential).} Here, the core radius is given by:\n\\begin{equation}\n\\label{eqn:core}\nr_{\\rm c} = \\frac{\\sigma_{\\rm 0}}{2{\\pi}G\\rho_{\\rm 0}}.\n\\end{equation}\n\nFor comparison, we also show our results assuming the density profile given by Equations 1 and 4 in \\citet{merritt10}, \nvia an additional inset in Figures~\\ref{fig:fig3} and~\\ref{fig:fig4}. As we will show, both of these assumptions for the \ndensity profile yield very comparable radial collision profiles. The central density is assumed to be \n10$^6$ M$_{\\odot}$ pc$^{-3}$, and we adopt a half-mass radius of \nr$_{\\rm h} =$ 2.5 pc \\citep{merritt13} along with a mass-to-light ratio of 2. We use Equation~\\ref{eqn:gamma10} \nto calculate all collision rates in the MW (with $\\Sigma =$ 0), appropriate for a \npressure-supported roughly isothermal nuclear environment, and transition smoothly between these formulae \nas gravitational-focusing becomes significant. Note that we set M$_{\\rm d} =$ 0, since there is no significant \nKeplerian disk component outside 0.1 pc. The results are shown in Figures~\\ref{fig:fig3} \nand~\\ref{fig:fig4} by the black lines.\n\nIn the MW, the collision rates are sufficiently high that all types of collision products should be \npresent, at least within $\\lesssim$ 1 pc of the Galactic Centre. This predicts \nnon-negligible numbers of MS+MS collision products and hence blue stragglers, as \nillustrated in Figure~\\ref{fig:fig4} and Table~\\ref{table:one}. More specifically, over a 1 Gyr period, \non the order of 710 $\\sim$ 10$^3$ MS+MS collisions should occur, which is roughly 0.1\\% of the total \nstellar mass in this region. \nAfter an initial rise from r $=$ 0, the number of collision products remains roughly constant with \nincreasing distance from the SMBH. As shown in Figure~\\ref{fig:fig3}, the \npredicted collision rates for a particular object show a near monotonic decrease with increasing distance \nfrom the cluster centre, after a relatively constant inner core that extends out to only $\\sim$ 0.2 pc.\nThe density profile taken from \\citet{merritt10}, while quantitatively very similar, shows a slightly steeper initial \ndrop followed by a very similar monotonic profile. The total collision rates for any pair of objects, however, show a \nrapid increase out to $\\sim$ 0.3 pc, \nfollowed by a monotonic decrease. This is the case for both of our assumed density profiles, which yield \nvery similar collision rate profiles, as shown in Figure~\\ref{fig:fig4}. Thus, we might expect a relatively \nweak central concentration of blue stars in the MW, assuming a collisional origin (ignoring any possible \nmass segregation of collision products). \n\nThe core radius given in Equation 4 of \\citet{merritt10} is much smaller \nthan the SMBH influence radius, such that the stellar orbits within this core should be roughly Keplerian. \nThis is what creates a central drop in some collision numbers within the core, as shown in Figure~\\ref{fig:fig4}. \nHowever, this inner core is such a small \nfraction of the total volume over which we integrate the collision rates that this correction should have a \nrelatively negligible impact on the total numbers of collision products. \n\nVery roughly, the numbers of 1+2 collisions that occur over a $\\gtrsim$ 10 Gyr period \nshould be comparable to the total number of binaries in the MW NSC, for our assumed binary fraction and \nconsidering only hard binaries. The hard-soft boundary in the MW NSC corresponds to the binary \ncomponents being in (or nearly in) contact, due to the very high velocity dispersion. Hence, the probability of \na direct collision or merger occurring during \nan encounter involving such a hard binary is nearly unity for small impact parameters \\citep[e.g.][]{leigh12}. The \ncollision product should expand post-collision by a factor of $\\sim$ a few \\citep[e.g.][]{sills97,fregeau04}, driving \nthe remaining binary components to merge when their radii overlap at periastron. Conversely, encounters \ninvolving soft binaries should have a high probability of being dissociative, or of significantly increasing the \nbinary separation post-encounter such that the time-scale for a subsequent encounter (with an even higher \nprobability of dissociation) becomes very short. \nThus, only (near) contact binaries should survive in the Galactic Centre for any significant amount of time although \nmany will also be destroyed, and our estimate for the binary fraction in the MW NSC \n(i.e., f$_{\\rm b} =$ 0.01) could be an over-estimate of the true binary fraction. \n\nWith that said, it seems unlikely that the disk of O- and B-type stars observed at $\\sim$ 0.1 pc from \nSgr A* has a collisional origin. First, although collisions should be dissipative, there is no obvious reason why the \ncollision products should \narrange themselves into a disk-like configuration in a spherical pressure-supported nucleus. Second, given \nthe average stellar mass in the \nMW NSC, multiple collisions would be required to form these massive O- and B-type stars, which must \nhave occurred over a very short interval of time, since O- and B-stars have very short lifetimes. Figure~\\ref{fig:fig3} \nsuggests that the rate of collisions involving individual MS stars is too low for this to be the case. \n\nInterestingly, however, the rate of MS+WD collisions is sufficiently high that a non-negligible supply of gas could be \nsupplied to the nucleus via this mechanism (especially assuming the density profile in \\citet{merritt10}). This is \nbecause collisions between stars and WDs should act to ablate \nthe star, nearly independent of the cluster velocity dispersion (see Section~\\ref{discussion} for more details regarding \nthe underlying physics responsible for this mechanism for gas liberation; \\citep{shara77,shara78,regev87}). This mechanism \ncould supply on the order of $\\lesssim$ 10$^3$ M$_{\\rm \\odot}$ in gas every $\\sim$ 1 Gyr in the MW NSC. This is because, using \nthe density profile of \\citet{merritt10}, on the \norder of $\\sim$ 100 MS\/RGB+WD collisions happen every 100 Myr (see Table~\\ref{table:one}). Assuming that most MS stars \nundergoing MS+WD collisions are close to the turn-off, which is of order $\\sim$ 1 M$_{\\odot}$ for an old population, then \nFigure~\\ref{fig:fig2} (see the dot-dashed line) suggests that the He core mass should comprise a small fraction of the total \nstellar mass ($\\lesssim$ 0.2 M$_{\\odot}$ or 20\\%) when the collision occurs. Hence, for our purposes, a reasonable \nassumption for the mean mass in gas liberated per MS\/RGB+WD collision is $\\sim$ 0.8 M$_{\\odot}$ (i.e., the envelope, or all \nmaterial outside of the degenerate core). Assuming 10$^3$ MS\/RGB+WD collisions per 1 Gyr, this predicts \n$\\sim$ 800 M$_{\\odot}$ $\\sim$ 10$^3$ M$_{\\odot}$ in gas should be supplied to the inner nucleus every Gyr due to \nMS+WD collisions alone. \n\nThe above back-of-the-envelope calculation suggests that MS+WD collisions alone cannot supply \nenough gas to form stars. This is because the time required to accumulate enough gas to form an \nactual disk is much longer than the gas cooling time (of order a few to a few hundred \nyears, for either optically thick or thin disks), which must be less than roughly 3 times the local dynamical \ntime-scale \\citep{nayakshin06,levin07,chang07}. Thus, if star formation were to occur only from gas supplied \nby MS+WD collisions, then it should only occur after nearly a Hubble Time. Given that stellar \nmass-loss in the surrounding old population should likely supply gas to the inner nucleus at a comparable \nor even higher rate \n\\citep[e.g.][]{chang07}, and the fact that the gas must not be accreted by the central SMBH before it \nfragments to form stars, we conclude that, while MS+WD collisions could contribute non-negligibly to the \ngas supply in the inner nucleus, this mechanism seems by itself insufficient to form stars and an \nadditional source of gas is needed.\n\nAs shown in Section~\\ref{ehbform}, nearly the entire RGB envelope must be stripped to significantly affect the \nphotometric appearance of an RGB star. Given that more than one collision is typically required to strip an RGB of its \nenvelope \\citep[e.g.][]{bailey99,dale09,amaroseoane14}, the \nderived rate for a given RGB star to encounter other MS stars is too low for collisions to contribute significantly \nto the observed paucity of giants (as shown by the dotted red line in Figure~\\ref{fig:fig3}). Assuming instead an \nisothermal density profile that starts to diverge near the SMBH, \nonly in the inner $\\lesssim$ 0.1 pc could the rate become sufficiently high for any given RGB star to encounter more \nthan a single MS star within $\\lesssim$ 1 Gyr (i.e. the typical lifetime of an RGB star). \nAlthough we expect \nBH+RGB collisions to be more effective at stripping the RGB envelope on a per collision basis \\citep{dale09}, \nwe find that the rate of BH+RGB collisions is likely too low for more than a handful of RGB stars to have been \nfully stripped. We conclude that it seems difficult to account for any missing RGB stars in the Galactic Centre \nvia collisions. \n\nThis is consistent with the general picture described in \\citet{merritt10}, which suggests that the stellar density \nof the underlying (not yet observed) old population traces that of the observed RGB stars, and has always been \nlow. In the absence of star formation, a low initial density remains low for a long time, since the relaxation time is \nvery long. Figures 4 and 5 in \\citet{merritt10} show that a model with a core can reproduce the observed number \ncounts of RGB stars in the MW. Our fiducial model for the MW also has a core, which is partly responsible for the \nlow predicted numbers of collisions involving RGB stars. Thus, the low observed density of RGB stars could be \nconsistent with being a natural consequence of a low (initial) density in the underlying old population. This picture is \nconsistent with the low rate of RGB collisions found here, which would not be able to account for any inferred \npaucity of RGB stars.\n\nTo summarize, the predicted rates of single-single collisions are too low to have significantly affected the \nphotometric appearance of the MW NSC. This is the case for stars belonging to any evolutionary stage. In particular, \nMS+MS collisions should have produced $\\lesssim$ 1\\% of the total stellar mass in collision products after a Hubble \ntime. Collisions involving individual RGB stars occur too infrequently for the RGB envelope to have been affected \nsignificantly, and any paucity of RGB stars to be observed. The rate of single-binary collisions, on the other hand, is \nsufficiently high that very few binaries should remain at the present day, if any. Finally, the rate of MS+WD collisions \nis insufficient to supply enough gas to form stars and, while this mechanism for gas accumulation could contribute \nnon-negligibly to a single burst of star formation, an additional source of gas is also needed.\n\n\\subsection{M31} \\label{m31}\n\nThe M31 nucleus is a distinct stellar system at the centre of its host late-type spiral galaxy, with a surface brightness profile that \nrises significantly above the background potential at r $<$ 10 pc. The nucleus shows an asymmetric double-lobed \nstructure within r $<$ 3 pc of the central SMBH, which is consistent with the diffuse eccentric disk model \nof \\citet{tremaine95}. Here, the nucleus consists of an old stellar population, but at even smaller radii (r $<$ 0.6 pc; \nrelative to the photocentre of the surrounding bulge component) there exists an UV-bright cluster of blue stars whose\norigins are unknown. Unlike the younger and hotter O- and B-stars observed in our Galactic Centre, these \nappear to be A-type stars \\citep{bender05,lauer12}. Below, we elaborate further on the details of the M31 nucleus.\n\nAt the centre of M31, there lurks an SMBH of mass $\\sim$ 1.4 $\\times$ 10$^8$ M$_{\\odot}$ \n\\citep{dressler84,dressler88,kormendy88,richstone90,kormendy99,bender05}. \nThe central blue cluster, called P3, is surrounded by two over-densities of stars, called P1 and P2 \n\\citep{lauer93}, which reside \non either side of P3 with a separation of $\\sim$ 1.8 pc \\citep{bender05}. These \ntwo ``lobes'' are distinct from P3 both in terms of their stellar content and \nkinematics \\citep[e.g.][]{lauer12}. P1 and P2 are much redder in \ncolour, while P3 contains a significant ultraviolet excess \n\\citep[e.g.][]{nieto86,king92,king95,lauer98,brown98}. A velocity dispersion of $\\sim$ 250 km\/s \nhas been measured in P2 along the same line of sight to P3, with a maximum \nof 373 $\\pm$ 48 km\/s on the anti-P1 side of the central blue cluster \\citep{bender05}. \nUsing Equation~\\ref{eqn:rinf}, the mean velocity dispersion gives an influence \nradius for the central SMBH of $\\sim$ 9.6 pc. P3, on the other hand, has the highest \nvelocity dispersion measured to date, with \na central dispersion of 1183 $\\pm$ 200 km\/s \\citep{bender05}.\n\nThe most likely explanation for the double-lobed structure of P1+P2 is \nan eccentric disk viewed in projection. This was originally proposed by \n\\citet{tremaine95}, who argued that P1 and P2 are unlikely to be two distinct \nstar clusters caught during the final stages of a merger, since the merger would \noccur within $\\lesssim$ 10$^8$ years by dynamical friction \\citep{lauer93,emsellem97}. \nTo reconcile this, \n\\citet{tremaine95} proposed that both nuclei or ``lobes'' are part of the same eccentric \ndisk of stars. The observations can be explained if the brighter lobe, P1, is located \nat a farther distance from the central SMBH and is the result of stars lingering near \napocentre \\citep[e.g.][]{statler99,kormendy99,bacon01}. Conversely, the fainter lobe, P2, \ncan be accounted for if it corresponds \napproximately to pericentre and the disk density increases toward the central SMBH. Hence, \nthe SMBH dominates the central potential, such that stars in the surrounding disk \nshould follow approximately Keplerian orbits.\\footnote{If the disk has a mass $\\gtrsim$ 10\\% that of the SMBH, \nthen self-gravity is required to keep the disk aligned \\citep{statler99}.} \n\\citet{peiris03} later refined the eccentric disk model, taking advantage of more recent \nground-based spectroscopy to help constrain their improved model. The models were \nused to predict the kinematics that should be observed via the Ca triplet in HST spectra \nof P1+P2, and the model predictions are in excellent agreement with the \ndata \\citep{bender05,lauer12}.\n\nOur collision rate estimates in Section~\\ref{collisions} are based on alterations to the \nformulation provided in \\citet{goldreich04}, originally derived to treat collisions within \nprotoplanetary disks. This approach can be almost directly applied in M31, \nsince the total mass of the Tremaine disk is M$_{\\rm d}$ $\\sim$ 3 $\\times$ 10$^7$ M$_{\\odot}$ \n\\citep{bender05} (assuming all 1.0 M$_{\\odot}$ stars and a mass-to-light ratio of \n5.7, which is appropriate for a bulge population \\citep{tremaine95}). This is nearly \nan order of magnitude smaller than the mass of the central SMBH. Hence, here we \ncan ignore any self-gravity within the disk \\citep[e.g.][]{statler99,emsellem07}.\n\nWe use Equation~\\ref{eqn:gamma10} to calculate the collision rates in M31 with $\\rho =$ 0, \nappropriate to a Keplerian nuclear disk (as in the MW, we transition smoothly between \nthese formulae as gravitational-focusing becomes significant). The disk has a scale length and \nheight of, respectively a $=$ 1.8 pc and h $=$ 0.7a.\\footnote{The M31 nuclear disk is observed \nto satisfy the relation 1 - h\/a $>$ 0.3 \\citep{lauer93}.} We assume a constant surface mass \ndensity $\\Sigma =$ M$_{\\rm d}$\/($\\pi$a$^2$), where M$_{\\rm d} =$ 3 $\\times$ 10$^7$ M$_{\\odot}$ \n\\citep{bender05}. This is because, as shown in Figure 17 of \\citet{lauer98}, the central surface \nbrightness profile in M31 cannot be described by a simple analytic function. \nCritically, this neglects the torus-like structure of the M31 nucleus, and over-estimates the surface \nmass density at small radii close to the SMBH (ignoring the inner blue disk). However, as we will show, \nthe collision rates are sufficiently high in this region that collisions would have transformed the inner \nnucleus over the last few Gyrs. Thus, the absence of a significant background of old stars in the inner \nnucleus at present is not necessarily indicative of an absence in the past. Consequently, we assume \nthat the presently-observed torus-like structure was once a diffuse disk extending further in toward the \ncentral SMBH, and include any collision products calculated to have occurred there in our estimates. However, \nwe note that if this assumption is incorrect, and the presently observed torus was always present, then \nthere is a true inversion in the stellar density at small radii and hence the collision rates here would be \nnegligible, dropping to zero at small r instead of continually rising all the way down to r $=$ 0, as shown \nin Figure~\\ref{fig:fig3}.\n\nAs in the MW, the collision rates in M31 are sufficiently high that all six types of collision products \nshould be present in non-negligible numbers over the entire extent of the disk (i.e. $\\gtrsim$ 2 pc), as \nillustrated in Figure~\\ref{fig:fig4}. The radial dependence of the collision rate is weak beyond $\\gtrsim$ 1pc, \nbut becomes more significant in the inner nucleus. \nSpecifically, the predicted collision rates at $\\sim$ 0.1 pc should outweigh those at $\\sim$ 1 pc by $\\lesssim$ an \norder of magnitude, for all six types of collision products, as shown in Figure~\\ref{fig:fig4}. \n\nOnly in the inner $\\lesssim$ 0.1 pc is the rate of MS+RGB collisions ever sufficiently high for individual RGB stars \nto have undergone enough collisions ($\\sim$ 100) over 100 Myr to unbind most of the RGB envelope, and \nhence to cause an observed paucity of RGB stars. This is the case for MS+RGB collisions alone, however, \nsince multiple collisions involving the same main sequence star and other MS stars are less common. \nThis suggests that any BSs present in M31 are likely only slightly more massive than the MS turn-off, \nand hence should appear as A-type stars, as observed for the inner blue disk \\citep{lauer12} (and not \nO- and B-type stars, unlike in M33 and M32; see below).\n\nThe numbers of 1+2 collisions over a $\\gtrsim$ 100 Myr interval should \nexceed the total numbers of binaries in M31, for our assumed binary fraction of f$_{\\rm b} =$ 0.01 and \nconsidering only hard binaries. The velocity dispersion in M31 is so high that the hard-soft boundary \ncorresponds to the binary components being in contact. Hence, for small impact parameters, the \nprobability of a direct collision or merger occurring during any direct 1+2 encounter involving a hard binary \nis approximately unity \\citep[e.g.][]{leigh12}. Conversely, the probability of dissociation during encounters \ninvolving soft binaries is very high. Thus, even (most) contact binaries should not have survived until the \npresent-day, and our estimate for the binary fraction in the M31 nucleus could be an over-estimate of the true \nbinary fraction. \n\nMS+WD collisions should liberate on the order of \n10$^5$ M$_{\\rm \\odot}$ in gas every $\\sim$ 1 Gyr, which can subsequently be used to form \nyoung (blue) stars. This is because on the \norder of $\\sim$ 1.1 $\\times$ 10$^4$ MS+WD collisions happen every 100 Myr (see Table~\\ref{table:one}). \nHence, following the same assumptions as in the preceding section for the MW, if \n1.1 $\\times$ 10$^5$ MS+WD collisions occur per 1 Gyr, this predicts \n$\\sim$ 8.8 $\\times$ 10$^4$ M$_{\\odot}$ $\\sim$ 10$^5$ M$_{\\odot}$ in gas should be supplied to the \ninner nucleus every Gyr due to MS+WD collisions alone. In M31, however, the critical mass needed for \nfragmentation is 10$^4 <$ M$_{\\rm crit}$\/M$_{\\odot} <$ 10$^5$ for a distance from the \ncluster centre 0.1 $\\le$ r\/pc $\\le$ 3 (see Figure 7 in \\citet{chang07}). As in the MW, the \ngas accumulation times needed to satisfy this critical mass requirement are much longer \nthan the gas cooling time. Thus, these results suggest that star formation should occur \non the order of every $\\sim$ 1 Gyr in M31, if the only supply of gas to the inner nucleus is \nMS+WD collisions. Once again, the rate of mass loss due to stellar evolution is, very roughly, \ncomparable to the rate at which gas is supplied by MS+WD collisions. Hence, the actual time-scale \nfor disk fragmentation could be much shorter than this simple calculation suggests, and \ngas from stellar evolution in the surrounding old population should contribute non-negligibly to the \ntotal gas mass available for fragmentation.\n\nWe note that, in M31, the central velocity dispersion rises toward r $=$ 0 and can even exceed \n1000 km\/s interior to $\\lesssim$ 0.1 pc. This should decrease the rates of MS+MS, MS+WD, MS+NS \nand BH+RGB collisions relative \nto what is shown in Figures~\\ref{fig:fig3} and~\\ref{fig:fig4}, by reducing the significance of gravitational \nfocusing. Additionally, given \nsuch high relative velocities at impact, some direct MS+MS collisions could \ncompletely unbind or ablate the collision product, leaving behind a puffed-up cloud of gas and \ndust in its place \\citep{spitzer66,spitzer67}. Similarly, grazing or off-centre collisions could \n(eventually) leave behind a very rapidly rotating collision product. In any event, the collisional velocities \nat impact are sufficiently high in M31 very close to the central SMBH that gas can be supplied to the \nNSC not only via MS+WD and WD+RGB collisions, but also via \nMS+MS and MS+RGB collisions (in addition to mass loss due to stellar evolution). Additionally, \nwe caution that, if a non-negligible fraction of MS+MS collisions ablate the product completely, then \nwe might not expect any central rise in the numbers of blue stragglers, but rather a dip in the numbers. \nAlternatively, if the products of MS+MS collisions are not ablated, then the deposited kinetic \nenergy at impact will serve to puff up the collision product, reducing the time-scale for subsequent \ncollisions to occur. Immediately after a collision, the product shoots up the Hayashi track before \ncontracting back down to the main-sequence \\citep[e.g.][]{sills97,sills01}. Hence, at least temporarily, \nthe collision product \nwill be brighter and redder than a normal MS star of the same mass. Interestingly, the photometric \nappearance of the object observed at the nominal centre of P3, called S11, is qualitatively consistent \nwith this general picture, since it is the brightest and reddest object in the central blue cluster \\citep{lauer12}.\n\nTo summarize, the predicted rate of MS+MS collisions is sufficiently high in the inner $\\lesssim$ parsec \nof the M31 nucleus that most, if not all, of the mass of the inner blue disk in M31 could be composed of MS+MS \ncollision products, or blue stragglers. Importantly, these collisions would have dissipated orbital angular \nmomentum and, given the disk-like nature of the M31 NSC, the thin disk-like structure of the inner blue excess \nis in general consistent with a collision origin. Note that the additional concentration of collision products within \nthe plane of the surrounding disk is unique to the M31 nucleus, relative to the other NSCs in our sample. \nWe caution that MS+MS collisions could become ablative in the inner \n$\\lesssim$ 0.1 pc due to the very high velocity dispersion here, which predicts impact velocities with enough \nkinetic energy to exceed the binding energy of any collision product. This would contribute to the total gas supply \nin the inner M31 nucleus, but likely not enough to form the inner blue disk in a burst of star formation. On the other \nhand, MS+WD collisions, which should be largely ablative nearly independent of the impact velocity, could \ncontribute significantly to the total gas reservoir in the inner nucleus. Together with stellar evolution-induced \nmass loss in the surrounding old population, this could supply enough gas to form stars in much less than a \nGyr. Finally, the rate at which individual RGB stars collide with other objects is sufficiently high in the inner \n$\\lesssim$ 0.1 pc that enough collisions could have occurred to completely unbind the RGB \nenvelope, causing an observed paucity of RGB stars in the inner M31 nucleus.\n\nWith the above results in mind, we note that \\citet{yu03} found from a similar analysis of analytic \ncollision rates calculated for the M31 nucleus that collisions alone cannot account for the blue excess observed \nat r $<$ 0.6 pc. We note, however, that \\citet{yu03} took the (currently) observed present-day density of the old \npopulation at face value, and did not account for the possibility that the Tremaine disk once extended inward \nto smaller radii. What's more, \\citet{yu03} did not have the more recent results of \\citet{bender05} and \\citet{lauer12}, \nwhich provide, respectively, better constraints for the properties of the blue cluster P3 and better spatial \nresolution for the entire inner M31 nucleus. Overall, our estimates for the collision rates in M31 are in decent \nagreement with those calculated by \\citet{yu03}, but the additional information provided by more recent \nobservational studies has allowed us to perform a slightly more focused analysis based on a more up-to-date \nunderstanding of the structure of the inner nucleus. This accounts for the different conclusions reached by \n\\citet{yu03} for the M31 nucleus, relative to those presented in this paper.\n\n\\subsection{M32} \\label{M32}\n\nThe nearby elliptical galaxy M32 is home to the smallest of the three SMBHs known to exist \nin the Local Group, with a mass $\\lesssim$ 2.5 $\\times$ 10$^6$ M$_{\\odot}$ \n\\citep{tonry84,verolme02,vandenbosch10}. The properties and even detection of this SMBH are, \nhowever, controversial \\citep{merritt13}. Surrounding this putative central SMBH is a rotating \ndisk of stars with a density $>$ 10$^7$ M$_{\\odot}$ pc$^{-3}$ at r $<$ 0.1 pc \n\\citep{walker62,lauer98}. Beyond this, an additional stellar component is \npresent, with an effective radius of $\\sim$ 6 pc and a total mass \n$\\sim$ 3 $\\times$ 10$^7$ M$_{\\odot}$ \\citep{kormendy99,graham09}. This represents \n$\\sim$ 10\\% of the total galaxy mass, such that the NSC in M32 contains a much larger \nfraction of the total galaxy luminosity than is typical for early-type galaxies \n\\citep[e.g.][]{cote06}. The stellar populations characteristic of the central NSC are younger \nthan is normal for the rest of the galaxy, with a mean age of $\\sim$ 4 Gyr in the central \nnuclear region and rising to $\\sim$ 8 Gyr at larger radii \\citep{worthey04,rose05,coelho09}. \nEven though the nucleus is a two-component system, with a dominant or primary disk component \nembedded in a more spherically-distributed pressure-supported secondary component, there is \nno observed break in the properties of these stellar populations as a function of radius \\citep{seth10}. \n\nThe typical velocity dispersion in the M32 nucleus is $\\sim$ 60 km\/s, but rises to $\\sim$ 120 km\/s \nvery close to the central SMBH \\citep{seth10}. Using Equation~\\ref{eqn:rinf}, this yields an \ninfluence radius for the central SMBH of only $\\lesssim$ 0.7 pc. Thus, we use Equation~\\ref{eqn:gamma10} \nwith $\\Sigma =$ 0 to calculate the collision rates in M32, appropriate to a dynamically hot pressure-supported \nnuclear cluster, and transition smoothly between these formulae as gravitational focusing becomes important. \nAs in the MW, however, we correct for the influence of the central SMBH by replacing the isothermal velocity dispersion \nwith the local velocity dispersion given by Equation~\\ref{eqn:sigloc}, with $\\sigma_{\\rm 0} =$ 60 km\/s and \nM$_{\\rm BH} =$ 2.5 $\\times$ 10$^6$ M$_{\\odot}$. \n\nFor the surface mass density profile, we adopt the best-fitting solution to \nthe V-band surface brightness profile from \\citet{lauer98}, which is a Nuker-law fit of the form:\n\\begin{equation}\n\\label{eqn:sbm32}\n\\Sigma(r) = 2^{(\\beta-\\gamma)\/\\alpha}{\\Upsilon}\\Sigma_{\\rm 0}\\Big( \\frac{r_{\\rm b}}{r} \\Big)^{\\gamma}\\Big[1 + \\Big( \\frac{r}{r_{\\rm b}} \\Big)^{\\alpha}\\Big]^{(\\gamma-\\beta)\/\\alpha},\n\\end{equation} \nwhere $\\Upsilon =$ 2.0 is the V-band mass-to-light ratio (in M$_{\\odot}$\/L$_{\\odot}$), $\\alpha =$ 1.39, \n$\\beta =$ 1.47, $\\gamma =$ 0.46, r$_{\\rm b} =$ 0\".47 and $\\Sigma_{\\rm 0}$ is the central surface mass \ndensity in L$_{\\odot}$pc$^{-2}$, calculated from the V-band surface brightness $\\mu_{\\rm 0} =$ 12.91 \ngiven in \\citet{lauer98} assuming a distance to M32 of 770 kpc. In calculating the collision rates using \nEquation~\\ref{eqn:gamma10}, we assume $\\Sigma$ $=$ 0 (since there is no Keplerian disk component). Assuming \nspherical symmetry, we can obtain \nthe stellar density distribution from the observed surface brightness profile using Equation 3.65a in \\citet{merritt13}:\n\\begin{equation}\n\\label{eqn:rhofromsb}\n\\rho(r) = -\\frac{1}{\\pi} \\int_r^{\\infty} \\frac{d\\Sigma}{dR} \\frac{dR}{\\sqrt{R^2 - r^2}}.\n\\end{equation}\nwhich is independent of any assumptions for the gravitational potential. Equation~\\ref{eqn:rhofromsb} is the \nclassical Abel inversion of a projected surface brightness profile \\citep[e.g.][]{bracewell65}. Note that, in M32, the break \nradius corresponds almost exactly to the outer radial limit we adopt for the calculated collision rates, \nor r$_{\\rm b} \\sim$ 2 pc. Inside the break radius, the approximation $\\rho$(r) $\\sim$ $\\Sigma$(r)\/r is \nreasonable for power-law galaxies, where \n$\\Sigma$(r) is the observed surface brightness profile, as given by Equation~\\ref{eqn:sbm32}. \n\n\nAs shown in Figure~\\ref{fig:fig3}, the predicted collision rates for a particular object show a rise toward \nr $=$ 0, increasing by $\\lesssim$ two orders of magnitude over the inner $\\sim$ 1 pc. For WD+RGB and 1+2 collisions, \nhowever, the central increase is much less significant. Upon integrating \nnumerically, the predicted numbers of collision products remain approximately constant with \nincreasing distance from the galaxy centre, as shown in Figure~\\ref{fig:fig4}. We see only a slight rise in \nthe numbers of collision products at very small radii $\\lesssim$ 0.1 pc, similar to what is predicted for M31. Thus, \nwe do not expect a significant central concentration of blue stars in M32 (albeit perhaps a mild one at \n$\\lesssim$ 0.1 pc that would be difficult to identify observationally, at least currently), assuming a collisional \norigin (ignoring any possible mass segregation of collision products). Interestingly, M32 is the only \ngalaxy in our sample for which this is the case, while also being the only galaxy in our sample \nthat does not show any observational evidence for a central blue excess. \n\nBeyond $\\gtrsim$ 0.1 pc, the integrated numbers of collision products are high \nrelative to the other galaxies in our sample (ignoring M31); the predicted \nnumbers of MS+MS, MS+WD and MS+RGB collisions produced over a Hubble time should be of order a few percent \nof the total number of stars \nin the M32 nucleus, as shown in Table~\\ref{table:one}. This naively suggests that the observed light distribution in \nthe M32 NSC, and hence its stellar mass function, could have been non-negligibly affected by collisions. However, The \nMS+MS collision rates for a given MS star are too low for any massive \nO- and B-type stars to have formed from multiple collisions. And yet, the collision rates are high enough that collisions \ncould be connected with the lack of recent star formation in M32. Collisions could perhaps act to inhibit star \nformation by, for example, colliding with protostars sufficiently early on in their formation that \nthe protostellar embryos are destroyed.\n\n \n\nAs in both the MW and M31, the rate of 1+2 collisions in the M32 \nnucleus is sufficiently high that most binaries should have been destroyed by the present-day (since the \nhard-soft boundary is at only 0.04 AU), \nand a total mass in gas of order $\\sim$ 10$^3$ M$_{\\odot}$ should be supplied to the inner nucleus \nevery $\\sim$ 100 Myr. This is because on the \norder of $\\sim$ 1.4 $\\times$ 10$^3$ MS+WD collisions happen every 100 Myr (see Table~\\ref{table:one}). \nHence, following the same assumptions as in the preceding sections, if \n1.4 $\\times$ 10$^4$ MS+WD collisions occur per 1 Gyr, this predicts \n$\\sim$ 0.7 $\\times$ 10$^4$ M$_{\\odot}$ $\\sim$ 10$^4$ M$_{\\odot}$ in gas should be supplied to the \ninner nucleus every Gyr due to MS+WD collisions alone. This is enough gas for star formation to occur, \nignoring any other potentially complicating effects. More generally, it predicts that non-negligible amounts of \ngas should be present in the nucleus at any given time, roughly independent of any recent star \nformation or the details of gas heating\/cooling.\n\n\nAdditionally, as shown in Figure~\\ref{fig:fig3}, the rates of MS+RGB collisions are \nonly sufficiently high in the inner r $\\lesssim$ 0.1 pc for any given RGB star to undergo multiple (up \nto $\\sim$ 100 or more) collisions within 100 Myr. This suggests that only in the very inner nucleus could RGB \nstars in M32 be fully stripped, and appear as EHB stars. This predicts a paucity of giants in the very inner \n$\\lesssim$ 0.1 pc of the M32 nucleus.\n\nIn summary, the single-single collision rates are sufficiently high that the numbers of collision products \nproduced over a Hubble time should be of order a few percent of the total number of stars \nin the M32 nucleus. These rates are roughly independent of distance from the cluster \ncentre, apart from a brief but sharp rise near the origin. Hence, only in the inner $\\lesssim$ 0.1 pc of the nucleus \ncould the collision rates ever be sufficiently high to create a (significant) blue excess via MS+MS collisions. \nIndividual RGB stars could have undergone multiple collisions only within r $<$ 0.1 pc, with \nsufficient frequency for their envelopes to become fully stripped and their photometric appearance significantly \naffected. Finally, the rates of MS+WD collisions are sufficiently high to supply \nsignificant ($\\lesssim$ 10$^4$ M$_{\\odot}$ every 1 Gyr) gas to the nucleus. This predicts non-negligible quantities of gas to \nbe present in the M32 nucleus at present (assuming they did not already form stars; but see Section~\\ref{sf}).\n\nFinally, we note that \\citet{yu03} obtained similar results to what we report here, namely that collisions \nalone are unlikely to produce any colour gradients in M32, consistent with observations. However, we point out that the \ncollision rates are sufficiently high for a significant fraction of stars in the cluster to have undergone at least one collision over a \nHubble Time. This could significantly impact the stellar mass function and hence luminosity profile. The weak \ndependence of the collision rate on clustercentric distance suggests that this effect would \naffect the overall colour of the M32 nucleus roughly uniformly (with only a slight central blue excess). This \ncould directly affect any age determination assigned to the M32 nucleus from stellar population synthesis models \n(likely causing an under-estimation). Only in \nthe inner $\\lesssim$ 0.1 pc could the collision rates ever become sufficiently high for individual objects to undergo \nmultiple collisions. Here alone, our results predict a paucity of RGB stars. Thus, similar to \\citet{yu03}, we \nconclude that collisions are unlikely to be able to cause the appearance of an age gradient in M32. \n\n\\subsection{M33} \\label{m33}\n\nM33 is a nearby spiral galaxy lacking a significant bulge component. Intriguingly, this could be connected \nwith the lack of any detected SMBH in M33. \\citet{gebhardt01} placed an upper limit of 1500 M$_{\\odot}$ on the \nmass of any SMBH that could reside in the nucleus, using three-integral dynamical models fit to Hubble Space \nTelescope WFPC2 photometry and Space Telescope Imaging Spectrograph spectroscopy. \nThe NSC at the centre of M33 is \nextremely compact, with a central density approaching that of M32 \\citep{kormendy93,lauer98}. The \ncentral velocity dispersion is only 21 km\/s, and reaches central mass densities of at least \n2 $\\times$ 10$^6$ M$_{\\odot}$ pc$^{-3}$ \\citep{lauer98}. Assuming an SMBH mass of 1500 M$_{\\odot}$, \nthis yields an influence radius of only 0.01 pc, using Equation~\\ref{eqn:rinf}. Importantly, this is only an \nupper limit for the SMBH mass. There could be no SMBH in M33. Indeed, the presence of an additional strong \nX-ray source \\citep{long96,dubus99} \nin the nucleus so close to the putative SMBH suggests against the presence of a massive \ncentral BH, since SMBHs are very effective at tidally disrupting X-ray binaries and even ejecting their compact remnant \ncompanions from the NSC and even host galaxy \\citep{leigh14,giersz15}.\n\nWe use Equation~\\ref{eqn:gamma10} with $\\Sigma =$ 0 to calculate the collision rates in M33, which are \nappropriate to a dynamically hot pressure-supported nuclear cluster. We note that in M33 we find \nv$_{\\rm esc} >$ $\\sigma$ for all types of collisions, and gravitational focusing must always be taken into \naccount. For the surface mass density profile, we again adopt the best-fitting solution to \nthe V-band surface brightness profile from \\citet{lauer98}, which includes an analytic core in this case:\n\\begin{equation}\n\\label{eqn:sbm33}\n\\Sigma(r) = {\\Upsilon}\\Sigma_{\\rm 0}\\Big[1 + \\Big( \\frac{r}{a} \\Big)^{2}\\Big]^{-0.745},\n\\end{equation} \nwhere $\\Upsilon =$ 0.4 is the V-band mass-to-light ratio (in M$_{\\odot}$\/L$_{\\odot}$), a $=$ 0.1 pc is the \napproximate half-power radius, $\\Sigma_{\\rm 0}$ is the central surface mass density in L$_{\\odot}$pc$^{-2}$, \ncalculated from the V-band surface brightness $\\mu_{\\rm 0} =$ 10.87 given in \\citet{lauer98} assuming a \ndistance to M33 of 785 kpc. Although Equation~\\ref{eqn:sbm33} is consistent with the observed surface \nbrightness profile in M33, we note that \\citet{lauer98} also considered a model with an inner cusp, and \n\\citet{carson15} adopted Sersic models.\n\nAs in the preceding section, when calculating the collision rates using \nEquation~\\ref{eqn:gamma10}, we assume $\\Sigma$ $=$ 0 (since there is no Keplerian disk component). We adopt \nthe stellar density distribution provided in Equation 2 of \\citet{merritt01b}. This yields results \nthat are nearly identical to what we find upon integrating over the observed surface brightness profile of the nucleus. \nThat is, assuming spherical symmetry, we can obtain \nthe stellar density distribution from the observed surface brightness profile using Equation~\\ref{eqn:rhofromsb}, as \nbefore, and requiring that the luminosity density at r $=$ 1 pc is 10$^6$ L$_{\\odot}$ pc$^{-3}$. Note that, in M33, \nthe core radius corresponds almost exactly to the inner radial limit we adopt for the \ncalculated collision rates, or a $\\sim$ 0.1 pc. Outside the core radius, the surface brightness profile approximately \nresembles a power-law form, and the approximation $\\rho$(r) $\\sim$ $\\Sigma$(r)\/r is reasonable. \n\n\nFigure~\\ref{fig:fig3} illustrates that the predicted collision rates for a particular object decrease rapidly with \nincreasing distance from the galaxy \ncentre, dropping by about three orders of magnitude over the inner $\\sim$ 1 pc. In Figure~\\ref{fig:fig4}, \nwe see a similar behaviour and the total rates for any pair of objects follow a monotonic decrease with increasing \ndistance from the galaxy centre, dropping by about two orders of magnitude over the inner $\\sim$ 1 pc. As shown \nin Table~\\ref{table:one}, the predicted total \nnumbers of MS+MS, MS+WD and MS+RGB collisions over a Hubble Time is greater than the total number of \nstars in the central (i.e., the core) NSC. The number of 1+2 collisions are sufficiently high that almost if not every binary \nshould have undergone \na direct encounter over the cluster lifetime. However, in general, a non-negligible number of binaries should survive \nin M33, given its low velocity dispersion of $\\sim$ 21 km\/s, which yields a hard-soft boundary of 0.6 AU. This is \nvery close to the hard-soft boundaries of typical Milky Way globular clusters \\citep{harris96}, which are known to \nhave binary fractions on the order of a few percent \\citep[e.g.][]{sollima07,sollima08,milone12}. \n\nA total mass in gas comparable to the total NSC mass in the core should be supplied \nto the inner nucleus every $\\sim$ 1 Gyr due to ablating MS+WD collisions. This is because on the \norder of $\\sim$ 4.9 $\\times$ 10$^3$ MS+WD collisions happen every 100 Myr (see Table~\\ref{table:one}). \nHence, following the same assumptions as in the preceding sections, if \n4.9 $\\times$ 10$^4$ MS+WD collisions occur per 1 Gyr, this predicts \n$\\sim$ 2.5 $\\times$ 10$^4$ M$_{\\odot}$ $\\sim$ 10$^4$ M$_{\\odot}$ in gas should be supplied to the \ninner nucleus every Gyr due to MS+WD collisions alone. This is less than the total mass of the M33 nucleus by about \nan order of magnitude. This also implies that stars should have formed recently if the gas is able to become sufficiently \ndense to cool. As in M32, this further predicts that significant gas \nshould be present in the nucleus at any given time, roughly independent of any recent star \nformation or the details of gas heating\/cooling. \nFinally, the collision rates are never high enough in M33 for individual RGB stars \nto undergo multiple MS+RGB collisions within 100 Myr. Thus, we do not expect a paucity of RGB stars in \nthe M33 nucleus. \n\n\nTo summarize, the photometric appearance of the M33 nucleus should have also been significantly \naffected by collisions; the total number of collision products should be of order $\\lesssim$ 10\\% of the total \nnumber of stars in the inner nucleus. An excess (by roughly an order of magnitude) of blue stars from MS+MS \ncollisions is expected at r $\\lesssim$ 0.1 pc. \nHowever, individual MS and RGB stars should not have undergone multiple collisions with any regularity. Since \nof order $\\sim$ 100 collisions are \nneeded to fully strip the RGB envelope and significantly affect its photometric appearance, our results do not predict any \nobserved paucity of RGB stars. Unlike the other nuclei in \nour sample, however, binary destruction due to single-binary encounters should be comparably inefficient, due to \nthe much lower velocity dispersion. This predicts a binary fraction of order $\\lesssim$ a few percent, and possibly a \nlarge number of X-ray binaries due to exchange interactions involving hard binaries and stellar remnants.\n\n\n\\section{Discussion} \\label{discussion}\n\nIn this section, we discuss the implications of our results for the origins of the centrally concentrated \nblue stars observed (or not) in the four Local Group galactic nuclei considered here and, more generally, for \nobserving exotic or enigmatic stellar populations in \ngalactic nuclei, including blue straggler stars, extreme horizontal branch stars, young or recently formed \nstars, X-ray binaries, etc. Our results suggest that collisional processes could \ncontribute significantly to the observed blue excesses in M31 and M33, via several different channels. Below, \nwe highlight the relevant physics for the development of improved theoretical models, which are needed for \ncomparison to existing and future observational data. \n\n\n\n\n\n\n\n\\subsection{Young stars and\/or recent star formation} \\label{sf}\n\nThe presence of very blue stars in the inner nuclear regions is typically argued to be evidence for a recent \n(in situ) burst of star formation. This is because the timescale for such a cluster of young stars to inspiral into \nthe nucleus via dynamical friction tends to exceed the age of the stars, for all but extremely close initial \ndistances from r $=$ 0. Many authors have proposed channels via which such in situ star formation might \noccur so close to a central SMBH \\citep[e.g.][]{chang07,nayakshin06}.\n\nGiven the current state of the observations, we can neither rule out nor confirm an in situ star formation origin for \nany blue excesses observed in our sample, at least when considering individual nuclei. Upon consideration \nof the entire sample at once, however, more stringent constraints can perhaps be placed. For example, \nif the star formation occurs in a single burst, some fine-tuning must be required to produce 3 out of 4 galactic \nnuclei with recent star formation \nwithin $\\lesssim$ 0.1 pc of the galactic centre at the current epoch. This is because, relative \nto the lifetimes of the older populations in these NSCs, the young blue stars would live for only a \nvery short time. Thus, it is unlikely that all three nuclei were caught in the act of recent star formation, unless \nthe process is continuous or episodic with a short recurrence time. If correct, then significant star formation \nmust have previously occurred in \nthe nucleus as well. It follows that the most recent burst would have to occur in the presence of a significant \npopulation of remnants orbiting within $\\lesssim$ 0.1 pc. This predicts that a high \nmass-to-light ratio could also be present along with any inner blue excess, and that significant X-rays could \nbe observable due to gas accretion onto these inner remnants (if gas is still being supplied to the inner nucleus; \nsee below).\n\nInterestingly, as shown in Section~\\ref{app}, the rates of MS+WD collisions are sufficiently high \nthat a significant and steady supply of gas could be supplied to the inner nuclei. This is because, unlike \nMS+MS collisions, collisions between MS stars and WDs do not require very high relative velocities to ablate the star \n\\citep{shara77,shara78}. The WD is sufficiently compact that, upon penetrating the star, a shock wave is sent \nthrough it, increasing the temperature in the outer layers by roughly an order of magnitude. This in turn triggers \nCNO burning, which liberates enough energy to unbind most of the envelope. The collision of a WD with a massive \nMS star is hence disruptive, with the energy source unbinding the MS star being due only in small part to the kinetic \nenergy of the non degenerate star \\citep{regev87}. Thus, MS+WD collisions can supply \ngas to the nucleus independent of its velocity dispersion, which sets the typical relative velocity at infinity for direct \ncollisions. In the MW and M31, enough gas could be supplied to the inner NSC via MS\/RGB+WD collisions to account \nfor the mass observed in blue stars via star formation every $\\lesssim$ 1 Gyr. In the inner $\\lesssim$ 0.1 pc of M31 \nand M32, the rates of MS+WD \ncollisions are the highest, such that the recurrence time for star formation in this scenario should be considerably \nshorter. For comparison, Equation 8 in \\citet{generozov15} gives the mass input from stellar winds, parameterized \nby the fraction $\\eta$ of the stellar density being recycled into gas. With a lower limit of $\\eta \\ge$ 0.02 (and taking into \naccount the star formation efficiency), our results \nsuggest that the rate of gas supplied by ablative WD collisions could be comparable to the rate of gas supplied by stellar \nwinds in the surrounding old population. This could represent a significant correction to steady-state models of \ngas inflow\/outflow in galactic nuclei \\citep[e.g.][]{quataert04,shcherbakov10,generozov15}.\n\nWe emphasize that MS+MS collisions should contribute little to the total gas supply in the nuclei in our sample. This is \nbecause the velocity dispersions are sufficiently low that the total kinetic energy at impact is only ever a small fraction of \nthe binding energy of the collision product. This is the case even upon consideration of the entire spectrum of possible \nrelative velocities at impact, appropriate for a Maxwellian velocity distribution. Only in the inner $\\lesssim$ 0.1 pc \nof the M31 nucleus does the velocity dispersion ever become sufficiently high for MS+MS collisions to ablate the collision \nproduct and contribute to the total gas reservoir.\n\nIf young stars are indeed formed from violently liberated gas during collisions then, in a sample of NSCs with \ninner blue stars, we would expect a correlation between the (integrated) NSC collision rate and the total \nmass in blue\/young stars. With this in mind, we can already say that M32 would be a significant outlier in such a \nrelation, ignoring any over-looked effects that could potentially inhibit star formation preferentially \nin the M32 nucleus (e.g. the M32 NSC is by far the densest in our sample, such that collisions could even \nserve to inhibit star formation). Additionally, \nthis scenario predicts approximately solar abundance for such collisionally-derived \nyoung stars, without any significant helium enrichment (assuming that the collisionally-destroyed MS stars \nare members of the original old stellar population). This is because only a small fraction of the total mass \nactually undergoes enhanced CNO burning during the collision.\n\n\\citet{generozov15} recently calculated steady-state one-dimensional hydrodynamic profiles for hot gas in \nthe immediate vicinity of an SMBH. The authors provide an analytic estimate for the \"stagnation radius\" in their \nEquation 14, defined as the distance from the SMBH at which the radial velocity of the gas passes through zero: \n\\begin{equation}\n\\label{eqn:rstag}\nr_{\\rm s} = \\frac{GM_{\\rm BH}}{\\nu(v_{\\rm w}^2 + \\sigma_{\\rm 0}^2)}\\Big( 4\\frac{M(r < r_{\\rm s})}{M_{\\rm BH}} + \\frac{13 + 8\\Gamma_{\\rm s}}{4 + 2\\Gamma_{\\rm s}} - \\frac{3\\nu}{2 + \\Gamma_{\\rm s}} \\Big),\n\\end{equation}\nwhere M(r $<$ r$_{\\rm s}$) is the total stellar mass inside the stagnation radius, $\\Gamma_{\\rm s}$ is the two-dimensional power-law \nslope of the stellar light distribution (not to be confused with the collision rate), $\\nu$ is the three-dimensional power-law \nslope of the gas inflow profile (defined by Equation 13 in \\citet{generozov15}) and v$_{\\rm w}$ is an effective heating \nparameter that describes the energy deposited from stellar winds, supernovae and BH feedback. For an old \nstellar population, we expect v$_{\\rm w} \\sim$ 100 km\/s. Inside the stagnation radius \ngas flows inward, whereas outside this radius gas flows outward. In the MW and M31, the velocity dispersion is sufficiently \nhigh that the stagnation radius lies outside the SMBH sphere of influence, and beyond our outer limit of integration when \ncalculating the integrated number of collision products (i.e. 2 pc). Thus, all of the gas liberated from MS\/RGB+WD collisions \nin our calculations should quasispherically flow inward, until it circularizes at its angular momentum barrier. In M32, \nwe expect r$_{\\rm s} \\gtrsim$ r$_{\\rm inf}$, and the numerical solution of the 1D hydro equations shows that r$_{\\rm s} =$ 2 pc \nfor v$_{\\rm w} =$ 100 km\/s (Aleksey Generozov, private communication). \nIn M33, assuming that no SMBH is present, the stagnation radius will be close to zero since v$_{\\rm w} \\gg$ $\\sigma_{\\rm 0}$, as \ncan be seen from taking the limit M$_{\\rm BH} \\rightarrow$ 0 in Equation~\\ref{eqn:rstag}. Numerical experiments confirm that r$_{\\rm s} \\ll$ 1 pc for v$_{\\rm w} \\sim$ 100 km\/s (Aleksey Generozov, private communication). Hence, any (hot) gas liberated by collisions should be lost to a quasispherical outflow. Finally, \nwe note that the preceding estimates for the stagnation radius assume that stellar winds dominate the energy injection rate. However, \nin the inner nuclei of M31 and M32, the energy injection due to collisions could dominate, causing an increase in v$_{\\rm w}$ and a subsequent decrease in the stagnation radius. \n\n\nIn all nuclei in our sample, our results suggest that the 1+2 collision rate is sufficiently high that most \nstellar binaries should be driven to merge, destroyed (via collisions) or dissociated very early on in the cluster lifetime, compared to \nthe age of an old stellar population ($>$ a few Gyr). It follows that most binaries \\textit{currently} present could be \nvery young (mostly likely formed due to recent star formation). This predicts that, if \nyoung binaries are observed in a galactic nucleus, this could be a smoking gun for recent star formation, since \nno other mechanism considered in this paper or presented in the literature should yield a bright blue \nstar with a binary companion. Star formation, on the other hand, is expected to yield very high \nbinary fractions for massive stars \\citep[e.g.][]{sana12}. Provided that star formation occurred sufficiently \nrecently, most of these binaries should still be present. Observationally, unresolved binaries should broaden\nthe color distribution (at a given magnitude) along the inferred main-sequence, relative to a coeval population \nof single stars. With that said, our results presented in Section~\\ref{bmkl} suggest that the time-scales for \nKozai-Lidov oscillations to operate are sufficiently short very close to a central SMBH that all binaries \nwith initially high inclinations should merge on relatively short time-scales, leaving both binaries and merged \nbinaries in young nuclear environments. \n\nTo summarize, our results suggest that collisions could offer an additional pathway toward re-fueling an inner \nnucleus with gas for recent star formation. MS+WD collisions could contribute significantly to the total gas supply needed \nto form the centrally concentrated blue stars in every nuclei in our sample, with the possible exception of M32 and \nperhaps M33 (due to a relatively small stagnation radius for gas inflow\/outflow). In nearly all cases, the relative velocities are \nnot high enough to ablate the stars during MS+MS collisions, except within the inner $\\sim$ 0.1 pc of the M31 nucleus where \nthe velocity dispersion reaches $\\sim$ 1000km\/s. Here, however, MS+MS collisions could also offer an additional source of \ngas for star formation, as originally suggested by \\citet{spitzer66} and \\citet{spitzer67}. During MS+WD collisions, on the other \nhand, the relative velocity at impact plays only a \nminor role in deciding how much gas is liberated. This is because the strong surface gravity of the WD is primarily responsible \nfor generating the high temperature shock front that triggers explosive CNO burning in the MS star, as opposed to the kinetic \nenergies of the objects at impact (see \\citet{shara77} and \\citet{shara78}). \n\n\\subsection{Missing giants and extreme horizontal branch stars} \\label{ehb}\n\nIf the envelope of an RGB star is stripped, the hot core is revealed and the star settles onto the extreme horizontal \nbranch in the cluster colour-magnitude diagram before dimming to join the white dwarf cooling sequence \n\\citep[e.g.][]{davies98,amaroseoane14}. If only a single grazing collision occurs, only a small fraction of the \nRGB envelope should end up unbound. Although the photometric appearance of the resulting RGB star can still be \nslightly affected by a single mass loss episode, our results presented in Section~\\ref{ehbform} suggest that almost all \nof the RGB envelope must be stripped before its subsequent evolution and photometric appearance are significantly \naffected, which should require multiple collisions. \n\nIt is not clear how many MS+RGB collisions are needed for an RGB star to be collisionally stripped by MS stars to \nthe point of forming an hot EHB star. The number could range from on the order of 10 to hundreds \\citep[e.g.][]{dale09}. \nThus, the relevant rate here is that for a \\textit{specific} RGB star to undergo collisions with MS stars (i.e. Figure~\\ref{fig:fig3}), \nwhich is lower than the rate for \\textit{any} RGB star to undergo collisions by a factor N$_{\\rm RGB}$, \nwhere N$_{\\rm RGB}$ is the total number of RGB stars. The rates \nof MS+RGB collisions are sufficiently high in only the inner $\\lesssim$ 0.1 pc of M31 and M32 \nthat this mechanism could potentially produce EHB stars (and hence a paucity of RGB stars) in the inner nuclear \nregions. Given that the \nEHB lifetime is only on the order of 10$^8$ years \\citep[e.g.][]{maeder09,leigh11b}, however, \nit seems unlikely that any observed blue excess is due to EHB stars, since the implied rate of EHB formation \nmust be unrealistically high. Although BHs are expected to \nbe able to remove a larger fraction of the envelope on a per encounter basis \\citep{dale09}, Figure~\\ref{fig:fig3} \nsuggests that only in perhaps M33 could the rate of BH+RGB collisions be of comparable significance \nto MS+RGB collisions for removing RGB envelopes. A \ndeficit of late-type giants can perhaps be looked for observationally using the presence or absence of CO bandhead \nabsorption to distinguish late-type from early-type stars (see \\citet{genzel96} for more details). \n\n\n\n\n\\subsection{Blue straggler stars} \\label{bss}\n\nBlue straggler stars are rejuvenated main-sequence stars, identifiable in the cluster colour-magnitude\ndiagram as being brighter and bluer than the main-sequence turn-off \\citep{sandage53}. BSs are thought \nto form from a number of channels in both open and globular clusters, including direct collisions \nbetween main-sequence stars \\citep[e.g.][]{shara97,sills97,sills01,leigh07,leigh11}, mass-transfer in \nbinary systems from an evolved donor onto a normal MS star \n\\citep[e.g.][]{mccrea64,mathieu09,geller11,geller13,knigge09,leigh11b} and mergers of MS-MS binaries \n\\citep[e.g.][]{chen09,perets09}. In M31, M32 and M33, the rate of MS+MS collisions is sufficiently high to supply, \nin as little as a few Gyrs or less, a total mass in MS+MS collision products that is a significant fraction of the total \nmass of the inner nuclear regions, and could thus contribute non-negligibly to the observed light distribution. \n\nWith the exception of M33, the relaxation times of the NSCs in our sample tend to be on the order of a few \nGyr or more \\citep{merritt13}. This is longer than the expected age of a typical collision product formed from an old \npopulation \\citep[e.g.][]{sills97,sills01}. It follows that mass segregation of collision products formed at larger radii \ninto the inner nuclear regions should have only a minor impact on deciding the observed numbers of collision \nproducts here.\n\nWith that said, any tension due to an over-abundance of predicted BSs can perhaps be reconciled by the \nfact that the products of direct MS+MS collisions should be puffed up post-collision due to the deposition of \nkinetic energy \\citep[e.g.][]{sills97,fregeau04}. This increases the collisional cross-section and reduces the \ntime-scale for subsequent collisions, which could act to strip the collision product of its inflated envelope. This \nwould serve to reduce the acquired mass in collision products. The envelope should contract on a \nKelvin-Helmholtz time-scale, which can be comparable to the time for subsequent collisions to occur. Finally, \nwe note that more massive collision products should have shorter lifetimes, which reduces their probability of \nactually being observed. \n\nIf BSs are produced abundantly, then we might naively also expect an over-abundance of RGB stars, since \nthe heavier BSs should evolve onto the giant branch on shorter time-scales than the rest of the MS \npopulation. As explained in the next section, these evolved BSs might be rapidly destroyed via MS+RGB \nand WD+RGB collisions, however we do not expect this to be the case in the MW and M33. And yet, in the \nMW NSC, Table~\\ref{table:one} suggests that, over a 1 Gyr period, we expect on the order of 10$^3$ MS+MS \ncollisions in the inner $\\sim$ 2 pc. Although this is only a small fraction of the total number of RGB stars, this excess \nis nonetheless puzzling in the MW NSC, since a paucity of giants has been observed, not an excess \n(but see the last paragraph in Section~\\ref{MW}). \n\nVery roughly, the rates of 1+2 collisions are sufficiently high in every NSC in our sample that most primordial binaries \nshould have been destroyed by the present-day (with the exception of M33). It follows that our assumed binary fractions \ncould be over-estimates, \nin spite of correcting for the fraction of hard binaries using the observed velocity dispersion. More importantly, this \nsuggests that not enough binaries should be present \nto contribute significantly to BS production via either binary coalescence or binary mass-transfer. Only those \nbinaries born in a near contact state have sufficiently small cross-sections (i.e. near contact) to survive for \n$\\gtrsim$ 1 Gyr. This is because the orbital separation corresponding to \nthe hard-soft boundary is near contact in nuclei with such high velocity dispersions. For \nvery close binaries near contact, the probability of a merger or direct collision occurring during a 1+2 \ninteraction approaches unity \\citep[e.g.][]{fregeau04,leigh12}. Hence, most direct 1+2 encounters that do not \nresult in a direct stellar collision should be dissociative, or at least widen the binary and reduce the time-scale for a \nsubsequent (likely dissociative) encounter to occur. It follows that the rate of direct 1+2 encounters \nshould be approximately equal to the rate of binary destruction very early on in the cluster lifetime. For those \nfew near contact binaries that are able to survive for the bulk of the cluster lifetime, if stable mass-transfer is \ninitiated then mass, energy and angular momentum conservation dictate that the binary orbit should \n(eventually) widen, significantly increasing the geometric cross-section and reducing the time-scale for a \ndirect 1+2 encounter \\citep[e.g.][]{leigh16}. Similarly, if unstable mass-transfer is initiated, then a common \nenvelope could form, \nwhich would also (at least temporarily) increase the geometric cross-section for collision and reduce the \n1+2 collision time. Thus, we expect that the very few BSs that might form from binary mass-transfer should \nquickly lose their binary companions post-formation, due to dissociative or destructive 1+2 encounters \\citep{leigh16}. \nThe simple calculations performed here should be verified and better quantified in future studies, using more \nrealistic binary fractions and distributions of binary orbital parameters.\n\n\nFinally, we consider one additional mechanism for BS formation, which has not yet been considered. This is \nthe accretion of significant amounts of gas from the interstellar medium (ISM) onto a normal old MS star. The gas \ncould be supplied to the ISM by MS+WD and even MS+MS collisions (if the velocity dispersion is very high, which \nis the case in only M31 and only in its inner $\\lesssim$ 0.1 pc; see Section~\\ref{sf}), and\/or \nmass loss from evolving stars in the surrounding old population. The gas collects at the \ntidal truncation radius (if an SMBH is present) due to the focusing of gas particle orbits in an axisymmetric potential \n\\citep{chang07}. This gas is then channeled inward on a viscous time-scale, where we are assuming it is accreted by \nold main-sequence stars already orbiting there. \nAs discussed by \\citet{nayakshin06} for a similar scenario, as the gas density builds up, feedback from stars can serve \nto stop fragmentation \nof the gas, while accretion onto pre-existing old stars could proceed at very high rates. Although this scenario \nrequires an initial old and centrally-concentrated stellar population to provide the seeds for accretion, it is \nappealing since it ultimately requires a much smaller gas reservoir than the recent burst of star \nformation scenario, while also avoiding the special conditions required to induce the fragmentation of \na giant molecular cloud in such an extreme environment so close to an SMBH. We caution that more work needs \nto be done to quantify the expected accretion rates in this scenario, which are poorly known.\n\nThus, accretion onto pre-existing old MS stars also predicts \nan inner disk of blue stars in the M31, M32 and M33 nuclei, assuming that the accretion rates of the ISM onto the \nold stars are sufficiently high. This mechanism is also appealing in that, unlike a recent burst of star formation, \nit is a \\textit{continuous} mechanism for rejuvenating old MS stars. Indeed, we might naively expect the mean BS mass \n(and hence luminosity) for BSs formed from this mechanism in the inner nucleus to correlate with the age of the \nsurrounding old population, since older stars will have had longer to expel their winds into the ISM, and to accrete \nthe polluted material. This mechanism also predicts \nlow binary fractions, a very low central mass-to-light ratio and (possible) surface abundance spreads throughout the \ncluster \\citep[e.g.][]{lauer98,seth10}, depending on the quantity and composition of the accreted material.\n\nWe conclude that all nuclei in our sample should harbour an abundance of blue stragglers, formed mostly from MS+MS \ncollisions. Accretion onto old MS stars already orbiting in the inner nuclear regions offers an additional possible pathway \ntoward BS formation. Our results suggest that the rates of MS+WD collisions are sufficiently high to contribute significantly \nto the gas reservoir \nrequired for this mechanism to explain the observed blue stars. These \"non-binary mass transfer BSs\" should have \nproperties tightly correlated with those of the underlying old stellar population.\n\n\\subsection{Implications for inferred age spreads} \\label{inferred}\n\nAs illustrated in Table~\\ref{table:one}, our results predict non-negligible numbers of collision products formed \nover only a 100 Myr interval that can be a significant fraction of the total number of stars in the nuclear cluster. To help \ndistinguish blue stars formed via collisions from young stars and test for a true age gradient in a given NSC, \nparticular attention can be put into deriving the radial dependence of the collision rate as a function of the host \ncluster environment. The collision number \nprofiles can be converted into corresponding surface brightness profiles, and from there into radial colour profiles. \nThese collisional colour profiles can then be subtracted from the observed colour profiles, after multiplying the collisional \ncolour profiles by an appropriate constant to ensure that no over- or under-subtraction occurs -- the normalization \nshould be applied at large radii, or where the predicted collision rates are at their lowest. If any \nresidual colour gradient is present, this suggests that its origin is a recent burst of star \nformation. If no residual colour gradient remains and the U-V or U-B colours turn out to be approximately \nconstant as a function of distance from r $=$ 0, then the observed colour gradient is indeed consistent with \na collisional origin. This test can be performed on the Local Group nuclei when future observations with \nsuperior resolution capabilities become available and, in particular, more accurate velocity information. \n\nA number of collisional mechanisms considered in this paper could contribute to the appearance of a colour gradient, \nand should be accounted for when looking for a true age gradient. \nIn every nucleus in our sample, our results predict non-negligible numbers of MS+MS collision products, or blue stragglers. \nIn the inner $\\lesssim$ 0.1 pc of M31 and M32, the collision rates are sufficiently high that individual \nRGB stars could undergo multiple \ncollisions within 100 Myr. If this were to result in a paucity of RGB stars and a corresponding excess of single EHB stars, then \nthe contribution of RGB-stripping to observed colour gradients is two-fold; the loss of RGB stars reduces the total amount of \nred light and the production of EHB stars can increase the total amount of blue light. Thus, the stripping of RGB \nstars could also contribute to the appearance of an age gradient, provided the numbers of MS+RGB, WD+RGB and \nBH+RGB collisions are extremely high and centrally concentrated. \n\nFinally, we note that future studies considering age gradients in nearby galactic nuclei should include the \nLocal Group dwarf spheroidal galaxy NGC 205, which is known to harbor bright blue (and possibly young) stars. We \nhave not done so in this paper, since the rate of collisions is expected to be much lower than in the other NSCs in our \nsample \\citep{valluri05}, due to a lower central surface brightness. \n\n\\subsection{Binary mergers due to Kozai-Lidov oscillations} \\label{bmkl}\n\nThe key point to take away from Equation~\\ref{eqn:taukl} is that, for any binary within a distance r$_{\\rm SC}$ \nfrom the centre of the galaxy with a sufficiently large angle of inclination relative to its orbital plane about the SMBH \n(i.e. i $\\gtrsim$ 40$^{\\circ}$), the \nKozai-Lidov time-scale is much shorter than the cluster age, for any old (i.e. $\\gtrsim$ a few Gyrs) stellar \npopulation. For example, for a binary with a separation of 1 AU and component masses \nof 1 M$_{\\rm \\odot}$ on a circular orbit about the SMBH with a semi-major axis equal to the \nscale radius of the nucleus, the Kozai-Lidov time-scale given by Equation~\\ref{eqn:taukl} is 4.8 $\\times$ 10$^4$, \n1.0 $\\times$ 10$^2$ and 1.5 $\\times$ 10$^5$ years in, respectively, the MW, M31 and M32. We also calculate \nthe critical distance r$_{\\rm SC}$ from the SMBH beyond which Kozai-Lidov oscillations are suppressed by general \nrelativistic precession. This is 0.04 pc, 0.15 pc and 0.04 pc for, respectively, the MW, M31 and M32.\n\nThese Kozai-Lidov time-scales are significantly shorter than the ages of the surrounding old stellar populations, \nas well as the collision times \npresented in Section~\\ref{app}. It follows that, over sufficiently long time-scales (much shorter than the cluster age), the \nrate of binary mergers due to Kozai-Lidov oscillations \nshould be determined by the time-scale for single stars to scatter the binary orbital plane (and reduce the distance from \nthe SMBH to $\\lesssim$ r$_{\\rm SC}$) into \nthe active Kozai-Lidov domain. Vector resonant relaxation acting to re-orient the binary orbital plane could contribute \nto either an increase or decrease in this rate (see \\citet{antonini12b} for more details). Regardless, in all nuclei in our \nsample, the rate of direct 1+2 collisions is \nsufficiently high relative to the age of the old stellar population that only very compact binaries should \nsurvive to the present-day, since the hard-soft boundary is near contact in these galactic nuclei due to their high velocity \ndispersions and the collision probability for a direct encounter with small impact parameter is close to unity for such compact \nbinaries \\citep{leigh12}. The \ncollision product should expand after the encounter by a factor of a few \\citep[e.g.][]{sills97,fregeau04}, causing it to merge \nwith its binary companion nearly independent of the semi-major axis or expansion factor. This is \nthe case in every NSC in our sample, with the exception of M33 due to its lower velocity dispersion. \nIf the binary progenitors are destroyed, then Kozai-Lidov oscillations cannot operate. \nTherefore, we \ndo not expect binary mergers due to Kozai-Lidov oscillations to have contributed significantly to any of the anomalous \nblue star populations currently observed in the inner regions of the (old) NSCs in our sample of galaxies. \n\nAlthough the mass segregation timescale for a massive population can approach the cluster age in the NSCs in our \nsample, heavy objects could still have been delivered via this mechanism to the inner nuclear regions in non-negligible \nnumbers by the present-day (provided they are born at sufficiently small clustercentric radii). BH-BH binary mergers \ninduced by Kozai-Lidov oscillations with the central SMBH \ncould still be relatively high at the present epoch relative to the rates of stellar binary mergers \n\\citep[e.g.][]{vanlandingham16}. However, these BH-BH binaries \nshould either be primordial or have formed very far out in the NSC and only recently migrated inward. This is because the rate \nof \\textit{stellar} binary destruction is expected to be very high. If few binaries exist at the present epoch, then BHs cannot \nbe exchanged in to them via dynamical encounters to form BH-BH binaries. Additionally, due to the presence of \nthe massive central BH, BH binary formation via tidal capture can be unlikely since the timescale for this is comparable \nto the timescale on which the BH binary will experience a strong encounter with the central SMBH and itself \nbe stripped of its companion \\citep[e.g.][]{leigh14}.\n\n\\subsection{X-ray binaries and millisecond pulsars} \\label{xrays}\n\nIn order to estimate the numbers of X-ray binaries using our derived collision rates (i.e. the number of dynamically formed \nX-ray binaries), we need to know the fractions of \nstellar remnants, namely white dwarfs f$_{\\rm wd}$, neutron stars f$_{\\rm ns}$ and black holes f$_{\\rm bh}$. Unfortunately, \nthese numbers are highly uncertain, due to uncertainties in the initial stellar mass function at the high-mass end (which are \nsignificant), neutron star kicks at birth which can eject them from the nucleus, etc. Regardless, we can make general \ncomments about the expected frequencies of X-ray binaries in our sample of nuclei, which is also highly sensitive \nto the cluster binary fraction f$_{\\rm b}$. Based on our results for the 1+2 collision rates, we expect very few binaries \nto have survived until the present-day. This is due both to the high 1+2 collision rates, but also to the separations \ncorresponding to the hard-soft boundaries in such high velocity dispersion environments, which are near contact. With \nthis in mind, it seems more likely that the nuclei with the lowest velocity dispersions should harbor the largest \nbinary fractions, namely M32 and M33, which have hard-soft boundaries of 0.04 AU and 0.6 AU, respectively. In both \nM32 and M33, the 1+2 collision rates are also the highest, such that exchange encounters should occur very \nfrequently. Thus, we might expect a larger fraction of cataclysmic variables, NS and BH X-ray binaries as well as \nmillisecond pulsars (MSPs) in the inner nuclear regions of M32 and especially M33, relative to the MW and \nM31 which are home to more hostile environments for long-term binary\nsurvival in the inner $\\lesssim$ 1 pc. \nWith that said, \\citet{leigh14} recently showed (which was subsequently confirmed by \\citet{giersz15}) that the \npresence of a massive black hole near the cluster centre can \nefficiently destroy X-ray binaries, lowering their numbers significantly relative to what would be predicted in the \nabsence of a massive BH. Thus, putting this all together, we expect M33 to have the largest fraction \nof X-ray binaries per unit cluster mass, since it has both the lowest velocity dispersion and the smallest upper limit on \nthe SMBH mass. Indeed, constraining the formation of X-ray binaries in the M33 nucleus \ncould help to place constraints on the possible presence of a central BH, which remains controversial and \nis at best, of very low mass (i.e. $\\lesssim$ 1500 M$_{\\rm \\odot}$). \n\nFinally, we note that various processes not considered in our study may effectively replenish the nuclear regions \nof galaxies with binaries and compact objects, possibly enhancing the formation of X-ray binaries and MSPs in NSCs.\nFor example, nuclear stellar clusters may result from the continuous infall of star clusters that disrupt close to the \nSMBH \\citep[e.g.,][]{2015ApJ...812...72A}. Such stellar clusters may harbor MSPs, X-ray binaries as well as\nlarge populations of stellar remnants. In addition, \\citet{2009ApJ...698.1330P} suggest that the disruption of triple stars \ncould leave behind a binary in a close orbit around an SMBH; the rate of triple disruptions (brought in to the inner \nnuclear regions at late times either via NSC infall or mass segregation) could be high enough to \nserve as a continuous source of binaries close to the SMBH.\n\n\\section{Summary} \\label{summary}\n\nIn this paper, we consider the origins of the enigmatic stellar populations observed in the \nnuclear star clusters of Local Group galaxies, specifically the Milky Way, M31, M32 and M33. \nThese curious populations are blue stars found in the inner $\\lesssim$ 0.1 pc of three out of \nthe four galactic nuclei considered here. The origins of these centrally-concentrated blue stars are not \nknown. Several candidates have been proposed in the literature, including blue straggler stars, extended horizontal \nbranch stars and young recently formed stars. Here, \nwe calculate rough order-of-magnitude estimates for various rates of collisions, as a \nfunction of the host nuclear cluster environment and distance from the cluster centre. We subsequently \nquantify the contributions to the enigmatic blue stars from BSs, extended horizontal branch stars \nand young recently formed stars. \n\nThe collision rates are sufficiently high that blue stragglers, formed via direct collisions between single main-sequence stars, \ncould contribute non-negligibly ($\\sim$ 1-10\\%) to the observed surface brightness profiles, in nearly (with the \nexception of the MW) every nucleus in our sample. However, \nthe radial profiles of the collision products do not always predict a strong central concentration of these objects, as \nin the MW (for our assumed density profile, which has a constant density core), M32 and, to a lesser extent, M31. \nIn M31 and M32, the rates of MS+RGB collisions might only be sufficiently high for \nindividual giants to undergo multiple collisions within $\\lesssim$ 0.1 pc from the galaxy \ncentre. Only here could a paucity of giants be observed, since the results of stellar evolution models presented in \nthis paper suggest that the envelope \nmust be nearly fully stripped to destroy an RGB star and leave an EHB star in its place. \n\nThe rates of collisions between white dwarfs and MS stars, which are expected to typically ablate \nthe MS stars, are sufficiently high that this could offer a steady supply of gas to the inner nucleus and \nsubsequently contribute non-negligibly to the total gas reservoir needed to seed star formation in every galactic \nnucleus in our sample. This scenario seems the most likely \n(dominant?) of those considered in this paper for explaining the origins of the inner blue stars, relative to a \ndirect collisional origin. However, we suggest that the gas might be more likely to simply accrete onto \npre-existing old stars rather than fragment and form new stars. This could offer a new channel for BS \nformation or the rejuvenation of old MS stars, by creating \"non-binary mass transfer blue stragglers\" in the \ninner nucleus. We emphasize, however, that more work needs to be done to better constrain the expected \naccretion rates in this scenario, which are poorly known.\n\nOur results suggest that collisional processes could contribute significantly to the observed blue excesses in \nM31 and M33, via several different channels. More sophisticated theoretical models are needed, however, in \norder to properly compliment existing and future observations. \nFor example, accurate collision surface brightness profiles, which can be derived from theoretical collision \nnumber profiles (such as those presented in this paper), can be converted into radial colour profiles, multiplied \nby a suitable constant (to ensure not under- or over-subtracting) and subtracted from the observed \nradial colour profiles. Any remaining colour gradient observed in the residuals should then be indicative \nof a real age gradient in any high-density unresolved stellar population. We caution that this method is best \nsuited to quantifying only the \\textit{gradient} in an observed age spread, and not the relative sizes of the underlying blue \nand red populations.\n\n\\section*{Acknowledgments}\n\nWe would like to kindly thank Barry McKernan and Saavik Ford for useful discussions and suggestions. \nNL acknowledges the generous support of an AMNH Kabfleisch Postdoctoral Fellowship. \nFA acknowledges support from a NASA Fermi Grant NNX15AU69G and from a CIERA postdoctoral fellowship at Northwestern\nUniversity. Financial support was provided to NS by NASA through Einstein Postdoctoral Fellowship Award Number PF5-160145. \nFA and DM acknowledge insightful conversations with Eugene Vasiliev that stimulated them to conduct the \ncalculation presented in Section~\\ref{ehbform}. Financial support was granted to DM by the National Science Foundation under \ngrant no. AST 1211602 and by the National Aeronautics and Space Administration under grant no. NNX13AG92G.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\\IEEEPARstart{T}{he} number of electric vehicles on the road is expected to reach 125 million by 2030, generating 404 TWh of additional electricity demand \\cite{bunsen2018global}.\nCharging these EVs cleanly, affordably, and without excessive stress on the grid will require advances in charging system design, hardware, monitoring, and control. Collectively we refer to these advances as smart charging. Smart charging will substantially reduce the environmental footprint of transportation while unlocking immense potential for demand-side management.\n\nSmart charging is especially crucial for large-scale charging facilities such as those in workplaces, apartment complexes, shopping centers, airports, and fleet charging facilities. Providing charging at these diverse sites is vital to the widespread adoption of electric vehicles. Doing so can reduce range anxiety and provide an alternative to personal charging ports for those who cannot install them at their homes. Since many of these sites will provide charging during daytime hours, they can make use of abundant solar energy production and enable EVs to provide grid services throughout the day. However, with current technology, most sites are unable to install more than a few charging ports due to limited infrastructure capacity and fear of high electricity bills. Smart charging allows sites to scale their port capacity without costly infrastructure upgrades. Moreover, scheduling algorithms can reduce operating costs by optimizing for time-of-use tariffs, demand charges, and on-site renewable generation.\n\\iflong{Algorithms can also enable additional revenue streams by providing grid services.}\n\nIn this paper, we report the design, implementation, and application of a smart charging system that we call an Adaptive Charging Network (ACN). In Section~\\ref{sec:overview}, we describe the architecture through the lens of the first ACN, which was built on the Caltech campus in 2016, shown in Fig.~\\ref{fig:ACN_photo}. The ACN enables real-time control and monitoring of charging systems at scale. It has spawned a company, PowerFlex Systems, which operates over 100 similar charging systems around the United States. \\iflong{These include national laboratories, universities, schools, businesses, apartment complexes, hotels, and public parking facilities.}\n\nThrough our experience building large-scale charging facilities, we find that common assumptions made in theoretical models do not hold in many practical systems. We describe these in Section~\\ref{sec:key-insights}. This makes it challenging to apply algorithms proposed in the literature directly. In order to develop practical and robust algorithms for the ACN, we present the Adaptive Scheduling Algorithm (ASA) framework for online (causal) smart charging algorithms based on convex optimization and model predictive control (MPC). We describe the ASA framework in Section~\\ref{sec:scheduling-framework}. Then, in Section~\\ref{sec:applications}, we demonstrate ASA's performance in the context of maximizing energy delivery in congested infrastructure and minimizing operating costs, including demand charge using simulations based on real data collected from the ACN.\n\n\\begin{figure}[!t]\n\\includegraphics[width = \\columnwidth]{figs\/ACN_photo}\n\\centering\n\\caption{The ACN smart EV charging testbed at Caltech.}\n\\label{fig:ACN_photo}\n\\end{figure}\n\nBeyond serving as a model for smart charging systems, the ACN has led to the creation of the ACN Research Portal. This portal has three parts: ACN-Data, a collection of real fine-grained charging data collected from the Caltech ACN and similar sites \\cite{lee_ACN-Data_2019}; ACN-Sim, an open-source simulator which uses data from ACN-Data and realistic models derived from actual ACNs to provide researchers with an environment to evaluate their algorithms and test assumptions \\cite{lee_acnsim_2019}; and ACN-Live, a framework for safely field testing algorithms directly on the Caltech ACN. Thus the ACN has proven to be a valuable tool in both commercial and academic environments. \n\\section{Acknowledgment}\nLarge-scale infrastructure projects like the ACN require herculean effort and coordination between researchers, administrators, private companies, and funding agencies. In addition to the authors, the ACN project would not have been possible without the efforts of John Onderdonk from Caltech Facilities, Stephanie Yankchinski and Neil Fromer from the Resnick Sustainability Institute, Jennifer Shockroo and Fred Farina from Caltech Office of Technology Transfer, and Roger Klemm from the JPL Green Club. Technology development was aided by data provided by Roff Schreiber of Google and generations of Caltech and international students since 2016, especially the work of Karl Fredrik Erliksson of Lund University. We would also like to thank PowerFlex Systems and EDF Renewables North America for their continued support, development, and commercialization of the ACN project.\n\\section{Case Studies} \\label{sec-case-studies}\nTo demonstrate the efficacy and flexibility of our online scheduling framework, we now consider a series of case studies. These case studies consider a variety of operator objectives which have been expressed by real site owners. All case studies are run using real data collected from the Caltech ACN. In addition, when considering profit maximization, we use realistic revenues and Southern California Edison's time-of-use rate schedule, making our case studies not only informative of our scheduling framework's performance, but also of the business potential of operating large-scale EV charging systems.\n\n\\subsection{Preliminaries}\n\\subsubsection{Data}\nFor these case studies we consider real data collected from the Caltech ACN as part of ACN-Data between May 1, 2018 and October 1, 2018. During this period our system served 10,415 charging sessions and delivered a total of 92.78 MWh of energy. Table \\ref{tab:summary_stats} shows relevant summary statistics for this dataset. We also show the distribution of arrivals and departures in the system in Fig.~\\ref{fig:arrival_departure_dist}. These distributions can help us understand when the system is likely to be highly loaded. In addition, the distribution of arrivals relative to the TOU rate schedule helps us to understand how much shifting is necessary to reduce energy costs in our system. All simulations were conducted using ACN-Sim \\cite{lee_acnsim_2019}.\n\n\\begin{table}[t!]\n\\centering\n\\captionsetup{justification=raggedright}\n\\caption{\\hspace{15pt} Average Statistics for EV Charging Test Cases Per Day\\newline\nMay 1, 2018 - Oct. 1, 2018}\\label{tab:summary_stats}\n\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{tabular}{l|c|c|c|c|c|}\n\\cline{2-6}\n & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Daily\\\\Sessions \\end{tabular} &\n \\begin{tabular}[c]{@{}c@{}}Mean\\\\Session\\\\Duration\\\\(hours)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Session\\\\Energy\\\\(kWh)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Daily\\\\Energy\\\\(kWh)\\end{tabular} &\n \\begin{tabular}[c]{@{}c@{}}Max\\\\Concurrent\\\\Sessions\\end{tabular}\\\\ \\hline\n\\multicolumn{1}{|l|}{Sun} & 41.32 & 3.94 & 415.06 & 10.05 & 18\\\\\n\\multicolumn{1}{|l|}{Mon} & 71.00 & 6.14 & 677.13 & 9.54 & 42\\\\\n\\multicolumn{1}{|l|}{Tues} & 76.73 & 6.24 & 685.79 & 8.94 & 47\\\\\n\\multicolumn{1}{|l|}{Wed} & 75.45 & 6.22 & 660.16 & 8.75 & 44\\\\\n\\multicolumn{1}{|l|}{Thurs} & 78.50 & 5.96 & 665.21 & 8.47 & 42\\\\\n\\multicolumn{1}{|l|}{Fri} & 77.18 & 6.71 & 697.41 & 9.04 & 43\\\\\n\\multicolumn{1}{|l|}{Sat} & 43.32 & 5.01 & 439.59 & 10.15 & 18\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}[t!]\n\\includegraphics[width = 0.8\\columnwidth]{figs\/Weekday_Arrival_Departure_Distribution_Oct_1.pdf}\n\\centering\n\\caption{Average arrivals and departures per hour for the period May 1, 2018 - Oct. 1, 2018. Background shading depicts the TOU rate at each time, with light grey, grey, and dark grey depicting off-peak, mid-peak, and peak pricing respectively. On weekdays we can see that most drivers arrive at the beginning of the morning mid-peak period and leave prior to the end of the peak period. Weekends, however have a much more uniform distribution of arrivals and departure owing to the heterogeneity of drivers weekend schedules.}\n\\label{fig:arrival_departure_dist}\n\\end{figure}\n\n\\subsubsection{Baseline Algorithms}\nFor comparison we consider a number of baseline algorithms which are included in ACN-Sim. These include Uncontrolled Charging, Round Robin (RR), Earliest Deadline First (EDF), Least-Laxity First (LLF), and Greedy Cost Minimization. In addition, we compare with the offline optimal to provide an upper bound on what any online algorithm could achieve. It is found by solving (\\ref{eq:SCH.1}) with the relevant utility function and perfect knowledge of future arrivals. \n\n\\subsubsection{Adaptive Scheduling Algorithm Setup}\nFor all case studies we consider the three-phase infrastructure present in the Caltech ACN. We set the length of each time slot, $\\delta$, to 5 minutes and consider a maximum optimization horizon of 12 hours. For all cases except those in Section~\\ref{sec:practical_considerations} we do not consider a maximum recompute period, $P$, as we use an ideal battery management system model for these tests and EVs are assumed to follow the control signal exactly. In Section~\\ref{sec:practical_considerations} we use more realistic models and set $P=5$ minutes.\n\n\\subsection{Energy Delivery in Oversubscribed Infrastructure}\nIn our first case study we consider the objective of maximizing total energy delivered. This objective is used when the operator's primary goal is driver satisfaction or when profit per unit energy at a site is constant. In these cases the primary purpose of adaptive scheduling is to meet driver demand with minimal infrastructure investment. To satisfy this operator objective we use the Adaptive Scheduling Algorithm (Alg. \\ref{alg:MPC}) with utility function\n\n\\begin{equation}\n U^\\mathrm{QC} := V^{QC} + 10^{-14}V^{ES}\n\\end{equation}\n\nHere $U^{QC}$ encourages the system to deliver as much energy as possible as quickly as possible which helps to free capacity for future arrivals.\nWe refer to this algorithm as SCH-Quick Charge (SCH-QC).\n\nIn order to demonstrate the effect of highly constrained infrastructure, we vary the capacity of transformer $t_1$ between 0 and 100 kW. For reference, the actual transformer in our system is 150 kW and a conventional system for \\numevse{} 7kW EVSEs would require 378 kW of capacity. We then measure what percent of the total energy demand can be met using SCH-QC as well as four baseline scheduling algorithms and the offline optimal. The results of this test are shown in Fig.~\\ref{fig:energy_delivered_const_infra}.\n\nFrom Fig.~\\ref{fig:energy_delivered_const_infra} we can see that SCH-QC performs near optimally in terms of the percent of demand met and with it we are able to meet 100\\% of driver demand with just 70 kW of transformer capacity even in the online setting. Meanwhile other baseline algorithms struggle in highly constrained infrastructure.\n\n\\zach{This section is interesting but less relevant, so we could remove it.}Among the baselines algorithms, we note that RoundRobin outperforms the other baselines for extremely low transformer capacities. This is because RoundRobin is able better balance between phases which the sorting based methods do not consider. As capacity increases, however, EDF and LLF outperform RoundRobin, as prioritizing drivers by their energy demands and deadline becomes more important than squeezing out capacity from the system. SCH-QC is able to factor in driver demands and deadlines while also balancing phases, which helps to explain its superior performance.\n\n\\begin{figure}[t!]\n\\includegraphics[width=\\columnwidth]{figs\/GreedyInfrastructureDerate}\n\\centering\n\\caption{Percentage of driver's energy demands which can be met at varying capacities for transformer $t_1$ for the month of Sept. 2018. Here demand met is defined as the ratio of total energy delivered to total energy requested.\n}\n\\label{fig:energy_delivered_const_infra}\n\\end{figure}\n\n\\subsection{Profit Maximization with TOU Rates}\\label{sec:case-no-dc}\nWe next consider the case where an operator would like to maximize their profit subject to time-of-use electricity tariffs. For this case study we adopt the Southern California Edison TOU rate schedule for separately metered EV charging systems between 20-500~kW which is shown in Table~\\ref{tab:tou_rates} \\cite{choi_general_2017}. We do not, however, consider a demand charge for this case, as we will do so in the next case study. We consider a fixed revenue of \\$0.30 \/ kWh.\\footnote{While this may seem high, it is actually reasonable as it can be sourced not only from the driver but also from subsidies and by selling credits in carbon markets such as the California Low-Carbon Fuel Standard program which is currently trading at the equivalent of \\$0.14~\/~kWh as of Oct. 2018.}\n\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{table}[t!]\n\\centering\n\\captionsetup{justification=raggedright}\n\\caption{\\hspace{15pt} SCE TOU Rate Schedule for EV Charging}\\label{tab:tou_rates}\n\\begin{tabular}{c|c|}\n\\cline{2-2}\n\\multicolumn{1}{l|}{} & Rates \\\\ \\hline\n\\multicolumn{1}{|c|}{On-Peak (12pm-6pm)} & \\$0.25 \/ kWh \\\\ \\hline\n\\multicolumn{1}{|c|}{Mid-Peak (8am-12pm, 6pm-11pm)} & \\$0.09 \/ kWh \\\\ \\hline\n\\multicolumn{1}{|c|}{Off-Peak (11pm-8am)} & \\$0.06 \/ kWh \\\\ \\hline\n\\multicolumn{1}{|c|}{Demand Charge} & \\$15.48 \/ kW \/ month \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nTo facilitate maximizing profits considering only energy charges, we propose two algorithms based on \\textbf{SCH}. The first, denoted SCH-Energy Charge (SCH-EC), uses the utility function\n\\begin{equation}\n U^\\mathrm{EC} := V^{EC} + 10^{-14}V^{ES}\n\\end{equation}\n\n\\noindent and the second, denoted SCH-Energy Charge Quick (SCH-ECQ) uses\n\n\\begin{equation}\n U^\\mathrm{ECQ} := V^{EC} + 10^{-4}V^{QC} + 10^{-14}V^{ES}\n\\end{equation}\n\nWe add the $V^{QC}$ term to SCH-ECQ in order to free capacity for future arrivals by encouraging charging early in each rate period. Since rates are flat during each period, we can front load charging within a period without increasing energy costs.\n\nWe consider the profit gained by our algorithms as well as the baselines for the five month period beginning in May 2018. The results of this test are shown in Fig.~\\ref{fig:profit_max_no_dc}. Here we can see that Uncontrolled, FCFS, and Round Robin perform poorly relative to the optimal.\\footnote{Somewhat surprisingly uncontrolled charging generates higher profits than FCFS and Round Robin. This is because uncontrolled charging does not have to contend with infrastructure limits. This, combined with the arrival patterns shown in Fig.~\\ref{fig:arrival_departure_dist}, means that uncontrolled charging is able to deliver more energy during mid-peak (rather than on-peak) hours than FCFS or Round Robin.} This is expected, as these algorithms do not consider pricing signals at all. The IndividualGreedy algorithm does quite well in this test, achieving 98.7\\% of the optimal profit available. However, the algorithms we propose based on our framework perform even better, achieving 99.58\\% and 99.99\\% for SCH-EC and SCH-ECQ respectively. As expected SCH-ECQ out-performs SCH-EC.\n\n\\begin{figure}[t!]\n\\includegraphics[width = \\columnwidth]{figs\/ProfitMaxNoDC.pdf}\n\\centering\n\\caption{Operator profit achieved via various scheduling approaches when using SCE's TOU rate schedule without demand charge.\n}\n\\label{fig:profit_max_no_dc}\n\\end{figure}\n\n\\subsection{Profit Maximization with Demand Charge}\\label{sec:pm_w_dc}\nWe next consider the more difficult task of maximizing profit in the presence of TOU rates and demand charge. We still consider the same rate schedule in Table~\\ref{tab:tou_rates}, but now with the additional demand charge component. Demand charges are assessed on the maximum power draw of a customer over the billing period which we assume to be one month. These charges can be very high, especially in uncontrolled EV charging systems where many drivers might arrive in close succession and charge at concurrently leading to high peak demand.\n\n\\begin{figure}[t!]\n\\includegraphics[width = \\columnwidth]{figs\/ProfitMaxDcCombined}\n\\centering\n\\caption{Operator profit achieved via various scheduling approaches when using SCE's TOU rate schedule with demand charge.}\n\\label{fig:profit_max_dc}\n\\end{figure}\n\nTo demonstrate the importance of considering demand charge when scheduling, we can evaluate the profit of scheduled found by SCH-EC and SCH-ECQ when including demand charge. This is shown in Fig.~\\ref{fig:profit_max_dc}. Here we can see that demand charge was reduced the profit obtained by SCH-EC and SCH-ECQ by 55\\% and 60\\% respectively. Moreover, these algorithms under perform the optimal by 35\\% and 45\\% respectively. Note that when demand charge is considered, SCH-ECQ performs worse than SCH-EC. This is because SCH-ECQ leads to higher peaks in demand. We also note that the baseline algorithms we consider also perform poorly when demand charge is considered. This is especially true of uncontrolled charging, which leads to high peaks in demand which severely reduces profitability.\n\nThis motivates us to consider demand charge in our scheduling problem. To do this we use the utility function\\footnote{Note that while SCH-ECQ fared worse than SCH-EC when considering demand charge, we still include $V^{QC}$ in this formulation. This is because $V^{DC}$ will enforce a low peak, so we can still use $V^{QC}$ to encourage charging quickly when possible.}\n\\begin{equation}\n U^\\mathrm{DC} := V^{DC} + 10^{-4}V^{QC} + 10^{-14}V^{ES}\n\\end{equation}\n\nHere $V^{DC}$ as two tunable parameters the demand charge proxy, $\\hat{\\Delta}$, and the previous\/projected peak demand, $L_0^{max}$. Careful selection of these parameters is necessary as demand charge is a ratcheting cost which must be considered over the entirety of the billing period. Meanwhile, the ASA framework is designed to work on small sub-intervals of this period. If we try to naively include demand charge into our objective by setting $\\hat{\\Delta}$ to the equivalent of \\$15.48 \/ kW and $L_0^{max}=0$, no charging would occur as the marginal cost of increasing peak demand would be far greater than the marginal benefit of charging the EVs.\n\nTo account for this we consider three parameter configurations. The first we call SCH-Demand Charge Proportional (SCH-DCP). In this configuration we scale the true demand charge, $\\Delta$, by the percentage of the billing period which is covered by the optimization horizon to arrive at $\\hat{\\Delta}$. In this case we begin the billing period with $L^{max}_0=0$ then update it, prior to each iteration, to be the highest peak so far in the billing cycle. We see in Fig.~\\ref{fig:profit_max_dc} that this results in a significant improvement over not considering demand charge at all, but still achieves only 84.5\\% of the optimal profit available. By examining the schedules produced by SCH-DCP, we see that while it achieves lower energy costs that the optimal, it has a much higher demand charge. This is likely because $\\hat{\\Delta}$ is too low.\n\nTo address this, we can instead prorate the demand charge on a daily basis, e.g. $\\hat{\\Delta} := \\Delta \/ 30$. We denote this configuration SCH-Demand Charge Daily (SCH-DCD). The intuition behind this change is that usage in our system is extremely low overnight. Thus the scheduling algorithm should at least consider the demand charge it will generate over a full 24 hour period. We see in Fig.~\\ref{fig:profit_max_dc} that this change increases performance to be within 93.6\\% of optimal.\n\nWhile SCH-DCD offers good performance, there is still room for improvement. We note that selecting the correct peak usage is extremely important to maximizing profit. So rather than using a demand charge proxy, we consider what would happen if we knew a-priori what the optimal peak from the offline case would be.\nWe could then set $L_0^{max}$ to be this optimal peak current and $\\hat{\\Delta}=\\Delta$. We denote this configuration SCH-Demand Charge Oracle (SCH-DCO) since we rely on an oracle to supply the optimal peak. We can see in Fig.~\\ref{fig:profit_max_dc} that SCH-DCO performs near optimally (within 99.6\\%).\n\nHowever, in practice, we do not have access to an oracle to provide the optimal peak demand, so we cannot use SCH-DCO directly. We must instead predict the optimal peak. One way to do so is to run OfflineOptimal over the previous 28 day period and use the resulting peak as an estimate of the optimal peak. We repeat this procedure daily (at midnight), so that the algorithm is able to adapt to changing usage patterns. At the beginning of each day, we set $L_0^{max}$ to be the max of the peak usage so far in that billing period and our estimate of the optimal peak. We denote this configuration SCH-Demand Charge Reflective (SCH-DCR). In Fig.~\\ref{fig:profit_max_dc} we can see that SCH-DCR is very close to optimal, achieving 97.7\\% of the available profit.\n\nWhile SCH-DCR is able to outperform SCH-DCD, there is still a trade-off between the two. First, SCH-DCD does not require access to historical system usage. This is important for new sites and for sites which do not wish to store past usage. Second, the two algorithms have very different behavior during high usage periods. Because SCH-DCR essentially fixes its peak using historical data, some vehicles will not be charged if doing so would violate this peak. Meanwhile, SCH-DCD has a much lower bar for raising its peak usage, meaning that more of the EV demand would be met but at higher cost over the entire billing period. Thus when choosing between the two, the site operator should consider which behavior better aligns with their goals.\\footnote{It is also possible to run both algorithms with an equality energy constraint, which would mean that the demand for all drivers would be met. However, doing this in practice could lead to incredibly high costs for the operator.}\n\n\\subsection{Practical Considerations} \\label{sec:practical_considerations}\nIn previous case studies we have ignored some of the more difficult issues which arise from practical EV charging systems, namely non-ideal battery behavior and additional (discrete) constraints on individual charging rates. In this section we address the effect of incorporating these constraints into our scheduling problem.\n\nWe once again consider the problem of profit maximization with demand charge. We use the best practical algorithm we have considered, SCH-DCR, as a base and measure performance relative to the profit achieved in the previous section where ideal EV and EVSE models were used. The effect of including these practical EV and EVSE considerations on the operator's profit are shown in Fig.~\\ref{fig:practical_considerations}.\n\n\\begin{figure}[t!]\n\\includegraphics[width = \\columnwidth]{figs\/PracticalConsiderations}\n\\centering\n\\caption{Operator profit achieved when considering practical EV charging models as a percent of profit when considering ideal EV and EVSE models. Profits are aggregated over June 1 through Oct. 1.\n}\n\\label{fig:practical_considerations}\n\\end{figure}\n\n\\subsubsection{Non-ideal battery behavior}\nWe first address the effects of non-ideal battery behavior. We consider a two-stage battery charging model. The first stage, referred to as \\emph{bulk} charging, occurs up to 80\\% state of charge. In this stage current draw, neglecting changes in pilot, is near constant. The second stage, called \\emph{absorption}, finishes charging the remaining 20\\%. In this stage, current decreases as the battery reaches full charge. To simulate this we use the Linear2Stage battery model included in ACN-Sim \\cite{lee_acnsim_2019}.\n\nWe account for this non-ideal battery behavior by computing a new solution to our scheduling problem periodically. This allows our algorithm to update its schedule to account for deviations from the assigned charging rates caused by battery behavior. For this test we set the minimum recompute period to be equal to the length of one period, 5 minutes. From Fig.~\\ref{fig:practical_considerations} we see that non-ideal battery behavior leads to approximately a 5.3\\% reduction in operator profit.\n\n\\subsubsection{Charging rate restrictions}\nWe next consider the limitations on charging rate imposed by EVSEs and battery management systems. As mentioned in Section \\ref{sec:key-insights} most EVSEs on the market, including those used in our ACN testbed, provide only a discrete set of allowable pilot signals. For this test we use $\\rho := \\{0, 6, 7, ..., 31, 32\\}$, which is the allowable rate set for AeroViroment EVSEs. In addition, the J1772 standard forbids charging rates in the range of 0-6 A and experience has shown us that some EVs do not recover after being assigned at 0 A pilot signal, meaning EVs should receive at least a 6 A pilot from when they plug in until their demand is met. These factors motivate the post-processing heuristic described in Section \\ref{sec:post_processing}.\n\nTo demonstrate the effect of accounting for these restrictions on operator profit, we first run SCH-DCR then run the resulting schedule through our post-processing steps (\\ref{eq:post_processing}) and (\\ref{eq:rate_projection}) with $\\gamma = 10^{-2}$. We once again set the maximum recompute period to be 5 minutes (1 period) so that SCH-DCR is able to account for deviations caused by the post-processing. From Fig.~\\ref{fig:practical_considerations} we see that accounting for charging rate restrictions using the post-processing approach we propose leads to approximately a 10\\% reduction in operator profit.\n\n\\subsubsection{Practical model}\nIn practice we must contend with both non-ideal battery behavior and rate restrictions. As such, in our final scenario we consider both factors simultaneously. The result, as seen in Fig.\\ref{fig:practical_considerations}, is an approximately 12.5\\% reduction in operator profit relative to SCH-DCR with the ideal EV\/EVSE model. This demonstrates that our proposed algorithms are robust when applied to more realistic models of EV and EVSE behavior and capabilities.\n\\zach{It would be nice to formulate the offline optimal problem as a mixed integer program and solve. But I don't know if this is feasible.}\n\\section{Related Work}\nSeveral smart EV charging systems have been developed, though usually at a smaller scale than the ACN. The Smart Energy Plaza (SEP) at Argonne National Laboratory consists of six controllable level-2 EVSEs\\cite{bohn_real_2016}. Likewise, the WinSmartEV system at UCLA consists of quad-port EVSEs capable of sharing a single oversubscribed circuit and multiple level-1 chargers with binary control \\cite{chynoweth_smart_2014, chung_master-slave_2014}. My Electric Avenue tested a control system to managed charging from 200 Nissan LEAFs to manage congestion in the distribution system \\cite{quiros-tortos_how_2018}. The Parker project utilized a testbed of 10 bi-directional EVSEs at a commercial site to investigate the potential of EVs to provide frequency regulation services and adapt to marginal emissions signals \\cite{andersen_parker_2019}.\n\nThere is also a tremendous literature on EV charging algorithms; see surveys \\cite{wang_smart_2016, mukherjee_review_2015} for extensive pointers to the literature. \nSome recent work specifically related to large-scale EV charging includes \\cite{chen_iems_2012,yu_intelligent_2016, wang_two-stage_2016, wang_predictive_2017, nakahira_smoothed_2017, zhang_optimal_2017, frendo_real-time_2019, alinia_online_2020}.\n\nSeveral works have been based on the ACN and data collected from it. \\cite{lee_ACN-Data_2019} uses data from the ACN to predict user behavior, size solar generation for EV charging systems, and evaluate the potential of EV charging to smooth the duck curve. \\cite{sun_electric_2020} clusters sessions based on their charging behavior using time series of charging current collected from the ACN. \\cite{lee_pricing_2020} proposes a pricing scheme to allocate costs (including demand charge) to charging sessions. \\cite{li_real-time_2020, schlund_flexability_2020} use data to quantify fleet-level flexibility within a charging facility, and \\cite{al_zishan_adaptive_2020} uses ACN-Sim to train reinforcement learning agents to schedule large-scale EV charging. \n\n\\iftoggle{long}{Our work adds to the existing body of research insights from building real-world EV charging systems. It}\n{This work}\nextends \\cite{lee_adaptive_2016, lee_large-scale_2018}, which describe the ACN in earlier stages of development. It provides a more thorough description of the ACN\n\\iflong{architecture and algorithms}{and ASA} \nand demonstrates ASA's effectiveness in realistic scenarios through simulation.\n\\section{Practical Challenges from the Testbed} \\label{sec:key-insights}\nBy building and operating the Caltech ACN, we have identified several important features of the physical system that have not been addressed in the EV charging literature but pose real problems for implementing practical EV scheduling algorithms. \nAmong these are proper modeling of the unbalanced three-phase electrical network, incorporation of EVSE quantization, and adaptation to non-ideal battery behavior.\nWe describe these models in this section and explain how we incorporate them into an MPC framework in Section~\\ref{sec:scheduling-framework}. These models also form the basis of the component models included in ACN-Sim \\cite{lee_acnsim_2019}.\n\n\\subsection{Infrastructure modeling}\\label{sec:three_phase_model}\nAs discussed in Section~\\ref{sec:physical_system}, the electrical infrastructure within the ACN is oversubscribed and often unbalanced. In our data, we observe that without proper control, these phase imbalances can be significant%\n\\iflong{, as seen in Fig.~\\ref{fig:phase_imbalance}}. \nWhile many algorithms have been proposed to handle charging with an aggregate power limit or even a hierarchy of limits, most previous work has focused on single-phase or balanced three-phase systems, making them inapplicable to charging systems like the ACN. An exception to this is the work of De Hoog et al. \\cite{de_hoog_optimal_2015}, which considers an unbalanced three-phase distribution system but only in the case of wye-connected EVSEs. In contrast, the EVSEs in the ACN and most large charging systems in the United States are connected line-to-line%\n\\iflong{, as shown in Fig.~\\ref{fig:circuit-diag}}. \n\n\\iflong{%\n\\begin{figure*}[htbp]\n\\includegraphics[width =0.88\\textwidth]{figs\/ACN_Circuit_Diagram}\n\\centering\n\\caption{Circuit diagram depicting the connection loads within the California Parking Garage. For simplicity, transformer $t_2$ is omitted and all EVSEs between phases A and B have been lumped together as $I_{ab}^{evse}$, and so forth for BC and CA.}\n\\label{fig:circuit-diag}\n\\end{figure*}\n}\n\n\\iftoggle{long}{%\n \\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\columnwidth]{figs\/LineCurrentsSept10}\n \\caption{Line currents from uncontrolled charging. We note significant current imbalances caused by differences in the allocation of EVSEs to phases and driver preferences. In the ACN, phase AB has 26 stations, whereas phases BC and CA each have 14. This imbalance is caused by the two 8-EVSE pods which are both on phase AB.}\n \\label{fig:phase_imbalance}\n\\end{figure}\n}\n\nIn general, infrastructure constraints can be expressed as upper bounds on the magnitudes of currents within the system. By Kirchoff's Current Law, we can express any current within the system as the sum of load currents. Let $\\mathcal{V}(t)$ denote the set of EVs which are available to be charged at time $t$. The current draw of EV $i$ at time $t$ can be expressed as a phasor in the form $r_i(t) e^{j\\phi_i}$ where $r_i(t)$ is the charging rate of the EV in amps, and $\\phi_i$ is the phase angle of the current sinusoid. We assume in this model that each charging EV has a unity power factor, so $\\phi_i$ is known based on how the EVSE is connected and the voltage phase angles (which we assume are separated by $\\pm 120\\degree$).\\footnote{This is reasonable since EVs' onboard charger generally includes power factor correction and voltage phase angles can be easily measured.} We can then model constraints within the electrical system as\n\\begin{equation}\\label{eq:three_phase_model}\n \\left| \\sum_{i \\in \\mathcal{V}} A_{li} r_i(t)e^{j\\phi_i} + L_l(t)\\right| \\leq \\ c_{lt} \\quad \\quad t \\in \\mathcal{T}, l \\in \\mathcal{L}\n\\end{equation}\n\\noindent Infrastructure limits of the network are indexed by resources $l\\in \\mathcal{L}$, e.g., $l$ may refer to a transformer or a breaker on a phase.\nFor each constraint $l$, $c_{lt}$ is a given capacity limit for time $t$ and $L_l(t)$ is the aggregate current draw from uncontrollable loads through resource $l$. $A=(A_{li}) \\in \\mathcal{R}^{|\\mathcal{L}|\\times|\\mathcal{V}|}$ is a matrix which maps individual EVSE currents to aggregate currents within the network. Matrix $A$ can account for both the connection of loads and lines as well as the effect of transformers, such as the delta-wye transformers in the Caltech ACN. The constraints in (\\ref{eq:three_phase_model}) are second-order cone constraints, which are convex and can be handled by many off-the-shelf solvers such as ECOS, MOSEK, and Gurobi. In some applications, however, these constraints could be too computationally expensive or difficult to analyze. Simpler, but more conservative constraints can be derived by observing\n\\begin{equation*}\n \\left| \\sum_{i \\in \\mathcal{V}} A_{li} r_i(t) e^{j\\phi_i} \\right| \\ \\leq \\sum_{i\\in\\mathcal{V}} |A_{li}| r_i(t)\n\\end{equation*}\nThis yields conservative affine constraints in the form\n\\begin{equation}\\label{eq:inf_constraints_affine}\n \\sum_{i \\in \\mathcal{V}} |A_{li}| r_i(t) + |L_l(t)| \\ \\leq \\ c_{lt} \\quad t \\in \\mathcal{T}, l \\in \\mathcal{L}\n\\end{equation}\n\nFor an example on how to find $A$ specifically for a subset of the Caltech ACN network, as well as a comparison of the performance of \\eqref{eq:three_phase_model} and \\eqref{eq:inf_constraints_affine}, see \\cite{lee_large-scale_2018}.\n\n\\iflong{%\n\\subsection{Battery Management System behavior} \\label{sec:bms_model}\nFor level-2 EVSEs, each EV's onboard charger and battery management system (BMS) controls its charging rate. As discussed in Section~\\ref{sec:evse}, the EV's actual charging current often deviates, sometimes significantly, from the pilot signal it receives. This requires us to develop algorithms that accurately model battery behavior or are robust against deviations from simpler models. While many tractable models for battery charging behavior exist, these models require information about the specific battery pack and initial state of charge of the vehicle \\cite{kazhamiaka_simple_2018, kazhamiaka_tractable_2019}. Other models rely on machine learning to learn the relationship between state of charge and current draw \\cite{frendo_data-driven_2020}.\nHowever, these ML models still require access to the state of charge of the vehicle. Since this information is not available with current charging hardware, we use a model-free approach to estimate battery behavior in real-time and use closed-loop control to account for modeling errors. This approach is described in Section~\\ref{sec:battery-tail-reclamation}.\n}\n\n\\subsection{EVSE limitations}\\label{sec:evse_limits}\nIn practice, EVSEs impose limits on the pilot signals which they support. For example, the J1772 standard does not allow pilot signals below 6 A (except 0). Also, most commercially available EVSEs only support a discrete set of pilot signals. Within the Caltech ACN, we have four types of EVSEs. EVSEs from ClipperCreek only support five pilot signals \\{0, 8, 16, 24, 32\\} for 32 amp EVSEs and \\{0, 16, 32, 48, 64\\} for 64 amp EVSEs. EVSEs from Webasto, Tesla, and OpenEVSE offer more control with 1 A (Webasto) or 0.1 A (Tesla and OpenEVSE) increments between 6 A and their maximum rate (32 A for Webasto and 80 A for Tesla and OpenEVSE). These limitations can be expressed mathematically as:\n\\begin{equation*} \nr_i(t) \\in \\rho_i(t) \\quad \\forall i,t\n\\end{equation*}\nwhere $\\rho_i(t)$ denote the set of allowable charging rates for EV $i$ at time $t$, which can depend on both the EVSE and our model of the EV's BMS%\n\\iflong{ described in Section~\\ref{sec:bms_model}}. \n\nWe also require the charging rate to be non-zero from when a car plugs in until its charging demand is met. \nThis constraint helps prevent contactor wear in the EVSEs and improves user experience, since most vehicles will notify their owner when charging stops.\nWe can encode this constraint as:\n\\begin{equation} \\label{eq:flapping-rate-const}\n r_i(t) \\in\n \\begin{cases}\n \\rho_i(t) \\setminus \\{0\\} & \\text{if $\\sum_{t=1}^{T} r_i(t) < \\, e_i$}\\\\\n \\{0\\} & \\text{otherwise}\n \\end{cases}\n\\end{equation}\nwhere $e_i$ is the energy request of EV $i$.\nUnfortunately, these constraints are discrete, making it difficult to incorporate them into optimization-based algorithms. In section \\ref{sec:post_processing} we propose heuristics to deal with these discrete constraints.\n\\section{Conclusions} \\label{sec-conclusions}\nIn this paper, we describe the Adaptive Charging Network (ACN), a framework for large-scale, managed electric vehicle charging facilities. The ACN and its scheduling algorithm (ASA) have been proven at scale through deployments around the United States, including the first ACN installed on the Caltech campus in 2016.\n\nThrough building the ACN, we have identified practical challenges, including unbalanced three-phase infrastructure, quantization of pilot signals, and non-ideal battery behavior, which require us to rethink classical scheduling approaches. To meet these challenges we propose ASA, a flexible model predictive control based algorithm along with pre- and post-processing heuristics, which can be easily configed to meet different operator objectives. \n\\iflong{We propose a collection of such objectives, including regularizers to promote desirable properties in the final schedule.}\n\n\\ifelselong{%\nThrough case studies, we consider the objective of delivering as much energy as possible in constrained infrastructure and maximizing profit, subject to time-of-use tariffs and demand charges. Using real workload data collected from the Caltech ACN and accurate models of ACN infrastructure, we demonstrate that ASA offers significant improvements in terms of energy delivered with constrained infrastructure when continuous pilots are allowed and performs comparably to baselines when pilots are restricted to a discrete set of values. We also note that by changing the objective function, we can easily modify ASA to maximize operator profit. Using real data from Sept. 2018, we achieve profits of \\$2,835 (98.1\\% of offline optimal) in an idealized setting, and \\$2,600 (90\\% of offline optimal) when considering non-ideal batteries and EVSEs. Compared to uncontrolled charging systems, our simulations show that an ACN like system can increase an EV charging system operator's profit by 3.4 times (see Section \\ref{sec:cost_min})\n}{%\nUsing real data from the ACN, we demonstrate that the ACN with ASA can reduce the infrastructure capacity necessary to meet charging demands and maximize profit subject to time-of-use tariffs and demand charges. In particular we find that ASA is able to consistently deliver more energy in highly constrained systems than baseline algorithms, and can increase an EV charging system operator's profit by 3.4 times over uncontrolled charging systems (see Section \\ref{sec:cost_min}).\n}\n\n\\iflong{\nBeyond its operational role of charging hundreds of EVs each week, the Caltech ACN and similar sites at research institutions like JPL, NREL, SLAC, and UC San Deigo, provide a valuable platform for research in managed EV charging. To facilitate this new frontier of research, ACN Research Portal provides open-access data from the Caltech and JPL ACNs and an open-source simulation environment based on the ACN architecture. In addition, a new project called ACN-Live will allow researchers from anywhere in the world to field test algorithms on the Caltech ACN.\n}\n\\section{Adaptive Charging Network Architecture}\\label{sec:overview}\nWe first describe the architecture of the ACN through the lens of the Caltech ACN. Since its installation in early 2016, the Caltech ACN has grown from 54 custom-built level-2 electric vehicle supply equipment (EVSEs) in a single garage \\cite{lee_adaptive_2016}, to 126 commercially available level-2 EVSEs, one 50 kW DC Fast Charger (DCFC), and four 25 kW DCFC, spread between three parking garages on campus. These EVSEs have delivered over 1,103 MWh as of July 7, 2020. The Caltech ACN is a cyber-physical system, as shown in Fig.~\\ref{fig:ACN_arch}, that consists of five interacting subsystems: (1) the information system which is responsible for collecting information and computing control actions; (2) the sensor system which gathers information from the physical system; (3) the actuation system (made up of EVSEs and the EVs' battery management systems) which controls each vehicle's charging rate; (4) the physical system (electrical infrastructure) which delivers power to the EVs and other loads within the system; (5) drivers who provide data to the system and decide when their vehicles are available to charge. \n\\ifelselong{In this section, we describe these subsystems and their interactions.}{We now focus on the components of the information and physical systems in detail.}\n\n\\begin{figure}[!t]\n\\includegraphics[width =\\columnwidth]{figs\/ACN_CPS_Diagram_small}\n\\centering\n\\vspace{-0.35in}\n\\caption{Architecture of the ACN. Blue and green arrows signify the flow of information and power, respectively. Sensors measure power flowing in the electrical network and convert this into information. Likewise, EVSEs and the EVs' onboard battery management system (BMS) work together as actuators to control the flow of power into each EVs battery based on signals from the information system. Drivers provide information to the system via a mobile app and directly control when EVs are plugged in or unplugged from the system (signified by the purple arrow).}\n\\label{fig:ACN_arch}\n\\end{figure}\n\\subsection{Information system}\nACN's information system collects and stores relevant data and computes control actions.\nIt consists of four components:\n\n\\textbf{Communication interface:}\nThe communication interface collects sensor data and passes it to the data storage layer. It also passes signals generated by the control algorithms to the corresponding EVSEs. An industrial computer within the parking garage controls this communication interface. It connects to the cloud-based components through a cellular internet connection and to sensors and EVSEs within the ACN via a Zigbee based mesh network.\n\n\\textbf{Data storage:}\nThe ACN utilizes a relational database to store information such as site configurations, driver profiles, and charging session parameters. A dedicated time-series database stores measurements like voltage, current, and power readings taken from sensors in the electrical network\n. \nThe data storage layer allows us to create visualizations for drivers and site operators%\n\\iflong{, which helps them understand the state of the system and their own EV's charging trajectory in real-time}, as seen at \\cite{caltech_dashboard_2019}.\n\n\\textbf{Mobile app:} \nOur mobile app collects data directly from drivers. After setting up an account, a driver scans a QR code on the EVSE, and then provides an estimated departure time and requested energy. If the driver's plans change, they can update these parameters throughout the charging session. The app also allows the site to collect payment and, if desired, implement access control. To ensure that drivers provide information through the app, an EV will only charge at 8 amps until the driver claims the session through the app. After 15 minutes, if the session is not claimed, it will be terminated, and the EVSE will cease charging.\n\n\\textbf{Control algorithms:}\nThe control layer takes inputs from the data layer and calculates a charging schedule for each EV in the system.\nWe use an event-based system to trigger the scheduling updates. The events considered include a vehicle plugging in or unplugging, a driver changing request parameters, or a demand response signal from the utility. Events are handled by a publish-subscribe model.\nWhenever an event occurs, or the time since the last charging schedule update exceeds a threshold (for example, 5 min), we compute a new charging schedule. These periodic computations close the control loop and account for discrepancies between the control signal sent to each EV and its actual current draw. We describe this model predictive control framework in detail in Section~\\ref{sec:scheduling-framework}.\n\n\\iflong{%\n\\subsection{EVSEs and Battery Management System}\\label{sec:evse}\nTo control charging rates, we use the pilot signal mechanism defined by the J1772 standard for level-2 EVSEs \\cite{sae_sae_2017}. According to this standard, the EVSE can communicate an upper bound to the EV's battery management system (BMS) that limits the amount of current it may draw from the EVSE. Because it is only an upper bound, the vehicle's BMS may choose to charge at a lower rate. This can occur for various reasons such as the pilot signal being higher than the vehicle's maximum charging rate or the BMS limiting current draw as the battery reaches a high state of charge. It can be difficult to diagnose why a car is charging below its allocated pilot signal since the J1772 standard does not provide a way to gather the EV's state of charge. Also, most EVSEs on the market today, including the ClipperCreek, Webasto, and Tesla EVSEs in the Caltech ACN, only support a finite set of pilot signal values and require quantization of the control signal.\n}\n\n\\iflong{%\n\\subsection{Sensors}\nSensors provide a bridge between the physical system and the information system. These sensors measure power, current, and voltage within the local electrical network, allowing us to monitor the system state and accurately track energy usage. The sensors also provide feedback for the control algorithm.\n}\n\n\\subsection{Physical system} \\label{sec:physical_system}\nThe physical system of the ACN includes the local electrical network (including transformers, lines, breakers, loads, and local generation), a connection to the grid, and the electric vehicles. Fig.~\\ref{fig:network-top} shows the topology of the local electrical network for one garage of the Caltech ACN. Power is delivered to the garage from the distribution transformer via three-phase service at 480 V\\textsubscript{LL}. \n\nFrom there, power is distributed throughout the garage via the main switch panel. The ACN is connected to this panel by two 150 kVA delta-wye transformers $t_1$ and $t_2$, which step the voltage down to 120 V\\textsubscript{LN}. Each level-2 EVSE is a single-phase load connected line-to-line (208 V\\textsubscript{LL}) with a maximum current draw between 32 A to 80 A depending its type. Because of unequal loading between phases,\n\\iflong{which is unavoidable due to the stochastic nature of driver demands on the system,}\nbalanced operation cannot be assumed. This makes protection of transformers $t_1$ and $t_2$ challenging which we discuss in Section~\\ref{sec:three_phase_model}. \n\\iflong{Another interesting feature of the Caltech ACN is the two pods of eight EVSEs. These pods are each fed by an 80 amp line. Since each EVSE in the pod has a maximum charging rate of 32 A, these lines are oversubscribed by 3.2 times. This demonstrates how smart charging can allow sites to scale EVSE capacity with existing infrastructure.}\n\n\\iflong{%\nIn addition to the 78 EVSEs in the garage, the ACN also includes a 50 kW DC fast charger (DCFC). This DCFC is a balanced three-phase load.\nWhile this garage does not have local generation, other garages in the Caltech ACN and other PowerFlex sites have on-site solar generation.}\n\\begin{figure}[!t]\n\\includegraphics[width = \\columnwidth]{figs\/ACN_tree.pdf}\n\\centering\n\\vspace{-0.35in}\n\\caption{System topology for the California Parking Garage in the Caltech ACN. The system consists of 78 EVSEs and one 50 kW DC Fast Charger. Switch Panel 1 is fed by a 150 kVA transformer and feeds 54 6.7 kW EVSEs, leading to a 2.4X over-subscription ratio. Nineteen of these lines feed pairs of two 6.7 kW AeroVironment EVSEs. Two additional 80 A lines feed pods of eight 6.7 kW EVSEs each, one pod of AeroVironment stations and the other of Clipper Creek stations. Switch Panel 2 is fed by an identical 150 kVA transformer and feeds 14 13.3 kW Clipper Creek EVSEs and 10 16.6 kW Tesla EVSEs. Each of these EVSEs has a dedicated 80 A line. All EVSEs in the system are connected line-to-line at 208 V. The 50kW DC Fast Charger from BTC Power is a balanced 3-phase load connected directly to the main switch panel. We do not directly control the DCFC at this time.\n}\n\\label{fig:network-top}\n\\end{figure}\n\n\\iflong{\n\\subsection{Drivers}\nHuman behavior can add significant randomness to the system. Drivers may arrive, depart, or change their input parameters at any time. Drivers are also difficult to model. Input through the mobile app can be highly inaccurate, as shown in \\cite{lee_ACN-Data_2019}. To combat this, we have explored using machine learning to predict driver parameters \\cite{lee_ACN-Data_2019} as well as pricing schemes that incentivize drivers to provide accurate estimates \\cite{lee_pricing_2020}.}\n\n\n\\section{Applications} \\label{sec:applications}\nWe now turn our attention to applications of the Adaptive Charging Network. We first examine the real-world operational data we have collected from the system. We then use this data to evaluate (through simulations) how the Adaptive Scheduling Algorithm proposed in Section~\\ref{sec:scheduling-framework} handles the practical challenges described in Section~\\ref{sec:key-insights}. To do this, we consider two practical objectives, charging users quickly in highly constrained systems, and maximizing operating profits. \n\\iflong{Due to limited space, we cannot address all possible use-cases of ACN and the ASA framework. For additional information about dynamic pricing and cost minimization using ACN see \\cite{lee_pricing_2020}.}\n\n\\subsection{Data Collected}\\label{sec:data_collected}\nThe ACN charging data includes actual arrival and departure times of EVs, estimated departure times and energy demands provided by drivers through the mobile app, and measurements of the actual energy delivered in each session. ACNs also record time series of the control signals passed to each EV and the EV's actual charging rate at 4-second resolution. This data is freely available, see \\cite{lee_ACN-Data_2019}, and can be used to evaluate new scheduling algorithms using ACN-Sim. Important workload features include the arrival and departure distribution of EVs, shown in Fig.~\\ref{fig:arrival_departure_dist}. \n\\iflong{Other statistics are summarized in Table~\\ref{tab:summary_stats}.}\nThis data was collected from May 1, 2018 - October 1, 2018 during which charging was free and only the 54 EVSEs connected to transformer $t_1$ were active. EVSEs on $t_2$ were added in early 2019, while the second and third garages in the ACN were added in late 2019. \n\\iflong{During this period, the system served 10,415 charging sessions and delivered a total of 92.78 MWh of energy.}\n\n\\iflong{%\nFrom Table \\ref{tab:summary_stats}, we observe a significant difference in system usage between weekdays and weekends. The total energy delivered is much higher on weekdays, but this energy is divided over far more charging sessions leading to a lower per session energy delivery. Also, charging sessions on weekends tend to be shorter than those during the week. This means that our system must be able to handle large numbers of flexible sessions on weekdays and smaller numbers of relatively inflexible sessions on weekends. This behavior precludes simple solutions such as installing large numbers of level-1 chargers, which would be too slow on weekends and for low laxity weekday sessions, or small numbers of level-2 chargers, which would be insufficient for the number of concurrent sessions on weekdays.}\n\n\\begin{figure}[t!]\n\\includegraphics[width =\\columnwidth]{figs\/arrivals_departures_2018.pdf}\n\\centering\n\\vspace{-0.2in}\n\\caption{Average arrivals and departures per hour for the period May 1, 2018 - October 1, 2018 (54 EVSEs). On weekdays we see a peak in arrivals between 7:00 - 10:00 followed by a peak in departures between 16:00 - 19:00. The Caltech ACN also has a much smaller peak in arrivals beginning around 18:00, which is made up of community members who use the site in the evening including some patrons of the nearby campus gym. Weekends, however have a more uniform distribution of arrivals and departures.}\n\\label{fig:arrival_departure_dist}\n\\end{figure}\n\n\n\\iflong{\n\\begin{table}[t!]\n\\centering\n\\captionsetup{justification=raggedright}\n\\caption{\\hspace{15pt} Average Statistics for EV Charging Test Cases Per Day\\newline\nMay 1, 2018 - Oct. 1, 2018 \n}\\label{tab:summary_stats}\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{lccccc}\n\\toprule\n & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Daily\\\\Sessions \\end{tabular} &\n \\begin{tabular}[c]{@{}c@{}}Mean\\\\Session\\\\Duration\\\\(hours)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Session\\\\Energy\\\\(kWh)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Mean\\\\Daily\\\\Energy\\\\(kWh)\\end{tabular} &\n \\begin{tabular}[c]{@{}c@{}}Max\\\\Concurrent\\\\Sessions\\end{tabular}\\\\ \\midrule\nSun & 41.32 & 3.94 & 10.05 & 415.06 & 18\\\\\nMon & 71.00 & 6.14 & 9.54 & 677.13 & 42\\\\\nTues & 76.73 & 6.24 & 8.94 & 685.79 & 47\\\\\nWed & 75.45 & 6.22 & 8.75 & 660.16 & 44\\\\\nThurs & 78.50 & 5.96 & 8.47 & 665.21 & 42\\\\\nFri & 77.18 & 6.71 & 9.04 & 697.41 & 43\\\\\nSat & 43.32 & 5.01 & 10.15 & 439.59 & 18\\\\ \n\\midrule\n\\end{tabular}\n}\n\\end{table}\n}\n\n\\subsection{Practical Scenarios}\n\\begin{table}[t]\n\\caption{Modeling Assumptions by Scenario}\n\\label{tab:scenarios}\n\\centering\n\\begin{tabular}{@{}lccccc@{}}\n\\toprule\n & I & II & III & IV & V \\\\ \\midrule\nPerfect Information? & \\cmark & \\xmark & \\xmark & \\xmark & \\xmark \\\\\nContinuous EVSE? & \\cmark & \\cmark & \\xmark & \\cmark & \\xmark \\\\\nIdeal Battery? & \\cmark & \\cmark & \\cmark & \\xmark & \\xmark \\\\ \\midrule\n\\end{tabular}%\n\\end{table}\n\nTo better understanding the effect of practical limitations such as limited information, non-ideal batteries, and pilot signal quantization, we consider each operator objective in the context of the five scenarios in Table~\\ref{tab:scenarios}. Here perfect information refers to having access to the arrival time, duration, and energy demand of all EVs in advance, allowing for offline optimization. Continuous EVSEs allow for continuous pilot control between 0 and the EVSE's upper bound, while quantized EVSEs only allow a discrete set of values and must keep the charging rate at or above 6 A until the EV is finished charging. Finally, ideal batteries are assumed to follow the pilot signal exactly. In contrast, non-ideal batteries follow the linear two-stage model described in \\cite{lee_acnsim_2019}, where the initial state of charge and battery capacity are fit to maximize tail behavior, and the tail begins at 80\\% state-of-charge.\n\nFor our simulations we use ACN-Sim, which includes realistic models for each of the scenarios above. In each case, we consider the three-phase infrastructure of the Caltech ACN. We set the length of each time slot to 5 minutes, the maximum time between scheduler calls to 5 minutes, and consider a maximum optimization horizon of 12 hours. \n\n\\subsection{Energy delivery with constrained infrastructure}\nWe first consider the objective of maximizing total energy delivered when infrastructure is oversubscribed. This is a common use case when electricity prices are static or when user satisfaction is the primary concern. To optimize for this operator objective, we use the Adaptive Scheduling Algorithm (ASA) (Alg. \\ref{alg:MPC}) with utility function\n\n\\begin{equation*}\n U^\\mathrm{QC}(r) := u^{QC}(r) + 10^{-12}u^{ES}(r)\n\\end{equation*}\n\nHere $U^{QC}$ encourages the system to deliver energy as quickly as possible, which helps free capacity for future arrivals. We include the regularizer $u^{ES}(r)$ to promote equal sharing between similar EVs and force a unique solution. We refer to this algorithm as ASA-QC.\n\nTo control congestion in the system, we vary the capacity of transformer $t_1$ between 20 and 150 kW. For reference, the actual transformer in our system is 150 kW, and a conventional system of this size would require 362 kW of capacity. We then measure the percent of the total energy demand met using ASA-QC as well as three baseline scheduling algorithms; least laxity first (LLF), earliest deadline first (EDF), and round-robin (RR), as implemented in ACN-Sim.\n\nResults from this experiment are shown in Fig.~\\ref{fig:energy_delivered_const_infra}, from which we observe the following trends. \n\\begin{enumerate}\n \\item In scenario II, ASA-QC performs near optimally (within 0.4\\%), and significantly outperforms the baselines (by as much as 14.1\\% compared to EDF with 30 kW capacity).\n \n \\item In almost all cases, ASA-QC performs better than baselines, especially so in highly congested settings.\n \\footnote{For scenarios III and V and transformer capacities less than 68 kW, it may sometimes be infeasible to allocate a minimum of 6 A to each active EV. When this is the case, we allocate 6 A to as many EVs as possible then allocate 0 A to the rest.\n %\n \\iflong{This allocation is done by first sorting EVs (by laxity for LLF and ASA-QC, deadline for EDF, and arrival time for RR) then allocating 6 A to each EV until the infrastructure constraints are binding.}}\n \n \\item Non-ideal EVSEs (scenarios III and V) have a large negative effect on ASA-QC, which we attribute to rounding of the optimal pilots and restriction of the feasible set.\n \n \\item Surprisingly, non-ideal EVSEs increase the performance of LLF and EDF for transformer capacities $<$60 kW. This may be because the minimum current constraint leads to better phase balancing.\n \n \\item Non-ideal batteries (scenarios IV and V) have relatively small effect on the performance of ASA-QC compared to baselines, indicating the robustness of the algorithm.\n\\end{enumerate}\n\\iflong{%\nTo understand why ASA-QC performs so much better than the baselines, especially in scenario II, we must consider what information each algorithm uses. RR uses no information aside from which EVs are currently present, and as such, performs the worst. Likewise, EDF uses only information about departure time, while LLF also makes use of the EVs energy demand. Only ASA-QC actively optimizes over infrastructure constraints, allowing it to better balance phases (increasing throughput) and prioritize EVs, including current and anticipated congestion. A key feature of the ASA framework is its ability to account for all available information cleanly.\\footnote{When even more information is available, i.e., a model of the vehicle's battery or predictions of future EV arrivals, this information can also be accounted for in the constraint set $\\mathcal{R}$ and objective $U(r)$. However, these formulations are outside the scope of this paper.}\\\\\n}\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.95\\columnwidth]{figs\/infrastructure_heatmap.pdf}\n\\centering\n\\vspace{-0.2in}\n\\caption{Percentage of driver's energy demands that can be met at varying capacities for transformer $t_1$ for Sept. 2018. Here demand met is defined as the ratio of total energy delivered to total energy requested.\n}\n\\label{fig:energy_delivered_const_infra}\n\\end{figure}\n\n\\subsection{Profit maximization with TOU tariffs and demand charge}\\label{sec:cost_min}\nNext, we consider the case where a site host would like to minimize their operating costs. Within this case, we will consider the Southern California Edison TOU EV-4 tariff schedule for separately metered EV charging systems between 20-500~kW, shown in Table~\\ref{tab:tou_rates} \\cite{choi_general_2017}. In each case, we assume that the charging system operator has a fixed revenue of \\$0.30\/kWh and only delivers energy when their marginal cost is less than this revenue. \n\\begin{table}[t!]\n\\caption{SCE EV TOU-4 Rate Schedule for EV Charging (Summer)}\n\\label{tab:tou_rates}\n\\centering\n\\begin{tabular}{lccc}\n\\toprule\nName & Time Range & Weekday & Weekend\\\\ \n\\midrule\nOff-Peak & 23:00 - 8:00 & \\$0.056 \/ kWh & \\$0.056 \/ kWh\\\\\n\\arrayrulecolor{black!30}\\midrule\nMid-Peak & \\begin{tabular}[c]{@{}l@{}} \\ 8:00 - 12:00\\\\18:00 - 23:00\\end{tabular}\n& \\$0.092 \/ kWh & \\$0.056 \/ kWh\\\\ \n\\arrayrulecolor{black!30}\\midrule\nPeak & 12:00 - 18:00 & \\$0.267 \/ kWh & \\$0.056 \/ kWh\\\\ \n\\arrayrulecolor{black!55}\\midrule\nDemand Charge & \\multicolumn{3}{c}{\\$15.51 \/ kW \/ month} \\\\ \n\\arrayrulecolor{black}\\bottomrule\n\\end{tabular}\n\\end{table}\nIn order to maximize profit, we use the objective:\n\\begin{align*}\n &U^\\mathrm{PM} := u^{EC} + u^{DC} + 10^{-4}u^{QC} + 10^{ -12}u^{ES}\n\\end{align*}\n\\noindent We denote the ASA algorithm with this objective ASA-PM.\n\\ifelselong{%\nThe revenue term $\\pi$ in $u^{EC}$ can have several interpretations. In the most straightforward case, $\\pi$ is simply the price paid by users. However, $\\pi$ can also include subsidies by employers, governments, automakers, or carbon credits through programs like the California Low-Carbon Fuel Standard (LCFS). For example, LCFS credits for EV charging have averaged between \\$0.13 - \\$0.16 \/ kWh in 2018-2019. In these cases, some energy demands might not be met if the marginal price of that energy exceeds $\\pi$. This is especially important when demand charge is considered since the marginal cost can be extremely high if it causes a spike above the previous monthly peak. Alternatively, $\\pi$ can be set to a very high value (greater than the maximum marginal cost of energy) and act as a non-completion penalty. When this is the case, the algorithm will attempt to minimize costs while meeting all energy demands (when it is feasible to do so).%\n}\n{%\nThe revenue term $\\pi$ in $u^{EC}$ can be interpreted as either the price paid by the user, a subsidy paid by the operator, or as a price on unmet demand.\n}\n\nIn $u^{DC}$, $\\hat{P}$ and $q'$ are tunable parameters. The demand charge proxy $\\hat{P}$ controls the trade-off between energy costs and demand charges in the online problem. In this case, we use the heuristic proposed in \\cite{lee_pricing_2020}, $\\hat{P} = P\/(D_p - d)$, where $D_p$ is the number of days in the billing period, and $d$ is the index of the current day. We will consider one version of the algorithm without a peak hint, e.g. $q'=0$, and one where the peak hint is 75\\% of the optimal peak calculated using data from the previous month. \n\\iflong{This percentage is chosen based on maximum historic month-to-month variability in the optimal peak (+11\\%\/-16\\%).}\n\n\nWe fix the transformer capacity to 150~kW and consider the previous baselines along with uncontrolled charging\n\\iflong{, which is the most common type of charging system today}.\nResults of the experiment are shown in Fig.~\\ref{fig:profit_max}, from which we observe:\n\\begin{enumerate}\n \\item Profits from both ASA-PM and ASA-PM w\/ Hint, are within 3.6\\% and 1.9\\% of the optimal respectively, and far exceed the profits of all baseline algorithms.\n \n \\item Uncontrolled, LLF and RR result in \\emph{lower} energy costs, but incur \\emph{very high} demand charges. These algorithms are not price aware. Instead low energy costs are a result of drivers arriving during off-peak and mid-peak times.%\n \\iflong{In particular, uncontrolled charging, which does not consider an infrastructure limit, leads to \\emph{extremely high} demand charges. On the other hand, both ASA-PM algorithms (and the offline optimal) trade-off higher energy costs for much lower peaks resulting in lower overall costs.}\n \n \\item Providing a peak hint to ASA-PM increases revenue by allowing more energy demands to be met. In this case, 97.8\\% vs. 95.6\\% without peak hints. Accurate hints allow the algorithm to utilize higher capacity earlier in the billing period, increasing throughput without increasing cost.\n \\iflong{Even with the peak hint, ASA-PM does not meet 100\\% of demands even though the offline optimal does. Since ASA-PM does not have knowledge of future arrivals, it must act conservatively in increasing the peak over time. It is, however, important that hints not be too large, as the algorithm can increase the peak as needed, but once a high peak is set, the demand charge cannot be lowered.}\n \n \\item While EVSE quantization and non-ideal batteries each reduce the operator's profit, even in scenario V, MPC w\/ Hint still produces 90\\% of the optimal profit.\n \n \\iflong{\n \\item Interestingly, revenue increases in scenarios with quantization (III and V). It can be hard to reason about exactly why this occurs, though it appears that the post-processing step leads to initial conditions for the next solve of \\textbf{OPT} to produce a higher revenue, higher cost solution. \n }\n \n \\item Because we use real tariffs structures, real workloads, and realistic assumptions (scenario V), we can conclude with reasonable certainty that a charging system operator could expect to net approximately \\$2,600~\/~month using an ACN like system, compared to just \\$763~\/~month in a conventional, uncontrolled system.\n\\end{enumerate}\n\n\\begin{figure}[t!]\n\\includegraphics[width = .85\\columnwidth]{figs\/profit_maximization.pdf}\n\\centering\n\\vspace{-0.25in}\n\\caption{Operator profit, costs, and revenue for various scheduling approaches when using SCE's EV TOU-4 tariff, $\\pi$ = \\$0.30, 150 kW transformer capacity, and data from Sept. 2018. In the middle panel we break out energy costs (darker, lower bar) from demand charge (lighter, upper bar). In each case, the offline optimal in the ideal setting is shown as a grey background. \n}\n\\label{fig:profit_max}\n\\end{figure}\n\\section{Online Scheduling Framework} \\label{sec:scheduling-framework}\nTo address these challenges, we have developed a practical and flexible framework for online scheduling of EV charging based on model predictive control and convex optimization. Within this\nframework, we introduce constraints that address unbalanced three-phase infrastructure and utilize feedback to account for inaccuracies in modeling, such as non-ideal battery behavior. Finally, we introduce a heuristic approach to account for discrete constraints arising from EVSE limitations efficiently.\n\n\\subsection{Model predictive control}\\label{sec:MPC}\nThe ACN computes charging rates using model predictive control, described in Alg. \\ref{alg:MPC}.\n\\begin{algorithm}[htbp] \\label{alg:MPC}\n\\SetAlgoLined\n\\SetNlSty{texttt}{(}{)}\n\\For{$k \\in \\mathcal{K}$}{\n \\nl $\\mathcal{V}_k := \\{i\\in \\hat{\\mathcal{V}}_k \\mid e_i(k) > 0$ \\textbf{AND} $d_i(k) > 0$\\} \\label{alg:active_set}\\\\\n \\nl \\If{event fired \\textbf{OR} time since last computation $> P$ \\label{alg:recomp_condition}}\n {\n \\nl $(r^*_i(1),...,r^*_i(T), i \\in \\mathcal{V}_k)$ := \\textbf{OPT}$(\\mathcal{V}_k, U_k, \\mathcal{R}_k)$ \\label{alg:scheduler.1} \\\\\n \\nl $r_i(k+t) := r^*_i(1+t), \\ t=0, \\dots, T-1$ \\label{alg:scheduler.2}\n }\n \\nl set the pilot signal of EV $i$ to $r_i(k)$, $\\forall i \\in \\mathcal{V}_k$ \\label{alg:pilot_signal}\\\\\n \\nl $e_i(k+1) := e_i(k) - \\hat{e}_i(k)$, $\\forall i \\in \\mathcal{V}_k$ \\label{alg:energy_update}\\\\ \\nl $d_i(k+1) := d_i(k) - 1$, $\\forall i \\in \\mathcal{V}_k$ \\label{alg:departure_update}\n }\n \\caption{Adaptive Scheduling Algorithm (ASA)}\n\\end{algorithm}\nWe use a discrete time model, with time indexed by $k$ in $\\mathcal{K} := \\{1,2,3,...\\}$. The length of each time period is $\\delta$ e.g. 5 minutes. At time $k$, $\\hat{\\mathcal{V}}_k$ is the set of all EVs present at the ACN and $\\mathcal{V}_k \\subseteq \\hat{\\mathcal{V}}_k$ is the subset of \\emph{active} EVs, i.e. the set of EVs whose energy demands have not been met.\nThe state of EV $i\\in\\mathcal V_k$ at time $k$ is described by a tuple ($e_i(k)$, $d_i(k)$, $\\bar{r}(k)$) where $e_i(k)$ is the remaining energy demand of the EV at the beginning of the period, $d_i(k)$ is the remaining duration of the session, and $\\bar{r}(k)$ is the maximum charging rates for EV $i$. In addition, we define $\\hat{e}(k)$ to be the measured energy delivered to the EV over time interval $k$. For simplicity of notation, we express $r_i(t)$ in amps and $e_i(t)$ and $\\hat{e}_i(t)$ in $\\delta$ $\\times$ amps assuming nominal voltage.\n\nWe now describe the MPC algorithm. In line \\ref{alg:active_set} we compute the active EV set\n$\\mathcal{V}_k$ by looking for all EVs currently plugged in which have non-zero remaining energy demand and are not already scheduled to depart. \nWe then check, in line \\ref{alg:recomp_condition}, if we should compute a new optimal schedule. This is done whenever an event-fired flag is True, or when the time since the last computed schedule exceeds $P$ periods.\n\nIf a new schedule is required, we call the optimal scheduling algorithm $\\textbf{OPT}(\\mathcal{V}_k, U_k, \\mathcal{R}_k)$ in line \\ref{alg:scheduler.1} that takes the form:\n\\begin{subequations}\n\\begin{eqnarray}\n\\max_{\\hat r} & & U_k(\\hat r)\n\\\\\n\\text{s.t.} & & \\hat r \\in \\mathcal R_k\n\\end{eqnarray}\n\\label{eq:SCH.1}\n\\end{subequations}\nThe set $\\mathcal V_k$ of active EVs defines the optimization variable $\\hat r:=(\\hat r_i(1), \\dots, \\hat r_i(T), i\\in\\mathcal V_k)$ for every active EV $i$ over the optimization horizon $\\mathcal T := \\{1, \\dots, T\\}$.\nThe utility function $U_k$ encodes the problem's objective while the feasible set $\\mathcal{R}_k$ encodes various constraints. They will be discussed in detail in the next two subsections.\nNote that \\textbf{OPT} does not have a notion of the current time $k$ and returns an optimal solution $r^*_i := (r_i^*(1),...,r_i^*(T))$ of \\eqref{eq:SCH.1} as a $T$-dimensional vector for each active EV $i$.\nThe algorithm then adjusts the indexing and sets the scheduled charging rates of EVs $i$ at time $k$ as $r_i(k+t) := r^*_i(1+t)$, $t = 0,...,T-1$ in line \\ref{alg:scheduler.2}.\nAt every time $k$, regardless of if a new schedule was produced, we set the pilot signal of each EV $i$ to $r_i(k)$ (line \\ref{alg:pilot_signal}) and update the system state (lines \\ref{alg:energy_update}, \\ref{alg:departure_update}) for the next time period.\n\nWe now describe how to design the utility function $U_k$ to achieve desirable features and how to model various constraints that define the feasible set $\\mathcal R_k$ for practical systems.\n\n\\subsection{Utility Functions $U_k$} \\label{sec:cost_funct}\nIn general, charging system operators may have many objectives they wish to achieve via smart charging, including charging vehicles as quickly as possible, maximizing their operating profit, \\ifelselong{}{or} utilizing renewable energy sources\\iflong{, or smoothing their total load profile}. Operators also have secondary objectives such as fairly distributing available capacity.\n\nTo allow operators to specify multiple objectives, our utility function $U_k(r)$ is a weighted sum of utility functions $u_k^v(r)$:\n\\begin{equation*}\nU_k(r) := \\sum_{v=1}^V \\alpha^{v}_k u_{k}^{v}(r)\n\\end{equation*}\nWe allow the utility function to change for each computation.\n Here $u_k^{v}(r)$,~$v=1,...,V$ are a set of utility functions which capture the system operator's objectives and promote desirable properties in the final schedule.\nMeanwhile, $\\alpha_k^{v} > 0$,~$v=1,...,V$ are time-dependent weights used to determine the relative priority of the various components.\nTo simplify notations, we will henceforth drop the subscript $k$ when we discuss the computation at time $k$.\n\n\\textbf{Charging quickly:}\nOne common operator objective is to charge all vehicles as quickly as possible. This can be done by specifying an objective such as\n\\begin{equation*}\nu^{QC}(r) := \\sum_{t \\in \\mathcal{T}} \\frac{T-t+1}{T} \\sum_{i \\in \\mathcal{V}} r_i(t)\n\\end{equation*}\nwhere the reward for delivering energy is strictly decreasing in time.\n\n\n\\textbf{Minimizing cost \/ maximizing profit:}\nAnother common objective for system operators is to maximize their operating profit. Let $\\pi$ be the per unit revenue from charging and $c(t)$ be the time-varying cost of one unit of energy. \nTo account for other loads and generation which share a meter with the ACN, we define the net load \n\\begin{equation*}\n N(t) := \\sum_{i \\in \\mathcal{V}} r_i(t) + L(t) - G(t)\n\\end{equation*}\n\n\n\\noindent where $L(t)$ denotes the net draw of the other loads while $G(t)$ denotes on-site generation such as PV. Since $L(t)$ and $G(t)$ are unknown for $t > 0$ this formulation relies on a prediction of these functions into the future. There are several methods for load\/generation prediction proposed in the literature, but these are outside the scope of this paper. We can express the objective of maximizing profit as:\n\n\\begin{equation*}\\label{eq:profit_max_util_energy_only}\nu^{EC}(r) := \\pi \\sum_{\\substack{t \\in \\mathcal{T}\\\\i \\in \\mathcal{V}}} r_i(t) - \\sum_{t \\in \\mathcal{T}} c(t)N(t)\n\\end{equation*}\nThis is equivalent to cost minimization when $\\pi = 0$. \n\n\\textbf{Minimizing demand charge:}\nIn addition to energy costs, utilities often impose a price on the maximum power draw in a billing period called demand charge. Since, demand charge is assessed over an entire month, while the optimization horizon is typically $<12$ hours, we replace the full demand charge $P$ with a proxy $\\hat{P} \\leq P$. We also introduce $q_0$ to be the highest peak so far in the billing period, and $q'$ as a prediction of the optimal peak. The demand charge can then be expressed as:\n\n\\begin{equation*} \\label{eq:profit_max_util}\n u^{DC}(r) \\: := -\\hat P\\cdot \\max \\left( \\max_{t\\in\\mathcal{T}} N(t),\\, q_0, q'\\right)\n\\end{equation*}\nNote that $\\hat{P}$ and $q'$ are tunable parameters. We describe the selection of these in Section~\\ref{sec:cost_min}.\n\n\\iflong{%\n\\textbf{Minimizing total load variations:}\nAnother common objective for EV charging operators is to minimize load variations. We can express this objective as:\n\n\\begin{equation*}\n u^{LV}(r) \\: := -\\sum_{t \\in \\mathcal{T}} N(t)^2 \n\\end{equation*}\n}\n\n\\textbf{Fairly distributing capacity:}\nThe utility functions described so far are not strictly concave in $r$ and hence the optimal solution, $r^*$, is generally non-unique. We can force a unique optimal solution by including the regularizer:\n\\begin{equation*}\\label{eq:sharing}\nu^{ES}(r) := -\\sum_{\\substack{t \\in \\mathcal{T}\\\\ i \\in \\mathcal{V}}} r_i(t)^2\n\\end{equation*}\nThis regularizer also promotes equal sharing among the EVs, which is desirable for the operator and drivers and minimizes line losses along the lines which feed each EVSE. This property comes from the fact that all things being equal, this component is maximized when all charging rates are as low as possible. Thus, it is sub-optimal to have one EV charging faster than another if both charging at an equal rate would result in the same optimal value for all other objective components. \n\n\\iflong{%\n\\textbf{Non-completion penalty:}\\label{sec:non-completion}\nA general goal of EV charging systems is to meet users' energy needs by their deadlines. While this can be accomplished by an equality constraint in $\\mathcal{R}_k$, doing so can lead to infeasibility. \nInstead, we can use the inequality constraint \\eqref{eq:Constraints.1c}, and add a non-completion penalty of the form:\n\n\\begin{equation*}\\label{eq:non-completion}\nu^{NC}(r) := - \\sqrt[p]{\\sum_{i \\in \\mathcal{V}} \\left|\\sum_{t \\in \\mathcal{T}} r_i(t) - e_i \\right|^p}\n\\end{equation*}\n\n\\noindent where $p \\geq 1$. This is the p-norm of the difference between the energy delivered to each EV and its requested energy. When $p=1$, this regularizer shows no preference between EVs. For $p > 1$, EVs with higher $e_i$ will be prioritized (given more energy) over those with lower $e_i$ when it is infeasible to meet all energy demands. Note that this regularizer is 0 whenever the energy demands of all EVs are fully met, e.g. $\\sum_{t \\in \\mathcal{T}} r_i(t) = e_i$. Thus, with sufficient weight on this component, \\eqref{eq:Constraints.1c} will be tight whenever feasible. \nLikewise, if \\eqref{eq:Constraints.1c} would have been tight without \\eqref{eq:non-completion}, this regularizer has no effect. \n}\n\n\\subsection{Feasible set $\\mathcal R_k$}\nThe feasible set $\\mathcal R_k$ is defined by a set of equality and inequality constraints that can depend on $k$, but for notational simplicity, we drop the\nsubscript $k$. These constraints then take the form:\n\\begin{subequations}\n\\begin{align}\n& 0 \\ \\leq \\ r_i(t) \\ \\leq \\ \\bar{r}_i(t) & t \\leq d_i, i \\in \\mathcal{V}\n\\label{eq:Constraints.1a} \\\\\n& r_i(t) \\ = \\ 0 & t > d_i, i \\in \\mathcal{V}\n\\label{eq:Constraints.1b} \\\\\n& \\sum_{t\\in\\mathcal{T}} r_i(t) \\ \\leq \\ e_i & i \\in \\mathcal{V}\n\\label{eq:Constraints.1c} \\\\\n& \\left| \\sum_{i \\in \\mathcal{V}} A_{li} r_i(t) e^{j\\phi_i} \\right| \\ \\leq \\ c_{lt}(t) & t \\in \\mathcal{T}, l \\in \\mathcal{L}\n\\label{eq:Constraints.1d}\n\\end{align}\n\\label{eq:Constraints.1}\n\\end{subequations}\n\n\\noindent Constraints (\\ref{eq:Constraints.1a}) ensure that the charging rate in each period is non-negative (we do not consider V2G) and less than its upper bound defined by the EV's BMS and the maximum pilot supported by the EVSE. This is a relaxation of the set of discrete rates allowed by the EVSE and is necessary to keep the scheduling problem convex. We discuss how to recover a feasible discrete solution in Section~\\ref{sec:post_processing}. \nConstraints (\\ref{eq:Constraints.1b}) ensure that an EV does not charge after its departure time. We use constraints (\\ref{eq:Constraints.1c}) to limit the total energy delivered to EV $i$ to at most $e_i$. To ensure feasibility, we do not require equality (the zero vector is always a feasible solution). This ensures that \\textbf{OPT} always returns a feasible schedule, which is important in practice. We can then craft the objective function to ensure this constraint is tight whenever possible%\n\\iflong{, see Section~\\ref{sec:non-completion}}.\n\n\\subsection{Quantization of pilot signal} \\label{sec:post_processing}\nThe pilot signal constraints imposed by EVSEs described in Section~\\ref{sec:evse_limits} are discrete and intractable in general for large problems. Because of this, we do not include \\eqref{eq:flapping-rate-const} in the definition of $\\mathcal R_k$, instead relaxing it to \\eqref{eq:Constraints.1a}. However, to account for our non-zero rate constraint, we add an additional constraint \n\\begin{equation*}\n r_i(0) \\geq \\min \\left( \\rho_i(0) \\setminus \\{0\\} \\right)\n\\end{equation*}\n\n\\noindent to \\eqref{eq:Constraints.1}.\\footnote{This constraint implicitly assumes that it is feasible to deliver a minimum charging rate to each EV, thus charging infrastructure should be designed with this constraint in mind if the operators want to ensure a minimum charging rate to each EV.} We denote the output of this optimization $r^* := (r^*_i(t), \\forall i\\in\\mathcal V \\ \\forall t\\in\\mathcal T)$. For simplicity, we assume that the maximum $P$ between scheduler calls (see Algorithm \\ref{alg:MPC})\nis set to the length of one period, so that only the first charging rate in $r^*$ will be applied.\n\nWe then round $r^*_i(0)$ down to the nearest value in $\\rho_i$:\n\\begin{equation*} \n\\tilde{r}_i(0) \\ \\leftarrow \\ \\left\\lfloor{r^*_i(0)}\\right\\rfloor_{\\rho_i}\n\\end{equation*}\n\\noindent This rounding may leave unused capacity which can be reclaimed. To reclaim this capacity, we first sort EVs in descending order by the difference between their originally allocated charging rate, $r^*_i(0)$, and the rate after rounding, $\\tilde{r}_i(0)$. We then iterate over this queue and increment each EV's charging rate to the next highest value in $\\rho_i(t)$, if the resulting current vector $\\tilde{r}(0) \\in \\mathcal{R}_k$ and $\\sum_i \\tilde{r}_i(0) \\leq \\sum_i r^*_i(0)$. We continue to loop over this queue until we cannot increment any EV's allocated rate.\n\n\\iflong{\n\\subsection{Battery tail capacity reclamation} \\label{sec:battery-tail-reclamation}\nAs discussed in Section~\\ref{sec:bms_model}, an EV's battery management system will sometimes limit the power draw of the battery as it approaches 100\\% state-of-charge.\nWhen this happens, the difference between the pilot signal and the vehicle's actual charging rate is wasted capacity. To reclaim this capacity, we use a simple algorithm which we call \\emph{rampdown}. Let $r^k_i(0)$ be the pilot signals sent to EV $i$ at time $k$, $m_i(k)$ be its measured charging current, and $\\bar{r}_i^k(0)$ be the upper bound on its charging rate. We define two thresholds, $\\theta_d$ and $\\theta_u$. If $r_i^k(0) - m_i(k) > \\theta_d$, we can reclaim some capacity by setting the upper limit on pilot signal of EV $i$ for the next period to be $m_i(k) + \\sigma$, where $\\sigma$ is typically around 1 A. In order to account for the possibility of the EV's BMS only limiting current temporarily, if $\\bar{r}_i^k(0) - m_i(k) < \\theta_u$, we increment the pilot signal upper bound by $\\sigma$ (clipping at the EV's BMS limit or the EVSE's pilot limit). With this scheme, we can quickly reclaim capacity during the tail region, while still allowing EVs to throttle back up if this reclamation was premature. Note that in our current implementation, the upper bound on the pilot signal, $\\bar r^k_i(t)$ is the same for all $t$ within the same sub-problem $k$. \nIn more advanced rampdown schemes, this bound could depend on $t$ or the decision variables $r(t)$.\n}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nFrom the very beginning of condensed matter physics, solid state theory relied on two complementary approaches: model-building and the use of general principles. The atomistic model was invoked by early crystallographers to explain the laws of crystal habit \\cite{hauy}; more than 400 years ago, Kepler speculated about the connection of the shape of a snowflake with the dense packing of spheres \\cite{kepler}. The simple Drude \\cite{drude} and Sommerfeld \\cite{sommerfeld} models of metals were followed by a more realistic band theory of solids \\cite{bloch,kane-hasan}. Later, improved models that incorporate electron interactions led to some of the greatest triumphs of condensed matter\nphysics including the Fermi-liquid theory \\cite{Gabriele} and the explanation of superconductivity \\cite{BCS-book}. Any recent issue of this journal contains articles on the Hubbard, Tomonaga-Luttinger, Kondo or other models of materials.\n\nIn some cases it is possible to obtain nontrivial predictions without the use of models, solely from the basic principles of quantum and statistical mechanics. Thermodynamics is a particularly powerful source of such predictions and Einstein famously said \\cite{einstein-quote}: ``A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression which classical thermodynamics made upon me.'' Einstein himself made some of the most brilliant predictions from general principles by his masterful use of detailed balance \\cite{einstein1,einstein2}. The fluctuation relations, addressed in this review, are a far-reaching development of his ideas.\n\nEarly milestones in the field, opened by Einstein, included the Nyquist formula \\cite{johnson,nyquist} and the Onsager reciprocity relations \\cite{onsager,casimir}. Their generalization led to the fluctuation-dissipation theorem \\cite{FDT} (FDT) and the linear response theory \\cite{Kubo}. Another breakthrough in the application of general principles to many-body systems came from the idea of universality \\cite{Kadanoff}. It allowed a precise quantitative description of critical phenomena from the symmetry considerations without any microscopic information \\cite{ZJ}.\n\nThese and other achievements greatly advanced the theory of condensed matter in and close to thermal equilibrium. On the other hand, far from equilibrium, little could be told without the resort to microscopic models \\cite{kynetics}. That situation changed in the 1990s, when the discovery of fluctuation relations brought a powerful general principle to nonequilibrium statistical mechanics \\cite{fl-3,fl-2,fl-1,fl0,fl1,fl2}.\n\nThe key idea behind fluctuation relations is similar to the principle of detailed balance. Consider a time-reversal-invariant system in an initial state $|\\psi_i\\rangle$. The system evolves during the time $t_0$. From the unitarity of quantum mechanics, the probability to find the system in the final state $|\\psi_f\\rangle$\nis exactly the same as the probability to find the system with the initial state $|\\psi(t=0)\\rangle=\\Theta|\\psi_f\\rangle$ in the final state $|\\psi(t=t_0)\\rangle=\\Theta|\\psi_i\\rangle$, where $\\Theta$ is the time-reversal operator. We now consider a process made of the following three steps:\n\n\n1) The initial state of the system is measured;\n\n2) The system undergoes unitary evolution over the time interval $t_0$;\n\n3) The final state is determined through measurement.\n\n\\noindent\nThe probability to observe the evolution from $|\\psi_i\\rangle$ into $|\\psi_f\\rangle$ is no longer the same as the probability to observe the evolution from $\\Theta|\\psi_f\\rangle$ to $\\Theta|\\psi_i\\rangle$ since the probabilities to find the system in the initial states\n$|\\psi_i\\rangle$ and $\\Theta|\\psi_f\\rangle$ are not the same away from thermal equilibrium. However, it is often possible to write a simple relation between the two probabilities \\cite{crooks}. This, in turn, allows a derivation of numerous relations between correlation functions of observables. In this review we will be particularly interested in the correlation functions of electric currents.\n\nOne might think that all this is of no use in the absence of time-reversal symmetry, e.g., when an external magnetic field is applied. Indeed, most work on fluctuation relations assumes the time-reversal symmetry. However, it transpired recently that a number of nontrivial fluctuation relations hold even without such symmetry. One result \\cite{saito} connects systems that transform into each other under the action of the time-reversal operator. A family of relations has been found\nfor nonlinear transport coefficients of a system, close to thermal equilibrium, at two opposite directions of the magnetic field. Another result \\cite{wang1,wang2} applies even far away from equilibrium and connects non-linear response and noise at a fixed direction of the magnetic field.\nThe latter result holds in chiral systems. Since the word ``chirality'' has all too many meanings, we need to explain ours.\n\nBy chiral we mean systems, where excitations can propagate in one direction only, that is, either all excitations can propagate only clockwise or all excitations can only propagate counterclockwise. Such transport is known to occur on the edges of certain quantum Hall liquids and in some other systems. There has been much recent interest in chiral transport in the quantum Hall effect (QHE). The interest comes, in part, from the search for elusive neutral modes \\cite{chang}. Besides, the question of chirality proved relevant for the ongoing search for non-Abelian anyons \\cite{anyons} in quantum Hall states in the second Landau level (see Section 4). The more powerful fluctuation relations \\cite{wang1,wang2} in chiral systems originate from a stronger form of the causality principle for chiral transport. The standard causality principle states that the past is not affected by the future events. In the chiral case, in addition, one of the following two alternatives holds: either what happens on the right is not affected by the past events on the left or what happens on the left is not affected by the past events on the right.\n\nIn this review we address theoretical and experimental work on fluctuation relations in the absence of the time-reversal symmetry in chiral and non-chiral systems. Many questions remain open and we discuss future directions in Section 6. We also address the ongoing controversy\nabout microreversibility without the time-reversal symmetry.\n\n\n\nThe paper is organized as follows. In the second section we derive the quantum fluctuation theorem \\cite{fl2,tobiska05,andrieux06,esposito07,andrieux09,campisi10} that serves as a foundation for all subsequent discussion. In Section 3, we extract from that theorem the Saito-Utsumi relations \\cite{saito} for the nonlinear transport coefficients of identical systems in the opposite magnetic fields. We also address the verification of the Saito-Utsumi relations in microscopic models and the experimental results \\cite{exp1,exp2}. In Section 4 we introduce chiral systems with the emphasis on QHE. We derive fluctuation relations \\cite{wang1,wang2} for chiral systems in Section 5. Finally, we address open problems and summarize in Section 6.\n\n\\section{Fluctuation theorems}\n\n\n\n\\begin{figure}[b]\n\\centering \n\\includegraphics{fig_ft1.eps}\n\\caption{Schematics of a quantum open system. The system can exchange energy and particles with $r$ equilibrium reservoirs. Chemical potential difference or temperature difference between any two reservoirs will induce particle current and$\/$or heat current.}\\label{fig1}\n\\end{figure}\n\nIn statistical mechanics, physical quantities of interest usually undergo random fluctuations. {\\it Fluctuation theorems} are a class of exact relations for the distribution functions of those random fluctuations, regardless whether the system under investigation is in an equilibrium or nonequilibrium state. There are fluctuation theorems for, e.g, entropy in driven isolated systems, work and heat in closed systems, {\\it etc}. Fluctuation theorems are typically of the form \\cite{fl1,fl2,fl3}\n\\begin{equation}\n P_F(x) = P_B(-x) \\exp[a(x-b)], \\label{ftgeneral}\n\\end{equation}\nwhere $x$ is the physical quantity or the collection of quantities under investigation, $ P_F(x)$ and $ P_B(x)$ are the distribution functions of $x$ in the so-called {\\it forward} and {\\it backward} processes respectively, and the constants $a$ and $b$ are determined by the initial conditions of the two processes. The definitions of the forward and backward processes may differ slightly for different systems, but in general they follow these lines: 1) Initially in both processes, the system obeys a Gibbs distribution or can be factorized into several subsystems that obey Gibbs distributions and 2) the dynamical equations (e.g., the Schr\\\"odinger equation) in the backward process is obtained from the dynamical equations in the forward process by the time-reversal operation. Generally speaking, the system in the forward process and the system in the backward process should not be considered as the same system because they follow different microscopic dynamical equations. However, for a system with the time-reversal symmetry, the dynamical equation is the same in the forward and backward processes. Hence, provided that the initial Gibbs distributions in the two processes are the same, no difference exists between the two processes for a time-reversal invariant system, and thereby the subindices $F$ and $B$ in (\\ref{ftgeneral}) can be dropped.\n\nIn this review, we consider systems without the time-reversal symmetry, and focus on a particular fluctuation theorem for energy and particle transport in quantum open systems. This fluctuation theorem is very useful for studying transport phenomena in systems without the time-reversal symmetry, and, particularly, in systems with chirality (see Sections 4 and 5). For a reader, interested in other fluctuation theorems, many excellent reviews exist, for example, Refs. \\refcite{fl1,fl2,fl3}.\n\nThe approach of this section builds on Refs. \\refcite{andrieux09,campisi10} and closely follows Supplementary information to Ref. \\refcite{wang2}.\n\n\n\nLet us discuss the specific fluctuation theorem that we are interested in. We consider a setup shown in Fig.~\\ref{fig1}: the system in the center is coupled to $r$ reservoirs, each being in equilibrium. The system serves as a bridge, so that energy and particles can be transported between the reservoirs. This setup is commonly used in transport experiments with mesoscopic systems, such as quantum dots, quantum Hall bars, {\\it etc}. A {\\it forward} process can then be defined as follows. Initially, at $t\\le 0$, the system and reservoirs are decoupled. An interaction $\\mathcal V(t)$ that allows particle and energy exchange with the reservoirs is turned on at the times $0\\le t\\le \\tau$. The interaction is turned off at $t\\ge \\tau$. At $t\\le 0$, reservoir $i$ is at equilibrium with the inverse temperature $\\beta_i=1\/T_i$ and the chemical potential $\\mu_i=qV_i$, where $q$ is the charge of a charge carrier and $V_i$ the electric potential. We use only one set of chemical potentials and thus assume that only one carrier type is present which is usually electron. We assume that the size of the system is much smaller than that of the reservoirs, so its initial state is irrelevant in the $\\tau\\rightarrow \\infty$ limit which we will take. It is convenient to regroup the system with one of the reservoirs\\cite{andrieux09}, for example, the $r$-th reservoir. The interaction $\\mathcal V(t)$ becomes a constant $\\mathcal V_0$ when fully turned on during $\\tau_0\\le t\\le \\tau-\\tau_0$. We assume that $\\tau_0\\ll\\tau $ and $\\tau$ is much longer than the relaxation time so that the system remains in a steady state during most of the time interval $\\tau$.\n\n\nWe now find the statistical distribution of the changes $\\Delta N_i=N_i(t=\\tau)-N_i(t=0)$ and $\\Delta E_i= E_i(t=\\tau) - E_i(t=0)$, where $N_i$ is the particle number and $E_i$ is the energy of the $i$-th reservoir. Let $\\mathcal H_i$ and $\\mathcal N_i$ be the Hamiltonian and particle number operators of the $i$-th reservoir ($\\mathcal H_r$ includes the system). The particle numbers conserve in the absence of $\\mathcal V(t)$, i.e., $[\\mathcal H_i, \\mathcal N_i]=0$. Thus, the initial density matrix factorizes into a product of Gibbs distributions in each reservoir,\n\\begin{equation}\n\\rho_{n}=\\frac{1}{Z_0^+}\\prod_{i} e^{-\\beta_i[E_{in}-\\mu_i N_{in}]}\\label{gibbs}\n\\end{equation}\nwhere $Z_0^+$ is the initial partition function and the index $n$ labels the quantum state $|\\psi_n\\rangle$ with the reservoir energies $E_{in}$ and particle numbers $N_{in}$. Here, the ``+'' sign reminds us that we are studying the forward process. An initial joint quantum measurement of $\\mathcal H_i$ and $\\mathcal N_i$ is performed at $t=0$, so that the quantum state of the system collapses to a common eigenstate $|\\psi_n\\rangle$ with the probability $\\rho_n$. The state $|\\psi_n\\rangle$ then evolves according to the evolution operator $ U(t;+)$, which satisfies\n\\begin{equation}\ni\\frac{d}{dt} U(t; +) = \\mathcal H (t; +) U(t;+), \\label{forwardpropagator}\n\\end{equation}\nwhere the Hamiltonian $\\mathcal H(t;+)=\\sum_{i} \\mathcal H_i +\\mathcal V(t)$, and the initial condition is $U(0;+)=1$. At $t=\\tau$, a second joint measurement is taken, leading to the collapse of the system to the state $|\\psi_m\\rangle$ with the reservoir energies $E_{im}$ and particle numbers $N_{im}$.\nThe probability to observe such process is\n\\begin{equation}\nP[m,n]=|\\langle\\psi_m|U(\\tau;+)|\\psi_n\\rangle|^2\\rho_{n}.\n\\end{equation}\nHence, repeating the forward process, we obtain the joint distribution function of the energy and particle changes $\\Delta E_{i,mn}=E_{im}-E_{in}$ and $\\Delta N_{i,mn}=N_{im}-N_{in}$\n\\begin{align}\nP[\\Delta\\mathbf E, \\Delta \\mathbf N;+] = &\\sum_{mn}\\prod_{i=1}^r\\delta(\\Delta E_i - \\Delta E_{i,mn})\\delta(\\Delta N_i - \\Delta N_{i,mn}) \\nonumber\\\\ &\\times|\\langle\\psi_m|U(\\tau;+)|\\psi_n\\rangle|^2\\rho_{n} \\label{forwarddistri},\n\\end{align}\nwhere the vectors $\\Delta\\mathbf E=(\\Delta E_1, \\dots, \\Delta E_r)$ and $\\Delta\\mathbf N=(\\Delta N_1, \\dots, N_r)$. Since the total energy and particle number are conserved, one finds that $\\sum_i\\Delta E_i=\\sum_i\\Delta N_i=0$. The energy conservation is an approximation due to the time-dependent interaction $\\mathcal V(t)$. However, in the limit $\\tau\\rightarrow \\infty$, the violation of the energy conservation is negligible since \nthe time-dependence is relevant only during a short period of time $\\tau_0$.\n\n\n\n\n\nWe now study the {\\it backward} process. In that process, the initial temperatures and chemical potentials of the reservoirs are the same as those in the forward process. Note that such initial temperatures and chemical potentials are not necessarily the same as the final thermodynamic parameters at $t= \\tau$ in the forward process for large but finite reservoirs. The time evolution operator $U(t;-)$ is determined by the equation\n\\begin{equation}\ni\\frac{d}{dt} U(t; -) = \\mathcal H (t; -) U(t;-), \\label{backwardpropagator}\n\\end{equation}\nwhere the Hamiltonian $\\mathcal H(t;-)=\\Theta \\mathcal H(\\tau -t;+)\\Theta^{-1}$ with $\\Theta$ being the time-reversal operator. Here, the ``$-$'' sign stands for the backward process. The $i$-th reservoir has the Hamiltonian $\\Theta\\mathcal H_i\\Theta^{-1}$. Clearly, $\\Theta|\\psi_m\\rangle$ is a common eigenstate of $\\Theta\\mathcal H_i\\Theta^{-1}$ and $\\mathcal N_i=\\Theta\\mathcal N_i\\Theta^{-1}$ with the eigenvalues $E_{im}$ and $N_{im}$ respectively.\nThe amplitude of the transition from the state $\\Theta|\\psi_m\\rangle$ to $\\Theta|\\psi_n\\rangle$ after the time $\\tau$ is $(\\langle\\psi_n|\\Theta^{-1}U(\\tau;-)\\Theta|\\psi_m\\rangle)^*$.\nHence, performing two quantum measurements at $t=0$ and $t=\\tau$ in the backward process, we find that the probability to observe the collapse to the state $\\Theta|\\phi_m\\rangle $ at $t=0$ and the collapse to the state $\\Theta |\\phi_n\\rangle$ at $t=\\tau$ is\n\\begin{equation}\nP[m,n]=|\\langle\\psi_n|\\Theta^{-1}U(\\tau;-)\\Theta|\\psi_m\\rangle|^2\\rho_{m},\n\\end{equation}\nwhere the initial density matrix $\\rho_{m}=\\prod_{i} e^{-\\beta_i[E_{im}-\\mu_i N_{im}]}\/Z_0^-$. It follows from the antiunitarity of $\\Theta$ that $Z_0^+=Z_0^-$, that is, the Gibbs distribution $\\rho_m$\nin the time-reversed basis is the same as Eq.~(\\ref{gibbs}). One then finds the distribution of the energy and particle number changes, similar to Eq. (\\ref{forwarddistri}),\n\\begin{align}\nP[\\Delta\\mathbf E, \\Delta \\mathbf N;-] = &\\sum_{mn}\\prod_{i=1}^r\\delta(\\Delta E_i - \\Delta E_{i,nm})\\delta(\\Delta N_i - \\Delta N_{i,nm}) \\nonumber\\\\ &\\times |\\langle\\psi_n|\\Theta^{-1}U(\\tau;-)\\Theta|\\psi_m\\rangle|^2\\rho_{m}.\\label{backwarddistri}\n\\end{align}\nwhere $\\Delta E_{i,nm}=E_{in}-E_{im}$ and $\\Delta N_{i,nm}=N_{in}-N_{im}$.\n\n\nWe want to relate the distribution functions (\\ref{forwarddistri}) and (\\ref{backwarddistri}), i.e., obtain the fluctuation theorem. The evolution operators have an important property\\cite{andrieux09}\n\\begin{equation}\n\\Theta^{-1}U(\\tau;-)\\Theta = U^\\dag(\\tau; +).\\label{eq-U}\n\\end{equation}\nThis can be seen by checking that both operators $\\Theta^{-1}U(t;-)\\Theta$ and $U(\\tau-t;+)$ satisfy the equation\n\\begin{align}\ni\\frac{d}{dt}V(t) = -\\mathcal H(\\tau -t;+)V(t). \\label{prop}\n\\end{align}\nIn terms of $\\Theta^{-1}U(t;-)\\Theta$, we find that given the initial condition $V(0)=\\Theta^{-1}U(0;-)\\Theta=1$, the final operator at $t=\\tau$ is $V(\\tau) = \\Theta^{-1}U(\\tau;-)\\Theta$. In terms of $U(\\tau-t;+)$, given the initial condition $V(0) = U(\\tau;+)$, we have $V(\\tau) = U(0; +)$. In other words, if we have the initial condition $V(0)=1=U^\\dagger(\\tau;+)U(\\tau;+)$, the final operator at $t=\\tau$ will be $V(\\tau ) = U^\\dag(\\tau;+)U(0;+)=U^\\dag(\\tau;+)$. Therefore, due to the uniqueness of the solution of Eq.~(\\ref{prop}), the property (\\ref{eq-U}) is obtained.\n\n\nNow, we combine Eqs.~(\\ref{forwarddistri}), (\\ref{backwarddistri}) and (\\ref{eq-U}), and obtain the fluctuation theorem\n\\begin{equation}\n\\frac{P[\\Delta\\mathbf E, \\Delta \\mathbf N;+]}{ P[-\\Delta\\mathbf E, -\\Delta \\mathbf N;-]}=\\prod_{i} e^{\\beta_i(\\Delta E_i-\\mu_i\\Delta N_i)} \\label{eq-fr}.\n\\end{equation}\nClearly, it is of a general form (\\ref{ftgeneral}). As mentioned above, in the case of time-reversal invariant systems, i.e., for $\\mathcal H(t;+)=\\Theta \\mathcal H(\\tau -t;+)\\Theta^{-1}$, the microscopic dynamical equations (\\ref{forwardpropagator}) and (\\ref{backwardpropagator}) are the same. Hence, the ``$+$'' and ``$-$'' can be dropped in the fluctuation theorem. However, in what follows, we will focus on systems without the time-reversal symmetry. Therefore, one has to keep in mind that the two distribution functions in (\\ref{eq-fr}) describe two different systems. In most of the following applications, the system in the backward process is realized by reversing the direction of the magnetic field $B$ which is present in the system in the forward process.\n\n\n\\section{Saito-Utsumi relations}\n\nThe Saito-Utsumi relations \\cite{saito} connect transport properties of a conductor in two opposite magnetic fields. They generalize quantum fluctuation relations for the electric current and noise in time-reversal invariant systems \\cite{fl2}.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{fig_2terminal.eps}\n\\caption{ A conductor is placed in a magnetic field and connects two terminals at the same temperature and the voltage difference $V$. }\n\\label{AAA}\n\\end{figure}\n\nWe consider a conductor, connected to two reservoirs with the voltage difference $V$ and the same temperature $T$, in the presence of a magnetic field, Fig. \\ref{AAA}. It is possible to generalize to a multi-terminal case. Below we will only investigate the simplest two-terminal situation. We will derive the relations between various current correlation functions for two opposite orientations of the magnetic field. An infinite number of relations can be derived for an infinite number of the correlation functions but we will focus on a few simplest relations that are most relevant for the experiment.\n\nWe will be interested in three quantities:\n\n1) The electric current\n\n\\begin{equation}\n\\label{dima1}\nI=e\\langle\\Delta N\\rangle\/\\tau;\n\\end{equation}\n\n\n2) The noise power\n\n\\begin{equation}\n\\label{dima2}\nS=2e^2(\\langle \\Delta N^2\\rangle-\\langle\\Delta N\\rangle^2)\/\\tau;\n\\end{equation}\n\n3) The third cumulant\n\n\\begin{equation}\n\\label{dima3}\nC=\\frac{2e^3}{\\tau}[\\langle \\Delta N^3\\rangle-3\\langle\\Delta N\\rangle\\langle\\Delta N^2\\rangle+2\\langle\\Delta N\\rangle^3],\n\\end{equation}\nwhere $e<0$ is one electron charge, $\\tau$ the duration of the protocol, Section 2, $\\Delta N$ the number of electrons transferred between the reservoirs, and the angular brackets denote the average over the distribution $P(\\Delta N, B)$, where $B$ is the magnetic field.\nNote that $\\langle\\Delta N\\rangle\\rightarrow 0$ at $V\\rightarrow 0$ since there is no current at zero voltage. We will only compute the noise $S$ with the accuracy up to the linear term in $V$ below. Thus, it is legitimate to omit the contribution $\\langle\\Delta N\\rangle^2$ in Eq. (\\ref{dima2}).\n\nThe fluctuation relations, considered in this section, were first derived in Ref. \\refcite{saito}. We follow a simpler derivation from Ref. \\refcite{exp2}.\n\n\n\n\\subsection{Symmetric and antisymmetric variables}\n\nThe results express in the simplest way in terms of the symmetrized and antisymmetrized combinations of the currents and noises $I_{\\pm}=I(B)\\pm I(-B)$, $S_{\\pm}=S(B)\\pm S(-B)$ in the opposite magnetic fields $\\pm B$.\nWe introduce the Taylor expansions in powers of the voltage $V$\n\n\\begin{equation}\n\\label{dima4}\nI_+=G_1V+\\frac{G_2V^2}{2}+\\dots;\n\\end{equation}\n\n\\begin{equation}\n\\label{dima5}\nI_-=G_1^AV+\\frac{G_2^AV^2}{2}+\\dots;\n\\end{equation}\n\n\\begin{equation}\n\\label{dima6}\nS_+=S_0+S_1V+\\dots;\n\\end{equation}\n\n\\begin{equation}\n\\label{dima7}\nS_-=S_0^A+S_1^AV+\\dots;\n\\end{equation}\n\n\\begin{equation}\n\\label{dima8}\nC_-=C_0^A+\\dots.\n\\end{equation}\n\nThe Saito-Utsumi relations connect the above Taylor coefficients.\n\nAccording to the Onsager reciprocity relations \\cite{onsager,casimir}, the linear conductance is an even function of the magnetic field, $G_1(B)=G_1(-B)$. Hence, $G_1^A=0$. Next, the Nyquist formula \\cite{nyquist} implies that\nthe equilibrium noise power $S_1=4G_1T$ does not depend on the direction of the magnetic field and $S_0^A=0$. We will also see that the symmetrized third cumulant $C_+=C(B)+C(-B)$ is zero at $v=0$.\n\n\\subsection{Fluctuation relations for symmetric variables}\n\nWe will use the notation $P(\\Delta N,B)$ for the probability to transfer $\\Delta N$ electrons from the left reservoir to the right one during the protocol, Section 2. According to the fluctuation theorem (\\ref{eq-fr}),\n\n\\begin{equation}\n\\label{dima9}\nP(\\Delta N,B)=P(-\\Delta N,-B)e^{v\\Delta N},\n\\end{equation}\nwhere $v=eV\/T$.\nWe next introduce the symmetrized and antisymmetrized probabilities $P_{\\pm}(\\Delta N)=P(\\Delta N,B)\\pm P(\\Delta N,-B)$. Eq. (\\ref{dima9}) then yields\n\n\\begin{equation}\n\\label{dima10}\nP_{\\pm}(\\Delta N)=\\pm P_{\\pm}(-\\Delta N)e^{v\\Delta N}.\n\\end{equation}\nWe also introduce the symmetrized and antisymmetrized averages $\\langle\\Delta N^k\\rangle_\\pm=\\sum P_{\\pm}(\\Delta N)\\Delta N^k$.\n\nEq. (\\ref{dima10}) for $P_+$ assumes exactly the same form as the fluctuation theorem for $P(\\Delta N,B=0)$. Thus, all fluctuation relations for the symmetric currents $I_+$ and noises $S_+$ are the same as the relations for the currents and noises in the absence of the magnetic field.\n\nNote that at $v=0$, Eq. (\\ref{dima10}) yields $P_+(\\Delta N)=P_+(-\\Delta N)$. Hence, $\\langle \\Delta N^{2k+1}\\rangle_+=\\sum P_{+}(\\Delta N)\\Delta N^{2k+1}$ is zero at $v=0$ for any odd $2k+1$.\nA trivial consequence of this relation is the absence of the average electric current in equilibrium, $I_+=e\\langle \\Delta N\\rangle\/\\tau=0$. We also find that $C_+=C(B)+C(-B)=0$.\n\nWe now expand in powers of $v$ the left and right hand sides of the relation\n\n\\begin{equation}\n\\label{dima11}\n\\langle\\Delta N\\rangle_+=\\sum_{\\Delta N}\\Delta N P_+(\\Delta N)=-\\sum_{\\Delta N} \\Delta N P_+(\\Delta N)e^{-v\\Delta N}.\n\\end{equation}\nAfter defining\n\n\\begin{equation}\n\\label{dima12}\n\\langle\\Delta N^k\\rangle_\\pm=N^\\pm_{k,0}+vN^\\pm_{k,1}+\\frac{v^2}{2}N^\\pm_{k,2}+\\dots\n\\end{equation}\nwe obtain $N^+_{2,0}=2N^+_{1,1}$ and $N^+_{1,2}=N^+_{2,1}$, where we used the fact that $N^+_{3,0}=0$. Comparing with Eqs. (\\ref{dima4}) and (\\ref{dima6}), we finally obtain\n\n\\begin{equation}\n\\label{dima13}\nS_0=4G_1T;\n\\end{equation}\n\\begin{equation}\n\\label{dima14}\nS_1=2TG_2.\n\\end{equation}\nThe first equation is nothing but the Nyquist formula. Eq. (\\ref{dima14}) goes beyond the standard fluctuation-dissipation relation since it contains the nonlinear transport coefficient $G_2$. Since that coefficient is nonzero in general, the noise $S=S_0+VS_1+\\dots$ is minimal at a nonzero voltage, Fig. \\ref{BBB}.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{fig_parabola.eps}\n\\caption{Noise is minimal at a nonzero $V$.}\n\\label{BBB}\n\\end{figure}\n\n\n\\subsection{Fluctuation relations for antisymmetric variables}\n\nWe now turn to the new results that distinguish systems in a magnetic field from the time-reversal invariant situation.\nFirst, consider the identity\n\n\\begin{equation}\n\\label{dima15}\n\\langle \\Delta N\\rangle_-=\\sum_{\\Delta N}P_-(\\Delta N)\\Delta N=\\sum_{\\Delta N}P_-(\\Delta N)\\Delta N e^{-v\\Delta N}.\n\\end{equation}\nThe Taylor expansion of the left and right hand sides yields\n\n\\begin{equation}\n\\label{dima16}\nN^-_{3,0}=2N^-_{2,1}.\n\\end{equation}\nThis is equivalent to\n\\begin{equation}\n\\label{dima17}\nS_1^A=\\frac{C_0^A}{2T}.\n\\end{equation}\n\nThe normalization of probability implies\n\n\\begin{equation}\n\\label{dima18}\n\\sum P_-(\\Delta N)=0=-\\sum P_-(\\Delta N)e^{-v\\Delta N}.\n\\end{equation}\nOne obtains from the expansion of the right hand side in powers of $v$\n\n\\begin{equation}\n\\label{dima19}\n3N^-_{1,2}-3N^-_{2,1}+N^-_{3,0}=0.\n\\end{equation}\nThis reduces to\n\n\\begin{equation}\n\\label{dima20}\nS_1^A-2TG_2^A=C_0^A\/3T.\n\\end{equation}\n\nEquations (\\ref{dima17}) and (\\ref{dima20}), first derived in Ref. \\refcite{saito}, are the main results of Section 3.\n\n\n\\subsection{Microreversibility}\n\nThe crucial assumption behind our derivation is microreversibility: we assume that after the magnetic field, the velocities of all particles and their spins are reversed, the system traces its evolution backwards.\nAs was pointed out in Refs. \\refcite{forster08,forster09}, such assumption is counterintuitive in mesoscopic systems without time-reversal symmetry.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{fig_box.eps}\n\\caption{Trajectories of charged particles connect different holes in opposite magnetic fields.}\n\\label{CCC}\n\\end{figure}\n\nConsider a mesoscopic conductor, connected to several infinite reservoirs, maintained at different voltages. A standard way to calculate currents and noises is based on the Landauer-B\\\"uttiker formalism \\cite{datta}. One first finds the self-consistent charge density $\\rho({\\bf r})$ as a function of the coordinates in the conductor. The charge distribution creates a self-consistent electrostatic potential. The currents and noises can be computed from the scattering theory for free particles, entering the mesoscopic conductor from the reservoirs, and moving in the self-consistent potential in the conductor. While this picture is compatible with microreversibility at zero magnetic field, there is a conflict at a finite $B$. Indeed, $\\rho({\\bf r},B)$ is not the same as the charge distribution $\\rho({\\bf r},-B)$ in the opposite field. Fig. \\ref{CCC} illustrates why. In that oversimplified example, charged particles can enter a square box through holes 1 and 3 only. At one direction of the magnetic field, particles from hole 1 move into hole 2 and particles from hole 3 move into hole 4. At the opposite field the particle trajectories connect hole 1 with 4 and 2 with 3. Since the charge density is nonzero only on those trajectories, $\\rho({\\bf r},B)\\ne\\rho({\\bf r},-B)$. The self-consistent electrostatic potentials depend on $\\rho$ and are different in the opposite magnetic fields. Consider now a scattering process in which a particle with the momentum ${\\bf k}_i$ acquires the momentum ${\\bf k}_f$ at the magnetic field B. The scattering amplitude depends on the self-consistent electrostatic potential $\\phi({\\bf r},B)$. The scattering probability for the time-reversed process at the field $-B$ describes the transition form the state with the momentum $-{\\bf k}_f$ into the state with the momentum $-{\\bf k_i}$. It depends on a different self-consistent potential $\\phi({\\bf r},-B)$ and hence is different from the scattering probability ${\\bf k}_i\\rightarrow{\\bf k}_f$ in the field $B$.\nSuch asymmetry of the self-consistent potential is closely related to the physics of rectification in mesoscopic conductors \\cite{rec1,rec2,rec3,rec4,rec5,rec6,rec7}.\n\nThe above argument does not by itself disprove microreversibility. Indeed, it is a mean-field argument which deals with single-particle states. On the other hand, the proof of the fluctuation theorem, Section 2, considers many-particle states. The unitarity of the evolution operator in quantum mechanics shows that the transition probability between the many-body states of the whole system, $P({\\rm initial~state~}\\rightarrow{\\rm~final~state},B)$, always equals $P({\\rm time-reversed~final~state~}\\rightarrow{\\rm~time-reversed~initial~state},-B)$. In each point, the charge densities in the forward and backward processes remain exactly the same in the corresponding moments of time. Hence, the electrostatic potentials do, in fact, remain the same. The mean-field argument is based on the different charge distributions in the steady states in the opposite magnetic fields. However, if one reverses time in a system in a steady state then the charge distribution does not change and hence does not become the steady state distribution of the electric charge in the opposite magnetic field.\n\nWe would like to emphasize that time-reversal in an interacting macroscopic system is not mathematical fiction. It was demonstrated experimentally long ago in the context of NMR \\cite{time1,time2}.\n\nSeveral groups have verified the fluctuation relations at a finite magnetic field without the use of microreversibility.\nThe Saito-Utsumi relations were confirmed by microscopic calculations beyond the Landauer-B\\\"uttiker formalism in two models \\cite{saito09,nasb10}. Note also that an approximate calculation beyond the mean-field theory in\nRef. \\refcite{lim10} agrees with Eq. (\\ref{dima14}). The fluctuation relations for a general chiral system from Section 5 below were derived both from microreversibility \\cite{wang2} and without its use \\cite{wang1}. Recent experiments \\cite{exp1,exp2} were interpreted as supporting the fluctuation relations \\cite{saito} (see the next subsection).\nWhile all this gives credibility to the approach, based on microreversibility, one must remember some subtleties.\n\nAll models in Refs. \\refcite{saito09,nasb10,lim10,wang1} assume a finite range of the electrostatic interaction. This implies the presence of screening gates. The gates are crucial for the fluctuation relations in chiral systems \\cite{wang1,wang2} since it is meaningful to speak about chiral transport only in systems with short-range interactions (Section 4). At the same time, the Saito-Utsumi relations do not make assumptions about the range of interactions. Yet, our derivation of the fluctuation theorem, Section 2, implicitly assumes short-range forces. Indeed, in the presence of the long-range Coulomb interaction, the reservoirs are not independent even in the beginning of the forward and backward processes and cannot be described by the Gibbs distribution. Certainly, this is a rather standard issue. It can be resolved by splitting the Coulomb force into a finite-range part with some large but finite interaction radius and the long-range part which must be treated in the mean-field approximation. This allows using the Gibbs distribution but returns the problem from the Landauer-B\\\"uttiker approach. Indeed, the average charge densities are not the same at the opposite field orientations and hence the mean-field effective long-range potentials are not the same in the forward and backward process. Fortunately, if the reservoirs are sufficiently large and the radius of the finite-range interaction is selected large enough, one can see that the difference of the long-range potentials can be neglected.\n\nAnother issue equally affects systems with and without the time-reversal symmetry. Our derivation, Section 2, assumes that the system is isolated. This may not be easy to accomplish in practice. Moreover, if this has been accomplished then no experiments can be performed since any measurement device disturbs the system. Fortunately, energy exchange with the outside world turns out not to be a problem provided the temperature of the environment is the same as the temperature of the system of interest. Indeed, let us include the environment as an additional reservoir and repeat the derivation of Eq. (\\ref{eq-fr}). The energy changes $\\Delta E_i$ of the reservoirs in the forward process enter Eq. (\\ref{eq-fr}) in the form $\\sum\\beta_i\\Delta E_i$. This combination is zero at equal $\\beta_i$ from the energy conservation.\nThus, all $\\Delta E_i$, including the energy absorbed by the environment, drop out from Eq. (\\ref{eq-fr}). On the other hand, the electrostatic potential of the environment drops out only in the absence of the particle exchange with the outside world. In fact, the fluctuation theorem Eq. (\\ref{eq-fr}) may break down even if charges are transfered between different regions of the environment in the absence of the particle exchange with the system of interest\n\\cite{exp3,exp4}. This issue has been a major difficulty in the experiments on fluctuation relations in mesoscopic conductors as we discuss in the next subsection.\n\n\n\\subsection{Experiment}\n\nNote that Eq. (\\ref{dima17}) crucially depends on the microreversibility while Eq. (\\ref{dima20}) does not require such assumption as shown in Ref. \\refcite{forster08}, see also Ref. \\refcite{s09}. Thus, the verification of Eq. (\\ref{dima17}) is particularly interesting. On the other hand, it is easier to measure noises and currents than the third cumulant \\cite{reznikov}. As a result, recent experiments have focused on the verification of a consequence of Eqs. (\\ref{dima17},\\ref{dima20}):\n\n\\begin{equation}\n\\label{dima21}\nS_1^A=6TG_2^A.\n\\end{equation}\n\nRefs. \\refcite{exp1,exp2} tested Eqs. (\\ref{dima21}) and (\\ref{dima14}) in a mesoscopic interferometer. Ref. \\refcite{exp1} observed $S_1^A\/[TG_2^A]=8.7^{+1.3}_{-0.7}$ and Ref. \\refcite{exp2} obtained $S_1^A\/[TG_2^A]=9.7^{+1.3}_{-1.2}$ in satisfactory agreement with Eq. (\\ref{dima21}). This was interpreted as a proof of microreversibility.\nOn the other hand, the results for $S_1$, $S_1\/TG_2=10.8^{+2.4}_{-1.4}$, Ref. \\refcite{exp1}, and $S_1\/TG_2=12.0^{+1.9}_{-2.0}$, Ref. \\refcite{exp2}, are incompatible with (\\ref{dima14}). The reasons for the discrepancy of the theory and experiment are unclear. A related experiment without a magnetic field may provide some hints.\n\nA violation of the fluctuation theorem (\\ref{eq-fr}) was found in a single electron tunneling experiment through a double quantum-dot system \\cite{exp3,exp4}. The violation has been explained by a careful analysis of the experimental circuit. The circuit included a quantum point contact\n(QPC) electrometer, used as a measurement device. When a finite voltage bias is applied to the quantum point contact, the fluctuation theorem (\\ref{eq-fr}) must be modified. The right hand side now contains an additional factor $\\exp(QV_{QPC}\/T)$, where $V_{QPC}$ is the bias at the QPC and $Q$ is the charge that travels through the QPC during the forward process. The derivation of the modified relation is exactly the same as our argument in Section 2. One just needs to include the QPC, connected to two reservoirs at different electrochemical potentials, into the system under consideration. This interpretation is supported by the fact that a modified fluctuation relation was found to hold in the experiment \\cite{exp1,exp2,sinitsyn}.\n\n\n\\section{Chiral systems}\n\nThe Saito-Utsumi relations from the previous section are very general and apply to any conductor in a magnetic field. This generality comes at a price. Indeed, the relations connect the coefficients in the expansions of the currents and noises in powers of the voltage. Thus, they only apply at low voltages, that is, close to equilibrium. One can overcome this limitation\nby looking at so-called chiral systems. In such systems all excitations propagate in one direction only, for example all excitations propagate to the right only.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3in]{fig_idealgas.eps}\n\\caption{Ideal gas in a reservoir with a tube (from Ref. 30).}\n\\label{idealgas}\n\\end{figure}\n\nThe simplest example of a chiral system is illustrated in Fig. \\ref{idealgas}. A box is filled with an ideal gas and placed in vacuum. A narrow tube with an open end and smooth walls is attached to the box. Particles can leave the box through the tube but do not come back. Thus, the transport in the tube is chiral.\n\nThe above example is too simple to be interesting other than as a toy model. Several more interesting examples of chiral transport are known. For example, statistical mechanics has been used to model traffic \\cite{helbig}. Transport is chiral on a network of one-way roads. In some biological systems, transport can only occur in one direction. Most importantly, chiral transport takes place on the edges and surfaces of some topological states of matter.\n\n\\subsection{Chiral transport in topological systems}\n\nChiral transport occurs in several topological systems without the time-reversal symmetry. For example, the surface of a 3D stack of integer quantum Hall liquids is chiral \\cite{3D1,3D2,3D3}. Transport is chiral on the edges of\n$p\\pm ip$ superconductors \\cite{anyons,green,ivanov}.\nBesides, many states of the two-dimensional electron gas in the conditions of the quantum Hall effect \\cite{perspectives} (QHE) are chiral. So far, most research on chiral transport has been focused on those QHE states.\n\nWe expect that the bulk of a quantum Hall system is gapped (see, however, Refs. \\refcite{4-3,heiblum14}). Thus, low-energy transport occurs on the edges. Wen's hydrodynamic theory \\cite{wen} predicts that in many cases the low-energy transport is chiral.\nThe integer QHE with the filling factor $\\nu=1$ is the simplest example. Wen's theory describes the edge physics in terms of a single field $\\phi$, where $\\partial_x\\phi$ is proportional to the linear charge density. The action assumes the form\n\n\\begin{equation}\n\\label{dima22}\nL=\\int dxdt[\\partial_t\\phi\\partial_x\\phi-v(\\partial_x\\phi)^2].\n\\end{equation}\nThe solution of the equation of motion, $\\phi=\\phi(x+vt)$, describes excitations that move only to the left.\n\nGeneralizations of the action (\\ref{dima22}) also predict \\cite{chang,wen} chiral transport at all other integer filling factors and in many fractional QHE states, including the states with the filling factors $\\nu=1\/3$ and $\\nu=2\/5$.\nSome other QHE states are not expected to be chiral. This point can be easily understood by considering $\\nu=2\/3$. The $\\nu=2\/3$ liquid can be described as the $\\nu=1\/3$ state of holes on top of the $\\nu=1$ state of electrons.\nThus, its edge theory contains two excitation branches, corresponding to $\\nu=1$ and $\\nu=1\/3$, with the opposite chirality of the electron and hole edges.\n\nSuch description of the $2\/3$ edge conflicts with the experiment that shows only one propagation direction for charged excitations. This was explained by Kane, Fisher and Polchinsky \\cite{kfp} who uncovered the nontrivial role of impurities which are inevitably present along QHE edges. Impurities promote tunneling between the contra-propagating edge channels and change the nature of the edge modes: all charge excitations move in the same direction, called downstream; in addition, a neutral excitation branch of the opposite, i.e., upstream, chirality emerges.\n\nA similar picture is expected to apply in several other QHE states. Detecting upstream neutral modes proved to be a great challenge and only recently has progress been reported in the field \\cite{36,deviatov,yacoby}. We will see that the fluctuation relations from Section 5 give a tool to test the presence of upstream neutral modes. Confusingly, there were recent reports of upstream modes in the $1\/3$ and $2\/5$ states and even at the integer filling factors \\cite{heiblum14,deviatov,yacoby}. Thus, an independent test of chirality on the edges is of great importance.\n\n\\subsection{Non-Abelian quantum Hall states}\n\nThe question of chirality on QHE edges also touches upon the ongoing search for non-Abelian anyons \\cite{anyons}. Quasiparticles in fractional QHE states are known to be anyons with a different exchange statistics from bosons and fermions \\cite{wen}. The simplest Abelian anyons accumulate nontrivial phases when encircle each other. The many-body quantum state of a system of Abelian anyons does not change after one of them makes a circle around another adiabatically. Hypothetical non-Abelian anyons \\cite{anyons,2} change their quantum state after one particle braids another. This does not involve a change of the internal quantum numbers of any particle and reflects the fact that the information about a quantum state of an anyonic system is distributed over the whole system. This property makes non-Abelian anyons attractive for topological quantum information processing, naturally protected from errors \\cite{anyons,6}. Indeed, local perturbations from the interaction with the environment cannot change or erase quantum information that is stored globally.\n\nThe most promising candidate for non-Abelian anyons is the QHE state \\cite{willett87} at $\\nu=5\/2$. At the same time, there are many competing Abelian and non-Abelian candidate states at that filling factor \\cite{2,3,4,5,8,19,BS,26,Overbosch,yang} and the existing body of experiments does not allow the determination of the right state at this point, see Ref. \\refcite{yang} for a review. Interferometry\n \\cite{Stern10,chamon97,fradkin98,mz,dassarma05,11,12,13,14,hou06,grosfeld06,15,willett09,bishara09,willett10,fp331,16,kang,double,rosenow12} is the most direct approach but its implementation has faced significant difficulties. This motivated the search for non-interferometric ways to obtain information about the $5\/2$ state\n \\cite{yang,feldman08,viola12,overbosch09,CS,seidel09,yang09,wang10a,mstern10,rhone11,tiemann12,mstern12,chickering13,20,21}.\nNeither non-interferometric method would provide a direct observation of anyonic statistics but their combination may be sufficient to identify the correct state. In particular, some of the proposed states are chiral while others are not. This makes a chirality test, based on the fluctuation relations from Section 5, a useful tool in the search for non-Abelian particles.\n\n\\subsection{Chirality and causality}\n\nIn chiral systems transport occurs in one direction only. This includes the transport of information. Thus, the causality principle is enhanced. In addition to the requirement that future events do not affect the past, we also expect that the downstream events do not affect upstream events even in the future. Such modified causality principle allows one to generalize the Nyquist formula for nonlinear transport far from equilibrium (Section 5). Indeed, the standard derivation of the fluctuation-dissipation theorem (FDT) is based on the combination of the Gibbs distribution and linear response theory. Causality is the key ingredient of the latter theory. The enhanced causality principle allows a derivation of a generalized FDT even without the use of the Gibbs distribution, that is, far from equilibrium.\n\nNote that our definition of chirality assumes the absence of long-range forces \\cite{wang1}. Otherwise, such forces could mediate instantaneous information exchange between distant points so that the upstream events would affect the events downstream. In particular, we assume the presence of screening gates in chiral systems of charged particles.\n\n\n\\section{Fluctuation relations in chiral systems}\n\nIn this section we focus on chiral systems. We discover that fluctuation relations assume a form \\cite{wang1,wang2}, similar to the equilibrium FDT, but hold for nonlinear transport away from equilibrium.\nWe start with a heuristic derivation in Subsection 5.1. In Subsection 5.2 we verify the nonequilibrium FDT in the toy model, Fig. ~\\ref{idealgas}. We then give a general proof in Subsection 5.3. Numerous generalizations \\cite{wang2} and possible applications are briefly addressed in Subsections 5.4 and 5.5.\n\nOne closely related fluctuation relation was derived in Refs. \\refcite{kf,FLS,FS} in an exactly solvable model. Interestingly, that model could be used to describe both a chiral system in the context of QHE physics and a nonchiral quantum wire.\nOur results show that the integrability of the Hamiltonian is not required for the existence of an infinite number of fluctuation relations in chiral systems. At the same time, chirality is crucial. The applicability of the results of Refs. \\refcite{kf,FLS,FS} in a nonchiral system is a peculiar feature of the exactly solvable model.\n\n\n\n\\subsection{Qualitative argument}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{fig_3terminal1.eps}\n\\caption{ From Ref. 30. Three-terminal setup of a chiral system. We consider a quantum Hall bar with the lower edge, coupled to terminal C. Charge can tunnel between terminal C and the lower edge. The\n solid lines represent chiral edge modes whose directions are shown by arrows and determined by the external magnetic field. The arrow on the dotted line represents our\nconvention about the positive direction of the tunneling current $I_{\\rm T}$. }\n\\label{3terminal}\n\\end{figure}\n\nIn the next three subsections, we only consider the simplest example of the nonequilibrium FDT\\cite{wang1,wang2}. The extended causality principle (Subsection 4.3)\nwill be extensively used in the derivation of these results. Many more generalized nonequilibrium fluctuation relations, e.g, fluctuation relations in a\nmulti-terminal setup and for heat transport, can be derived\\cite{wang2} but their discussion is postponed to section 5.4.\n\nLet us study the nonequilibrium FDT in the three-terminal system shown in Fig.~\\ref{3terminal}. It is a Hall bar with one of the two chiral edges coupled to a\nthird terminal. The three terminals [the source (S), the drain (D) and the third contact (C)] are assumed to be ideal reservoirs that have zero impedance\nand infinitely large capacitance. The bulk of the Hall bar is gapped and the edges are chiral. The strength of the coupling\nbetween the Hall bar and the third contact is unimportant. We assume that the two chiral edges are fully absorbed by source S at the left end\nand drain D at the right end, and they are far apart so that they do not interfere with each other.\nSource S is biased at the voltage $V_{\\rm S}$, contact C is at $V_{\\rm C}$, and the drain is grounded. Let the\ntemperature of the system be $T$. Steady currents flow between the terminals, in particular, charge tunnels into terminal C, leading to a tunneling current\n$I_{\\rm T}$. We will show that the following relation holds\n\\begin{equation}\n\\label{chenjie1}\nS_{\\rm D} = S_{\\rm C} - 4T\\frac{\\partial I_{\\rm T}}{\\partial V_{\\rm S}} + 4GT,\n\\end{equation}\nwhere $S_{\\rm D}=\\int dt \\langle \\Delta I_{\\rm D}(t) \\Delta I_{\\rm D}(0) + \\Delta I_{\\rm D}(0) \\Delta I_{\\rm D}(t)\\rangle$ is the zero-frequency noise of\nthe drain current $I_{\\rm D}$, $S_{\\rm C}=\\int dt \\langle \\Delta I_{\\rm T}(t) \\Delta I_{\\rm T}(0) + \\Delta I_{\\rm T}(0) \\Delta I_{\\rm T}(t)\\rangle$ is the\nzero-frequency noise of the tunneling current $I_{\\rm T}$, and $G$ is the quantized Hall conductance in the absence of contact C. The FDT (\\ref{chenjie1})\nholds for arbitrary $T$, $V_{\\rm S}$ and $V_{\\rm C}$ as long as they are far below the QHE gap. Note that the system is far from equilibrium when $T\n\\lesssim V_{\\rm C}, V_{\\rm S}$.\n\nBelow we give a heuristic derivation of the nonequilibrium FDT (\\ref{chenjie1}) following Ref.~\\refcite{wang1}. As shown in Fig.~\\ref{3terminal},\nthe current $I_{\\rm D} = I_{\\rm L} - I_{\\rm U}$, where $I_{\\rm L}$ is the current, entering the drain along the lower edge, and $I_{\\rm U}$ is the current,\nemitted from the drain along the upper edge. Since the system is in a steady state and no charge is accumulated on the lower edge, the low-frequency part\nof $I_{\\rm L}$ can be written as $I_{\\rm L} = I_{\\rm S} - I_{\\rm T}$, where $I_{\\rm S}$ is the current, emitted from the source. Because the two edges are\nuncorrelated, the low-frequency noises obey the relation\n\\begin{equation}\n\\label{chenjie2}\nS_{\\rm D} = S_{\\rm C} -2 S_{\\rm ST} + S_{\\rm S} + S_{\\rm U},\n\\end{equation}\nwhere $S_{\\rm ST}=\\int dt \\langle \\Delta I_{\\rm T}(t) \\Delta I_{\\rm S}(0) + \\Delta I_{\\rm S}(0) \\Delta I_{\\rm T}(t)\\rangle$ is the cross noise of $I_{\\rm\nS}$ and $I_{\\rm T}$, and $S_{\\rm S}$ and $S_{\\rm U}$ are the noises of $I_{\\rm S}$ and $I_{\\rm U}$ respectively. To derive (\\ref{chenjie1}) from\n(\\ref{chenjie2}), let us first find $S_{\\rm S}$ and $S_{\\rm U}$. To do this, it is enough to consider a simplified case, where contact C is absent.\nIn that case, we have an obvious result: both $S_{\\rm S}$ and $S_{\\rm U}$ are equal to one half of the standard Nyquist noise, that is\n\\begin{equation}\n\\label{chenjie3}\nS_{\\rm S}= S_{\\rm U}= 2GT.\n\\end{equation}\nCrucially, in the presence of contact C, Eq.(\\ref{chenjie3}) still holds. We notice that adding contact C does not affect the noise of $I_{\\rm S}$\nbecause of the extended causality principle in chiral systems. Also, it does not affect the noise of $I_{\\rm U}$ because of the assumption that the two edges are\nuncorrelated. Hence, the noises $S_{\\rm S}$ and $S_{\\rm U}$ are given by Eq. (\\ref{chenjie3}) even in the presence of terminal C.\n\nWe are left with the calculation of the cross noise $S_{\\rm ST}$. Let us analyze the dependence of the tunneling current $I_{\\rm T}(t)$ on the emitted\ncurrent $I_{\\rm S}$. The tunneling current $I_{\\rm T}$ depends on the average emitted current $\\bar I_{\\rm S}$ and its fluctuations $I_{\\omega}$, where\n$\\omega$ denotes the fluctuation frequency. According to the extended causality principle, the average emitted current $\\bar I_{\\rm S}$\ndepends only on $V_{\\rm S}$ and is not affected by terminal C. In other words, $\\bar I_{\\rm S}= G V_{\\rm S}$. We now assume that the central part of the lower edge has a relaxation time $\\tau$, so that\nthe instantaneous value of $I_{\\rm T}(t)$ depends only on the emitted current within the time interval $\\tau$. It is convenient to separate the\nfluctuations of $I_{\\rm S}$ into the fast part $I^>$ which contains the fluctuations of the frequencies above $1\/\\tau$, and the slow part $I^<$ which contains\nthe fluctuations of the frequencies below $1\/\\tau$. Within the time interval $\\tau$, $I^<$ does not exhibit a time-dependence. Hence $I^<$ enters the expression for\nthe tunneling current $I_{\\rm T}(t)$ in the combination $\\bar I_{\\rm S}+I^<$ only, $I_{\\rm T} = \\langle I(\\bar I_{\\rm S}+I^<, I^>)\\rangle$, where the\nbrackets denote the average with respect to the fluctuations of $I_{\\rm S}$. According to the Nyquist formula for the emitted current, its harmonics with\ndifferent frequencies have zero correlations $\\langle I_{\\omega} I_{-\\omega'} \\rangle\\sim \\delta(\\omega-\\omega')$. For the sake of the heuristic argument,\nwe will assume a Gaussian distribution of $I_{\\rm S}$, hence, independence of the high- and low-frequency fluctuations. With this assumption, we\naverage over the fast fluctuations and write $I_{\\rm T} = \\langle J(\\bar I_{\\rm S}+I^<)\\rangle$, where $J$ is obtained by averaging over $I^>$. $I^<$\ncorresponds to a narrow frequency window and can be neglected in comparison with $\\bar I_{\\rm S}$, i.e., the average tunneling current $I_T= J(\\bar I_{\\rm\nS})$. For the calculation of the cross noise, we expand $J(\\bar I_{\\rm S}+I^<)$ to the first order in $I^<$ and obtain\n\\begin{equation}\n\\label{chenjie4}\nS_{\\rm ST} = \\langle I_{\\rm T,\\omega}I_{-\\omega} + I_{\\rm T, -\\omega} I_{\\omega}\\rangle = \\frac{\\delta J(\\bar I_{\\rm S})}{\\delta \\bar I_{\\rm S}} \\langle\nI_\\omega I_{-\\omega} + I_{-\\omega} I_{\\omega}\\rangle = 2T \\frac{\\partial I_{\\rm T}}{\\partial V_{\\rm S}},\n\\end{equation}\nwhere we have used the result $S_{\\rm S} =\\langle I_\\omega I_{-\\omega} + I_{-\\omega} I_{\\omega}\\rangle=2GT$, and $\\delta \\bar I_{\\rm S}= G\\delta V_{\\rm\nS}$. Combining the results (\\ref{chenjie2})-(\\ref{chenjie4}), the nonequilibrium FDT (\\ref{chenjie1}) is easily obtained.\n\nWe have seen that chirality plays an important role in the derivation of Eqs. (\\ref{chenjie3}) and (\\ref{chenjie4}). In this heuristic derivation, an unnecessary\nassumption of Gaussian fluctuations of $I_{\\rm S}$ has been made. In Subsection 5.3, the nonequilibrium FDT will be derived from the fluctuation theorem (\\ref{eq-fr}) without that\nassumption.\n\n\\subsection{Toy model}\n\n\nBefore moving to a general proof of the nonequilibrium FDT (\\ref{chenjie1}), let us verify it in a toy model of an ideal gas\n(Fig.~\\ref{idealgas}). Our discussion will follow the appendix to Ref. \\refcite{wang1}.\n\nConsider a large reservoir of an ideal gas of noninteracting molecules at the temperature $T$ and chemical potential $\\mu$.\nMolecules can leave the reservoir through a narrow tube with smooth walls. By smooth, we mean such walls that the collisions of the molecules with the walls are elastic and\ndo not change the velocity projection along the tube axis. Thus, molecules can only leave the reservoir but never come back. The system is chiral. Imagine\nnow that molecules can escape through a side hole in the wall of the tube (Fig. ~\\ref{idealgas}). We can derive a relation, similar to Eq.~(\\ref{chenjie1}):\n\\begin{equation}\n\\label{chenjie6}\nS_{\\rm D} = S_{\\rm C} - 4T \\frac{\\partial I_{\\rm T}}{\\partial \\mu} + S_{\\rm S},\n\\end{equation}\nwhere $I_{\\rm T}$ is the particle current through the side hole in the tube wall and $S_{\\rm C}$ is its noise, Fig. ~\\ref{idealgas}, $S_{\\rm D}$ is the noise of the current $I_{\\rm\nD}$ at the open end of the tube, and $S_{\\rm S}$ is the noise of the particle current $I_{\\rm S}$ at the opposite end, attached to the box. The noise $S_{\\rm S}$ can be determined\nfrom the measurement of $S_{\\rm D}$ in the absence of the side hole in the tube wall. Note that the above relation is almost the same as the nonequilibrium FDT\n(\\ref{chenjie1}) in the QHE setup, Fig.~\\ref{3terminal}.\n\nThe proof of the above expression is rather simple and builds on Ref.~\\refcite{martin}.\nIt is most convenient to work with a Fermi gas with\na high negative chemical potential.\nOther cases, such as a Bose gas, can be considered in a similar way but will not be addressed below.\nLet $f=1\/\\{\\exp[(E-\\mu)\/T]+1\\}$ be the Fermi distribution in the box and $T_{E}$ the transmission\ncoefficient through the side hole in the tube wall for a particle of the energy $E$. According to Ref.~\\refcite{martin}, for the particles within the energy window $(E,\nE+dE)$, the current through the side hole is $T_{E}f dE$ and the noises $S_{\\rm S}=2f(1-f)dE$, $S_{\\rm C} = 2T_{E}f(1-T_{E}f)dE$, and $S_{\\rm\nD}=2(1-T_{E})f[1-(1-T_{E})f]dE$. One needs to integrate over the energy to obtain the overall current and noises. It is then easy to obtain the\nexpression (\\ref{chenjie6}).\n\n\n\\subsection{General derivation}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{fig_3terminal2.eps}\n\\caption{ The time-reversed setup. The only differences from Fig.~\\ref{3terminal} are the reversed directions of the magnetic field and the chiral edge modes.}\n\\label{3terminal-tr}\n\\end{figure}\n\n\nIn this subsection, we give a derivation of the nonequilibrium FDT (\\ref{chenjie1}) based on the fluctuation theorem (\\ref{eq-fr}). In the\nheuristic argument, Subsection 5.1, an unnecessary assumption was only made in the derivation of the expression for the cross noise (\\ref{chenjie4}). Thus, below we\nfocus on proving the relation (\\ref{chenjie4}).\n\nFig.~\\ref{3terminal} is a three-terminal case of the most general setup Fig.~\\ref{fig1}, so the fluctuation theorem (\\ref{eq-fr}) applies. Let us follow\nthe protocol in Section 2 and assume that after the time period $\\tau$, the changes of the particle numbers in the three terminals are $\\Delta N_{\\rm S}$,\n$\\Delta N_{\\rm C}$, and $\\Delta N_{\\rm D} = -\\Delta N_{\\rm S} - \\Delta N_{\\rm C}$, respectively, and the distribution function is $P(\\Delta N_{\\rm S},\n\\Delta N_{\\rm C}; B)$. Note that only two out of the three $\\Delta N$'s are independent due to the charge conservation. Then\nthe fluctuation theorem (\\ref{eq-fr}) becomes\n\n\\begin{equation}\n\\label{chenjie7}\n\\frac{P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};+ B)}{P(-\\Delta N_{\\rm S}, -\\Delta N_{\\rm C}; -B)}= e^{-\\beta e(V_{\\rm S}\\Delta N_{\\rm S} + V_{\\rm C} \\Delta\nN_{\\rm C})}\n\\end{equation}\nwhere $\\beta =1\/T$, $e$ is the electron charge, and $P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};-B)$ is the distribution function for the backward process (see Section 2) in the time-reversed version\nof Fig.~\\ref{3terminal} as illustrated in Fig. ~\\ref{3terminal-tr}. Note that we could write the drain voltage $V_{\\rm D}$ explicitly in the fluctuation theorem instead\nof setting $V_{\\rm D}=0$. That, however, would only burden our notation without providing any advantage. Fig.~\\ref{3terminal-tr} shows the\ntime-reversed setup which has the opposite chirality, compared to Fig. ~\\ref{3terminal}. It is worth emphasizing that the observables $I_{\\rm T}$, $S_{\\rm ST}$, {\\it etc.}, that we are\ninterested in are defined in the setup from Fig.~\\ref{3terminal}. The role of the time-reversed setup is only to help us prove the nonequilibrium\nFDT (\\ref{chenjie1}). In terms of the distribution function $P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C}; B)$, the observables of interest can be written as\n\n\\begin{equation}\n\\quad I_{\\rm T} =\\frac{e}{\\tau}\\langle\\Delta N_{\\rm C}\\rangle,\n\\end{equation}\n\n\\begin{equation}\nI_{\\rm R} = I_{\\rm U} - I_{\\rm S}= \\frac{e}{\\tau}\\langle\\Delta N_{\\rm S}\\rangle,\n\\end{equation}\n\n\\begin{equation}\nS_{\\rm ST} = -S_{\\rm R T} = -2\\frac{e^2}{\\tau}(\\langle\\Delta N_{\\rm S}\\Delta N_{\\rm C}\\rangle - \\langle\\Delta N_{\\rm S}\\rangle\\langle\\Delta N_{\\rm\nC}\\rangle),\n\\end{equation}\nwhere\n\\begin{equation}\n\\langle x \\rangle = \\sum_{\\Delta N_{\\rm S}, \\Delta N_{\\rm C}} x P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C}; B).\n\\end{equation}\nWe have defined the current $I_{\\rm R}$ as the overall current flowing into source S. The cross noise $S_{\\rm RT}$ of $I_{\\rm R}$ and $I_{\\rm T}$\nequals $-S_{\\rm ST}$ since $I_{\\rm U}$ is uncorrelated with $I_{\\rm T}$.\n\nIt is convenient to define the cumulant generating functions\n\n\\begin{equation}\n\\label{chenjie8}\nQ(x,y;\\pm B) = \\lim_{\\tau\\rightarrow \\infty} \\frac{1}{\\tau} \\ln\\left\\{\\sum_{\\Delta N_{\\rm S}, \\Delta N_{\\rm C}} e^{-x e\\Delta N_{\\rm S} - y e\\Delta N_{\\rm\nC}} P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};\\pm B)\\right\\}.\n\\end{equation}\nWith these generating functions, the quantities of interest can be expressed as\n\n\\begin{equation}\n\\label{def1}\nI_{\\rm R} = -\\left.\\frac{\\partial Q(x,y; +B)}{\\partial x}\\right|_{x= y= 0},\n\\end{equation}\n\n\\begin{equation}\n\\label{def2}\nI_{\\rm T} = -\\left.\\frac{\\partial Q(x,y;+B)}{\\partial y}\\right|_{x=y=0},\n\\end{equation}\n\n\\begin{equation}\n\\label{def3}\nS_{\\rm ST} = -2\\left.\\frac{\\partial^2 Q(x,y; +B)}{\\partial x \\partial y}\\right|_{x=y=0}.\n\\end{equation}\n\nInserting the fluctuation theorem (\\ref{chenjie7}) into the definition (\\ref{chenjie8}), we find that the generating functions $Q(x,y;\\pm B)$ have a very\nnice property:\n\\begin{equation}\n\\label{chenjie5}\nQ(x, y; +B) = Q(-\\beta V_{\\rm S} - x, -\\beta V_{\\rm C} -y; -B).\n\\end{equation}\nNote that this is an equation, relating two generating functions, $Q(x, y; +B)$ and $Q(x, y; -B)$. Also, because the distribution functions $P(\\Delta\nN_{\\rm S}, \\Delta N_{\\rm C};\\pm B)$ are normalized, we have $Q(x=0, y=0; \\pm B)=0$. Since the distribution functions depend on the biases $V_{\\rm S}$\nand $V_{\\rm C}$, the generating functions are also functions of $V_{\\rm S}$ and $V_{\\rm C}$\n\\begin{equation}\nQ = Q(x, y, V_{\\rm S}, V_{\\rm C}; \\pm B).\n\\end{equation}\n\n We now prove the relation (\\ref{chenjie4}) and hence also the nonequilibrium FDT\n(\\ref{chenjie1}) from the property (\\ref{chenjie5}) and enhanced causality. We first apply the differential operator\n\n\\begin{equation}\n\\hat D = \\left(\\frac{d}{dx} - T \\frac{d}{d V_{\\rm S}}\\right)\\left(\\frac{d}{dy} - T \\frac{d}{d V_{\\rm C}}\\right)\n\\end{equation}\non both sides of Eq.~(\\ref{chenjie5}). A straightforward calculation using the expressions (\\ref{def1})-(\\ref{def3}) yields\n\n\\begin{align}\n\\label{chenjie9}\n\\frac{1}{2}S_{\\rm ST} = & T\\frac{\\partial I_{T}}{\\partial V_{\\rm S}} +T\\frac{\\partial I_{R}}{\\partial V_{\\rm C}}+ T^2\\left.\\frac{\\partial^2Q(x, y, V_{\\rm\nS}, V_{\\rm C};+B)}{\\partial V_{\\rm S}\\partial V_{\\rm C}}\\right|_{x=y=0} \\nonumber \\\\\n& -T^2 \\left.\\frac{\\partial^2Q(x, y, V_{\\rm S}, V_{\\rm C};-B)}{\\partial V_{\\rm S}\\partial V_{\\rm C}}\\right|_{x=-\\beta V_{\\rm S}, y=-\\beta V_{\\rm C}}.\n\\end{align}\n\n\\noindent\nThis equation does not look the same as Eq.~(\\ref{chenjie4}), which is simply $S_{\\rm ST} = 2T\\frac{\\partial I_{T}}{\\partial V_{\\rm S}} $. However, we will\nshow that only the first term on the right-hand-side of Eq.~(\\ref{chenjie9}) is nonzero.\n\nThe third term on the right-hand-side of Eq.~(\\ref{chenjie9}) is\nzero simply because $Q(x=0, y=0, V_{\\rm S}, V_{\\rm C}; B)=0$.\n\nNow the chirality-enhanced causality principle enters the game as a key tool to prove that the second and fourth terms vanish. First, it is easy to\nsee that the second term is zero, because: (1) $I_{\\rm R} = I_{\\rm U} - I_{\\rm S}$; (2) $I_{\\rm S}$ does not depend on $V_{\\rm C}$ due to extended causality and\n(3) the upper and lower edges are assumed to be uncorrelated, so $I_{\\rm U}$ does not depend on $V_{\\rm C}$.\n\nTo prove that the last term is zero, we need a more sophisticated argument from extended causality. Note that the last term comes from the\ntime-reversed setup, Fig.~\\ref{3terminal-tr}. Since the upper and lower edges in Fig.~\\ref{3terminal-tr} are uncorrelated, we can write the\ndistribution function $P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};- B)$ as\n\n\\begin{equation}\nP(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};- B)= \\sum_{\\Delta N_{\\rm S}'} P_1(\\Delta N_{\\rm S}';- B) P_2(\\Delta N_{\\rm S}-\\Delta N_{\\rm S}', \\Delta N_{\\rm C};-\nB),\n\\end{equation}\nwhere $P_1(\\Delta N_{\\rm S}';- B)$ is the probability that $-\\Delta N_{\\rm S}'$ particles leave source $S$ through the {\\it\nupper} edge during the time interval $\\tau$, and $P_2(\\Delta N_{\\rm S}-\\Delta N_{\\rm S}', \\Delta N_{\\rm C};- B)$ is the probability that $\\Delta\nN_{\\rm S}-\\Delta N_{\\rm S}'$ particles enter S through the {\\it lower} edge while $\\Delta N_{\\rm C}$ particles enter contact\nC. This property of the distribution function $P(\\Delta N_{\\rm S}, \\Delta N_{\\rm C};- B)$ allows one to decompose the generating function $Q(x,y;- B)$ as\n\n\n\\begin{equation}\n\\label{dimaQ}\nQ(x,y;- B) = Q_1(x,y;- B)+Q_2(x,y;- B),\n\\end{equation}\n\n\\begin{equation}\nQ_1(x,y;- B) = \\lim_{\\tau\\rightarrow \\infty} \\frac{1}{\\tau} \\ln\\left\\{\\sum_{\\Delta N_{\\rm S}'} e^{-x e\\Delta N_{\\rm S}'} P_1(\\Delta N_{\\rm S}';-\nB)\\right\\},\\nonumber\n\\end{equation}\n\n\\begin{equation}\nQ_2(x,y;- B) = \\lim_{\\tau\\rightarrow \\infty} \\frac{1}{\\tau} \\ln\\left\\{\\sum_{\\Delta N_{\\rm S}'', \\Delta N_{\\rm C}} e^{-x e\\Delta N_{\\rm S}'' - y e\\Delta\nN_{\\rm C}} P_2(\\Delta N_{\\rm S}'', \\Delta N_{\\rm C};- B)\\right\\}. \\nonumber\n\\end{equation}\nWith the above decomposition (\\ref{dimaQ}), we look at the dependences of $Q_1$ and $Q_2$ on the voltages $V_{\\rm S}$ and $V_{\\rm C}$. Since the upper edge is chiral, the\ndistribution $P_1$ depends on $V_{\\rm S}$ but not on $V_{\\rm C}$. Similarly, since the lower edge is chiral, the distribution $P_2$ does not depend on\n$V_{\\rm S}$ while it does depend on $V_{\\rm C}$. Therefore, $Q_{1}$ depends on $V_{\\rm S}$ but not on $V_{\\rm C}$, while $Q_{2}$ depends on $V_{\\rm C}$\nbut not on $V_{\\rm S}$. Hence,\n\\begin{equation}\n\\frac{\\partial^2Q(x, y, V_{\\rm S}, V_{\\rm C};-B)}{\\partial V_{\\rm S}\\partial V_{\\rm C}}= \\frac{\\partial^2Q_1}{\\partial V_{\\rm S}\\partial V_{\\rm C}}\n+\\frac{\\partial^2Q_2}{\\partial V_{\\rm S}\\partial V_{\\rm C}} =0.\n\\end{equation}\nThus, the last term on the right-hand-side of Eq.~(\\ref{chenjie9}) is zero.\n\nTo sum up, we have proved that three of the terms on the right-hand-side of Eq.~(\\ref{chenjie9}) are zero, leaving the equation to be simply\n\\begin{equation}\nS_{\\rm ST} = 2 T\\frac{\\partial I_{T}}{\\partial V_{\\rm S}},\n\\end{equation}\nwhich is exactly the relation (\\ref{chenjie4}). Therefore, the nonequilibrium FDT (\\ref{chenjie1}) is indeed true for chiral systems.\n\n\n\n\\subsection{Generalizations}\n\nAbove, we studied the simplest example of the nonequilibrium FDT for charge transport. Let us mention some generalizations.\n\nOne of the generalizations is to study chiral heat transport in the three-terminal setup in Fig.~\\ref{3terminal}. In this case, the three terminals are at different temperatures, $T_{\\rm S}$ in S, $T_{\\rm C}$ in C, and $T_{\\rm D}$ in D. We can assume that they have the same chemical potential. According to Ref.\\refcite{wang2}, the cross noise $S_{\\rm ST}^h$ of the heat currents $I_{\\rm S}^h$ and $I_{\\rm T}^h$ satisfies\n\\begin{equation}\nS_{\\rm ST}^h =2T_{\\rm S}^2 \\frac{\\partial I^h_{\\rm T}}{\\partial T_{\\rm S}},\n\\end{equation}\nwhere $I_{\\rm T}^h$ is the heat current flowing into contact C and $I^h_{\\rm S}$ is the heat current flowing out of source S. This expression, valid for nonequilibrium states, resembles the standard equilibrium FDT which has the form $S^h = 4T^2 G^h$ with $S^h$ being the noise of the heat current in equilibrium, $G^h$ being the thermal conductance and $T$ the temperature.\n\nNonequilibrium fluctuation relations of higher-order cumulants can also be obtained in chiral systems. For example, the following relation holds\\cite{wang2} for the three-terminal setup in Fig.~\\ref{3terminal}\n\\begin{equation}\n\\label{chenjie10}\nC_{\\rm TT S} = T\\frac{\\partial S_{\\rm TT}}{\\partial V_{\\rm S}},\n\\end{equation}\nwhere $S_{\\rm TT}$ is the noise of the current $I_{\\rm T}$, and $C_{\\rm TTS}$ is the third cumulant defined as\n\\begin{equation}\nC_{\\rm TTS} = -\\frac{2e^3}{\\tau}\\langle(\\Delta N_{\\rm C}-\\langle\\Delta N_{\\rm C}\\rangle)\\cdot(\\Delta N_{\\rm C}-\\langle\\Delta N_{\\rm C}\\rangle) \\cdot (\\Delta N_{\\rm S}-\\langle\\Delta N_{\\rm S}\\rangle)\\rangle.\n\\end{equation}\n\nIt is also possible to generalize nonequilibrium fluctuation relations to multi-terminal systems and to higher-order cumulants for heat transport. The reader may consult Ref.~\\refcite{wang2} for details.\n\n\n\\subsection{Applications}\n\nOur main result, Eq. (\\ref{chenjie1}), as well as its generalization from the previous subsection apply to chiral systems only. Thus, the nonequilibrium FDT can be used to test edge chirality. If the FDT is satisfied both in and beyond equilibrium this is compatible with chirality. If it is broken then the edge is not chiral.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3in]{fig_expsetup.eps}\n\\caption{From Ref. 30. A possible experimental setup. Charge carriers, emitted\nfrom the source, can either tunnel through constriction Q and\ncontinue toward the drain or are absorbed by Ohmic contact C.}\n\\label{EEE}\n\\end{figure}\n\nA possible experimental setup is shown in Fig. \\ref{EEE}.\nOne of the mechanisms how our FDT gets broken in nonchiral systems is illustrated in Fig. \\ref{DDD}. The downstream charged mode dissipates energy in the hot spot \\cite{klass}, where it enters the drain. The upstream neutral mode carries the dissipated energy back to the tunneling contact and the point, where the charged mode exits the drain, and heats them. This affects both the tunneling current and the noise and breaks the theorem (\\ref{chenjie1}).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3in]{fig_neutralmode.eps}\n\\caption{From Ref. 30. A nonchiral system. The solid line along the lower edge\nillustrates the downstream mode, propagating from the source to\nthe drain. The dashed line shows a counter-propagating upstream\nmode}\n\\label{DDD}\n\\end{figure}\n\n\nRecently there has been much interest in neutral modes on QHE edges. They were first reported \\cite{36} in the particle-hole conjugated QHE states, where the theory predicts upstream neutral modes. Latter, upstream neutral modes were also reported in some states where they had not been expected \\cite{heiblum14,deviatov,yacoby}. In such situation a new experimental test, based on the FDT (\\ref{chenjie1}), will be helpful.\n\nOur results can also be used to narrow the range of the candidate QHE states at the filling factor $5\/2$. Some candidates (e.g., the Pfaffian state \\cite{2} or the 331 state \\cite{8}) have chiral edges. Others, most notably the anti-Pfaffian state \\cite{3,4}, are not chiral.\nSome evidence of an upstream neutral mode on the $5\/2$ edge has been reported recently \\cite{36}. Obviously, verification with a different method is highly desirable. Our FDT (\\ref{chenjie1}) provides such a method.\n\n\\section{Conclusion}\n\nIn this review we considered fluctuation relations in the absence of the time-reversal symmetry. The Saito-Utsumi relations\\cite{saito} apply to any conductor in a magnetic field and connect nonlinear transport coefficients in the opposite magnetic fields close to equilibrium. The fluctuation theorem for chiral systems \\cite{wang1,wang2} applies even far from equilibrium and connects currents and noises at the same direction of the magnetic field. This relation can be used to test the chirality of QHE edges. This is relevant in the ongoing search for neutral modes on QHE edges and in the search for non-Abelian anyons.\n\nMany questions remain open. New experimental tests of the fluctuation relations \\cite{saito} beyond Refs. \\refcite{exp1,exp2} would be important. The conflict between the existing experimental data and Eq. (\\ref{dima14}) has not been understood yet.\nOn the theory side, fluctuation relations for electric currents can be generalized for any other conserving quantity. In particular, Ref. \\refcite{wang2} addresses fluctuation relations for heat currents in chiral systems. Fluctuation relations for spin currents is another interesting question.\nIn that context time-reversal symmetry may be broken by an external magnetic field or by the spontaneous magnetization of the leads. Some work, based on weaker fluctuation relations \\cite{forster08}, has been published in Refs. \\refcite{lopez12,lim13}. Ref. \\refcite{spin-utsumi} attempted to apply stronger fluctuation relations to spintronics but overlooked the correct transformation law for spin currents under time reversal.\n\nExternal magnetic fields and spontaneous magnetization are not the only ways how the time-reversal symmetry gets broken. It is interesting to extend the results, discussed in this review, to systems with time-dependent Hamiltonians which are not invariant with respect to time reversal. Research in that direction includes Ref. \\refcite{altland10}. Ref. \\refcite{yi11} considers a system with a time-dependent magnetic field.\nThe ideas from Refs. \\refcite{Maes1,Maes2} may be useful for the class of problems considered in this review.\n\nMost research in the field has focused on quantum systems but there is nothing inherently quantum about time-reversal symmetry breaking. A discussion of fluctuation relations in classical systems with time-reversal symmetry breaking can be found in Refs. \\refcite{class1,class2}.\nAn intriguing question involves possible applications of the results to biological systems with unidirectional transport.\n\n\n\\section*{Acknowledgments}\n\nWe thank M. Kardar and K. Saito for helpful discussions. C.W. acknowledges support from the NSF under Grant No. DMR-1254741. D.E.F. was supported in part by the NSF under Grant No. DMR-1205715.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe class $\\mathbf{N}_1$ of \\textit{generalized Nevanlinna functions with one negative square} \nis the set of all scalar functions $Q$ which are meromorphic on ${\\dC \\setminus \\dR}$, which satisfy\n $Q(\\overline{z})=\\overline{Q(z)}$, and for which the kernel\n\\begin{equation}\\label{1.0}\n\\frac{Q(z)-\\overline{Q(w)}}{z-\\bar{w}},\n\\end{equation}\nhas one negative square, see \\cite{KL73, KL77, KL81, L}. The class $\\mathbf{N}_0$ of ordinary Nevanlinna\nfunctions consists of functions holomorphic on ${\\dC \\setminus \\dR}$, which satisfy $Q(\\overline{z})=\\overline{Q(z)}$, and for which the above\nkernel has no negative squares, i.e., ${\\rm Im\\,} Q(z) \/ {\\rm Im\\,} z \\ge 0$, $z \\in {\\dC \\setminus \\dR}$; cf. \\cite{donoghue}. Let $Q(z)$\nbelong to $\\mathbf{N}_1$. A point $z_0\\in\\dC^+\\cup\\dR\\cup\\{\\infty\\}$ is a \\textit{generalized zero of\nnonpositive type} (\\textit{GZNT}) of $Q(z)$ if either $z_0\\in\\dC^+$ and\n\\begin{equation}\nQ(z_0)=0,\n\\end{equation}\nor $z_0\\in\\dR$ and\n\\begin{equation}\\label{x_0}\n\\lim_{z \\wh\\to z_0}\\frac{Q(z)}{z-z_0}\\in (-\\infty,0],\n\\end{equation}\nor $z_0=\\infty$ and\n\\begin{equation}\\label{x_00}\n\\lim_{z \\wh\\to \\infty} zQ(z) \\in [0,\\infty),\n\\end{equation}\nsee \\cite[Theorem 3.1, Theorem 3.1']{L}.\nHere the symbol $\\wh\\to$ denotes the non-tangential limit.\nIf $Q(z)$ is holomorphic in a neighborhood of\n$z_0\\in\\dR$, then \\eqref{x_0} simplifies to $Q(z_0)=0$ and\n$Q'(z_0)\\leq0$. Any function $Q(z) \\in \\mathbf{N}_1$ has precisely one GZNT in\n$\\dC^+\\cup\\dR\\cup\\{\\infty\\}$; cf. \\cite{KL81}.\n\nEach generalized Nevanlinna function with one negative square is the Weyl function of a closed symmetric\noperator or relation with defect numbers $(1,1)$ in a Pontryagin space with one negative square, see\n\\cite{BDHS,BHSWW,DHS1}. The selfadjoint extensions of the symmetric operator or relation are parametrized\nover $\\dR \\cup \\{\\infty\\}$; in fact, each selfadjoint extension corresponding to $\\tau \\in \\dR \\cup\n\\{\\infty\\}$ has a Weyl function of the form:\n\\begin{equation}\\label{bil}\nQ_\\tau(z)=\\frac{Q(z)-\\tau}{1+\\tau Q(z)}, \\quad \\tau \\in \\dR,\\quad \\mbox{and} \\quad Q_\\infty(z)=-\\frac1{Q(z)}.\n\\end{equation}\nThe transform \\eqref{bil} takes the class $\\mathbf{N}_1$ onto itself as follows from calculating the kernel\ncorresponding to \\eqref{1.0}. Hence, for each $\\tau \\in \\dR \\cup \\{\\infty\\}$ the function $Q_\\tau(z)$ has a\nunique GZNT, denoted by $\\alpha(\\tau)$. The study of the path $\\tau \\mapsto \\alpha(\\tau)$, $\\tau \\in \\dR \\cup\n\\{\\infty\\}$, was initiated in \\cite{DHS1}. Some simple examples of functions in $\\mathbf{N}_1$ may help to\nillustrate the various possibilities; cf. Theorem \\ref{factor}.\n\nFirst consider the $\\mathbf{N}_1$--function $Q(z)=-z$. It has a GZNT at the origin. For $\\tau \\in \\dR$ the equation $Q_\\tau(z)=0$ has one solution; hence the\npath $\\alpha(\\tau)$ of the GZNT is given by\n\\[\n \\alpha(\\tau)=-\\tau, \\quad \\tau \\in \\dR.\n\\]\nTherefore $\\alpha(\\tau)$ stays on the (extended) real line; cf. Section \\ref{ontherealline}.\n\nThe function $Q(z)=z^2$ provides another simple example of a function belonging to $\\mathbf{N}_1$. Observe that it\nhas a GZNT at the origin, hence $\\alpha(0)=0$. For $\\tau>0$ the equation $Q_\\tau(z)=0$ has two solutions, namely\n$-\\sqrt{\\tau}$ and $\\sqrt{\\tau}$. Since $Q_\\tau'(z)$ exists and is negative (positive) on the negative\n(positive) half-axes, it follows that the path $\\alpha(\\tau)$ of the GZNT is given by\n \\[\n \\alpha(\\tau)=-\\sqrt\\tau, \\quad \\tau>0.\n \\]\nFor $\\tau<0$ the equation $Q_\\tau(z)=0$ has precisely one solution in $\\dC^+$, so that the path\n$\\alpha(\\tau)$ of the GZNT is given by\n \\[\n \\alpha(\\tau)=\\ii\\sqrt{-\\tau}, \\quad \\tau<0.\n \\]\nHence the path approaches the real line\nvertically at the origin and continues along the negative axis; see Figure 1. Finally, note that $\\alpha(\\infty)=\\infty$.\n\nThe function $Q(z)=z^3$ also belongs to $\\mathbf{N}_1$ with a GZNT at the origin. For $\\tau \\in \\dR$\nthe equation $Q_\\tau(z)=0$ has three solutions. An argument similar to the one above shows that the path of\nthe GZNT of $\\alpha(\\tau)$ is given by\n\\[\n\\alpha(\\tau)=\\frac{ -{\\rm sgn\\,}{\\tau} + \\sqrt{3}\\ii }2\\,\n \\sqrt[3]{\\tau}, \\quad \\tau\\in\\dR.\n\\]\nTherefore the path approaches the real line under an angle $\\pi\/3$ and leaves the real line under an angle\n$2\\pi\/3$; see Figure 1.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{picture}(105,20)(0,0)\n\\linethickness{0.05mm} \\put(0,5){\\line(1,0){50}}\n \\put(55,5){\\line(1,0){50}}\n \\thicklines \\put(25,5){\\vector(-1,0){10}}\n \\put(15,5){\\line(-1,0){15}}\n \\put(25,17){\\vector(0,-1){9}}\n \\put(25,14){\\line(0,-1){9}}\n \\put(28,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}2$}}\n \\put(20,10){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(25,02){\\makebox(0,0)[cc]{$0$}}\n\\thicklines \\put(88,17){\\vector(-2,-3){4}}\n \\put(84,11){\\line(-2,-3){4}}\n \\put(80,5){\\vector(-2,3){4}}\n \\put(76,11){\\line(-2,3){4}}\n \\put(65,15){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(80,02){\\makebox(0,0)[cc]{$0$}}\n \\put(80,9){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n \\put(84,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n \\put(76,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n\\end{picture}\n\\end{center}\n\\caption{The path of $\\alpha(\\tau)$ for $Q(z)=z^2$ and for $Q(z)=z^3$.}\n\\end{figure}\n\nThe previous examples were about functions $Q(z) \\in \\mathbf{N}_1$ which have a GZNT on the real axis and\nsuch that $Q(z)$ is holomorphic in a neighborhood of the GZNT. If $Q(z)$ is not holomorphic at its GZNT, the behaviour may be quite different. For instance, consider the function $Q(z)=z^{2+\\rho}$\nwhere $0<\\rho<1$ and the branch is chosen to make $Q(z)$\nholomorphic and positive on the positive axis. Then $Q(z)$ belongs to $\\mathbf{N}_1$\nwith a GZNT at the origin. The path $\\alpha(\\tau)$ now approaches $z=0$ via the angles\n$\\pi\/(2+\\rho)$ and $2\\pi\/(2+\\rho)$, since\n\\[\n \\alpha(\\tau)=(-\\tau)^{1\/(2+\\rho)} e^{i \\pi \/(2+\\rho)}, \\quad \\tau<0, \\quad\n \\alpha(\\tau)=(\\tau)^{1\/(2+\\rho)} e^{i 2\\pi \/(2+\\rho)}, \\quad \\tau>0,\n\\]\nsee Figure 2.\n\n\\begin{figure}[hbt]\n \\begin{center}\n\\begin{picture}(100,20)(0,0)\n\\linethickness{0.05mm} \\put(0,5){\\vector(1,0){100}}\n \\put(55,4){\\line(0,1){2}}\n \\put(55,2){\\makebox(0,0)[cc]{$\\alpha(0)=0$}}\n \\linethickness{0.6mm} \\put(0,5){\\line(1,0){55}}\n \\put(5,2){\\makebox(0,0)[cc]{${\\rm supp\\,}\\sigma$}}\n \\thicklines \\put(63,17){\\vector(-2,-3){4}}\n \\put(59,11){\\line(-2,-3){4}}\n \\put(55,5){\\vector(-2,3){4}}\n \\put(55,5){\\line(-2,3){8}}\n \\put(40,15){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(55,11){\\makebox(0,0)[cc]{\\tiny $\\frac\\pi{2+\\rho}$}}\n \\put(61,7){\\makebox(0,0)[cc]{\\tiny $\\frac\\pi{2+\\rho}$}}\n\\end{picture}\n\\end{center}\n\\caption{The path of $\\alpha(\\tau)$ for $Q(z)=z^{2+\\rho}$, $0< \\rho <1$. }\\label{nons}\n\\end{figure}\n\nAs a final example, consider the function\n\\begin{equation}\\label{notonR}\n Q(z)= \\frac{z^2+4}{z^2+1 }\\,\\,\\ii.\n\\end{equation}\nThis function belongs to $\\mathbf{N}_1$ and it has a GZNT at $z=2\\ii$. The equation $Q(z)=\\tau$ has a nonreal\nsolution for each $\\tau\\in\\dR\\cup\\{\\infty\\}$. In fact, the path of the GZNT $\\alpha(\\tau)$ is a simple closed\ncurve bounded away from the real axis.\n\nIn general the path of the GZNT has a complicated behaviour. The path may come from $\\dC^+$, be part of the\nreal line, and then leave again to $\\dC^+$, it may approach the real line in different ways, it may stay\ncompletely on the real line, or it may stay away boundedly from the real line. In the present paper, some\naspects of the path are treated. Especially, the local behavior of $\\alpha(\\tau)$ is completely determined in\nthe domain of holomorphy of the function $Q(z)$. Furthermore under certain holomorphy conditions it is shown\nthat a small interval of the real line is part of the path of $\\alpha(\\tau)$. Finally, the case where the\npath stays on the extended real line is completely characterized.\n\nThe contents of this paper are now described. Section 2 contains some preliminary observations concerning\ngeneralized Nevanlinna functions with one negative square. Furthermore a useful version of the inverse\nfunction theorem is recalled. Some elementary notions concerning the path $\\alpha(\\tau)$ of the GZNT of\n$Q_\\tau(z)$ are presented in Section 3. In Section 4 the function $Q(z)$ is assumed to be holomorphic in a\nneighborhood of a real GZNT and the path $\\alpha(\\tau)$ of the GZNT is studied in such a neighborhood. In\nSection 5 necessary and sufficient conditions are given so that a left or right neighborhood of a GZNT of\n$Q(z)$ belongs to the path $\\alpha(\\tau)$. Section \\ref{ontherealline} is devoted to the case where the GZNT stays on the\nextended real line. A complete characterization is given.\n\nThe present paper has points of contact with \\cite{JL83,JL95} and \\cite{HSSW}.\nAn example of a path as described in the present paper can be found in\n\\cite{KL71}. Furthermore, it should be pointed out that there are strong\nconnections to the recent perturbation analysis\nin \\cite{MMRR1,MMRR2,RanWojtylak}.\nThe authors thank Vladimir Derkach and Seppo Hassi, who have influenced \nthis paper in more than one way.\n\n\n\\section{Preliminaries}\n\n\\subsection{Nevanlinna functions}\n\nLet $M(z)$ belong to $\\mathbf{N}_0$, i.e. $M(z)$ is a Nevanlinna function (without any negative squares).\nThen $M(z)$ has the usual integral representation\n\\begin{equation}\\label{nev'}\nM(z)= a+ b z+ \\int_\\dR \\left(\\frac{1}{s-z}-\\frac{s}{s^2+1}\\right) \\,d\\sigma(s),\\quad\nz\\in\\dC\\setminus\\mathbb{R},\n\\end{equation}\nwhere $a \\in \\dR$, $b \\ge 0$, and $\\sigma$ is a nondecreasing function with\n\\begin{equation}\n\\label{int'} \\int_\\dR \\frac{d \\sigma(s)}{s^2+1} < \\infty.\n\\end{equation}\nSince the function $\\sigma$ can possess jump discontinuities the following normalization is used:\n\\[\n \\sigma(t)=\\frac{\\sigma(t+0)+\\sigma(t-0)}{2}.\n\\]\nIn addition, it is assumed that $\\sigma(0)=0$. Note that $a$ and $b$ can be recovered from the function\n$M(z)$ by\n\\begin{equation}\\label{nev+}\n a=\\Re M(i), \\quad b = \\lim_{ z \\wh\\to \\infty} \\frac{M(z)}{z}.\n\\end{equation}\n Likewise, the function $\\sigma$ can be recovered from the\nfunction $M(z)$ by the Stieltjes inversion formula:\n\\[\n \\sigma(t_2)-\\sigma(t_1)=\\lim_{\\varepsilon \\downarrow\n 0} \\frac{1}{\\pi} \\int_{t_1}^{t_2} {\\rm Im\\,}\n M(x+i\\varepsilon)\\,dx, \\quad t_1 \\leq t_2,\n\\]\ncf. \\cite{donoghue}, \\cite{KK}.\nIf $\\sigma(s)$ is constant on $(\\gamma,\\delta)\\subseteq\\dR$, then $(\\gamma,\\delta)$ will be called a\n\\textit{gap} of $\\sigma$ or of $M(z)$. Note that in this case $M(z)$ given by \\eqref{nev'} is well--defined\nfor $z\\in(\\gamma,\\delta)$. By the Schwarz reflection principle $M(z)$ is also holomorphic on\n$\\dC^+\\cup\\dC^-\\cup(\\gamma,\\delta)$. Conversely, if the function $M(z)$ given by \\eqref{nev'} is holomorphic\non some interval $(\\gamma,\\delta)\\subseteq\\dR$ then $\\sigma(s)$ is constant on that interval. Furthermore,\nobserve that if $M(z)$ is holomorphic at $z\\in\\dC$ then\n\\begin{equation}\\label{gapp}\nM'(z)=b+\\int_{\\dR}\\frac{d\\,\\sigma(t)}{(t-z)^2}.\n\\end{equation}\nThe symbols $z \\downarrow \\gamma$ and $z \\uparrow \\delta$ will stand for the approximation of $\\gamma$ and $\\delta$ along $\\dR$ from above and below, respectively. So\n\\begin{equation}\\label{rreff}\nM(\\gamma+)=\\lim_{z\\downarrow\\gamma} M(z) \\in [-\\infty,\n\\infty),\\quad M(\\delta-)=\\lim_{z\\uparrow\\delta}M(z)\\in (-\\infty, \\infty].\n\\end{equation}\nRecall that these limits are equal to the nontangential limits at $\\gamma$ and $\\delta$, respectively;\nsee \\cite{donoghue}.\n\nThe function $\\sigma(s)$ introduces in a natural way a measure on $\\mathbb{R}$, which is denoted by the same symbol. The formula for the point mass\n\\begin{equation}\\label{nev++}\n \\sigma(\\{c\\})=\\sigma(c+0)-\\sigma(c-0)= \\lim_{z \\wh \\to c}\\, (c-z)M(z),\\quad c \\in\\dR,\n\\end{equation}\n complements the limit formula in \\eqref{nev+}.\n\nThe following result is based on a careful analysis of the relationship of the limiting behaviour of the\nimaginary part of $M(z)$ and the behaviour of the spectral function $\\sigma(s)$ in \\eqref{nev'}; see\n\\cite{donoghue} for details.\n\n\\begin{theorem}\\label{limits}\nLet $M(z)$ be a Nevanlinna function and let $(\\gamma,\\delta) \\subset \\dR$ be a finite interval. If\n\\[\n \\lim_{y \\downarrow 0} \\,\\Im M(x+ \\ii y)=0,\\quad x\\in(\\gamma,\\delta),\n\\]\nthen $M(z)$ is holomorphic on $(\\gamma,\\delta)$.\n\\end{theorem}\n\n\\begin{proof}\nLet the function $M(z)$ be of the form \\eqref{nev'} and let $x \\in (\\gamma,\\delta)$. It follows from\n\\eqref{nev'} that\n\\begin{equation}\\label{ilm}\n {\\rm Im\\,} M(x+iy)=by+\\int_\\dR \\frac{y}{(s-x)^2+y^2}\\,d\\sigma(s).\n\\end{equation}\nIt is known that if the limit of the integral in \\eqref{ilm} is $0$ as $y \\downarrow 0$, then $\\sigma$ is\ndifferentiable at $x$; see \\cite[Theorem IV.II]{donoghue}. An application of \\cite[Theorem IV.I]{donoghue}\nshows that $\\sigma'(x)=0$.\n\nHence, by assumption, it follows that $\\sigma$ is differentiable on $(\\gamma,\\delta)$ and that $\\sigma'(x)=0$\nfor all $x \\in (\\gamma,\\delta)$. Therefore $\\sigma$ is constant on $(\\gamma,\\delta)$ and, hence, $M(z)$ is\nholomorphic on $(\\gamma,\\delta)$.\n\\end{proof}\n\n\\subsection{Generalized Nevanlinna functions with one negative square}\n\nAssume that $Q(z) \\in \\mathbf{N}_1$. A point $z_0\\in\\dC^+\\cup\\dR\\cup\\{\\infty\\}$ is a \\textit{generalized pole\nof nonpositive type} (GPNT) of $Q(z)$ if $z_0$ is a GZNT for the function $-1\/Q(z)$ (which automatically\nbelongs to $\\mathbf{N}_1$). A function in $\\mathbf{N}_1$ has precisely one GPNT in\n$\\dC^+\\cup\\dR\\cup\\{\\infty\\}$, just as it has precisely one GZNT in $\\dC^+\\cup\\dR\\cup\\{\\infty\\}$. For the\nfollowing result, see \\cite{DHS1,DHS3,DLLSh}.\n\n\\begin{theorem}\\label{factor}\nAny function $Q(z) \\in \\mathbf{N}_1$ admits the following factorization\n\\begin{equation}\\label{fack}\n Q(z)=R(z) M(z),\n\\end{equation}\nwhere $M(z) \\in \\mathbf{N}_0$ and $R(z)$ is a rational function of the form\n\\begin{equation}\\label{einz}\n \\frac{(z-\\alpha)(z-\\overline{\\alpha})}{(z-\\beta)(z-\\overline{\\beta })},\n \\quad\n \\frac{ 1}{(z-\\beta)(z-\\bar{\\beta)}}, \\quad\n \\mbox{or}\n \\quad\n (z-\\alpha)(z-\\bar{\\alpha}).\n\\end{equation}\nHere $\\alpha, \\beta \\in \\dC^+ \\cup \\dR \\cup \\{\\infty\\}$ stand for the GZNT and GPNT of $Q(z)$, respectively;\nin the first case $\\alpha$ and $\\beta$ are finite, in the second case $\\infty$ is a GZNT and $\\beta$ is\nfinite, and in the third case $\\alpha$ is finite and $\\infty$ is a GPNT.\n\\end{theorem}\n\nFor the function $Q(z) \\in \\mathbf{N}_1$ the function $M(z) \\in \\mathbf{N}_0$ and the factors in\n\\eqref{einz} are uniquely determined. Note that $\\alpha\\ne\\beta$, otherwise $Q(z)$ would not have any negative\nsquares.\n\n\\begin{corollary}\\label{factor+}\nLet $Q(z) \\in \\mathbf{N}_1$ and let $z_0 \\in \\dC^+$. If $Q(z_0)=0$, then $Q'(z_0) \\ne 0$.\n\\end{corollary}\n\n\\begin{proof}\nSince $Q(z_0)=0$, it follows that $z_0 \\in \\dC^+$ is a GZNT of $Q(z)$. Therefore, according to Theorem\n\\ref{factor}, $Q(z)=(z-z_0)(z-\\bar{z}_0) H(z)$ where $H(z)$ is holomorphic in a neighborhood of $z_0$ and\n$H(z_0) \\neq 0$ (since $M(z_0) \\neq 0$). Differentiation of the identity leads to\n\\[\n Q'(z)=(z-z_0)H(z)+(z-\\bar z_0)H(z)+(z-z_0)(z-\\bar z_0)H'(z),\n\\]\nwhich implies that\n\\[\nQ'(z_0)=2 \\text{i} \\, ({\\rm Im\\,} z_0)H(z_0).\n\\]\nSince $z_0 \\in \\dC^+$ and $H(z_0)\\neq 0$, this implies that $Q'(z_0) \\ne 0$.\n\\end{proof}\n\nLet $Q(z) \\in \\mathbf{N}_1$ and assume that $Q(z)$ is holomorphic in a neighborhood of $z_0 \\in \\dR$. The\nfollowing is a classification of $Q(z_0)=0$; see \\cite{DHS3}. A proof is included for completeness.\n\n\\begin{proposition}\\label{zeros>0}\nLet $Q(z) \\in \\mathbf{N}_1$ and assume that $Q(z)$ is holomorphic in a neighborhood of $z_0 \\in \\dR$. If\n$Q(z_0)=0$, then precisely one of the following possibilities occurs:\n \\begin{itemize}\n \\item[(0)] $Q'(z_0) > 0$;\n \\item[(1)] $Q'(z_0) < 0$;\n \\item[(2a)] $Q'(z_0) = 0$ and $Q''(z_0) > 0$;\n \\item[(2b)] $Q'(z_0) = 0$ and $Q''(z_0) < 0$;\n \\item[(3)] $Q'(z_0) = 0$ and $Q''(z_0) = 0$ (in which case $Q'''(z_0) > 0$).\n \\end{itemize}\nIn the cases {\\rm(1)--(3)} the point $z_0 \\in \\dR$ is necessarily a GZNT of the function $Q(z)$.\n\\end{proposition}\n\n\\begin{proof}\nSince $Q(z) \\in \\mathbf{N}_1$ it is of the form $Q(z)=R(z)M(z)$, where $R(z)$ is of the form \\eqref{einz}\nand $M(z)$ is a Nevanlinna function; see Theorem \\ref{factor}. Assume that $Q(z)$ is holomorphic in a\nneighborhood of $z_0 \\in \\dR$ and that $Q(z_0)=0$.\n\n\\textit{Case 1}. Consider the case $R(z_0) \\neq 0$. Then $M(z)$ must be holomorphic around $z_0 \\in \\dR$ and\n$M(z_0)=0$. Observe that $M'(z_0)>0$, otherwise $M(z)$ would be constant, see relation \\eqref{gapp}, and\nhence identical zero, which would imply $Q(z)\\equiv 0\\notin \\mathbf{N}_1$.\n It follows from $Q'(z)=R'(z)M(z)+R(z)M'(z)$ that $Q'(z_0)=R(z_0) M'(z_0)$. Observe from\n\\eqref{einz} that $R(z_0) >0$. Therefore $Q'(z_0)>0$, so that (0) occurs.\n\n\\textit{Case 2}. It remains to consider the case $R(z_0) = 0$. This implies that $z_0=\\alpha \\in \\dR$. Hence\n\\[\n Q(z)=(z-\\alpha)^2M(z) \\quad \\mbox{or} \\quad Q(z)=\\frac{(z-\\alpha)^2}{(z-\\beta)(z-\\bar{\\beta})}\\,M(z).\n\\]\nSince $Q(z)$ is holomorphic around $\\alpha \\in \\dR$, it follows that the Nevanlinna function $M(z)$ has the\nexpansion\n\\begin{equation}\\label{nevv}\nM(z)=\\frac{m_{-1}}{z-\\alpha}+m_0+m_1 (z-\\alpha)+ \\cdots\n\\end{equation}\nwhere $m_{-1} \\le 0$ and $m_i \\in \\dR$, $i \\in \\dN$. Moreover\n\\[\n\\begin{split}\n Q(z)&=c_0m_{-1} (z-\\alpha)+(c_0m_0+c_1m_{-1})(z-\\alpha)^2 \\\\\n &\\hspace{2cm} +(c_0m_1+c_1m_0+c_2m_{-1})(z-\\alpha)^3+\\cdots,\n\\end{split}\n\\]\nwhere $c_0>0$ and $c_i \\in \\dR$, $i \\in \\dN$, stand for the coefficients of the power series expansion of the\nfunction $[(z-\\beta)(z-\\bar{\\beta})]^{-1}$ or of the function $1$ if $\\beta=\\infty$. In particular, the\nfollowing identities are clear:\n\\begin{equation}\\label{id1}\n Q'(\\alpha)=c_0m_{-1},\n\\end{equation}\n\\begin{equation}\\label{id2}\n2Q''(\\alpha)=c_0m_0+c_1m_{-1},\n\\end{equation}\nand\n\\begin{equation}\\label{id3}\n 6Q'''(\\alpha)=c_0m_1+c_1m_0+c_2m_{-1}.\n\\end{equation}\nIt follows from \\eqref{id1} that $Q'(\\alpha) \\le 0$. There is the following subdivision:\n\\begin{itemize}\n\\item $Q'(\\alpha) < 0$. Then (1) occurs.\n\\item $Q'(\\alpha)=0$ and $Q''(\\alpha)>0$ or $Q''(\\alpha)<0$. Then (2a) or (2b) occur.\n\\item $Q'(\\alpha)=0$ and $Q''(\\alpha)=0$. In this case it follows from \\eqref{id1} and \\eqref{id2}\nthat $m_{-1}=0$ and $m_0=0$. Note that \\eqref{nevv} with $m_{-1}=m_0=0$ implies that $m_1>0$. According to\n\\eqref{id3}, it follows that $Q'''(\\alpha)=c_0m_1$.\nTherefore $Q'''(\\alpha) > 0$, so that (3) occurs.\n\\end{itemize}\n\nTo conclude observe that it follows from \\eqref{x_0} that $Q(z_0)=0$ and $Q'(z_0) \\le 0$ imply that $z_0 \\in\n\\dR$ is a GZNT of $Q(z)$.\n\\end{proof}\n\n\\subsection{The inverse function theorem}\n\nThe following consequence of the usual inverse function theorem will be useful. It specifies branches of\nsolutions of an equation involving holomorphic functions; cf. \\cite[Theorem 9.4.3]{H}.\n\n\\begin{theorem}\\label{inver'}\nLet $Q(z)$ be a function which is holomorphic at $z_0$ and assume that $Q^{(i)}(z_0)=0$, $0 \\le i \\le n-1$,\nand $Q^{(n)}(z_0)\\ne 0$ for some $n \\ge 1$ so that\n\\[\n Q(z)=\\frac{Q^{(n)}(z_0)}{n!}(z-z_0)^n+\\frac{Q^{(n+1)}(z_0)}{(n+1)!}(z-z_0)^{n+1} + \\cdots.\n\\]\nThen there is a neighborhood of $z_0$ where the equation\n\\begin{equation}\\label{inv'}\nQ(z)=w^n\n\\end{equation}\nhas $n$ solutions $z=\\phi^{i}(w)$, $1\\le i\\le n$. The functions $\\phi^{i}(w)$ are holomorphic at $0$, of the\nform\n\\[\n \\phi^{i}(w)=z_0+\\phi^{i}_1 w+ \\phi^{i}_2 w^2+\\cdots,\n\\]\nand their first order coefficients $\\phi^{i}_1$, $1\\le i\\le n$, are the $n$ distinct roots of the equation\n\\begin{equation}\n\\label{crit'} (\\phi_1^i)^n= \\frac{n!}{Q^{(n)}(z_0)},\\quad 1\\le i\\le n.\n\\end{equation}\n\\end{theorem}\n\n\n\\section{Elementary properties of $\\alpha(\\tau)$ }\\label{basicalpha}\n\nLet $Q(z) \\in \\mathbf{N}_1$. The fractional linear transform $Q_\\tau(z)$, $\\tau \\in \\dR \\cup \\{\\infty\\}$, is\ndefined in \\eqref{bil}, so that the derivative of $Q_\\tau$ is given by\n\\begin{equation}\\label{QQ'}\n Q'_\\tau(z)=(1+\\tau^2)\\frac{Q'(z)}{(1+\\tau Q(z))^2}, \\quad \\tau \\in \\dR, \\quad \\mbox{and}\n \\quad Q'_\\infty(z)=\\frac{1}{Q(z)^2}.\n\\end{equation}\nSince $Q_\\tau(z)$ also belongs to $\\mathbf{N}_1$ for $\\tau\\in\\dR\\cup\\{\\infty\\}$, Theorem~\\ref{factor} may be\napplied.\n\n\\begin{corollary}\nLet $Q(z) \\in \\mathbf{N}_1$. Then $Q_\\tau(z)$, $\\tau\\in\\dR\\cup\\{\\infty\\}$, has the factorization\n\\begin{equation}\\label{QQ+}\n Q_\\tau(z)=R_{(\\tau)}(z) M_{(\\tau)}(z),\n\\end{equation}\nwhere $M_{(\\tau)}(z) \\in \\mathbf{N}_0$ and $R_{(\\tau)}(z)$ is a rational function of the form\n\\begin{equation}\\label{einzt}\n \\frac{(z-\\alpha(\\tau))(z-\\overline{\\alpha(\\tau)})}{(z-\\beta(\\tau))(z-\\overline{\\beta(\\tau}))},\n \\quad\n \\frac{ 1}{(z-\\beta(\\tau))(z-\\overline{\\beta(\\tau)})},\n \\quad\n \\mbox{or}\n \\quad\n (z-\\alpha(\\tau))(z-\\overline{\\alpha(\\tau)}).\n\\end{equation}\nHere $\\alpha(\\tau), \\beta(\\tau) \\in \\dC^+ \\cup \\dR \\cup \\{\\infty\\}$ stand for the GZNT and GPNT of\n$Q_\\tau(z)$, respectively; in the first case $\\alpha(\\tau)$ and $\\beta(\\tau)$ are finite, in the second case\n$\\infty$ is a GZNT and $\\beta(\\tau)$ is finite, and in the third case $\\alpha(\\tau)$ is finite and $\\infty$\nis a GPNT.\n\\end{corollary}\n\nNote that $\\alpha(0)=\\alpha$ and $\\beta(0)=\\beta$, and that $\\alpha(\\tau)\\ne\\beta(\\tau)$ for all\n$\\tau\\in\\dR\\cup\\{\\infty\\}$. The paths $\\alpha(\\tau)$ and $\\beta(\\tau)$ are related. To see this, observe that\nit follows from the form of the fractional linear transform \\eqref{bil} that\n\\[\n Q_{-1\/\\tau}(z)=\\frac{Q(z)+1\/\\tau}{1-Q(z)\/\\tau}\n =-Q_\\tau(z)^{-1}, \\quad \\tau \\in \\dR \\cup \\{\\infty\\}.\n\\]\nTherefore, \\eqref{QQ+} and \\eqref{einzt} lead to\n\\begin{equation}\\label{alphbet}\n \\alpha(\\tau)=\\beta({-1\/\\tau}), \\quad \\beta(\\tau)=\\alpha(-1\/\\tau),\\quad\\tau\\in\\dR\\setminus\\{0\\},\n\\end{equation}\nand, in particular, to $\\alpha(\\infty)=\\beta$ and $\\beta(\\infty)=\\alpha$.\n\nAccording to the identities in \\eqref{alphbet} it suffices to describe the function $\\alpha(\\tau)$. The\n\\textit{path} of the GZNT of the function $Q(z) \\in \\mathbf{N}_1$ is defined by\n\\[\n\\mathcal{F}_Q:=\\{\\,\\alpha(\\tau):\\tau\\in\\dR\\cup\\{\\infty\\}\\,\\}.\n\\]\nThe following result concerning the parametrization of the path may be useful.\n\n\\begin{lemma}\nLet $Q(z) \\in \\mathbf{N}_1$ and let $\\tau_0 \\in \\dR \\cup \\{\\infty\\}$. Then the path of the GZNT of $Q(z)$\ncoincides with the path of the GZNT of $Q_{\\tau_0}(z)$, i.e.\n \\begin{equation}\\label{FQt}\n\\mathcal{F}_Q=\\mathcal{F}_{Q_{\\tau_0}},\\quad \\tau_0\\in\\dR\\cup\\{\\infty\\}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFirst consider the case $\\tau \\in \\dR$. Then a simple calculation shows that\n\\begin{equation}\\label{tr}\n (Q_\\tau)_\\rho(z)= Q_{\\frac{\\tau+\\rho}{1-\\rho\\tau}}(z),\n \\quad \\rho\\in\\dR,\n \\quad (Q_\\tau)_\\infty(z)=Q_{-1\/\\tau}(z).\n\\end{equation}\nHence, if the GZNT of $(Q_\\tau)_\\rho(z)$ is denoted by $\\alpha_\\tau(\\rho)$, then it follows from \\eqref{tr}\nthat\n\\begin{equation}\\label{tr1}\n\\alpha_\\tau(\\rho)=\\alpha\\left(\\frac{\\tau+\\rho}{1-\\rho\\tau}\\right), \\quad \\rho\\in\\dR,\\quad\n\\alpha_\\tau(\\infty)=\\alpha_{-1\/\\tau}.\n\\end{equation}\nThe identity \\eqref{tr1} shows that \\eqref{FQt} is valid. The case $\\tau=\\infty$ can be treated similarly.\n\\end{proof}\n\nThe points of $\\mathcal{F}_Q$, i.e. the solutions of the equation $\\alpha(\\tau_0)=z_0$, will be\ncharacterized in terms of the function $Q(z)$. But first observe the following. If $Q(z)$ is holomorphic in a\nneighborhood of $\\alpha(\\tau_0)$ and if $Q'(\\alpha(\\tau_0))\\neq0$, then clearly the function $\\alpha(\\tau)$\nis holomorphic in a neighborhood of $\\tau_0$; this follows from the usual inverse function theorem (see\nTheorem \\ref{inver'} with $n=1$).\n\n\\begin{theorem}\\label{charact}\nLet $Q(z) \\in \\mathbf{N}_1$ and let $\\tau_0 \\in \\dR$.\n\\begin{enumerate}\\def\\rm (\\roman{enumi}) {\\rm (\\roman{enumi})}\n\\item Let $z_0 \\in \\dC^+$. Then $\\alpha(\\tau_0)=z_0$ if and only if\n\\begin{equation}\\label{un}\n Q(z_0)=\\tau_0.\n\\end{equation}\nIn this case, the function $\\alpha(\\tau)$ is holomorphic in a neighborhood of $\\tau_0$.\n\n\\item Let $z_0\\in\\dR$. Then $\\alpha(\\tau_0)=z_0$ if and only if\n\\begin{equation}\\label{deux}\n\\lim_{z \\wh\\to z_0}\\frac{Q(z)-\\tau_0}{z-z_0} \\in (-\\infty,0].\n\\end{equation}\n\n\\item Let $z_0=\\infty$. Then $\\alpha(\\tau_0)=\\infty$ if and only if\n\\begin{equation}\\label{deux+}\n\\lim_{z \\wh\\to \\infty} z (Q(z)-\\tau_0) \\in [0,\\infty).\n\\end{equation}\n\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n(i) Let $z_0 \\in \\dC^+$ and let $\\tau_0 \\in \\dR$.\n\n($\\Leftarrow$) If \\eqref{un} holds, then $Q_{\\tau_0}(z_0)=0$. Hence, $z_0 \\in \\dC^+$ is a zero of\n$Q_{\\tau_0}(z) \\in \\mathbf{N}_1$, which implies that $z_0$ is a GZNT of $Q_{\\tau_0}(z)$. By uniqueness it\nfollows that $z_0=\\alpha(\\tau_0)$.\n\n($\\Rightarrow$) If $\\alpha(\\tau_0)=z_0$, then $Q_{\\tau_0}(z_0)=Q_{\\tau_0}(\\alpha(\\tau_0))=0$, so that\n\\eqref{un} holds.\n\nIf $z_0:=\\alpha(\\tau_0)\\in\\dC^+$, then $Q_{\\tau_0}(z_0)=0$ and hence\n $Q'_{\\tau_0}(z_0) \\ne 0$; cf. Corollary \\ref{factor+}.\nIt follows from \\eqref{QQ'} that $Q'(z_0)\\ne 0$. Therefore $\\alpha(\\tau)$ is holomorphic at $\\tau_0$.\n\n(ii) Let $z_0 \\in \\dR$ and let $\\tau_0 \\in \\dR$.\nIt follows from the fractional linear transform \\eqref{bil} that\n\\begin{equation}\\label{deux--}\n\\frac{Q_{\\tau_0}(z)}{z-z_0}=\\frac{1}{1+\\tau_0 Q(z)} \\,\\,\\frac{Q(z)-\\tau_0}{z-z_0}.\n\\end{equation}\n\n($\\Leftarrow$) Assume that \\eqref{deux} holds.\nSince \\eqref{deux} implies $\\lim_{z\\wh\\to z_0} Q_{\\tau_0}(z)=0$, it follows from \\eqref{deux--} that\n\\[\n \\lim_{z \\wh\\to z_0}\\frac{Q_{\\tau_0}(z)}{z-z_0}\n =\\frac1{1+\\tau_0^2}\\,\\lim_{z \\to z_0} \\frac{Q(z)-\\tau_0}{z-z_0} \\in (-\\infty,0].\n\\]\nHence, by \\eqref{x_0} $z_0$ is the GZNT of $Q_{\\tau_0}(z)$.\n\n($\\Rightarrow$)\nSince $Q_{\\tau_0}(z)\/(z-z_0) \\to 0$ as $z \\wh\\to z_0$, it follows that $Q_{\\tau_0}(z) \\to 0$ as $z \\wh\\to z_0$ and, therefore, $Q(z) \\wh\\to \\tau_0$. Hence $\\alpha(\\tau_0)=z_0$ follows from \\eqref{deux--}.\n\n(iii) Let $z_0 = \\infty$ and let $\\tau_0 \\in \\dR$.\nIt follows from the fractional linear transform \\eqref{bil} that\n\\begin{equation}\nz Q_{\\tau_0}(z) =\\frac{1}{1+\\tau_0 Q(z)} \\,\\,z( Q(z)-\\tau_0).\n\\end{equation}\nThe proof now uses \\eqref{x_00} with similar arguments as in (ii).\n\\end{proof}\n\nThe case $\\tau_0=\\infty$ is not explicitly mentioned in Theorem \\ref{charact}.\nRecall that $\\alpha(\\infty)=\\beta$. Hence, the identity $z_0=\\alpha(\\infty)$ means actually that $z_0$ is a generalized pole of nonpositive type of the function $Q(z)$.\n\n\\begin{corollary}\\label{quh}\nLet $Q(z) \\in \\mathbf{N}_1$, then\n\\[\n\\{\\,z\\in \\dC^+ : {\\rm Im\\,} Q(z)=0\\,\\} \\subseteq\\mathcal{F}_Q.\n\\]\n\\end{corollary}\n\n\\begin{proof}\nLet $z_0 \\in \\dC^+$ and assume that ${\\rm Im\\,} Q(z_0)=0$. Then\n\\[\n Q(z_0)-\\tau_0= {\\rm Re\\,} Q(z_0)-\\tau_0,\n\\]\nso that the lefthand side equals zero, if $\\tau_0$ is defined as ${\\rm Re\\,} Q(z_0)$. In this case $Q_{\\tau_0}(z)$\nhas a zero at $z_0$.\n\\end{proof}\n\n\\begin{corollary}\\label{alphainjective}\nLet $Q(z) \\in \\mathbf{N}_1$, then the function\n\\[\n\\dR\\cup\\{\\infty\\}\\ni\\tau\\mapsto \\alpha(\\tau)\\in\\dC^+\\cup\\dR\\cup\\{\\infty\\}\n\\]\nis injective.\n\\end{corollary}\n\n\\begin{proof}\nAssume that $\\alpha(\\tau_1)=\\alpha(\\tau_2)$ with $\\tau_1, \\tau_2 \\in \\dR \\cup \\{\\infty\\}$.\n\nFirst consider $\\tau_1, \\tau_2 \\in \\dR$ and let $z_0= \\alpha(\\tau_1)=\\alpha(\\tau_2)$.\nIf $z_0$ is in $\\dC^+$, then (i) of Theorem \\ref{charact}\nimplies $\\tau_1=Q(\\alpha(\\tau_1)=Q(\\alpha(\\tau_2)=\\tau_2$.\nIf $z_0$ is in $\\dR\\cup\\{\\infty\\}$, then\n(ii) and (iii) of Theorem \\ref{charact} imply\n\\[\n\\tau_1=\\lim_{z\\wh\\to \\alpha(\\tau_1)} Q(z)=\\lim_{z\\wh\\to \\alpha(\\tau_2)} Q(z) =\\tau_2.\n\\]\n\nNext consider $\\tau_1 \\in \\dR$ and $\\tau_2=\\infty$ and let $z_0= \\alpha(\\tau_1)=\\alpha(\\infty)$.\nThen $\\alpha(\\infty)=z_0$ means that $z_0$ is a GPNT, so that $Q(z) \\to \\infty$ as $z \\wh\\to z_0$. Furthermore, $\\alpha(\\tau_1)=z_0$ implies, by Theorem \\ref{charact}),\nthat $Q(z) \\to \\tau_1$ as $z \\wh\\to z_0$, a contradiction.\n\nHence, $\\alpha(\\tau_1)=\\alpha(\\tau_2)$ with $\\tau_1, \\tau_2 \\in \\dR \\cup \\{\\infty\\}$, implies that\n$\\tau_1=\\tau_2$. This completes the proof.\n\\end{proof}\n\nThe following result can be seen as a consequence of Theorem \\ref{limits}.\n\n\\begin{theorem}\\label{tauonreal}\nLet $Q(z) \\in \\mathbf{N}_1$ and let the interval $(\\gamma,\\delta) \\subset \\dR$ be contained in\n$\\mathcal{F}_Q$. Then $Q(z)$ is holomorphic on $(\\gamma,\\delta)$ except possibly at the GPNT of $Q(z)$,\nwhich is then a pole of $Q(z)$.\n\\end{theorem}\n\n\\begin{proof}\nSince $Q(z) \\in \\mathbf{N}_1$, it can be written as $Q(z)=R(z)M(z)$ as in \\eqref{fack} where $R(z)$ is of the\nform as in \\eqref{einz} with GZNT $\\alpha$ and GPNT $\\beta$. By assumption each $z_0 \\in (\\gamma,\\delta)$ is\nof the form $z_0=\\alpha(\\tau_0)$. Hence by Theorem \\ref{charact} it follows that\n\\begin{equation}\\label{frid}\n \\lim_{z\\wh\\to z_0} Q(z)=\\tau_0.\n\\end{equation}\nClearly, if $z_0=\\alpha(\\tau_0)$ with $\\tau_0=\\infty$ then $z_0=\\beta$ so that $z_0$ is a GPNT of $Q(z)$.\nThere are three cases to consider:\n\n\\textit{Case 1:} $\\beta \\not\\in (\\gamma,\\delta)$ and $\\alpha \\not\\in (\\gamma,\\delta)$. Then $\\lim_{z\\wh\\to\nz_0} R(z) \\in \\dR\\setminus \\{0\\}$, so that it follows from \\eqref{frid} that\n\\begin{equation}\\label{frid1}\n \\lim_{z\\wh\\to z_0} {\\rm Im\\,} M(z)=0.\n\\end{equation}\nHence, by Theorem \\ref{limits}, it follows from \\eqref{frid1} that $M(z)$ and therefore $Q(z)$ is\nholomorphic on $(\\gamma,\\delta)$.\n\n\\textit{Case 2:} $\\beta \\not\\in (\\gamma,\\delta)$ and $\\alpha \\in (\\gamma,\\delta)$. Then by Case~1 it follows\nthat $Q(z)$ is holomorphic on $(\\gamma,\\alpha)$ and on $(\\alpha, \\delta)$. Hence, either $Q(z)$ is\nholomorphic on $(\\gamma,\\delta)$ or $Q(z)$ has an isolated singularity at $\\alpha$. However, this last case\ncannot occur due to the representation \\eqref{einz}.\n\n\\textit{Case 3:} $\\beta \\in (\\gamma,\\delta)$. Then by Case 1 and Case 2 it follows that $Q(z)$ is holomorphic\non $(\\gamma,\\beta)$ and $(\\beta,\\delta)$. This implies that $\\beta$ is an isolated singularity of $Q(z)$; in\nother words the GPNT $\\beta$ is a pole of $Q(z)$.\n\\end{proof}\n\n\\section{Local behavior of $\\alpha(\\tau)$ in a gap of $Q(z)$.}\\label{Gap}\n\nLet the function $Q(z) \\in \\mathbf{N}_1$ be holomorphic in a neighborhood of $z_0 \\in \\dR$. Assume that\n$Q(z_0)=0$ and that $z_0$ is in fact a GZNT of $Q(z)$, so that $Q'(z_0) \\le 0$; cf. Proposition\n\\ref{zeros>0}. The local form of the path $\\alpha(\\tau)$ in a neighborhood of $\\tau=0$ will now be described.\nThe items in the following theorem correspond to the classification in Proposition~\\ref{zeros>0}.\n\n\\begin{theorem}\\label{mainth}\nLet $Q(z) \\in \\mathbf{N}_1$ be holomorphic in a neighborhood of $z_0 \\in \\dR$ and let $z_0$ be a GZNT of\n$Q(z)$. Then precisely one of the following possibilities hold.\n\\begin{itemize}\n\\item[(1)] $Q'(z_0)<0$:\nThere exists $\\varepsilon > 0$ such that the function $\\alpha(\\tau)$ is real-valued and holomorphic with\n$\\alpha'(\\tau)<0$ on $(-\\varepsilon, \\varepsilon)$.\n\n\\begin{figure}[htb]\n\n\\begin{center}\n\\begin{picture}(100,12)(0,0)\n\\linethickness{0.05mm} \\put(0,5){\\vector(1,0){100}}\n \\put(55,4){\\line(0,1){2}}\n \\put(55,2){\\makebox(0,0)[cc]{$\\alpha(0)$}}\n \\linethickness{0.6mm} \\put(0,5){\\line(1,0){30}}\n \\put(70,5){\\line(1,0){20}}\n \\put(15,2){\\makebox(0,0)[cc]{${\\rm supp\\,}\\sigma$}}\n \\thicklines \\put(60,5){\\vector(-1,0){15}}\n \\put(45,5){\\line(-1,0){5}}\n \\put(45,8){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n\\end{picture}\n\\caption{Case (1)}\n\\end{center}\n\\end{figure}\n\n\\item[(2a)] $Q'(z_0)=0$ and $Q''(z_0) >0$:\nThere exist $\\varepsilon_1 > 0$ and $\\varepsilon_2 > 0$ such that the function $\\alpha(\\tau)$ is continuous\non $(-\\varepsilon_1,\\varepsilon_2)$, and holomorphic on each of the intervals $(-\\varepsilon_1,0)$ and\n$(0,\\varepsilon_2)$. Moreover $\\alpha(\\tau) \\in \\dC^+$ for $\\tau \\in (-\\varepsilon_1,0)$ and $\\arg\n(\\alpha(\\tau)-z_0)\\to \\pi\/2$ as $\\tau \\uparrow 0$ and $\\alpha(\\tau) \\in \\dR$ for $\\tau \\in\n(0,\\varepsilon_2)$.\n\n\\item[(2b)] $Q'(z_0)=0$ and $Q''(z_0) < 0$:\nThere exist $\\varepsilon_1 > 0$ and $\\varepsilon_2 > 0$ such that the function $\\alpha(\\tau)$ is continuous\non $(-\\varepsilon_1,\\varepsilon_2)$, and holomorphic on each of the intervals $(-\\varepsilon_1,0)$ and\n$(0,\\varepsilon_2)$. Moreover $\\alpha(\\tau) \\in \\dR$ for $\\tau \\in (-\\varepsilon_1,0)$ and $\\alpha(\\tau) \\in\n\\dC^+$ for $\\tau \\in (0,\\varepsilon_2)$ and $\\arg (\\alpha(\\tau)-z_0)\\to \\pi\/2$ as $\\tau \\downarrow 0$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{picture}(105,22)(0,0)\n\\linethickness{0.05mm} \\put(0,5){\\line(1,0){50}}\n \\put(55,5){\\line(1,0){50}}\n \\linethickness{0.6mm}\n \\put(35,5){\\line(1,0){15}}\n \\put(45,2){\\makebox(0,0)[cc]{${\\rm supp\\,}\\sigma$}}\n \\put(90,5){\\line(1,0){15}}\n \\put(100,2){\\makebox(0,0)[cc]{${\\rm supp\\,}\\sigma$}}\n \\thicklines \\put(25,5){\\vector(-1,0){10}}\n \\put(15,5){\\line(-1,0){5}}\n \\put(25,11){\\vector(0,-1){3}}\n \\put(25,14){\\line(0,-1){9}}\n \\put(20,14){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(25,2){\\makebox(0,0)[cc]{$\\alpha(0)$}}\n \\put(28,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}2$}}\n \\qbezier(25,14)(25,17)(26,20)\n\n \\put(80,5){\\vector(-1,0){10}}\n \\put(70,5){\\line(-1,0){5}}\n \\put(65,9){\\vector(0,1){3}}\n \\put(65,14){\\line(0,-1){9}}\n \\put(70,14){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(65,2){\\makebox(0,0)[cc]{$\\alpha(0)$}}\n \\put(68,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}2$}}\n \\qbezier(65,14)(65,17)(66,20)\n\\end{picture}\n\\caption{Cases (2a) and (2b)}\n\\end{center}\n\\end{figure}\n\n\\item[(3)]\n$Q'(z_0)=Q''(z_0)=0$, and $Q'''(z_0)>0$: There exist $\\varepsilon_1 > 0$ and $\\varepsilon_2 > 0$ such that\nthe function $\\alpha(\\tau)$ is continuous on $(-\\varepsilon_1,\\varepsilon_2)$, and holomorphic on each of the\nintervals $(-\\varepsilon_1,0)$ and $(0,\\varepsilon_2)$. Moreover $\\alpha(\\tau) \\in \\dC^+$ for $\\tau \\in\n(-\\varepsilon_1,0)$ and $\\arg (\\alpha(\\tau)-z_0)\\to \\pi\/3$ as $\\tau \\uparrow 0$; and $\\alpha(\\tau) \\in \\dC^+$\nfor $\\tau \\in (0,\\varepsilon_2)$ and $\\arg (\\alpha(\\tau)-z_0)\\to 2\\pi\/3$ as $\\tau \\downarrow 0$.\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{picture}(100,20)(0,0)\n\\linethickness{0.05mm} \\put(0,5){\\vector(1,0){100}}\n \\put(55,4){\\line(0,1){2}}\n \\put(55,2){\\makebox(0,0)[cc]{$\\alpha(0)$}}\n \\linethickness{0.6mm} \\put(0,5){\\line(1,0){30}}\n \\put(70,5){\\line(1,0){20}}\n \\put(15,2){\\makebox(0,0)[cc]{${\\rm supp\\,}\\sigma$}}\n \\thicklines \\put(63,17){\\vector(-2,-3){4}}\n \\put(59,11){\\line(-2,-3){4}}\n \\put(55,5){\\vector(-2,3){4}}\n \\qbezier(51,11)(48,17)(42,18)\n \\put(40,15){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(55,9){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n \\put(59,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n \\put(51,7){\\makebox(0,0)[cc]{\\tiny $\\frac{\\pi}3$}}\n\\end{picture}\n\\caption{Case (3) }\n\\end{center}\n\\end{figure}\n\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nThe assumption is that $Q(z) \\in \\mathbf{N}_1$ is holomorphic in a neighborhood of $z_0 \\in \\dR$ and that\n$Q(z)$, possibly together with some derivatives, vanishes at $z_0 \\in \\dR$, as described in Proposition\n\\ref{zeros>0}. The theorem will be proved via Theorem \\ref{inver'} (implicit function theorem).\n\n\\textit{Case (1): $Q(z_0)=0$ and $Q'(z_0)<0$.} According to Theorem \\ref{inver'} with $n=1$ there is some neighborhood of $w=0$ where the equation\n\\begin{equation}\\label{Qphi}\n Q(\\phi(w))=w\n\\end{equation}\nhas a unique holomorphic solution $\\phi(w)$ which satisfies $\\phi(0)=z_0$ and is real-valued for real $w$. It\nfollows from \\eqref{Qphi} that $Q'(\\phi(w))\\phi'(w)=1$, so that the condition $Q'(z_0)<0$ implies that\n$\\phi'(0)<0$, and thus $\\phi'(w)<0$ on some neighborhood $(-\\varepsilon, \\varepsilon)$. It follows from\nTheorem \\ref{charact} that $\\alpha(\\tau)=\\phi(\\tau)$.\n\n\\textit{Case (2a): $Q(z_0)=0$, $Q'(z_0)=0$, and $Q''(z_0) >0$.} According to Theorem \\ref{inver'} with $n=2$\n there is some neighborhood of $0$ where the equation\n\\[\nQ(\\phi^\\pm(w))=w^2\n\\]\nhas holomorphic solutions $\\phi^+(w)$ and $\\phi^-(w)$ with $\\phi^\\pm(0)=z_0$, and which have expansions\n\\[\n\\phi^\\pm(w)=\\phi_1^\\pm w+\\phi_2^\\pm w^2+ \\cdots ,\n\\]\nwhere $\\phi_1^\\pm=\\pm (Q''(z_0)\/2)^{-1\/2}$. Note that all the coefficients $\\phi_i^\\pm$ in the above\nexpansions are real.\n\nLet $\\tau>0$. Put $w=\\tau^{1\/2}$ such that\n\\begin{equation}\\label{pmtau}\nQ(\\phi^\\pm(\\tau^{1\/2}))=\\tau.\n\\end{equation}\nSince $\\phi_1^-<0$, there is some $\\varepsilon_2>0$ such that $\\phi^{-'}(\\tau^{1\/2})<0$ for\n$\\tau\\in(0,\\varepsilon_2)$. Now the relation \\eqref{pmtau} implies by taking the derivative with respect to\n$\\tau$ that $Q'(\\phi^-(\\tau^{1\/2}))<0$ for $\\tau\\in(0,\\varepsilon_2)$. Hence, by relation \\eqref{pmtau} and\nTheorem \\ref{charact} one finds that $\\alpha(\\tau)=\\phi^-(\\tau^{1\/2})$ for $\\tau\\in(0,\\varepsilon_2)$.\n\nLet $\\tau<0$. Put $w=i|\\tau|^{1\/2}$ such that\n\\begin{equation}\\label{pmtaui}\nQ(\\phi^\\pm(i|\\tau|^{1\/2}))=\\tau.\n\\end{equation}\nSince $\\phi_1^+>0$, there is some $\\varepsilon_1>0$ such that $(\\phi^{+})'(i|\\tau|^{1\/2})\\in\\dC^+$ for\n$\\tau\\in(-\\varepsilon_1,0)$. Hence, by relation \\eqref{pmtaui} and Theorem \\ref{charact} one finds that\n$\\alpha(\\tau)=\\phi^+(i|\\tau|^{1\/2})$ for $\\tau\\in(-\\varepsilon_1,0)$.\n\nThe expansion of $\\phi^+(i|\\tau|^{1\/2})$ implies that $\\arg (\\alpha(\\tau)-z_0)\\to \\pi\/2$ as $\\tau\\uparrow 0$.\n\n\\textit{Case (2b): $Q(z_0)=0$, $Q'(z_0)=0$, and $Q''(z_0)<0$.} This case can be treated similarly as the case\n(2a).\n\n\\textit{Case (3): $Q(z_0)=0$, $Q'(z_0)=0$, $Q''(z_0)=0$, and $Q'''(z_0) > 0$.} According to Theorem\n\\ref{inver'} there is a neighborhood of $0$, where the equation\n$$\nQ(\\phi(w))=w^3\n$$\n has three solutions $\\phi^{(j)}(w)$, $j=1,2,3$, determined by\n\\[\n\\phi^{(1)}_1=\\sqrt[3]{r},\\quad\n \\phi^{(2)}_1=\\left(-\\frac12+\\frac{\\sqrt{3}}2\\ii\\right)\\sqrt[3]{r},\\quad\n \\phi^{(3)}_1 = \\left(-\\frac12-\\frac{\\sqrt{3}}2\\ii\\right)\\sqrt[3]{ r},\n\\]\nwith $r=6\/Q'''(z_0)$.\n\nLet $\\tau>0$. Then Theorem \\ref{inver'} implies that there is some $\\varepsilon_2>0$ such that\n$\\phi^{(2)}(\\tau^{1\/3})$ is in $\\dC^+$ for $\\tau\\in( 0,\\varepsilon_2)$, and that\n$Q(\\phi^{(2)}(\\tau^{1\/3}))=\\tau$. Hence, by Theorem \\ref{charact}\n\\[\n\\alpha(\\tau)=\\phi^{(2)}(\\tau^{1\/3}),\\qquad \\tau\\in(0,\\varepsilon_2).\n\\]\n\nLet $\\tau<0$. Then the Theorem \\ref{inver'} implies that there is some $\\varepsilon_1>0$ such that\n$\\phi^{(3)}(-|\\tau|^{1\/3})$ is in $\\dC^+$ for $\\tau\\in (-\\varepsilon_1,0)$, and that\n$Q(\\phi^{(3)}(-|\\tau|^{1\/3}))=-|\\tau|=\\tau$. Hence, by Theorem \\ref{charact}\n\\[\n\\alpha(\\tau)=\\phi^{(3)}(-|\\tau|^{1\/3})),\\qquad \\tau\\in(-\\varepsilon_1,0)<0.\n\\]\nMoreover, using Theorem \\ref{inver'} one finds that\n\\[\n\\lim_{\\tau\\uparrow0}\\tan(\\arg(\\alpha(\\tau)-z_0))=\\lim_{\\tau\\uparrow0}\\frac{{\\rm Im\\,}\\alpha(\\tau)}{{\\rm Re\\,}\\alpha(\\tau)-z_0}=\n \\frac{{\\rm Im\\,}\\phi_1^{(3)}}{{\\rm Re\\,}\\phi_1^{(3)}}\n =\\sqrt{3},\n\\]\nand\n\\[\n\\lim_{\\tau\\downarrow0}\\tan(\\arg(\\alpha(\\tau)-z_0))=\\lim_{\\tau\\downarrow0}\\frac{{\\rm Im\\,}\\alpha(\\tau)}{{\\rm Re\\,}\\alpha(\\tau)-z_0}=\n \\frac{{\\rm Im\\,}\\phi_1^{(2)}}{{\\rm Re\\,}\\phi_1^{(2)}}\n =-\\sqrt{3},\n\\]\nwhich shows that the angles are indeed $\\pi\/3$ and $2\\pi\/3$, respectively.\n\\end{proof}\n\n\\begin{remark}\\label{R2}\nIn Case (2) and Case (3) other zeros of $Q_\\tau(z)$ will occur, but they need not be GZNT.\nBelow each case will be considered separately.\n\nFor Case (2) it suffices to restrict to Case (2a) as Case (2b) is similar. When $\\tau>0$ the\nfunction $\\alpha_+(\\tau):=\\phi^+(\\tau^{1\/2})$ is also a local solution of $Q(\\alpha_+(\\tau))=\\tau$. However,\n$Q'(\\alpha_+(\\tau))>0$ for $\\tau>0$, so that $\\alpha_+(\\tau)$ is not a GZNT. Moreover, $\\alpha_+(\\tau)$ is\nlocally increasing in $\\tau$. A different situation happens when $\\tau<0$. Then clearly $\\phi^-(-\\ii\n\\tau^{1\/2})=\\phi^+(\\ii \\tau^{1\/2})$, and locally $\\bar\\alpha(\\tau)=\\phi^+(-\\ii \\tau^{1\/2})$ is the GZNT of\n$Q_\\tau(z)$ in the lower half plane conjugate to the GZNT $\\alpha(\\tau)$, cf \\cite{KL71}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{picture}(105,25)(0,10)\n\\linethickness{0.05mm} \\put(0,20){\\line(1,0){50}}\n \\put(55,20){\\line(1,0){50}}\n \\thicklines \\put(25,20){\\vector(-1,0){10}}\n \\put(15,20){\\line(-1,0){5}}\n \\put(25,20){\\vector(1,0){10}}\n \\put(35,20){\\line(1,0){5}}\n \\put(25,26){\\vector(0,-1){3}}\n \\put(25,29){\\line(0,-1){9}}\n \\put(25,14){\\vector(0,1){3}}\n \\put(25,11){\\line(0,1){9}}\n\\put(20,29){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}} \\put(13,17){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n\\put(37,17){\\makebox(0,0)[cc]{$\\alpha_+(\\tau)$}} \\put(30,11){\\makebox(0,0)[cc]{$\\bar\\alpha(\\tau)$}}\n \\put(65,20){\\vector(1,0){10}}\n \\put(75,20){\\line(1,0){5}}\n \\put(95,20){\\vector(-1,0){10}}\n \\put(85,20){\\line(-1,0){5}}\n \\put(80,24){\\vector(0,1){3}}\n \\put(80,29){\\line(0,-1){9}}\n \\put(80,16){\\vector(0,-1){3}}\n \\put(80,11){\\line(0,1){9}}\n\\put(75,29){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}} \\put(68,17){\\makebox(0,0)[cc]{$\\alpha_+(\\tau)$}}\n\\put(92,17){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}} \\put(85,11){\\makebox(0,0)[cc]{$\\bar\\alpha(\\tau)$}}\n\\end{picture}\n\\caption{Cases (2a) and (2b) - positive and negative zeros}\n\\end{center}\n\\end{figure}\n\nIn Case (3) there are locally two other zeros of the function $Q_\\tau(z)$ crossing $z_0$ at $\\tau=0$. The\nfirst of them is real and given by\n\\[\n \\alpha_+(\\tau)=\\phi^{(1)}({\\rm sgn\\,}\\tau |\\tau|^{1\/3}).\n\\]\nThe second one lies in the lower half plane and is conjugate to $\\alpha(\\tau)$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{picture}(100,27)(0,0)\n\\linethickness{0.05mm} \\put(20,12){\\line(1,0){60}}\n \\thicklines\\put(58,24){\\vector(-2,-3){4}}\n \\put(54,18){\\line(-2,-3){4}}\n \\put(50,12){\\vector(-2,3){4}}\n \\put(46,18){\\line(-2,3){4}}\n \\put(50,22){\\makebox(0,0)[cc]{$\\alpha(\\tau)$}}\n \\put(58,0){\\vector(-2,3){4}}\n \\put(54,6){\\line(-2,3){4}}\n \\put(50,12){\\vector(-2,-3){4}}\n \\put(46,6){\\line(-2,-3){4}}\n \\put(50,2){\\makebox(0,0)[cc]{$\\bar\\alpha(\\tau)$}}\n \\put(25,12){\\vector(1,0){10}}\n \\put(35,12){\\vector(1,0){35}}\n \\put(70,12){\\line(1,0){5}}\n \\put(35,9){\\makebox(0,0)[cc]{$\\alpha_+(\\tau)$}}\n \\put(70,9){\\makebox(0,0)[cc]{$\\alpha_+(\\tau)$}}\n\\end{picture}\n\\caption{Case (3) - positive and negative zeros}\n\\end{center}\n\\end{figure}\n\\end{remark}\n\n\\begin{example}\\label{drie}\nLet $\\delta_1, \\delta_2 >0$ and consider the function\n\\[\n Q(z)=z^2 \\left( \\frac{\\delta_1}{-1-z}+\\frac{\\delta_2}{1-z} \\right), \\quad z \\in \\dC \\setminus\\{-1,1\\}.\n\\]\nThen $Q(z)$ is of the form \\eqref{fack} with\n\\[\n R(z)=z^2, \\quad M(z)=\\frac{\\delta_1}{-1-z}+\\frac{\\delta_2}{1-z},\n\\]\nwith $M(z) \\in \\mathbf{N}_0$. Hence $Q(z) \\in \\mathbf{N}_1$ with GZNT at $\\alpha=0$ and GPNT at\n$\\beta=\\infty$. The function $Q(z)$ is holomorphic in a neighborhood of $z=0$. Clearly, $Q(0)=0$, $Q'(0)=0$, and $Q''(0)=2(\\delta_2-\\delta_1)$. Hence,\n\\[\nQ''(0)=0 \\quad \\Leftrightarrow \\quad \\delta_1=\\delta_2,\n\\]\nin which case $Q'''(0)=12 \\delta_1=12 \\delta_2>0$. Therefore with regard to Theorem \\ref{mainth}\none obtains\n\\begin{enumerate}\n\\item $\\delta_2 > \\delta_1$ implies Case 2a;\n\\item $\\delta_2 < \\delta_1$ implies Case 2b;\n\\item $\\delta_2 = \\delta_1$ implies Case 3.\n\\end{enumerate}\n\\end{example}\n\n\\section{Part of the path on the real line.}\n\nLet $Q(z) \\in \\mathbf{N}_1$. The path of the GZNT may hit the real line coming from $\\dC^+$ and immediately\nreturn to $\\dC^+$ as the example $Q(z)=z^3$ shows. However it is also possible that the path will have a part\nof the real line in common as the example $Q(z)=z^2$ shows. In this section it is assumed that the GZNT\n$\\alpha$ belongs to the real line and the interest is in the existence of $\\varepsilon > 0$ such that\n\\[\n[\\alpha-\\varepsilon,\\alpha]\\subset\\mathcal{F}_Q \\quad \\mbox{or} \\quad\n[\\alpha,\\alpha+\\varepsilon]\\subset\\mathcal{F}_Q.\n\\]\nAccording to Theorem \\ref{tauonreal} this implies that $M(z)$ is holomorphic on\n\\[\n(\\alpha-\\varepsilon,\\alpha) \\quad \\mbox{or} \\quad (\\alpha,\\alpha+\\varepsilon),\n\\]\nrespectively. Recall that if the function $M(z) \\in \\mathbf{N}_0$ is holomorphic on an interval\n$(\\alpha,\\beta)$, then $M(z)$ has (possibly improper) limits at the end points of the interval and $M(\\alpha+) \\in [-\\infty, \\infty)$ and $M(\\beta-) \\in (-\\infty, \\infty]$. The fact that those limits are equal to the nontangential limits at $\\alpha$ and $\\beta$, respectively, will be extensively used in the proof below.\n \n\\begin{theorem}\\label{realpath}\nLet $Q(z)\\in\\mathbf{N}_1$ be of the form\n\\[\n Q(z)=(z-\\alpha)^2M(z) \\quad \\mbox{or} \\quad\n Q(z)=\\frac{(z-\\alpha)^2}{(z-\\beta)(z-\\bar{\\beta})} M(z),\n\\]\nwith $\\alpha \\in \\dR$, $\\beta \\in \\dC^+ \\cup \\dR$, $\\beta \\neq \\alpha$, and $M(z) \\in \\mathbf{N}_0$. Then\nthe following statements are valid:\n\\begin{enumerate}\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\\item\nThere exists $\\varepsilon>0$ such that $(\\alpha-\\varepsilon,\\alpha]\\subset \\mathcal{ F}_Q$ if and only if\n$M(z)$ is holomorphic on $(\\alpha-\\gamma,\\alpha)$ for some $\\gamma >0$\nand $M(\\alpha-) \\in (0,\\infty]$. \n\\item\nThere exists $\\varepsilon>0$ such that $[\\alpha,\\alpha+\\varepsilon)\\subset \\mathcal{F}_Q$ if and only if \n$M(z)$ is holomorphic on $(\\alpha,\\alpha+\\gamma)$ for some $\\gamma>0$\n and $M(\\alpha+) \\in [-\\infty, 0)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nSince the proofs of the statements (i) and (ii) are analogous, only (i) will be shown. By a translation the\nstatement can be easily reduced to the case $\\alpha=0$. Hence, from now on it is assumed that $M(z)$ is\nholomorphic on some interval $(-\\gamma,0)$, with $\\gamma>0$. The proof will be carried out for the functions\n\\[\n z^2M(z) \\quad \\mbox{and} \\quad \\frac{z^2M(z)}{(z-\\beta)(z-\\bar{\\beta})},\n\\]\nrespectively, in three steps. \\\\\n\n\\textit{Step 1.} The function $R(z)=z^2M(z)$ is considered. Since $M(z) \\in \\mathbf{N}_0$ it has the\nrepresentation \\eqref{nev'}. Recall that\n\\begin{equation}\\label{hash}\n \\lim_{z \\uparrow 0} -z M(z)=\\sigma(\\{0\\}),\n\\end{equation}\ncf. \\eqref{nev++}. Note that if $\\sigma(\\{0\\})=0$, dominated convergence on $(-\\infty,-\\gamma)$ and monotone convergence on $(0, \\infty)$ implies that\n\\begin{equation}\\label{ooo}\nM(0-)=a+\\int_{\\dR } \\frac{d\\sigma(s)}{s (s^2+1)} \\in \\dR \\cup \\{\\infty\\},\n\\end{equation}\ncf. \\eqref{rreff}.\n\nIn the general case $\\sigma(\\{0\\}) \\neq 0$\nthe function $R(z)= z^2M(z)$ can be written as\n\\[\n R(z)=az^2+bz^3 -z \\sigma(\\{0\\})\n +z^2 \\int_{\\dR \\setminus \\{0\\}} \\left( \\frac{1}{s-z}-\\frac{s}{s^2+1} \\right) d\\sigma(s)\n\\]\nand it is holomorphic on $(-\\gamma, 0)$. A straightforward calculation implies that\n\\begin{equation}\\label{kuhstrich}\nR'(z)=-\\sigma(\\{0\\})+2az+3bz^2+zT(z),\n\\end{equation}\nwhere the function $T(z)$ is given by\n$$\nT(z)=\\int_{\\dR\\setminus \\{0\\}} h(s,z) \\,\\frac{d\\sigma(s)}{s-z},\n$$\nwith\n$$\nh(s,z)=\\frac{2+2sz}{s^2+1}+\\frac{z}{s-z}.\n$$\nIt will be shown that\n\\begin{equation}\\label{??}\n\\lim_{z\\uparrow 0}zT(z)=0.\n\\end{equation}\nIn order to see this, observe that\n\\begin{equation}\\label{limzh}\n \\lim_{z\\uparrow 0} z \\left( h(s,z) \\frac{1}{s-z} \\right) =0,\\quad s\\in\\dR\\setminus\\{0\\}.\n\\end{equation}\nThe argument will be finished via dominated convergence. For this purpose\nwrite the function $T(z)$ as $T(z)=T_1(z)+T_2(z)$ with\n$$\nT_1(z)=\\int_{(0,1\/2)} h(s,z) \\,\\frac{d\\sigma(s)}{s-z},\n\\quad \nT_2(z)= \\int_{\\dR\\setminus(-\\gamma,1\/2)} h(s,z) \\,\\frac{d\\sigma(s)}{s-z}.\n$$\nNote that for $s\\in (0,1\/2)$, $z\\in(-1\/2,0)$ the following\nestimations hold:\n\\begin{equation}\\label{est1}\n -1< \\frac{z}{s-z}<0\n\\end{equation}\nand\n\\begin{equation}\\label{est2}\n\\frac{6}{5}< \\frac{2-s}{s^2+1}<\\frac{2+2sz}{s^2+1}<2.\n\\end{equation}\nIt follows from \\eqref{est1} and \\eqref{est2} that for $s \\in (0,1\/2)$ and $z \\in (-1\/2,0)$\none has\n\\begin{equation}\\label{hschaetz}\n \\frac{1}{5}0.\n\\end{equation}\nThen the identity \\eqref{kuhstrich} together with \\eqref{??} implies that there is some $\\varepsilon>0$\nsuch that $R(z)$ is holomorphic with $R'(z)<0$ on $(-\\varepsilon,0)$. By Theorem \\ref{mainth} the interval\n$(-\\varepsilon,0)$ is contained in $\\mathcal{F}_Q$. Note that in this case it follows from \\eqref{hash} that\n\\begin{equation}\\label{1a'}\n M(0-)=+\\infty.\n\\end{equation}\n\n\\textit{Case (b)}. Assume that\n\\begin{equation}\\label{1b}\n \\sigma(\\{0\\})=0 \\quad \\mbox{and} \\quad \\int_{(0,1\/2)} \\frac{d\\sigma (s)}{s}=+\\infty.\n\\end{equation}\nBy the estimate \\eqref{hschaetz} and monotone convergence it follows that\n$$\n\\int_{(0,1\/2)} h(s,z) \\,\\frac{d\\sigma(s)}{s-z} \\ge \\frac{1}{5} \\int_{(0,1\/2)}\\frac{d\\sigma(s)}{s-z}\\to +\\infty,\n \\quad z\\uparrow 0.\n$$\nConsequently, $T_1(0-)=+\\infty$. Since $T_2(z)$ is holomorphic on the interval $(-\\gamma,1\/2)$,\none obtains\n$T(0-)=+\\infty$. As $T$ is holomorphic on $(-\\gamma,0)$, the identity \\eqref{kuhstrich} implies\nagain that $R'(z)<0$ on some interval $(-\\varepsilon,0)$, which is therefore contained in $\\mathcal{F}_Q$.\nNote that in this case it follows from \\eqref{ooo} that\n\\begin{equation}\\label{1b'}\nM(0-)=+\\infty.\n\\end{equation}\n\n\\textit{Case (c)}. Assume that\n\\begin{equation}\\label{1c}\n\\sigma(\\{0\\})=0 \\quad \\mbox{and} \\quad \\int_{(0,1\/2)} \\frac{d\\sigma (s)}{s} < + \\infty.\n\\end{equation}\n Then\n\\begin{equation}\\label{altkuhstrich}\nR'(z)=2z\\left(a+\\int_{\\dR\\setminus \\{0\\}} \\frac{d\\sigma(s)}{s(s^2+1)}\\right) + z^2S(z),\n\\end{equation}\nwith\n\\begin{equation}\\label{bbb}\nS(z)=3b+ \\int_{\\dR\\setminus \\{0\\}} \\frac{3s-2z}{s(s-z)^2} \\, d\\sigma(s).\n\\end{equation}\nIt will be shown by dominated convergence that\n\\begin{equation}\\label{limzS}\n\\lim_{z \\uparrow 0} zS(z)=0.\n\\end{equation}\nTo do this, it suffices to find an integrable upper bound when the integrand in \\eqref{bbb}\nis multiplied by $z$.\nFor $s>0$ and $z<0$ one has $0< 3s-2z < 3(s-z)$, so that\n\\[\n 0 \\le \\frac{3s-2z}{s(s-z)^2} \\le \\frac{3}{s(s-z)}.\n\\]\nSince $|z| < s-z$ for $s >0$ and $z<0$, it follows that\n\\begin{equation}\\label{hen1}\n \\left| z \\frac{3s-2z}{s(s-z)^2} \\right| \\le \\frac{3}{s}, \\quad s>0, \\quad z<0,\n\\end{equation}\nand in particular this estimate holds for $00$. Then the identities \\eqref{altkuhstrich} and \\eqref{limzS} imply that\n$R'(z)<0$ on some interval $(-\\varepsilon,0)$, which is therefore contained in $\\mathcal{F}_Q$.\n\n\\item\n$M(0-)< 0$. Similarly, one obtains $R'(z)>0$ on some interval $(-\\varepsilon,0)$, which is\ntherefore not contained in $\\mathcal{F}_Q$.\n\n\\item\n$M(0-)=0$.\nIt follows from \\eqref{ooo} and \\eqref{altkuhstrich}\nthat $R'(z)=z^2S(z)$. The identity \\eqref{r1lim} implies that $\\lim_{z\\uparrow 0} S(z)\\in(0,+\\infty]$\n(otherwise $b$ and $\\sigma$ and by \\eqref{ooo} also $a$ would vanish, so that $M(z) \\equiv 0$,\ncontradicting the fact that $R(z)\\in\\mathbf{N}_1$).\nHence, $R'(z)>0$ on some interval $(-\\varepsilon,0)$, which is therefore not contained in $\\mathcal{F}_Q$. \\\\\n\\end{itemize}\n\n\\textit{Step 2.} In order to treat the general case it will now be proved that\n\\begin{equation}\\label{rprim}\n \\lim_{z \\uparrow 0} \\frac{R'(z)}{R(z)} = -\\infty,\n\\end{equation}\nin each of the cases (a), (b), and (c) in Step 1. \\\\\n\n\\textit{Case (a)}.\nRecall that $R'(z) <0$ on some interval $(-\\varepsilon,0)$ with $\\lim_{z \\uparrow 0} R'(z)=-\\sigma(\\{0\\})$ according to \\eqref{kuhstrich}.\nSince $R(z) >0$ on some interval $(-\\varepsilon_1,0)$ by \\eqref{1a'} with\n$\\lim_{z \\uparrow 0} R(z)=0$ (see \\eqref{hash}), it\nfollows that \\eqref{rprim} holds. \\\\\n\n\\textit{Case (b)}.\nRecall that $T(0-) =+ \\infty$, so that\n\\eqref{kuhstrich} implies that\n\\[\n\\lim_{z \\uparrow 0}\\frac{R'(z)}{z} \\to +\\infty.\n\\]\nFurthermore\n\\[\n\\lim_{z \\uparrow 0} \\frac{R(z)}{z} = \\lim_{z \\uparrow 0} zM(z) =0,\n\\]\nand note that $z M(z) \\uparrow 0$ for $z \\uparrow 0$ (see \\eqref{1b'}).\nThus \\eqref{rprim} follows. \\\\\n\n\\textit{Case (c)}.\nIt follows from \\eqref{ooo}, \\eqref{altkuhstrich}, and\n\\eqref{limzS} that\n\\[\n\\lim_{z \\uparrow 0} \\frac{R'(z)}{z}=2M(0-).\n\\]\n\nIf $M(0-) \\neq 0$, then\n\\begin{equation}\n \\lim_{z \\uparrow 0} \\frac{z R'(z)}{R(z)} =\n \\lim_{z \\uparrow 0} \\frac{R'(z)}{z} \\frac{1}{M(z)} =2,\n\\end{equation}\nfrom which \\eqref{rprim} follows.\n\nIf $M(0-)=0$, then $M(z) \\uparrow0$ as $z \\uparrow 0$, so that\n\\begin{equation}\n \\lim_{z \\uparrow 0} \\frac{R'(z)}{R(z)} = \\lim_{z \\uparrow 0} \\frac{S(z)}{M(z)}=-\\infty,\n\\end{equation}\nand, again, \\eqref{rprim} follows. \\\\\n\n\\textit{Step 3.} Consider the function $Q(z)$ defined by\n\\[\n Q(z)=\\frac{z^2M(z)}{(z-\\beta)(z-\\bar{\\beta})}=\\frac{R(z)}{D(z)},\n\\]\nwhere $R(z)=z^2M(z)$ as in Step 1 and $D(z)=(z-\\beta)(z-\\bar{\\beta})$. Then\n\\begin{equation}\\label{deriv}\n Q'(z)=\\frac{R'(z)D(z)-R(z)D'(z)}{D(z)^2}=M(z)\\,\\frac{z^2}{D(z)} \\left( \\frac{R'(z)}{R(z)}-\\frac{D'(z)}{D(z)} \\right).\n\\end{equation}\nNote that $D(z) >0$ for $z \\in \\dR$ and $z \\neq \\beta$, and that\n\\[\n \\frac{D'(0)}{D(0)}=- \\frac{{\\rm Re\\,} \\beta}{|\\beta|^2}.\n\\]\nThe identity \\eqref{deriv} and the limit in \\eqref{rprim} now imply that\n\\[\nQ'(z) <0,\\,\\, z \\in (-\\varepsilon,0), \\mbox{ for some $\\varepsilon > 0$}\n\\quad \\Leftrightarrow \\quad 0< M(0-) \\le \\infty,\n\\]\nand\n\\[\nQ'(z) \\ge 0, \\,\\, z \\in (-\\varepsilon,0), \\mbox{ for some $\\varepsilon > 0$} \\quad \\Leftrightarrow \\quad M(0-) \\le 0.\n\\]\nThis completes the proof of the theorem.\n\\end{proof}\n\n\\begin{example}\nIf $Q(z)=z^{2+\\rho}$, $0 < \\rho<1$, then $Q(z)=z^2M(z)$ with $M(z)=z^\\rho \\in \\mathbf{N}_0$. Hence $Q(z)\n\\in \\mathbf{N}_1$ and $z=0$ \nis the GZNT of $Q(z)$. With the interpretation of $z^\\rho$\nas in the introduction, it follows that $M(z)$ is holomorphic on\n$(0,+\\infty)$. It is not\ndifficult to see that $\\lim_{z \\downarrow 0} M(z)=0$. Hence, the conditions of Theorem \\ref{realpath} are not\nsatisfied. The path is indicated in Figure 2.\n\\end{example}\n\n\\begin{example}\nIf $Q(z)=z^2 \\log z$, then $Q(z)=z^2 M(z)$ with $M(z) =\\log z$. Hence $Q(z) \\in \\mathbf{N}_1$ and $z=0$ is the GZNT of $Q(z)$. Clearly $\\lim_{z \\downarrow 0} \\log z=-\\infty$, so that by Theorem\n\\ref{realpath}, there exists $\\gamma >0$ such that $(0, \\gamma) \\subset \\mathcal{F}_Q$. It can be shown that\n$\\gamma=1\/\\sqrt{e}$ is the maximal possible value.\n\\end{example}\n\nIn the special case that $Q(z) \\in \\mathbf{N}_1$ is holomorphic in a neighborhood\nof the GZNT $\\alpha \\in \\dR$, a combination of Theorem \\ref{realpath} and\nProposition \\ref{zeros>0} leads to the following description, which\nagrees with Theorem \\ref{mainth}.\n\n\\begin{corollary}\\label{zeros>0+}\nLet $Q(z) \\in \\mathbf{N}_1$ be holomorphic in a neighborhood\nof the GZNT $\\alpha \\in \\dR$. Then the following cases appear:\n\\begin{itemize}\n\\item[(1)] $Q'(\\alpha) < 0$: There exists $\\varepsilon>0$ such that $(\\alpha-\\varepsilon, \\alpha+\\varepsilon)\\subset \\mathcal{ F}_Q$.\n\n\\item[(2a)] $Q'(\\alpha) = 0$, $Q''(\\alpha) > 0$: There exists $\\varepsilon>0$ such that $(\\alpha-\\varepsilon,\\alpha]\\subset \\mathcal{ F}_Q$.\n\n \\item[(2b)] $Q'(\\alpha) = 0$, $Q''(\\alpha) < 0$: There exists $\\varepsilon>0$ such that $[\\alpha,\\alpha+\\varepsilon ) \\subset \\mathcal{ F}_Q$.\n\n\\item[(3)] $Q'(\\alpha) = 0$, $Q''(\\alpha) = 0$: There exists no $\\varepsilon>0$ such that $(\\alpha-\\varepsilon,\\alpha]\\subset \\mathcal{ F}_Q$ or $[\\alpha,\\alpha+\\varepsilon ) \\subset \\mathcal{ F}_Q$.\n\\end{itemize}\n\\end{corollary}\n\nIn order to show how this follows from Theorem \\ref{realpath}\nassume for simplicity that $Q(z)$ is of the form $Q(z)=(z-\\alpha)^2M(z)$ with $M(z) \\in \\mathbf{N}_0$.\nThen the condition that $Q(z)$ is holomorphic around $\\alpha$ implies that\n$M(z)$ has an isolated first order pole at $\\alpha$ or that $M(z)$ is holomorphic\naround $\\alpha$.\n\nIf $M(z)$ has a first order pole at $\\alpha$, then\n\\begin{equation}\\label{pol}\n M(z)=\\frac{c}{z-\\alpha} + \\varphi(z),\n\\end{equation}\nwith $c<0$ and $\\varphi(z)$ holomorphic around $\\alpha$. Hence, in this case\n\\[\n Q'(\\alpha) =c <0.\n\\]\n\nIf $M(z)$ is holomorphic at $\\alpha$, then\n\\[\nQ'(\\alpha)=0, \\quad Q''(\\alpha)=2 M(\\alpha), \\quad Q'''(\\alpha)=6M'(\\alpha).\n\\]\n\nIf Case (1) prevails, then $Q'(\\alpha)<0$. Hence the function $M(z)$ must be of the form \\eqref{pol}.\n(It also follows from \\eqref{pol} and Theorem \\ref{realpath} that the assertion in (1) holds).\n\nIf Cases (2a), (2b), or (3) prevail, then $Q'(\\alpha)=0$. Hence the function $M(z)$ must be holomorphic at $\\alpha$.\nIn particular in Case (2a) $M(\\alpha)>0$ and in Case (2b) $M(\\alpha)<0$, so that assertions follows from Theorem \\ref{realpath}. Finally, in Case (3) $M(\\alpha)=0$ and the assertion in (3) follows from Theorem \\ref{realpath}.\n\n\\begin{remark}\\label{53}\nIt may be helpful to reconsider some of the concrete simple examples given earlier\nin terms of Theorem \\ref{realpath} and Corollary \\ref{zeros>0}.\n\nIf $Q(z)=-z$, then $Q'(0)=-1$. Since $Q(z)=z^2M(z)$ with $M(z)=-1\/z \\in \\mathbf{N}_0$,\none has $M(0+) =-\\infty$, $M(0-)=\\infty$.\n \nIf $Q(z)=z^2$, then $Q'(0)=0$, $Q''(0)=2$. Since\n$Q(z)=z^2M(z)$ with $M(z)=1 \\in \\mathbf{N}_0$, one has $M(0)=1$.\n \nIf $Q(z)=z^3$, then $Q''(0)=0$, and $Q'''(0)=6$. Since\n$Q(z)=z^2M(z)$ with $M(z)=z \\in \\mathbf{N}_0$, one has $M(0)=0$\n \nIf $Q(z)$ is as in Example \\ref{drie}, then\n$Q'(0)=0$, $Q''(0)=2(\\delta_2-\\delta_1)$, and $Q'''(0)=2(\\delta_1+\\delta_2)$, and one has\n$M(0)= \\delta_2-\\delta_1$.\n\\end{remark}\n\n\n\\section{The path associated with a special function}\\label{twe}\n\nEach of the factors\nin \\eqref{einz} belongs to $\\mathbf{N}_1$ and has \n$\\alpha \\in \\dC^+ \\cup \\dR \\cup{\\infty}$ as GZNT and $\\beta\n\\in \\dC^+ \\cup \\dR \\cup{\\infty}$ as GPNT. \nIn this section the path $\\alpha(\\tau)$ of the GZNT for this factor\nwill be described. The extreme case\nwith $\\beta=\\infty$ can be treated as in the introduction (after a shift). The other extreme case with\n$\\alpha=\\infty$ can be treated similarly.\nTherefore it suffices to consider the function $Q(z)$ defined by\n\\begin{equation}\n\\label{einz-}\n Q(z)=\\frac{(z-\\alpha)(z-\\overline{\\alpha})}{(z-\\beta)(z-\\overline{\\beta })},\n \\quad \\alpha, \\beta \\in \\dC^+ \\cup \\dR.\n\\end{equation}\n\nObserve that the equation ${\\rm Im\\,} Q(z)=0$ where $z=x+ \\ii y$, is equivalent to either the equation\n\\begin{equation}\\label{EINS}\ny=0,\n\\end{equation}\nor the equation\n\\begin{equation}\\label{ZWEI}\n ({\\rm Re\\,} \\alpha-{\\rm Re\\,} \\beta) (x^2+y^2)+(|\\beta|^2-|\\alpha|^2)x+|\\alpha|^2 {\\rm Re\\,} \\beta-|\\beta|^2 {\\rm Re\\,} \\alpha=0.\n\\end{equation}\nIf ${\\rm Re\\,} \\alpha \\ne {\\rm Re\\,} \\beta$, then the equation \\eqref{ZWEI} is equivalent to \n\\[\n\\left( x-\\frac{|\\beta|^2-|\\alpha|^2}{2\\Re(\\beta-\\alpha)}\\right)^2+y^2= \\left(\n\\Re\\alpha-\\frac{|\\beta|^2-|\\alpha|^2}{2\\Re(\\beta-\\alpha)}\\right)^2 +\\left(\\Im\\alpha\\right)^2.\n\\]\nThis defines a circle $C$ which can be also described by the condition that it contains $\\alpha$ and $\\beta$ and\nthat its center lies on $\\dR$. If ${\\rm Re\\,} \\alpha = {\\rm Re\\,} \\beta$, then the equation \\eqref{ZWEI} is equivalent to\n\\[\nx={\\rm Re\\,} \\alpha,\n\\]\nwhich defines a vertical line $C$ through $x ={\\rm Re\\,} \\alpha$.\nNote that by Theorem \\ref{charact} and Corollary \\ref{quh}:\n\\[\n\\mathcal{ F}_Q \\subset (C\\cap \\dC^+)\\cup\\dR\\cup\\{\\infty\\}.\n\\]\nIt is straightforward to check that the sign of $Q'(x)$, $x \\in \\dR$ (with the possible exception of $\\beta$\nwhen $\\beta \\in \\dR$), is given by the sign of the polynomial\n\\begin{equation}\\label{DREI}\n ({\\rm Re\\,} \\alpha-{\\rm Re\\,} \\beta) x^2+(|\\beta|^2-|\\alpha|^2)x+|\\alpha|^2 {\\rm Re\\,} \\beta-|\\beta|^2 {\\rm Re\\,} \\alpha.\n\\end{equation}\n\nDenote for ${\\rm Re\\,} \\alpha \\neq {\\rm Re\\,} \\beta$ the intersections of $C$ with $\\dR$ by $P_l$ and $P_r$ in such way that $P_l {\\rm Re\\,} \\beta$. Then it follows from \\eqref{ZWEI} and \\eqref{DREI} that\n$Q'(x)<0$ for $x\\in(P_l,P_r)$ and $Q'(x)>0$ for $x\\in\\dR\\setminus(P_l,P_r)$.\nHence,\n \\[\n \\mathcal{F}_Q=(C\\cap \\dC^+)\\cup[P_l,P_r].\n\\]\nThe direction of the path is indicated in Figure 8.\n\n\\begin{figure}[hbt]\\label{circle1}\n\\begin{center}\n\\begin{picture}(95,32)(0,13)\n\\linethickness{0.05mm} \\put(0,20){\\line(1,0){90}} \\thicklines\n \\put(65,20){\\vector(-1,0){10}}\n \\put(55,20){\\line(-1,0){30}}\n \\multiput(64.99,20.5)(0.01,-0.5){1}{\\line(0,-1){0.5}}\n\\multiput(64.98,21)(0.02,-0.5){1}{\\line(0,-1){0.5}}\n \\multiput(64.94,21.49)(0.03,-0.5){1}{\\line(0,-1){0.5}}\n\\multiput(64.9,21.99)(0.04,-0.5){1}{\\line(0,-1){0.5}}\n 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\\multiput(32.14,35.32)(0.13,0.11){3}{\\line(1,0){0.13}}\n\\multiput(31.77,35)(0.13,0.11){3}{\\line(1,0){0.13}}\n \\multiput(31.4,34.66)(0.12,0.11){3}{\\line(1,0){0.12}}\n\\multiput(31.04,34.32)(0.12,0.11){3}{\\line(1,0){0.12}}\n \\multiput(30.68,33.96)(0.12,0.12){3}{\\line(1,0){0.12}}\n \\multiput(30.34,33.6)(0.11,0.12){3}{\\line(0,1){0.12}}\n \\multiput(30,33.23)(0.11,0.12){3}{\\line(0,1){0.12}}\n\\multiput(29.68,32.86)(0.11,0.13){3}{\\line(0,1){0.13}}\n \\multiput(29.36,32.47)(0.11,0.13){3}{\\line(0,1){0.13}}\n\\multiput(29.06,32.08)(0.1,0.13){3}{\\line(0,1){0.13}}\n \\multiput(28.76,31.67)(0.15,0.2){2}{\\line(0,1){0.2}}\n\\multiput(28.48,31.27)(0.14,0.2){2}{\\line(0,1){0.2}}\n \\multiput(28.2,30.85)(0.14,0.21){2}{\\line(0,1){0.21}}\n\\multiput(27.93,30.43)(0.13,0.21){2}{\\line(0,1){0.21}}\n \\multiput(27.68,30)(0.13,0.21){2}{\\line(0,1){0.21}}\n\\multiput(27.44,29.57)(0.12,0.22){2}{\\line(0,1){0.22}}\n \\multiput(27.2,29.12)(0.12,0.22){2}{\\line(0,1){0.22}}\n\\multiput(26.98,28.68)(0.11,0.22){2}{\\line(0,1){0.22}}\n \\multiput(26.77,28.23)(0.11,0.23){2}{\\line(0,1){0.23}}\n\\multiput(26.57,27.77)(0.1,0.23){2}{\\line(0,1){0.23}}\n \\multiput(26.38,27.31)(0.09,0.23){2}{\\line(0,1){0.23}}\n\\multiput(26.21,26.84)(0.18,0.47){1}{\\line(0,1){0.47}}\n \\multiput(26.04,26.37)(0.16,0.47){1}{\\line(0,1){0.47}}\n\\multiput(25.89,25.9)(0.15,0.47){1}{\\line(0,1){0.47}}\n \\multiput(25.75,25.42)(0.14,0.48){1}{\\line(0,1){0.48}}\n\\multiput(25.62,24.94)(0.13,0.48){1}{\\line(0,1){0.48}}\n \\multiput(25.5,24.45)(0.12,0.48){1}{\\line(0,1){0.48}}\n\\multiput(25.4,23.96)(0.1,0.49){1}{\\line(0,1){0.49}}\n \\multiput(25.3,23.47)(0.09,0.49){1}{\\line(0,1){0.49}}\n\\multiput(25.22,22.98)(0.08,0.49){1}{\\line(0,1){0.49}}\n \\multiput(25.16,22.49)(0.07,0.49){1}{\\line(0,1){0.49}}\n\\multiput(25.1,21.99)(0.06,0.5){1}{\\line(0,1){0.5}}\n \\multiput(25.06,21.49)(0.04,0.5){1}{\\line(0,1){0.5}}\n\\multiput(25.02,21)(0.03,0.5){1}{\\line(0,1){0.5}}\n \\multiput(25.01,20.5)(0.02,0.5){1}{\\line(0,1){0.5}}\n\\multiput(25,20)(0.01,0.5){1}{\\line(0,1){0.5}}\n \\thinlines \\put(35,36){\\line(0,1){2}}\n\\put(61,31){\\line(0,1){2}}\n \\put(36,35){$\\beta$}\n \\put(58,30){$\\alpha$}\n\\put(25,19){\\line(0,1){2}}\n \\put(65,19){\\line(0,1){2}}\n \\put(25,17){\\makebox(0,0)[cc]{$P_l$}}\n\\put(65,17){\\makebox(0,0)[cc]{$P_r$}}\n\n\\end{picture}\n\\caption{The path of $\\alpha(\\tau)$ for ${\\rm Re\\,} \\alpha > {\\rm Re\\,} \\beta$.}\n\\end{center}\n\\end{figure}\n\n\n\\textit{Case 2}. ${\\rm Re\\,} \\alpha < {\\rm Re\\,} \\beta$. Then it follows from \\eqref{ZWEI} and \\eqref{DREI} that\n$Q'(x)>0$ for $x\\in(P_l,P_r)$ and $Q'(x)<0$ for $x\\in\\dR\\setminus(P_l,P_r)$.\nHence,\n \\[\n \\mathcal{F}_Q=(C\\cap \\dC^+)\\cup(-\\infty, P_l] \\cup [P_r, \\infty).\n\\]\nThe direction of the path is indicated in Figure 9.\n\n\n\\begin{figure}[hbt]\\label{circle2}\n\\begin{center}\n\\begin{picture}(90,32)(0,13)\n\\linethickness{0.05mm} \\put(0,20){\\line(1,0){90}} \\thicklines\n \\put(25,20){\\vector(-1,0){10}}\n \\put(15,20){\\line(-1,0){15}}\n \\put(90,20){\\vector(-1,0){10}}\n \\put(80,20){\\line(-1,0){15}}\n \\multiput(64.99,20.5)(0.01,-0.5){1}{\\line(0,-1){0.5}}\n\\multiput(64.98,21)(0.02,-0.5){1}{\\line(0,-1){0.5}}\n \\multiput(64.94,21.49)(0.03,-0.5){1}{\\line(0,-1){0.5}}\n\\multiput(64.9,21.99)(0.04,-0.5){1}{\\line(0,-1){0.5}}\n \\multiput(64.84,22.49)(0.06,-0.5){1}{\\line(0,-1){0.5}}\n\\multiput(64.78,22.98)(0.07,-0.49){1}{\\line(0,-1){0.49}}\n\\multiput(64.7,23.47)(0.08,-0.49){1}{\\line(0,-1){0.49}}\n\\multiput(64.6,23.96)(0.09,-0.49){1}{\\line(0,-1){0.49}}\n\\multiput(64.5,24.45)(0.1,-0.49){1}{\\line(0,-1){0.49}}\n\\multiput(64.38,24.94)(0.12,-0.48){1}{\\line(0,-1){0.48}}\n\\multiput(64.25,25.42)(0.13,-0.48){1}{\\line(0,-1){0.48}}\n\\multiput(64.11,25.9)(0.14,-0.48){1}{\\line(0,-1){0.48}}\n\\multiput(63.96,26.37)(0.15,-0.47){1}{\\line(0,-1){0.47}}\n\\multiput(63.79,26.84)(0.16,-0.47){1}{\\line(0,-1){0.47}}\n\\multiput(63.62,27.31)(0.18,-0.47){1}{\\line(0,-1){0.47}}\n\\multiput(63.43,27.77)(0.09,-0.23){2}{\\line(0,-1){0.23}}\n\\multiput(63.23,28.23)(0.1,-0.23){2}{\\line(0,-1){0.23}}\n\\multiput(63.02,28.68)(0.11,-0.23){2}{\\line(0,-1){0.23}}\n\\multiput(62.8,29.12)(0.11,-0.22){2}{\\line(0,-1){0.22}}\n\\multiput(62.56,29.57)(0.12,-0.22){2}{\\line(0,-1){0.22}}\n\\multiput(62.32,30)(0.12,-0.22){2}{\\line(0,-1){0.22}}\n\\multiput(62.07,30.43)(0.13,-0.21){2}{\\line(0,-1){0.21}}\n\\multiput(61.8,30.85)(0.13,-0.21){2}{\\line(0,-1){0.21}}\n\\multiput(61.52,31.27)(0.14,-0.21){2}{\\line(0,-1){0.21}}\n\\multiput(61.24,31.67)(0.14,-0.2){2}{\\line(0,-1){0.2}}\n \\multiput(60.94,32.08)(0.15,-0.2){2}{\\line(0,-1){0.2}}\n\\multiput(60.64,32.47)(0.1,-0.13){3}{\\line(0,-1){0.13}}\n\\multiput(60.32,32.86)(0.11,-0.13){3}{\\line(0,-1){0.13}}\n\\multiput(60,33.23)(0.11,-0.13){3}{\\line(0,-1){0.13}}\n \\multiput(59.66,33.6)(0.11,-0.12){3}{\\line(0,-1){0.12}}\n\\multiput(59.32,33.96)(0.11,-0.12){3}{\\line(0,-1){0.12}}\n\\multiput(58.96,34.32)(0.12,-0.12){3}{\\line(1,0){0.12}}\n\\multiput(58.6,34.66)(0.12,-0.11){3}{\\line(1,0){0.12}}\n \\multiput(58.23,35)(0.12,-0.11){3}{\\line(1,0){0.12}}\n\\multiput(57.86,35.32)(0.13,-0.11){3}{\\line(1,0){0.13}}\n\\multiput(57.47,35.64)(0.13,-0.11){3}{\\line(1,0){0.13}}\n\\multiput(57.08,35.94)(0.13,-0.1){3}{\\line(1,0){0.13}}\n \\multiput(56.67,36.24)(0.2,-0.15){2}{\\line(1,0){0.2}}\n\\multiput(56.27,36.52)(0.2,-0.14){2}{\\line(1,0){0.2}}\n \\multiput(55.85,36.8)(0.21,-0.14){2}{\\line(1,0){0.21}}\n\\multiput(55.43,37.07)(0.21,-0.13){2}{\\line(1,0){0.21}}\n \\multiput(55,37.32)(0.21,-0.13){2}{\\line(1,0){0.21}}\n\\multiput(54.57,37.56)(0.22,-0.12){2}{\\line(1,0){0.22}}\n\\multiput(54.12,37.8)(0.22,-0.12){2}{\\line(1,0){0.22}}\n\\multiput(53.68,38.02)(0.22,-0.11){2}{\\line(1,0){0.22}}\n\\multiput(53.23,38.23)(0.23,-0.11){2}{\\line(1,0){0.23}}\n\\multiput(52.77,38.43)(0.23,-0.1){2}{\\line(1,0){0.23}}\n\\multiput(52.31,38.62)(0.23,-0.09){2}{\\line(1,0){0.23}}\n\\multiput(51.84,38.79)(0.47,-0.18){1}{\\line(1,0){0.47}}\n\\multiput(51.37,38.96)(0.47,-0.16){1}{\\line(1,0){0.47}}\n\\multiput(50.9,39.11)(0.47,-0.15){1}{\\line(1,0){0.47}}\n\\multiput(50.42,39.25)(0.48,-0.14){1}{\\line(1,0){0.48}}\n\\multiput(49.94,39.38)(0.48,-0.13){1}{\\line(1,0){0.48}}\n\\multiput(49.45,39.5)(0.48,-0.12){1}{\\line(1,0){0.48}}\n \\multiput(48.96,39.6)(0.49,-0.1){1}{\\line(1,0){0.49}}\n\\multiput(48.47,39.7)(0.49,-0.09){1}{\\line(1,0){0.49}}\n\\multiput(47.98,39.78)(0.49,-0.08){1}{\\line(1,0){0.49}}\n\\multiput(47.49,39.84)(0.49,-0.07){1}{\\line(1,0){0.49}}\n \\multiput(46.99,39.9)(0.5,-0.06){1}{\\line(1,0){0.5}}\n\\multiput(46.49,39.94)(0.5,-0.04){1}{\\line(1,0){0.5}}\n \\multiput(46,39.98)(0.5,-0.03){1}{\\line(1,0){0.5}}\n\\multiput(45.5,39.99)(0.5,-0.02){1}{\\vector(-1,0){0.5}}\n \\multiput(45,40)(0.5,-0.01){1}{\\line(1,0){0.5}}\n\\multiput(44.5,39.99)(0.5,0.01){1}{\\line(1,0){0.5}}\n \\multiput(44,39.98)(0.5,0.02){1}{\\line(1,0){0.5}}\n\\multiput(43.51,39.94)(0.5,0.03){1}{\\line(1,0){0.5}}\n \\multiput(43.01,39.9)(0.5,0.04){1}{\\line(1,0){0.5}}\n\\multiput(42.51,39.84)(0.5,0.06){1}{\\line(1,0){0.5}}\n \\multiput(42.02,39.78)(0.49,0.07){1}{\\line(1,0){0.49}}\n\\multiput(41.53,39.7)(0.49,0.08){1}{\\line(1,0){0.49}}\n \\multiput(41.04,39.6)(0.49,0.09){1}{\\line(1,0){0.49}}\n\\multiput(40.55,39.5)(0.49,0.1){1}{\\line(1,0){0.49}}\n \\multiput(40.06,39.38)(0.48,0.12){1}{\\line(1,0){0.48}}\n\\multiput(39.58,39.25)(0.48,0.13){1}{\\line(1,0){0.48}}\n \\multiput(39.1,39.11)(0.48,0.14){1}{\\line(1,0){0.48}}\n\\multiput(38.63,38.96)(0.47,0.15){1}{\\line(1,0){0.47}}\n \\multiput(38.16,38.79)(0.47,0.16){1}{\\line(1,0){0.47}}\n\\multiput(37.69,38.62)(0.47,0.18){1}{\\line(1,0){0.47}}\n \\multiput(37.23,38.43)(0.23,0.09){2}{\\line(1,0){0.23}}\n\\multiput(36.77,38.23)(0.23,0.1){2}{\\line(1,0){0.23}}\n \\multiput(36.32,38.02)(0.23,0.11){2}{\\line(1,0){0.23}}\n\\multiput(35.88,37.8)(0.22,0.11){2}{\\line(1,0){0.22}}\n \\multiput(35.43,37.56)(0.22,0.12){2}{\\line(1,0){0.22}}\n\\multiput(35,37.32)(0.22,0.12){2}{\\line(1,0){0.22}}\n \\multiput(34.57,37.07)(0.21,0.13){2}{\\line(1,0){0.21}}\n\\multiput(34.15,36.8)(0.21,0.13){2}{\\line(1,0){0.21}}\n \\multiput(33.73,36.52)(0.21,0.14){2}{\\line(1,0){0.21}}\n\\multiput(33.33,36.24)(0.2,0.14){2}{\\line(1,0){0.2}}\n \\multiput(32.92,35.94)(0.2,0.15){2}{\\line(1,0){0.2}}\n\\multiput(32.53,35.64)(0.13,0.1){3}{\\line(1,0){0.13}}\n \\multiput(32.14,35.32)(0.13,0.11){3}{\\line(1,0){0.13}}\n\\multiput(31.77,35)(0.13,0.11){3}{\\line(1,0){0.13}}\n \\multiput(31.4,34.66)(0.12,0.11){3}{\\line(1,0){0.12}}\n\\multiput(31.04,34.32)(0.12,0.11){3}{\\line(1,0){0.12}}\n \\multiput(30.68,33.96)(0.12,0.12){3}{\\line(1,0){0.12}}\n \\multiput(30.34,33.6)(0.11,0.12){3}{\\line(0,1){0.12}}\n \\multiput(30,33.23)(0.11,0.12){3}{\\line(0,1){0.12}}\n\\multiput(29.68,32.86)(0.11,0.13){3}{\\line(0,1){0.13}}\n \\multiput(29.36,32.47)(0.11,0.13){3}{\\line(0,1){0.13}}\n\\multiput(29.06,32.08)(0.1,0.13){3}{\\line(0,1){0.13}}\n \\multiput(28.76,31.67)(0.15,0.2){2}{\\line(0,1){0.2}}\n\\multiput(28.48,31.27)(0.14,0.2){2}{\\line(0,1){0.2}}\n \\multiput(28.2,30.85)(0.14,0.21){2}{\\line(0,1){0.21}}\n\\multiput(27.93,30.43)(0.13,0.21){2}{\\line(0,1){0.21}}\n \\multiput(27.68,30)(0.13,0.21){2}{\\line(0,1){0.21}}\n\\multiput(27.44,29.57)(0.12,0.22){2}{\\line(0,1){0.22}}\n \\multiput(27.2,29.12)(0.12,0.22){2}{\\line(0,1){0.22}}\n\\multiput(26.98,28.68)(0.11,0.22){2}{\\line(0,1){0.22}}\n \\multiput(26.77,28.23)(0.11,0.23){2}{\\line(0,1){0.23}}\n\\multiput(26.57,27.77)(0.1,0.23){2}{\\line(0,1){0.23}}\n \\multiput(26.38,27.31)(0.09,0.23){2}{\\line(0,1){0.23}}\n\\multiput(26.21,26.84)(0.18,0.47){1}{\\line(0,1){0.47}}\n \\multiput(26.04,26.37)(0.16,0.47){1}{\\line(0,1){0.47}}\n\\multiput(25.89,25.9)(0.15,0.47){1}{\\line(0,1){0.47}}\n \\multiput(25.75,25.42)(0.14,0.48){1}{\\line(0,1){0.48}}\n\\multiput(25.62,24.94)(0.13,0.48){1}{\\line(0,1){0.48}}\n \\multiput(25.5,24.45)(0.12,0.48){1}{\\line(0,1){0.48}}\n\\multiput(25.4,23.96)(0.1,0.49){1}{\\line(0,1){0.49}}\n \\multiput(25.3,23.47)(0.09,0.49){1}{\\line(0,1){0.49}}\n\\multiput(25.22,22.98)(0.08,0.49){1}{\\line(0,1){0.49}}\n \\multiput(25.16,22.49)(0.07,0.49){1}{\\line(0,1){0.49}}\n\\multiput(25.1,21.99)(0.06,0.5){1}{\\line(0,1){0.5}}\n \\multiput(25.06,21.49)(0.04,0.5){1}{\\line(0,1){0.5}}\n\\multiput(25.02,21)(0.03,0.5){1}{\\line(0,1){0.5}}\n \\multiput(25.01,20.5)(0.02,0.5){1}{\\line(0,1){0.5}}\n\\multiput(25,20)(0.01,0.5){1}{\\line(0,1){0.5}}\n \\thinlines \\put(35,36){\\line(0,1){2}}\n\\put(61,31){\\line(0,1){2}}\n \\put(36,35){$\\alpha$}\n \\put(58,29){$\\beta$}\n \\put(3,22){\\makebox(0,0)[cc]{$\\infty$}}\n\\put(25,19){\\line(0,1){2}}\n \\put(65,19){\\line(0,1){2}}\n \\put(25,17){\\makebox(0,0)[cc]{$P_l$}}\n\\put(65,17){\\makebox(0,0)[cc]{$P_r$}}\n\n\\end{picture}\n\\caption{The path of $\\alpha(\\tau)$ for ${\\rm Re\\,} \\alpha < {\\rm Re\\,} \\beta$.}\n\\end{center}\n\\end{figure}\n\n\\textit{Case 3.} ${\\rm Re\\,} \\alpha = {\\rm Re\\,} \\beta$. Then it follows from \\eqref{ZWEI} and \\eqref{DREI} that\nin the case ${\\rm Im\\,} \\beta < {\\rm Im\\,} \\alpha$\n\\[\n Q'(x) <0, \\quad x > P; \\quad Q'(x) > 0, \\quad x < P,\n\\]\nand in the case ${\\rm Im\\,} \\beta > {\\rm Im\\,} \\alpha$\n\\[\n Q'(x) >0, \\quad x > P; \\quad Q'(x) < 0, \\quad x < P.\n\\]\nThe direction of the path is indicated in each of these cases in Figure 10.\n\n\\begin{figure}[hbt]\\label{nocircles}\n\\begin{center}\n\\begin{picture}(105,32)(0,13)\n\n\\thinlines \\put(0,20){\\line(1,0){50}}\n\n \\put(24,27){\\line(2,0){2}}\n \\put(23,27){\\makebox(0,0)[cc]{$\\beta$}}\n \\put(24,33){\\line(2,0){2}}\n \\put(23,33){\\makebox(0,0)[cc]{$\\alpha$}}\n \\thicklines\n \\put(50,20){\\vector(-1,0){15}}\n \\put(35,20){\\line(-1,0){10}}\n \\put(25,20){\\vector(0,1){4}}\n \\put(25,24){\\vector(0,1){13}}\n \\put(25,37){\\line(0,1){3}}\n\n\\thinlines\n \\put(55,20){\\line(1,0){50}}\n\n \\put(79,27){\\line(2,0){2}}\n \\put(78,27){\\makebox(0,0)[cc]{$\\alpha$}}\n \\put(79,33){\\line(2,0){2}}\n \\put(78,33){\\makebox(0,0)[cc]{$\\beta$}}\n \\thicklines\n \\put(80,40){\\vector(0,-1){4}}\n \\put(80,36){\\vector(0,-1){13}}\n \\put(80,23){\\line(0,-1){3}}\n \\put(70,20){\\line(-1,0){15}}\n \\put(80,20){\\vector(-1,0){10}}\n\n \\put(25,17){\\makebox(0,0)[cc]{$P$}}\n \\put(80,17){\\makebox(0,0)[cc]{$P$}}\n\\end{picture}\n\\caption{The path of $\\alpha(\\tau)$ for ${\\rm Re\\,} \\alpha ={\\rm Re\\,} \\beta$ when ${\\rm Im\\,} \\alpha > {\\rm Im\\,} \\beta$ and\nwhen ${\\rm Im\\,} \\beta > {\\rm Im\\,} \\alpha$.}\n\\end{center}\n\\end{figure}\n\n\\begin{remark}\nIt follows from the above three cases that the point $\\infty$ belongs to $\\mathcal{F}_Q$ if and only if ${\\rm Re\\,}\n\\alpha \\le {\\rm Re\\,} \\beta$. This can be seen directly. It follows from \\eqref{x_00} \nthat $Q_\\tau(z)$ has $\\infty$ as GZNT if and only if\n\\[\n\\lim_{z \\wh\\to \\infty} z Q_\\tau(z) \\in [0,\\infty).\n\\]\nIn this case $Q_\\tau(z) \\to 0$ when $z \\wh\\to \\infty$. Since $Q(z) \\to 1$ as $z \\wh\\to \\infty$ this can only take place for $\\tau =1$, in which case\n\\[\n\\lim_{z \\wh\\to \\infty} z Q_\\tau(z)={\\rm Re\\,} \\beta -{\\rm Re\\,} \\alpha,\n\\]\nfrom which the assertion follows.\n\\end{remark}\n\n\\begin{remark}\nThe present treatment of the function $Q(z)$ in \\eqref{einz-} can be seen as an illustration of Theorem \\ref{mainth} and Theorem \\ref{realpath}.\nFor this purpose assume (without loss of generality) that $\\alpha \\in \\dR$. When ${\\rm Re\\,} \\beta < \\alpha$ the point $P_r$ coincides with $\\alpha$ and when ${\\rm Re\\,} \\beta > \\alpha$ the point $P_l$ coincides with $\\alpha$; if ${\\rm Re\\,} \\beta=\\alpha$, then\n the point $P$ coincides with $\\alpha$ (and ${\\rm Im\\,} \\beta > 0$).\nHence the path hits $\\dR$ at $\\alpha$ in a perpendicular way (cf. Theorem \\ref{mainth}). Afterwards it stays on the real line (cf. Theorem \\ref{realpath} with $M(z)=1 \\in \\mathbf{N}_0$). \n\\end{remark}\n\n\n\\section{Path entirely on the extended real line}\\label{ontherealline}\n\nThere exist functions $Q(z) \\in \\mathbf{N}_1$ whose GZNT $\\alpha(\\tau)$ stays on the (extended) real line.\nThese functions will be classified in this section.\n\n\\begin{example}\nLet $\\alpha, \\beta \\in \\dR$ and $c \\in \\dR$ such that $c(\\alpha-\\beta) < 0$ and define the function $Q(z)$ by\n\\begin{equation}\\label{R0}\n Q(z)=c\\,\\frac{z-\\alpha}{z-\\beta}, \n \\quad z \\neq \\beta. \n\\end{equation}\nThen $Q(z)$ belongs to $\\mathbf{N}_1$; actually, one has\n\\begin{equation}\\label{R0+}\n Q(z) =\\frac{(z-\\alpha)^2}{(z-\\beta)^2}\\,\\,M(z),\n \\quad\n M(z)=c + \\frac{c(\\alpha-\\beta)}{z-\\alpha},\n\\end{equation}\nwhere $M(z)$ belongs to $\\mathbf{N}_0$\n\nIt follows from \\eqref{R0+} that\n$M(\\alpha+)=\\infty$, $M(\\alpha-)=-\\infty$.\nHence, according to Theorem \\ref{realpath}, there exists $\\varepsilon > 0$ such that the intervals\n$(\\alpha-\\varepsilon, \\alpha)$ and $(\\alpha, \\alpha+\\varepsilon)$ belong to $\\cF_Q$. However, in this case one can obtain stronger results. \nObserve that the GZNT $\\alpha(\\tau)$ of $Q_\\tau(z)$, $\\tau \\in \\dR \\cup \\{\\infty\\}$, is given by\n\\[\n \\alpha(\\tau)=\\frac{c \\alpha-\\tau \\beta}{c-\\tau},\n \\quad \\tau \\in \\dR\\setminus\\left\\{c\\right\\},\n\\]\nand, in the remaining cases, by \n\\[\n \\alpha(\\infty)=\\beta, \\quad \\alpha(c)=\\infty. \n\\]\nTherefore the path $\\mathcal{F}_Q$ covers the extended real line.\nMoreover, note that for the functions \n\\begin{equation}\\label{Rc}\nQ(z) =\\frac{d}{z-\\gamma}, \n\\quad \\gamma\\in\\dR,\\quad d>0,\n\\end{equation}\nand\n\\begin{equation}\\label{R1c}\nQ(z)=\\frac{\\gamma-z}{d}, \n\\quad \\gamma\\in\\dR,\\quad d>0,\n\\end{equation}\none has $\\mathcal{F}_Q=\\dR\\cup\\{\\infty\\}$ as well.\n\\end{example}\n\nIn fact, the functions in \\eqref{R0}, \\eqref{Rc}, and \\eqref{R1c} are the only functions in $\\mathbf{N}_1$\nwhose path of the GZNT stays on the (extended) real line. In order to prove this, the following lemma is\nneeded.\n\n\\begin{lemma}\\label{kuhkuh}\nLet $\\alpha, \\beta \\in \\dR$ with $\\alpha \\not=\\beta$. Assume that $U(z)$ and $V(z)$ are nontrivial \nNevanlinna functions which\nsatisfy one of the following relations \n\\begin{equation}\\label{q1q0}\nU(z)=-\\frac{(z-\\alpha)^2 }{(z-\\beta)^{2}}\\,V(z),\n\\end{equation}\n\\begin{equation}\\label{q1q0a}\nU(z)=-\\frac{1}{(z-\\beta)^{2}}\\,V(z),\n\\end{equation}\nor\n\\begin{equation}\\label{q1q0b}\nU(z)=-(z-\\alpha)^2 \\,V(z).\n\\end{equation}\n Then the functions $U(z)$ and $V(z)$ are of the form\n\\begin{equation}\\label{q1q0rep}\nU(z)=c\\left(1-\\frac{\\beta-\\alpha}{\\beta-z}\\right), \\quad V(z)=c\\left(-1+\\frac{\\alpha-\\beta}{\\alpha-z}\\right), \\quad\nc(\\beta-\\alpha) < 0,\n\\end{equation}\n\\begin{equation}\\label{q1q0repa}\nU(z)=\\frac{d}{\\beta-z}\\, \\quad V(z)=d (z-\\beta), \\quad d> 0,\n\\end{equation}\nor\n\\begin{equation}\\label{q1q0repb}\nU(z)= e (z-\\alpha), \\quad V(z)= \\frac{e}{\\alpha-z}, \\quad e> 0,\n\\end{equation}\nrespectively. Conversely, all functions of the form \\eqref{q1q0rep}, \\eqref{q1q0repa}, or \\eqref{q1q0repb}\nare Nevanlinna functions and they satisfy \\eqref{q1q0}, \\eqref{q1q0a}, and \\eqref{q1q0b}, respectively.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $U(z)$ and $V(z)$ are Nevanlinna functions. Let the integral representations of $U(z)$ and $V(z)$ be of the\nform \\eqref{nev'} with spectral measures $\\sigma_1$, $\\sigma_0$, and constants $a_1 \\in \\dR$, $b_1 \\ge 0$, $a_0\n\\in \\dR$, $b_0 \\ge 0$, respectively.\n\nAssume that $U(z)$ and $V(z)$ satisfy \\eqref{q1q0}. Then the identity \\eqref{nev+} implies\nthat \n$$\nb_1=\\lim_{z\\wh\\to\\infty}\\frac{U(z)}{z} =-\\lim_{z\\wh\\to\\infty}\\frac{V(z)}{z} =-b_0.\n$$\nSince both $b_0$ and $b_1$ are nonnegative, it follows that\n\\begin{equation}\\label{b0b1=0}\nb_0=b_1=0.\n\\end{equation}\nThe Stieltjes inversion formula implies that\n$$\n(s-\\beta)^2 d\\sigma_1(s)=-(s-\\alpha)^2 d\\sigma_0(s).\n$$\nIt follows that ${\\rm supp\\,} \\sigma_1\\subset\\{\\beta\\}$ and ${\\rm supp\\,} \\sigma_0\\subset\\{\\alpha\\}$. If $\\sigma_1=0$ then by\n\\eqref{b0b1=0} the function $V(z)$ is equal to a real constant, and the identity \\eqref{q1q0} implies that\n$U(z)=V(z)=0$. \nThe same conclusion holds if $\\sigma_0=0$. Therefore, it may be assumed that $U(z)$ and $V(z)$\nhave representations of the form\n$$\nU(z)=a_1+\\frac{d_1}{\\beta-z}, \\quad V(z)=a_0+\\frac{d_0}{\\alpha-z},\n$$\nwith $d_0,~d_1>0$. The identity \\eqref{q1q0} implies further that $U(\\alpha)=V(\\beta)=0$, so that\n$$\na_1+\\frac{d_1}{\\beta-\\alpha}=a_0+\\frac{d_0}{\\alpha-\\beta}=0.\n$$\nThus $a_1(\\beta-\\alpha) < 0$, $a_0(\\beta-\\alpha) > 0$, and\n$$\nU(z)=a_1\\left(1-\\frac{\\beta-\\alpha}{\\beta-z}\\right), \\quad V(z)=a_0\\left(1-\\frac{\\alpha-\\beta}{\\alpha-z}\\right).\n$$\nThis and $\\eqref{q1q0}$ imply that \n$$\na_1=\\lim_{z\\wh\\to\\infty}U(z)=-\\lim_{z\\wh\\to\\infty}V(z)=-a_0,\n$$\nand the representations in $\\eqref{q1q0rep}$ follow with $c=a_1$.\n\nNow assume that $U(z)$ and $V(z)$ satisfy \\eqref{q1q0a}. The identity \\eqref{nev+} \nimplies that\n$$\nb_1=\\lim_{z \\wh\\to\\infty}\\frac{U(z)}{z} = 0.\n$$\nThe Stieltjes inversion formula implies that\n$$\n(s-\\beta)^2 d\\sigma_1(s)=-d\\sigma_0(s),\n$$\nwhich leads to ${\\rm supp\\,} \\sigma_1\\subset\\{\\beta\\}$ and ${\\rm supp\\,} \\sigma_0= \\emptyset$. \nTherefore, it follows that $U(z)$ and $V(z)$ have representations of the form\n$$\nU(z)=a_1+\\frac{d_1}{\\beta-z}, \\quad V(z)=b_0z+a_0,\n$$\nwith $d_1>0$. The identity \\eqref{q1q0a} implies that $V(\\beta)=0$, so that $b_0 \\beta+a_0=0$. Hence\n$V(z)=b_0(z-\\beta)$ and, again by \\eqref{q1q0a}, \n\\[\n a_1+\\frac{d_1}{\\beta-z}=-\\frac{b_0}{z-\\beta}.\n\\]\nHence $a_1=0$ and\n\\[\n U(z)=\\frac{d}{\\beta-z}, \\quad V(z)=-(z-\\beta)^2 U(z)=d (z-\\beta), \\quad d>0.\n\\]\n\nThe treatment of the case where the functions $U(z)$ and $V(z)$ satisfy \\eqref{q1q0b} is similar to what has been\njust shown. It also follows by symmetry from the previous case.\n\nAs to the converse statement: it is straightforward to check that the functions in $\\eqref{q1q0rep}$,\n$\\eqref{q1q0repa}$, or $\\eqref{q1q0repb}$, are Nevanlinna functions which satisfy the identities\n\\eqref{q1q0}, \\eqref{q1q0a}, or \\eqref{q1q0b}, respectively.\n\\end{proof}\n\nThe case $\\alpha=\\beta$ excluded in Lemma \\ref{kuhkuh} concerns Nevanlinna functions\n$U(z)$ and $V(z)$ which satisfy $U(z)=-V(z)$. Of course, this implies that $U(z)=c$ and $V(z)=-c$, where $c$\nis a real constant.\n\n\\begin{theorem}\\label{ontheline:t}\nLet $Q(z) \\in \\mathbf{N}_1$ have the property that the path $\\mathcal{F}_Q$ is contained in the extended real\nline.\n\\begin{enumerate}\\def\\rm (\\roman{enumi}){\\rm (\\roman{enumi})}\n\n\\item If both the GZNT and the GPNT of $Q(z)$ are finite, then $Q(z)$ is of the form\n\\eqref{R0} with some $\\alpha,\\beta\\in\\dR$ and $c\\in\\dR$ such that $c(\\alpha-\\beta) < 0$.\n\n\\item If the GZNT of $Q(z)$ is at $\\infty$ and the GPNT of $Q(z)$ is finite,\nthen $Q(z)$ is of the form \\eqref{Rc}. \n\n\\item If the GZNT of $Q(z)$ is finite and the GPNT of $Q(z)$ is at $\\infty$,\nthen $Q(z)$ is of the form \\eqref{R1c}. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n(i) \nThe assumption that $Q_{\\tau}(z)$ has no GZNT in $\\dC^+$ for all $\\tau\\in\\dR\\cup\n\\{\\infty\\}$ implies that $\\Im Q(z)\\not=0$ in $\\dC^+$; cf. Corollary \\ref{quh}. Since $Q(z) \\in\\mathbf{N}_1$,\nthere is at least one point $z_0\\in\\dC^+$ with $\\Im Q(z_0)<0$. However, $\\Im Q(z)$ is a continuous function\non $\\dC^+$, so that $\\Im Q(z)<0$ on $\\dC^+$. Observe that the functions $U(z)=-Q(z)$ and $V(z)=M(z)$ are\nnontrivial Nevanlinna functions which satisfy $\\eqref{q1q0}$. Therefore, by Lemma \\ref{kuhkuh}\n\\[\n Q(z)=-U(z)=c \\frac{z-\\alpha}{z-\\beta},\n\\]\nwhich completes the proof of (i). \n\n(ii) \\& (iii) The remaining parts of the theorem can be proved in a similar way, using the\nrespective parts of Lemma \\ref{kuhkuh}. \n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}