diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpoin" "b/data_all_eng_slimpj/shuffled/split2/finalzzpoin" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpoin" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThis paper and its sequel explores a literal interpretation of the standard analogy between the Frobenius in the absolute Galois group of a finite field, and the generator of the fundamental group of a circle.\n\nWe define an ``$F$-field'' on a manifold $M$ to be a locally constant sheaf of algebraically closed fields of positive characteristic. The best example is the $F$-field $\\underline{k}$ on a circle whose fiber at a base point is an algebraic closure $k$ of $\\mathbf{F}_p$, and whose monodromy around the circle is the $p$th power map --- we write $\\SF{p}$ to indicate the circle along with such an $F$-field. Every other good example is pulled back from this one along a map $\\mathfrak{f}:M \\to \\SF{p}$.\n\nThe pullback of $\\underline{k}$ along $\\mathfrak{f}$ is a sheaf of rings, and we can consider sheaves of modules over it. For example, the category of locally free $\\underline{k}$-modules of finite rank on $\\SF{p}$ is equivalent to the category of finite-dimensional $\\mathbf{F}_p$-vector spaces. In this paper we will prove the following:\n\\begin{prop*}\nIf $M$ is the complement of a knot in $S^3$, $p$ is a sufficiently large prime, and $\\mathfrak{f}:M \\to \\SF{p}$ has degree $n$ on $H_1(M)$, then the set of isomorphism classes of invertible $\\mathfrak{f}^*\\underline{k}$-modules is a finite commutative group of cardinality $\\Delta_K(p^n)$.\n\\end{prop*}\n\nIn the Proposition, $\\Delta_K(x) = c_0 + c_1 x + \\cdots + c_n x^n$ denotes the classical Alexander polynomial of $K$, with Alexander's normalization that $c_0$ should be a positive integer. The primes $p$ that must be excluded are the divisors of $c_0$.\n\nThe Proposition gives a combinatorial interpretation of the Alexander polynomial that seems to be new, though it can be proved by unwinding the sheaf-theoretic definitions until one arrives at the cokernel of Alexander's matrix, evaluated at $x = p^n$. We will express this ``unwinding of definitions'' as a brief development of theory in \\S\\ref{sec:two}, and give the proof in \\S\\ref{sec:three}.\n\nA sequel paper will make a more detailed study of the influence of $F$-fields on categories of constructible sheaves and other categories associated to symplectic manifolds. The Proposition perhaps gives some motivation for doing so.\n\nIn contrast with knot complements and most other 3-manifolds, when $M$ is two-dimensional the locally constant sheaves of $\\mathfrak{f}^*\\underline{k}$-modules come in positive-dimensional moduli over the ground field $k$. Local systems over a nontrivial $F$-field are much more stable than traditional local systems, in the GIT sense that their automorphism groups are finite. When $M$ is symplectic, one can study a Fukaya category whose objects are Lagrangian submanifolds $L \\subset M$ equipped with locally constant sheaves of $(\\mathfrak{f}^*\\underline{k})\\vert_L$-modules. The sequel will develop some examples related to Deligne-Lusztig varieties.\n\n\n\n\n\\subsection{Notation}\n\\label{intro:notation}\nThroughout, $p$ is a prime number. We let $\\mathbf{F}_p$ denote the finite field with $p$ elements, and fix an algebraic closure $k$ of $\\mathbf{F}_p$ once and for all. If $q$ is a power of $p$, we write $\\mathbf{F}_q$ for the subfield of $k$ with $q$ elements, fixed by the $q$th-power automorphism of $k$. \n\n\n\\section{$F$-fields and local systems}\n\\label{sec:two}\n\nFor each prime power $q$ we fix an oriented circle $\\SF{q}$, endowed with a base point. For definiteness, we put $\\SF{q} = \\mathbf{R}\/\\log(q) \\mathbf{Z}$, oriented in the positive direction and with the base point at the coset of $0$. We call these the ``reference circles.''\n\nWe endow each reference circle with a sheaf of rings $\\underline{k} = \\underline{k}_{\\SF{q}}$. It is locally constant with fiber $k$ (as in \\S\\ref{intro:notation}), and the monodromy from $0$ to $\\log(q)$ is given by $a \\mapsto a^q$. More formally, define the \\'etal\\'e space of $\\underline{k}$ to be the quotient of $\\mathbf{R} \\times k$ by the equivalence relation\n\\begin{equation}\n\\label{eq:espace-etale}\n\\underline{k} = (\\mathbf{R} \\times k) \/ \\sim \\qquad (t,x) \\sim (t + \\log(q),x^q)\n\\end{equation}\n\n\n\\subsection{$F$-fields}\n\\label{subsec:2.1}\nWe define an \\emph{$F$-field} of characteristic $p$ on a space $X$ to be a continuous map $\\mathfrak{f}_X:X \\to \\SF{p}$. When $q = p^\\nu$ is a power of $p$ we have a canonical $F$-field $\\SF{q} \\to \\SF{p}$ sending $t+ \\log(q) \\mathbf{Z}$ to $t + \\log(p)\\mathbf{Z}$.\n\n\nWhen $X$ carries an $F$-field $\\mathfrak{f}:X \\to \\SF{p}$, it carries a locally constant sheaf of rings $\\mathfrak{f}^* \\underline{k}$ as well, the pullback of \\eqref{eq:espace-etale} along $\\mathfrak{f}$. If the $F$-field is clear from context we will sometimes write $\\underline{k}_X$ or just $\\underline{k}$ in place of $\\mathfrak{f}^* \\underline{k}$. \nWe write $\\mathrm{Loc}(X,\\mathfrak{f}^* \\underline{k})$ for the category of locally free sheaves of $\\mathfrak{f}^* \\underline{k}$-modules on $X$. It is an abelian category but it is not usually $k$-linear. For example, $\\mathrm{Loc}(\\SF{q},\\underline{k})$ is equivalent to the category of finite-dimensional $\\mathbf{F}_q$-modules. The equivalence is given by the global sections functor $L \\mapsto \\Gamma(\\SF{q},L)$, whose $\\mathbf{F}_q$-module structure comes from $\\Gamma(\\SF{q},\\underline{k}) \\cong \\mathbf{F}_q$.\n\n\n\n\n\n\\subsection{Variants of $\\underline{k}$}\nAny automorphism at all can be repurposed as a locally constant sheaf on a circle; e.g. we could replace $k$ with any perfect ring $R$ along with its $p$th power automorphism. If we let $\\underline{R}$ denote the corresponding sheaf of rings on $\\SF{p}$, then studying $\\mathrm{Loc}_n(X,\\mathfrak{f}^* \\underline{R})$ as $R$ runs through perfect $k$-algebras gives a moduli functor that is represented by a perfect Deligne-Mumford stack. We will study these and similar moduli problems in the sequel paper --- in the case of a knot complement these moduli stacks are zero-dimensional. Here are three other significant examples of a different flavor:\n\n\\subsubsection{Isocrystals} \nLet $\\mathbf{Q}_{p^{\\infty}}$ be the maximal unramified extension of $\\mathbf{Q}_p$, whose residue field is $k$. The Frobenius automorphism of $k$ lifts to a field automorphism of $\\mathbf{Q}_{p^{\\infty}}$ that we denote by $\\sigma$. Then $\\mathbf{Q}_{p^{\\infty}}$ is the fiber of a sheaf of rings on $\\SF{p}$ whose monodromy is $\\sigma$, we denote it by $\\underline{\\mathbf{Q}_{p^{\\infty}}}$. A locally constant sheaf of $\\underline{\\mathbf{Q}_{p^{\\infty}}}$-modules is the same data as an isocrystal, studied in \\cite{Dieudonne}. \n\n\\subsubsection{The Tate motive}\n\\label{subsec:Tate}\nThe action of $\\pi_1(\\SF{p})$ on the multiplicative group $k^*$ of $k$ corresponds to a local system of abelian groups that we denote by $\\underline{k^*}$. There is a closely related local system of $\\mathbf{Z}[1\/p]$-modules, which we denote by $\\mathbf{Z}[1\/p](1)$ --- the fiber is $\\mathbf{Z}[1\/p]$ and the generator of $\\pi_1(\\SF{p})$ acts by multiplication by $p$. The two sheaves are related by \n\\[\n\\text{(sheaf hom)}(\\mathbf{Z}[1\/p](1),\\text{const. sheaf with fiber }k^*) \\cong \\underline{k^*}\n\\]\n\n\\subsubsection{Finite Chevalley groups}\n\\label{subsec:Chevalley}\nIf $\\mathbf{G}$ is an algebraic group over $k$, and $\\sigma:\\mathbf{G} \\to \\mathbf{G}$ is the Frobenius isogeny coming from an $\\mathbf{F}_q$-rational structure on $\\mathbf{G}$, then $\\sigma$ induces an automorphism on $k$-points $\\mathbf{G}(k) \\to \\mathbf{G}(k)$. We denote the corresponding locally constant sheaf of groups on $\\SF{q}$ by $\\underline{(\\mathbf{G}(k),\\sigma)}$. \nThe groupoid of $\\underline{(\\mathbf{G}(k),\\sigma)}$-torsors over $\\SF{q}$ is equivalent to the classifying groupoid of torsors over the finite group $\\mathbf{G}(k)^{\\sigma}$.\n\nThere is a similar construction for the Suzuki or Ree isogenies of $\\mathbf{G} = \\mathrm{Sp}_4$ when $p = 2$ and $\\mathbf{G} = \\mathrm{G}_2$ when $p = 3$ --- the square of one of these isogenies is a Frobenius map for an $\\mathbf{F}_q$-rational structure. Thus they again induce bijections on $k$-points and it is natural to regard $\\underline{(\\mathbf{G},\\sigma)}$ as a sheaf of groups on a reference circle of circumference $\\log(\\sqrt{q})$.\n\n\\subsection{Framings} \nWe write $\\mathrm{Loc}_n(X,\\mathfrak{f}^* \\underline{k})$ for the groupoid of locally free sheaves of rank $n$ in $\\mathrm{Loc}(X,\\mathfrak{f}^*\\underline{k})^{\\simeq}$ and all isomorphisms between them. There are two ways to present the groupoid as a quotient:\n\\subsubsection{Point framings} Let $x \\in X$ be a base point and let $E \\in \\mathrm{Loc}_n(X,\\mathfrak{f}^*\\underline{k})$. A \\emph{point framing} of $E$ at $x$ is a $(\\mathfrak{f}^* \\underline{k})_x$-basis in $E_x$. Note $(\\mathfrak{f}^* \\underline{k})_x$ is canonically identified with $\\underline{k}_{\\mathfrak{f}(x)}$. The set of isomorphism classes of point-framed local systems of rank $n$ is in bijection with the set of homomorphisms\n\\begin{equation}\n\\label{eq:point-framing}\n\\rho:\\pi_1(X,x) \\to \\mathrm{GL}_n(\\underline{k}_{\\mathfrak{f}(x)}) \\rtimes \\pi_1(\\SF{p}) \n\\end{equation}\nthat commute with the projections to $\\pi_1(\\SF{p})$. In the semidirect product in \\eqref{eq:point-framing}, the distinguished generator of $\\pi_1(\\SF{p})$ acts on $\\mathrm{GL}_n(\\underline{k}_{\\mathfrak{f}(x)})$ by raising each matrix entry to the $p$th power.\n\nIf we write $\\rho(\\gamma) = \\rho_1(\\gamma) \\rtimes \\mathfrak{f}(\\gamma)$, the bijection sends $\\rho$ to the locally constant sheaf of abelian groups whose fiber above $x$ is $\\underline{k}_{X,x}^n$, and whose monodromy around the loop $\\gamma$ acts on the vector $(v_1,\\ldots,v_n)$ by\n\\[\n\\rho_1(\\gamma)(v_1^{p^{\\mathfrak{f}(\\gamma)}},\\ldots,v_n^{p^{\\mathfrak{f}(\\gamma)}})\n\\]\nThe groupoid $\\mathrm{Loc}_n(X,\\mathfrak{f}^*\\underline{k})$ is equivalent to the quotient of the set of \\eqref{eq:point-framing} by the $\\mathrm{GL}_n(\\underline{k}_{\\mathfrak{f}(x)})$-conjugation action.\n\n\\subsubsection{Section framings}\nWith $q = p^\\nu$, we shall call the data of a map $s:\\SF{q} \\to X$ commuting with the projections to $\\SF{p}$ a \\emph{$\\nu$-sheeted multisection} of the $F$-field $\\mathfrak{f}$. Each $E \\in \\mathrm{Loc}_n(X,\\mathfrak{f}^*\\underline{k})$ restricts to a locally constant sheaf on such a multisection, or equivalently an $n$-dimensional $\\mathbf{F}_q$-vector space, $\\Gamma(s,E)$. A \\emph{section framing} of $E$ at $x$ is an $\\mathbf{F}_q$-basis for this vector space.\n\nA section framing induces a point framing, at the image of the base point of $\\SF{q}$ under $s$. Using $g$ to denote the distinguished generator of $\\pi_1(\\SF{p})$, we say that homomorphism \\eqref{eq:point-framing} preserves the section-framing if it carries $s$ to $1 \\rtimes g^{\\nu}$. These homomorphisms are in bijection with the set of section-framed local systems. A change of basis in $\\Gamma(s,E)$ corresponds to conjugating $\\rho$ by an element in the finite group $\\mathrm{GL}_n(\\underline{k}_{\\mathbf{F}_{p^\\nu}}) \\subset \\mathrm{GL}_n(\\underline{k}_{\\mathfrak{f}(x)})$. In particular this shows that $\\mathrm{Loc}_n(X,\\mathfrak{f}^*\\underline{k})$ has finite isotropy groups whenever $X$ is connected and the $F$-field is nontrivial (i.e. whenever $\\pi_1(X) \\to \\pi_1(\\SF{p})$ is nonzero).\n\n\\subsection{Invertible modules}\n\nWhen $n = 1$, the homomorphisms \\eqref{eq:point-framing} do not necessarily factor through an abelian quotient of $\\pi_1(X,x)$. Nevertheless the tensor product over $\\mathfrak{f}^*\\underline{k}$ endows the set of isomorphism classes in $\\mathrm{Loc}_1(X,\\mathfrak{f}^*\\underline{k})$ with the structure of commutative group. The groupoid $\\mathrm{Loc}_1(X,\\mathfrak{f}^*\\underline{k})$ has a symmetric monoidal structure --- it is a commutative $2$-group. \n\nIndeed, if we regard the abelian group $\\mathrm{GL}_1(\\underline{k}_{\\mathfrak{f}(x)})$ as a $\\pi_1(X,x)$-module through the homomorphism $\\pi_1(X,x) \\to \\pi_1(\\SF{p})$, and $\\rho_1:\\pi_1(X,x) \\to \\mathrm{GL}_1(\\underline{k}_{\\mathfrak{f}(x)})$ is a $1$-cocycle on $\\pi_1(X,x)$ with coefficients in this module, then $\\rho_1 \\rtimes \\mathfrak{f}$ is a homomorphism of the form \\eqref{eq:point-framing}. This is a bijection between such cocycles and point-framed local systems. If we write $Z^1 := Z^1(\\pi_1(X,x),\\mathrm{GL}_1(\\underline{k}_{\\mathfrak{f}(x)}))$ for this group of cocycles, the commutative $2$-group structure on $\\mathrm{Loc}_1(X,\\mathfrak{f}^*\\underline{k})$ can be encoded by the two-term chain complex\n\\begin{equation}\n\\label{eq:GL1Z1}\n\\mathrm{GL}_1(\\underline{k}_{\\mathfrak{f}(x)}) \\to Z^1\n\\end{equation}\nwhere the differential is the usual differential in group cohomology. If $\\pi_1(X,x) \\to \\pi_1(\\SF{p})$ is nonzero, the kernel is a finite group (it is $\\mathrm{GL}_1$ of a finite subfield of $\\underline{k}_{\\mathfrak{f}(x)}$).\n\n\n\\section{Proof of the Proposition}\n\\label{sec:three}\n\nFix a diagram $D$ for a knot with $v$ crossing points and $v+2$ regions. Let us label the regions $0,\\ldots,v+1$, and orient $K$. Then there is a Dehn presentation of $\\pi_1(S^3 - K)$ with $v+1$ generators and $v$ relations \\cite[\\S 8]{Alexander}. The generators $g_i$ correspond to regions $1,\\ldots,v+1$ of $D$, and we put $g_0 = 1$. Each crossing gives a relation between the generators associated to the four regions incident with it, as in the following diagram,\n\\begin{center}\n\\begin{tikzpicture}\n\\node at (-3,0) {$g_j g_k^{-1} g_{\\ell} g_m^{-1} = 1$};\n\\draw[thick] (-1,0)--(1,0);\n\\draw[thick] (0,-1)--(0,-.2);\n\\draw[thick, ->] (0,.2)--(0,1);\n\\node at (-.25,-.25) {$\\cdot$};\n\\node at (-.25,.25) {$\\cdot$}; \n\\node at (.5,.5) {$m$};\n\\node at (-.5,.5) {$j$};\n\\node at (-.5,-.5) {$k$};\n\\node at (.5,-.5) {$\\ell$};\n\\end{tikzpicture}\n\\end{center}\n(In \\cite{Alexander}, knot diagrams are drawn with two dots on the left side of the underpass crossing --- we have put them in the diagram above as well.)\n\nThe ``index'' of a region defined by Alexander \\cite[Fig. 2]{Alexander} gives a homomorphism \n\\[\nI:\\pi_1(S^3 - K) \\to \\mathbf{Z}\n\\] \nA choice of prime $p$ and prime power $q = p^\\nu$ turns the index homomorphism into an $F$-field $\\mathfrak{f}$, by identifying it with a homotopy class of maps $S^3 - K \\to \\SF{q}$. A rank one point-framed local system of $\\mathfrak{f}^* \\underline{k}$-modules on $S^3 -K$ is completely specified by a family of scalars $z_j \\in \\mathrm{GL}_1(k)$, one for each region. In terms of these scalars, the action of $g_j$ (and its inverse, recorded for convenience) on $x \\in k$ is given by\n\\[\ng_j(x) = x^{q^{I(j)}} z_j \\qquad \\text{(and }g_j^{-1}(x) = x^{q^{-I(j)}} z_j^{-q^{-I(j)}} \\text{)}\n\\]\nThe scalars $z_j$ must obey $z_0 = 1$ and an additional relation for each crossing indicident with regions $j,k,\\ell,m$ as above, which reduces to \n\\[\nz_j z_k^{-q} z_{\\ell}^{q} z_m^{-1} = 1 \\qquad \\text{ or } \\qquad z_j z_k^{-q^{-1}} z_{\\ell}^{q^{-1}} z_m^{-1} = 1\n\\]\naccording to whether the crossing is left-handed or right-handed, respectively. \nAn element $y \\in \\mathrm{GL}_1(k)$ acts on the point-framing by sending $(z_j)_{j = 1}^{v+1}$ to $(y^{I(j) - 1} z_j)_{j = 1}^{v+1}$. Thus, the group of isomorphism classes of objects in $\\mathrm{Loc}_1(X,\\mathfrak{f}^* \\underline{k})$ is isomorphic to the middle cohomology of the chain complex\n\\begin{equation}\n\\label{eq:vacuum}\n\\mathrm{GL}_1(k) \\to \\mathrm{GL}_1(k)^{v+1} \\to \\mathrm{GL}_1(k)^{v}\n\\end{equation}\nwhere the first and second differentials are\n$\ny \\mapsto (y^{q^{I(j)} - 1})_{j = 1}^{v+1}$ and $(z_j)_{j = 1}^{v+1} \\mapsto (z_j z_k^{-q^{\\pm 1}} z_{\\ell}^{q^{\\pm 1}} z_m^{-1})_c$. \n\nConsider the $v \\times (v+1)$-matrix whose rows are indexed by the crossings, whose columns are indexed by non-null regions, and whose $(c,j)$-entry is indicated by the following diagram if $j$ is incident with $c$, and is otherwise $0$:\n\\begin{center}\n\\begin{tikzpicture}\n\\draw[thick, ->] (-1,0)--(1,0);\n\\draw[thick] (0,-1)--(0,-.2);\n\\draw[thick, ->] (0,.2)--(0,1);\n\\node at (-.25,-.25) {$\\cdot$};\n\\node at (-.25,.25) {$\\cdot$}; \n\\node at (.5,.6) {$-1$};\n\\node at (-.5,.6) {$1$};\n\\node at (-.7,-.6) {$-x$};\n\\node at (.65,-.6) {$x$};\n\\end{tikzpicture}\n\\qquad\n\\begin{tikzpicture}\n\\draw[thick,<-] (-1,0)--(1,0);\n\\draw[thick] (0,-1)--(0,-.2);\n\\draw[thick, ->] (0,.2)--(0,1);\n\\node at (-.25,-.25) {$\\cdot$};\n\\node at (-.25,.25) {$\\cdot$}; \n\\node at (.5,.6) {$-1$};\n\\node at (-.5,.6) {$1$};\n\\node at (-.75,-.6) {$-x^{-1}$};\n\\node at (.75,-.6) {$x^{-1}$};\n\\end{tikzpicture}\n\\end{center}\nLet $D'$ denote the diagram obtained from $D$ by switching the sense of over and under at every crossing. If we multiply each row corrsponding to a right-hand crossing by $x$, we obtain the usual Alexander matrix (i.e. the coefficient matrix of the system of equations \\cite[Eq. 3.3]{Alexander}) for the diagram $D'$, with the column corresponding to the null region left off. In particular $A(x)$ is elementary equivalent to the usual Alexander matrix for both $D'$ and (using the mirror invariance of the Alexander invariants) $D$.\n\nBy evaluating $A(x)$ at $x = q = p^\\nu$, we obtain a matrix $A(q) \\in \\mathbf{Z}[1\/p]^{v \\times (v+1)}$. Then \\eqref{eq:vacuum} is obtained by taking $\\mathrm{Hom}(C,\\mathrm{GL}_1(k))$, where $C$ is a complex of free $\\mathbf{Z}[1\/p]$-modules of the form\n\\[\n\\mathbf{Z}[1\/p]^v \\xrightarrow{A(q)} \\mathbf{Z}[1\/p]^{v+1} \\xrightarrow{} \\mathbf{Z}[1\/p]\n\\]\n\nIn particular, the order of the middle cohomology of \\eqref{eq:vacuum} is equal to the order of the middle cohomology of this complex, which is equal to the order of the torsion subgroup of the cokernel of $A(q)$. This in turn is equal to the prime-to-$p$ part of the greatest common divisor of all the $v \\times v$-minors of $A(q)$. If $p$ does not divide the constant term of $\\Delta_K(x)$, that divisor is just $\\Delta_K(q)$.\n\n\n\\section{Plausible generalizations}\nIt is natural to consider the size of $\\mathrm{Loc}_1(M;\\mathfrak{f}^*\\underline{k})$ ``as an orbifold'', i.e. to weight each isomorphism class by the reciprocal of the order of its automorphism group. In the case of a knot complement these automorphism groups are not very sensitive to $\\mathfrak{f}$ so that the orbifold count is $\\Delta_K(q)\/(q-1)$. In the case of a more general $3$-manifold both the kernel and cokernel of \\eqref{eq:GL1Z1} are more irregular as $\\mathfrak{f}$ varies, but the orbifold count is likely to have a clean relation to the multivariable Alexander polynomial.\n\nOne can study local systems of modules over $\\mathfrak{f}^* \\underline{\\mathbf{Q}_{p^{\\infty}}}$. By restricting such a local system $L$ to a meridian of the knot, one gets a rank one isocrystal whose slope is a discrete invariant of $L$. The set of $L$ of a fixed slope make a $p$-adic analytic manifold (in fact a torsor for a commutative $p$-adic analytic group) whose dimension over $\\mathbf{Q}_{p^n}$ is the degree of the Alexander polynomial. I suspect that these $p$-adic manifolds carry natural measures, perhaps up to powers of $p$, of volume $\\Delta_K(p^n)$.\n\nOne might obtain interesting invariants by counting nonabelian local systems of $\\mathfrak{f}^* \\underline{k}$-modules, or more generally torsors for the sheaves of groups $\\mathfrak{f}^* (\\underline{\\mathbf{G}},\\sigma)$. The problem of doing so can be expressed as the problem of solving high-degree equations in the entries of a matrix in $\\mathbf{G}(k)$. For example (using the Wirtinger presentation of $\\pi_1(S^3 - K)$ in place of the Dehn presentation) a $\\mathfrak{f}^*\\underline{(\\mathbf{G},\\sigma)}$-torsor on the complement of a trefoil is determined by an element $g \\in \\mathbf{G}(k)$ subject to the equation\n\\[\ng \\sigma^2(g) = \\sigma(g)\n\\] \nand taken up to the conjugation action by the finite group $\\mathbf{G}^{\\sigma}$. But even in the simplest examples I have not been able to solve these equations directly when $\\mathbf{G}$ is not commutative.\n\n\n\\subsection*{Acknowledgments}\nI am grateful to the Institute for Advanced Study where this paper was written. My stay at the IAS was supported by a Sloan fellowship, a von Neumann fellowship, and a Boston College faculty fellowship, and I was also supported by NSF-DMS-1510444.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\nThrough the paper, let $G$ be a finite abelian group, written additively, $p$ be the smallest prime dividing $|G|$\nand $\\mathsf r(G)$ denote the rank of $G$. Let $S$ be a sequence over $G$. We say that $S$ is an {\\sl additive\nbasis} of $G$ if every element of $G$ can be expressed as the sum over a nonempty subsequence of $S$. For every\nsubgroup $H$ of $G$, let $S_H$ denote the subsequence of $S$ consisting of all terms of $S$ contained in $H$. We\nsay that $S$ is a regular sequence over $G$ if $|S_H|\\leq |H|-1$ holds for every subgroup $H \\subsetneq G$. Let\n$\\mathsf c_0(G)$ denote the smallest integer $t$ such that every regular sequence over $G$ of length at least $t$\nis an additive basis of $G$. The problem of determining $\\mathsf c_0(G)$ was first proposed by Olson and it was\nconjectured that $\\mathsf c_0(C_p\\oplus C_p)=2p-1$. In 1987, Peng proved this conjecture and further determined\n$\\mathsf c_0(G)$ for all the finite elementary abelian $p$-groups (\\cite{Peng1,Peng2}). Recently, the problem\nrelated to the additive basis of a finite abelian group has been investigated by several authors (\\cite{GHQQZ, GQZ,\nQH1, QH2}). In particular, $\\mathsf c_0(G)$ has been determined for any of the following finite abelian groups:\n\\begin{enumerate}\n\\item $G$ is cyclic;\n\n\\item $|G|$ is even;\n\n\\item $\\mathsf r(G)\\geq 4$ and $G\\neq C_3^3\\oplus C_{3n}$ where $n>3$ is odd and is not a power of $3$ with $|G|<\n 3.72\\times 10^7$ ;\n\n\\item $\\mathsf r(G)=3$ and either $p\\geq 11$ or $3\\leq p \\leq 7$ with $|G| \\geq 3.72\\times 10^7$;\n\n\\item $\\mathsf r(G)\\geq 2$ and $G$ is a $p$-group.\n\n\\end{enumerate}\n\n\n\n In this paper, we focus our investigation on the remaining case when $G$ is of rank $2$. It was conjectured in\n \\cite{QH2} that $\\mathsf c_0(G)=pn+2p-3$ where $G=C_p\\oplus C_{pn}$ with $n\\geq 3$. We remark that the existing\n methods used to compute this invariant for groups of rank greater than $2$ cannot be applied directly to calculate\n $\\mathsf c_0(G)$ when $G$ is a group of rank 2. We adopt a new method (i.e., use group algebras as a tool) and we\n are able to confirm the above conjecture for the case when $p=3$ and $n=q \\,(\\geq 5)$ is a prime.\n\n\n\\begin{theorem} \\label{mainthm} Let $G=C_3\\oplus C_{3q}$ be a finite abelian group with a prime $q\\geq 5$. Then\n$\\mathsf c_0(G)=3q+3$.\n\\end{theorem}\n\n\\section{Notations and Preliminaries}\nSuppose that $G_0\\subseteq G$ is a subset of $G$ and $\\mathcal{F}(G_0)$ is the multiplicatively written, free\nabelian monoid with basis $G_0$. The elements of $\\mathcal{F}(G_0)$ are called {\\it sequences} over\n$G_0$. A sequence $S$ over $G_0$ will be written in the form\n$$\nS = g_1 \\cdot \\ldots \\cdot g_{\\ell}=\\Pi_{i\\in [1,\\ell]}g_i, $$\nwhere $g_i \\in G_0$ for all $1\\leq i\\leq {\\ell}$. We say $T=\\Pi_{i\\in I}g_i$ a subsequence of $S$ and denote by\n$T|S$, where $I\\subseteq [1, \\ell]$. For a subsequence $T|S$, let $I_T=\\{i\\in [1, \\ell]~|~g_i|T\\}$. We set $T=1$ if\n$I_T=\\emptyset$. We call\n\\begin{eqnarray*}\n|S|&=&\\ell \\in \\mathbb N_0 \\quad \\text{the {\\it length} of } \\ S,\\\\\n\\sigma (S)&=&\\sum_{i=1}^{\\ell}g_i\\in G\\quad \\text{the {\\it sum} of} \\ S.\n\\end{eqnarray*}\nDefine\n $$\n \\sum(S)=\\{\\sigma(T): \\ 1 \\neq T\\mid S\\},\n $$\n\\noindent and $$\n \\sum\\nolimits_0(S)=\\sum(S)\\cup\\{0\\}.\n $$\n\\noindent We call a sequence $S$ a zero-sum sequence if $\\sigma(S)=0$, and a zero-sumfree sequence if $0\\notin\n\\sum(S)$.\n\nLet $\\mathsf D(G)$ denote the Davenport constant of $G$, which is defined as the smallest integer $t$ such that\nevery sequence $S$ over $G$ of length $|S|\\geq t$ contains a nonempty zero-sum subsequence. Let $\\mathsf d(G)$\ndenote the maximal length of a zero-sumfree sequence over $G$. Then $\\mathsf d(G)=\\mathsf D(G)-1$. The exact value\nof $\\mathsf D(G)$ has been determined only for a few classes of groups, such as finite abelian $p$-groups, abelian\ngroups rank not exceeding $2$, and certain very special abelian groups of rank $3$ (\\cite{GL}). Notice that\n$\\mathsf D(C_{n_1}\\oplus C_{n_2})=n_1+n_2-1$, where $1\\leq n_1|n_2$ (\\cite[Theorem 5.8.5]{GeK}).\n\n\nFor each subset $A$ of $G$, denote by $\\langle A \\rangle$ the subgroup generated by $A$. Let ${\\rm st}(A)=\\{g\\in\nG: g+A=A\\}$. Then ${\\rm st}(A)$ is the maximal subgroup $H$ of $G$ such that $H+A=A$. The following is the well\nknown Kneser's theorem and a proof of it can be found in \\cite{Na}.\n\n\n\\begin{lemma}(Kneser) \\cite[Theorem 4.4]{Na}\\label{Kneser} Let $A_1, \\ldots, A_r$ be nonempty finite subsets of an\nabelian group $G$, and let $H={\\rm st}(A_1+\\cdots +A_r)$. Then,\n$$\n|A_1+\\cdots +A_r|\\geq |A_1+H|+\\cdots +|A_r+H|-(r-1)|H|.\n$$\n\\end{lemma}\n\nWe note that if $H=G$ then $A_1+\\cdots +A_r=G$. So when using the above Kneser's Theorem to prove $A_1+\\cdots\n+A_r=G$, we need only consider the case when $H\\neq G$.\n\nThe following three lemmas provide some results concerning the additive basis and its inverse problem, which will be\nneeded in sequel.\n\n\\begin{lemma}\\cite[Theorem 1.1]{QH1}\\label{cyclic}\nLet $G$ be a cyclic group with order $n$, and let $S$ be a regular sequence of length $|S|=n-1$ over $G$. If\n$\\sum(S)\\neq G$, then $S=g^{n-1}$ where $g$ is a generator of $G$.\n\\end{lemma}\n\n\\begin{lemma}\\cite[Lemma 2.3(1)]{GHQQZ}\\label{st=0}\nLet $G$ be a finite abelian group, and let $p$ be the smallest prime dividing $|G|$. Let $S$ be a regular sequence over $G$ of length $|S|\\geq \\mbox{max}\\{\\frac{|G|}{p}+p-2, \\mathsf D(G)\\}$. If $\\sum(S)\\neq G$, then $\\mbox{st}(\\sum(S))=\\{0\\}$.\n\\end{lemma}\n\n\n\\begin{lemma}\\cite[Theorem 2]{Peng1}\\cite[Lemma 3.12]{GPZ2013}\\label{rank2}\nLet $G=C_p\\oplus C_p$ and $S$ be a regular sequence over $G$. Then $\\mathsf c_0(G)=2p-1$. Moreover, if $|S|=2p-2$,\nthen $|\\sum_0(S)|\\geq |G|-1$.\n\\end{lemma}\n\nFor a field $\\mathbf{F}$, let $\\mathbf{F}G$ denote the group algebra of $G$ over $\\mathbf{F}$ and $d(G, \\mathbf{F})$ be the largest integer $\\ell\\in \\mathbb{N}$ having the following\nproperty:\n\nThere exists some sequence $S=g_1\\cdot \\ldots \\cdot g_{\\ell}$ over $G$ of length $\\ell$ such that\n$$(X^{g_1}-a_1)\\cdot \\ldots \\cdot (X^{g_{\\ell}}-a_{\\ell})\\neq 0 \\in \\mathbf{F}G \\mbox{ for all }\na_1,\\ldots,a_{\\ell}\\in \\mathbf{F}^{\\times}.$$\n\n\\begin{lemma}\\cite[Theorem 3.3]{GL}\\cite[Theorem 1.1]{S}\\label{d(G,F)}\nLet $G$ be a finite abelian group and $\\mathbf{F}$ be a splitting field of $G$. Then,\n\\begin{enumerate}\n\n\\item if $G=C_2\\oplus C_{2n}$, then $d(G, \\mathbf{F})=d(G)=2n$;\n\n\\item if $G=C_3\\oplus C_{3n}$, then $d(G, \\mathbf{F})=d(G)=3n+1$.\n\\end{enumerate}\n\\end{lemma}\n\n\nFor any $\\alpha\\in \\mathbf{F}G$, by $L_{\\alpha}$ we denote the set of elements $g\\in G$ such that $\\alpha\n(X^g-a)=0$ holds for some $a\\in \\mathbf{F}^{\\times}$. We note that the statement in the following lemma is slightly\nmore general than that in \\cite[Lemma 5]{QH2}; however, the same proof carries over.\n\n\\begin{lemma}\\cite[Lemma 5]{QH2}\\label{coset}\nLet $G$ be a finite abelian group and $S=g_1\\cdot \\ldots \\cdot g_\\ell$ be a sequence over $G$. Suppose that\n$\\alpha=(X^{g_1}-a_1)\\cdot \\ldots \\cdot (X^{g_{\\ell}}-a_{\\ell})\\neq0$ for some $a_1,\\ldots,a_{\\ell}\\in\n\\mathbf{F}^{\\times}$ and $H=L_{\\alpha}$, then $\\sum(S)\\supseteq (g_0+H)\\setminus\\{0\\}$ for some $g_0\\in G$.\nMoreover, $\\sum(Sh)\\supseteq g_0+H$ for any $h\\in H$.\n\\end{lemma}\n\n\n\n\n\n\\section{Proof of Theorem \\ref{mainthm}}\nLet $G=C_3\\oplus C_{3q}=H\\oplus K$, where $H\\cong C_3\\oplus C_3$, $K\\cong C_q$ and $q \\geq 5$ is a prime. Let\n$S={0\\choose1}^{3q-2}{1\\choose-1}^{4}$ be a sequence over $G$ with length $3q+2$. Then $S$ is regular and\n${2\\choose 3q-3}\\notin\\sum(S)$. It follows from this example that $\\mathsf c_0(C_3\\oplus C_{3q})\\geq 3q+3$. To show\nthe equality holds it is sufficient to prove every regular sequence over $G$ with length $3q+3$ forms an additive\nbasis of $G$. Let $S=g_1\\cdot \\ldots \\cdot g_\\ell$ be a regular sequence over $G$ with length $\\ell=3q+3$,\n$S_H=g_1\\cdot \\ldots \\cdot g_t$ and $S_K=g_{t+1}\\cdot \\ldots \\cdot g_{t+r}$. We first prove the following crucial\nlemma, which provides some sufficient conditions for a regular sequence $S$ of length $3q+3$ to be an additive basis.\n\n\\begin{lemma}\\label{basic}\nLet $G=C_3\\oplus C_{3q}$, and $S$ be a regular sequence over $G$ with length $\\ell=3q+3$, where $q \\geq 5$ is a\nprime. Then $\\sum(S)=G$ if any of the following conditions holds:\n\\begin{itemize}\n\\item[(i)] There exists a nontrivial subgroup $H'$ of $G$ such that $\\sum_0(S_{H'})=H'$;\n\n\\item[(ii)] There exist a subsequence $S'|S$ and a nontrivial subgroup $H'$ of $G$ such that $\\sum_0(S')\\supseteq\n a+H'$ for some $a\\in G$ and $(\\ell-|I_{S_M}\\cup I_{S'}|+1)|M|\\geq 9q $ where\n $M=st((a+H')+\\sum_0(SS'^{-1}))$;\n\n\\item[(iii)] There exist some $b_i\\in \\mathbf{F}^{\\times}$ for all $i\\in I_{S_H}\\cup I_{S_K}$ such that\n $\\Pi_{i\\in I_{S_{H}}\\cup I_{S_K}}(X^{g_i}-b_i)=0$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n(i) Since $\\sum_0(S_{H'})=H'$ for some nontrivial subgroup $H'$ of $G$ by assumption, we have $\\sum(S)+H'= \\sum(S)$, so $st(\\sum(S))\\supseteq H'\\neq \\{0\\}$. Since $S$ is a regular sequence and $|S|\\geq \\mbox{max}\\{\\frac{|G|}{3}+3-2=3q+1, \\mathsf D(G)=3q+2\\}$, by Lemma~\\ref{st=0} we have $\\sum(S)=G$.\n\n(ii) Let $A=(a+H')+\\sum_0(SS'^{-1})$. By assumption, we have $\\sum(S)\\supseteq A$ and $M\\supseteq H'$. By\nLemma~\\ref{Kneser}, we have\n\\begin{align*}\n|\\sum(S)|&\\geq |a+H'+M|+\\sum_{i\\in I_S\\setminus (I_{S_M}\\cup I_{S'})}|\\{0, g_i\\}+M|-(\\ell-|I_{S_M}\\cup\nI_{S'}|)|M|\\\\\n&\\geq (\\ell-|I_{S_M}\\cup I_{S'}|+1)|M|\\\\\n&\\geq |G|.\n\\end{align*}\nThus $\\sum(S)=G$.\n\n(iii) Recall that $S_H=g_1\\cdot \\ldots \\cdot g_t$ and $S_K=g_{t+1}\\cdot \\ldots \\cdot g_{t+r}$. Since $\\Pi_{i\\in\n[1,t+r]}(X^{g_i}-b_i)=0$ for some $b_i\\in \\mathbf{F}^{\\times}$ and $G=H\\oplus K$, we conclude that either\n$\\Pi_{i\\in [1,t]}(X^{g_i}-b_i)=0$ or $\\Pi_{i\\in [t+1,t+r]}(X^{g_i}-b_i)=0$. Otherwise, if both $\\Pi_{i\\in\n[1,t]}(X^{g_i}-b_i)\\neq 0$ and $\\Pi_{i\\in [t+1,t+r]}(X^{g_i}-b_i)\\neq 0$ hold, then let\n$$\\Pi_{i\\in [1,t]}(X^{g_i}-b_i)=\\sum_{h\\in H}c_hX^h, \\ \\ \\ \\ \\Pi_{i\\in [t+1,t+r]}(X^{g_i}-b_i)=\\sum_{k\\in K}c_kX^k,$$\nand $$\\Pi_{i\\in I_{S_{H}}\\cup I_{S_K}}(X^{g_i}-b_i)=\\sum_{g\\in G}e_gX^g.$$\nWe have $\\Pi_{i\\in I_{S_{H}}\\cup I_{S_K}}(X^{g_i}-b_i)=(\\sum_{h\\in H}c_hX^h)(\\sum_{k\\in K}c_kX^k)=\\sum_{h\\in H, k\\in\nK}c_hc_kX^{h+k}$, and thus $e_g=\\sum_{g=h+k, h\\in H, k\\in K}c_hc_k$. Since $G=H\\oplus K$, for each $g\\in G$,\n$e_g=c_hc_k$ for unique $h\\in H$, $k\\in K$ with $g=h+k$. Since $\\Pi_{i\\in [1,t]}(X^{g_i}-b_i)\\neq 0$ and $\\Pi_{i\\in\n[t+1,t+r]}(X^{g_i}-b_i)\\neq 0$, we have $c_{h_0}\\neq 0$ and $c_{k_0}\\neq 0$ for some $h_0\\in H$ and $k_0\\in K$.\nTherefore, $e_{h_0+k_0}=c_{h_0}c_{k_0}\\neq 0$, yielding a contradiction to the assumption.\n\n\n\nIf $\\Pi_{i\\in [t+1,t+r]}(X^{g_i}-b_i)=0$, then by Lemma~\\ref{coset}, $\\sum_0(S_K)=K$. By~(i), $\\sum(S)=G$.\n\nNext assume that $\\Pi_{i\\in [1,t]}(X^{g_i}-b_i)=0$. Since $S$ is regular, $|S_{H}|\\leq |H|-1=8$. If $|S_{H}|\\geq\n5$, then by Lemma~\\ref{rank2} $\\sum(S_{H})=H$. By (i), $\\sum(S)=G$.\n\nWe now consider the case when $|S_{H}|\\leq 4$. Since $\\Pi_{i\\in [1,t]}(X^{g_i}-b_i)=0$, by Lemma~\\ref{coset},\n$\\sum_0(S_{H})\\supseteq a+N$ for some nontrivial subgroup $N\\subseteq H$. Let $A=(a+N)+\\sum_0(SS_{H}^{-1})$ and\n$st(A)=M$. Then $\\sum(S)\\supseteq A$ and $M\\supseteq N$. If $M=N$, then $M\\subseteq H$. Thus $S_M|S_{H}$.\nTherefore, $(\\ell-|I_{S_M}\\cup I_{S_H}|+1)|M|\\geq(\\ell-3)|M|\\geq 9q$. By~(ii), $\\sum(S)=G$.\n\n\nSuppose that $M\\supsetneq N$. If $M\\supseteq H$, then $S_{H}|S_M$. Since $|S_M|\\leq |M|-1$, we have\n$(\\ell-|I_{S_M}\\cup I_{S_H}|+1)|M|=(\\ell-|I_{S_M}|+1)|M| \\geq(3q-|M|+5)|M|\\geq 9q$. By~(ii), $\\sum(S)=G$.\n\n\nIf $M\\nsupseteq H$, then $M\\cong C_{3q}$. If $|S_M|\\geq |M|-1$, then by Lemma~\\ref{cyclic} $\\sum_0(S_M)=M$. Thus\nby~(i), $\\sum(S)=G$ and we are done. So we may assume that $|S_M|\\leq |M|-2$. If there exists a subgroup $N_0$ of\norder $3$ such that $|S_{N_0}|\\geq 2$, then $|S_{N_0}|=2$ and $\\sum_0(S_{N_0})=N_0$. By~(i), $\\sum(S)=G$. Next we\nmay always assume $|S_{N_0}|\\leq 1$ for every subgroup $N_0$ of order $3$. Then $|I_{S_H}\\setminus I_{S_{H'}}|\\leq\n3$ where $H'$ is a cyclic subgroup with order $3$ or $3q$. In particular, $|I_{S_{H}}\\setminus I_{S_M}|\\leq 3$.\nThus $(\\ell-|I_{S_M}\\cup I_{S_{H}}|+1)|M| \\geq(\\ell-|I_{S_M}|-2)|M| \\geq(3q-|M|+3)|M|\\geq 9q$. Hence by~(ii), again\nwe obtain $\\sum(S)=G$. This completes the proof. \\end{proof}\n\nWe are now in position to give a proof for the main result.\\\\\n\n\\noindent {\\bf Proof of Theorem \\ref{mainthm}.}\\\\\n\nLet $S$ be a regular sequence over $G$ with length $3q+3$. We need to show that $\\sum(S)=G$. Assume to the contrary\nthat $\\sum(S)\\neq G$. By Lemma~\\ref{basic}~(iii), we can find a subsequence $S_1|S$ with maximal length such that\n$S_HS_K|S_1$ and $\\Pi_{i\\in I_{S_1}}(X^{g_i}-a_i)\\neq 0$ for all $a_i\\in \\mathbf{F}^{\\times}$ where $i\\in I_{S_1}$ and $\\mathbf{F}$ is a splitting field of $G$.\nNotice that every element of $SS_1^{-1}$ has order $3q$. Without loss of generality, we may assume that\n$S_1=\\Pi_{i=1}^{m}g_i$ where $m\\in[t+r, \\ell]$. We distinguish the proof into the following two cases:\\\\\n\n\\noindent {\\bf Case 1.} $m\\geq 3q+1$.\\\\\n\nBy Lemma~\\ref{d(G,F)}, $\\mathsf d(G,\\mathbf{F})=3q+1$, so $m=3q+1$. Then for any $g\\in G$, there exists a subsequence $S_g|S_1$ with minimal length such that $(X^g-b_g)\\Pi_{i\\in I_{S_g}}(X^{g_i}-b_i)=0$ for some $b_g, b_i\\in \\mathbf{F}^{\\times}$ where $i\\in I_{S_g}$. By Lemma \\ref{coset}, $\\sum_0(S_1)$ contains a coset of $\\langle\ng\\rangle$. Since there are exactly $4$ distinct subgroups of $G$ of order $3q$, and both $g_{3q+2}$ and $ g_{3q+3}$\nhave order of $3q$, we can find a subgroup $\\langle g_0\\rangle$ of order $3q$ such that $g_{3q+2}, g_{3q+3}\\notin\n\\langle g_0\\rangle$. Thus, $(\\ell-|I_{S_{\\langle g_0\\rangle}}\\cup I_{S_1}|+1)|M|\\geq 3|M|\\geq 9q$. By\nLemma~\\ref{basic}~(ii), $\\sum(S)=G$, yielding a contradiction.\\\\\n\n\\noindent {\\bf Case 2.} $m\\leq 3q$.\\\\\n\nFor any $g|SS_1^{-1}$, there exists a subsequence $S_g|S_1$ with minimal length such that $S_HS_K|S_g$ and\n$(X^{g}-b_{g})\\Pi_{i\\in I_{S_g}}(X^{g_i}-b_i)=0$ for some $b_g, b_{i}\\in \\mathbf{F}^{\\times}$ where $i\\in I_{S_g}$.\nSince both $\\Pi_{i\\in I_{S_g}}(X^{g_i}-b_i)\\neq 0$ and $(X^g-b_g)\\Pi_{i\\in I_{S_g}\\setminus\\{j\\}}(X^{g_i}-b_i)\\neq\n0$ where $j\\notin I_{S_H}\\cup I_{S_K}$, it follows from Lemma~\\ref{coset} that\n\\begin{center}\n$\\sum_0(S_g)$ contains a complete coset of $\\langle g\\rangle$ \\ \\ \\ \\ \\ (*)\n\\end{center}\nand\n\\begin{center}\n$\\sum_0(S_gg)$ contains a complete coset of $\\langle g_j\\rangle$ for any $j\\in I_{S_g}\\setminus (I_{S_H}\\cup\nI_{S_K})$. \\ \\ \\ (**)\n\\end{center}\nIf $\\langle g_{3q+1}, g_{3q+2}, g_{3q+3}\\rangle$ is not a cyclic group, then without loss of generality, we may\nassume that both $g_{3q+2}, g_{3q+3}$ are not in $\\langle g_{3q+1}\\rangle$. This together with (*), proves that\n$(\\ell-|I_{S_{\\langle g_{3q+1}\\rangle}}\\cup I_{S_1}|+1)|M|\\geq 3|M|\\geq 9q$. By Lemma~\\ref{basic} (ii),\n$\\sum(S)=G$, yielding a contradiction.\n\nNext we assume that $\\langle g_{3q+1}, g_{3q+2}, g_{3q+3}\\rangle=H_1$ is a cyclic group. We distinguish the rest\nof the proof into the following two subcases:\\\\\n\n\\noindent {\\bf Subcase 2.1.} Every element of $S_{g_{3q+1}}g_{3q+1}(S_HS_K)^{-1}$ is contained in the subgroup $H_1$.\\\\\n\nLet $T=S_{g_{3q+1}}$. Then $I_T\\setminus I_{S_{H_1}} \\subseteq I_{S_H}$. By (*), $\\sum_0(T)$ contains a complete coset\nof $\\langle g_{3q+1}\\rangle$. If $|S_{H_1}|\\geq 3q-1$, by Lemma \\ref{cyclic}, $\\sum_0(S_{H_1})=H_1$. By Lemma\n\\ref{basic} (i), $\\sum(S)=G$, yielding a contradiction.\n\nIf $|S_{H_1}|\\leq 3q-2$, since $\\sum(S)\\neq G$, as in the proof of lemma~\\ref{basic}~(iii), we obtain\n$|I_{S_H}\\setminus I_{S_{H_1}}|\\leq 3$. Since $I_T\\setminus I_{S_{H_1}} \\subseteq I_{S_H}$, we have\n$(\\ell-|I_{S_{H_1}}\\cup I_T|+1)|M|\\geq (\\ell-|I_{S_{H_1}}|-2)|M|\\geq 3|M|\\geq 9q$. By Lemma \\ref{basic} (ii),\n$\\sum(S)=G$, yielding a contradiction.\\\\\n\n\\noindent {\\bf Subcase 2.2.} There exists an element $g_j$ of $S_{g_{3q+1}}g_{3q+1}(S_HS_K)^{-1}$ such that $g_j\\notin H_1$ for some $j\\in I_{S_{g_{3q+1}}}\\setminus (I_{S_H}\\cup I_{S_K})$. \\\\\n\nBy (**), $\\sum_0(S_{g_{3q+1}}g_{3q+1})$ contains a\ncomplete coset of $\\langle g_j\\rangle$. Since $\\langle g_{3q+2}\\rangle=\\langle g_{3q+3}\\rangle=H_1$, we have\n$g_{3q+2}, g_{3q+3}\\notin \\langle g_j\\rangle$. Then $(\\ell-|I_{S_{\\langle g_j\\rangle}}\\cup I_{S_1}|+1)|M|\\geq\n3|M|\\geq 9q$. By Lemma \\ref{basic} (ii), $\\sum(S)=G$, yielding a contradiction.\n\nIn all cases we have found contradictions. Thus $\\sum(S)=G$ holds as desired.\\\\\n\n\\subsection*{Acknowledgements}\nThis work was carried out during a visit by the first author to Brock University as an international visiting scholar. He would like to sincerely thank the host institution for its hospitality and for providing an excellent atmosphere for research. This work was supported in part by the National Science Foundation of China (Grant No. 11701256, 11871258), the Youth Backbone Teacher Foundation of Henan's University (Grant No. 2019GGJS196), the China Scholarship Council (Grant No. 201908410132), and it was also supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN 2017-03903).\n\n\n\n\\normalsize\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Appendix: }\nHere, we list all the textures of these new triangular mass matrices \nwhere we have dropped the flavor indices $u,d$. \\\\\n~\\\\\n{\\bf I--weak basis such that $H_{12} = H_{21} = 0$ }, \\\\\n~\\\\\n\\underline{ 1--lower triangular : } \\\\\n~\\\\\na)~~$t_{21} = 0$, \\\\\n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\nt_{11} & 0 & 0 \\\\\n0 & t_{22} & 0 \\\\\nt_{31} & t_{32} & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\nb) singular matrix with $t_{11} = 0$, \\\\\n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\n0 & 0 & 0 \\\\\nt_{21} & t_{22} & 0 \\\\\nt_{31} & t_{32} & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\n\\underline{ 2--upper triangular : } \\\\\n~\\\\\n\\begin{eqnarray*}\nT_h & = & \\left( \\begin{array}{ccc}\nt_{11} & - \\frac{t_{22} t_{23}^{\\star} t_{13}}{|t_{22}|^2} & t_{13} \\\\\n0 & t_{22} & t_{23} \\\\\n0 & 0 & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\n{\\bf II--weak basis such that $H_{23} = H_{32} = 0$}, \\\\\n~\\\\\n\\underline{ 1--lower triangular : } \\\\\n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\nt_{11} & 0 & 0 \\\\\nt_{21} & t_{22} & 0 \\\\\nt_{31} & - \\frac{t_{22} t_{21}^{\\star} t_{31}}{|t_{22}|^2} & t_{33} \n\\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\n\\underline{2-- upper--triangular matrix :} \\\\\n~\\\\\na)~~$t_{23} = 0$, \\\\\n\\begin{eqnarray*}\nT_h & = & \\left( \\begin{array}{ccc}\nt_{11} & t_{12} & t_{13} \\\\\n0 & t_{22} & 0 \\\\\n0 & 0 & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\nb) singular matrix with $t_{33} = 0$, \\\\\n\\begin{eqnarray*}\nT_h & = & \\left( \\begin{array}{ccc}\nt_{11} & t_{12} & t_{13} \\\\\n0 & t_{22} & t_{23} \\\\\n0 & 0 & 0 \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\n{\\bf III--weak basis such that $H_{13} = H_{31} = 0$ }, \\\\\n~\\\\\n\\underline{ 1--lower triangular :} \\\\\n~\\\\\na) $t_{31} = 0$, \\\\\n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\nt_{11} & 0 & 0 \\\\\nt_{21} & t_{22} & 0 \\\\\n0 & t_{23} & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\nb) singular matrix for $t_{11} = 0$, \\\\ \n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\n0 & 0 & 0 \\\\\nt_{21} & t_{22} & 0 \\\\\nt_{31} & t_{32} & t_{33} \\end{array} \\right)\n\\end{eqnarray*}\n~\\\\\n\\underline{ 2--upper triangular :} \\\\ \n~\\\\\na) $t_{13} = 0$, \\\\ \n\\begin{eqnarray*}\nT_h & = & \\left( \\begin{array}{ccc}\nt_{11} & t_{12} & 0 \\\\\n0 & t_{22} & t_{23} \\\\\n0 & 0 & t_{33} \\end{array} \\right)\n\\end{eqnarray*} \n~\\\\\nb) singular matrix for $t_{33} = 0$, \\\\\n\\begin{eqnarray*}\nT_b & = & \\left( \\begin{array}{ccc}\nt_{11} & t_{12} & t_{13} \\\\\n0 & t_{22} & t_{23} \\\\\n0 & 0 & 0 \\end{array} \\right)\n\\end{eqnarray*}\n\\section*{Acknowlegment :}\nI would like to thank the DAAD for its financial \nsupport. Special thanks go to Prof. Scheck for reading the manuscript \nand giving me his comments.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\\label{dgfjhngjk}\n\nIn this paper, we will be focussed on $\\T{1}$, $\\T{i}\\overset{d}{=}T$,\n$i=2,3,\\dots$, with p.d.f. $f_{\\T{1}}$ and $f_{T}$, which are independent\npositive random variables, called intervals between renewals, and on\n$\\Y{i}\\overset{d}{=}Y$, $i=1,2,\\dots$, with p.d.f. $f_{Y}$, which are\nindependent positive random variables called jump sizes at the moments of\nrenewals. We assume that these sequences are mutually independent and\n$\\T{1}\\overset{d}{=}T$, i.e., we confine ourselves to the ordinary renewal\nprocess $\\homN{s}:={}\\max\\big\\{n>0:\\sum_{i=1}^{n}\\T{i}\\leqslant s\\big\\}$,\nwith $\\homN{s}:={} 0$, if $\\T{1}>s$. Moreover, we focus on p.d.f.\n$f_{\\T{1}}$, $f_{T}$, and $f_{Y}$ bounded above by a finite constant; these\nrestriction may be relaxed, but not in this paper.\n\nLet us introduce the random process\n\\begin{equation}\\label{sdghtyjrtYY}\n\\homR{s}:={} u+cs-\\homV{s},\\quad s\\geqslant 0,\n\\end{equation}\nwhere $u\\geqslant 0$ and $c\\geqslant 0$ are constants,\n$\\homV{s}:={}\\sum_{i=1}^{\\homN{s}}\\Y{i}$, with $\\homV{s}:={} 0$, if\n$\\T{1}>s$. The random process $\\homV{s}$, $s\\geqslant 0$, is called compound\n(ordinary) renewal processes; its trajectories are piecewise linear.\n\nBy the \\emph{direct} level crossing problem we call the study of probability\n$\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$, where\n\\begin{equation}\\label{wq4t5g4ywe}\n\\begin{aligned}\n\\Tich{u,c}&:={}\\inf\\left\\{s>0:\\homV{s}-cs>u\\right\\}\n\\\\\n&=\\inf\\left\\{s>0:\\homR{s}<0\\right\\},\n\\end{aligned}\n\\end{equation}\nor $+\\infty$, if $\\homV{s}-cs\\leqslant u$ for all $s>0$, whereas the\n\\emph{inverse} level crossing problem is focussed on the study of a solution\n(with respect to $u$) to the equation\n\\begin{equation}\\label{rtyujtkjtyk}\n\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}=\\alpha,\n\\end{equation}\nwhere $\\alpha$ is positive and reasonably small, e.g., $\\alpha=0{.}05$. This\nsolution is denoted by $u_{\\alpha,t}(c)$, $c\\geqslant 0$, and is called\nfixed-probability level. It is easily seen that $\\P\\{\\Tich{u,c}\\leqslant\nt\\}=\\P\\big\\{\\inf_{0\\leqslant s\\leqslant t}\\homR{s}<0\\big\\}$, and the left-hand\nside of \\eqref{rtyujtkjtyk} can be rewritten accordingly.\n\nThe fixed-probability level defined by equation \\eqref{rtyujtkjtyk} is an\nimplicit function. Its analysis is based on a detailed study of the probability\nin the left-hand side of \\eqref{rtyujtkjtyk}, i.e., on the direct level\ncrossing problem. When the random variables $T$ and $Y$ are exponentially\ndistributed with parameters $\\delta>0$ and $\\rho>0$, the random process\n$\\homN{s}$, $s\\geqslant 0$, is a Poisson process with intensity $\\delta$. In\nthis case, $\\P\\big\\{\\homV{s}\\leqslant x\\big\\}$, $\\mathsf{E}\\hskip 1pt\\big(\\homV{s}\\big)$,\n$\\mathsf{D}\\hskip 1pt\\big(\\homV{s}\\big)$, and $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ are expressed\nin a closed form, using elementary and special functions such as the modified\nBessel functions $\\BesselI{n}(z)$ of the first kind of order $n$. Consequently,\nequation \\eqref{rtyujtkjtyk} is written explicitly and the study of an implicit\nfunction $u_{\\alpha,t}(c)$, $c\\geqslant 0$, is carried out in\n\\cite{[Malinovskii=2012]}, \\cite{[Malinovskii=2014b]}; it goes along the road\nmap set before in \\cite{[Malinovskii=2009]}, \\cite{[Malinovskii=2014a]} in the\ndiffusion model.\n\nIn the case of generally distributed random variables $T$ and $Y$, to find a\nsolution to the direct (let alone inverse) level crossing problem in terms of\nelementary and special functions seems impossible, except for a few very\nspecial cases, whence our attention to approximations of\n$\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$, as $u\\to\\infty$ and $u,t\\to\\infty$, and\nof $u_{\\alpha,t}(c)$, $c\\geqslant 0$, as $t\\to\\infty$.\n\nWe proceed investigating $u_{\\alpha,t}(c)$, $c\\geqslant 0$, from the inverse Gaussian\napproximation\\footnote{This stands out from a number of previously known\napproximations, of which Cram{\\'e}r's and diffusion, obtained by means of the\ninvariance principle, are the most famous.} obtained in\n\\cite{[Malinovskii=2018=dan=1]},\n\\cite{[Malinovskii=2017a]}--\\cite{[Malinovskii=2018]}. In a nutshell, we use\nKendall's identity for $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$, which expresses\nthis probability through convolution powers of $f_{T}$ and $f_{Y}$; the central\nlimit theory is then applied to them. This method is widely applicable, e.g.,\nit allows us to find approximations for the first-order derivatives\n$\\frac{\\partial}{\\partial c}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ and\n$\\frac{\\partial}{\\partial u}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$, and even\nfor higher-order derivatives, such as $\\frac{\\partial^2}{\\partial\nc^2}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ and $\\frac{\\partial^2}{\\partial\nu^2}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$. This issue is central (see\nTheorem~\\ref{etrtjt}) in the study of, e.g., monotony and convexity of the\nimplicitly defined function $u_{\\alpha,t}(c)$, $c\\geqslant 0$.\n\nThis approach is aimed at obtaining a large set of results using standard\ntechniques. Such results include approximations and estimates of the rate of\nconvergence, as well as various refinements, e.g., asymptotic expansions. The\nmain focus is on the diversity and accuracy of the results, rather than the\nminimality of technical conditions, although the conditions in these results\nare close to minimal\\footnote{Compare with \\cite{[Borovkov=2015]}, where\nminimization of conditions and a more general level are focussed.}.\n\nIn applications, both direst and inverse level crossing problems are of a great\nimportance. In risk theory, the function $u_{\\alpha,t}(c)$, $c\\geqslant 0$, models a\nnon-ruin capital\\footnote{Apparently, this mathematical concept, viewed as an\nimplicit function, was focussed straightforwardly for the first time in\n\\cite{[Malinovskii=2012]}, where it was referred to as ``level capital'', or\n``$\\alpha$-level initial capital'' (see \\cite{[Malinovskii=2012]},\nDefinition~3.1). In \\cite{[Malinovskii=2014b]}, the term ``ruin capital'',\nemphasizing its role in matters of solvency, was used instead; if one seeks to\nescape ruin, the term ``non-ruin capital'' sounds more appropriate.} that makes\nthe probability of ruin over time $t$ equal to a predetermined value $\\alpha$,\nchosen as an acceptable degree of insolvency. This academic concept is related\nto fundamental methods of insurance solvency's regulation (see, e.g.,\n\\cite{[Beard-et-al.-1984]}, \\cite{[Daykin-et-al.-1996]},\n\\cite{[Pentikainen-et-al.-1989]}, \\cite{[Sandstrom-2006]},\n\\cite{[Sandstrom=2011]}); in practice, they are mainly implemented by\nsimulation. Analytically, as a problem of collective risk theory, the inverse\nlevel crossing problem was first investigated in \\cite{[Malinovskii=2012]} (see\nalso \\cite{[Malinovskii=2014b]}), where equitable solvent controls in a\nmulti-period game model of risk were considered; as a partial single-period\nmodel, Lunderg's model with exponentially distributed claim size was focussed\nin \\cite{[Malinovskii=2012]}; similar issue in the diffusion risk model was\ninvestigated in \\cite{[Malinovskii=2014a]}.\n\nThe rest of this paper is arranged as follows. In Section~\\ref{srteyrjfr}, we\nrecall Kendall's identity. In Section~\\ref{rgerhyryttryjh}, we derive similar\nidentities for derivatives $\\frac{\\partial}{\\partial\nc}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ and $\\frac{\\partial}{\\partial\nu}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$. In Section~\\ref{asrgrhger}, we\noutline the inverse Gaussian approximation (see\n\\cite{[Malinovskii=2018=dan=1]}) in the direct level crossing problem; this\nresult is presented in detail in \\cite{[Malinovskii=2017a]},\n\\cite{[Malinovskii=2017b]}, \\cite{[Malinovskii=Malinovskii=2017]}. In\nSection~\\ref{wqstgerhs}, we establish approximations for derivatives, using the\nsame stages as in the proof of inverse Gaussian approximation. In\nSection~\\ref{srgterjer}, which is core of this paper, we focus on\napproximations in the inverse level crossing problem: first, we deal with\nstructural results, then with monotony and convexity\\footnote{Recall that a\ndifferentiable function is convex (i.e., has the form $\\smile$) if its second\nderivative is positive.} results, and finally with heuristic fixed-probability\nlevel and with elementary bounds on the fixed-probability level, which are a\ntool for numerical calculations.\n\n\\section{Kendall's identity: a keystone result}\\label{srteyrjfr}\n\nLet us introduce\\footnote{The $\\inf$-definition for $\\homM{x}$, $x>0$, in\ncontrast to equivalent $\\max$-definition for $\\homN{s}$, $s\\geqslant 0$, is a\nhint on the difference between these renewal processes.}\n\\begin{equation}\\label{erthtjthew}\n\\homM{x}:={}\\inf\\left\\{k\\geqslant 1:\\sum_{i=1}^{k}Y_{i}>x\\,\\right\\}-1,\\quad\nx>0,\n\\end{equation}\nwhich is a renewal process generated by the random variables $\\Y{i}$,\n$i=1,2,\\dots$. The following result is known as Kendall's identity (see first\n\\cite{[Kendall=1957]}, and then\n\\cite{[Borovkov=1965]}--\\cite{[Borovkov=Dickson=2008]}, \\cite{[Keilson=1963]},\n\\cite{[Rogozin=1966]}, \\cite{[Skorohod=1991]}, \\cite{[Zolotarev=1964]}).\n\n\\begin{assertion}[Kendall's identity]\\label{23456uy7}\nWith $0z\\}\\,dz.\n\\end{aligned}\n\\end{equation*}\nMaking the change of variables $y=z-v$ in \\eqref{dfgbehredf}, we rewrite it as\n\\begin{equation}\\label{wertyrjhn}\n\\begin{aligned}\n\\P\\big\\{v<\\Tich{u,c}\\leqslant t\\mid\\T{1}=v\\big\\}\n&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{u+cv}{u+cv+cy}\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\n\\\\[2pt]\n&\\times f_{Y}^{*n}(u+cv+cy-z)\\,f_{T}^{*n}(y)\\,dy\\,dz.\n\\end{aligned}\n\\end{equation}\n\nEquality \\eqref{dfgbehredf}, or its copy \\eqref{wertyrjhn}, and equality\n\\eqref{ewrktulertye} are fundamental in a series of approximations and\nclosed-form results presented in \\cite{[Malinovskii=2012]},\n\\cite{[Malinovskii=2014b]}--\\cite{[Malinovskii=Malinovskii=2017]}. In\nparticular, when $T$ and $Y$ are exponentially distributed with parameters\n$\\delta>0$ and $\\rho>0$, closed-form expressions for\n$\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ follow from the next corollary of\nAssertion~\\ref{23456uy7}.\n\n\\begin{corollary}\\label{etyjhtrjrt}\nFor $Y$ exponentially distributed with parameter $\\rho>0$, we have\n\\begin{equation}\\label{wedteyjtkj}\n\\begin{aligned}\n\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}&=\\int_{0}^{t}e^{-\\rho\\,(u+cs)}\\,\\Bigg(f_{\\T{1}}(s)\n+\\frac{1}{u+cs}\\sum_{n=1}^{\\infty}\\frac{\\big(\\rho\\,(u+cs)\\big)^n}{n!}\n\\\\[0pt]\n&\\times\\int_{0}^{s}(u+cv)f_{T}^{*n}(s-v)f_{\\T{1}}(v)\\,dv\\Bigg)\\,ds.\n\\end{aligned}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}[Proof of Corollary~\\ref{etyjhtrjrt}]\\label{eartfygul}\nFor $\\Y{i}\\overset{d}{=}Y$, $i=1,2,\\dots$, when $Y$ is exponentially\ndistributed with parameter $\\rho$, we have\n$\\P\\big\\{u+cv-Y<0\\big\\}=e^{-\\rho\\,(u+cv)}$ and\n\\begin{equation*}\n\\P\\big\\{\\homM{u+cs}=n\\big\\}=e^{-\\rho\\,(u+cs)}\\,\\frac{\\big(\\rho\\,(u+cs)\\big)^n}{n!},\\quad\nn=1,2,\\dots,\n\\end{equation*}\nwhence equality \\eqref{ewrktulertye} rewrites as \\eqref{wedteyjtkj}.\n\\end{proof}\n\n\\section{Derivatives of $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ via Kendall's identity}\\label{rgerhyryttryjh}\n\nLet p.d.f. $f_{T}$ and $f_{Y}$ be differentiable. Kendall's identity\n\\eqref{dfgbehredf}, or its copy \\eqref{wertyrjhn}, and equality\n\\eqref{ewrktulertye} allow us to express the derivatives of\n$\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ with respect to $c$ and $u$ in a similar\nway.\n\n\\subsection{Derivative $\\frac{\\partial}{\\partial c}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$}\\label{sdtyjfer}\n\nLet us start with the derivative of $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ with\nrespect to $c$ and introduce the following expressions:\n\\begin{equation}\\label{sdfghfghnfgn}\n\\begin{aligned}\n\\DerC{u,c}{1}(t\\mid\nv)&=-\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{uy}{(u+cv+cy)^{\\,2}}\n\\\\[0pt]\n&\\times\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\nf_{Y}^{*n}\\big(u+cv+cy-z\\big)\\,dz\\,f_{T}^{*n}(y)\\,dy,\n\\\\[0pt]\n\\DerC{u,c}{2}(t\\mid\nv)&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{(u+cv)\\,(v+y)}{u+cv+cy}\n\\\\[0pt]\n&\\times\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,\\Bigg\\{\\,f_{Y}(0)f_{Y}^{*(n-1)}\\big(u+cv+cy-z\\big)\n\\\\[0pt]\n&+\\int_{0}^{u+cv+cy-z}f_{Y}^{\\,\\prime}(\\xi)\nf_{Y}^{*(n-1)}\\big(u+cv+cy-z-\\xi\\big)\\,d\\xi\\,\\Bigg\\}\\,dz\\,\n\\\\[0pt]\n&\\times f_{T}^{*n}(y)\\,dy,\n\\\\[0pt]\n\\DerC{u,c}{3}(t\\mid\nv)&=\\sum_{n=1}^{\\infty}f_{Y}^{*n}(0)\\int_{0}^{t-v}\\frac{(u+cv)\\,(v+y)}{u+cv+cy}\\,\n\\P\\big\\{\\Y{n+1}>u+cv+cy\\big\\}\\,\n\\\\[0pt]\n&\\times f_{T}^{*n}(y)\\,dy.\n\\end{aligned}\n\\end{equation}\n\n\\begin{lemma}\\label{yukiyulu}\nFor $c>0$, $u>0$, $t>v>0$, we have\n\\begin{equation}\\label{wrgrhrtrr}\n\\begin{aligned}\n\\frac{\\partial}{\\partial c}\\,\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}&=-\\int_{0}^{t}f_{Y}(u+cv)\\,v\\,f_{\\T{1}}(v)\\,dv\n\\\\[0pt]\n&+\\int_{0}^{t}\\frac{\\partial}{\\partial c}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}\\,f_{\\T{1}}(v)\\,dv,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{srdtyrjrtj}\n\\frac{\\partial}{\\partial c}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}=\\DerC{u,c}{1}(t\\mid v)+\\DerC{u,c}{2}(t\\mid v)\n+\\DerC{u,c}{3}(t\\mid v).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{yukiyulu}]\nThe proof is based on identities \\eqref{ewrktulertye} and \\eqref{wertyrjhn},\ni.e.,\n\\begin{equation}\\label{ws4d5uhjtd}\n\\begin{aligned}\n\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}&=\\int_{0}^{t}\\P\\big\\{u+cv-Y<0\\big\\}\\,f_{\\T{1}}(v)\\,dv\n\\\\[0pt]\n&+\\int_{0}^{t}\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}\\,f_{\\T{1}}(v)\\,dv\n\\end{aligned}\n\\end{equation}\nand\n\\begin{equation}\\label{adsgfsfbdfnb}\n\\begin{aligned}\n\\P\\big\\{v<\\Tich{u,c}\\leqslant t\\mid\\T{1}=v\\big\\}\n&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{u+cv}{u+cv+cy}\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\n\\\\[2pt]\n&\\times f_{Y}^{*n}\\big(u+cv+cy-z\\big)\\,f_{T}^{*n}(y)\\,dy\\,dz.\n\\end{aligned}\n\\end{equation}\n\nDifferentiating \\eqref{adsgfsfbdfnb}, we have\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial c}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant t\\mid\\T{1}=v\\big\\}\n&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{\\partial}{\\partial\nc}\\,\\Bigg(\\frac{u+cv}{u+cv+cy} \\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\n\\\\[2pt]\n&\\times f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg)\\,f_{T}^{*n}(y)\\,dy.\n\\end{aligned}\n\\end{equation*}\nThe integrand is\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial c}&\\,\\Bigg(\\frac{u+cv}{u+cv+cy}\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg)\n\\\\[0pt]\n&=\\frac{\\partial}{\\partial c}\\,\\Bigg(\\frac{u+cv}{u+cv+cy}\\,\\Bigg)\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\n\\\\[0pt]\n&+\\frac{u+cv}{u+cv+cy}\\,\\frac{\\partial}{\\partial\nc}\\,\\Bigg(\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg),\n\\end{aligned}\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\dfrac{\\partial}{\\partial\nc}\\,\\Bigg(\\dfrac{u+cv}{u+cv+cy}\\,\\Bigg)=-\\dfrac{uy}{(u+cv+cy)^2}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial\nc}&\\,\\Bigg(\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg)\n\\\\[0pt]\n&=\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,\\Bigg(\\frac{\\partial}{\\partial\nc}\\,f_{Y}^{*n}(u+cv+cy-z)\\Bigg)\\,dz\n\\\\[0pt]\n&+(v+y)\\,\\P\\big\\{\\Y{n+1}>u+cv+cy\\big\\}\\,f_{Y}^{*n}(0).\n\\end{aligned}\n\\end{equation*}\nFor $n\\geqslant 2$, differentiation of $n$-fold convolutions yields\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial c}\\,f_{Y}^{*n}(u+cv+cy-z)&=\\frac{\\partial}{\\partial\nc}\\int_{0}^{u+cv+cy-z}f_{Y}\\left(\\left(u+cv+cy-z\\right)-\\zeta\\right)f_{Y}^{*(n-1)}(\\zeta)\\,d\\zeta\n\\\\[0pt]\n&=(v+y)\\underbrace{\\int_{0}^{u+cv+cy-z}f_{Y}^{\\,\\prime}\\left(\\left(u+cv+cy-z\\right)-\\zeta\\right)\nf_{Y}^{*(n-1)}(\\zeta)\\,d\\zeta}_{\\int_{0}^{u+cv+cy-z}f_{Y}^{\\,\\prime}(\\xi)f_{Y}^{*(n-1)}(\\left(u+cv+cy-z\\right)-\\xi)\\,d\\xi}\n\\\\[0pt]\n&+(v+y)\\,f_{Y}(0)f_{Y}^{*(n-1)}(u+cv+cy-z),\n\\end{aligned}\n\\end{equation*}\nwhence, by elementary calculations, the result.\n\\end{proof}\n\n\\subsection{Derivative $\\frac{\\partial}{\\partial u}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$}\\label{asrdthdgjfg}\n\nLet proceed with the derivative of $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ with\nrespect to $u$ and introduce the following expressions:\n\\begin{equation*}\n\\begin{aligned}\n\\DerU{u,c}{1}(t\\mid v)&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\n\\frac{cy}{(u+cv+cy)^{\\,2}}\n\\\\[-2pt]\n&\\times\\,\\int_{0}^{u+cv+cy} \\P\\big\\{\\Y{n+1}>z\\big\\}\\,\nf_{Y}^{*n}(u+cv+cy-z)\\,f_{T}^{*n}(y)\\,dy\\,dz,\n\\\\[0pt]\n\\DerU{u,c}{2}(t\\mid v)&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{u+cv}{u+cv+cy}\n\\\\[-2pt]\n&\\times\\,\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,\n\\Bigg\\{\\,f_{Y}(0)f_{Y}^{*(n-1)}(u+cv+cy-z)\n\\\\[0pt]\n&+\\int_{0}^{u+cv+cy-z}\nf_{Y}^{\\,\\prime}(\\xi)f_{Y}^{*(n-1)}(\\left(u+cv+cy-z\\right)-\\xi)\\,d\\xi\\,\\Bigg\\}\n\\\\[0pt]\n&\\times f_{T}^{*n}(y)\\,dy\\,dz,\n\\\\[0pt]\n\\DerU{u,c}{3}(t\\mid v)&=\\sum_{n=1}^{\\infty}f_{Y}^{*n}(0)\\int_{0}^{t-v}\n\\frac{u+cv}{u+cv+cy}\\,\\P\\big\\{\\Y{n+1}>u+cv+cy\\big\\}\\,\n\\\\[0pt]\n&\\times f_{T}^{*n}(y)\\,dy.\n\\end{aligned}\n\\end{equation*}\n\n\\begin{lemma}\\label{ytuktyktly}\nFor $c>0$, $u>0$, $t>v>0$, we have\n\\begin{equation}\\label{srdtyutryitk}\n\\begin{aligned}\n\\frac{\\partial}{\\partial u}\\,\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}&=-\\int_{0}^{t}f_{Y}(u+cv)\\,v\\,f_{\\T{1}}(v)\\,dv\n\\\\[0pt]\n&+\\int_{0}^{t}\\frac{\\partial}{\\partial u}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}\\,f_{\\T{1}}(v)\\,dv,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{serdtgfjkyfg}\n\\frac{\\partial}{\\partial u}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}=\\DerU{u,c}{1}(t\\mid v)+\\DerU{u,c}{2}(t\\mid v)\n+\\DerU{u,c}{3}(t\\mid v).\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{ytuktyktly}]\nDifferentiating identity \\eqref{adsgfsfbdfnb}, we have\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial u}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant t\\mid\\T{1}=v\\big\\}\n&=\\sum_{n=1}^{\\infty}\\int_{0}^{t-v}\\frac{\\partial}{\\partial\nu}\\,\\Bigg(\\frac{u+cv}{u+cv+cy} \\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\n\\\\[0pt]\n&\\times f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg)\\,f_{T}^{*n}(y)\\,dy.\n\\end{aligned}\n\\end{equation*}\nThe integrand is\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial u}&\\,\\Bigg(\\frac{u+cv}{u+cv+cy}\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg)\n\\\\[0pt]\n&=\\frac{\\partial}{\\partial u}\\,\\Bigg(\\frac{u+cv}{u+cv+cy}\\,\\Bigg)\n\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\n\\\\[0pt]\n&+\\frac{u+cv}{u+cv+cy}\\,\\frac{\\partial}{\\partial\nu}\\,\\Bigg(\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\,\\Bigg),\n\\end{aligned}\n\\end{equation*}\nwhere $\\dfrac{\\partial}{\\partial\nu}\\Bigg(\\dfrac{u+cv}{u+cv+cy}\\Bigg)=\\dfrac{cy}{(u+cv+cy)^2}$ and\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial\nu}\\,\\Bigg(\\int_{0}^{u+cv+cy}&\\P\\{\\Y{n+1}>z\\}\\,f_{Y}^{*n}(u+cv+cy-z)\\,dz\\Bigg)\n\\\\[0pt]\n&=\\int_{0}^{u+cv+cy}\\P\\big\\{\\Y{n+1}>z\\big\\}\\,\\Bigg(\\frac{\\partial}{\\partial\nu}\\,f_{Y}^{*n}(u+cv+cy-z)\\Bigg)\\,dz\n\\\\[0pt]\n&+\\P\\big\\{\\Y{n+1}>u+cv+cy\\big\\}\\,f_{Y}^{*n}(0),\n\\end{aligned}\n\\end{equation*}\nFor $n\\geqslant 2$, differentiation of $n$-fold convolutions yields\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\partial}{\\partial u}f_{Y}^{*n}(u+cv+cy-z)&=\\frac{\\partial}{\\partial\nu}\\int_{0}^{u+cv+cy-z}f_{Y}\\big((u+cv+cy-z)-\\zeta\\big)f_{Y}^{*(n-1)}(\\zeta)\\,d\\zeta\n\\\\[0pt]\n&=\\underbrace{\\int_{0}^{u+cv+cy-z}f_{Y}^{\\,\\prime}\\left(\\left(u+cv+cy-z\\right)-\\zeta\\right)\nf_{Y}^{*(n-1)}(\\zeta)\\,d\\zeta}_{\\int_{0}^{u+cv+cy-z}\nf_{Y}^{\\,\\prime}(\\xi)f_{Y}^{*(n-1)}(\\left(u+cv+cy-z\\right)-\\xi)\\,d\\xi}\n\\\\[0pt]\n&+f_{Y}(0)f_{Y}^{*(n-1)}(u+cv+cy-z).\n\\end{aligned}\n\\end{equation*}\nwhence, by elementary calculations, the result.\n\\end{proof}\n\n\\section{Approximations in direct level crossing problem}\\label{asrgrhger}\n\nThe probability density function (p.d.f.) and cumulative distribution function\n(c.d.f.) of a Gaussian distribution with mean $a$ and variance $b^2$ are\ndenoted by $\\UGauss{a}{b^2}(x)$ and $\\Ugauss{a}{b^2}(x)$.\n\n\\subsection{Core integral expressions}\\label{srdthrdjh}\n\nFor $t$, $u$, $c$, $M$, and $D^{2}$ fixed positive constants, the elementary\nintegral expressions are defined as\n\\begin{equation}\\label{rterherher}\n\\Elem{u,c}{k}(t):={}\\int_{0}^{\\frac{ct}{u}}\\frac{1}{(x+1)^{k}}\\,\n\\Ugauss{cM(x+1)}{\\frac{c^{2}D^{2}}{u}(x+1)}(x)\\,dx,\\quad k=0,1,2,\\dots.\n\\end{equation}\nWe write $c^{\\ast}:={} M^{-1}$ and $\\Elem{u,0}{k}(t):={}\\lim_{c\\to\n0}\\Elem{u,c}{k}(t)$,\n$\\Elem{u,\\infty}{k}(t):={}\\lim_{c\\to\\infty}\\Elem{u,c}{k}(t)$, and the like.\n\nThese integral expressions can be expressed through c.d.f.\n$F\\big(x;\\mu,\\lambda,p\\big)$ of a generalized inverse Gaussian\ndistribution\\footnote{There are some differences in terminology. In\n\\cite{[Zigangirov=1962]}, this distribution is called Wald's distribution.\nSeveral authors (see \\cite{[Morlat=1956]}, \\cite{[Seshadri=1997]},\n\\cite{[Perreault=1999]}) attribute the invention of generalized inverse\nGaussian distributions to E.\\,Halphen and use the term ``Halphen Distribution\nSystem'' or ``Halphen's laws''. The others, e.g., M.A.\\,Chaudry and\nS.M.\\,Zubair \\cite{[Chaudry=2002]}, refer to B.\\,J{\\o}rgensen\n\\cite{[Jorgensen=1982]} and attribute the invention of generalized inverse\nGaussian distribution to I.J.\\,Good \\cite{[Good=1953]}.}, which depends on\nparameters $\\mu>0$, $\\lambda>0$, and $p\\in\\mathsf{R}$, and whose p.d.f.\nis\\footnote{Note that the choice $p=-{1}\/{2}$ yields the ``ordinary'' inverse\nGaussian distribution.}\n\\begin{equation}\\label{45t34y34}\n\\begin{aligned}\nf\\big(x;\\mu,\\lambda,p\\big)\n&:={}\\frac{e^{-\\frac{\\lambda}{\\mu}}}{2\\mu^p\\BesselK{p}\\Big(\\frac{\\lambda}{\\mu}\\Big)}\\,x^{p-1}\n\\exp\\Bigg\\{-\\frac{\\lambda(x-\\mu)^{2}}{2\\mu^{\\,2} x}\\Bigg\\}\n\\\\[0pt]\n&=\\frac{\\sqrt{2\\pi}\\,e^{-\\frac{\\lambda}{\\mu}}}{2\\mu^p\\BesselK{p}\\Big(\\frac{\\lambda}{\\mu}\\Big)}\\,x^{p-1}\n\\,\\Ugauss{0}{1}\\left(\\sqrt{\\frac{\\lambda}{x}}\\,\\Bigg(\\frac{x}{\\mu}-1\\Bigg)\\right),\\quad\nx>0,\n\\end{aligned}\n\\end{equation}\nwhere $\\BesselK{p}(z)$, $z>0$, with $p\\in\\mathsf{R}$, denotes the modified Bessel\nfunction of the second kind. In particular, for $u>0$, $t>0$, we have\n\\begin{equation}\\label{drgerherhe}\n\\Elem{u,c}{1}(t)=\\begin{cases}\n\\Big(F\\big(x+1;\\mu,\\lambda,-\\tfrac{1}{2}\\big)\n\\\\[0pt]\n\\hskip 30pt -F\\big(1;\\mu,\\lambda,-\\tfrac{1}{2}\\big)\n\\Big)\\,\\big|_{x=\\frac{ct}{u},\\mu=\\frac{1}{1-cM},\\lambda=\\frac{u}{c^{2}D^{2}}},&\n0c^{\\ast},\n\\end{cases}\n\\end{equation}\nand\n\\begin{equation}\\label{asdfgreherXX}\n\\begin{aligned}\n\\Elem{u,0}{1}(t)&=\\UGauss{0}{1}\\left(\\frac{M\\sqrt{u}}{D}\\right)-\\UGauss{0}{1}\\left(\\frac{Mu-t}{D\\sqrt{u}}\\right),\n\\\\[0pt]\n\\Elem{u,c^{\\ast}}{1}(t)&=2\\left(\\UGauss{0}{1}\\left(\\dfrac{M\\sqrt{u}}{D}\\right)-\\UGauss{0}{1}\\left(\\dfrac{Mu}{D\\sqrt{u+c^{\\ast}\nt}}\\right)\\right),\n\\\\[0pt]\n\\Elem{u,\\infty}{1}(t)&=0.\n\\end{aligned}\n\\end{equation}\n\nThe c.d.f. of generalized inverse Gaussian distribution can be represented in\nterms of c.d.f. and p.d.f. of a standard Gaussian distribution, e.g.,\n\\begin{equation}\\label{xzfvbdbf}\nF\\big(x;\\mu,\\lambda,-\\tfrac{1}{2}\\big)=\n\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\lambda}{x}}\\,\\Bigg(\\frac{x}{\\mu}-1\\Bigg)\\right)\n+\\exp\\Bigg\\{\\frac{2\\lambda}{\\mu}\\Bigg\\}\n\\,\\UGauss{0}{1}\\left(-\\sqrt{\\frac{\\lambda}{x}}\\,\\Bigg(\\frac{x}{\\mu}+1\\Bigg)\\right),\\quad\nx>0,\n\\end{equation}\nand\\footnote{We do not present here all the expressions for\n$F\\big(x;\\mu,\\lambda,\\tfrac{1}{2}\\big)$,\n$F\\big(x;\\mu,\\lambda,-\\tfrac{3}{2}\\big)$, for the derivatives like\n$\\frac{\\partial}{\\partial\\lambda}F\\big(x;\\mu,\\lambda,-\\tfrac{1}{2}\\big)$\nand\n$\\frac{\\partial}{\\partial\\lambda}F\\big(x;\\mu,\\lambda,-\\tfrac{1}{2}\\big)$,\nand for $\\Elem{u,c}{0}(t)$, $\\Elem{u,c}{2}(t)$, though they are available by\nmeans of direct calculations and similar to these presented: this would require\ndramatically more space. We leave this to the reader.}\n\\begin{equation*}\n\\begin{aligned}\nF\\big(x;\\mu,\\lambda,-\\tfrac{5}{2}\\big)\n&=\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\lambda}{x}}\\,\\bigg(\\frac{x}{\\mu}-1\\bigg)\\right)\n+\\frac{\\lambda^{2}-3\\lambda\\mu+3\\mu^{\\,2}}{\\lambda^{2}+3\\lambda\\mu+3\\mu^{\\,2}}\n\\exp\\Bigg\\{\\frac{2\\lambda}{\\mu}\\Bigg\\}\n\\\\[0pt]\n&\\times\\UGauss{0}{1}\\left(-\\sqrt{\\frac{\\lambda}{x}}\\,\\bigg(\\frac{x}{\\mu}+1\\bigg)\\right)\n+\\frac{2\\sqrt{\\lambda}\\,\\mu^{\\,2}\\,\\big(\\lambda+3x\\big)}{x^{3\/2}\n\\,\\big(\\lambda^{2}+3\\lambda\\mu+3\\mu^{\\,2}\\big)}\\,\n\\Ugauss{0}{1}\\left(\\sqrt{\\frac{\\lambda}{x}}\\,\\bigg(\\frac{x}{\\mu}-1\\bigg)\\right),\\quad\nx>0.\n\\end{aligned}\n\\end{equation*}\n\n\\subsection{Inverse Gaussian approximation, as $u$ tends to infinity}\\label{aesrfgeh}\n\nThe inverse Gaussian approximation for $\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ in\nthe direct level crossing problem was studied in\n\\cite{[Malinovskii=2017a]}--\\cite{[Malinovskii=2018=dan=1]},\n\\cite{[Malinovskii=Malinovskii=2017]}. For $c\\geqslant 0$, $u>0$, $00$ and $\\rho>0$, we have\n$\\mathsf{E}\\hskip 1pt{T}=\\delta^{-1}$, $\\mathsf{E}\\hskip 1pt{Y}=\\rho^{-1}$, $\\mathsf{D}\\hskip 1pt{T}=\\delta^{-2}$,\n$\\mathsf{D}\\hskip 1pt{Y}=\\rho^{-2}$, and $M=\\rho\/\\delta$,\n$D^{\\,2}=2\\,\\rho\/\\delta^{\\,2}$, whence\n$D\/M^{\\,3\/2}=\\sqrt{2\\delta}\/\\rho$.}\n\\begin{equation}\\label{adsfgherfjnfnm}\nM:={}\\mathsf{E}\\hskip 1pt{T}\/\\mathsf{E}\\hskip 1pt{Y},\\quad\nD^{\\,2}:={}\\big((\\mathsf{E}\\hskip 1pt{T})^{2}\\mathsf{D}\\hskip 1pt{Y}+(\\mathsf{E}\\hskip 1pt{Y})^{2}\\mathsf{D}\\hskip 1pt{T}\\big)\/(\\mathsf{E}\\hskip 1pt{Y})^{3}\n\\end{equation}\nwe write\n\\begin{equation*}\n\\AInt{M}{u,c}(t\\mid v):={}\\int_{0}^{\\frac{c(t-v)}{cv+u}}\\frac{1}{x+1}\n\\,\\Ugauss{cM(x+1)}{\\frac{c^{2}D^{\\,2}}{cv+u}(x+1)}(x)\\,dx\n\\end{equation*}\nand note that $\\AInt{M}{u,c}(t):={}\\AInt{M}{u,c}(t\\mid 0)$ equals\n$\\Elem{u,c}{1}(t)=\\int_{0}^{\\frac{ct}{u}}\\frac{1}{x+1}\\,\n\\Ugauss{cM(x+1)}{\\frac{c^{2}D^{\\,2}}{u}(x+1)}(x)\\,dx$.\n\nThe following theorem for conditional distribution of $\\Tich{u,c}$ (see the\nleft-hand side of \\eqref{dfgbehredf}) is fundamental.\n\n\\begin{theorem}\\label{srdthjrfX}\nIn the renewal model, let p.d.f. $f_{T}$ and $f_{Y}$ be bounded above by a\nfinite constant, $D^{\\,2}>0$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$, $\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$. Then\nfor any fixed $c\\geqslant 0$ and $0v}\\,\\left|\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}-\\AInt{M}{u,c}(t\\mid v)\\,\\right|\n=\\underline{O}\\left(\\frac{\\ln\\left(u+cv\\right)}{u+cv}\\right),\n\\end{equation*}\nas\\footnote{With $c$ and $v$ fixed, $u+cv\\to\\infty$ is trivially equivalent to\n$u\\to\\infty$.} $u+cv\\to\\infty$.\n\\end{theorem}\n\nThe following results for non-conditional distribution of $\\Tich{u,c}$ is an\neasy corollary of Theorem~\\ref{srdthjrfX} and equality \\eqref{ewrktulertye}.\n\n\\begin{theorem}\\label{er6u6rti76}\nSuppose that conditions of Theorem~\\ref{srdthjrfX} are satisfied. Then\n\\begin{equation*}\n\\sup_{t>0}\\,\\bigg|\\,\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}-\\int_{0}^{t}\\AInt{M}{u,c}(t\\mid\nv)\\,f_{\\T{1}}(v)\\,dv\\,\\bigg|=\\underline{O}\\left(\\frac{\\ln u}{u}\\right),\\quad\nu\\to\\infty.\n\\end{equation*}\n\\end{theorem}\n\nWe can replace the integral $\\int_{0}^{t}\\AInt{M}{u,c}(t\\mid\nv)\\,f_{\\T{1}}(v)\\,dv$ by the integral\n$\\int_{0}^{t}\\AInt{M}{u,c}(t-v)\\,f_{\\T{1}}(v)\\,dv$, which is a convolution. It\nagrees with the probabilistic intuition about the role which plays the first\ntime interval $\\T{1}$ in the event of crossing a high level $u$ within finite\ntime $t$: given $\\T{1}=v$, the whole time length becomes $t-v$, with no other\nchanges.\n\n\n\\begin{theorem}\\label{rtutujrtirt6}\nSuppose that conditions of Theorem~\\ref{srdthjrfX} are satisfied, and that\n$\\mathsf{E}\\hskip 1pt\\T{1}<\\infty$. Then\n\\begin{equation*}\n\\sup_{t>0}\\,\\bigg|\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}\n-\\int_{0}^{t}\\AInt{M}{u,c}(t-v)\\,f_{\\T{1}}(v)\\,dv\\,\\bigg|=\\underline{O}\\left(\\frac{\\ln\nu}{u}\\right),\\quad u\\to\\infty.\n\\end{equation*}\n\\end{theorem}\n\n\\subsection{Approximation, as $u$ and $t$ tend to infinity}\\label{wqreterjh}\n\nWhereas the influence of $\\T{1}$ in Theorem~\\ref{rtutujrtirt6} can not be\neliminated for $t$ small and moderate, it becomes negligible for $t$ large.\nGiven that $t\\to\\infty$, the integral\n$\\int_{0}^{t}\\AInt{M}{u,c}(t-v)\\,f_{\\T{1}}(v)\\,dv$ in\nTheorem~\\ref{rtutujrtirt6} can be approximated by $\\AInt{M}{u,c}(t)$, whence\nthe following result.\n\n\\begin{theorem}\\label{esrytrf}\nSuppose that conditions of Theorem~\\ref{srdthjrfX} are satisfied, and that\n$\\mathsf{E}\\hskip 1pt\\T{1}<\\infty$. Then\n\\begin{equation*}\n\\sup_{t>0}\\,\\Big|\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}\n-\\AInt{M}{u,c}(t)\\,\\Big|=\\underline{O}\\left(\\frac{\\ln\nu}{u}\\right)+\\underline{O}\\left(\\frac{1}{t^{1\/2}}\\right),\\quad t,u\\to\\infty.\n\\end{equation*}\n\\end{theorem}\n\n\\subsection{An outline of the proof}\\label{srdtytujrt}\n\nThe proofs in \\cite{[Malinovskii=2017a]}, \\cite{[Malinovskii=2017b]} are\nconducted in a uniform manner. In the proof of Theorem~\\ref{srdthjrfX}, Step~0\nis the identity \\eqref{dfgbehredf}, or its copy \\eqref{wertyrjhn}, for the\nprobability $\\P\\big\\{v<\\Tich{u,c}\\leqslant t\\mid\\T{1}=v\\big\\}$. Step~1 is a\nreduction of the range of integration in \\eqref{dfgbehredf}. This cutting off\nof unlikely events, such as when $\\Y{n+1}$ is excessively large compared to the\nwhole sum $\\sum_{i=1}^{n}\\Y{i}$, complies with intuition. Step~2 is a reduction\nof the range of summation in \\eqref{dfgbehredf}. This applies Nagaev's\ninequalities for sums used to reject the summands in the range $00$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$,\n$\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$. Then for any fixed $c>0$ and $0v}\\;\\Bigg|\\;\\frac{\\partial}{\\partial c}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}\n\\\\[0pt]\n&\\hskip 60pt-\\frac{M(u+cv)}{c^2D^2}\\left((1-cM)\\,\\Elem{u,c}{0}(t\\mid\nv)-\\Elem{u,c}{1}(t\\mid v)\\right)\n\\\\[0pt]\n&\\hskip 60pt+\\frac{Mu}{c^2D^2}\\left((1-cM)\\,\\Elem{u,c}{1}(t\\mid\nv)-\\Elem{u,c}{2}(t\\mid v)\\right)+\\frac{1}{c}\\,\\Elem{u,c}{1}(t\\mid v)\n\\\\[0pt]\n&\\hskip 60pt-\\frac{1}{c}\\,\\Elem{u,c}{2}(t\\mid\nv)\\;\\Bigg|=\\underline{O}\\left(\\frac{\\ln\\left(u+cv\\right)}{u+cv}\\right),\n\\end{aligned}\n\\end{equation}\nas $u+cv\\to\\infty$.\n\\end{theorem}\n\nThe starting point in the proof of Theorems~\\ref{srdthjrfX} was Kendall's\nidentity, whereas the starting point in the proof of Theorem~\\ref{wertyujkl} is\nLemma~\\ref{yukiyulu}. The proof of Theorem~\\ref{wertyujkl} follows the scheme\noutlined in Section~\\ref{srdtytujrt}. We leave the details to the reader.\n\nIn the same way as Theorem~\\ref{esrytrf} follows from Theorem~\\ref{srdthjrfX},\nas $u,t\\to\\infty$, it follows from Theorem~\\ref{wertyujkl} that the derivative\n$\\frac{\\partial}{\\partial c}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ is\napproximated, as $u,t\\to\\infty$, by the expression\n\\begin{equation}\\label{dsfgrehgdfhd}\n\\begin{aligned}\nF_{u,c}(t)&=\\frac{Mu}{c^2D^2}\\,\\left((1-cM)\\,\\Elem{u,c}{0}(t)-\\Elem{u,c}{1}(t)\\right)\n\\\\[0pt]\n&-\\frac{Mu}{c^2D^2}\\left((1-cM)\\,\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t)\\right)\n-\\frac{1}{c}\\,\\Elem{u,c}{1}(t)+\\frac{1}{c}\\,\\Elem{u,c}{2}(t).\n\\end{aligned}\n\\end{equation}\n\nUsing the representation of $\\Elem{u,c}{0}(t)$, $\\Elem{u,c}{1}(t)$,\n$\\Elem{u,c}{2}(t)$, such as \\eqref{drgerherhe}, we can verify that\n\\begin{equation}\\label{wqe4ryttjfyjk}\n(1-cM)\\,\\Elem{u,c}{0}(t)-\\Elem{u,c}{1}(t)\n=\\underline{O}\\left(u^{-1}\\right),\\quad u\\to\\infty,\n\\end{equation}\nand that\n\\begin{equation}\\label{esrtgeyheje}\n(1-cM)\\,\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t)\n=\\underline{O}\\left(u^{-1}\\right),\\quad u\\to\\infty.\n\\end{equation}\n\n\\begin{figure}[t]\n\\center{\\includegraphics[scale=.8]{Fig=1}}\n\\caption{\\small Graphs ($X$-axis is $c$) of the functions $F_{u,c}(t)$ (red),\n$\\AInt{M}{u,c}^{\\,(0,1)}(t)$ (blue). Here $t=100$, $u=40$, $M=1$,\n$D^2=6$.}\\label{asrtdyujkl}\n\\end{figure}\n\n\\begin{remark}\\label{wqe45r6u7rui}\nLet us compare \\eqref{dsfgrehgdfhd} with\n$\\AInt{M}{u,c}^{\\,(0,1)}(t):={}\\frac{\\partial}{\\partial c}\\AInt{M}{u,c}(t)$.\nIn other words, let us compare the ``approximation of derivative'' with the\n``derivative of approximation''.\n\nFor $u>0$, $t>0$, $c>0$, we have by straightforward differentiation\n\\begin{equation*}\n\\begin{aligned}\n\\AInt{M}{u,c}^{\\,(0,1)}(t)&=\\frac{u\\,(1-cM)}{c^{3}D^{2}}\\,\\Elem{u,c}{\\,0}(t)\n-\\frac{1}{c}\\,\\Bigg(\\frac{u\\,(2-cM)}{c^{2}D^{2}}+1\\Bigg)\\,\\Elem{u,c}{1}(t)\n\\\\[-2pt]\n&+\\frac{u}{c^{3}D^{2}}\\,\\Elem{u,c}{\\,2}(t)+\\frac{t}{u+ct}\\,\n\\Ugauss{cM(1+\\frac{ct}{u})}{\\frac{c^{2}D^{2}}{u}(1+\\frac{ct}{u})}\\bigg(\\frac{ct}{u}\\bigg).\n\\end{aligned}\n\\end{equation*}\nAlternatively, this equality is written as\n\\begin{equation}\\label{dfyjhkghkgh}\n\\begin{aligned}\n\\AInt{M}{u,c}^{\\,(0,1)}(t)&=\\frac{u}{c^{3}D^{2}}\\left((1-cM)\\,\\Elem{u,c}{0}(t)-\\Elem{u,c}{1}(t)\\right)\n\\\\[2pt]\n&-\\frac{u}{c^{3}D^{2}}\\left((1-cM)\\,\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t)\\right)\n\\\\[0pt]\n&-\\frac{1}{c}\\,\\Elem{u,c}{1}(t)+\\frac{t}{u+ct}\\,\n\\Ugauss{cM(1+\\frac{ct}{u})}{\\frac{c^{2}D^{2}}{u}(1+\\frac{ct}{u})}\\bigg(\\frac{ct}{u}\\bigg).\n\\end{aligned}\n\\end{equation}\nThis equality\\footnote{Bear in mind the asymptotic relations\n\\eqref{wqe4ryttjfyjk} and \\eqref{esrtgeyheje}.} is suitable for comparison with\nequality \\eqref{dsfgrehgdfhd}. Both are illustrated in Fig.~\\ref{asrtdyujkl}.\n\\end{remark}\n\nWe conclude this analysis with the following summary. The proximity between\n$F_{u,c}(t)$, i.e., ``approximation of derivative'', and\n$\\AInt{M}{u,c}^{\\,(0,1)}(t)$, i.e., ``derivative of approximation'',\nillustrated numerically in Fig.~\\ref{asrtdyujkl}, can be proved rigorously\nusing equalities \\eqref{dsfgrehgdfhd} and \\eqref{dfyjhkghkgh}, evaluated\nanalytically. However, the ``approximation of derivative'' is one thing and the\n``derivative of approximation'' is another. Their study requires a separate\nanalysis; the naive belief that one can replace the other is largely\ngroundless.\n\n\\subsection{Approximation for derivative $\\frac{\\partial}{\\partial\nu}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$}\\label{tyjtrujrtjurtj}\n\n\\begin{theorem}\\label{qewrweytery}\nIn the renewal model, let p.d.f. $f_{T}$ and differentiable $f_{Y}$ be bounded\nabove by a finite constant, $D^{\\,2}>0$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$,\n$\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$. Then for any fixed $c>0$ and $0v}\\;\\Bigg|\\;\\frac{\\partial}{\\partial u}\\,\\P\\big\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\big\\}\n\\\\[0pt]\n&\\hskip 60pt-\\frac{M}{c\\,D^2}\\left((1-cM)\\,\\Elem{u,c}{1}(t\\mid\nv)-\\Elem{u,c}{2}(t\\mid v)\\right)\n\\\\[0pt]\n&\\hskip 60pt-\\frac{1}{u}\\left(\\Elem{u,c}{1}(t\\mid v)-\\Elem{u,c}{2}(t\\mid\nv)\\right)\n\\,\\Bigg|=\\underline{O}\\left(\\frac{\\ln\\left(u+cv\\right)}{(u+cv)^2}\\right),\n\\end{aligned}\n\\end{equation}\nas $u+cv\\to\\infty$.\n\\end{theorem}\n\nIn the same way as in Section~\\ref{adsfgvre}, we can deduce from\nTheorem~\\ref{qewrweytery} that the derivative $\\frac{\\partial}{\\partial\nu}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ is approximated, as $u,t\\to\\infty$, by\nthe expression\\footnote{Bear in mind the asymptotic relation\n\\eqref{esrtgeyheje}.}\n\\begin{equation}\\label{fgyjfgkmhh}\nG_{u,c}(t)=\\frac{M}{c\\,D^2}\\left((1-cM)\\,\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t)\\right)\n+\\frac{1}{u}\\left(\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t)\\right).\n\\end{equation}\n\n\\begin{figure}[t]\n\\center{\\includegraphics[scale=.8]{Fig=2}\\\\\n\\includegraphics[scale=.8]{Fig=3}}\n\\caption{\\small Graphs ($X$-axis is $u$) of the functions $G_{u,c}(t)$ (red),\n$\\AInt{M}{u,c}^{\\,(1,0)}(t)$ (blue). Here $t=100$, $M=1$, $D^2=6$,\n$c=0{.}8c^{\\ast}$ (below), where $c^{\\ast}=1$.}\\label{yt7yiktyyu}\n\\end{figure}\n\n\\begin{remark}\\label{aesrgegher}\nLet us compare \\eqref{fgyjfgkmhh} with\n$\\AInt{M}{u,c}^{\\,(1,0)}(t):={}\\frac{\\partial}{\\partial u}\\AInt{M}{u,c}(t)$.\nIn other words, let us compare the ``approximation of derivative'' with the\n``derivative of approximation''.\n\nFor $u>0$, $t>0$, $c>0$, we have by straightforward differentiation\n\\begin{equation*}\n\\begin{aligned}\n\\AInt{M}{u,c}^{\\,(1,0)}(t)&=-\\frac{(1-cM)^{2}}{2c^{2}D^{2}}\\,\\Elem{u,c}{\\,0}(t)+\\frac{1}{2u}\\,\\Elem{u,c}{1}(t)\n+\\frac{(1-cM)}{c^2D^{2}}\\,\\Elem{u,c}{1}(t)\n\\\\[0pt]\n&-\\frac{1}{2c^{2}D^{2}}\\,\\Elem{u,c}{2}(t)-\\frac{ct}{u\\,(u+ct)}\\,\\Ugauss{cM(1+\\frac{ct}{u})}{\\frac{c^{2}D^{2}}{u}\n(1+\\frac{ct}{u})}\\bigg(\\frac{ct}{u}\\bigg).\n\\end{aligned}\n\\end{equation*}\nAlternatively, this equality is written as\n\\begin{equation}\\label{qwertyerjk}\n\\begin{aligned}\n\\AInt{M}{u,c}^{\\,(1,0)}(t)&=\\frac{1}{c^2D^{2}}\\left((1-cM)\\,\\Elem{u,c}{1}(t)-\\Elem{u,c}{2}(t\\mid\nv)\\right)\n\\\\[2pt]\n&-\\frac{1}{2c^2D^{2}}\\left((1-cM)^{2}\\,\\Elem{u,c}{\\,0}(t)-\\Elem{u,c}{\\,2}(t)\\right)\n\\\\[-2pt]\n&+\\frac{1}{2u}\\,\\Elem{u,c}{1}(t)-\\frac{ct}{u\\,(u+ct)}\\,\\Ugauss{cM(1+\\frac{ct}{u})}{\\frac{c^{2}D^{2}}{u}\n(1+\\frac{ct}{u})}\\bigg(\\frac{ct}{u}\\bigg).\n\\end{aligned}\n\\end{equation}\nThis equality is suitable for comparison with equality \\eqref{fgyjfgkmhh}. Both\nare illustrated in Fig.~\\ref{yt7yiktyyu}.\n\\end{remark}\n\n\\begin{remark}\\label{w45ey6u5}\nAt the beginning of Section~\\ref{wqstgerhs}, we noted that the derivative\n$\\frac{\\partial}{\\partial u}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$ is negative\n\\emph{for all} $u>0$. But neither the approximation $G_{u,c}(t)$ nor the\nexpression $\\AInt{M}{u,c}^{\\,(1,0)}(t)$ are negative for all $u>0$; for $u$\nsmall and moderate (see Fig.~\\ref{yt7yiktyyu}), both these expressions take\npositive values. This is not a flaw, because the approximation of\nTheorem~\\ref{qewrweytery} only works for $u$ large.\n\\end{remark}\n\n\\subsection{Approximation for higher-order derivatives}\n\nThe approximations for higher-order derivatives, such as\n$\\frac{\\partial^2}{\\partial c^2}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$,\n$\\frac{\\partial^2}{\\partial u^2}\\,\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}$, and\n$\\frac{\\partial^2}{\\partial c\\,\\partial u}\\,\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}$ used in \\eqref{ewryjhnmfm} are carried out as described above, and\nleft to the reader.\n\n\\section{Approximations in inverse level crossing problem}\\label{srgterjer}\n\n\\subsection{Structural results for fixed-probability level}\\label{werdtyjmt}\n\nThe inverse level crossing problem when $T$ and $Y$ are exponentially\ndistributed with parameters $\\delta>0$ and $\\rho>0$ was studied in\n\\cite{[Malinovskii=2012]} and \\cite{[Malinovskii=2014b]}. In a diffusion model,\nthe similar analysis was done in \\cite{[Malinovskii=2014a]}; see also\n\\cite{[Malinovskii=2009]}. When $T$ and $Y$ are non-exponentially distributed,\nsimulation analysis in the inverse level crossing problem was done in\n\\cite{[Malinovskii=Kosova=2014]}.\n\nGoing along the road map set in \\cite{[Malinovskii=2009]}--\n\\cite{[Malinovskii=2014b]}, we focus on the asymptotic structure of the\nfixed-probability level. We start with the following simple result.\n\n\\begin{theorem}\\label{gyuiytik}\nAssume that $f_{T}(x)$ and $f_{Y}(x)$ are bounded above by a finite constant,\n$D^{2}>0$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$, and $\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$. Then for\n$t\\to\\infty$, we have\n\\begin{equation}\\label{3etu6juh}\nu_{\\alpha,t}(0)=\\frac{t}{M}+\\frac{D}{M^{\\,3\/2}}\\,\\NQuant{\\alpha}\\sqrt{t}\\,\\big(1+\\overline{o}(1)\\big).\n\\end{equation}\n\\end{theorem}\n\nSince $u_{\\alpha,t}(0)$ is a solution to the equation $\\P\\big\\{\\Tich{u,0}\\leqslant\nt\\big\\}=\\alpha$, where $\\Tich{u,0}:={}\\inf\\left\\{s>0:\\homV{s}>u\\right\\}$, and\nsince the trajectories of the compound renewal process $\\homV{s}$, $s\\geqslant\n0$, are (a.s.) step functions with only jumps up, this equation rewrites as\n$\\P\\left\\{\\homV{t}>u\\right\\}=\\alpha$. Therefore, Theorem~\\ref{gyuiytik} is a\ndirect corollary of the normal approximation for the distribution of compound\nrenewal process $\\homV{t}$, which is well known: as $t\\to\\infty$, the\nprobability $\\P\\left\\{\\homV{t}>u\\right\\}$ is approximated by\n\\begin{equation*}\n1-\\UGauss{0}{1}\\left(\\frac{u-\\mathsf{E}\\hskip 1pt\\homV{t}}{\\sqrt{\\,\\mathsf{D}\\hskip 1pt\\homV{t}}}\\right),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{sdrthrjh}\n\\begin{aligned}\n\\mathsf{E}\\hskip 1pt{\\homV{t}}&=(\\mathsf{E}\\hskip 1pt{Y}\/\\mathsf{E}\\hskip 1pt{T})\\,t+\\mathsf{E}\\hskip 1pt{Y}(\\mathsf{D}\\hskip 1pt{T}-(\\mathsf{E}\\hskip 1pt{T})^{\\,2})\/(2(\\mathsf{E}\\hskip 1pt{T})^{\\,2})+\\overline{o}(1),\n\\\\[0pt]\n\\mathsf{D}\\hskip 1pt{\\homV{t}}&=(((\\mathsf{E}\\hskip 1pt{Y})^{\\,2}\\mathsf{D}\\hskip 1pt{T}+(\\mathsf{E}\\hskip 1pt{T})^{\\,2}\\mathsf{D}\\hskip 1pt{Y})\/(\\mathsf{E}\\hskip 1pt{Y})^3)\\,t+\\overline{o}(t),\n\\end{aligned}\n\\end{equation}\nwhence \\eqref{3etu6juh}.\n\nThe following theorem is a generalization of Theorem~2.2 in\n\\cite{[Malinovskii=2014b]}; it is worthwhile to compare it with Theorem~1 in\n\\cite{[Malinovskii=2014a]}.\n\n\\begin{theorem}\\label{dtfyjfgjk}\nAssume that $f_{T}(x)$ and $f_{Y}(x)$ are bounded above by a finite constant,\n$D^{2}>0$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$, and $\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$. Then for\n$t\\to\\infty$, we have\n\\begin{equation}\\label{eryhjmgh}\nu_{\\alpha,t}(c^{\\ast})=\\dfrac{D}{M^{\\,3\/2}}\\,\\NQuant{\\alpha\/2}\\sqrt{t}\\,\\big(1+\\overline{o}(1)\\big).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{dtfyjfgjk}]\\label{yukryikt}\nThe left-hand side of equation \\eqref{rtyujtkjtyk} is (see\n\\eqref{ewrktulertye})\n\\begin{equation*}\n\\P\\left\\{\\Tich{u,c}\\leqslant\nt\\right\\}=\\int_{0}^{t}\\P\\left\\{u+cv-Y<0\\right\\}\\,f_{\\T{1}}(v)\\,dv\n+\\int_{0}^{t}\\P\\left\\{v<\\Tich{u,c}\\leqslant\nt\\mid\\T{1}=v\\right\\}\\,f_{\\T{1}}(v)\\,dv,\n\\end{equation*}\nwhence for $c=c^{\\ast}$ \\eqref{rtyujtkjtyk} rewrites as\n\\begin{equation}\\label{ewtyjhjmfg}\n\\int_{0}^{t}\\P\\left\\{u+c^{\\ast} v<\\Y{1}\\right\\}f_{\\T{1}}(v)\\,dv\n+\\int_{0}^{t}\\P\\left\\{v<\\Tich{u,c^{\\ast}}\\leqslant\nt\\mid\\T{1}=v\\right\\}f_{\\T{1}}(v)\\,dy=\\alpha.\n\\end{equation}\nBearing in mind the second equality in \\eqref{asdfgreherXX}, the probability\n$\\P\\left\\{v<\\Tich{u,c^{\\ast}}\\leqslant t\\mid\\T{1}=v\\right\\}$ is approximated by\n\\begin{equation*}\n2\\left(\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\mathsf{E}\\hskip 1pt{T}(v\\,\\mathsf{E}\\hskip 1pt{Y}+u\\,\\mathsf{E}\\hskip 1pt{T})}{(\\mathsf{E}\\hskip 1pt{Y})^{2}D^{2}}}\\,\\right)\n-\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\mathsf{E}\\hskip 1pt{T}}{(\\mathsf{E}\\hskip 1pt{Y})^{2}D^{2}}}\n\\frac{u\\,\\mathsf{E}\\hskip 1pt{T}+v\\,\\mathsf{E}\\hskip 1pt{Y}}{\\sqrt{u\\,\\mathsf{E}\\hskip 1pt{T}+t\\,\\mathsf{E}\\hskip 1pt{Y}}}\\right)\\right).\n\\end{equation*}\n\nLet us show that $u_{\\alpha,t}(c^{\\ast})$ in \\eqref{eryhjmgh} is an asymptotic solution to\nequation \\eqref{ewtyjhjmfg}. First, bearing in mind that $\\mathsf{E}\\hskip 1pt{T}^{3}<\\infty$,\n$\\mathsf{E}\\hskip 1pt{Y}^{3}<\\infty$, it is easily seen that\n\\begin{equation*}\n\\int_{0}^{t}\\P\\{u+c^{\\ast} v-\\Y{1}<0\\}f_{\\T{1}}(v)\\,dv\\to 0,\\quad t\\to\\infty,\\\nu\\to\\infty.\n\\end{equation*}\nSecondly, it is easy to see that\n\\begin{equation*}\n\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\mathsf{E}\\hskip 1pt{T}(v\\,\\mathsf{E}\\hskip 1pt{Y}+u\\,\\mathsf{E}\\hskip 1pt{T})}{(\\mathsf{E}\\hskip 1pt{Y})^{2}D^{2}}}\\,\\right)\\to\n1,\\quad u\\to\\infty.\n\\end{equation*}\nSelecting in \\eqref{eryhjmgh} $u$ as $\\underline{O}\\left(t^{1\/2}\\right)$, we\nhave\n\\begin{equation*}\n\\frac{u\\,\\mathsf{E}\\hskip 1pt{T}+v\\,\\mathsf{E}\\hskip 1pt{Y}}{\\sqrt{u\\,\\mathsf{E}\\hskip 1pt{T}+t\\,\\mathsf{E}\\hskip 1pt{Y}}}\n=\\frac{u\\,\\mathsf{E}\\hskip 1pt{T}}{\\sqrt{t\\,\\mathsf{E}\\hskip 1pt{Y}}}\\,\\big(1+\\overline{o}(1)\\big),\\quad\nt\\to\\infty.\n\\end{equation*}\nTherefore, equation \\eqref{ewtyjhjmfg} reduces to\n\\begin{equation*}\n2\\left(1-\\UGauss{0}{1}\\left(\\sqrt{\\frac{\\mathsf{E}\\hskip 1pt{T}}{(\\mathsf{E}\\hskip 1pt{Y})^{2}D^{2}}}\n\\frac{u\\,\\mathsf{E}\\hskip 1pt{T}}{\\sqrt{t\\,\\mathsf{E}\\hskip 1pt{Y}}}\\right)\\right)=\\alpha,\n\\end{equation*}\nwhence the result.\n\\end{proof}\n\nThe following theorem is a generalization of Theorem~2.2 in\n\\cite{[Malinovskii=2014b]}. It is useful to compare it with Theorem~2 in\n\\cite{[Malinovskii=2014a]}, or Theorem~4.4 in \\cite{[Malinovskii=2009]}.\n\n\\begin{theorem}\\label{ergtrewghwerg}\nAssume that differentiable $f_{T}(x)$ and $f_{Y}(x)$ are bounded above by a\nfinite constant, and $D^{2}>0$, $\\mathsf{E}\\hskip 1pt({T}^{3})<\\infty$, $\\mathsf{E}\\hskip 1pt({Y}^{3})<\\infty$.\nThen for $c^{\\ast}=M^{-1}$ we have\n\\begin{equation}\\label{asdfgrfjhf}\nu_{\\alpha,t}(c)=\\begin{cases} \\big(c^{\\ast}-c\\big)\\,t+\\dfrac{D}{M^{\\,3\/2}}\\,\\funU{\\alpha,t}\n\\left(\\dfrac{M^{\\,3\/2}(c^{\\ast}-c)}{D}\\sqrt{t}\\,\\right)\\sqrt{t}, &0\\leqslant\nc\\leqslantc^{\\ast},\n\\\\[10pt]\n\\dfrac{D}{M^{\\,3\/2}}\\,\\funU{\\alpha,t}\n\\left(\\dfrac{M^{\\,3\/2}(c^{\\ast}-c)}{D}\\sqrt{t}\\,\\right)\\sqrt{t},&c>c^{\\ast},\n\\end{cases}\n\\end{equation}\nwhere for $t$ sufficiently large the function $\\funU{\\alpha,t}(y)$,\n$y\\in\\mathsf{R}$, is continuous and monotone increasing, as $y$ increases from\n$-\\infty$ to $0$, and monotone decreasing, as $y$ increases from $0$ to\n$\\infty$, and such that\\footnote{We recall that\n$0<\\NQuant{\\alpha}<\\NQuant{\\alpha\/2}$ for $0<\\alpha<\\frac12$.}\n\\begin{equation*}\n\\lim_{y\\to-\\infty}\\funU{\\alpha,t}(y)=0,\\\n\\lim_{y\\to\\infty}\\funU{\\alpha,t}(y)=\\NQuant{\\alpha}\n\\end{equation*}\nand $\\funU{\\alpha,t}(0)=\\NQuant{\\alpha\/2}\\,\\big(1+\\overline{o}(1)\\big)$, as\n$t\\to\\infty$.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{ergtrewghwerg}]\\label{fgjhrtjkrkj}\nThis proof is carried out in two stages. In each stage we make a suitable\nchange of variables. Its aim is to focus on the function $\\funU{\\alpha,t}(y)$,\n$y\\in\\mathsf{R}$, and to check its monotony using the standard criterion based on\nthe sign of its derivative; this is calculated by means of (see\nTheorem~\\ref{etrtjt}) the implicit function derivative theorem.\n\n\\par\\textbf{\\emph{Step~1.}}\nLet us consider the case $00.\n\\end{equation}\n\nThe original equation \\eqref{rtyujtkjtyk} rewrites as\n\\begin{equation*}\n\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}\\,\\big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y} =\\alpha.\n\\end{equation*}\n\nTo prove that $\\funU{\\alpha,t}(y)$, $y>0$, is monotone decreasing, we have to\nprove that $\\frac{d}{dy}\\,\\funU{\\alpha,t}(y)<0$, $y>0$. Referring to the\nimplicit function derivative theorem (see Theorem~\\ref{etrtjt}), we have\n\\begin{equation}\\label{retyjrkrt}\n\\left.\\frac{d}{dy}\\,\\funU{\\alpha,t}(y)=-\\left(\\frac{\\frac{\\partial}{\\partial\ny}\\Bigg(\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\Big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y} \\Bigg)}{\\frac{\\partial}{\\partial\nz}\\Bigg(\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\Big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\Bigg)}\\right)\n\\;\\right|_{z=\\funU{\\alpha,t}(y)}.\n\\end{equation}\nThe numerator is\n\\begin{equation}\\label{yujmyhmk}\n\\begin{aligned}\n&\\frac{\\partial}{\\partial y}\\left(\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\right)\n\\\\\n&=\\Bigg(\\frac{\\partial}{\\partial u}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\Bigg)\\,\\Bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\,\\underbrace{\\frac{\\partial}{\\partial\ny}\\,\\Bigg(\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y)\\Bigg)}_{\\frac{D\\sqrt{t}}{M^{\\,3\/2}}}\n\\\\\n&+\\Bigg(\\frac{\\partial}{\\partial c}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\Bigg)\\,\\bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\underbrace{\\frac{\\partial}{\\partial\ny}\\,\\Bigg(\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}\\,y\\,\\Bigg)}_{-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}},\n\\end{aligned}\n\\end{equation}\nwhose approximation, as $t\\to\\infty$, follows from \\eqref{fgyjfgkmhh},\n\\eqref{dsfgrehgdfhd}, \\eqref{esrtgeyheje}, \\eqref{wqe4ryttjfyjk}; for large $t$\nthis is obviously negative. The denominator is\n\\begin{equation}\\label{ergtewgwergy}\n\\begin{aligned}\n&\\frac{\\partial}{\\partial z}\\left(\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\right)\n\\\\\n&=\\Bigg(\\frac{\\partial}{\\partial u}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\Bigg)\\,\\Bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y),\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\,\\underbrace{\\frac{\\partial}{\\partial\nz}\\,\\Bigg(\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,(z+y)\\Bigg)}_{\\frac{D\\sqrt{t}}{M^{\\,3\/2}}},\n\\end{aligned}\n\\end{equation}\nwhose approximation, as $t\\to\\infty$, follows from \\eqref{fgyjfgkmhh},\n\\eqref{esrtgeyheje}, \\eqref{wqe4ryttjfyjk}; for large $t$ this is obviously\nnegative, whence the result.\n\n\\par\\textbf{\\emph{Step~2.}}\nWe continue the proof with investigating the case $c>c^{\\ast}=M^{-1}$. We switch\nfrom the variables $u$ and $c$ to the variables\n\\begin{equation}\\label{drtgrfjhnrtjhntr}\nz=\\dfrac{u}{DM^{-3\/2}\\sqrt{t}}>0,\\quad\ny=\\frac{(M^{-1}-c)\\sqrt{t}}{DM^{-3\/2}}<0,\n\\end{equation}\n\nLet us rewrite the original equation \\eqref{rtyujtkjtyk} as\n\\begin{equation*}\n\\P\\big\\{\\Tich{u,c}\\leqslant t\\big\\}\\,\\Big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}z,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}=\\alpha.\n\\end{equation*}\n\nTo prove that $\\funU{\\alpha,t}(y)$, $y<0$, is monotone increasing, we have to\nprove that $\\frac{d}{dy}\\,\\funU{\\alpha,t}(y)>0$, $y<0$. Referring to the\nimplicit function derivative theorem (see Theorem~\\ref{etrtjt}), we have\n\\begin{equation*}\n\\left.\\frac{d}{dy}\\,\\funU{\\alpha,t}(y)=-\\frac{\\frac{\\partial}{\\partial\ny}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\Big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\\,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}}{\\frac{\\partial}{\\partial\nz}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\Big|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\\,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}}\\;\\right|_{z=\\funU{\\alpha,t}(y)}.\n\\end{equation*}\nThe numerator is\n\\begin{equation}\\label{tryjutikty}\n\\begin{aligned}\n&\\frac{\\partial}{\\partial y}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\n\\\\\n&=\\Bigg(\\frac{\\partial}{\\partial c}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\Bigg)\\,\\bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\\,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\frac{\\partial}{\\partial\ny}\\,\\Bigg(\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}\\,y\\,\\Bigg),\n\\end{aligned}\n\\end{equation}\nwhose approximation, as $t\\to\\infty$, follows from \\eqref{dsfgrehgdfhd},\n\\eqref{esrtgeyheje}, \\eqref{wqe4ryttjfyjk}; for large $t$ this is obviously\nnegative. The denominator is\n\\begin{equation}\\label{yuktykt}\n\\begin{aligned}\n&\\frac{\\partial}{\\partial z}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\,\\bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\\,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\n\\\\\n&=\\Bigg(\\frac{\\partial}{\\partial u}\\P\\big\\{\\Tich{u,c}\\leqslant\nt\\big\\}\\Bigg)\\,\\Bigg|_{u=\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z,\\,\nc=\\frac{1}{M}-\\frac{D}{M^{\\,3\/2}\\sqrt{t}}y}\\,\\frac{\\partial}{\\partial\nz}\\,\\Bigg(\\frac{D\\sqrt{t}}{M^{\\,3\/2}}\\,z\\,\\Bigg).\n\\end{aligned}\n\\end{equation}\nwhose approximation, as $t\\to\\infty$, follows from \\eqref{fgyjfgkmhh},\n\\eqref{esrtgeyheje}, \\eqref{wqe4ryttjfyjk}; for large $t$ this is obviously\nnegative, whence the result.\n\\end{proof}\n\n\\subsection{Monotony and convexity of fixed-probability level}\\label{defyguyl}\n\nThe fixed-probability level $u_{\\alpha,t}(c)$, $c\\geqslant 0$, \\emph{for all} $t$\nmonotone decreases, as $c$ increases. The following result, called weak-form\nconvexity, differs in that it is established by our means only for $t$ large.\n\n\\begin{theorem}[Weak-form convexity]\\label{ew5tuy65}\nSuppose that conditions of Theorem \\ref{ergtrewghwerg} are satisfied. Then for\n$t>0$ sufficiently large, the function $u_{\\alpha,t}(c)$, $c\\geqslantc^{\\ast}$, is convex.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{ew5tuy65} requires dramatically large space and is\nleft to the reader. Nevertheless, it is quite clear\\footnote{In the diffusion\nmodel, the proof of convexity was carried out with complete details in\n\\cite{[Malinovskii=2014a]}.}: one should check that for $c\\geqslantc^{\\ast}$ the\ninequality $\\frac{d^2}{dc^2}u_{\\alpha,t}(c)>0$ holds for $t$ sufficiently large. This\nstarts with equality \\eqref{ewryjhnmfm}, proceeds with, first, calculation of\nthe second-order derivatives as it is done in Section~\\ref{rgerhyryttryjh},\nand, second, approximating them as it is done in Section~\\ref{wqstgerhs}. The\nproof of positivity of the approximation for $\\frac{d^2}{dc^2}u_{\\alpha,t}(c)>0$, when\n$t$ is large, brings the proof to a close.\n\n\\subsection{Heuristic fixed-probability level}\\label{therhrher}\n\nLet us introduce $u_{\\alpha,t}^{[\\mathcal{M}]}(c)$, $c\\geqslant 0$, which is a positive solution to\nthe equation\\footnote{Recall that $\\AInt{M}{u,c}(t)$ is an alternative notation\nfor $\\Elem{u,c}{1}(t)=\\int_{0}^{\\frac{ct}{u}}\\frac{1}{x+1}\\,\n\\Ugauss{cM(x+1)}{\\frac{c^{2}D^{\\,2}}{u}(x+1)}(x)\\,dx$.}\n\\begin{equation}\\label{edthr6j}\n\\AInt{M}{u,c}(t)=\\alpha,\n\\end{equation}\nwhose right-hand side is expressed\\footnote{See \\eqref{rterherher} with\n$p=-{1}\/{2}$, \\eqref{drgerherhe}, and \\eqref{xzfvbdbf}.} in a closed form.\nPlainly, to get \\eqref{edthr6j}, we replaced the left-hand side of the original\nequation \\eqref{rtyujtkjtyk} by an approximation found in\nTheorem~\\ref{esrytrf}.\n\n\\begin{theorem}\\label{drtyryhrXX}\nFor\\, $0\\leqslant cKc^{\\ast}$, $K>1$, we have\n\\begin{equation}\\label{aegwege}\nu_{\\alpha,t}^{[\\mathcal{M}]}(c)=\\frac{D^{\\,2}}{M^{\\,2}}\\,x_{\\alpha}(c)\\big(1+\\overline{o}(1)\\big),\\quad\nt\\to\\infty,\n\\end{equation}\nwhere $x_{\\alpha}(c)$ is a positive solution to the equation\n\\begin{equation}\\label{sdfgbfdnbf}\n\\left(1-\\UGauss{0}{1}\\left(\\frac{cM-2}{cM}x^{1\/2}\\,\\right)\\right)\n\\exp\\left\\{-2\\,\\frac{cM-1}{c^2M^{\\,2}}\\,x\\,\\right\\}\n+\\UGauss{0}{1}\\left(x^{1\/2}\\,\\right)=1+\\alpha.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{tyurtutXX}]\nFor\\, $c>Kc^{\\ast}$, $K>1$, and $u_{t}(c)=\\frac{D^{\\,2}}{M^{\\,2}}\\,z_{t}(c)$, where\n$z_{t}(c)$ is a solution to \\eqref{sdfgbfdnbf}, we have\n$\\AInt{M}{u,c}(t)\\,\\big|_{\\,u=u_{t}(c)}\\sim\\alpha$, $t\\to\\infty$, whence the\nresult.\n\\end{proof}\n\nThe following result is an alternative (or addition) to\nTheorem~\\ref{ergtrewghwerg} for $0\\leqslant c\\leqslantc^{\\ast}$, when both\n$u_{\\alpha,t}(c)$ and $u_{\\alpha,t}^{[\\mathcal{M}]}(c)$ tend to infinity, as $t\\to\\infty$. Of particular\ninterest is the right $t^{-1\/2}$-neighborhood of the point $c^{\\ast}$.\n\n\\begin{figure}[t]\n\\center{\\includegraphics[scale=.8]{Fig=4}}\n\\caption{\\small Graphs ($X$-axis ic $c$) of $u_{\\alpha,t}^{[\\mathcal{M}]}(c)$ and simulated values of\n$u_{\\alpha,t}(c)$ for $T$ exponentially distributed with parameter $\\delta=4\/5$, $Y$\nPareto with parameters $\\parA{Y}=10$, $\\parB{Y}=0{.}05$, $\\alpha=0{.}05$, and\n$t=200$. Vertical grid line: $c^{\\ast}=1{.}7778$. Horizontal grid line: simulated\n$u_{\\alpha,t}(c^{\\ast})=80$.}\\label{w4ytrujrtjrY}\n\\end{figure}\n\n\\begin{theorem}\\label{asdgshbndfnf}\nSuppose that conditions of Theorem~\\ref{ergtrewghwerg} are satisfied. Then for\n$0Kc^{\\ast}$, $K>1$, i.e., when they do not tend to infinity, as $t\\to\\infty$.\n\n\\subsection{Elementary asymptotic bounds for the fixed-probability level}\\label{srtyhtrj}\n\n\\begin{figure}[t]\n\\center{\\includegraphics[scale=.8]{Fig=5}}\n\\caption{\\small Graphs ($X$-axis is $c$) of two-sided bounds\n\\eqref{adsfgsdbsb}, when $0\\leqslant c\\leqslantc^{\\ast}$, of the upper bound, when\n$c>c^{\\ast}$, and of simulated values of $u_{\\alpha,t}(c)$, drawn for $T$ which is Erlang\nwith parameters $\\delta=8\/5$, $k=2$ and $Y$ exponentially distributed with\nparameter $\\rho=3\/5$, and $\\alpha=0{.}05$, $t=200$. Vertical grid line:\n$c^{\\ast}=4\/3$. Horizontal grid line: $u_{\\alpha,t}(c^{\\ast})=48$.}\\label{egrrgd}\n\\end{figure}\n\nFor $0\\leqslant c\\leqslantc^{\\ast}$, elementary asymptotic bounds\n\\begin{equation}\\label{adsfgsdbsb}\n\\begin{aligned}\n(c^{\\ast}-c)\\,t+\\dfrac{D}{M^{\\,3\/2}}\\,&\\NQuant{\\alpha}\\sqrt{t}\\,(1+{o}(1))\\leqslantu_{\\alpha,t}(c)\n\\\\[0pt]\n&\\leqslant(c^{\\ast}-c)\\,t+\\dfrac{D}{M^{\\,3\/2}}\\,\\NQuant{\\alpha\/2}\\sqrt{t}\\,(1+{o}(1)),\\quad\nt\\to\\infty,\n\\end{aligned}\n\\end{equation}\nfollow straightforwardly from \\eqref{asdfgrfjhf}.\n\nFor $c>c^{\\ast}$, elementary upper bounds, quite satisfactory for $c>Kc^{\\ast}$ with\n$K>1$ large enough, are also straightforward in many cases of interest. In\nparticular, when $T$ and $Y$ are exponentially distributed with parameters\n$\\delta$ and $\\rho$, we have $c^{\\ast}=\\delta\/\\rho$ and (see, e.g.,\n\\cite{[Rolski=et=al.=1999]})\n$\\P\\{\\Tich{u,c}<\\infty\\}=(1-\\varkappa\/\\rho)\\,e^{-\\varkappa\\,u}$ for all\n$u\\geqslant 0$, where $\\varkappa=\\rho-\\delta\/c$. This rewrites as\n$\\P\\{\\Tich{u,c}<\\infty\\}=\\left(\\delta\/(c\\rho)\\right)\n\\,\\exp\\left\\{-(\\rho-\\delta\/c)\\,u\\right\\}$; by simple calculations we have\n\\begin{equation*}\nu_{\\alpha,t}(c)\\leqslant\\max\\left\\{\\,0,-\\frac{\\ln\\left(\\alpha\nc\\rho\/\\delta\\right)}{\\rho-\\delta\/c}\\right\\},\\quad c>c^{\\ast}.\n\\end{equation*}\n\nWhen $T$ is exponentially distributed with parameter $\\delta$ and the\ndistribution of $Y$ is light-tailed, but non-exponential, we have\n$c^{\\ast}=\\delta\\,\\mathsf{E}\\hskip 1pt{Y}$ and (see, e.g., \\cite{[Rolski=et=al.=1999]})\n$\\P\\{\\Tich{u,c}<\\infty\\}\\leqslant e^{-\\varkappa\\,u}$ for all $u\\geqslant 0$,\nwhere $\\varkappa$ is a positive solution to the equation\n$\\mathsf{E}\\hskip 1pt\\exp\\{\\varkappa\\,Y\\}=1+c\\varkappa\/\\delta$. Therefore, we have\n\\begin{equation*}\nu_{\\alpha,t}(c)\\leqslant-{\\ln\\alpha}\/{\\varkappa},\\quad c>c^{\\ast}.\n\\end{equation*}\n\nWhen $Y$ is exponentially distributed with parameter $\\rho$ and the\ndistribution of $T$ is arbitrary, we have $c^{\\ast}=1\/(\\rho\\,\\mathsf{E}\\hskip 1pt{T})$ and (see,\ne.g., \\cite{[Rolski=et=al.=1999]})\n$\\P\\{\\Tich{u,c}<\\infty\\}=(1-\\varkappa\/\\rho)\\,e^{-\\varkappa u}$ for all\n$u\\geqslant 0$, where $\\varkappa$ is a positive solution to the equation\n$\\mathsf{E}\\hskip 1pt\\exp\\{-\\varkappa\\,c\\,T\\}=1-\\varkappa\/\\rho$. Bearing in mind that\n$1-\\varkappa\/\\rho\\leqslant 1$, we have\n\\begin{equation*}\nu_{\\alpha,t}(c)\\leqslant-{\\ln\\alpha}\/{\\varkappa},\\quad c>c^{\\ast}.\n\\end{equation*}\nIn this case, the elementary bounds for the fixed-probability level are shown\nin Fig.~\\ref{egrrgd}.\n\n\\section{Derivatives of implicit function}\n\nThe derivatives of an implicit function defined by the equation $F(x,y)=0$,\n$x,y\\in\\mathsf{R}$, can be obtained (see, e.g., \\cite{[Widder=1947]}, Chapter~I,\n\\S~5.2 and \\S~5.3) without finding this implicit function in closed form.\n\n\\begin{theorem}\\label{etrtjt}\nAssume that the function $F(x,y)$, $x,y\\in\\mathsf{R}$, possesses partial\nderivatives up to second order, which are continuous in some neighborhood of a\nsolution $(x_0,y_0)$ of the equation $F(x,y)=0$. If $\\frac{\\partial}{\\partial\ny}F(x_0,y_0)\\ne 0$, then there exists an $\\epsilon>0$ and a unique continuously\ndifferentiable function $f$ such that $f(x_0)=y_0$ and $F(x,f(x))=0$ for\n$|x-x_0|<\\epsilon$. Moreover, for $|x-x_0|<\\epsilon$ we have\n\\begin{equation}\\label{ewqrfghwre}\n\\left.f^{\\,\\prime}(x)=-\\frac{\\frac{\\partial}{\\partial\nx}F(x,y)}{\\frac{\\partial}{\\partial y}F(x,y)}\\;\\right|_{\\,y=f(x)},\n\\end{equation}\nand\n\\begin{equation}\\label{ewryjhnmfm}\n\\begin{aligned}\nf^{\\,\\prime\\prime}(x)=&-\\left(\\frac{\\frac{\\partial^{2}}{\\partial\nx^{2}}F(x,y)}{\\frac{\\partial}{\\partial\ny}F(x,y)}-\\frac{2\\,\\frac{\\partial^{2}}{\\partial x\\,\\partial y}F(x,y)\\,\n\\frac{\\partial}{\\partial x}F(x,y)}{\\left(\\frac{\\partial}{\\partial\ny}F(x,y)\\right)^{2}}\\left.+\\frac{\\frac{\\partial^{2}}{\\partial\ny^{2}}F(x,y)\\left(\\frac{\\partial}{\\partial x}F(x,y)\\right)^{2}}\n{\\left(\\frac{\\partial}{\\partial\ny}F(x,y)\\right)^{3}}\\right)\\,\\right|_{\\,y=f(x)}.\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\n\nRecently, the fullerenes C$_N$ which have the hollow cage\nstructures of carbons have been intensively investigated.\nThere are several experimental indications that the doped\nfullerenes show polaronic properties due to the Jahn-Teller\ndistortion, for example: (1) The electron spin resonance (ESR)\nstudy [1] on the radical anion of C$_{60}$ has revealed the\nsmall $g$-factor, $g=1.9991$, and this is associated with\nthe residual orbital angular momentum due to the Jahn-Teller\ndistortion. (2) Photoemission studies [2] of C$_{60}$ and\nC$_{70}$ doped with alkali metals have shown peak structures,\nwhich cannot be described by a simple band-filling picture.\n(3) When poly(3-alkylthiophene) is doped with ${\\rm C}_{60}$ [3],\ninterband absorption of the polymer is remarkably suppressed\nand the new absorption peak evolves in the low energy range.\nThe Jahn-Teller splitting of LUMO in C$_{60}^-$ state and\/or\nthe Coulomb attraction of positively charged polaron to\nC$_{60}^-$ might occur. (4) The luminescence of neutral ${\\rm C}_{60}$\nhas been measured [4]. There are two peaks around 1.5 and\n1.7eV below the gap energy 1.9eV, interpreted by the effect\nof the polaron exciton. In addition, the experiments on the\ndynamics of photoexcited states have shown the interesting\nroles of polarons [5].\n\n\nSeveral authors [6,7] have proposed an interacting\nelectron-phonon model in order to describe the polarons\nin doped ${\\rm C}_{60}$. The Su-Schrieffer-Heeger (SSH) model\nof conjugated polymers [8] has been extended to fullerenes\nin these works [6,7]. The $\\pi$-electrons hop between nearest\nneighbor sites. The hopping integral depends on the change\nof the bond length linearly. The bond modelled by the classical\nharmonic spring is the contribution from the $\\sigma$ bonding.\nIn the previous paper [9], we have calculated\nlattice distortion and electronic structures of the molecules,\nwhere one to six electrons are added, or one to ten electrons\nare removed. When ${\\rm C}_{60}$ is doped with one or two electrons\n(or holes) (the lightly doped case), the additional charges\naccumulate at twenty carbon atoms along almost an equatorial\nline of the molecule. The dimerization becomes the weakest\nalong the same line. Two energy levels, the occupied state\nand the empty state, intrude largely in the gap. These are\nthe polaron effects. The changes of the electronic structures\nof the molecules with more charges (the heavily doped case)\nhave been reported in Ref. 9. However, the complex changes of\nlattice geometries and electron density distributions have not\nshown yet. Section III of this paper will be devoted\nto this purpose.\n\n\nIn the study [9], the molecule has been assumed as isolated\nand the calculations have been done within the adiabatic\napproximation. However, it has been discussed [10] that the\nwidth of the zero point motion in conjugated polymers and\nfullerene tubules is of the order of 0.01\\AA. The same order\nof magnitude would be expected in ${\\rm C}_{60}$ [11]. This is also\nof the same order as the difference between the short and\nbond lengths: 0.05\\AA{\\ } [12]. Therefore, the polaronic\ndistortion described in the adiabatic approximation might\nchange its structures by thermal fluctuations. The thermal\nfluctuation effects can be simulated by introducing the bond\ndisorder potentials. The doped SSH system with gaussian\nbond disorders will be studied in Sec. IV.\n\n\nIt is known that ${\\rm C}_{60}$ molecules contain a small amount of\n${\\rm C}_{70}$ as impurities [4]. Sometimes the ${\\rm C}_{60}$ films and\nsolids are contaminated with oxygens [4,13]. There would\nremain misalignment of molecules in ${\\rm C}_{60}$ solids. These\neffects would be good origins of additional potentials acting\non $\\pi$-electrons at the carbon sites. They can be modelled\nby random site disorders. Site disorder effects will be\ninvestigated in Sec. V.\n\n\nSample average is performed over sufficient number of independent\ndisorder configurations. The distribution functions of bond\nlengths and electron densities are calculated with changing\nthe disorder strength and the additional electron number.\nWe mainly consider stability of polaron excitations as well\nas dimerization patterns. We find that polarons and\ndimerizations in lightly doped cases are rather stable against\ndisorders. This property is common to bond and site disorder\neffects. In more heavily doped cases, the several peaks in\ndistribution functions merge into a single peak. This\nindicates that polaron structures are broken while the\ndimerization strength decreases, owing to doped charges\nand disorders.\n\n\nThis paper is organized as follows. We explain the model in the next\nsection. In Sec. III, the lattice and electronic patterns of\nelectron-doped ${\\rm C}_{60}$ are extensively reported. In Sec. IV,\nbond disorder effects are studied. Sec. V is devoted to site\ndisorder effects. We close this paper with summary and discussion\nin Sec. VI.\n\n\n\\section{MODEL}\n\n\nThe extended SSH hamiltonian for the fullerene ${\\rm C}_{60}$ [9],\n\\begin{equation}\nH_{\\rm SSH} = \\sum_{\\langle i,j \\rangle, \\sigma} ( - t_0 + \\alpha y_{i,j} )\n( c_{i,\\sigma}^\\dagger c_{j,\\sigma} + {\\rm h.c.} )\n+ \\frac{K}{2} \\sum_{\\langle i,j \\rangle} y_{i,j}^2,\\\\\n\\end{equation}\nis studied with gaussian bond disorders,\n\\begin{equation}\nH_{\\rm bond} = \\alpha \\sum_{\\langle i,j \\rangle, \\sigma} \\delta y_{i,j}\n( c_{i,\\sigma}^\\dagger c_{j,\\sigma} + {\\rm h.c.} ),\n\\end{equation}\nas well as with gaussian site disorders,\n\\begin{equation}\nH_{\\rm site} = \\sum_{i,\\sigma} U_i c_{i,\\sigma}^\\dagger c_{i,\\sigma}.\n\\end{equation}\nIn $H_{\\rm SSH}$, $c_{i,\\sigma}$ is an annihilation operator of a\n$\\pi$-electron; the quantity $t_0$ is the hopping integral of the ideal\nundimerized system; $\\alpha$ is the electron-phonon coupling; $y_{i,j}$\nindicates the bond variable which measures the length change of the bond\nbetween the $i$- and $j$-th sites from that of the undimerized system;\nthe sum is taken over nearest neighbor pairs $\\langle i j \\rangle$; the\nsecond term is the elastic energy of the lattice; and the quantity $K$\nis the spring constant. The part $H_{\\rm bond}$ is the disorder\npotential due to the gaussian modulation of transfer integrals. The\ndisorder strength is measured by the standard deviation $y_s$ of the\nbond variable modulations $\\delta y_{i,j}$. The mean value of $\\delta\ny_{i,j}$ is assumed to be zero. The term $H_{\\rm site}$ is the site\ndisorder potential. The quantity $U_i$ is the strength of the disorder\nat the $i$th site, with the standard deviation $U_s$. The model is\nsolved with the assumption of the adiabatic approximation and by an\niteration method used in [9].\n\n\n\\section{POLARONS IN C$_{\\rm 60}$}\n\n\nIn this section, we present detailed discussion on the\npolarons of the electron doped ${\\rm C}_{60}$. Further data of the\nlattice and electron density structures are reported.\nIn Ref. 9, only the electronic energy levels and averaged\ndimerization strengths have been discussed for heavily doped ${\\rm C}_{60}$.\n\n\nIn Fig. 1, the unfolded figure of the truncated icosahedron\nis shown. When we make the paper model of ${\\rm C}_{60}$, we cut\nthe figure along the outer edges of white hexagons. After\nfolding edges between neighboring hexagons and combining\nseveral edges, we obtain a closed structure of the molecule.\nThe shadowed hexagons in Fig. 1 become pentagons.\nThe symbols, A-L, specify different pentagons.\n\n\nThe model Eq. (1) is numerically solved for the parameters:\n$t_0 = 2.1$eV, $\\alpha = 6.0$eV\/\\AA, and $K = 52.5$eV\/\\AA$^2$.\nThese give the characteristic scales for the neutral ${\\rm C}_{60}$:\nthe total $\\pi$-band width $6t_0 = 12.6$eV, the energy gap\n1.904eV, and the difference of the bond length between the\nshort and long bonds 0.4557\\AA. These values are typical.\nThey are slightly different from those in Ref. 9, but the\nqualitative features of solutions are not affected by the\nslight modifications. Quantitative differences are small, too.\nThe total electron number is changed within $N \\leq N_{\\rm el}\n\\leq N+6$, $N = 60$ being the number of sites.\n\n\nIn Fig. 2, the magnitudes of bond variables are shown in upper\nfigures by changing the electron number. The bond is represented\nby a bold line when it is shorter and $y_{i,j} < 0$. The bond\nis represented by a dashed line when it is longer and\n$y_{i,j} > 0$. The thickness of the line indicates\n$|y_{i,j}|$. The valence number (the negative of the\nadditional electron number) of the doped molecule is\naccompanied with each figure. The lower figures show the\nadditional electron density. The area in each circle is\nproportional to the absolute value.\n\n\nIn ${\\rm C}_{60}^{-1,-2}$, the most (about 70-80 percent) of the\nadditional charges accumulate at twenty carbons along an\nequatorial line of the molecule. At the same time, the\ndifference between the lengths of the short and long bonds\nbecomes smallest at sites along the same equatorial line.\nThis means that the dimerization strength is weakest. Two\nnondegenerate energy levels intrude largely in the energy\ngap as shown in Fig. 2(a) of Ref. 9. These lattice and\nelectronic structures are the same as those of polarons in\nconjugated polymers, so we named the changes as polaron\nexcitations. A fivefold axis penetrates between the centers\nof the pentagons, A and H.\n\n\nWhen ${\\rm C}_{60}$ is doped with three electrons, the symmetry is highly\nreduced. There is only an inversion symmetry. Only two sites have\nthe same electron density. Only two bonds have the same length.\nAs shown in Fig. 2, additional electron densities have large values\nat the twenty carbons along the equatorial line as well as\nat the other sites near the pentagons, A and H. The energy levels\nare shown in Ref. 9. The 31th wavefunction has large amplitudes\nat sites along the equatorial line, while the 32th one has larger\namplitudes at the other sites. The distribution of the electron\ndensity reflects this property.\n\n\nWhen the molecule is further doped and the additional electron\nnumber is four, the dimerization pattern changes qualitatively.\nThe symmetry becomes higher and there is a threefold axis\nwhich penetrates the center of the hexagon surrounded by the\npentagons, B, C, and L. The molecule doped with five electrons\nhas the similar bond alternation pattern and the same symmetry\n(the figures are not shown for the simplicity).\nFinally, in the molecule doped with six electrons, the dimerization\nalmost disappears and is rather negligible. The icosahedral symmetry\nrecovers again.\n\n\n\\section{BOND-DISORDER EFFECTS}\n\n\nThe SSH model $H_{\\rm SSH}$ is solved with the bond disorders\n$H_{\\rm bond}$. The additional electron number,\n$n \\equiv N_{\\rm el} - N$, is changed within $0 \\leq n \\leq 6$.\nThe most realistic origin for bond disorders is the thermal\nfluctuation of phonons. So, we shall change the strength of\nthe disorder in the range comparable with that of the amplitude\nof thermal fluctuations. It has been discussed [10] that the width\nof the zero point motion of phonons is 0.03-0.05\\AA{\\ } in\nconjugated polymers and is of the same order in fullerenes.\nSo, it is reasonable to assume that the maximum value\nof $y_s$ is of the similar magnitude. We take $y_s = 0.01,\n0.03,$ and $0.05$\\AA.\n\n\nA fairly large number of mutually independent samples of\ndisorders are generated, and the model, Eqs. (1) and (2),\nis solved for each sample. The bond variable and electron\ndensity are counted in order to draw histograms of distributions.\nThe sample number 5000 yields good convergence.\n\n\nFigure 3 shows the distributions of bond variables $D(y)$.\nThe thick, thin, and dashed lines are for $y_s = 0.01, 0.03$,\nand $0.05$\\AA, respectively. The ordinate is normalized so\nthat the area between the curve and the abscissa is unity.\nFigure 3(a) for the neutral ${\\rm C}_{60}$ shows the two peak\nstructure, related with presence of the dimerization: the\npositive $y_{i,j}$ corresponds to the longer bond, while\nthe negative one to the short bond. The magnitude of the\nzero point motion would be of the order 0.01\\AA.\nFor example, the treatment of the $H_g$-type phonon within\nthe framework of the SSH model by Friedman and Harigaya has\nresulted that the magnitude is about 0.02-0.03\\AA{\\ } as the\nadiabatic energy curve in Ref. 11 indicates. The curve of\n$y_s = 0.01$\\AA{\\ } has two peaks which are clearly separated.\nThe curve of $y_s = 0.03$\\AA{\\ } still shows distinct peaks.\nTherefore, the dimerization survives thermal fluctuations\nin the neutral molecule. Actually, the nuclear magnetic\nresonance (NMR) [12] shows the existence of the dimerization. When doped\nfurther up to ${\\rm C}_{60}^{-1,-2}$, the dimerization seems to remain\nagainst disorders: the two peaks can be identified. However,\na new peak emerges between the two peaks for $y_s = 0.01$\\AA{\\ }\nwhen doped with two electrons. This peak corresponds to the\nsmall bond variables which have been located along the\nequatorial line in the impurity free case. Thus,\n{\\sl the polaronic distortion seems to persist}, too.\n\n\nFigure 4 shows the distributions of the electron density per site\n$D(\\rho)$. In $\\rho$, the electron density of the impurity-free\nhalf-filled system is subtracted. The notations of the lines\nare the same as in Fig. 3. Figure 4(a) shows the nearly unform\nelectron density. When doped with one or two electrons,\na shoulder develops at the positive-$\\rho$ side of the curve.\nThis is owing to the accumulation of the extra charge in the\nlimited carbon sites of the molecule as found in Fig. 2.\nThe dimerization begins to be broken in the same portion.\nAnd also, the central peak around $\\rho = 0$ still remains,\ndue to the smaller changes of the electron density at sites\nof the pentagons, A and H, of Fig. 2. Thus, {\\sl the polaronic\ncharge distribution persists in the presence of bond disorders}.\n\n\nNext, we discuss heavily doped cases with $n \\geq 3$.\nIn the distribution function of the bond variables, the three peaks\nmerge into a single peak centered around $y = 0$\\AA. This is\nowing to the dimerization, the strength of which became very\nsmaller. The polaronic distortion becomes smaller also, as indicated\nby the averaged dimerization $\\langle | y_{i,j} | \\rangle $\npresented in Table III of Ref. 9. The electron distribution\nfunctions shown in Figs. 4(d) and (e) have a largest peak\ncentering the value $n \/ N$ which is the\nresult of the uniform doping. This is also related with the breakdown\nof the bond alternation pattern.\n\n\n\\section{SITE-DISORDER EFFECTS}\n\n\nThe model Eqs. (1) and (3) is solved for each sample of site\ndisorder potentials. Taking the number of samples up to 5000\nyields nice convergence of distribution functions. We assume three\nvalues for disorder strength: $U_s = 0.5, 1.0,$ and $2.0$eV.\nThe calculation of Madelung potential in the solid ${\\rm C}_{60}$ [14]\nhas yielded the variation of the potential on the surface\nof ${\\rm C}_{60}$ within 0.5eV. Therefore, the misalignment of ${\\rm C}_{60}$\nin the solid gives rise to the similar strength of site\ndisorders. The other intercalated impurities (remaining\nC$_{\\rm 70}$ [4], oxygens [4,13], dopants [2], and so on)\nmight yield site disorder potentials of the order of 1eV.\nThese are the realistic origins of site disorders.\nThe additional electron number $n$ is changed up to six.\n\n\nFig. 5 shows the distribution functions of bond variables $D(y)$.\nThe thick, thin, and dashed lines are for $U_s = 0.5, 1.0$,\nand $2.0$eV, respectively. While the molecule is weakly doped\n($n = 1,2$), the dimerization tends to survive disorder potentials.\nThis is easily seen by thick and thin lines in Figs. 5(b) and (c).\nThe dashed lines have a single peak. This is due to the strong\ndisorder potential comparable to the size of the energy gap of the undoped\nmolecule; the dimerization becomes undisernible, when energies of the\noccupied and unoccupied electronic states are closer. The actual\nsite potentials would not so strong as that of the dashed line,\nin view of the screening effects due to the $\\pi$-electrons spread\nover the surface of the molecule. Therefore, {\\sl the dimerization\npersists strongly when site disorders are present}.\n\n\nWhen the doping proceeds further ($3 \\leq n \\leq 6$), the two major\npeaks join into a single peak in $D(y)$. This shows that the dimerization\nis easy to disappear due to the disorder potentials as well as\nthe densely accumulated extra charges.\n\n\nFigure 6 shows charge density distributions $D(\\rho)$.\nWe show results only for $n = 0, 3,$ and 6, because the qualitative\nfeatures are the same for all $n$. The notation of the lines is the same\nas in Fig. 5. The charge density is directly modulated by the\nsite disorders. So, each curve has the shape near the gaussian\ndistribution. The value of $\\rho$ at the peak is close to $n\/N$.\n\n\n\\section{SUMMARY AND DISCUSSION}\n\n\nEffects on C$_{\\rm 60}$ by thermal fluctuations of phonons\nhave been simulated by bond disorder potentials. Next,\nmisalignment of C$_{\\rm 60}$ molecules in a crystal, and\nother intercalated impurities (remaining C$_{\\rm 70}$,\noxygens, dopants, and so on) have been studied with site\ndisorder potentials. The extended SSH model for doped\nC$_{\\rm 60}$ has been solved with the assumption of the\nadiabatic approximation. The distributions of bond lengths\nand electron densities, $D(y)$ and $D(\\rho)$, have been\nshown as functions of the disorder strength and the\nadditional electron number. Stability of polaron excitations\nas well as dimerization patterns have been considered.\n\n\nMain conclusions are common to bond and site disorder effects.\nPolarons and dimerizations in lightly doped cases\n(C$_{\\rm 60}^{-1,-2}$) are relatively stable against disorders.\nThis property has been indicated by peak structures in\ndistribution functions. In more heavily doped cases, the\nseveral peaks merge into a single peak, showing the\nbreakdown of polaron structures as well as the decrease\nof the dimerization strength.\n\n\nHowever, there exist qualitative differences between bond and site\ndisorder effects. In the bond disorder problem, the bond length is\naffected directly by the disorder potentials, but the charge density\nis modulated indirectly. In the site disorder problem, the charge\ndensity is modulated directly by the disorder potentials. Therefore,\nthe distribution function of charge density shows the apparent peak\nstructure related with the dimerization and polaronic distribution in\nthe bond disorder problem, but it has only one peak in the site\ndisorder problem.\n\n\nThen, how is our finding related with experiments? The NMR\ninvestigation [12] gives the evidence that there are two bond lengths\nin ${\\rm C}_{60}$. The single molecule will be always in the presence\nof some kinds of external potentials. These potentials could be\neffectively regarded as disorders. The presence of the two bond\nlengths in actual molecules agrees with our result that the\ndimerization is relatively stable against disorders. The ESR\nstudy [1] of ${\\rm C}_{60}$ monoanion in the solvent shows the reduced\n$g$-factor. This is interpreted as the result of the Jahn-Teller\ndistortion, in other words, the polaronic distortion. The molecules\nin the solvent would be affected by the strong site disorders as\nwell as bond disorders. Nevertheless, the effect related with\npolarons is observed. This is again in accord with our\nconclusion that polarons are stable in lightly doped ${\\rm C}_{60}$ in\ndisordered external potentials.\n\n\nThe electrochemical experiment [15] can produce ${\\rm C}_{60}$ anions doped with\nup to six electrons. The molecule can be doped with six electrons\nin the solid also. The photoemission experiments [2] show that the\nmaximumly doped ${\\rm C}_{60}$ solid is an insulator. This accords with\nthe present calculation. However, it is not certain whether the\ndimerizations still remain or not in heavily doped ${\\rm C}_{60}$.\nIn view of the fact that dimerizations are very small in the heavily\ndoped ${\\rm C}_{60}$ and the width of zero point motion of phonons is the\norder of 0.01\\AA{\\ } [11], it is certainly possible that the dimerization\nwould not be observed in actual samples.\n\n\n{}~~~~~~\n\n\n\\noindent\n{\\bf ACKNOWLEDGEMENTS}\\\\\nFruitful discussion with Prof. G. A. Gehring, Dr. M. Fujita,\nand Dr. Y. Asai is acknowledged. Useful correspondences with\nProf. B. Friedman and Dr. S. Abe are also acknowledged.\nThe author is grateful to Dr. M. Fujita\nfor providing him with Figs. 1 and 2 of this\npaper. Numerical calculations have been performed on FACOM\nM-1800\/30 of the Research Information Processing System,\nAgency of Industrial Science and Technology, Japan.\n\n\n\\pagebreak\n\n\n\\begin{flushleft}\n{\\bf REFERENCES}\n\\end{flushleft}\n\n\n\\noindent\n$*$ electronic mail address: harigaya@etl.go.jp.\\\\\n$**$ permanent address.\\\\\n$[1]$ T. Kato, T. Kodama, M. Oyama, S. O\\-ka\\-za\\-ki, T. Shida, T. Nakagawa,\nY. Matsui, S. Suzuki, H. Shiromaru, K. Yamauchi, and Y. Achiba,\nChem. Phys. Lett. {\\bf 180}, 446 (1991).\\\\\n$[2]$ T. Takahashi, S. Suzuki, T. Morikawa, H. Katayama-Yoshida,\nS. Hasegawa, H. Inokuchi, K. Seki, K. Kikuchi, S. Suzuki, K. Ikemoto,\nand Y. Achiba, Phys. Rev. Lett. {\\bf 68}, 1232 (1992);\nC. T. Chen, L H. Tjeng, P. Rudolf, G. Meigs, L. E. Rowe, J. Chen, J. P.\nMcCauley Jr., A. B. Smith III, A. R. McGhie, W. J. Romanow,\nand E. W. Plummer, Nature {\\bf 352}, 603 (1991).\\\\\n$[3]$ S. Morita, A. A. Zakhidov, and K. Yoshino, Solid State Commun. {\\bf 82},\n249 (1992); S. Morita, A. A. Zakhidov, T. Kawai, H. Araki, and K. Yoshino,\nJpn. J. Appl. Phys. {\\bf 31}, L890 (1992).\\\\\n$[4]$ M. Matus, H. Kuzmany, and E. Sohmen, Phys. Rev. Lett. {\\bf 68},\n2822 (1992).\\\\\n$[5]$ P. A. Lane, L. S. Swanson, Q. X. Ni, J. Shinar, J. P. Engel,\nT. J. Barton, and L. Jones, Phys. Rev. Lett. {\\bf 68}, 887 (1992).\\\\\n$[6]$ F. C. Zhang, M. Ogata, and T. M. Rice, Phys. Rev. Lett.\n{\\bf 67}, 3452 (1991).\\\\\n$[7]$ K. Harigaya, J. Phys. Soc. Jpn. {\\bf 60}, 4001 (1991);\nB. Friedman, Phys. Rev. B {\\bf 45}, 1454 (1992).\\\\\n$[8]$ W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B {\\bf 22},\n2099 (1980).\\\\\n$[9]$ K. Harigaya, Phys. Rev. B {\\bf 45}, 13676 (1992).\\\\\n$[10]$ McKenzie and Wilkins, Phys. Rev. Lett. {\\bf 69}, 1085 (1992).\\\\\n$[11]$ B. Friedman and K. Harigaya, Phys. Rev. B {\\bf 47},\n(1993) (February issue; in press).\\\\\n$[12]$ C. S. Yannoni, P. P. Bernier, D. S. Bethune,\nG. Meijer and J. R. Salem, J. Am. Chem. Soc. {\\bf 113}, 3190 (1991).\\\\\n$[13]$ T. Arai, Y. Murakami, H. Suematsu, K. Kikuchi, Y. Achiba,\nand I. Ikemoto, Solid State Commun. {\\bf 84}, 827 (1992).\\\\\n$[14]$ K. Harigaya, (preprint).\\\\\n$[15]$ Q. Xie, E. P\\'{e}rez-Cordero, and L. Echegoyen,\nJ. Am. Chem. Soc. {\\bf 114}, 3978 (1992).\\\\\n\n\\pagebreak\n\n\\begin{flushleft}\n{\\bf Figure Captions}\n\\end{flushleft}\n\n\n\\noindent\nFIG. 1. Unfolded pattern of the paper model of ${\\rm C}_{60}$.\nSee the text for the notations.\n\n{}~\n\n\\noindent\nFIG. 2. Bond variables and excess electron densities shown on the\nunfolded pattern of doped ${\\rm C}_{60}$ without disorders. The upper figures\nshow the bond variables, while lower figures display excess electron\ndensities. The number at the top is $N - N_{\\rm el}$. Notations\nare explained in the text.\n\n{}~\n\n\\noindent\nFIG. 3. Distribution function $D(y)$ of bond variables $y$ of\nthe doped ${\\rm C}_{60}$ in the presence of bond disorders.\nThe ordinate is normalized so that the area between the curve\nand the abscissa is unity. The thick, thin, and dashed lines\nare for $y_s = 0.01, 0.03$, and $0.05$\\AA, respectively.\n\n{}~\n\n\\noindent\nFIG. 4. Distribution function $D(\\rho)$ of the excess electron density\nof the doped ${\\rm C}_{60}$ in the presence of bond disorders.\nThe ordinate is normalized so that the area between the curve\nand the abscissa is unity. The thick, thin, and dashed lines\nare for $y_s = 0.01, 0.03$, and $0.05$\\AA, respectively.\n\n{}~\n\n\\noindent\nFIG. 5. Distribution function $D(y)$ of bond variables $y$ of\nthe doped ${\\rm C}_{60}$ in the presence of site disorders.\nThe ordinate is normalized so that the area between the curve\nand the abscissa is unity. The thick, thin, and dashed lines are for\n$U_s = 0.5, 1.0$, and $2.0$eV, respectively.\n\n\n\n{}~\n\n\\noindent\nFIG. 6. Distribution function $D(\\rho)$ of the excess electron density\nof the doped ${\\rm C}_{60}$ in the presence of site disorders.\nThe ordinate is normalized so that the area between the curve\nand the abscissa is unity. The thick, thin, and dashed lines are for\n$U_s = 0.5, 1.0$, and $2.0$eV, respectively.\n\n\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}