Inverse spectral theory of finite Jacobi matrices
Peter
C.
Gibson
4703-4749
Abstract: We solve the following physically motivated problem: to determine all finite Jacobi matrices $J$ and corresponding indices $i,j$ such that the Green's function \begin{displaymath}\langle e_j,(zI-J)^{-1}e_i\rangle \end{displaymath} is proportional to an arbitrary prescribed function $f(z)$. Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map \begin{displaymath}(J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle] \end{displaymath} (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter
Kusano
Takasi;
Manabu
Naito
4751-4767
Abstract: In this paper the following half-linear ordinary differential equation is considered: $\alpha > 0$ is a constant, $\lambda > 0$ is a parameter, and $p(t)$ is a continuous function on $[a, \infty)$, $a > 0$, and $p(t) > 0$ for $t \in [a, \infty)$. The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions $x(t; \lambda)$ of (H $_{\lambda})$ such that \begin{displaymath}\lim_{t\to\infty}\frac{x(t; \lambda)}{\sqrt{t}} = 0. \end{displaymath} It is shown that, if $\alpha \geq 1$ and if (H $_{\lambda})$ is strongly nonoscillatory, then there exists a sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ such that $0=\lambda_{0}<\lambda_{1}<\cdots< \lambda_{n}<\cdots$, $\lambda_{n} \to +\infty$ as $n \to \infty$; and $x(t; \lambda)$ with $\lambda = \lambda_n$ has exactly $n-1$ zeros in the interval $(a,\infty)$ and $x(a; \lambda_n) = 0$; and $x(t; \lambda)$ with $\lambda \in (\lambda_{n-1}, \lambda_n)$ has exactly $n-1$ zeros in $(a,\infty)$ and $x(a; \lambda_n) \neq 0$. For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.
Sets of uniqueness for spherically convergent multiple trigonometric series
J.
Marshall
Ash;
Gang
Wang
4769-4788
Abstract: A subset $E$ of the $d$-dimensional torus $\mathbb{T} ^{d}$ is called a set of uniqueness, or $U$-set, if every multiple trigonometric series spherically converging to $0$ outside $E$ vanishes identically. We show that all countable sets are $U$-sets and also that $H^{J}$ sets are $U$-sets for every $J$. In particular, $C\times\mathbb{T} ^{d-1}$, where $C$ is the Cantor set, is an $H^{1}$ set and hence a $U$-set. We will say that $E$ is a $U_{A}$-set if every multiple trigonometric series spherically Abel summable to $0$ outside $E$ and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for $U_{A}$ sets. In addition, every $U_{A}$-set has measure $0$, and a countable union of closed $U_{A}$-sets is a $U_{A}$-set.
Ribbon tilings and multidimensional height functions
Scott
Sheffield
4789-4813
Abstract: We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A ribbon tile of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of height functions from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear (in the area of $R$) algorithm for determining whether $R$ can be tiled with ribbon tiles of order $n$ and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order-$n$ ribbon tilings of $R$ can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order-$2$ ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.
Lines tangent to $2n-2$ spheres in ${\mathbb R}^n$
Frank
Sottile;
Thorsten
Theobald
4815-4829
Abstract: We show that for $n \ge 3$there are $3 \cdot 2^{n-1}$ complex common tangent lines to $2n-2$ general spheres in $\mathbb{R}^n$ and that there is a choice of spheres with all common tangents real.
Automorphisms of finite order on Gorenstein del Pezzo surfaces
D.-Q.
Zhang
4831-4845
Abstract: In this paper we shall determine all actions of groups of prime order $p$ with $p \ge 5$ on Gorenstein del Pezzo (singular) surfaces $Y$of Picard number 1. We show that every order-$p$ element in $\operatorname{Aut}(Y)$ ( $= \operatorname{Aut}({\widetilde Y})$, ${\widetilde Y}$ being the minimal resolution of $Y$) is lifted from a projective transformation of ${\mathbf{P}}^{2}$. We also determine when $\operatorname{Aut}(Y)$ is finite in terms of $K_{Y}^{2}$, $\operatorname{Sing} Y$ and the number of singular members in $\vert-K_{Y}\vert$. In particular, we show that either $\vert\operatorname{Aut}(Y)\vert = 2^{a}3^{b}$ for some $1 \le a+b \le 7$, or for every prime $p \ge 5$, there is at least one element $g_{p}$ of order $p$ in $\operatorname{Aut}(Y)$ (hence $\vert\operatorname{Aut}(Y)\vert$ is infinite).
Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields
Claus
Mazanti
Sorensen
4847-4869
Abstract: In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke $L$-series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind $\xi$-function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.
Hilbert transforms and maximal functions along variable flat curves
Jonathan
M.
Bennett
4871-4892
Abstract: We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their $L^{2}(\mathbb{R} ^{2})$ boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.
Nonisotropic strongly singular integral operators
Bassam
Shayya
4893-4907
Abstract: We consider a class of strongly singular integral operators which include those studied by Wainger, and Fefferman and Stein, and extend the results concerning the $L^p$ boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp.
Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence
Donald
R.
King
4909-4920
Abstract: Let $G$ be a connected, linear semisimple Lie group with Lie algebra $\mathfrak g$, and let ${K_{{}_{\mathbf C}}}~\rightarrow~{\operatorname{Aut} (\mathfrak p_{{}_{\mathbf C}})}$ be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent $K_{{}_{\mathbf C}}$-orbits in $\mathfrak p_{{}_{\mathbf C}}$ and the nilpotent $G$-orbits in $\mathfrak g$. We show that this correspondence associates each spherical nilpotent $K_{{}_{\mathbf C}}$-orbit to a nilpotent $G$-orbit that is multiplicity free as a Hamiltonian $K$-space. The converse also holds.
A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula
K.
S.
Ryu;
M.
K.
Im
4921-4951
Abstract: In this article, we consider a complex-valued and a measure-valued measure on $C [0,t]$, the space of all real-valued continuous functions on $[0,t]$. Using these concepts, we establish the measure-valued Feynman-Kac formula and we prove that this formula satisfies a Volterra integral equation. The work here is patterned to some extent on earlier works by Kluvanek in 1983 and by Lapidus in 1987, but the present setting requires a number of new concepts and results.
Detection of renewal system factors via the Conley index
Jim
Wiseman
4953-4968
Abstract: Let $N$ be an isolating neighborhood for a map $f$. If we can decompose $N$ into the disjoint union of compact sets $N_1$ and $N_2$, then we can relate the dynamics on the maximal invariant set $\operatorname{Inv} N$ to the shift on two symbols by noting which component of $N$ each iterate of a point $x\in \operatorname{Inv} N$ lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to $N$. In essence, we measure the difference between the Conley index of $\operatorname{Inv}N$and the sum of the indices of $\operatorname{Inv} N_1$ and $\operatorname{Inv} N_2$.
Thick points for intersections of planar sample paths
Amir
Dembo;
Yuval
Peres;
Jay
Rosen;
Ofer
Zeitouni
4969-5003
Abstract: Let $L_n^{X}(x)$ denote the number of visits to $x \in \mathbf{Z} ^2$ of the simple planar random walk $X$, up to step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 \pi)$, there are $n^{1-2\pi b+o(1)}$ points $x \in \mathbf{Z}^2$ for which $\limsup_{r \rightarrow 0} \mathcal{I} (x,r)/(r^2\vert\log r\vert^4)=a^2$, is almost surely $2-2a$. Here $\mathcal{I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run until time $1$. The proofs rely on a ``multi-scale refinement'' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$centered at $x$ by $x+rK$ for general sets $K$.
Equilibrium existence and topology in some repeated games with incomplete information
Robert
S.
Simon;
Stanislaw
Spiez;
Henryk
Torunczyk
5005-5026
Abstract: This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals. This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping $F$ on a compact convex set $P\subset \mathbb R^n$, we give further conditions which imply that every point $p_0\in P$ belongs to the convex hull of a finite subset $P _0$ of the domain of $F$satisfying $\bigcap_{x\in P_0} F(x)\ne \emptyset$.
Location of the Fermat-Torricelli medians of three points
Carlos
Benítez;
Manuel
Fernández;
María
L.
Soriano
5027-5038
Abstract: We prove that a real normed space $X$ with $\dim X\ge 3$ is an inner product space if and only if, for every three points $u,v,w\in X$, the set of points at which the function $x\in X\to \Vert u-x\Vert+\Vert v-x\Vert+\Vert w-x\Vert$attains its minimum (called the set of Fermat-Torricelli medians of the three points) intersects the convex hull of these three points.
Uniform and Lipschitz homotopy classes of maps
Sol
Schwartzman
5039-5047
Abstract: If $X$ is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line $R$ into $X$ an element of $H_{1} (X, U/U_{0}),$ where $U$ is the space of uniformly continuous functions from $R$ to $R$ and $U_{0}$ is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to $H_{1}(X,U/U_{0})$ is surjective. If $X$ is the $n$-dimensional torus, it is bijective, while if $X$ is a compact orientable surface of genus $>1$, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds $P$ to compact smooth manifolds $X.$ With each such Lipschitz homotopy class we associate an element of $H_{n} (X, B^+/B_{0}^+),$ where $n$ is the dimension of $P,$ $B$ is the space of bounded continuous functions from the positive real axis to $R,$ and $B_{0}^+$ is the set of all $f\in B^+$ such that $\lim_{t \rightarrow \infty} f(t) = 0.$
Spin structures and codimension two embeddings of $3$-manifolds up to regular homotopy
Osamu
Saeki;
Masamichi
Takase
5049-5061
Abstract: We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into $5$-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of $3$-manifolds $M^3$, the following phenomenon occurs: there exists a codimension two immersion of the $3$-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of $M^3$. This involves introducing and studying an equivalence relation on the set of spin structures on $M^3$. Their associated $\mu$-invariants also play an important role.
Discrete morse theory and the cohomology ring
Robin
Forman
5063-5085
Abstract: In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].
On the blow-up of heat flow for conformal $3$-harmonic maps
Chao-Nien
Chen;
L.
F.
Cheung;
Y.
S.
Choi;
C.
K.
Law
5087-5110
Abstract: Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from $D^2$ (a unit ball in ${\mathbf R}^2$) to $S^2$ (a unit sphere in ${\mathbf R}^3$) under certain initial and boundary conditions. We generalize this result to the case of $3$-harmonic map heat flow from $D^3$ to $S^3$. In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.
Diffusions on graphs, Poisson problems and spectral geometry
Patrick
McDonald;
Robert
Meyers
5111-5136
Abstract: We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function.