Standard realizations of crystal lattices via harmonic maps
Motoko
Kotani;
Toshikazu
Sunada
1-20
Abstract: An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.
The Lipschitz continuity of the distance function to the cut locus
Jin-ichi
Itoh;
Minoru
Tanaka
21-40
Abstract: Let $N$ be a closed submanifold of a complete smooth Riemannian manifold $M$ and $U\mbox{{$\nu$}}$ the total space of the unit normal bundle of $N$. For each $v \in U\mbox{{$\nu$}}$, let $\rho(v)$ denote the distance from $N$ to the cut point of $N$ on the geodesic $\gamma_v$ with the velocity vector $\dot\gamma_v(0)=v.$ The continuity of the function $\rho$ on $U\mbox{{$\nu$}}$ is well known. In this paper we prove that $\rho$ is locally Lipschitz on which $\rho$is bounded; in particular, if $M$ and $N$ are compact, then $\rho$ is globally Lipschitz on $U\mbox{{$\nu$}}$. Therefore, the canonical interior metric $\delta$ may be introduced on each connected component of the cut locus of $N,$ and this metric space becomes a locally compact and complete length space.
Projective sets and ordinary differential equations
Alessandro
Andretta;
Alberto
Marcone
41-76
Abstract: We prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution is $\boldsymbol\Sigma_{1}^{1}$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $\boldsymbol\Pi_{1}^{1}$-complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution even if we perturb a bit the initial condition is $\boldsymbol\Pi_{2}^{1}$-complete.
Induced formal deformations and the Cohen-Macaulay property
Phillip
Griffith
77-93
Abstract: The main result states: if $A/B$ is a module finite extension of excellent local normal domains which is unramified in codimension two and if $S/\varkappa S \simeq \hat B$ represents a deformation of the completion of $B$, then there is a corresponding $S$-algebra deformation $T/\varkappa T \simeq \hat A$ such that the ring homomorphism $S \hookrightarrow T$ represents a deformation of $\hat B \hookrightarrow \hat A$. The main application is to the ascent of the arithmetic Cohen-Macaulay property for an étale map $f : X \to Y$ of smooth projective varieties over an algebraically closed field.${}^*$
Degree of strata of singular cubic surfaces
Rafael
Hernández;
María
J.
Vázquez-Gallo
95-115
Abstract: We determine the degree of some strata of singular cubic surfaces in the projective space $\mathbf{P}^3$. These strata are subvarieties of the $\mathbf{P}^{19}$ parametrizing all cubic surfaces in $\mathbf{P}^3$. It is known what their dimension is and that they are irreducible. In 1986, D. F. Coray and I. Vainsencher computed the degree of the 4 strata consisting on cubic surfaces with a double line. To work out the case of isolated singularities we relate the problem with (stationary) multiple-point theory.
Connectivity at infinity for right angled Artin groups
Noel
Brady;
John
Meier
117-132
Abstract: We establish sufficient conditions implying semistability and connectivity at infinity properties for CAT(0) cubical complexes. We use this, along with the geometry of cubical $K(\pi,1)$'s to give a complete description of the higher connectivity at infinity properties of right angled Artin groups. Among other things, this determines which right angled Artin groups are duality groups. Applications to group extensions are also included.
Extension theory of separable metrizable spaces with applications to dimension theory
Alexander
Dranishnikov;
Jerzy
Dydak
133-156
Abstract: The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let $L$ be a countable CW complex. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$ for some CW complex $K$, then for any map $f:A\to K$, $A$ closed in $X$, there is an extension $f':U\to K$ of $f$ over an open set $U$such that $L\in AE(X-U)$. Theorem. Let $K,L$ be countable CW complexes. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$, then there is a subset $Y$ of $X$ such that $K\in AE(Y)$ and $L\in AE(X-Y)$. Theorem. Suppose $G_{i},\ldots ,G_{n}$ are countable, non-trivial, abelian groups and $k>0$. For any separable metrizable space $X$ of finite dimension $\dim X>0$, there is a closed subset $Y$ of $X$ with $\dim _{G_{i}} Y=\max (\dim _{G_{i}} X-k,1)$ for $i=1,\ldots ,n$. Theorem. Suppose $W$ is a separable metrizable space of finite dimension and $Y$ is a compactum of finite dimension. Then, for any $k$, $0<k<\dim W-\dim Y$, there is a closed subset $T$ of $W$such that $\dim T=\dim W-k$ and $\dim (T\times Y)=\dim (W\times Y)-k$. Theorem. Suppose $X$ is a metrizable space of finite dimension and $Y$ is a compactum of finite dimension. If $K\in AE(X)$ and $L\in AE(Y)$ are connected CW complexes, then $K\wedge L\in AE(X\times Y).$
Endofiniteness in stable homotopy theory
Henning
Krause;
Ulrike
Reichenbach
157-173
Abstract: We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.
$(Z_{2})^{k}$-actions whose fixed data has a section
Pedro
L. Q.
Pergher
175-189
Abstract: Given a collection of $2^{k}-1$ real vector bundles $\varepsilon _{a}$ over a closed manifold $F$, suppose that, for some $a_{0}, \varepsilon _{a_{0}}$ is of the form $\varepsilon _{a_{0}}^{\prime }\oplus R$, where $R\to F$ is the trivial one-dimensional bundle. In this paper we prove that if $\bigoplus _{a} \varepsilon _{a} \to F$ is the fixed data of a $(Z_{2})^{k}$-action, then the same is true for the Whitney sum obtained from $\bigoplus _{a} \varepsilon _{a}$ by replacing $\varepsilon _{a_{0}}$ by $\varepsilon _{a_{0}}^{\prime }$. This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible $(Z_{2})^{k}$-actions fixing the disjoint union of an even projective space and a point.
Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation
Yue
Liu
191-208
Abstract: In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form \begin{equation*}( u_{t} + u_{xxx} + u^{p} u_{x} )_{x} = u_{yy} \quad (x, y) \in R^{2}, t \ge 0. \end{equation*} It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity $p \ge 4/3$ with $p$ the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if $2 < p < 4$; that is, the solutions blow up in finite time provided they start near an unstable solitary wave.
On the invariant faces associated with a cone-preserving map
Bit-Shun
Tam;
Hans
Schneider
209-245
Abstract: For an $n\times n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of $\{ 1,\cdots,n\}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\mathbb{R}^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If $A$ leaves invariant a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$ for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant join-irreducible faces of $K$, each containing in its relative interior a generalized eigenvector of $A$corresponding to $\lambda$ (referred to as semi-distinguished $A$-invariant faces associated with $\lambda$), where $m_\lambda$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $\lambda$, but there is no such chain with more than $m_\lambda$ members. We introduce the important new concepts of semi-distinguished $A$-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding $n$ that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Spectral theory and hypercyclic subspaces
Fernando
León-Saavedra;
Alfonso
Montes-Rodríguez
247-267
Abstract: A vector $x$ in a Hilbert space $\mathcal{H}$ is called hypercyclic for a bounded operator $T: \mathcal{H} \rightarrow \mathcal{H}$ if the orbit $\{T^{n} x : n \geq 1 \}$ is dense in $\mathcal{H}$. Our main result states that if $T$ satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is true even if $T$ is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator $T$ to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator $T:f(z)\rightarrow f(z+1)$ defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.
Sharp Sobolev inequalities with lower order remainder terms
Olivier
Druet;
Emmanuel
Hebey;
Michel
Vaugon
269-289
Abstract: Given a smooth compact Riemannian $n$-manifold $(M,g)$, this paper deals with the sharp Sobolev inequality corresponding to the embedding of $H_1^2(M)$ in $L^{2n/(n-2)}(M)$ where the $L^2$ remainder term is replaced by a lower order term.
The number of planar central configurations is finite when $N-1$ mass positions are fixed
Peter
W.
Lindstrom
291-311
Abstract: In this paper, it is proved that for $n>2$ and $n\not=4$, if $n-1$ masses are located at fixed points in a plane, then there are only a finite number of $n$-point central configurations that can be generated by positioning a given additional $n$th mass in the same plane. The result is established by proving an equivalent isolation result for planar central configurations of five or more points. Other general properties of central configurations are established in the process. These relate to the amount of centrality lost when a point mass is perturbed and to derivatives associated with central configurations.
Local derivations on $C^*$-algebras are derivations
B.
E.
Johnson
313-325
Abstract: Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any $C^*$-algebra $\mathfrak{A}$ into any Banach $\mathfrak{A}$-bimodule $\mathfrak{X}$. Most of the work is involved with establishing this result when $\mathfrak{A}$ is a commutative $C^*$-algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra $C^1[0,1]$ of continuously differentiable functions on $[0,1]$. We also give an automatic continuity result, that is, we show that local derivations on $C^*$-algebras are continuous even if not assumed a priori to be so.
Convergence of the Ruelle operator for a function satisfying Bowen's condition
Peter
Walters
327-347
Abstract: We consider a positively expansive local homeomorphism $T\colon X\to X$ satisfying a weak specification property and study the Ruelle operator $\mathcal{L}_\varphi$ of a real-valued continuous function $\varphi$satisfying a property we call Bowen's condition. We study convergence properties of the iterates $\mathcal{L}_\varphi^n$ and relate them to the theory of equilibrium states.
A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I
Donald
I.
Cartwright;
Gabriella
Kuhn;
Paolo
M.
Soardi
349-364
Abstract: Let $G=\mathrm{Aut}(T)$ be the group of automorphisms of a homogeneous tree $T$, and let $\Gamma$ be a lattice subgroup of $G$. Let $\pi$ be the tensor product of two spherical irreducible unitary representations of $G$. We give an explicit decomposition of the restriction of $\pi$ to $\Gamma$. We also describe the spherical component of $\pi$ explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.
Dade's invariant conjecture for general linear and unitary groups in non-defining characteristics
Jianbei
An
365-390
Abstract: This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups. The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.
On modules of finite upper rank
Dan
Segal
391-410
Abstract: For a group $G$ and a prime $p$, the upper $p$-rank of $G$ is the supremum of the sectional $p$-ranks of all finite quotients of $G$. It is unknown whether, for a finitely generated group $G$, these numbers can be finite but unbounded as $p$ ranges over all primes. The conjecture that this cannot happen if $G$ is soluble is reduced to an analogous `relative' conjecture about the upper $p$-ranks of a `quasi-finitely-generated' module $M$for a soluble minimax group $\Gamma$. The main result establishes a special case of this relative conjecture, namely when the module $M$ is finitely generated and the minimax group $\Gamma$ is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)
Optimal filtrations on representations of finite dimensional algebras
Lieven
Le Bruyn
411-426
Abstract: We present a representation theoretic description of the non-empty strata in the Hesselink stratification of the nullcone of representations of quivers. We use this stratification to define optimal filtrations on representations of finite dimensional algebras. As an application we investigate the isomorphism problem for uniserial representations.