Corrigendum to: ``Embedding graphs into colored graphs''
A.
Hajnal;
P.
Komjáth
A law of large numbers for fast price adjustment
H. Jerome
Keisler
1-51
Abstract: The purpose of this paper is to prove a law of large numbers for certain Markov processes involving large sets of weakly interacting particles. Consider a large finite set $ A$ of "particles" which move about in $m$-dimensional Euclidean space ${R^m}$. The particles interact with each other indirectly by means of an auxiliary quantity $ p$ in $d$-dimensional Euclidean space $ {R^d}$. At each time $ t$, a particle $a \in A$ is randomly selected and randomly jumps to a new location in $ {R^m}$ with a distribution depending on $p$ and its old location. At the same time, the value of $p$ changes to a new value depending on these same arguments. The parameter $p$ moves by a small amount at each time but moves fast compared to the average position of the particles. Under appropriate hypotheses on the rules of motion, we shall prove the following law of large numbers. For sufficiently large $A$, the value of $p$ will be close to its expected value with large probability, and the average position of the particles will be close to its expected value with large probability. The work was motivated by the problem of modelling the adjustment of prices in mathematical economics, where the particles $a \in A$ are agents in an exchange economy, the position of $a$ at time $t$ is the commodity vector held by agent $ a$ at time $t$, and $p$ is the price vector at time $t$.
Pseudo-isotopies of irreducible $3$-manifolds
Jeff
Kiralis
53-78
Abstract: It is shown that a certain subspace of the space of all pseudo-isotopies of any irreducible $3$-manifold is connected. This subspace consists of those pseudo-isotopies corresponding to $ 1$-parameter families of functions which have nondegenerate critical points of index $1$ and $2$ only and which contain no slides among the $ 2$-handles. Some of the techniques developed are used to prove a weak four-dimensional $h$-cobordism theorem.
Rational fibrations in differential homological algebra
Aniceto
Murillo
79-91
Abstract: In this paper, a result of [6] is generalized as follows: Given a fibration $F \to E\xrightarrow{p}B$ of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and $\rho$ induces a surjection in rational homotopy, we construct an explicit isomorphism, \begin{displaymath}\begin{array}{*{20}{c}} {\varphi :\operatorname{Ext}_{{C^\ast... ...(E;{\mathbf{Q}})}(Q,{C^\ast}(E;{\mathbf{Q}})).} \end{array} \end{displaymath} This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.
Grothendieck groups of quotient singularities
Eduardo do Nascimento
Marcos
93-119
Abstract: Given a quotient singularity $R = {S^G}$ where $S = {\mathbf{C}}[[{x_1}, \ldots ,{x_n}]]$ is the formal power series ring in $n$-variables over the complex numbers ${\mathbf{C}}$, there is an epimorphism of Grothendieck groups $\psi :{G_0}(S[G]) \to {G_0}(R)$, where $ S[G]$ is the skew group ring and $\psi$ is induced by the fixed point functor. The Grothendieck group of $S[G]$ carries a natural structure of a ring, isomorphic to $ {G_0}({\mathbf{C}}[G])$. We show how the structure of ${G_0}(R)$ is related to the structure of the ramification locus of $V$ over $V/G$, and the action of $G$ on it. The first connection is given by showing that $\operatorname{Ker}\;\psi $ is the ideal generated by $ [{\mathbf{C}}]$ if and only if $G$ acts freely on $V$. That this is sufficient has been proved by Auslander and Reiten in [4]. To prove the necessity we show the following: Let $U$ be an integrally closed domain and $ T$ the integral closure of $ U$ in a finite Galois extension of the field of quotients of $U$ with Galois group $G$. Suppose that $\vert G\vert$ is invertible in $ U$, the inclusion of $ U$ in $T$ is unramified at height one prime ideals and $T$ is regular. Then ${G_0}(T[G]) \cong Z$ if and only if $U$ is regular. We analyze the situation $V = {V_1}{\coprod} _{\mathbf{C}[G]}{V_2}$ where $G$ acts freely on $ {V_1},{V_1} \ne 0$. We prove that for a quotient singularity $R,{G_0}(R) \cong {G_0}(R[[t]])$. We also study the structure of ${G_0}(R)$ for some cases with $\dim R = 3$.
Sharp estimate of the Laplacian of a polyharmonic function and applications
Ognyan Iv.
Kounchev
121-133
Abstract: The classical sharp inequality of Markov estimates the values of the derivative of the polynomial of degree $n$ in the interval $[a,b]$ through the uniform norm of the polynomial in the same interval multiplied by $2{n^2}/(b - a)$. In the present paper we provide an exact estimate for the values of the Laplacian of a polyharmonic function of degree $m$ by the uniform norm of the polyharmonic function multiplied by $2{(m - 1)^2}/{R^2}(x)$ where $R(x)$ is the distance from the point $ x$ to the boundary of the domain. The inequality of Markov (and the similar inequality of Bernstein about trigonometric polynomials) finds many applications in approximation theory for functions of one variable. We prove analogues to some of these results in the multivariate case.
Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side
Alexander
Nabutovsky
135-166
Abstract: We consider a semilinear boundary value problem $- \Delta u + f(u,x) = 0$ in $\Omega \subset {\mathbb{R}^N}$ and $u = 0$ on $\partial \Omega$. We assume that $f$ is a $ {C^\infty }$-smooth function and $\Omega$ is a bounded domain with a smooth boundary. For any $ {C^\alpha }$-smooth perturbation $h(x)$ of the right-hand side of the equation we consider the function ${N_h}(S)$ defined as the number of ${C^{2 + \alpha }}$-smooth solutions $ u$ such that $\left\Vert u\right\Vert _{{C^0}(\Omega )} \leq S$ of the perturbed problem. How "small" $ {N_h}(S)$ can be made by a perturbation $h(x)$ such that $ \left\Vert h\right\Vert _{{C^0}(\Omega )} \leq \varepsilon ?$ We present here an explicit upper bound in terms of $\varepsilon$ , $S$ and $\displaystyle \mathop {\max }\limits_{\vert u\vert \leq S,x \in \bar \Omega } \left\Vert D_u^i f(u,x)\right\Vert \quad (i \in \{ 0,1,2\} ).$ If $S$ is fixed then $h$ can be chosen by such a way that the upper bound persists under small in ${C^0}$-topology perturbations of $ h$ . We present an explicit lower bound for the radius of the ball of such admissible perturbations.
The center of $\mathbb{Z}[S^{n+1}]$ is the set of symmetric polynomials in $n$ commuting transposition-sums
Gadi
Moran
167-180
Abstract: Let ${S_{n + 1}}$ be the symmetric group on the $n + 1$ symbols $0,1,2, \ldots ,n$. We show that the center of the group-ring $\mathbb{Z}[{S_{n + 1}}]$ coincides with the set of symmetric polynomials with integral coefficients in the $ n$ elements ${s_1}, \ldots ,{s_n} \in \mathbb{Z}[{S_{n + 1}}]$, where ${s_k} = \sum\nolimits_{0 \leq i < k} {(i,k)}$ is a sum of $k$ transpositions, $ k = 1, \ldots ,n$. In particular, every conjugacy-class-sum of ${S_{n + 1}}$ is a symmetric polynomial in ${s_1}, \ldots ,{s_n}$.
Families of sets of positive measure
Grzegorz
Plebanek
181-191
Abstract: We present a combinatorial description of those families $\mathcal{P}$ of sets, for which there is a finite measure $\mu$ such that $\inf \{ \mu (P):P \in \mathcal{P}\} > 0$. This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions.
L'espace des arcs d'une surface
Robert
Cauty
193-209
Abstract: We prove that, for any surface $M$, the space of arcs contained in $M$, with the topology induced by the Hausdorff distance, is homeomorphic to $M \times {\sum}^\infty$, where $ \sum = \{ ({x_i}) \in {l^2}/\sum\nolimits_{i = 1}^\infty {{{(i{x_i})}^2} < \infty \} }$.
Excluding subdivisions of infinite cliques
Neil
Robertson;
P. D.
Seymour;
Robin
Thomas
211-223
Abstract: For every infinite cardinal $k$ we characterize graphs not containing a subdivision of ${K_k}$.
Iterating maps on cellular complexes
Stephen J.
Willson
225-240
Abstract: Let $K$ be a finite simplicial complex and $ f:K \to K$ be a "skeletal" map. A digraph $D$ is defined whose vertices correspond to the simplexes of $K$ and whose arcs give information about the behavior of $f$ on the simplexes. For every walk in $D$ there exists a point of $ K$ whose iterates under $ f$ mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are characterized; the corresponding maps $f$ exhibit complicated behavior.
The structure of the space of coadjoint orbits of an exponential solvable Lie group
Bradley N.
Currey
241-269
Abstract: In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group $G$. We construct a partition $\wp$ of the dual ${\mathfrak{g}^{\ast} }$ of the Lie algebra $\mathfrak{g}$ of $G$ into finitely many $\operatorname{Ad}^{\ast} (G)$-invariant algebraic sets with the following properties. For each $\Omega \in \wp$, there is a subset $\Sigma$ of $\Omega$ which is a cross-section for the coadjoint orbits in $\Omega$ and such that the natural mapping $\Omega /\operatorname{Ad}^{\ast} (G) \to \Sigma$ is bicontinuous. Each $ \Sigma$ is the image of an analytic $ \operatorname{Ad}^{\ast}(G)$-invariant function $P$ on $\Omega$ and is an algebraic subset of $ {\mathfrak{g}^{\ast}}$. The partition $\wp$ has a total ordering such that for each $\Omega \in \wp $, $ \cup \{ \Omega \prime:\Omega \prime \leq \Omega \}$ is Zariski open. For each $ \Omega$ there is a cone $ W \subset {\mathfrak{g}^{\ast} }$, such that $\Omega$ is naturally a fiber bundle over $ \Sigma$ with fiber $ W$ and projection $ P$. There is a covering of $ \Sigma$ by finitely many Zariski open subsets $O$ such that in each $O$, there is an explicit local trivialization $ \Theta :{P^{ - 1}}(O) \to W \times O$. Finally, we show that if $\Omega$ is the minimal element of $ \wp$ (containing the generic orbits), then its cross-section $\Sigma$ is a differentiable submanifold of $ {\mathfrak{g}^{\ast} }$. It follows that there is a dense open subset $ U$ of $G\hat \emptyset$ such that $U$ has the structure of a differentiable manifold and $G\widehat\emptyset \sim U$ has Plancherel measure zero.
Stability for an inverse problem in potential theory
Hamid
Bellout;
Avner
Friedman;
Victor
Isakov
271-296
Abstract: Let $D$ be a subdomain of a bounded domain $\Omega$ in $ {\mathbb{R}^n}$ . The conductivity coefficient of $D$ is a positive constant $k \ne 1$ and the conductivity of $\Omega \backslash D$ is equal to $1$. For a given current density $ g$ on $\partial \Omega$ , we compute the resulting potential $u$ and denote by $f$ the value of $u$ on $ \partial \Omega$. The general inverse problem is to estimate the location of $ D$ from the known measurements of the voltage $f$. If ${D_h}$ is a family of domains for which the Hausdorff distance $ d(D,{D_h})$ equal to $ O(h)$ ($h$ small), then the corresponding measurements ${f_h}$ are $O(h)$ close to $f$. This paper is concerned with proving the inverse, that is, $d(D,{D_h}) \leq \frac{1}{c}\left\Vert {f_h} - f\right\Vert$ , $c > 0$ ; the domains $D$ and ${D_h}$ are assumed to be piecewise smooth. If $ n \geq 3$ , we assume in proving the above result, that ${D_h} \supset D$ (or ${D_h} \subset D$) for all small $h$ . For $n = 2$ this monotonicity condition is dropped, provided $g$ is appropriately chosen. The above stability estimate provides quantitative information on the location of ${D_h}$ by means of ${f_h}$ .
Algorithmic procedures
Harvey
Friedman;
Richard
Mansfield
297-312
Abstract: We consider the state of elementary recursion theory when the familiar $0, 1, +, \times, =, <$ of ordinary arithmetic are replaced by constants, functions, and relations from an arbitrary model.
Free actions on $\mathbb{R}$-trees
Frank
Rimlinger
313-329
Abstract: We characterize the free minimal actions of finitely generated groups on $\mathbb{R}$-trees in terms of certain equivalence relations on compact metric graphs.
Boundary tangential convergence on spaces of homogeneous type
Patricio
Cifuentes;
José R.
Dorronsoro;
Juan
Sueiro
331-350
Abstract: We study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity. Our results contain those already known for the Euclidean case and give new ones for stratified nilpotent Lie groups and for solutions of the Dirichlet problem on Lipschitz domains.
Nonnegatively curved submanifolds in codimension two
Maria Helena
Noronha
351-364
Abstract: Let $M$ be a complete noncompact manifold with nonnegative sectional curvatures isometrically immersed in Euclidean spaces with codimension two. We show that $M$ is a product over its soul, except when the soul is the circle ${S^1}$ or $M$ is $3$-dimensional and the soul is the Real Projective Plane. We also give a rather complete description of the immersion, including the exceptional cases.
Convolution and hypergroup structures associated with a class of Sturm-Liouville systems
William C.
Connett;
Clemens
Markett;
Alan L.
Schwartz
365-390
Abstract: Product formulas of the type $\displaystyle {u_k}(\theta ){u_k}(\phi ) = \int_0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi$ are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on $ [0,\pi ]$ by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When $D(\xi ,\theta ,\phi )$ is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which $M[0,\pi ]$ becomes a Banach algebra with the functions $ {u_k}(\xi )/{u_0}(\xi )$ as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.
Loewy series of certain indecomposable modules for Frobenius subgroups
Zong Zhu
Lin
391-409
Abstract: We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic $p > 0$. By studying the translation functors from $p$-singular weights to $p$-regular weights, we obtain some results on Loewy series of certain indecomposable modules.
Degree one maps between geometric $3$-manifolds
Yong Wu
Rong
411-436
Abstract: Let $M$ and $N$ be two compact orientable $3$-manifolds, we say that $M \geq N$, if there is a degree one map from $M$ to $N$. This gives a way to measure the complexity of $ 3$-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called "geometric $3$-manifolds" must eventually become homotopy equivalences.
Free $\alpha$-extensions of an Archimedean vector lattice and their topological duals
Anthony J.
Macula
437-448
Abstract: Arch denotes the category of Archimedean vector lattices with vector lattice homomorphisms, and $\alpha$ denotes an uncountable cardinal number or the symbol $\infty$. $\operatorname{Arch}(\alpha )$ denotes the category of Arch objects with $\alpha$-complete Arch morphisms.
Nonexistence of nodal solutions of elliptic equations with critical growth in $\mathbb{R}^2$
Adimurthi;
S. L.
Yadava
449-458
Abstract: Let $f(t) = h(t){e^{b{t^2}}}$ be a function of critical growth. Under a suitable assumption on $h$, we prove that \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = f(u)} & {{\text... ...} & {{\text{on}}\;\partial B(R),} \end{array} \end{displaymath} does not admit a radial solution which changes sign for sufficiently small $ R$.
Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces
David E.
Barrett
459-474
Abstract: We consider various examples of compact Levi-flat hypersurfaces in two-dimensional complex manifolds, exploring the interplay between geometric properties of the induced foliation, behavior of the tangential Cauchy-Riemann equations along the hypersurface, and pseudoconvexity properties of small neighborhoods of the hypersurface.