Two-Phase Obstacle Problem

On the basis of the monotonicity formula due to Alt, Caffarelli, and Friedman, the boundedness of the second-order derivatives D ² u of solutions to the equation

Critical star multigraphs

A star-multigraphG is a multigraph in which there is a vertexv ⁺ which is incident with each non-simple edge. It is critical if it is connected, Class 2 and(G\e) < (G) for eache E(G). We show that, ifG is any star multigraph, then(G) (G) + 1. We investigate the edge-chromatic class of star multigraphs with at most two vertices of maximum degree. We also obtain a number of results on critical star multigraphs. We shall make use of these results in later papers.     

Opérateurs différentiels associés à certaines représentations unitaires d'un groupe de Lie résoluble exponentiel

Soient G = exp g un groupe de Lie résoluble exponentiel et H = exp h un sous-groupe connexe de G. Soient un caractère unitaire de H et = Ind_H ^G. Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalente à la commutativité de D (G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre D (G) ^H , des opérateurs différentiels qui laissent l'espace des vecteurs C de invariant et qui commuttent avec l'action de H sur cet espace.     

On a problem of B. I. Golubov

. . . : s_n(x) — , _n-(, 1)- n L ². , . , : a _k l ² _n () [0,1] , (*) , (**) . a _k l ² u _n () [0,1] , (**), (*) .     

The search for chromatically unique graphs

The number of vertex-colourings of a simple graphG in not more than colours is a polynomial in. This polynomial, denoted byP(G, ), is called the chromatic polynomial ofG. A graphG is said to be chromatically unique, in short-unique, ifH G for any graphH withP(H, ) = P(G, ). Since the appearance of the first paper on-unique graphs by Chao and Whitehead in 1978, various families of and several results on such graphs have been obtained successively, especially during the last five years. It is the aim of this expository paper to give a survey on most of the works done on-unique graphs. A number of related problems and conjectures are also included.1980 Mathematical Subject Classification. Primary 05C15This work was done while the author was visiting the Department of Mathematics, National University of Singapore.     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

Infinite dimensional multipoint methods and the solution of two point boundary value problems

Consider the problem of determining the roots of an equation of the formF() =0 whereF maps the Banach spaceX into itself. Convergence theorems for the iterative solution ofF() =0 are proved for multipoint algorithms of the form _n+1= _n - ( _n ), 1, where and ₀()=0. The theorems are applied to the solution of two point boundary value problems of the form =f (y, t), g(y(0))+h(y(1))=c. A set {A(t),B,C} of matrices is called boundary compatible if the linear two point boundary value problem =A(t)) y+k (t),B y (0) + C y (1) = d has a unique solution for allk (t) andd. Then, under certain conditions, there are boundary compatible sets such that the problem =f (y, t),g (y (0) ) +h (y (1)) =c has the equivalent integral representation where and are Green's matrices for the linear problem =A(t)y +k(t),B y (0) +C y (1) =d. Eq. (i) is viewed as an operator equation of the formF (x) =(I-T) (x) = 0 and convergence conditions for the iterative solution of (i) are deduced from the general theorems. Explicit interpretations of the convergence results are given in terms off, g, h and some illustrative numerical examples are presented.This research has been supported by the National Aeronautics and Space Administration under Grant No. NGR-40-002-015.This research has been supported by the National Science Foundation under Grant No. GK-2788.     

Nonresonant acoustic absorption of Ising magnet

Zubarev's nonequilibrium statistical operator method is used to construct a theory of the nonresonant acoustic absorption of the linear Ising magnet, the basic Hamiltonian of which is of the two-particle kind and is treated without approximations. The frequency dependence of the absorption is determined and found to differ analytically from the well-known Debye curve _D () but for reasonable values of the physical parameters gives an envelope that agrees qualitatively with _D ().Kazan Physicotechnical Institute, Russian Academy of Sciences; Moscow Institute of Instrument Manufacture. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 149–159, January, 1994.     

A Periodicity Theorem in K-Theory for Finite Covariant Systems

Let A be a complex Banach algebra. In this paper, we prove a periodicity theorem that K(S ² A, G, )K(A, G,), where (A,G,) is a finite covariant system and K(A,G,) is an Abelian group associated with (A,G,).     

Plane Partitions and Characters of the Symmetric Group

In this paper we show that the existence of plane partitions, which are minimal in a sense to be defined, yields minimal irreducible summands in the Kronecker product of two irreducible characters of the symmetric group S(n). The minimality of the summands refers to the dominance order of partitions of n. The multiplicity of a minimal summand equals the number of pairs of Littlewood-Richardson multitableaux of shape (, ), conjugate content and type . We also give lower and upper bounds for these numbers.     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

A class of strongly nonlinear functional differential equations

Let H be a real Hilbert space, :H [0, + ] a proper l.s.c., convex function with L_k:={u H; u² + (u) k} compact for every k > 0, let > 0 be a given constant and . We prove an existence result for strong solutions to a class of functional differential equations of the form

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Some Non-trivial Symmetry Classes of Tensors Associated with Certain Characters

Let G be a finite group and a set of n elements. Assume that G acts faithfully on and let V be a vector space over the complex field , with dim V = m 2. It is shown that for each irreducible constituent of permutation character of G, the symmetry class of tensors associated with G and is non-trivial. This extends a result of Merris and Rashid (see [6, Theorem 2]).1995 AMS subject classification primary 20C30 secondary 15A69This research was in part supported by a grant from IPM.     

Pseudo-mesures p- adiques associées aux fonctions L deQ

It is shown that some standard results concerning the p-adic L- functions, L_p(), ofQ(p-divisibilities of 1/2L_p(, s), and congruences for 1/2L_p(, t)–1/2L_p(, s), s, t_p) are direct consequences of a general structural theorem, based only on the functional properties of the p-adic pseudo-measures and distributions attached to these L_p-functions (essentially the eulerian ones). The method suggests that all such divisibilities and congruences are obtained systematically by this way, and are the best possible (in a standard point of view). In particular, these results improve significantly all the known ones.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Topological algebras of kernels

Given a locally convex spaceE we define thelocally convex algebra of kernels , in such a way that the set of all its proper closed 2-sided ideals coincides with the set of all closed vector subspaces of ker(h), whereh is a continuous algebra morphism of into _b (E). Moreover,h is a strictly irreducible representation such that every representation: ; _b (F) (F a locally convex space) is of the form =oh, where stands for a continuous isomorphis ofIm(h) into _b (F), for a suitable topology onIm(h).     

γ-Subdifferential and γ-convexity of functions on a normed space

In this paper, the -subdifferential is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX attains its global minimum onD atx ^*, then 0 f(x ^*). This necessary condition always holds, even iff is not continuous orx ^* is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable ₊, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called -convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a -convex function. There are -convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two -convex functions. For all that, -convex functions still have properties similar to those of convex functions. For instance, each -local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its -boundary.This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition of a natural number n, the partition h() of n is defined so as to obtain a certain set of zeros in the table of characters for S_n. Namely, h() is the greatest (under the lexicographic ordering ) partition among P(n) such that (g) 0. Here, is an irreducible character of S_n, indexed by a partition , and g is a conjugacy class of elements in S_n, indexed by a partition . We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition P(n), the partition f() of n is defined so that f() is greatest among the partitions of n which are opposite in sign to h() and are such that (g) 0 (Thm. 1). Also, for any self-associated partition of n > 1, we construct a partition () P(n) such that () is greatest among the partitions of n which are distinct from h() and are such that (g) 0 (Thm. 2).Supported by RFBR grant No. 04-01-00463 and by RFBR-BRFBR grant No. 04-01-81001.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 24–43, January–February, 2005.