Set-valued mappings arise quite naturally in optimization and nonsmooth analysis. In optimization, typically one has a family of optimization problems that depend on some parameter. One can then associate to this family of problems the set-valued mappings that assign to the parameter the set of optimal solutions, the set of feasible solutions or the set of multipliers. Many of these set-valued mappings encountered in optimization have been shown to be “proto-differentiable” (see Rockafellar [16]) i.e., in some sense these set-valued mappings are “differentiable”. Using estimates provided by the proto-derivatives, see Proposition 2.1, one can then obtain information on how the sets depend on the parameter. The concept of proto-differentiation is described in Section 2.

The Cauchy functional equations have been studied recently for Schwartz distributions by Koh in [3]. When the solutions are locally integrate functions, the equations reduce to the classical Cauchy equations (see [1]):

(1) f(x+y)=f﹛x)+f(y)

(2) f(x+y)=f(x)f(y)

(3) f(xy)=f(x)+f(y)

(4) f(xy)=f(x)f(y).

Earlier efforts to study functional equations in distributions were given by Fenyö [2]for the Hosszu’ equations

f(x + y - xy) +f(xy) =f(x) +f (y ),

by Neagu [4]for the Pompeiu equation

f(x+y+xy)=f(x)+f(y)+f(x)f(y)

and by Swiatak [6].

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Dans ce travail, nous construisons des solutions pour une certaine classe d'équations semi-linéaires complexes dans le plan. Plus précisément on considère prés d'un point XQde R2 l'équation

où Pest un opérateur différentiel d'ordre m(m≧ 1) à coefficients C∞ complexes, et o ù ƒest à valeurs complexes, analytique en et seulement C∞ en x.En supposant alors que Pest de type principal près de xoet vérifie la condition de Nirenberg-Trêves sous elliptique (que nous noterons ((P), voir [5]), nous construisons une solution locale de (*), de classe C∞ (Théorème 2.1).

Ce résultat échappe évidemment aux théorèmes classiques d'existence de Hamilton-Jacobi et de Cauchy-Kowalevsky.

We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.

A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

By P(ω) we denote the Boolean algebra of all subsets of the set ω of natural numbers. We identify each natural number with the set of its predecessors and define: the ideal of sets of density zero, and the ideal of sets of logarithmic density zero.

An analytic isomorphism of C*-algebras is a C*-isomorphism which maps one distinguished subalgebra, the analytic subalgebra, onto another. A strict partial order of a topological group acting on a topological space determines the analytic subalgebra of the transformation group C*-algebra as a certain non-self-adjoint subalgebra of the C*-algebra. When the group action is free and locally parallel, this analytic subalgebra is locally a subfield of compact operators contained in a reflexive algebra whose subspace lattice is determined by the group order. If in addition the group has the dominated convergence property, an analytic isomorphism of such transformation group C*-algebras induces a homeomorphism of the transformation spaces which maps orbits to orbits. In particular, the C*-algebras for two regular foliations of the plane are analytically isomorphic only if the foliations are topologically conjugate. In the case of parallel actions, a quotient of the group of analytic automorphisms is isomorphic to the second Čech cohomology of a transversal for the action.

Let be a cusp form of weight 2k and trivial character for Γ0(N), where N is prime, which is orthogonal with respect to the Petersson product to all forms g(dz), where g is of level L < N, dL\N. Let K be an imaginary quadratic field of discriminant — D where the prime N is inert. Denote by ∊ the quadratic character of determined by ∊(p) = (—D/p) for primes p not dividing D. For A an ideal class in K, let rA(m) be the number of integral ideals of norm m in A. We will be interested in the Dirichlet series L(f,A,s) defined by

Let H be a finite set; we can suppose H = { 1, 2, …,d } . Consider Hn, the set of length n words over the alphabet H. For every A ∈ Hn define the waiting time for A as the number of experiments needed till A appears as a connected sub-sequence of random elements of H. Formally, let X1, X2,... be i. i. d. random variables, P(X1 = i) = d−1, 1 ≤ i ≤ d then

Let be a sequence of independent Ornstein-Uhlenbeck processes with coefficients lk and ƛk,i.e., Xk(.)is a Gaussian process with EXk(t) =0 and

The process Y(.)was first studied by Dawson (1972) as the stationary solution of the infinite array of stochastic differential equations

where are independent Wiener processes (cf. also [6],[19],and [1]).

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

In this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigroups determined by laws of the form xn = xn + 1. Covering space methods are used to study the free objects in a variety of the form ∨ where is a variety of inverse semigroups and is the variety of groups.

The two Virasoro superalgebras, known as the Neveu- Schwarz algebra and the Ramond algebra, each have two unitary irreducible lowest weight modules with central charge . In this paper, I show how tensor products of these modules decompose into finite direct sums of irreducible modules with central charge .

We define the cover of an RG-module V to consist of an RG lattice Ṽ and a homomorphism π : Ṽ→ V such that π induces an isomorphism on Ext*RG(M, —) for any RG-lattice M. Here G is a finite group and, for simplicity in this introduction, R is a complete discrete valuation ring of characteristic zero with prime element p and perfect valuation class field. Let pn(G) be the highest power of p that divides |G| and, given an RG-lattice M, let pn(M) be the smallest power of p such that pn(M) idM : M→M factors through a projective lattice: n(M)≦n(G).

In this part, we shall be concerned with the structure of projections in a simple σ-unital C*-algebra with the FS property, and in the associated multiplier and corona algebras. We shall also consider the closed ideal structure of the corona algebra. Most of results appear to be new even for separable simple AF algebras, and are technically independent of the previous parts I and II ([37] and [38]). The whole work develops after finding a new property of a σ-unital (nonunital) simple C*-algebra with FS, which was not known even for a separable simple AF algebra. We relate this new property to the structure of the multiplier and corona algebras from vairous points of view.

Recently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

There is a body of results for lattices known as “Decomposition Theory” which is aimed at proving certain existence and uniqueness theorems concerning irredundant representations of elements of a compactly generated lattice. The motivation for these results is certainly the quest for sufficient conditions on congruence lattices to insure irredundant subdirect representations of algebras. These theorems usually include some kind of modularity or distribut i v e hypothesis (for uniqueness) and some atomicity hypothesis (for existence); the precise details can be found in [3]. The atomicity condition is usually the hypothesis that the lattice in question is strongly atomic or at least atomic. Now, it is well-known that every algebra has a weakly atomic congruence lattice.

The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Only §5 of this article contains original material, the rest of it is due to R. Langlands. Very briefly, we reduce the problem of proving certain orbital integral identities for “matching” functions in the respective Hecke algebras to two counting problems on the buildings. We give Langlands’ solution of one of these problems in the case of the unit elements of the respective Hecke algebras and §5 provides the solution to the other one, again, in the unit element case. The main results assume p ≠ 2.