Multivalued Exponentiation Analysis. Part I: Maclaurin Exponentials

The exponentiation theory of linear continuous operators on Banach spaces can be extended in manifold ways to a multivalued context. In this paper we explore the Maclaurin exponentiation technique which is based on the use of a suitable power series. More precisely, we discuss about the existence and characterization of the Painlevé–Kuratowski limit

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

On Auslander–Reiten Components and Heller Lattices for Integral Group Rings

Let be a finite group, a complete discrete valuation ring of characteristic zero with residue class field of characteristic , and a block of the group ring . Suppose that is of infinite representation type and is sufficiently large to satisfy certain conditions. Let be the Auslander–Reiten quiver of and a connected component of . In this paper, we show that if contains some Heller lattices then the tree class of the stable part of is . Also, we show that has infinitely many components of type if a defect group of is neither cyclic nor a Klein four group.Presented by Jon Carlson.     

Two-parameter p, q-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter -variation path integrals. Our condition of locally bounded -variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time pathwise and then give generalized It’s formula when is only of bounded -variation in . In the case that is of locally bounded variation in , the integral is the Lebesgue–Stieltjes integral and was used by Elworthy, Truman and Zhao. When is of only locally -variation, where , , and , the integral is a two-parameter Young integral of -variation rather than a Lebesgue–Stieltjes integral. In the special case that is independent of , we give a new condition for Meyer's formula and is defined pathwise as a Young integral. For this we prove the local time is of -variation in for each , for each almost surely (-variation in the sense of Lyons and Young, i.e. ).     

Regularity of Solution Maps of Differential Inclusions Under State Constraints

Consider a differential inclusion under state constraints

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

The Zaremba Problem with Singular Interfaces as a Corner Boundary Value Problem

We study mixed boundary value problems for an elliptic operator on a manifold with boundary , i.e., in on , where is subdivided into subsets with an interface and boundary conditions on that are Shapiro–Lopatinskij elliptic up to from the respective sides. We assume that is a manifold with conical singularity . As an example we consider the Zaremba problem, where is the Laplacian and Dirichlet, Neumann conditions. The problem is treated as a corner boundary value problem near which is the new point and the main difficulty in this paper. Outside the problem belongs to the edge calculus as is shown in Bull. Sci. Math. (to appear).With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.     

A Dual Criterion for Maximal Monotonicity of Composition Operators

In this paper we present a dual criterion for the maximal monotonicity of the composition operator , where is a maximal monotone (set-valued) operator and is a continuous linear map with the adjoint , and are reflexive Banach spaces, and the product notation indicates composition. The dual criterion is expressed in terms of the closure condition involving the epigraph of the conjugate of Fitzpatrick function associated with , and the operator As an easy application, a dual criterion for the maximality of the sum of two maximal monotone operators is also given. The work of this author was completed while at the School of Mathematics, University of New South Wales, Sydney, Australia.     

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.     

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

In [2] the subconstruct of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category of approach spaces. The main result of this paper states that moreover is firmly -reflective in for the class of epimorphic embeddings. ‘Firm -reflective’ is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category of Hausdorff uniform spaces with uniformly continuous maps. It means that is -reflective in and that the reflector is such that belongs to if and only if is an isomorphism. Firm -reflectiveness implies uniqueness of completion in the sense that whenever is a map with and sober, the associated is an isomorphism. Our result generalizes the fact that in the category the subconstruct of sober topological spaces is firmly reflective for the class of b-dense embeddings in . Also firmness in some other subconstructs of will be easily obtained.A. Gerlo and C. Van Olmen are research assistants at the Fund of Scientific Research Vlaanderen (FWO). E. Vandersmissen is a research assistant supported by the FWO-grant G.0244.05.     

Range of the Fractional Weak Discrepancy Function

In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy of a poset to be the minimum nonnegative for which there exists a function satisfying (1) if then and (2) if then . This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function is the set of rational numbers that are either at least one or equal to for some nonnegative integer . Moreover, is a semiorder if and only if , and the range taken over all semiorders is the set of such fractions .The third author's work was supported in part by a Wellesley College Brachman Hoffman Fellowship.     

Involutions of \operatorname{{SL}} (nk), (n>2)

A first characterization of the isomorphism classes of -involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of with any field of characteristic not equal to . We classify the isomorphism classes of involutions for algebraically closed, the real numbers, the -adic numbers and finite fields. We also determine in which cases the corresponding fixed point group is -anisotropic. In those cases the corresponding symmetric -variety consists of semisimple elements.Aloysius G. Helminck was partially supported by N.S.F. Grant DMS-9977392.     

Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds

Let be a one-parameter family of positive integral operators on a locally compact space . For a possibly non-uniform partition of define a finite measure on the path space by using a) for the transition between any two consecutive partition times of distance and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary normalize the result to get a probability measure. We prove a version of Chernoff's theorem of semigroup theory and tightness results which yield convergence in law of such measures as the partition gets finer. In particular let be a closed smooth submanifold of a manifold . We prove convergence of Brownian motion on , conditioned to visit at all partition times, to a process on whose law has a density with respect to Brownian motion on which contains scalar, mean and sectional curvatures terms. Various approximation schemes for Brownian motion on are also given.      

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra

We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space and Colombeau vector space      

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals of a deterministic function with respect to a random Fréchet () sup-measure. The measure is sup-additive rather than additive and is defined over a general measure space , where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary , and the metric metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with Fréchet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of stable processes, whereas the extremal integrals correspond to the usual stable stochastic integrals. We prove that essentially any strictly stable process belongs to the domain of max-stable attraction of an Fréchet, max-stable process. Moreover, we express the corresponding Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the stable process. The close correspondence between the max-stable and stable frameworks yields new examples of max-stable processes with non-trivial dependence structures.This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS-0505747 at Boston University.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Recursively enumerable sets of polynomials over a finite field are Diophantine

We construct a Diophantine interpretation of over . Using this together with a previous result that every recursively enumerable (r.e.) relation over is Diophantine over , we will prove that every r.e. relation over is Diophantine over . We will also look at recursive infinite base fields , algebraic over . It turns out that the Diophantine relations over are exactly the relations which are r.e. for every recursive presentation.     

Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.