Optimal Approximation of the Second Iterated Integral of Brownian Motion

Abstract

In this article, a theorem is proved that describes the optimal approximation (in the L ²(?)-sense) of the second iterated integral of a standard two-dimensional Wiener process, W, by a function of finitely many elements of the Gaussian Hilbert space generated by W. This theorem has some interesting corollaries: First of all, it implies that Euler's method has the optimal rate of strong convergence among all algorithms that depend solely on linear functionals of the Wiener process, W; second, it shows that the approximation of the second iterated integral based on Karhunen–Loève expansion of the Brownian bridge is asymptotically optimal.     

A Girsanov Type Theorem on the Path Space Over a Compact Riemannian Manifold

Abstract

This article establishes a Girsanov type theorem on path spaces over compact Riemannian manifolds, generalizing the classical Girsanov theorem for Euclidean spaces.     

The Validity Range of Two Complex Analysis Theorems

In this article we give complete characterizations of shift-invariant uniform algebras A_S on compact abelian groups, in which two of the classical theorems for analytic functions hold, namely, Radó's theorem for analytic extension and Riemann's theorem for removable singularities. Our characterization is in terms of algebraical properties of the semigroup S of non-zero Fourier coefficients of the functions in A_S .     

Meromorphic Functions on a Riemann Surface to a Locally Semi-convex Space

In this article, vector-valued holomorphic and meromorphic functions on a Riemann surface to a complete Hausdorff locally semi-convex space are discussed. By introducing the concepts of vector-valued holomorphic and meromorphic differential forms, Cauchy's theorem and the Residue theorem of a vector-valued differential form on a Riemann surface are proved. Using the theory on the operator and the theory of a cohomology of a sheaf, we give a proof of the Mittag-Leffler theorem for vector-valued meromorphic functions on a non-compact Riemann surface to a complete Hausdorff locally semi-convex space.     

Stochastic Convolution-Type Heat Equations with Nonlinear Drift

Abstract

In this article, we study the solution of a class of stochastic convolution-type heat equations with nonlinear drift. For general initial condition and coefficients, we prove existence and uniqueness by using the characterization theorem and Banach's fixed-point theorem. We also give an implicit solution, which is a well-defined generalized stochastic process in a suitable distribution space. Finally, we investigate the continuous dependence of the solution on the initial data as well as the dependence on the coefficient.     

Some necessary and some sufficient trace inequalities for Euclidean distance matrices

In this article, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg's theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM . We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure.     

The Complete Asymptotic Expansion for the Left Quasi Interpolants of Müller's Gamma Operators

Abstract

Recently, Müller introduced and studied left quasi interpolants of his Gamma operators. In this article we present the complete asymptotic expansion of these operators. As a special case we obtain a Voronovskaja type theorem.     

The Method of Upper and Lower Solutions of Stochastic Differential Equations and Applications

Abstract

The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance.     

How Fast Is the Bandit?

Abstract

In this article we investigate the rate of convergence of the so-called two-armed bandit algorithm. The behavior of the algorithm turns out to be highly non standard: no central limit theorem, possible occurrence of two different rates of convergence with positive probability.     

An abstract version of the Morse index theorem and its application to hypersurfaces of constant mean curvature

In [5] Smale generalized the Morse index theorem (originally proved by Morse in [3]) to elliptic partial differential systems in several independent variables. Smale's result was used by Simons in [4] to obtain the index theorem for minimal submanifolds. The purpose of this paper is to give an abstract version of the Morse index theorem and use it to prove an index theorem for hypersurfaces of constant mean curvature. This was sugested by Barbosa and do Carmo in [1].     

Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.     

On Complete Convergence for Arrays of Random Elements

Abstract

A complete convergence theorem for arrays of rowwise independent random variables was obtained by Kruglov, Volodin, and Hu (Statistics and Probability Letters 2006, 76:1631–1640). In this article, we extend the result to a Banach space without any additional conditions. No assumptions are made concerning the geometry of the underlying Banach space.     

Existence of Solution of Nonlinear Neutral Stochastic Differential Inclusions with Infinite Delay

Abstract

This article is concerned with the existence of solution of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert Space. Sufficient conditions for the existence are obtained by using a fixed point theorem for condensing maps due to Martelli.     

Some Weyl modules and cohomology for algebraic groups

Abstract

In this article, we give a generalization of O’Halloran’s theorem that gives rise to families of Weyl modules with simple radicals. This generalization is then applied to simply connected, semisimple algebraic groups of types B, C, and D. We also construct families of Weyl modules with formal characters of length 4, and compute the extensions and cohomology of simple modules associated with these Weyl modules.     

The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

Abstract

In this article, we consider a stochastic integral inclusion driven by semimartingale with discontinuous multivalued right hand side. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered sets. The presented studies extend some recent results both for deterministic differential inclusions and stochastic differential equations for increasing operators.     

A Reflection Principle and an Orthogonal Decomposition in Clifford Algebras

In this article we investigate spaces of functions defined in a domain Ω ? R with values in the Clifford algebra R _n. According to an inner product an orthogonal decomposition is proved. By this decomposition, we obtain a subspace A ₂(Ω) of regular functions with respect to the Dirac operator. In the orthogonal complement the Dirac equation with homogeneous boundary values is solvable. The decomposition can be proved in two ways: by a reflection principle and by Sobolev's regularity theorem. It will turn out, that the existence of the orthogonal decomposition and Sobolev's theorem is equivalent. So also a reflection principle will be proved, which describes the jump behavior of a Cauchy type integral. By the reflection principle, a countable dense subset of A ₂(Ω) can be obtained. Further considerations lead to a minimal generating system, by which the Bergman kernel function can be obtained. As a conclusion we also obtain Runge's theorem.     

A new version of a theorem of Kaplansky

Abstract

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the w-version of this theorem, where w is a hereditary torsion theory for modules over a commutative ring.

Communicated by Silvana Bazzoni     

An upper bound for the representation dimension of group algebras with elementary abelian Sylow p-subgroups

Abstract

In this article, we present an upper bound on the representation dimension of the group algebra of a group with an elementary abelian Sylow p-subgroup. Specifically, if k is a field of characteristic p and G is a group with elementary abelian Sylow p-subgroup P, we prove that the representation dimension of kG is bounded above by the order of P. Key to proving this theorem is the separable equivalence between the two algebras and some nice properties of Mackey decomposition.     

A RADON-NIKODYM THEOREM FOR RIESZ SPACE VALUED MEASURES

ABSTRACT

In this note, as a consequence of the Radon-Nikodym theorem for positive operators given by Luxemburg and Schep ([6]), we obtain a Radon-Nikodym theorem for Riesz space valued measures similar to Wright's theorem ([11]).