Weak homogeneity in generalized function algebras

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized stochastic processes in algebras of generalized functions

Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an algebra of generalized functions are characterized by their first and second moments and an application to stochastic differential equations is given.     

Nonsmooth differential geometry and algebras of generalized functions

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in nonsmooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.     

Strongly internal sets and generalized smooth functions

Based on a refinement of the notion of internal sets in Colombeau's theory, so-called strongly internal sets, we introduce the space of generalized smooth functions, a maximal extension of Colombeau generalized functions. Generalized smooth functions as morphisms between sets of generalized points form a sub-category of the category of topological spaces. In particular, they can be composed unrestrictedly.     

Non-linear expectations in spaces of Colombeau generalized functions

We introduce the notion of a non-linear expectation in spaces of Colombeau generalized functions and provide its characterization in terms of the upper expectation over a set of probability measures. We then study a fully non-linear backward stochastic differential equation in the Colombeau setting via its connection with the corresponding fully non-linear partial differential equation.     

Positivity and positive definiteness in generalized function algebras

The definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions. Versions of Bochner's theorem for a positive definite Colombeau generalized function are given.     

Suppleness of the sheaf of algebras of generalized functions on manifolds

We show that the sheaves of algebras of generalized functions Ω→G(Ω) and Ω→G^∞(Ω), Ω are open sets in a manifold X, are supple, contrary to the non-suppleness of the sheaf of distributions.     

Generalized group actions in a global setting

We study generalized group actions on differentiable manifolds in the Colombeau framework, extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions. As an application, we analyze group invariant generalized functions in this setting.     

Canonical bases for quantum generalized Kac–Moody algebras

We construct canonical bases for quantum generalized Kac–Moody algebras using semisimple perverse sheaves.     

Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras

After reformulating the representation theory of 0-Hecke algebras in an appropriate family of Yang-Baxter bases, we investigate certain specializations of the Ariki-Koike algebras, obtained by setting q=0 in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe restrictions, induction products, Cartan invariants and decomposition matrices. This allows us to identify the Grothendieck rings of the towers of algebras in terms of certain graded Hopf algebras known as the Mantaci-Reutenauer descent algebras, and Poirier quasi-symmetric functions. We also describe the Ext-quivers, and conclude with numerical tables.     

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93-106, J. Math. Anal. Appl. 142 (1989) 452-467].     

On the characterization of p ‐adic Colombeau–Egorov generalized functions by their point values

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so‐called p ‐adic Colombeau–Egorov algebra ??(?ⁿ _p ) uniquely. We further show in a more general way that for an Egorov algebra ??(M, R) of generalized functions on a locally compact ultrametric space (M, d) taking values in a nontrivial ring, a point value characterization holds if and only if (M, d) is discrete. Finally, following an idea due to M. Kunzinger and M. Oberguggenberger, a generalized point value characterization of ??(M, R) is given. Elements of ??(?ⁿ _p ) are constructed which differ from the p ‐adic δ ‐distribution considered as an element of ??(?ⁿ _p ), yet coincide on point values with the latter. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group

It is shown that for amenable groups, all finite-dimensional extensions of A_p(G) algebras split strongly. Furthermore, each extension of A_p(G) which splits algebraically also splits strongly. We also show that if G is an almost connected locally compact group, or a subgroup of GL_n(V) (V being a finite-dimensional vector space), and if for a fixed p∈(1,∞), all finite-dimensional singular extensions of A_p(G) split strongly, then G is amenable. Continuous order isomorphisms for the pointwise order of A_p(G) algebras, are characterized as weighted composition maps. Similarly, order isomorphisms for the pointwise order of B_p(G) algebras, are characterized as ∗-algebra isomorphisms followed by multiplication by an invertible positive multiplier. In addition, it is shown that for amenable groups, an order isomorphism for the pointwise order between A_p(G) algebras that preserve cozero sets is necessarily continuous, and hence induces an algebra isomorphism.     

Extremal generalized centralizers in matrix algebras

We describe matrices with extremal generalized centralizers over algebraically closed fields.     

Special subalgebras of Boolean algebras

We consider eight special kinds of subalgebras of Boolean algebras. In Section 1 we describe the relationships between these subalgebra notions. In succeeding sections we consider how the subalgebra notions behave with respect to the most common cardinal functions on Boolean algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Commutative combinatorial Hopf algebras

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its noncommutative dual is realized in three different ways, in particular, as the Grossman–Larson algebra of heap-ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees, and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.     

Inequalities on generalized trigonometric and hyperbolic functions

In this paper, we prove some inequalities involving the generalized trigonometric and hyperbolic functions.     

Geometrical embeddings of distributions into algebras of generalized functions

We use spectral theory to produce embeddings of distributions into algebras of generalized functions on a closed (compact without boundary) Riemannian manifold. These embeddings are invariant under isometries and preserve the singularity structure of the distributions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)