Unifying order structures for Colombeau algebras

We define a general notion of set of indices which, using concepts from pre‐ordered sets theory, permits to unify the presentation of several Colombeau‐type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big‐O such that its usual properties continue to hold. Using this generalized notion of big‐O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.     

New versions of the Colombeau algebras

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half‐space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions . The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one‐dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and [50]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The category of Colombeau algebras

Since the beginning of Colombeau’s the theory of algebras of generalized functions, the role of its characteristic polynomial growth versus a more general condition has been explored. Recently, we introduced the notion of asymptotic gauge (AG), and we used it to study Colombeau AG-algebras. This construction concurrently generalizes many different algebras used in Colombeau’s theory and, at the same time, allows for more general growth scales. In this paper, we study the categorical properties of Colombeau AG-algebras with respect to the choice of the AG. The main aim of the paper is to study suitable functors to relate differential equations framed in algebras having different growth scales.     

Internal sets and internal functions in Colombeau theory

This paper is devoted to the study of generalized functions as pointwise functions (so-called internal functions) on certain sets of generalized points (so-called internal sets). We treat the case of the Colombeau algebras of generalized functions, for which these notions have turned out to constitute a fundamental technical tool. We provide general foundations for the notion of internal functions and internal sets and prove a saturation principle. Various applications to Colombeau algebras are given.     

An algebraic approach to manifold-valued generalized functions

We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of smooth parametrization. As a consequence, and to underline the consistency and validity of this approach, we see that this generalized version on algebra isomorphisms in turn implies the classical result on algebras of smooth functions.     

Homogeneity in generalized function algebras

We investigate homogeneity in the special Colombeau algebra on Rd as well as on the pierced space Rd?{0}. It is shown that strongly scaling invariant functions on Rd are simply the constants. On the pierced space, strongly homogeneous functions of degree α admit tempered representatives, whereas on the whole space, such functions are polynomials with generalized coefficients. We also introduce weak notions of homogeneity and show that these are consistent with the classical notion on the distributional level. Moreover, we investigate the relation between generalized solutions of the Euler differential equation and homogeneity.     

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.     

Isomorphisms of algebras of generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the algebras of generalized functions (in the sense of Colombeau) on X, resp. Y is given by composition with a unique generalized function from Y to X. We also characterize the multiplicative linear functionals from the Colombeau algebra on X to the ring of generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.     

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)     

Point value characterizations and related results in the full Colombeau algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${{\mathcal {G}}^e(\Omega )}$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${{\mathcal {G}}^d(\Omega )}$\end{document}

We present a point value characterization for elements of the elementary full Colombeau algebra ${\mathcal {G}}^e(\Omega )$ and the diffeomorphism invariant full Colombeau algebra $\mathcal {G}^d(\Omega )$. Moreover, several results from the special algebra ${\mathcal {G}}^s(\Omega )$ about generalized numbers and invertibility are extended to the elementary full algebra.     

Modules and ideals of algebras of associative type

In this paper, we study some properties of algebras of associative type introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

Hugoniot–Maslov chains of a shock wave in conservation law with polynomial flow

In this paper we give a theoretical foundation to the asymptotical development proposed by V. P. Maslov for shock type singular solutions of conservations laws, in the framework of Colombeau theory of generalized functions. Indeed, operating with Colombeau differential algebra of simplified generalized functions, we proof that Hugoniot–Maslov chains are necessary conditions for the existence of shock waves in conservation laws with polynomial flows. As a particular case, these equations include the Hugoniot–Maslov chains for shock waves in the Hopf equation. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].Research partially supported by CNPq (Proc 300652/95-0).     

广义a-结合BCI-代数

引入了广义ａ－结合ＢＣＩ－代数的概念，研究了ＢＣＩ－代数的ｐ－半单部分与广义ａ－结合部分的关系．并将ｐ－半单ＢＣＩ－代数的若干重要性质推广到广义ａ－结合ＢＣＩ－代数上．最后我们证明了每个广义ａ－结合ＢＣＩ－代数可确定一个交换偏序幺半群．本文结果表明文［１］的正则ＢＣＩ－代数与ｐ－半单ＢＣＩ－代数是一致的．     

Full algebra of generalized functions and non-standard asymptotic analysis

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions.We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution.We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.     

Jump conditions in stress relaxation and creep experiments of Burgers type fluids: a study in the application of Colombeau algebra of generalized functions

We discuss stress relaxation and creep experiments of fluids that are generalizations of the classical model due to Burgers by allowing the material moduli such as the viscosities and relaxation and retardation times to depend on the stress. The physical problem, which is cast within the context of one dimension, leads to an ordinary differential equation that involves nonlinear terms like product of a function with a jump discontinuity and the derivative of a function with a jump discontinuity. As the equations are nonlinear, standard techniques that are used to study problems concerning linear viscoelastic fluids such as Laplace transforms and the theory of distributions are not applicable. We find it necessary to seek the solution in a more general setting. We discuss the mathematical and physical issues concerning the jump discontinuities and nonlinearity of the governing equation, and we show that the solution to the governing equation can be found in the sense of the generalized functions introduced by Colombeau. In the framework of Colombeau algebra we, under certain assumptions, derive jump conditions that shall be used in stress relaxation and creep experiments of fluids of the Burgers type. We conclude the paper with a discussion of the physical relevance of these assumptions.     

A note on ideals in synaptic algebras

The notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.     

On some class of periodic-discrete homogeneous difference equations via Fibonacci sequences

The intent of this paper is to solve the homogeneous linear difference equations with periodic coefficients. For this purpose, we turn our study in solving a class of discrete-time linear systems, in the algebra of square matrices. The tools used repose predominately on the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. Some explicit solutions are established and special cases are discussed. Illustrative examples are given.