Asymptotic gauges: Generalization of Colombeau type algebras

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general “growth condition” formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved.     

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.     

The functional analytic foundation of Colombeau algebras

Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a functional analytic approach placing these algebras in a unifying hierarchy, which clarifies their structural properties as well as their relation to each other.     

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)     

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.     

Approximating the number of integers without large prime factors

denotes the number of positive integers and free of prime factors . Hildebrand and Tenenbaum gave a smooth approximation formula for in the range , where is a fixed positive number . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate . The computational complexity of this algorithm is . We give numerical results which show that this algorithm provides accurate estimates for and is faster than conventional methods such as algorithms exploiting Dickman's function.

     

An estimate for the number of integers without large prime factors

denotes the number of positive integers and free of prime factors y$">. Hildebrand and Tenenbaum provided a good approximation of . However, their method requires the solution to the equation , and therefore it needs a large amount of time for the numerical solution of the above equation for large . Hildebrand also showed approximates for , where and is the unique solution to . Let be defined by for 0$">. We show approximates , and also approximates , where . Using these approximations, we give a simple method which approximates within a factor in the range , where is any positive constant.

     

General twisting of algebras

We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A,μ,u) in a monoidal category, as a morphism satisfying a list of axioms ensuring that (A,μ○T,u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich's braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.     

On linear versions of some addition theorems

Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dim _K ?AB? and dim _K ?B ⁿ ? in terms of dim _K A, dim _K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.     

Star product algebras of test functions

We prove that the Gelfand-Shilov spaces S _α ^β are topological algebras under the Moyal *-product if and only if α ≥ β. These spaces of test functions can be used to construct a noncommutative field theory. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in S _α ^β if and only if β < 1/2. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 3–17, October, 2007.     

On Asymptotic Proximity of Distributions

We consider some general facts concerning the convergence

Structural theorems for quasiasymptotics of distributions at the origin

An open question concerning the quasiasymptotic behavior of distributions at the origin is solved. The question is the following: Suppose that a tempered distribution has quasiasymptotic at the origin in S ′(?), then the tempered distribution has quasiasymptotic in D ′(?), does the converse implication hold? The second purpose of this article is to give complete structural theorems for quasiasymptotics at the origin. For this purpose, asymptotically homogeneous functions with respect to slowly varying functions are introduced and analyzed (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A computational procedure for the asymptotic analysis of homogeneous semi-Markov processes

A new algorithm for classifying the states of a homogeneous Markov chain having finitely many states is presented, which enables the investigation of the asymptotic behavior of semi-Markov processes in which the Markov chain is embedded. An application of the algorithm to a social security problem is also presented.     

Arithmetical distributions and the sequential extension of binary relations

With the help of an iterative process, we define a fairly broad class of arithmetical distributions on which continuous operations of addition, multiplication, and differentiation are defined. The multiplication of arithmetical distributions that we introduce is made consistent with the known definitions of the product of distributions. The results obtained can be used to justify the passage to the limit in the study of nonlinear problems of mathematical physics. Formalizing our approach, we describe a construction of an extension of binary relations, which is called sequential extension, from a dense subset to the whole topological space. These results are extended to operations of the first order. We show that the sequential extension of differentiation from the set of infinitely differentiable functions to the set of distributions coincides up to isomorphism with the generalized differentiation of distributions. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 836–853, June, 1999.     

求解第一类Fredholm积分方程的一种新的正则化算法

应用一种新的正则化方法建立了一类新的求解第一类Fredholm积分方程的正则化算法, 并借助Matlab软件给出了数值算例.数值结果与理论分析基本一致,而且表明文中建立的正则化比通常的Tikhonov正则化更精确.     

On linear functions of certain noncentral versions of independent gamma variables

A general Laplace transform and its inverse and their generalizations are considered in this article. This inverse contains the density functions of quadratic expressions in nonsingular as well as singular normal variables and a non-central version of linear functions of gamma variables, among others. Various representations of the inverse in power series, in gamma series, in Laguerre polynomials, in hypergeometric functions and in zonal polynomials are also discussed.     

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.     

Biderivations on tensor products of algebras

Let A be a noncommutative unital prime algebra and let S be a commutative unital algebra over a field 𝔽. We describe the form of biderivations of the algebra A?S. As an application, we determine the form of commuting linear maps of A?S.