Multiplication product formula for associated homogeneous distributions on R

The set of associated homogeneous distributions (AHDs) on R, ??^′(R), consists of the distributional analogues of power‐log functions with domain in R. This set contains the majority of the (one‐dimensional) distributions one typically encounters in physics applications. The recent work done by the author showed that the set ??^′(R) admits a closed convolution structure (??^′(R), *). By combining this structure with the generalized convolution theorem, a distributional multiplication product was defined, resulting in also a closed multiplication structure (??^′(R), .). In this paper, the general multiplication product formula for this structure is derived. Multiplication of AHDs on R is associative, except for critical triple products. These critical products are shown to be non‐associative in a simple and interesting way. The non‐associativity is necessary and sufficient to circumvent Schwartz's impossibility theorem on the multiplication of distributions. Copyright © 2011 John Wiley & Sons, Ltd.     

Convolution product formula for associated homogeneous distributions on R

The set of Associated Homogeneous Distributions (AHDs) on R, ??′(R), consists of distributional analogues of power‐log functions with domain in R. This set contains the majority of the (one‐dimensional) distributions typically encountered in physics applications. In earlier work of the author it was shown that ??′(R) admits a closed convolution structure, provided that critical convolution products are defined by a functional extension process. In this paper, the general convolution product formula is derived. Convolution of AHDs on R is found to be associative, except for critical triple products. Critical products are shown to be non‐associative in a minimal and interesting way. Copyright © 2010 John Wiley & Sons, Ltd.     

Structure theorems for associated homogeneous distributions based on the line

Associated homogeneous distributions (AHDs) with support in the line R are the distributional generalizations of one‐dimensional power‐log functions. In this paper, we derive a number of practical structure theorems for AHDs based on R and being complex analytic with respect to their degree of homogeneity in some region of the complex plane. Each theorem gives a representation that is designed to have a distinct advantage for calculating either convolution products, multiplication products, generalized derivatives and primitives, Fourier transforms or Hilbert transforms of AHDs. Copyright © 2008 John Wiley & Sons, Ltd.     

The convolution of associated homogeneous distributions on R–Part II

This is the second in a series of two papers in which we construct a convolution product for the set ?′ (R) of associated homogeneous distributions (AHDs) with support in R. In Part I we showed that if f ^a and g ^b are AHDs with degrees of homogeneity a ? 1 and b ? 1, the convolution f ^a * g ^b exists as an AHD, if the resulting degree of homogeneity a + b?1 ? N. In this article, we develop a functional extension process, based on the Hahn–Banach theorem, to give a meaning to the convolution product of two AHDs of degrees a ? 1 and b ? 1, in the critical case that a + b ? 1 ∈ N. With respect to this construction, the structure (?′(R), *) is shown to be closed.     

Some New Class of Absolute Valued Algebras with Left Unit

Let A be an absolute valued algebra with left unit. We prove that if A contains a nonzero central element, then A is finite dimensional and is isomorphic to \mathbb R, \mathbb C{\mathbb {R}, \mathbb {C}} or new classes of four and eight–dimensional absolute valued algebras with left unit. This is more general than those results in [2] and [3].     

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)     

Multiphase Transport in the Space of Stochastic Tempered Distributions

We construct a stochastic distributional theory for multiscale,multiphase transport. Field variables are viewed as stochastictempered distributions on Rⁿ. The field variables are made operationalvia convolution with a deterministic compact distribution whichis a representation of the measurement device. The correlationover scales of a field variable is analysed in a stochasticfunctional setting. Examples of functional transport equations,as represented by spectral measures, are presented     

Lie algebras and recurrence relations I

We study representations of the Heisenberg-Weyl algebra and a variety of Lie algebras, e.g., su(2), related through various aspects of the spectral theory of self-adjoint operators, the theory of orthogonal polynomials, and basic quantum theory. The approach taken here enables extensions from the one-variable case to be made in a natural manner. Extensions to certain infinite-dimensional Lie algebras (continuous tensor products, q-analogs) can be found as well. Particularly, we discuss the relationship between generating functions and representations of Lie algebras, spectral theory for operators that lead to systems of orthogonal polynomials and, importantly, the precise connection between the representation theory of Lie algebras and classical probability distributions is presented via the notions of quantum probability theory. Coincidentally, our theory is closed connected to the study of exponential families with quadratic variance in statistical theory.     

Prime ideals in Hopf galois extensions

For a finite-dimensional Hopf algebraH, we study the prime ideals in a faithfully flatH-Hopf-Galois extensionR ⊂A. One application is to quotients of Hopf algebras which arise in the theory of quantum groups at a root of 1. For the Krull relations betweenR andA, we obtain our best results whenH is semisolvable; these results generalize earlier known results for crossed products for a group action and for algebras graded by a finite group. We also show that ifH is semisimple and semisolvable, thenA is semiprime providedR isH-semiprime.     

Semi-algebraic Absolute Valued Algebras with an Involution

Abstract

Let A be an absolute valued algebra. In El-Mallah (El-Mallah,M. L. (1988). Absolute valued algebras with an involution. Arch. Math. 51: 39–49) we proved that,if A is algebraic with an involution,then A is finite dimensional. This result had been generalized in El-Amin et al. (El-Amin,K.,Ramirez,M. I.,Rodriguez,A. (1997). Absolute valued algebraic algebras are finite dimensional. J. Algebra 195:295–307),by showing that the condition “algebraic” is sufficient for A to be finite dimensional. In the present paper we give a generalization of the concept “algebraic”,which will be called “semi-algebraic”,and prove that if A is semi-algebraic with an involution then A is finite dimensional. We give an example of an absolute valued algebra which is semi-algebraic and infinite dimensional. This example shows that the assumption “with an involution” cannot be removed in our result.     

Boolean products of R0‐algebras

In this paper, the (weak) Boolean representation of R₀‐algebras are investigated. In particular, we show that directly indecomposable R₀‐algebras are equivalent to local R₀‐algebras and any nontrivial R₀‐algebra is representable as a weak Boolean product of local R₀‐algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).     

The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras

Let H be a Hopf algebra over the field k which is a finite module over a central affine sub-Hopf algebra R. Examples include enveloping algebras of finite dimensional k-Lie algebras in positive characteristic and quantised enveloping algebras and quantised function algebras at roots of unity. The ramification behaviour of the maximal ideals of Z(H) with respect to the subalgebra R is studied, and the conclusions are then applied to the cases of classical and quantised enveloping algebras. In the case of for semisimple a conjecture of Humphreys [28] on the block structure of is confirmed. In the case of for semisimple and an odd root of unity we obtain a quantum analogue of a result of Mirković and Rumynin, [35], and we fully describe the factor algebras lying over the regular sheet, [9]. The blocks of are determined, and a necessary condition (which may also be sufficient) for a baby Verma -module to be simple is obtained. Received: 24 June 1999; in final form: 30 March 2000 / Published online: 17 May 2001     

More examples of bicrossproduct and double cross product Hopf algebras

We construct examples of bicrossproducts and double cross products of quantum groupsă(R) associated to general matrix solutionsR of the Quantum Yang-Baxter Equations. We also describe iterated double cross products of quantum groups. In the course of constructingă(R) we are led to introduce a suitable notion of mutually dual Hopf algebras and a dual quantum groupŬ(R). Work supported by SERC Research Assistantship.     

The double sign of a real division algebra of finite dimension greater than one

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_nFor any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_n$ of all n‐dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of $\mathscr {D}_n$.     

On the Structure of Coboundary R-Matrices of Classical Series

Coboundary R-matrices for quantum algebras associated with simple Lie algebras of classical series are computed. In the fundamental representations, these R-matrices are equal to the exponents of the related classical r-matrices. Bibliography: 11 titles.     

Some Homological Properties of Skew PBW Extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well-known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.     

A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.