Non-Gaussian Malliavin calculus on real Lie algebras

The non-commutative Malliavin calculus on the Heisenberg-Weyl algebra is extended to the affine algebra. A differential calculus and a non-commutative integration by parts are established. As an application we obtain sufficient conditions for the smoothness of Wigner-type laws of non-commutative random variables with gamma or continuous binomial marginals.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

On the algebra of multidimensional integral operators with homogeneous SO(n)-invariant kernels and weakly radially oscillating coefficients

In this paper, we study the Banach algebra B generated by multidimensional integral operators whose kernels are homogeneous functions of degree (?n) invariant with respect to the rotation group SO(n) and by the operators of multiplication by radial weakly oscillating functions. A symbolic calculus is developed for the algebra 25. The Fredholm property and the formula for calculating the index are described in terms of this calculus.     

An algebra generated by multiplicative discrete convolution operators

We consider a Banach algebra generated by multiplicative discrete convolution operators. We construct a symbolic calculus for this algebra and in terms of this calculus we describe criteria for the Noetherian property of operators and obtain a formula for their index.     

中介命题演算系统$MP^{M}$的代数系统

MP~M系统是在中介逻辑系统的基础上建立起来的,用于处理数据库中不完全信息的三值逻辑命题演算系统．本文通过在MP~M系统上建立一个代数系统,对MP~M系统进行了代数抽象,讨论了MP~M系统的代数性质．本文还研究了该代数系统的次直积,以及与其它一些代数系统之间的关系．     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.     

Explicit algebras with the Leibniz–Mellin translation product

A sub‐calculus of the calculus (“algebra”) of all complete conormal symbols arising in the edge pseudodifferential calculus is constructed. This calculus of complete conormal symbols is suitable for constructing sub‐calculi of the general edge pseudodifferential calculus, for which the edge‐degenerate pseudodifferential operators involved map conormal asymptotics of distributions near the edges in a prescribed manner. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Formal power series, operator calculus, and duality on Lie algebras

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.     

线性代数与微积分学问题与解法的渗透

通过典型的线性代数问题与微积分学问题,就其问题解法上的相互渗透进行探讨,初步揭示大学理工科各专业的两门重要的基础课线性代数与微积分学它们之间密切的联系.     

A Banach algebra of generalized functions

An algebraic isomorphism from a convolution algebra of Laplace transformable functions with support on the half-line to a complete discrete normed convolution algebra of sequences is used to construct generalized functions. The extension of this function-to-sequence map to a commutative Banach algebra of generalized functions is shown to be a Banach algebra isomorphism which can be utilized to establish a discrete formulation of a Mikusiński-type operational calculus and to construct algorithms for the numerical solution of half-line convolution equations.     

Application of CAS for teaching of integral-transforming theorems

Teaching vector calculus (the integral-transforming theorems, in particular) to students of engineering has got many difficulties. On the one hand complex and deep knowledge of vector algebra (in three dimensions) and calculus (in one and many variables) is necessary for understanding, on the other hand it is difficult to illustrate the concepts of vector calculus exclusively by means of blackboard and chalk. In this paper we demonstrate how the line integral, the surface integral, the theorems of Gauss and Stokes are taught with the help of a MAPLE computer algebra system (CAS). Application of this pedagogical method supports the hypothesis that using of CAS makes it possible to teach those kinds of concepts and theorems which are often used in engineering practice but couldn’t be presented because of lack of time and appropriate preliminary training.     

Two Projection Theorems and Symbol Calculus for Operators with Massive Local Spectra

In this paper we construct a symbol calculus for Banach algebras generated by two idempotents and a coefficient algebra. This, combined with local principles for ?embedding algebras”?, leads to a symbol calculus for singular integral operators on spaces with Muckenhoupt weight and for singular integral operators with measurable coefficients.     

Invitation to composition

In 1963 [Ann. of Math. 78, 267-288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus.     

有关解线性方程组的一些思考

解线性方程组是线性代数课程的最重要内容之一,目前工科线性代数的大纲和教材一般不包括不相容方程组,其实这部分内容具有广泛的应用.本文用微积分方法给出不相容方程组的最小二乘解以及极小范数最小二乘解,可供线性代数课程的教学改革作参考.建议待条件成熟时,将不相容方程组的最小二乘解纳入工科线性代数的教学大纲和教材.     

Smoothness of Wigner densities on the affine algebra

The non-commutative Malliavin calculus on the Heisenberg–Weyl algebra (see (i) C. R. Acad. Sci. Paris, Sér. I 328 (11) (1999) 1061–1066, (ii) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (1) (2001) 11–38) is extended to the affine algebra. A differential calculus is established, which generalizes the corresponding commutative integration by parts formulas. As an application we obtain sufficient conditions for the smoothness of Wigner type laws of non-commutative random variables with gamma and continuous binomial marginals. To cite this article: U. Franz et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Doi-Hopf模以及其上的平坦联络

对于任意Hopf代数,在其上建立了一种新的具有非交换微分几何形式的代数结构,并论证了Doi-hopf模与具有平坦联络的模之间是同构的,其中,联络对应于我们所给的微分算子.     

一般多项式有界算子的泛函表示定理

该文在圆盘代数A(D)中引入了一个数乘变换, 找到了多项式有界算子的多项式演算与Riesz函数演算之间的联系, 得到了Banach空间X上的一般多项式有界算子的泛函表示定理.     

The umbral calculus on logarithmic algebras

D. E. Loeb and G.-C. Rota, using the operator of differentiation D, constructed the logarithmic algebra that is the generalization of the algebra of formal Laurent series. They also introduced Appell graded logarithmic sequences and binomial (basic) graded logarithmic sequences as sequences of elements of the logarithmic algebra and extended the main results of the classical umbral calculus on such sequences. We construct an algebra by an operator d that is defined by the formula (1.1). This algebra is an analog of the logarithmic algebra. Then we define sequences analogous to Boas-Buck polynomial sequences and extend the main results of the nonclassical umbral calculus on such sequences. The basic logarithmic algebra constructed by the operator of q-differentiation is considered. The analog of the q-Stirling formula is obtained.