Elimination in Weyl Algebra and Identities

In paper[1], Zeilberger presented an approach to sprcial functions identities, besdes     

The Algebra of Estimation in Linear Econometric Systems∗

In econometrics it is common for variables to be related together in a set of linear, multilateral and causal interdependencies. This type of system generally has properties which are unsatisfactory for application of classical regression techniques. Consequently, alternative estimation methods have been developed. This paper explores the relations between several such methods in terms of symmetric idempotents of predetermined variables and their orthogonal complements. Generalizations of two‐ and three‐stage least squares and instrumental variables are considered, including Wicken's estimator.² The relative efficiencies of the estimators are also discussed.

     

Virasoro Algebra,Dedekind η-function and Specialized Macdonald Identities

We motivate and prove a series of identities which form a generalization of the Euler pentagonal number theorem, and are closely related to specialized Macdonald identities for powers of the Dedekind -function. More precisely, we show that what we call denominator formula for the Virasoro algebra has higher analogue for all c_s,t-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as fusion and modular invariance that require all the irreducible modules of the series. In particular, in the case of c_2,2k+1-minimal models we give a new proof of a family of specialized Macdonald identities associated with twisted affine Lie algebras of type A⁽²⁾ _2k k 2 (i.e., BC_k affine root system) which involve (2k² - k)-th powers of the Dedekind -function.     

When Is a Graded PI Algebra a PI Algebra?

Abstract

Let G be a monoid with cancellation and let ? be a strongly G-graded algebra with finite G-grading satisfying a graded polynomial identity. We show that ? is a PI algebra.     

BRESEARCH ANNOUCEMENTS On an Ideal of a Nest Algebra

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Vertex-Compressed Subalgebras of a Graph von Neumann Algebra

In Cho (Acta Appl. Math. 95:95–134, 2007 and Complex Anal. Oper. Theory 1:367–398, 2007), we introduced Graph von Neumann Algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via graph-representations, which are groupoid actions. In Cho (Acta Appl. Math. 95:95–134, 2007), we showed that such crossed product algebras have the amalgamated reduced free probabilistic properties, where the reduction is totally depending on given directed graphs. Moreover, in Cho (Complex Anal. Oper. Theory 1:367–398, 2007), we characterize each amalgamated free blocks of graph von Neumann algebras: we showed that they are characterized by the well-known von Neumann algebras: Classical group crossed product algebras and (operator-valued) matricial algebras. This shows that we can provide a nicer way to investigate such groupoid crossed product algebras, since we only need to concentrate on studying graph groupoids and characterized algebras. How about the compressed subalgebras of them? i.e., how about the inner (cornered) structures of a graph von Neumann algebra? In this paper, we will provides the answer of this question. Consequently, we show that vertex-compressed subalgebras of a graph von Neumann algebra are characterized by other graph von Neumann algebras. This gives the full characterization of the vertex-compressed subalgebras of a graph von Neumann algebra, by other graph von Neumann algebras.     

On the Radical of a Hecke–Kiselman Algebra

Algebras and Representation Theory - The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ is an oriented cycle, it is shown that the algebra is...     

The minimal polynomial of a matrix ∗

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A (λ) of A over F. By using q_A (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

A Note on Ideals of a Three-Dimensional Associative Algebra

In this paper, we shall apply the theory of nilpotent radical to derive some results about ideals of a there-dimensional associative algebra over a field F and of an associative ring of order p~3 where p is a prime. These results are useful in studying the structure of the algebra and the ring.     

Generalizations of Several Identities

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Solutions of Equations Involving Analytic Functions in a Banach Algebra

By the properties of univalent analytic functions,we have discussed the exi-stence and uniqueness of eqation f(x)=a in a Banach algebra.We have the follo-wing fundamental lemmas. Lemma 1. Let A be a Banach algebra with identity e,W,U be two opensubsets of C, U W,f be an analytic function in W, and univalent in U, V=f(U),a∈A.If σ(a) V,then there exists unique x∈A such that σ(x) U {λ∈W|f(λ) V}and f(x)=a     

Nonrelativistic Contraction of a Canonically Deformed Super-Poincaré Hopf Algebra

We consider the nonrelativistic contraction of a canonically deformed N = 1{\mathcal{N}} = 1 super-Poincaré Hopf algebra. In such a way we get the new quantum supergroup – a canonically twisted super-Galilei Hopf algebra equipped with the Clifford-like spinorial sector of the corresponding super-space. Besides, we also propose the new Lie-algebraic twist deformation of a super-Galilei algebra leading to classical time and quantum space as well as to the Clifford-like fermionic part of the twisted N = 1{\mathcal{N}} = 1 superspace.     

The Socle and Semisimplicity of a Kumjian–Pask Algebra

The Kumjian–Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λ and is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left ideals of KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian–Pask algebra. As a consequence of this structure theorem, every semisimple Kumjian–Pask algebra can be obtained as a Leavitt path algebra of a directed graph.     

Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three

In this paper, we present some modules over the rank-three quantized Weyl algebra, which are closely related to modules over some vertex algebras. The isomorphism classes among these modules are also determined.     

Decomposable Multipliers on a Banach Algebra

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Representations of q-analogue of the Virasoro Algebra

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Two Wider Classes of Combinatorial Identities

Here introduced are two general classes of combinatorial identities that may be defined via inverse relations due to Gould and Hsu(cf.Duke Math J. 40(1973), 885 891). Classes(∑) and(∑~*). Any identity is said to belong to the class(∑) if it can be embedded in either of the forms(reciprocal relations)     

Changes of Working Styles in a Computer Algebra Environment – The Case of Functions

This study is an empirical investigation of 11th graders at a German high school (Gymnasium). Working over a 24-hour period in a computer lab, we investigated students' use of quadratic functions with `Derive', and trigonometric functions with `Mathplus' (or `Theorist' for Macintosh). We were particularly interested in the working styles of students while they solved problems and looked for changes in these styles, as compared to traditional paper and pencil activities. While students worked on the computer, their activities (such as inputs from the keyboard, menu choices or mouse movements) were saved by a special program, which ran in the `background'. We are interested in the possibilities of developing a research method based on these `computer protocols'. The study should be seen as an exploratory study for developing hypotheses for further empirical investigations.This revised version was published online in September 2005 with corrections to the Cover Date.     

Corrigendum To: “The Odd part of a N-Koszul Algebra”

The paper: The odd part of an N-Koszul algebra, Comm. in Algebra, 33:101–108, 2005, contains some missprints that make it difficult to read. We correct here these mistakes.