Identités Multilinéaires de Degré 4 pour les Algèbres de Bernstein et de Mutation Non Commutatives

We give the identities generating the spaces of degree 4 multilinear identities satisfied by the non commutative 0th-order Bernstein algebras and the non commutative mutation algebras. For it, we use a method reducing the search for multilinear identities to the resolution of linear systems.     

Associative Dialgebras from a Structural Viewpoint

In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some “perfection” property (simpleness, primitiveness, primeness, or semiprimeness) imply that the dialgebra comes from an associative algebra with both products ? and ? identified. We also describe the class of zero-cubed algebras and apply its study to that of dialgebras. Finally, we describe two-dimensional associative dialgebras.     

An Algebra of Exponential Polynomials as the Cofinite Dual of the Hopf Algebra of Polynomials

We prove that the cofinite dual of the Hopf algebra of polynomials in several variables can be represented as a Hopf algebra ? of exponential polynomials that contains the polynomials as a Hopf subalgebra. We also present some algebras isomorphic to ? whose elements are rational functions or multi-sequences.     

Research Article: On Extending Primal-Dual Interior-Point Method for Linear Optimization to Convex Quadratic Symmetric Cone Optimization

Disjoint frames are interesting frames in Hilbert spaces, which were introduced by Han and Larson in [4 D. Han and D. R. Larson ( 2000 ). Frames, Basis and Group Representations . Memoirs of the American Mathematical Society, No. 679. AMS, Providence, RI.  [Google Scholar]]. In this article, we use disjoint frames to construct frames. In particular, we obtain some conditions for the linear combinations of frames to be frames where the coefficients in the combination may be operators. Our results generalize the corresponding results obtained by Han and Larson. Finally, we provide some examples to illustrate our constructions.     

Automating the Partition of Indexes

Abstract

In this article we show how attention to the structure of a particular algebraic calculation can lead to the simple implementation of powerful computer algebra tools. The creation of partitions for a set of indexes is required for the implementation of many theoretical structures. This may be difficult to do by hand even when the number of indexes is only moderately large. These partitions arise through the action of differentiation and so we mimic differentiation in a computer algebra package to create partitions of indexes. The strategies employed are extended to the creation of complementary set partitions, their reduction to equivalence classes, and the implementation of Edgeworth expansions and the exlog relations.     

Orbits and Test Elements in Free Leibniz Algebras of Rank Two

Let F be the free Leibniz algebra of rank two over a field K of characteristic zero freely generated by x ₁ and x ₂. In this article we show that an endomorphism of F which preserves the orbit of a nontrivial element of F is an automorphism. Using this result, we determine some test elements of F.     

Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus

This article investigates the irreducibility of certain representations for the Lie algebra of divergence zero vector fields on a torus. In [2 Eswara Rao, S. (1996). Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus. J. Algebra 182(2):401–421.[Crossref], [Web of Science ®] , [Google Scholar]] Rao constructs modules for the Lie algebra of polynomial vector fields on an N-dimensional torus, and determines the conditions for irreducibility. The current article considers the restriction of these modules to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie algebra of divergence zero vector fields.