A simple characterization of convex functions as indefinite integrals of non-decreasing ones is obtained, using only Riemann integrals.

We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ_∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. Moreover, these spaces of finite matrices are naturally imbedded in Σ_∞.

In the present paper we study Riemannian and canonical geodesics in a real flag manifold M, considered as curves in the ambient Euclidean space of the natural embedding of M.

Nelson algebras arise naturally in algebraic logic as the algebraic models of Nelson's constructive logic with strong negation. This note gives two characterisations of the variety of Nelson algebras up to term equivalence, together with a characterisation of the finite Nelson algebras up to polynomial equivalence. The results answer a question of Blok and Pigozzi and clarify some earlier work of Brignole and Monteiro.

In this paper, we study a qualitative uncertainty principle for completely solvable Lie groups.

In the present paper, we study some direct results in simultaneous approximation for linear combinations of Lupaş-Beta type operators.

We prove a theorem of existence, uniqueness and positivity of the solution for the Boltzmann equation with force term and initial data near the vacuum.

A natural inference mechanism is presented: the Black Box problem is transformed into a Dirichlet problem on the closed cube. Then it is solved in closed polynomial form, together with a Mean-Value Theorem and a Maximum Principle. An algorithm for calculating the solution is suggested. A special feedforward neural net is deducted for each polynomial.

The purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finitely dimensional representations of finitely generated, and especially, finitely dimensional algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether.

Let R be a ring (associative, with 1). A non-zero module M is said to be a Prüfer module provided there exists a surjective, locally nilpotent endomorphism with kernel of finite length. The aim of this note is to construct Prüfer modules starting from a pair of module homomorphisms w,v : U₀ → U₁, where w is injective and its cokernel is of finite length. For R=Z the ring of integers, one can construct in this way the ordinary Prüfer groups considered in abelian group theory. Our interest lies in the case that R is an artin algebra.

Las álgebras de Koszul fueron inventadas por Priddy [P] y han tenido un enorme desarrollo durante los últimos diez años, el artículo de Beilinson, Ginsburg y Soergel [BGS] ha sido muy influyente. En estas notas veremos los teoremas básicos de Álgebras de Koszul usando métodos de teoría de anillos y módulos, como se hizo en los artículos [GM1],[GM2], después nos concentraremos en el estudio de las álgebras Koszul autoinyectivas, primero las de radical cubo cero y posteriormente el caso general y por último aplicaremos los resultados obtenidos al estudio de las gavillas coherentes sobre el espacio proyectivo.

In this notes we start with the basic definitions of derived categories, derived functors, tilting complexes and stable equivalences of Morita type. Our aim is to show via several examples that this is the best framework to do homological algebra, We also exhibit their usefulness for getting new proofs of well known results. Finally we consider the Morita invariance of Hochschild cohomology and other derived functors.

The structure of cluster categories [BMRRT] is well suited for the combinatorics of cluster algebras [FZ1] with the main correspondence being between tilting objects and clusters. Furthermore it was shown in [IOTW] that there is a close relation between domains of virtual semi-invariants and simplicial complexes associated to cluster categories. Also the same simplicial complexes associated to cluster categories are related to the Igusa-Orr pictures in the homology of nilpotent groups.

This is a joint work with Victor Ginzburg [4] in which we study a class of associative algebras associated to finite groups acting on a vector space. These algebras are non-homogeneous N-Koszul algebra generalizations of symplectic reflection algebras. We realize the extension of the N-Koszul property to non-homogeneous algebras through a Poincaré-Birkhoff-Witt property.

We classify finite-dimensional complex pointed Hopf algebras with group of group-like elements isomorphic to A₅. We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of π ∈ A_n is infinite-dimensional if the order of π is odd except for π = (1 2 3) in A₄. We also study pointed Hopf algebras over the dihedral groups.

In our work [9], we complete Jans' classification of TTF-triples [8] by giving a precise description of those two-sided ideals of a ring associated to one-sided split TTF-triples in the corresponding module category.

In this work we extend, to the path algebras context, some results obtained in the commutative context, [2]. The main result is that one can extend the Gröbner bases of an ungraded ideal to one possible definition of homogenization for the non commutative case.