Closure Łukasiewicz algebras

In this paper, the variety of closure n-valued Łukasiewicz algebras, that is, Łukasiewicz algebras of order n endowed with a closure operator, is investigated. The lattice of subvarieties in the particular case in which the open elements form a three-valued Heyting algebra is obtained.     

Noncommutative graded domains with quadratic growth

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z^–1; ], for some automorphism of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when ||= and thatR is a finite module over its centre when ||<. IfR is not generated in degree one, thenR will still be Noetherian and primitive if has infinite order, butR need not be Noetherian when has finite order.Dedicated to the memory of Shimshon AmitsurOblatum 5-XI-1994 & 28-III-1995This research was supported in part by NSF grants     

Sets with two associative operations

In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.     

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which is co-finite in ind A, and derive some consequences. Dedicated to Stanislaw Balcerzyk on the occation of his 70th birthday     

A pathwise solution for nonlinear parabolic equations with stochastic perturbations

We analyse here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient ∂ _xu with respect to the state variable, ∈ ℝⁿ. A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite dimensional Lie algebra generated by the given diffusion vector fields.     

Tensor products of symmetric functions over ℤ<Subscript>2</Subscript>

We calculate the homology and the cycles in tensor products of algebras of symmetric function over ℤ₂     

Left-sided quasi-invertible bimodules over Nakayama algebras

Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence.     

Chebyshev Centers in Hyperplanes of c 0

We give a full characterization of the closed one-codimensional subspaces of c ₀, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented.     

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.     

An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

The famous 1960's construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.

     

Singular quasilinear equations with quadratic growth in the gradient without sign condition

Given a bounded domain Ω in RN, and a function a∈Lq(Ω) with q>N/2, we study the existence of a positive solution for the quasilinear problem

     

Backward stochastic partial differential equations with quadratic growth

This paper is concerned with the weak solution (in analytic sense) to the Cauchy–Dirichlet problem of a backward stochastic partial differential equation when the nonhomogeneous term has a quadratic growth in both the gradient of the first unknown and the second unknown. Existence and uniqueness results are obtained under separate conditions.     

The asymptotic behavior of quadratic forms in -mixing random variables

Let {Z_i,i≥1} be a linear process defined by with {d_j,j≥0} being a regular varying sequence of real numbers and {ξ_t,−∞<t<∞} being a sequence of -mixing random variables. The present paper studies the asymptotic behavior of the quadratic form under some mild assumptions on d_j and ξ_t. Meanwhile, the similar results of α-mixing random variables are presented.     

On some sequence of graded Lie algebras associated to manifolds

A simple construction associates to any linear mapping a short exact sequence of graded Lie algebras. The sequence associated to the de Rham differential of an arbitrary smooth manifold is never split. Combined with a sort of algebraic Chern-Weil homomorphism adapted from [1] to the graded case, this leads to a family of cohomology classes of the Nijenhuis-Richardson algebra of the space of functions of the manifold. Some of these characteristic classes of degree 2 are computed. They are the classes constructed by hand in [2] and used in the theory of star-products.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Geometric classes of Goursat flags and the arithmetics of their encoding by small growth vectors

Goursat distributions are subbundles, of codimension at least 2, in the tangent bundles to manifolds having the flag of consecutive Lie squares of ranks not depending on a point and growing—very slowly—always by 1. The length of a flag thus equals the corank of the underlying distribution.After the works of, among others, Bryant&;Hsu (1993), Jean (1996), and Montgomery&;Zhitomirskii (2001), the local behaviours of Goursat flags of any fixed length r≥2 are stratified into geometric classes encoded by words of length r over the alphabet {G,S,T} (Generic, Singular, Tangent) starting with two letters G and having letter(s) T only directly after an S, or directly after another T.It follows from [6] that the Goursat germs sitting in any fixed geometric class have, up to translations by rk D?2, one and the same small growth vector (at the reference point) that can be computed recursively in terms of the G,S,T code. In the present paper we give explicit solutions to the recursive equations of Jean and show how, thanks to a surprisingly neat underlying arithmetics, one can algorithmically read back the relevant geometric class from a given small growth vector. This gives a secondary, Gödel-like super-encoding of the geometric classes of Goursat objects (rather than just a 1-1 correspondence between those classes and small growth vectors).     

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

The Hurwitz determinants and the signatures of irreducible representations of simple real Lie algebras

The paper deals with the real classical Lie algebras and their finite dimensional irreducible representations. Signature formulae for Hermitian forms invariant relative to these representations are considered. It is possible to associate with the irreducible representation a Hurwitz matrix of special kind. So the calculation of the signatures is reduced to the calculation of Hurwitz determinants. Hence it is possible to use the Routh algorithm for the calculation.