On Homogeneous Exact Categories

In the present paper we prove that a certain subcategory of the module category over some infinite-dimensional algebra R has almost split sequences and strongly homogeneous property; i.e., for each indecomposable module M in , there is an almost split sequence starting and also ending at M. It is also proved that except for a trivial case, is of wild representation type.     

On Identity Bases of Epigroup Varieties

Let C _n and C _n be the varieties of all completely regular and of all completely simple semigroups, respectively, whose idempotent generated subsemigroups are periodic with period n. We use Ol'shanski 's theory of geometric group presentations to show that for large odd n these varieties (and similarly defined varieties of epigroups) do not have finitely axiomatizable equational theories.     

Identities on Units of Algebraic Algebras

Let be an algebraic algebra over an infinite field K and let ( ) be its group of units. We prove a stronger version of Hartley's conjecture for , namely, if a Laurent polynomial identity (LPI, for short) f = 0 is satisfied in ( ), then satisfies a polynomial identity (PI). We also show that if is non-commutative, then is a PI-ring, provided f = 0 is satisfied by the non-central units of . In particular, is locally finite and, thus, the Kurosh problem has a positive answer for K-algebras whose unit group is LPI. Moreover, f = 0 holds in ( ) if and only if the same identity is satisfied in . The last fact remains true for generalized Laurent polynomial identities, provided that is locally finite.     

An Infinite Analogue of Rings with Stable Rank One

Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings) and include examples like End (V), the ring of endomorphisms of a vector space V over some field , and ( ), the ring of all row- and column-finite matrices over . We show that the category of QB-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the C*-algebras that are QB-rings are exactly the extremally rich C*-algebras studied by L. G. Brown and the second author.     

Generalized Green Classes

Generalized Green classes are introduced; some basic properties of members in a generalized Green class are studied. Finally, we apply the results to (Λ), the Ringel–Hall algebra of a finite-dimensional hereditary algebra Λ over a finite field. In particular, it is proved that (Λ) belongs to a suitable generalized Green class, and that there is direct decomposition of spaces (Λ) =  (Λ)  J, where (Λ) is the composition algebra of Λ and J is a twisted Hopf ideal of (Λ), which is exactly the orthogonal complement of (Λ).     

-Modules on Smooth Toric Varieties

Let X be a smooth toric variety. Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal . Let A denote the ring of differential operators on Spec(S). We show that the category of -modules on X is equivalent to a subcategory of graded A-modules modulo -torsion. Additionally, we prove that the characteristic variety of a -module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic -modules correspond to holonomic A-modules.     

Universal Functions on Complex General Linear Groups

In 1929, Birkhoff proved the existence of an entire function F on with the property that for any entire function f there exists a sequence {a_k} of complex numbers such that {F(ζ+a_k)} converges to f (ζ) uniformly on compact sets. Luh proved a variant of Birkhoff's theorem and the second author proved a theorem analogous to that of Luh for the multiplicative group *. In this paper extensions of the above results to the multi-dimensional case are proved. Let M(n,  ) be the set of all square matrices of degree n with complex coefficients, and let G=GL(n,  ) be the general linear group of degree n over . We denote by (G) the set of all holomorphic functions on G. Similarly, we define ( ). Let K be the (G)-hull of a compact set K in G. Finally we denote by B(G) the set of all compact subsets K of G with K=K such that there exists a holomorphic function f on M(n,  ) with f(0)(f(K)), where (f(K)) is the ( )-hull of f(K). Our main result is the following. There exists a holomorphic function F on G such that for any KB(G), for any function f holomorphic in some neighbourhood of K, and for any >0, there exists CG with max_ZK |F(CZ)−f(Z)|<.     

Young measures generated by a class of integrands: a narrow epicontinuity and applications to Homogenization

Given a subset E of convex functions from into which satisfy growth conditions of order p>1 and an open bounded subset of , we establish the continuity of a map μΦ_μ from the set of all Young measures on equipped with the narrow topology into a set of suitable functionals defined in and equipped with the topology of Γ-convergence. Some applications are given in the setting of periodic and stochastic homogenization.     

Commuting Varieties of Lie Algebras over Fields of Prime Characteristic

Let K be an algebraically closed field of positive characteristic and let G be a reductive group over K with Lie algebra . This paper will show that under certain mild assumptions on G, the commuting variety ( ) is an irreducible algebraic variety.     

Piecewise Smooth Spaces in Duality: Application to Blossoming

Given an (n+1)-dimensional space of piecewise smooth functions in which each basis has a non-vanishing Wronskian, and its dual space *, a canonical bilinear form is defined on × *, which provides a simple characterization of a contact of order rn. An intrinsic reproducing function is introduced, leading to Marsden-type identities. In the case of Chebyshev spaces connected with totally positive matrices, the bilinear form yields a general notion of blossom which can be extended to Chebyshev splines.     

An upper estimate in Turán's pure power sum problem

Let z₁, z₂, …, z_n be complex numbers, and write for their power sums. Let where the minimum is taken under the condition that . In this paper we prove that .     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

Abelian fourfold of Mumford-type and Kuga-Satake varieties

Let X be a complex abelian fourfold of Mumford-type and let V = H¹(X, ). The complex Mumford-Tate group of X is isogenous to SL(2)³. We recover information about the Hodge structure of X using representations of the Lie algebras (2)³ and (8) acting on V . Using these techniques we show that there is a Kuga-Satake variety A associated to X in such a way that A is isogenous to X³².     

Markov inequality for polynomials of degree n with m distinct zeros

Let be the collection of all polynomials of degree at most n with real coefficients that have at most m distinct complex zeros. We prove thatfor every . This is far away from what we expect. We conjecture that the Markov factor 32·8^mn above may be replaced by cmn with an absolute constant c>0. We are not able to prove this conjecture at the moment. However, we think that our result above gives the best-known Markov-type inequality for on a finite interval when mc log n.     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

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in C[−1, 1], where the nonreal poles in {a_k}^∞_k=1 \[−1, 1] are paired by complex conjugation. This extends an Achiezer's result.     

Topological semi-abelian algebras

Given an algebraic theory whose category of models is semi-abelian, we study the category of topological models of and generalize to it most classical results on topological groups. In particular, is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite -algebras.     

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|_n|²dσ}_n0, denoted by Lim(σ). Here {_n}_n0 are orthogonal polynomials in L²(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zⁿ=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zⁿ, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J₀ (including J₀=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).     

A Class of Polynomials over Finite Fields

Generalizing the norm and trace mappings for _q^r/ _q, we introduce an interesting class of polynomials over finite fields and study their properties. These polynomials are then used to construct curves over finite fields with many rational points.     

Invariants of 2×2-Matrices over Finite Fields

Let _q be the finite field with q elements, q=p^ν, p a prime, and Mat_2.2( _q) the vector space of 2×2-matrices over . The group GL(2, ) acts on Mat_2,2( _q) by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields.     

Nilpotent Lie Algebras of Maximal Rank and of Kac–Moody Type: E6

Let be the Kac–Moody algebra associated to the affine Cartan matrix E₆⁽¹⁾. Each nilpotent Lie algebra of type E₆⁽¹⁾ is isomorphic to a quotient of the positive part of . We determine the isomorphism classes of nilpotent Lie algebras of type E₆⁽¹⁾.