Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

The maximal conjugate and Hilbert operators on real Hardy spaces

Summary We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H¹ to L¹, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T² and R².     

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra W_n⋉g⊗O _n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl _n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n⋉g⊗O _n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230.     

Compatible actions of Lie algebras

Abstract

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object L from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in XMod_L(Lie_R).     

Characterization of Linear Structures

We study the notionof linear structure of a function defined from F ^mto F ⁿ, and in particular of a Boolean function.We characterize the existence of linear structures by means ofthe Fourier transform of the function. For Boolean functions,this characterization can be stated in a simpler way. Finally,we give some constructions of resilient Boolean functions whichhave no linear structure.     

Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

We consider some subsets of the Euclidean plane R ², having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on R ².     

The Disjunction Property in the Class of Paraconsistent Extensions of Minimal Logic

We consider the disjunction property, DP, in the class of extensions of minimal logic L _j . Conditions are described under which DP is translated from the class PAR of properly paraconsistent extensions of the logics of class L _j into the class INT of intermediate extensions and the class NEG of negative extensions, and conditions for its being translated back into PAR. The logic L _F in PAR, which specifies conditions for DP to be translated from PAR into NEG, is defined and is characterized in terms of j-algebras and Kripke frames. Moreover, we show that L _F is decidable and possesses the disjunction property.     

Relaxed chromatic numbers of graphs

Given any family of graphsP, theP chromatic number _p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.Research supported in part by ONR Grant N00014-85K0570 and by a grant from the University of Illinois Research Board.     

On Optimal Feedback Control for Stationary Linear Systems

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems A X+B U=Z with output Y=C X+D U from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.     

Exact Laws for Sums of Order Statistics from a Generalized Pareto Distribution

Abstract

We select the kth order statistic from each row from a sequence of independent and identically distributed random variables from a distribution that generalizes the Pareto distribution. We then examine weighted sums of these order statistics to see whether or not Laws of Large Numbers with nonzero limits exist.     

Varieties of two-dimensional cylindric algebras.¶Part I: Diagonal-free case

We investigate the lattice of all subvarieties of the variety Df ₂ of two-dimensional diagonal-free cylindric algebras. We prove that a Df ₂-algebra is finitely representable if it is finitely approximable, characterize finite projective Df ₂-algebras, and show that there are no non-trivial injectives and absolute retracts in Df ₂. We prove that every proper subvariety of Df ₂ is locally finite, and hence Df ₂ is hereditarily finitely approximable. We describe all six critical varieties in , which leads to a characterization of finitely generated subvarieties of Df ₂. Finally, we describe all square representable and rectangularly representable subvarieties of Df ₂. Received May 25, 2000; accepted in final form November 2, 2001.     

Irrationality of linear combinations of eigenvectors

A givenn ×n matrix of rational numbers acts onC ^π and onQ ^π. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors forC ^π with the standard basis forQ ^π. Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.     

Spectra of degrees of some structures

We argue for the existence of structures with the spectrum {x : x ≰ a} of degrees, where a is an arbitrary low degree. Also it is stated that there exist structures with the spectrum of degrees, {x : x ≰ a} ⋃ {x : x ≰ b}, for any low degrees a and b. Supported by RFBR grant No. 05-01-00605. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 729–744, November–December, 2007.     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

On one class of modules over integer group rings of locally solvable groups

We study a Z G-module A in the case where the group G is locally solvable and satisfies the condition min–naz and its cocentralizer in A is not an Artinian Z-module. We prove that the group G is solvable under the conditions indicated above. The structure of the group G is studied in detail in the case where this group is not a Chernikov group. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 44–51, January, 2009.     

Projective functors and the restriction of a Verma module to a subalgebra of Levi type

We observe that the restriction of a Verma module over a semi-simple Lie algebra to a subalgebra of Levi type may be viewed as a projective functor. By simple arguments we prove that this restriction can be decomposed into a direct sum of standard indecomposables in the category O. For the restriction problem from sl(n+1) to gl(n) we describe the complete answer. We study the properties of the modules with Verma flag also and prove that any module with Verma flag is a submodule of some projective.     

Cohomology of the Grothendieck construction

We consider cohomology of small categories with coefficients in a natural system in the sense of Baues and Wirsching. For any functor L : K → CAT, we construct a spectral sequence abutting to the cohomology of the Grothendieck construction of L in terms of the cohomology of K and of L(k), for k ∈ ObK.The first author was supported by DFG at University of BielefeldThe second author is a researcher from CONICET, Argentina     

A Note on the Double Affine Hecke Algebra of Type GL2

We express the double affine Hecke algebra ? associated to the general linear group GL₂(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪_q((k^×)³). With an eye towards proving ring-theoretic results pertaining to ?, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ? to the amalgamated free product ring of quadratic extensions over 𝒪_q((k^×)³), a ring known to be noetherian. Therefore, it follows that ? is noetherian.     

On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.