Definability of closure operations in the <Emphasis Type="Italic">h</Emphasis>-quasiorder of labeled forests

We prove that natural closure operations on quotient structures of the h-quasiorder of finite and (at most) countable k-labeled forests (k ≥ 3) are definable provided that minimal nonsmallest elements are allowed as parameters. This strengthens our previous result which holds that each element of the h-quasiorder of finite k-labeled forests is definable in the first-order language, and each element of the h-quasiorder of (at most) countable k-labeled forests is definable in the language L _ω1ω; in both cases k ≥ 3 and minimal nonsmallest elements are allowed as parameters. Similar results hold true for two other relevant structures: the h-quasiorder of finite (resp. countable) k-labeled trees and k-labeled trees with a fixed label on the root element.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

The spectrum of the Dirac operator on SU<Subscript>2</Subscript>/Q<Subscript>8</Subscript>

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU₂/Q₈, where Q₈ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär’s upper bound for the smallest Dirac eigenvalue in the particular case where SU₂/Q₈ is a homogeneous minimal hypersurface of S ⁴.     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Definable Ramsey and definable Erdös ordinals

This paper studies the relation between definable Ramsey ordinals and constructible sets which have a certain set of indiscernibles. It is shown that an ordinal κ is Σ₁-Ramsey if and only if κ is ∑_ω-Ramsey. Similar results are obtained for definable Erdös ordinals.     

Definability and nondefinability results for certain o‐minimal structures

The main goal of this note is to study for certain o‐minimal structures the following propriety: for each definable C^∞ function g₀: [0, 1] → ? there is a definable C^∞ function g: [–ε, 1] → ?, for some ε > 0, such that g (x) = g₀(x) for all x ∈ [0, 1] (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Restrictions of Modular Representations of Special Linear Groups to A<Subscript>1</Subscript>×A<Subscript>1</Subscript> Subgroups

We study the restrictions of infinitesimally irreducible modular representations of algebraic groups of type A_n to subsystem subgroups of type A₁ × A₁. Under some restrictions on the highest weight of a representation we find their composition factors.     

The universal covering homomorphism in o‐minimal expansions of groups

Suppose G is a definably connected, definable group in an o‐minimal expansion of an ordered group. We show that the o‐minimal universal covering homomorphism : → G is a locally definable covering homomorphism and π₁(G) is isomorphic to the o‐minimal fundamental group π (G) of G defined using locally definable covering homomorphisms. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SK<Subscript>1</Subscript> of graded division algebras

The reduced Whitehead group SK₁ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK₁ of a tame valued division algebra over a henselian field coincides with SK₁ of its associated graded division algebra. Furthermore, it is shown that SK₁ of a graded division algebra is isomorphic to SK₁ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK₁ for generic abelian crossed products.     

Semi-self-decomposable distributions on Z<Subscript>+</Subscript>

We present a notion of semi-self-decomposability for distributions with support in Z ₊. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z ₊-valued random variables. Multiple semi-self-decomposability on Z ₊ is also discussed.     

Mutual definability does not imply definitional equivalence,a simple example

We give two theories, Th₁ and Th₂, which are explicitly definable over each other (i.e. the relation symbols of one theory are explicitly definable in the other, and vice versa), but are not definitionally equivalent. The languages of the two theories are disjoint. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homology of GL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>: injectivity conjecture for GL<Subscript>4</Subscript>

The homology of GL _n (R) and SL _n (R) is studied, where R is a commutative ‘ring with many units’. Our main theorem states that the natural map H ₄(GL₃(R), k) → H ₄(GL₄(R), k) is injective, where k is a field with char(k) ≠ 2, 3. For an algebraically closed field F, we prove a better result, namely, is injective. We will prove a similar result replacing GL by SL. This is used to investigate the indecomposable part of the K-group K ₄(R).     

Non-low<Subscript>2</Subscript>-ness and computable Lipschitz reducibility

In this paper, we prove that if a c.e. Turing degree d is non-low₂, then there are two left-c.e. reals β ₀, β ₁ in d, such that, if β ₀ is wtt-reducible to a left-c.e. real α, then β ₁ is not computable Lipschitz (cl-) reducible to α. As a corollary, d contains a left-c.e. real which is not cl-reducible to any complex (wtt-complete) left-c.e. real.     

A Differential Calculus on the Z<Subscript>3</Subscript>-graded Quantum Group GL<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

In this work, we introduce Z₃-graded quantum group GL\({_q(2,{\mathbb C})}\) with the help of Z₃-graded quantum plane and a Z₃-graded bicovariant differential calculus on the Z₃-graded quantum group GL _q (2). The corresponding Z₃-graded quantum Lie superalgebra is obtained.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

Coherent Functors and Covariantly Finite Subcategories

Given a locally presentable additive category A, we study a class of covariantly finite subcategories which we call definable. A definable subcategory arises from a set of coherent functors F _i on A by taking all objects X in A such that F _i X=0 for all i. We give various characterizations of definable subcategories, demonstrating that all covariantly finite subcategories which arise in practice are of this form. This is based on a filtration of the category of all coherent functors on A.     

The F<Subscript>4</Subscript>-algorithm for Euclidean rings

In this short note, we extend Faugére’s F₄-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F₄-algorithm is based on the simultaneous reduction of several polynomials.     

P<Subscript>4k-1</Subscript>-factorization of complete bipartite graphs

Let K_m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P_v-factorization of K_m,n is a set of edge-disjoint P_v-factors of K_m,n which partition the set of edges of K_m,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for the existence of P_v-factorization of K_m,n. When v is an odd number, Ushio in 1993 proposed a conjecture. However, up to now we only know that Ushio Conjecture is true for v = 3. In this paper we will show that Ushio Conjecture is true when v = 4k - 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P_4k-1-factorization of K_m,n is (1) (2k - 1)m ⩽ 2kn, (2) (2k - 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k - 1), (4) (4k - 1)mn/[2(2k - 1)(m + n)] is an integer.     

Towards the Hurwitz Generation of G<Subscript>sc</Subscript>(E<Subscript>6</Subscript>, <Emphasis Type="Italic">q</Emphasis>)

The paper is devoted to the problem concerning the Hurwitz generation of the group G_sc(E₆, q). All possibilities for Hurwitz generators, except for just one, are excluded. Bibliography: 25 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 228–237.