Small weight codewords in the dual code of points and hyperplanes in PG(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

The main theorem of this article gives a classification of the codewords in \({C^{\bot}_{n-1}(n,q)}\) , the dual code of points and hyperplanes in PG(n, q), q even, with weight smaller than \({q+\sqrt[3]{q^{2}}+1}\). In the proof, we rely on the classification of the small blocking sets in PG(2, q), q even.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

A Characterization of PSL(3, <Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis>=2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>

The order components of a finite group are introduced in [12]. In [9], it is proved that the group PSL(3,q), where q is an odd prime power, is uniquely determined by its order components. In this paper, we show that the group PSL(3, q), where q=2 ^m , is also uniquely determined by its order components. Received December 15, 2000, Revised August 15, 2001, Accepted November 13, 2001     

Lifting endo-<Emphasis Type="Italic">p</Emphasis>-permutation modules

We prove that all endo-p-permutation modules for a finite group are liftable from characteristic \(p>0\) to characteristic 0.     

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

A new characterization of <Emphasis Type="Italic">PSU</Emphasis>(<Emphasis Type="Italic">p,q</Emphasis>)

Summary Let G be a finite group. Order components of G were introduced in Chen [5]. Let OC(G) be the set of order components of G. Some finite groups are characterizable by their order components. This assertion was proved for the simple groups PSU(p,q), where p=3, 5, 7 and 11. In this paper, we prove that the simple groups PSU(p,q) can be uniquely determined by their order components, where p&ge;13 is a prime number. Main consequences of our results are the validity of a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Structure of a code related to Sp(4, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

We determine the socle and the radical series of the binary code associated with a finite regular generalized quadrangle of even order, considered as a module for the commutator of each of the orthogonal subgroups in the corresponding symplectic group.     

Lower semicontinuity and relaxation for integral functionals with <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)- and <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x,u</Emphasis>)-growth

We consider the questions of lower semicontinuity and relaxation for the integral functionals satisfying the p(x)- and p(x, u)-growth conditions. Presently these functionals are actively studied in the theory of elliptic and parabolic problems and in the framework of the calculus of variations. The theory we present rests on the following results: the remarkable result of Kristensen on the characterization of homogeneous p-gradient Young measures by their summability; the earlier result of Zhang on approximating gradient Young measures with compact support; the result of Zhikov on the density in energy of regular functions for integrands with p(x)-growth; on the author’s approach to Young measures as measurable functions with values in a metric space whose metric has integral representation.     

Spectra of the finite simple groups <Emphasis Type="Italic">E</Emphasis> <Subscript>7</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We describe the spectra of simple and universal groups of Lie type E₇.     

Properties of element orders in covers for L<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>) and U<Subscript>n</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

We show that if a finite simple group G, isomorphic to PSL_n(q) or PSU_n(q) where either n ≠ 4 or q is prime or even, acts on a vector space over a field of the defining characteristic of G; then the corresponding semidirect product contains an element whose order is distinct from every element order of G. We infer that the group PSL_n(q), n ≠ 4 or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka Notebook.     

Properties of <Emphasis Type="Italic">P</Emphasis>-coherent and Baer modules

M is called a P-coherent (resp. PP) module if its every principal submodule is finitely presented (resp. projective). M is said to be a Baer module if the annihilator of its every subset is a direct summand of R. In this paper, we investigate the properties of P-coherent, PP and Baer modules. Some known results are extended.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

Charakterisierung der <Emphasis Type="Italic">q</Emphasis>-Orthogonalpolynome in <Emphasis Type="Italic">x</Emphasis>

Characterization of q-Orthogonal Polynomials. Im Anschluß an die Arbeit Orthogonalpolynome in x und q^–x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung (Monatsh. Math. 132, 123–140 (2001); im folgenden als [4] zitiert) werden alle Möglichkeiten für q-Orthogonalpolynome in x als Lösungen von q-Operatorgleichungen zweiter Ordnung angegeben (Orthogonalität im positiv definiten Sinne). Dabei erfolgt die Numerierung der Abschnitte und die Angabe der Formel-nummern unter Einbeziehung von [4].     

SOME RESULTS ABOUT BEST COAPPROXIMATION IN L^P（S,X）

As a counterpart to best approximation in normed linear spaces, the best coapproximation was introduced by Franchetti and Furi. In this paper, we shall consider the relation between coproximinality M in X and L ^p (S,M) in L ^p (S,X). Finally we give some results in cochebyshev subspaces and additional subspaces.     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

On constant <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript>2</Subscript>)-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We consider the spectral resolution of a Uq (sl ₂)-invariant solution R of the constant Yang–Baxter equation in the braid group form. It is shown that if the two highest coefficients in this resolution are not equal, then R is either the Drinfeld R-matrix or its inverse. Bibliography: 13 titles.