<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-convergence of Lagrange interpolation on the semiaxis

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L ^p -spaces.      

Phenomenologically symmetric local lie groups of transformations of the space <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.     

On the family of diophantine triples {<Emphasis Type="Italic">k</Emphasis> + 1, 4<Emphasis Type="Italic">k</Emphasis>, 9<Emphasis Type="Italic">k</Emphasis> + 3}

We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k ³ + 192k ² + 76k + 8.      

On edge-antipodal <Emphasis Type="Italic">d</Emphasis>-polytopes

A convex d-polytope in ℝ ^d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.      

Hyperbolic <Emphasis Type="Italic">n</Emphasis>-dimensional manifolds with automorphism group of dimension <Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>

We obtain a complete classification of complex Kobayashihyperbolic manifolds of dimension n ≥ 2, for which the dimension of the group of holomorphic automorphisms is equal to n². Received: May 2005 Accepted: November 2005     

The <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-saturation for linear combination of Bernstein-Kantorovich operators

We study the L _p -saturation for the linear combination of Bernstein-Kantorovich operators. As a result we obtain the saturation class by using K-functional as well as some modulus of smoothness. Research supported by National Natural Science Foundation of China (10671019) and Zhejiang Provincial Natural Science Foundation of China (102005).     

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Hypersurfaces of conformally and <Emphasis Type="Italic">h</Emphasis>-conformally related Finsler spaces

Certain geometrical properties of the hypersurfaces of two conformally and h-conformally related Finsler spaces have been discussed. The terminologies and notations are referred to Matsumoto [6]. Supported by UGC, Government of INDIA.     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.     

The mixed problem in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> for some two-dimensional Lipschitz domains

We consider the mixed problem,

<Emphasis Type="Italic">P</Emphasis>-adic sets of range uniqueness

We study sets of range uniqueness (SRU’s) in a complete, ultrametric, algebraically closed fieldK for analytic elements. We find monotonic distances sequences which appear to be SRU’s completely different from those known in ©. On the other hand, most of open closed sets cannot be SRU’s.     

Local volume estimate for manifolds with<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-bounded curvature

We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L² norm in dimension 4.     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Zeta functions of<Emphasis Type="Italic">q</Emphasis>-perfect numbers

We introduce a notion ofq-analogue of the perfect numbers. We also define a new zeta function which we call a zeta function ofq-perfect numbers. In this paper, the properties of theq-perfect numbers and the zeta functions are studied. Especially, we determine theq-perfect numbers whenq is a root of unity.     

Line partitions of internal points to a conic in <Emphasis Type="Italic">PG</Emphasis>(2,<Emphasis Type="Italic">q</Emphasis>)

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all non-tangent lines to C through an external point to C, the set of all non-tangent lines to C through a point in C, and, for square q, the set of all non-tangent lines to C belonging to a Baer subplane PG(2,√q) with √q+1 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmáros [9] characterizing the pencil of lines through an internal point to C as the unique set of lines, up to projectivities, which provides a partition of the set of all non-internal points to C. However, the proof is not analogous, since it does not rely on the famous Lemma of Tangents of Segre which was the main ingredient in [9]. The main tools in the present paper are certain partitions in conics of the set of all internal points to C, together with some recent combinatorial characterizations of blocking sets of non-secant lines, see [2], and of blocking sets of external lines, see [1].