New examples of <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-saturated Banach spaces

For every p ∈ (1, ∞), an isomorphically polyhedral Banach space E _p is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact metric space K. Moreover, E _p admits a quotient isomorphic to ℓ _p .     

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.     

On eigenvalues of <Emphasis Type="Italic">p</Emphasis>-adic curvature

We study the eigenvalues of the p-adic curvature transformationson buildings. In particular, we determine the maximal eigenvalues ofthese transformations.     

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     

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].     

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.     

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,

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

The limit of <Emphasis Type="Italic">W</Emphasis><Superscript>1,1</Superscript> homeomorphisms with finite distortion

We show that the limit f of a weakly convergent sequence of W ^1,1 homeomorphisms f _j with finite distortion has finite distortion as well, provided that it is a homeomorphism. Moreover, the lower semicontinuity of the distortions is deduced both in case of outer and inner distortion.     

On <Emphasis Type="Italic">pq</Emphasis>-hyperelliptic Klein surfaces

In this paper we study compact Klein surfaces of algebraic genus d > 1 admitting p- and q-hyperelliptic involutions by which we mean involutions with the orbit spaces having algebraic genera p and q. We give necessary and sufficient conditions for p, q and d to exist such surfaces. It turns out that these conditions are also sufficient for the existence of such surfaces with commuting involutions what allow us to study this class also. We study the spectrum of hyperellipticity degrees of the product of these involutions and topological type of these surfaces. G. Gromadzki was supported by the grant SAB 2005-0049 of the Spanish Ministry of Education and Sciences. E. Tyszkowska was supported by BW 5100-5-0198-6.     

On the unipotent characters of the Ree groups of type <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

This note is concerned with the unipotent characters of the Ree groups of type G ₂. We determine the roots of unity associated by Lusztig and Digne-Michel to each unipotent character for and prove that the Fourier matrix of defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G ₂.     

Note edge-colourings of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">n,n</Emphasis></Subscript> with no long two-coloured cycles

Consider the set of all proper edge-colourings of a graph G with n colours. Among all such colourings, the minimum length of a longest two-coloured cycle is denoted L(n, G). The problem of understanding L(n, G) was posed by Häggkvist in 1978 and, specifically, L(n, K _n,n ) has received recent attention. Here we construct, for each prime power q ≥ 8, an edge-colouring of K _n,n with n colours having all two-coloured cycles of length ≤ 2q ², for integers n in a set of density 1 ? 3/(q ? 1). One consequence is that L(n, K _n,n ) is bounded above by a polylogarithmic function of n, whereas the best known general upper bound was previously 2n ? 4.     

Multilinear Operator Ideals Associated to Saphar Type (<Emphasis Type="Italic">n</Emphasis> + 1)-tensor Norms

We study an (n + 1)-tensor norm generalizing Saphar’s classic norm to (n+1)-fold tensor products.We characterize the maps in the minimal and maximal multilinear operator ideals related to in the sense of Defant and Floret. Partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

<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.     

Note making a <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free graph bipartite

We show that every K ₄-free graph G with n vertices can be made bipartite by deleting at most n ²/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

A free-boundary problem for the evolution <Emphasis Type="Italic">p</Emphasis>-Laplacian equation with a combustion boundary condition

We study the existence, uniqueness and regularity of solutions of the equation f _t  = Δ _p f = div (|Df| ^p-2 Df) under over-determined boundary conditions f = 0 and |Df| = 1. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until vanishing identically. Furthermore, the free-boundary of the support of f is smooth for all positive time.     

Square-free orders for CM elliptic curves modulo <Emphasis Type="Italic">p</Emphasis>

Let E be an elliptic curve defined over , of conductor N, and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes , p ≤ x, for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of E modulo p is cyclic, first studied by Serre (1977). They are also related to the stronger problem about primitive points on E modulo p, formulated by Lang and Trotter (Bull Am Math Soc 83:289–292, 1977), and the one about the primality of the order of E modulo p, formulated by Koblitz [Pacific J. Math. 131(1):157–165, 1988].     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Invariants of finite aspherical CW-complexes

Let X be a finite aspherical CW-complex whose fundamental group π ₁(X) possesses a subnormal series with a non-trivial elementary amenable group G ₀. We investigate the L ²-invariants of the universal covering of such a CW-complex X. The main result is the proof of the vanishing of the L ²-torsion under the condition that π ₁(X) has semi-integral determinant. We further show that the Novikov–Shubin invariants are positive.     

Whittaker functions for generalized principal series representations of <Emphasis Type="Italic">SL</Emphasis>(3, R)

We study Whittaker functions for generalized principal series representations of the real special linear group SL(3, R) of degree 3. From the Capelli elements and Dirac-Schmid operators, we give the system of partial differential equations which is satisfied by Whittaker functions. We give six formal power series solutions of this system, which are called secondary Whittaker functions. We also give the Mellin-Barnes type integral expressions of primary Whittaker functions, i.e. the solutions having the moderate growth property.