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.     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

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">k</Emphasis>-noncrossing and <Emphasis Type="Italic">k</Emphasis>-nonnesting graphs and fillings of ferrers diagrams

We give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of Chen, Deng, Du, Stanley, and Yan, and extends results of Klazar to k > 2. Moreover, this correspondence reinforces the links recently discovered by Krattenthaler between fillings of diagrams and the results of Chen et al.     

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.     

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.     

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.     

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 = ∞.     

On the stability of <Emphasis Type="Italic">t</Emphasis>-convex functions

Summary.  A real-valued function f defined on an open convex set is called (d, t)-convex if it satisfies

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     

A unified theory for modifications of<Emphasis Type="Italic">g</Emphasis>-closed sets

We introduce a new set calledmg-closed which is defined on a family of sets satisfying some minimal conditions. This set enables us to unify certain kind of modifications of generalized closed sets due to Levine [17].     

<Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-free subgraphs of random graphs revisited

In Combinatorica 17(2), 1997, Kohayakawa, ?uczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G _{n, p} . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K ₄, thereby providing a conceptually simple proof for the main result in the paper cited above.     

<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.     

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.     

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

<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.     

<Emphasis Type="Italic">L</Emphasis>-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _n+1, s) be the L-function of the k-fold symmetric product of Kl _n+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L _p (G _{m, F p} , Sym ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).     

Rough blowup solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> critical NLS

We study the singularity formation for the cubic focusing L ²-critical nonlinear Schrödinger equation on \({\mathbb{R}^{2}}\) . In a series of recent works, Merle and Raphaël have completely described the so called log–log blowup regime and proven its stability in the energy space H ¹. Our aim in this paper is to investigate the stability of this blowup regime under rough perturbations in the direction of developing a theory at the level of the critical space L ². By blending the Merle, Raphaël techniques with the quantitative I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao for the study of the Cauchy problem for rough data, we obtain the stability of the log–log regime in H ^s for all s > 0.