Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Let be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of acting regularly on the points of . Then either t≡2±od s+1 or is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).     

On the graph of a function in two variables over a finite field

We show that if the number of directions not determined by a pointset of , of size q² is at least p^e q then every plane intersects in 0 modulo p^e+1 points and apply the result to ovoids of the generalised quadrangles and .     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

Base subsets of symplectic Grassmannians

Let V and V′ be 2n-dimensional vector spaces over fields F and F′. Let also Ω: V× V→ F and Ω′: V′× V′→ F′ be non-degenerate symplectic forms. Denote by Π and Π′ the associated (2n−1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π′ will be denoted by and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect and by so-called base subsets. We show that every mapping of to sending base subsets to base subsets is induced by a symplectic embedding of Π to Π′.     

On the VC-dimension of uniform hypergraphs

Let be a k-uniform hypergraph on [n] where k−1 is a power of some prime p and n≥ n ₀(k). Our main result says that if , then there exists E ₀∊ such that {E∩ E ₀: E∊ } contains all subsets of E ₀. This improves a longstanding bound of due to Frankl and Pach [7].Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship.Research supported in part by NSA grant H98230-05-1-0079. Part of this research was done while working at University of Illinois at Chicago.     

Quantum automorphism groups of vertex-transitive graphs of order ≤ 11

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups , symmetric groups S _n and quantum symmetric groups , by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B _n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e ₁ , ..., e _n denoting the standard basis vectors of ⁿ and letting x _k = e ₁ + ··· + e _k (k = 1, 2, ..., n), the set

The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, a nontrivial additive character, and a character on . Then ψ defines an Artin-Schreier sheaf on the affine line , and χ defines a Kummer sheaf on the n-dimensional torus . Let be a Laurent polynomial. It defines a k-morphism . In this paper, we calculate the weights of under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

Endomorphisms of Jacobians of modular curves

Let be the modular curve associated to a congruence subgroup Γ of level N with , and let be its canonical model over . The main aim of this paper is to show that the endomorphism algebra of its Jacobian is generated by the Hecke operators T _p , with , together with the “degeneracy operators” D _M,d , D ^t _M,d , for . This uses the fundamental results of Ribet on the structure of together with a basic result on the classification of the irreducible modules of the algebra generated by these operators. Received: 18 December 2007     

The combinatorial properties of the hyperplanes of DW(5, q) arising from embedding

In De Bruyn Discrete math(to appear), one of the authors proved that there are six isomorphism classes of hyperplanes in the dual polar space DW(5, q), q even, which arise from its Grassmann-embedding. In the present paper, we determine the combinatorial properties of these hyperplanes. Specifically, for each such hyperplane H we calculate the number of quads Q for which is a certain configuration of points in Q and the number of points for which is a certain configuration of points in . By purely combinatorial techniques, we are also able to show that the set of hyperplanes of DW(5, q), q odd, which arise from its Grassmann-embedding can be divided into six subclasses if one takes only into account the above-mentioned combinatorial properties. A complete classification of all hyperplanes of DW(5, q), q odd, which arise from its Grassmann-embedding, i.e. the division of the above-mentioned six classes into isomorphism classes, will unlike in De Bruyn (to appear) most likely need a group-theoretical approach. Postdoctoral Fellow of the Research Foundation—Flanders (Belgium).     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Degrees of difficulty of generalized r.e. separating classes

Important examples of classes of functions are the classes of sets (elements of ^ω 2) which separate a given pair of disjoint r.e. sets: . A wider class consists of the classes of functions f ∈ ^ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let denote the Medvedev degrees of those such that no m + 1 sets among A ₀,...,A _k-1 have a nonempty intersection. It is shown that each is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions is the greatest element of . If 2 ≤ l < k, then 0 _M is the only degree in which is below a member of . Each is densely ordered and has the splitting property and the same holds for the lattice it generates. The elements of are exactly the joins of elements of for . Supported by National Science Foundation grants DMS 0554841, 0532644 and 0652732.     

Normal subgroups of nonstandard symmetric and alternating groups

Let be a nonstandard model of Peano Arithmetic with domain M and let be nonstandard. We study the symmetric and alternating groups S _n and A _n of permutations of the set internal to , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A _n and S _n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an -valued metric on and (where B _S , B _A are the maximal normal subgroups of S _n and A _n identified earlier) making these groups into topological groups, and by showing that if is -saturated then and are complete with respect to this metric.      

Carleson Measures for the Bloch Space

In this paper we study the positive Borel measures μ on the unit disc in for which the Bloch space is continuously included in , 0 < p < ∞. We call such measures p-Bloch-Carleson measures. We give two conditions on a measure μ in terms of certain logarithmic integrals one of which is a necessary condition and the other a sufficient condition for μ being a p-Bloch-Carleson measure. We also give a complete characterization of the p-Bloch-Carleson measures within certain special classes of measures. It is also shown that, for p > 1, the p-Bloch-Carleson measures are exactly those for which the Toeplitz operator , defined by , maps continuously into the Bergman space A ¹, . Furthermore, we prove that if p > 1, α >-1 and ω is a weight which satisfies the Bekollé-Bonami -condition, then the measure defined by is a p-Bloch-Carleson-measure. We also consider the Banach space of those functions f which are analytic in and satisfy , as . The Bloch space is contained in . We describe the p-Carleson measures for and study weighted composition operators and a class of integration operators acting in this space. We determine which of these operators map continuously to the weighted Bergman space and show that they are automatically compact. This research is partially supported by several grants from “the Ministerio de Educación y Ciencia, Spain” (MTM2005-07347, MTM2007-60854, MTM2006-26627-E, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from “La Junta de Andalucía” (FQM210 and P06-FQM01504); from “the Academy of Finland” (210245) and from the European Networking Programme “HCAA” of the European Science Foundation.     

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S ³ and Λ(G) is the union of all complex projective lines in which are tangent to S ³ at a point in L(G).     

Periodic subgroups of projective linear groups in positive characteristic

We classify the maximal irreducible periodic subgroups of PGL(q, ), where is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and ^× has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, ) containing the centre ^×1 _q of GL(q, ), such that G/ ^×1 _q is a maximal periodic subgroup of PGL(q, ), and if H is another group of this kind then H is GL(q, )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, ) is self-normalising.