Moduli spaces of quadratic complexes and their singular surfaces

We construct the coarse moduli space of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.      

Ranks of abelian varieties over infinite extensions of the rationals

Let S be an infinite set of rational primes and, for some p ∈ S, let be the compositum of all extensions unramified outside S of the form , for . If , let be the intersection of the fixed fields by , for i = 1, . . , n. We provide a wide family of elliptic curves such that the rank of is infinite for all n ≥ 0 and all , subject to the parity conjecture. Similarly, let be a polarized abelian variety, let K be a quadratic number field fixed by , let S be an infinite set of primes of and let be the maximal abelian p-elementary extension of K unramified outside primes of K lying over S and dihedral over . We show that, under certain hypotheses, the -corank of sel _p ∞(A/F) is unbounded over finite extensions F/K contained in . As a consequence, we prove a strengthened version of a conjecture of M. Larsen in a large number of cases.     

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

The Cauchy-Dirichlet problem for the heat equation in Besov spaces

The solvability in anisotropic spaces , σ ∈ ℝ₊, p, q ∈ (1, ∞), of the heat equation u_t − Δu = f in Ω^T ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on S^T, u|_t=0 = u₀ in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝⁿ. The existence of a unique solution of the above problem is proved under the assumptions that and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence, the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation. Bibliography: 26 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 40–97.     

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy

A conic bundle degenerating on the Kummer surface

Let C be a genus 2 curve and the moduli space of semi-stable rank 2 vector bundles on C with trivial determinant. In Bolognesi (Adv Geom 7(1):113–144, 2007) we described the parameter space of non stable extension classes of the canonical sheaf ω of C by ω⁻¹. In this paper, we study the classifying rational map that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up along a certain cubic surface S and at the point p corresponding to the bundle , then the induced morphism defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in . Furthermore we construct the -bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

On the Laplacian in the halfspace with a periodic boundary condition

We study spectral and scattering properties of the Laplacian H ^(σ)=-Δ in corresponding to the boundary condition with a periodic function σ. For non-negative σ we prove that H ^(σ) is unitarily equivalent to the Neumann Laplacian H ⁽⁰⁾. In general, there appear additional channels of scattering due to surface states. We prove absolute continuity of the spectrum of H ^(σ) under mild assumptions on σ.     

On the Homological Dimension of Lattices

Let L be a finite lattice and let . It is shown that if the order complex satisfies then |L| ≥ 2 ^k . Equality |L| = 2 ^k holds iff L is isomorphic to the Boolean lattice {0,1} ^k . Research supported by the Israel Science Foundation.     

A two-to-one map and abelian affine difference sets

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. Set = H \ {1, ω}, where H = G/N and . Using D we define a two-to-one map g from to N. The map g satisfies g(σ ^m ) = g(σ) ^m and g(σ) = g(σ ⁻¹) for any multiplier m of D and any element σ ∈ . As applications, we present some results which give a restriction on the possible order n and the group theoretic structure of G/N.      

A nonstandard characterization of sequential equilibrium,perfect equilibrium,and proper equilibrium

New characterizations of sequential equilibrium, perfect equilibrium, and proper equilibrium are provided that use nonstandard probability. It is shown that there exists a belief system μ such that is a sequential equilibrium in an extensive game with perfect recall iff there exist an infinitesimal and a completely mixed behavioral strategy profile σ′ (so that assigns positive, although possibly infinitesimal, probability to all actions at every information set) that differs only infinitesimally from such that at each information set I for player i, σ _i is an -best response to conditional on having reached I. Note that the characterization of sequential equilibrium does not involve belief systems. There is a similar characterization of perfect equilibrium; the only difference is that σ _i must be a best response to conditional on having reached I. Yet another variant is used to characterize proper equilibrium. This work was supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, and by AFOSR under grant FA9550-05-1-0055.     

Homogeneous factorisations of Johnson graphs

For a graph Γ, subgroups , and an edge partition of Γ, the pair is a (G, M)-homogeneous factorisation if M is vertex-transitive on Γ and fixes setwise each part of , while G permutes the parts of transitively. A classification is given of all homogeneous factorisations of finite Johnson graphs. There are three infinite families and nine sporadic examples. This paper forms part of an ARC Discovery grant of the last two authors. The second author holds an Australian Research Council Australian Research Fellowship.     

On the number of moduli of plane sextics with six cusps

Let be the variety of irreducible sextics with six cusps as singularities. Let be one of irreducible components of . Denoting by the space of moduli of smooth curves of genus 4, we consider the rational map sending the general point [Γ] of Σ, corresponding to a plane curve , to the point of parametrizing the normalization curve of Γ. The number of moduli of Σ is, by definition the dimension of Π(Σ). We know that , where ρ(2, 4, 6) is the Brill–Noether number of linear series of dimension 2 and degree 6 on a curve of genus 4. We prove that both irreducible components of have number of moduli equal to seven.      

Scales for co-compact embeddings of virtually free groups

Let Γ be a group which is virtually free of rank at least 2 and let be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family .      

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3).     

On formulas for the index of the circular distributions

A circular distribution is a Galois equivariant map ψ from the roots of unity μ _∞ to an algebraic closure of such that ψ satisfies product conditions, for ϵ ∈ μ _∞ and , and congruence conditions for each prime number l and with (l, s) = 1, modulo primes over l for all , where μ _l and μ _s denote respectively the sets of lth and sth roots of unity. For such ψ, let be the group generated over by and let be , where U _s denotes the global units of . We give formulas for the indices and of and inside the circular numbers P _s and units C _s of Sinnott over . This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).     

Riemannian geometry on the diffeomorphism group of the circle

The topological group of diffeomorphisms of the unit circle of Sobolev class H ^k , for k large enough, is a Banach manifold modeled on the Hilbert space . In this paper we show that the H ¹ right-invariant metric obtained by right-translation of the H ¹ inner product on defines a smooth Riemannian metric on , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H ¹ right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.     

Ore extensions which are GPI-rings

Let R be a prime ring and δ a σ-derivation of R, where σ is an automorphism of R. It is proved that the skew polynomial ring is a GPI-ring (PI-ring resp.) if and only if R is a GPI-ring (PI-ring resp.), δ is quasi-algebraic, and σ is quasi-inner. If is a GPI-ring then soc , where Q is the symmetric Martindale quotient ring of R and where denotes the extended centroid of . If is a PI-ring, its PI-degree is determined as follows: if δ is X-outer, and if δ is X-inner.     

Schatten-von Neumann Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the joint membership of Hφ and in the Schatten-von Neumann p-class when φ ∈ L∞(Ω) and Ω is a planar domain. We use a result of K. Zhu and the localization near the boundary to solve the problem. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.      

On a spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by: