Gelfand–Zeitlin actions and rational maps

We observe that the analogue of the Gelfand–Zeitlin action on , which exists on any symplectic manifold M with an Hamiltonian action of , has a natural interpretation as a residual action, after we identify M with a symplectic quotient of . We also show that the Gelfand–Zeitlin actions on and on the regular part of can be identified with natural Hamiltonian actions on spaces of rational maps into full flag manifolds, while the Gelfand–Zeitlin action on the whole corresponds to a natural action on a space of rational maps into the manifold of half-full flags in . The research of the first author is supported by the Alexander von Humboldt Foundation.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.     

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.     

A spectral characterization of the uniform continuity of strongly continuous groups

Let X be a Banach space and a strongly continuous group of linear operators on X. Set and where is the unit circle and denotes the spectrum of T(t). The main result of this paper is: is uniformly continuous if and only if is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007     

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).     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

The Kostant–García quantization of the bundle of connections

We shall call quantum states of a principal bundle π : P → M with structure group a semi-simple Lie group G, the elements of certain space of sections of the adjoint bundle , associated to the G-bundle of connections . An inner product of sections of is defined for which is a Hilbert space such that the Gauge group gau(P) of the given bundle represents in a family of self-adjoint operators. This work crystallizes some heuristic considerations, on the unitary representations of Gauge algebras, of Garcia in the already a classical article (J. Differ. Geom. 12, 209–227, 1977).     

$$(O( V \oplus F), O( V))$$ is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let with the form Q extending q with Q(e) = 1. Consider the standard embedding and the two-sided action of on . In this note we show that any -invariant distribution on is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of we have A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).     

All superconformal surfaces in $${\mathbb R^4}$$ in terms of minimal surfaces

We give an explicit construction of any simply connected superconformal surface in Euclidean space in terms of a pair of conjugate minimal surfaces . That is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs (g, h) of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in and to images of superminimal surfaces in either a sphere or a hyperbolic space by an stereographic projection. We also determine the relation between the pairs (g, h) of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in .     

Sobolev estimates for the Cauchy–Riemann complex on <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> pseudoconvex domains

Let be a bounded pseudoconvex domain with C ^k boundary, k ≥ 1. In this paper, we will prove that the Cauchy–Riemann operator has a bounded solution operator in the Sobolev space for all .     

Uniform central limit theorems for kernel density estimators

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript>-estimates for Sobolev functions in terms of ∇ and Δ

Several L ^∞-estimates are obtained for in terms of and , where are determined by m. If p = 2, then the estimates are given with explicit constants. However, if p ≠ 2, it is difficult to derive explicit constants except in two simple cases. Applicability to PDE’s is illustrated.     

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.     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Distances to spaces of Baire one functions

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function belongs to B ₁(X, E): our result here says that for an arbitrary we have

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .