Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

On weakly bounded empirical processes

Let F be a class of functions on a probability space (Ω, μ) and let X ₁,...,X _k be independent random variables distributed according to μ. We establish an upper bound that holds with high probability on for every t > 0, and that depends on a natural geometric parameter associated with F. We use this result to analyze the supremum of empirical processes of the form for p > 1 using the geometry of F. We also present some geometric applications of this approach, based on properties of the random operator 〈X _i , ·〉e _i , where are sampled according to an isotropic, log-concave measure on .     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006     

Parameterized partition relations on the real numbers

We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738 of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully acknowledge the support provided by these institutions.     

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain

We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u ₀(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold x of dimension n > 1, a discrete sequence , and a discrete sequence where , there exists a proper holomorphic embedding satisfying f(a _j ) = b _j for every j = 1,2,... Forstnerič and Prezelj supported by grants P1-0291 and J1-6173, Republic of Slovenia. Kutzschebauch supported by Schweizerische National fonds grant 200021-107477/1. Ivarsson supported by The Wenner-Gren Foundations.     

Semicrossed products of simple C*-algebras

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

Derived McKay correspondence via pure-sheaf transforms

In most cases where it has been shown to exist the derived McKay correspondence can be written as a Fourier–Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in . We give a sufficient condition for to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of . Along the way we extract more geometrical meaning out of the Intersection Theorem and learn to compute θ-stable families of G-constellations and their direct transforms.     

KK-theoretic duality for proper twisted actions

Let be a smooth continuous trace algebra, with a Riemannian manifold spectrum X, equipped with a smooth action by a discrete group G such that G acts on X properly and isometrically. Then is KK-theoretically Poincaré dual to , where is the inverse of in the Brauer group of Morita equivalence classes of continuous trace algebras equipped with a group action. We deduce this from a strengthening of Kasparov’s duality theorem. As applications we obtain a version of the above Poincaré duality with X replaced by a compact G-manifold M and Poincaré dualities for twisted group algebras if the group satisfies some additional properties related to the Dirac dual-Dirac method for the Baum- Connes conjecture. This research was supported by the EU-Network Quantum Spaces and Noncommutative Geometry (Contract HPRN-CT-2002-00280) and the Deutsche Forschungsgemeinschaft (SFB 478) and by the National Science and Engineering Research Council of Canada Discovery Grant program.     

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier–Stokes equations

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier–Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as t goes to the possible time of singularity T. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, . For the Navier–Stokes equations the convergence of the velocity to the self-similar singularity is in L ^q (B(z,r)) for some , where the ball of radius r is shrinking toward a possible singularity point z at the order of as t approaches to T. In the convergence case with we present a simple alternative proof of the similar result in Hou and Li in arXiv-preprint, math.AP/0603126. This work was supported partially by KRF Grant(MOEHRD, Basic Research Promotion Fund) and the KOSEF Grant no. R01-2005-000-10077-0.     

On the Structure of Solutions of the Moisil-Théodoresco System in Euclidean Space

Let be open, let be the Dirac operator in and let be the Clifford algebra constructed over the quadratic space . If for fixed, denotes the space of r-vectors in , then an -valued smooth function W = W ^r  + W ^r+2 in Ω is said to satisfy the Moisil-Théodoresco system if . In terms of differential forms, this means that the corresponding - valued smooth form w = w ^r  + w ^r+2 satisfies in Ω the system d ^* w ^r = 0, dw ^r  + d ^* w ^r+2 = 0; dw ^r+2 = 0. Based on techniques and results concerning conjugate harmonic functions in the framework of Clifford analysis, a structure theorem is proved for the solutions of the Moisil-Théodoresco system.      

Rate of Type II blowup for a semilinear heat equation

A solution u of a Cauchy problem for a semilinear heat equation

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

$${\mathcal{R}}$$ -boundedness,pseudodifferential operators,and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class () and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with -bounded symbols, yielding by an iteration argument the -boundedness of λ(A−λ)⁻¹ in for some . To this end, elements of a symbolic and operator calculus of pseudodifferential operators with -bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on with operator valued coefficients.     

Partial regularity for polyconvex functionals depending on the Hessian determinant

We prove a C ^2,α partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth.     

Stably isomorphic dual operator algebras

We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Δ-equivalent (Eleftherakis in J Pure Appl Algebra, ArXiv:math. OA/0607489v4, 2007), if and only if they have completely isometric normal representations α,β on Hilbert spaces H, K respectively and there exists a ternary ring of operators such that and This project is cofunded by European Social Fund and National Resources—(EPEAEK II) “Pyhtagoras II” grant No. 70/3/7997.     

Extensions of linking systems with <Emphasis Type="Italic">p</Emphasis>-group kernel

We study extensions of p-local finite groups where the kernel is a p-group. In particular, we construct examples of saturated fusion systems which do not come from finite groups, but which have normal p-subgroups such that is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system. B. Oliver is partially supported by UMR 7539 of the CNRS. J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.