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.     

A spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric conformal to g, we denote by the first positive eigenvalue of the Dirac operator on . We show that

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

Asymptotic Nonlinearity of Boolean Functions

Boolean functions on the space are not only important in the theory of error-correcting codes, but also in cryptography. In these two cases, the nonlinearity of these functions is a main concept. Carlet, Olejár and Stanek gave an asymptotic lower bound for the nonlinearity of most of them, and I gave an asymptotic upper bound which was strictly larger. In this article, I improve the bounds and get an exact limit for the nonlinearity of most of Boolean functions. This article is inspired by a paper of G. Halász about the related problem of real polynomials with random coefficients. AMS Classification (2000) Primary: 11T71 · Secondary: 06E30 · 42A05 · 94C10     

Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals

We consider local minimizers of variational integrals , where F is of anisotropic (p, q)-growth with exponents . If F is in a certain sense decomposable, we show that the dimensionless restriction together with the local boundedness of u implies local integrability of for all exponents . More precisely, the initial exponents for the integrability of the partial derivatives can be increased by two, at least locally. If n = 2, then we use these facts to prove -regularity of u for any exponents .     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Asymptotic Properties of the Maximum Likelihood Estimate in Generalized Linear Models with Stochastic Regressors

For generalized linear models （GLM）, in case the regressors are stochastic and have different distributions, the asymptotic properties of the maximum likelihood estimate （MLE） β^n of the parameters are studied. Under reasonable conditions, we prove the weak, strong consistency and asymptotic normality of β^n     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

On the Flux-Across-Surfaces Theorem for Short-Range Potentials

The Flux-Across-Surfaces theorem is established for three-dimensional Schrödinger equation with short-range potentials V satisfying the decay condition for some . Exceptional cases are treated as well and the required decay index is . Explicit conditions for the initial states are found. A stationary representation for the integrated flux is obtained and exploited in the proof. Spatial asymptotic expansion of the resolvent is employed to calculate the limit of the integrated flux at spatial infinity. Communicated by Gian Michele Grafsubmitted 01/03/03, accepted 30/05/03     

On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \mathbb{R}^{N}

In this paper we establish a Serrin’s type regularity criterion on the gradient of pressure for weak solutions to the Navier–Stokes equations in It is proved that if the gradient of pressure belongs to L^{α, γ} with then the weak solution actually is regular and unique. Received: May 4, 2004     

Bounds for a constrained optimal stopping problem

In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem over stopping times τ subject to a constraint , where x(.) is a geometric Brownian motion coupled with an arbitrary diffusion process y(.), θ(., .) and c(.) are given positive, continuous functions and β > 0 is a fixed constant. The present result is derived from a corresponding Lagrangian dual problem, and using a recent result of Makasu (Seq Anal 27:435–440, 2008). Examples are given to illustrate our main result. Partial results of this note were obtained when the author was holding a postdoc grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway.     

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     

Refined asymptotic expansions for nonlocal diffusion equations

We study the asymptotic behavior for solutions to nonlocal diffusion models of the form u _t = J * u – u in the whole with an initial condition u(x, 0) = u ₀(x). Under suitable hypotheses on J (involving its Fourier transform) and u ₀, it is proved an expansion of the form

The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices

Norm Inequalities for Commutators of Self-adjoint Operators

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with and for some real numbers a ₁, a ₂, b ₁, and b ₂, then for every unitarily invariant norm|||·|||,

Polar Decompositions of C 0(N) Contractions

Let A be a bounded linear operator on a complex separable Hilbert space H. We show that A is a C₀(N) contraction if and only if , where U is a singular unitary operator with multiplicity and x₁, . . . , x_d are orthonormal vectors satisfying . For a C₀(N) contraction, this gives a complete characterization of its polar decompositions with unitary factors.     

On a certain exponential inequality for Gaussian processes

If X=(X _j ) _j=1 ^m is a zero-mean Gaussian stochastic process and $\sigma _{j}=\left( E{\big[} X_{j}^{2}{\big]} \right) ^{1/2},$ j=1,...,m, Tsirel’son (Theory Probab. Appl., 30, 820–828, 1985) and more explicitly Vitale (Ann. Probab., 24, 2172–2178, 1996 and A log-concavity proof for a Gaussian exponential bound. In: Hill, T.P., Houdré, C. (eds.) Advances in Stochastic Inequalities, Contemporary Mathematics, vol. 234, pp. 209–212. AMS, Providence, RI, 1999) applied results from Brunn–Minkowski theory to show that X satisfies the following inequality: $$ E\left[ \exp \left( \max_{1\leq j\leq m}{\bigg(}X_{j}-\frac{\sigma _{j}^{2}}{2} {\bigg)}\right) \right] \leq \exp \left( E\left[ \max_{1\leq j\leq m}X_{j}\right] \right). $$ In this paper a more general inequality will be derived using a known formula for Gaussian integrals. In particular, it also follows that $$ {\small \ }E\left[ \exp \left( \min_{1\leq j\leq m}{\bigg(}X_{j}-\frac{\sigma _{j}^{2}}{2}{\bigg)}\right) \right] \leq \exp \left( E\left[ \min_{1\leq j\leq m}X_{j}\right] \right) . $$ In the last section of this article the above exponential inequalities are combined with a well known variant of the Slepian lemma to compare certain option prices in the Black–Scholes and Bachelier models.