On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems

For the Hermitian inexact Rayleigh quotient iteration(RQI),we consider the local convergence of the in exact RQI with the Lanczos method for the linear systems involved.Some attractive properties are derived for the residual,whose norm is ξk,of the linear system obtained by the Lanczos method at outer iteration k+1.Based on them,we make a refned analysis and establish new local convergence results.It is proved that(i) the inexact RQI with Lanczos converges quadratically provided that ξk≤ξ with a constant ξ1 and (ii) the method converges linearly provided that ξk is bounded by some multiple of1/||rk|| with rkthe residual norm of the approximate eigenpair at outer iteration k.The results are fundamentally diferent from the existing ones that always require ξk<1,and they have implications on efective implementations of the method.Based on the new theory,we can design practical criteria to control ξkto achieve quadratic convergence and implement the method more efectively than ever before.Numerical experiments confrm our theory and demonstrate that the inexact RQI with Lanczos is competitive to the inexact RQI with MINRES.     

Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold (M,ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x). In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate an open book decomposition of M with any holomorphic function f:(X,x)→(C,0), with isolated singularity at x and we verify that all these open books carry the contact structure ξ of (M,ξ)—generalizing results of Milnor and Giroux.     

Diffusions with holding and jumping boundary

Consider a family of probability measures {νξ} on a bounded open region D R d with a smooth boundary and a positive parameter set {βξ},all indexed by ξ∈D.For any starting point inside D,we run a diffusion until it first exits D,at which time it stays at the exit point ξ for an independent exponential holding time with rate βξ and then leaves ξ by a jump into D according to the distribution νξ.Once the process jumps inside,it starts the diffusion afresh.The same evolution is repeated independently each time the process jumped into the domain.The resulting Markov process is called diffusion with holding and jumping boundary(DHJ),which is not reversible due to the jumping.In this paper we provide a study of DHJ on its generator,stationary distribution and the speed of convergence.     

Dirichlet units and critical points of closed 1-forms

S.P. Novikov developed an analog of the Morse theory for closed 1-forms. In this paper we suggest an analog of the Lusternik-Schnirelman theory for closed 1-forms. For any cohomology class ξ ε H¹(X,R) we define an integer cl(ξ) (the cup-length associated with ξ); we prove that any closed 1-form representing ξ has at least cl(ξ) - 1 critical points. The number cl(ξ) is defined using cup-products in cohomology of some flat line bundles, such that their monodromy is described by complex numbers, which are not Dirichlet units.     

On the distance from a rational power to the nearest integer

We prove that for any non-zero real number ξ the sequence of fractional parts {ξ(3/2)ⁿ}, n=1,2,3,…, contains at least one limit point in the interval [0.238117…,0.761882…] of length 0.523764…. More generally, it is shown that every sequence of distances to the nearest integer ||ξ(p/q)ⁿ||, n=1,2,3,…, where p/q>1 is a rational number, has both ‘large’ and ‘small’ limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue-Morse sequence and, for irrational ξ, are best possible for every pair p>1, q=1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence ||ξαⁿ||, n=1,2,3,…, where α>1 is an algebraic number and where ξ≠0 is an arbitrary real number satisfying ξ∉Q(α) in case α is a Pisot or a Salem number.     

Estimating a periodicity parameter in the drift of a time inhomogeneous diffusion

We consider a diffusion (ξ _t ) _t≥0 whose drift contains some deterministic periodic signal. Its shape being fixed and known, up to scaling in time, the periodicity of the signal is the unknown parameter ? of interest. We consider sequences of local models at ? corresponding to continuous observation of the process ξ on the time interval [0, n] as n → ∞, with suitable choice of local scale at ?. Our tools - under an ergodicity condition — are path segments of ξ corresponding to the period ?, and limit theorems for certain functionals of the process ξ, which are not additive functionals. When the signal is smooth, with local scale n ^?3/2 at ?, we have local asymptotic normality (LAN) in the sense of Le Cam [21]. When the signal has a finite number of discontinuities, with local scale n ^?2 at ?, we obtain a limit experiment of different type, studied by Ibragimov and Khasminskii [14], where smoothness of the parametrization (in the sense of Hellinger distance) is Hölder 1/2.     

Bifurcation of critical periods from the rigid quadratic isochronous vector field

This paper is concerned with the study of the number of critical periods of perturbed isochronous centers. More concretely, if X₀ is a vector field having an isochronous center of period T₀ at the point p and X? is an analytic perturbation of X₀ such that the point p is a center for X? then, for a suitable parameterization ξ of the periodic orbits surrounding p, their periods can be written as T(ξ,?)=T₀+T₁(ξ)?+T₂(ξ)?²+?. Firstly we give formulas for the first functions Tl(ξ) that can be used for quite general vector fields. Afterwards we apply them to study how many critical periods appear when we perturb the rigid quadratic isochronous center , inside the class of centers of the quadratic systems or of polynomial vector fields of a fixed degree.     

Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y=G(ξ^TX,ε) where G(⋅) is an unknown link function, ε is an independent error, and ξ is a p_n×1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G(⋅); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n=o(n^1/4) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

Pointwise Fourier inversion of distributions on spheres

Given a distribution T on the sphere we define, in analogy to the work of ?ojasiewicz, the value of T at a point ξ of the sphere and we show that if T has the value τ at ξ, then the Fourier-Laplace series of T at ξ is Abel-summable to τ.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

Autocorrelation measures for the quadratic assignment problem

In this article we provide an exact expression for computing the autocorrelation coefficient ξ and the autocorrelation length ? of any arbitrary instance of the Quadratic Assignment Problem (QAP) in polynomial time using its elementary landscape decomposition. We also provide empirical evidence of the autocorrelation length conjecture in QAP and compute the parameters ξ and ? for the 137 instances of the QAPLIB. Our goal is to better characterize the difficulty of this important class of problems to ease the future definition of new optimization methods. Also, the advance that this represents helps to consolidate QAP as an interesting and now better understood problem.     

Conditional Monte Carlo method for dynamic systems with random properties

A method is developed for calculating moments and other properties of states X(t) of dynamic systems with random coefficients depending on semi-Markov processes ξ(t) and subjected to Gaussian white noise. Random vibration theory is used to find probability laws of conditional processes X(t)∣ξ(·). Unconditional properties of X(t) are estimated by averaging conditional statistics of this process corresponding to samples of ξ(t). The method is particularly efficient for linear systems since X(t)∣ξ(·) is Gaussian during periods of constant values of ξ(t), so that and its probability law is completely defined by the process mean and covariance functions that can be obtained simply from equations of linear random vibration. The method is applied to find statistics of an Ornstein-Uhlenbeck process X(t) whose decay parameter is a semi-Markov process ξ(t). Numerical results show that X(t) is not Gaussian and that the law of this process depends essentially on features of ξ(t). A version of the method is used to calculate the failure probability for an oscillator with degrading stiffness subjected to Gaussian white noise.     

Ideals and free Pairs in the semigroup β ℤ

We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ? for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ?.     

How to determine the maximum genus of a graph

The maximum genus, γ_M(G), of a connected graph G is the largest genus γ(S) for orientable surfaces S in which G has a 2-cell embedding. In this paper, we define a new combinatorial invariant ξ(G), the Betti deficiency of G, to be ξ(C) = min_C?G{ξ(C) 6 ξ(C) = number of odd components of a cotree C of G (by odd component we mean one with an odd number of edges). We formalize a new embedding technique to obtain the formula:

$\text{γ}{}_{\mathrm{M}}\text{(G)=}\text{1}\text{2}\text{(β(G)?ξ(G))}$

where β(G) denotes the Betti number of G.In a further paper, various consequences will be given.     

Boundary Multifractal Behaviour for Harmonic Functions in the Ball

It is well known that if h is a nonnegative harmonic function in the ball of $\mathbb R^{d+1}$ or if h is harmonic in the ball with integrable boundary values, then the radial limit of h exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any β?∈?[0,d], the Hausdorff dimension of the set of points ξ on the sphere such that h(rξ) looks like (1???r)^???β is equal to d???β.     

Integro-local theorems for sums of independent random vectors in the series scheme

Let S(n) = ξ(1)+?+ξ(n) be a sum of independent random vectors ξ(i) = ξ _(n)(i) with general distribution depending on a parameter n. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability P(S(n) ∈ Δ[x), where Δ[x) is a cube with edge Δ and vertex at a point x.     

Approximation simultanée d'un nombre et de son carré

In 1969, H. Davenport and W.M. Schmidt established a measure of the simultaneous approximation for a real number ξ and its square by rational numbers with the same denominator, assuming only that ξ is not rational nor quadratic over $\text{Q}$. Here, we show by an example, that this measure is optimal. We also indicate several properties of the numbers for which this measure is optimal, in particular with respect to approximation by algebraic integers of degree at most three. To cite this article: D. Roy, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On Beurling’s boundary differential relation

We prove that for an arbitrary positive continuous function Φ on the complex plane ? there exists an injective disc algebra function φ and n ∈ ? such that ξ ? φ(ξ ⁿ ) solves Beurling’s boundary differential relation |f′(ξ)| = Φ(f(ξ)) on ?Δ. Moreover, if the growth of Φ is sublinear, the existence of univalent solutions of Beurling’s boundary differential relation is shown.     

Upper bounds for the forwarding indices of communication networks

The decision version of the forwarding index problem is, given a connected graph G and an integer ξ, to find a way of connecting each ordered pair of vertices by a path so that every vertex is an internal point of at most such paths. The optimization version of the problem is to find the smallest ξ for which a routing of this kind exists. Such a problem arises in the design of communication networks and distributed architectures. A model of parallel computation is represented by a network of processors, or machines processing and forwarding (synchronous) messages to each other, subject to physical constraints bearing on either the number of messages that can be processed by a single machine or the number of messages that can be sent through a connection. It was in this context that the problem was first introduced by Chung et al. (IEEE Trans. Inform. Theory 33 (1987) 224). The aim of this paper is to establish upper bounds for the optimal ξ as a function of the connectivity of the graph.