A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

Wavelet characterization of Hörmander symbol classS ρ,δ m and applications

In this paper, we characterize the symbol in Hormander symbol classS _ρ ^m ,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS _ρ ^m ,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L²-continuity which is better than Calderón and Vaillancourt’s result, and establishL ^p (1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients in phase space.     

Application of Hilbert＇s Projective Metric on Symmetric Cones

Let Ω be a symmetric cone. In this note, we introduce Hilbert＇s projective metric on Ω in terms of Jordan algebras and we apply it to prove that, given a linear invertible transformation g such that g（Ω） = Ω and a real number p, ｜p｜ 〉 1, there exists a unique element x ∈ Ω satisfying g（x） = x^p.     

Analogues of Euler and Poisson summation formulae

Euler-Maclaurin and Poisson analogues of the summations ε _{a <n ≤b} χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula in several variables.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

Some zero-sum constants with weights

For an abelian group G, the Davenport constant D(G) is defined to be the smallest natural number k such that any sequence of k elements in G has a nonempty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to (ℤ/nℤ) ^d , more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of (ℤ/nℤ)² where n is an odd integer.     

Normal Approximations for Descents and Inversions of Permutations of Multisets

Normal approximations for descents and inversions of permutations of the set {1,2,…,n} are well known. We consider the number of inversions of a permutation π(1),π(2),…,π(n) of a multiset with n elements, which is the number of pairs (i,j) with 1≤i<j≤n and π(i)>π(j). The number of descents is the number of i in the range 1≤i<n such that π(i)>π(i+1). We prove that, appropriately normalized, the distribution of both inversions and descents of a random permutation of the multiset approaches the normal distribution as n→∞, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than α n times in the multiset for a fixed α with 0<α<1. Both normal approximation theorems are proved using the size bias version of Stein’s method of auxiliary randomization and are accompanied by error bounds. This work was supported by a research fellowship from the Sloan Foundation.     

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.     

Application of Newton’s method to a homogenization problem

The homogenization of a family (P _ε) of uniformly elliptic semilinear partial differential equations of second order is studied. The main result is that any non-singular solutionu of the homogenized problem (P) is the limit of non-singular solutions of (P _ε). The method consists of specifying a functionw _ε starting from which the Newton iterates converge to a solutionu _ε ofP _ε. These solutionsu _ε converge to the given solutionu of (P).     

Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

Singular values, quasiconformal maps and the Schottky upper bound

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ϕdimtortion function ϕ _K (r) and the Agard η-distortion function η _K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.     

Product Gauss cubature over polygons based on Green’s integration formula

We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n-1 using N∼mn ² nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss–Legendre quadrature via Green’s integral formula. Several numerical tests are presented. AMS subject classification (2000)  65F20     

Darboux-integrable discrete systems

We extend Laplace’s cascade method to systems of discrete “hyperbolic” equations of the form u_i+1,j+1 = f(u_i+1,j, u_i,j+1 , u_i,j), where u_ij is a member of a sequence of unknown vectors, i, j ∊ ℤ. We introduce the notion of a generalized Laplace invariant and the associated property of the system being “Liouville.” We prove several statements on the well-definedness of the generalized invariant and on its use in the search for solutions and integrals of the system. We give examples of discrete Liouville-type systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 207–219, August, 2008.     

Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations

In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method.      

On the discounted global CLT for some weakly dependent random variables

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑ _r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ _n1:= ∫ _−∞ ^∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 584–597, October–December, 2006.     

On the frequency of Titchmarsh’s phenomenon for ζ(s)-VIII

For suitable functionsH = H(T) the maximum of|(ζ(σ + it)) ^z | taken overT≤t≤T + H is studied. For fixed σ(1/2≤σ≤l) and fixed complex constantsz “expected lower bounds” for the maximum are established.     

Stability of an expanding bubble in the Rayleigh model

A bubble expands adiabatically in an incompressible, inviscid liquid. The variation of its radiusR with time is given by the Rayleigh’s equation. We find that the bubble is stable at the equilibrium point in this model.