On the Trace Formula of Perturbation Theory. II

In a previous paper for a class of pairs of operators in a Hilbert space with nuclear difference or under a more general nuclearity condition there was introduced a spectral shift functional. Here we consider the local integrability of this spectral shift functional, i.e., the problem when this functional can locally be represented by an integrable function. The general results are then applied to pairs of definitizable and locally definitizable unitary operators in a Krein space.     

关于ρ-混合序列对数律的收敛速度

本文研究了ρ-混合序列对数律的收敛速度，在较弱的矩条件下得到了与独立同分布实随机变量类似的结果，并获得了ρ－混合序列满意对数律的一个充分性结果；讨论了ρ－混合序列重对数律的收敛速度的问题，得到了一个重对数律的充分性条件。     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

We consider a boundary value problem for the Sturm–Liouville equation with piecewise‐constant leading coefficient. We prove that some integral representations for the solutions of the considered equation can be obtained by using classical transformation operators for the Sturm–Liouville operator at the end points of a finite interval. We also investigate the spectral characteristics of the boundary value problem, prove the completeness and expansion theorem. Copyright © 2012 John Wiley & Sons, Ltd.     

On the second conjugate of several convex functions in general normed vector spaces

When dealing with convex functions defined on a normed vector space X the biconjugate is usually considered with respect to the dual system (X, X ^*), that is, as a function defined on the initial space X. However, it is of interest to consider also the biconjugate as a function defined on the bidual X ^**. It is the aim of this note to calculate the biconjugate of the functions obtained by several operations which preserve convexity. In particular we recover the result of Fitzpatrick and Simons on the biconjugate of the maximum of two convex functions with a much simpler proof.      

On the Finite Element Analysis of Problems with Nonlinear Newton Boundary Conditions in Nonpolygonal Domains

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition considered in a two-dimensional nonpolygonal domain with a curved boundary. The existence and uniqueness of the solution of the continuous problem is a consequence of the monotone operator theory. The main attention is paid to the effect of the basic finite element variational crimes: approximation of the curved boundary by a polygonal one and the evaluation of integrals by numerical quadratures. With the aid of some important properties of Zlamal's ideal triangulation and interpolation, the convergence of the method is analyzed.     

On the simultaneous determination of the zeros of an analytic function inside a simple smooth closed contour in the complex plane

A family of higher-order iterative methods for the simultaneous determination of all simple or multiple zeros of an analytic function (inside a simple smooth closed contour) is obtained using earlier results of the author. With the help of circular arithmetic, the interval variant of this family is proposed. Many parallel iterative methods of the literature are special cases of this family.     

On the Critical Exponents of Certain Nonlinear Boundary-Value Problems with Biharmonic Operator in the Exterior of a Ball

We establish sufficient conditions for the absence of global solutions of the differential inequality Δ² u≥|u|^q in the exterior of a ball. We consider various boundary conditions and show that the critical exponents depend on these conditions. The proofs are based on the test function method developed by Mitidieri and Pokhozhaev.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.