A Modified Convergence Analysis For Solving Fredholm Integral Equations of The First Kind

In this paper, we consider a modified convergence analysis for solving Fredholm integral equations of the first kind in Hilbert space setting using Tikhonov regularization. We follow a general approach which not only includes, as special case, the results of Groetsch [2] but also obtain the same with weaker assumptions.     

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).     

Positive solutions of Schrödinger equations in two-dimensional exterior domains

Sufficient conditions for the existence of a positive solution to the semilinear Schrödinger equation u+f(x,u)=0 in a two-dimensional exterior domain are given.     

Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm

We show that the Yang-Mills equation in three dimensions in the temporal gauge is locally well-posed in H^s for if the H^s norm is sufficiently small. The temporal gauge is slightly less convenient technically than the more popular Coulomb gauge, but has the advantage of uniqueness even for large initial data, and does not require solving a nonlinear elliptic problem. To handle the temporal gauge correctly we project the connection into curl- and divergence-free components, and develop some new bilinear estimates of X^s,b type which can handle integration in the time direction.     

A note on the exceptional set for Goldbach's problem in short intervals

Assume the Generalized Riemann Hypothesis and suppose thatHlog^–6 xt8. Then we prove that all even integers in any interval of the form (x, x, +H) butO(H^1/2log³x) exceptions are a sum of two primes.Partially supported by Consiglio Nazionale delle Ricerche, Visiting Professor Program.Partially supported by Hungarian National Foundation for Scientific Research, Grant n. 1901.     

Bayesian prediction for the two-parameter exponential distribution based on type II doubly censoring

The two-parameter exponential distribution is proposed to be an underlying model, and prediction bounds for future observations are obtained by using Bayesian approach. Prediction intervals are derived for unobserved lifetimes in one-sample prediction and two-sample prediction based on type II doubly censored samples. A numerical example is given to illustrate the procedures, prediction intervals are investigated via Monte Carlo method, and the accuracy of prediction intervals is presented. Supported by the National Natural Science Foundation of China (79970022) and Aviation Fund (02J53079).     

Multiple supersingular elliptic fibers on elliptic surfaces

Elliptic surfaces over an algebraically closed field in characteristic p>0 with multiple supersingular elliptic fibers, that is, multiple fibers of a supersingular elliptic curve, are investigated. In particular, it is shown that for an elliptic surface with q=g+1 and a supersingular elliptic curve as a general fiber, where q is the dimension of an Albanese variety of the surface and g is the genus of the base curve, the multiplicities of the multiple supersingular elliptic fibers are not divisible by p². As an application of this result, the structure of false hyperelliptic surfaces is discussed on this basis.     

Accelerated Landweber iterations for the solution of ill-posed equations

Summary In this paper, the potentials of so-calledlinear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.If the problem's right hand side data are contaminated by noise, semiiterative methods may be used asregularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results onfast decreasing polynomials are applied to answer an open question of Brakhage.Numerical examples are given including a comparison to the method of conjugate gradients.This research was sponsored by the Deutsche Forschungsgemeinschaft (DFG).     

A martingale decomposition for quadratic forms of Markov chains (with applications)

We develop a martingale-based decomposition for a general class of quadratic forms of Markov chains, which resembles the well-known Hoeffding decomposition of U$U$-statistics of i.i.d. data up to a reminder term. To illustrate the applicability of our results, we discuss how this decomposition may be used to studying the large-sample properties of certain statistics in two problems: (i) we examine the asymptotic behavior of lag-window estimators in time series, and (ii) we derive an asymptotic linear representation and limiting distribution of U$U$-statistics with varying kernels in time series. We also discuss simplified examples of interest in statistics and econometrics.     

The cutoff phenomenon for Ehrenfest chains

We consider families of Ehrenfest chains and provide a simple criterion on the L^p$L^p$-cutoff and the L^p$L^p$-precutoff with specified initial states for 1≤p<∞$1\le p<\infty$. For the family with an L^p$L^p$-cutoff, a cutoff time is described and a possible window is given. For the family without an L^p$L^p$-precutoff, the exact order of the L^p$L^p$-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.     

A complex version of the baer-krull theorems

In the setting of the complex spectrum of a field of characteristic zero we give complex analogs of the celebrated (real) Baer-Krull theorems relat-ing the orderings of a field compatible with a given valuation ring with the orderings of the residue class field of that valuation ring.     

Rayleigh-type isoperimetric inequality with a homogeneous magnetic field

We prove that the two dimensional free magnetic Schrödinger operator, with a fixed constant magnetic field and Dirichlet boundary conditions on a planar domain with a given area, attains its smallest possible eigenvalue if the domain is a disk. We also give some rough bounds on the lowest magnetic eigenvalue of the disk.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

On sums of subsets of a set of integers

Forr2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noBA and an integery with b=y ^r. It is shown that for any>0 andn>n(), (1+o(1))2^1/(r+1) n ^{(r–1)/(r+1)}p(n, r)n ^+2/3 for allr5, and that for every fixedr6,p(n, r)=(1+o(1))·2^1/(r+1) n ^{(r–1)/(r+1)} asn. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noBA the sum of whose elements ism. It is proved that for 3n ^6/3+mn ²/20 log² n andn>n(), f(n, m)=[n/s]+s–2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erds and Graham.Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

Small mass asymptotic for the motion with vanishing friction

We consider the small mass asymptotic (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.     

A correction to the paper “On minima of radially symmetric functionals of the gradient”

We prove a theorem for the existence of solutions to a variational problem, under assumptions that do not require the convexity of the integrand.     

Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation

ut+uxxx+(u⁴x₎=0     

Spectral computations for birth and death chains

We consider the spectrum of birth and death chains on an n$n$-path. An iterative scheme is proposed to compute any eigenvalue with exponential convergence rate independent of n$n$. This allows one to determine the whole spectrum in order n²$n^2$ elementary operations. Using the same idea, we also provide a lower bound on the spectral gap, which is of the correct order on some classes of examples.     

Relativistic Kronig-Penney-type Hamiltonians

The relativistic Kronig-Penney model is extended to a wide class of point interactions introduced in the nonrelativisfic case by Paul Chernoff and the author. A generalized Kronig-Penney relation, which converges to the standard Kronig-Penney relation in the nonrelativistic limit, is shown to determine the spectrum of the corresponding Hamiltonian. Several examples are considered, including the well-known special cases of electrostatic and Lorentz scalar point interactions, as well as several new solvable models.