矩阵空间之间的保幂线性映射

在本文中,设C是复数域,n和m是正整数,k为固定的自然数,且k≥2．设Mm(C)为C上m阶全矩阵空间,Sn(C)为C上n阶对称矩阵空间．本文分别刻画了从Sn(C)到Mm(C)和Sn(C)到Sm(C)上的保矩阵k次幂的线性映射．     

Fixed Point Theorem of $\{a,b,c\}$ Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let $C$ be a closed convex weakly Cauchy subset of a normed space $X$. Then we define a new $\{a,b,c\}$ type nonexpansive and $\{a,b,c\}$ type contraction mapping $T$ from $C$ into $C$. These types of mappings will be denoted respectively by $\{a,b,c\}$-$n$type and $\{a,b,c\}$-$c$type. We proved the following:1. If $T$ is $\{a,b,c\}$-$n$type mapping, then $\inf\{\|T(x)-x\|:x\in C\}=0$, accordingly $T$ has a unique fixed point. Moreover, any sequence $\{x_{n}\}_{n\in \mathcal{N}}$ in $C$ with $\lim_{n\to \infty}\|T(x_{n})-x_{n}\|=0$ has a subsequence strongly convergent to the unique fixed point of $T$.2. If $T$ is $\{a,b,c\}$-$c$type mapping, then $T$ has a unique fixed point. Moreover, for any $x\in C$ the sequence of iterates $\{T^{n}(x)\}_{n\in \mathcal{N}}$ has subsequence strongly convergent to the unique fixed point of $T$.This paper extends and generalizes some of the results given in [2,4,7] and [13].     

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

We investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.     

Bayesian Inversion of Piecewise Affine Operators in a Gaussian Framework

Piecewise affine inverse problems form a general class of nonlinear inverse problems. In particular inverse problems obeying certain variational structures, such as Fermat's principle in travel time tomography, are of this type. In a piecewise affine inverse problem a parameter is to be reconstructed when its mapping through a piecewise affine operator is observed, possibly with errors. A piecewise affine operator is defined by partitioning the parameter space and assigning a specific affine operator to each part. A Bayesian approach with a Gaussian random field prior on the parameter space is used. Both problems with a discrete finite partition and a continuous partition of the parameter space are considered.

The main result is that the posterior distribution is decomposed into a mixture of truncated Gaussian distributions, and the expression for the mixing distribution is partially analytically tractable. The general framework has, to the authors' knowledge, not previously been published, although the result for the finite partition is generally known.

Inverse problems are currently of large interest in many fields. The Bayesian approach is popular and most often highly computer intensive. The posterior distribution is frequently concentrated close to high-dimensional nonlinear spaces, resulting in slow mixing for generic sampling algorithms. Inverse problems are, however, often highly structured. In order to develop efficient sampling algorithms for a problem at hand, the problem structure must be exploited.

The decomposition of the posterior distribution that is derived in the current work can be used to develop specialized sampling algorithms. The article contains examples of such sampling algorithms. The proposed algorithms are applicable also for problems with exact observations. This is a case for which generic sampling algorithms tend to fail.     

On the Convergence of Successive Substitution Sampling

Abstract

The problem of finding marginal distributions of multidimensional random quantities has many applications in probability and statistics. Many of the solutions currently in use are very computationally intensive. For example, in a Bayesian inference problem with a hierarchical prior distribution, one is often driven to multidimensional numerical integration to obtain marginal posterior distributions of the model parameters of interest. Recently, however, a group of Monte Carlo integration techniques that fall under the general banner of successive substitution sampling (SSS) have proven to be powerful tools for obtaining approximate answers in a very wide variety of Bayesian modeling situations. Answers may also be obtained at low cost, both in terms of computer power and user sophistication. Important special cases of SSS include the “Gibbs sampler” described by Gelfand and Smith and the “IP algorithm” described by Tanner and Wong. The major problem plaguing users of SSS is the difficulty in ascertaining when “convergence” of the algorithm has been obtained. This problem is compounded by the fact that what is produced by the sampler is not the functional form of the desired marginal posterior distribution, but a random sample from this distribution. This article gives a general proof of the convergence of SSS and the sufficient conditions for both strong and weak convergence, as well as a convergence rate. We explore the connection between higher-order eigenfunctions of the transition operator and accelerated convergence via good initial distributions. We also provide asymptotic results for the sampling component of the error in estimating the distributions of interest. Finally, we give two detailed examples from familiar exponential family settings to illustrate the theory.     

Asymptotic Normality of Kernel Type Density Estimators for Random Fields

Kernel type density estimators are studied for random fields. It is proved that the estimators are asymptotically normal if the set of locations of observations become more and more dense in an increasing sequence of domains. It turns out that in our setting the covariance structure of the limiting normal distribution can be a combination of those of the continuous parameter and the discrete parameter cases. The proof is based on a new central limit theorem for α-mixing random fields. Simulation results support our theorems. Final version 29 October 2004     

A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of a modified random walk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions. A subclass GIT_3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT_3,1 extends Minkova’s inflated-parameter binomial and negative binomial. A bivariate model which has the GIT as a marginal distribution is also proposed.     

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.     

The Conditional-Potts Clustering Model

This article presents a Bayesian kernel-based clustering method. The associated model arises as an embedding of the Potts density for class membership probabilities into an extended Bayesian model for joint data and class membership probabilities. The method may be seen as a principled extension of the super-paramagnetic clustering. The model depends on two parameters: the temperature and the kernel bandwidth. The clustering is obtained from the posterior marginal adjacency membership probabilities and does not depend on any particular value of the parameters. We elicit an informative prior based on random graph theory and kernel density estimation. A stochastic population Monte Carlo algorithm, based on parallel runs of the Wang–Landau algorithm, is developed to estimate the posterior adjacency membership probabilities and the parameter posterior. The convergence of the algorithm is also established. The method is applied to the whole human proteome to uncover human genes that share common evolutionary history. Our experiments and application show that good clustering results are obtained at many different values of the temperature and bandwidth parameters. Hence, instead of focusing on finding adequate values of the parameters, we advocate making clustering inference based on the study of the distribution of the posterior adjacency membership probabilities. This article has online supplementary material.     

Inverse semigroups on graphs

For each [directed] graph we construct an inverse semigroup. Our main application is a simple proof of the characterization of partially ordered sets ofJ-classes of finite semigroups, and some generalizations; our proof avoids using the inductive construction of the previous method by one of the authors [4]. For a connected graph in which each vertex has index at least two, our construction gives a congruence free inverse semigroup. In the final section we describe how a slight modification bf the construction yields the polycyclic monoids.     

Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback–Leibler divergence can be used as the Lyapunov function for the IMRT planning system.Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems.The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.     

区组长为奇素数的自反MD设计 (英)

本文利用差方法对自反MD设计SCMD$(4mp, p,1)$的存在性给出了构造性证明, 这里$p$为奇素数, $m$为正整数.     

Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well-posed inference problems independently of the finite-dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimension reduction in functional regression with categorical predictor

In the present paper, we consider dimension reduction methods for functional regression with a scalar response and the predictors including a random curve and a categorical random variable. To deal with the categorical random variable, we propose three potential dimension reduction methods: partial functional sliced inverse regression, marginal functional sliced inverse regression and conditional functional sliced inverse regression. Furthermore, we investigate the relationships among the three methods. In addition, a new modified BIC criterion for determining the dimension of the effective dimension reduction space is developed. Real and simulation data examples are then presented to show the effectiveness of the proposed methods.     

An alternating method for an inverse Cauchy problem

In this paper, an iterative boundary element method based on our relaxed algorithm introduced in [8] is used to solve numerically a class of inverse boundary problems. A dynamical choice of the relaxation parameter is presented and a stopping criterion based on our theoretical results is used. The numerical results show that the algorithm produces a reasonably approximate solution and improves the rate of convergence of Kozlov's scheme [10]. This revised version was published online in June 2006 with corrections to the Cover Date.     

An Efficient Variant of the Restarted Shifted GMRES Method for Solving Shifted Linear Systems

We investigate the restart of the Restarted Shifted GMRES method for solving shifted linear systems.Recently the variant of the GMRES(m) method with the unfixed update has been proposed to improve the convergence of the GMRES(m) method for solving linear systems,and shown to have an efficient convergence property.In this paper,by applying the unfixed update to the Restarted Shifted GMRES method,we propose a variant of the Restarted Shifted GMRES method.We show a potentiality for efficient convergence within the variant by some numerical results.     

A non-standard representation for Brownian Motion and Itô integration

In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and Itô integration. We first develop a number of new tools about Loeb spaces. We then show that Brownian Motion can be obtained as the Loeb process corresponding to a non-standard random walk obtained from a*-finite number of coin tosses. This permits a very constructive proof of a special case of Donsker's Theorem. The Itô integral with respect to this Brownian Motion is a non-standard Stieltjes integral with respect to the random walk. As a consequence, an easy proof of Itô’s Lemma is possible. The results in this paper were announced in [1].     

Integration of singular braid invariants and graph cohomology

We prove necessary and sufficient conditions for an arbitrary invariant of braids with double points to be the `` derivative' of a braid invariant. We show that the ``primary obstruction to integration' is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on which works for invariants with values in any abelian group.

We find that to prove our theorem, we must show that every relation among four-term relations satisfies a certain geometric condition. To find the relations among relations we show that of a variant of Kontsevich's graph complex vanishes. We discuss related open questions for invariants of links and other things.

     

On the asymptotics of fault probability in least‐recently‐used caching with Zipf‐type request distribution

We consider the least‐recently‐used cache replacement rule with a Zipf‐type page request distribution and investigate an asymptotic property of the fault probability with respect to an increase of cache size. We first derive the asymptotics of the fault probability for the independent‐request model and then extend this derivation to a general dependent‐request model, where our result shows that under some weak assumptions the fault probability is asymptotically invariant with regard to dependence in the page request process. In a previous study, a similar result was derived by applying a Poisson embedding technique, where a continuous‐time proof was given through some assumptions based on a continuous‐time modeling. The Poisson embedding, however, is just a technique used for the proof and the problem is essentially on a discrete‐time basis; thus, it is preferable to make assumptions, if any, directly in the discrete‐time setting. We consider a general dependent‐request model and give a direct discrete‐time proof under different assumptions. A key to the proof is that the numbers of requests for respective pages represent conditionally negatively associated random variables. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

An elementary proof of the strong law of large numbers

Summary In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].