A note on the inclusion of the kernel in the core of the bilateral assignment game

In this note we prove that the kernel of a bilateral assignment game is always included in the core. This solves an outstanding open problem for bilateral assignment games. Received January 1997/Final version June 1997     

Resolvent bounds for jump generators

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially, we prove a number of lower and upper bounds for the resolvent of such operators. In particular we focus on sharp estimates of the resolvent kernel for small values of the spectral parameter. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schrödinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviour of its solutions.     

New Interior-Point Algorithm for Symmetric Optimization Based on a Positive-Asymptotic Barrier Function

Abstract

We define a new interior-point method (IPM), which is suitable for solving symmetric optimization (SO) problems. The proposed algorithm is based on a new search direction. In order to obtain this direction, we apply the method of algebraically equivalent transformation on the centering equation of the central path. We prove that the associated barrier cannot be derived from a usual kernel function. Therefore, we introduce a new notion, namely the concept of the positive-asymptotic kernel function. We conclude that this algorithm solves the problem in polynomial time and has the same complexity as the best known IPMs for SO.     

W_2~m[a，b]空间中再生核的计算(Ⅰ)

本文用Green函数与伴随函数方法讨论由一般线性微分算子确定的再生核的具体计算.提出了基本Green函数与基本再生核的概念,它们是由微分算子和初值点唯一确定的;指出基本再生核的计算可转化为求解微分方程的初值问题,一般的再生核可由基本再生核的投影而得到;最后用例子说明了所给方法.     

Analytic continuation and identities involving heat,Poisson, wave and Bessel kernels

In this article we use classical formulas involving the K–Bessel function in two variables to express the Poisson kernel on a Riemannian manifold in terms of the heat kernel. We then use the small time asymptotics of the heat kernel on certain Riemannian manifolds to obtain a meromorphic continuation of the associated Poisson kernel to all values of complex time with identifiable singularities. This result reproves in a different setting by different means a well–known theorem due to Duistermaat and Guillemin [DG 75]. Also, we develop analytic expressions for the heat kernel beyond asymptotic expansions. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonlocal parabolic model for type‐I superconductors

A vectorial nonlocal linear parabolic problem on a bounded domain with applications in superconductors of type‐I is studied. The nonlocal term is represented by a (space) convolution with a singular kernel (arising in Eringen's model). The well‐posedness of the problem is discussed under low regularity assumptions, and the error estimates for an implicit and semiimplicit time‐discrete scheme (based on backward Euler approximation) are established. It is shown that the solution of the problem satisfies a simpler nonlocal problem with a positive definite kernel if the normal component of the unknown vector field equals zero on the boundary of the domain. Numerical experiments support the obtained theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1821–1853, 2014     

Optimal Domains for Kernel Operators via Interpolation

The problem of finding optimal lattice domains for kernel operators with values in rearrangement invariant spaces on the interval [0,1] is considered. The techniques used are based on interpolation theory and integration with respect to C([0, 1])–valued measures.     

KERNEL IDEALS AND CONGRUENCES ON MS-ALGEBRAS

In this article, the authors describe the largest congruence induced by a kernel ideal of an MS-algebra and characterize those MS-algebras on which all the congruences are in a one-to-one correspondence with the kernel ideals.     

Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications

Kernel methods and rough sets are two general pursuits in the domain of machine learning and intelligent systems. Kernel methods map data into a higher dimensional feature space, where the resulting structure of the classification task is linearly separable; while rough sets granulate the universe with the use of relations and employ the induced knowledge granules to approximate arbitrary concepts existing in the problem at hand. Although it seems there is no connection between these two methodologies, both kernel methods and rough sets explicitly or implicitly dwell on relation matrices to represent the structure of sample information. Based on this observation, we combine these methodologies by incorporating Gaussian kernel with fuzzy rough sets and propose a Gaussian kernel approximation based fuzzy rough set model. Fuzzy T-equivalence relations constitute the fundamentals of most fuzzy rough set models. It is proven that fuzzy relations with Gaussian kernel are reflexive, symmetric and transitive. Gaussian kernels are introduced to acquire fuzzy relations between samples described by fuzzy or numeric attributes in order to carry out fuzzy rough data analysis. Moreover, we discuss information entropy to evaluate the kernel matrix and calculate the uncertainty of the approximation. Several functions are constructed for evaluating the significance of features based on kernel approximation and fuzzy entropy. Algorithms for feature ranking and reduction based on the proposed functions are designed. Results of experimental analysis are included to quantify the effectiveness of the proposed methods.     

Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in Cd is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d=2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z₁=z₂ as subvariety) and the ball (with z₂=0 as subvariety), as well as for a weighted Bargmann-Fock space over C² (with the diagonal z₁=z₂ as subvariety).     

A new method for solving singular fourth-order boundary value problems with mixed boundary conditions

In [1], [2], [3], [4], [5], [6], [7] and [8], it is very difficult to get reproducing kernel space of problem (1). This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space. The numerical results are compared with both the exact solution and its n-order derived functions in the example. It is demonstrated that the new method is quite accurate and efficient for fourth-order problems.     

Local Linear Estimation of Jump-Diffusion Models by Using Asymmetric Kernels

This article addresses the problem of nonparametric estimation of the first and second infinitesimal moments by using the local linear method of the underlying jump-diffusion models. The motivation behind the study is to use the asymmetric kernels instead of standard kernel smoothing. The basic idea relies on replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the nonparametric estimation for jump-diffusion models. We prove that the estimators based on the local linear method for jump-diffusion models are consistent and asymptotically follow normal distribution under the condition of recurrence and stationarity.     

Approximation of kernel matrices by circulant matrices and its application in kernel selection methods

This paper focuses on developing fast numerical algorithms for selection of a kernel optimal for a given training data set. The optimal kernel is obtained by minimizing a cost functional over a prescribed set of kernels. The cost functional is defined in terms of a positive semi-definite matrix determined completely by a given kernel and the given sampled input data. Fast computational algorithms are developed by approximating the positive semi-definite matrix by a related circulant matrix so that the fast Fourier transform can apply to achieve a linear or quasi-linear computational complexity for finding the optimal kernel. We establish convergence of the approximation method. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.     

Reproducing kernel method of solving singular integral equation with cosecant kernel

In the paper, a reproducing kernel method of solving singular integral equations (SIE) with cosecant kernel is proposed. For solving SIE, difficulties lie in its singular term. In order to remove singular term of SIE, an equivalent transformation is made. Compared with known investigations, its advantages are that the representation of exact solution is obtained in a reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, the representation of reproducing kernel becomes simple by improving the definition of traditional inner product and requirements for image space of operators are weakened comparing with traditional reproducing kernel method. The final numerical experiments illustrate the method is efficient.     

椭圆型界面问题的破裂再生核方法

本文基于一维椭圆型界面问题提出了一种有效的数值方法.首先,根据模型构建一个崭新的破裂再生核空间.其次,应用破裂再生核方法给出了此类界面问题的近似解,并讨论该方法的收敛性.最后,通过几个有效的数值算例来说明该方法的精确性和稳定性.     

Bootstrapping the mode

The problem of constructing bootstrap confidence intervals for the mode of a density is considered. Estimates of the mode are derived from kernel density estimates based on fixed and data-dependent bandwidths. The asymptotic validity of bootstrap techniques to estimate the sampling distribution of the estimates is investigated. In summary, the results are negative in the sense that a straightforward application of a naive bootstrap yields invalid inferences. In particular, the bootstrap fails if resampling is done from the kernel density estimate. On the other hand, if one resamples from a smoother kernel density estimate (which is necessarily different from the one which yields the original estimate of the mode), the bootstrap is consistent. The bootstrap also fails if resampling is done from the empirical distribution, unless the choice of bandwidth is suboptimal. Similar results hold when applying bootstrap techniques to other functionals of a density.     

New concepts for completely regular semigroups

Let 𝒞? denotes the variety of completely regular semigroups considered with the unary operation of inversion. The global study of the lattice of subvarieties of 𝒞? depends heavily on various decompositions. Some of the most fruitful among these are induced by the kernel and the trace relations. In their turn, these relations are induced by the kernel and the trace relations on the lattice of congruences on regular semigroups. These latter admit the concepts of kernel and trace of a congruence. The kernel and the trace relations for congruences were transferred to kernel and trace relations on varieties but the kernel and trace got no analogue for varieties.

We supply here the kernel and the trace of a variety which induce the relations of their namesake. For the local and core relations, we also define the local and core of a variety. All the new concepts are certain subclasses of 𝒞?. In this way, we achieve considerable similarity of the new concepts with those for congruences. We also correct errors in two published papers.     

基于区间效率的贷款定价研究

以存款利息支出率、费用支出率、预期违约损失率、目标利润率等4个定价指标作为输入,以贷款利率作为输出,采用支持向量机回归算法建立基于区间效率的贷款定价模型.创新与特色一是通过区间数形式来反映预期违约损失率、目标利润率、费用支出率等,改变了现有研究将目标利润、贷款费用、违约损失等变量看作确定性数值来定价的现状.使贷款利率更具有竞争性.二是通过比较不同核函数、核参数下的训练样本的贷款效率区间与合理的贷款效率区间的匹配程度,确定贷款定价模型最优的核函数和核参数,进而建立了贷款定价模型.间接解决了在考虑贷款价格能否被银行、客户接受情况下贷款利率确定的问题.     

Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers

A numerical method is proposed for solving singularly perturbed turning point problems exhibiting twin boundary layers based on the reproducing kernel method (RKM). The original problem is reduced to two boundary layers problems and a regular domain problem. The regular domain problem is solved by using the RKM. Two boundary layers problems are treated by combining the method of stretching variable and the RKM. The boundary conditions at transition points are obtained by using the continuity of the approximate solution and its first derivatives at these points. Two numerical examples are provided to illustrate the effectiveness of the present method. The results compared with other methods show that the present method can provide very accurate approximate solutions.     

Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body. The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.