On the eigenvalues of the

We study the nonlinear eigenvalue problem

where , is a bounded smooth domain in . We prove that the first and the second variational eigenvalues of (1) are continuous functions of . Moreover, we obtain the asymptotic behavior of the first eigenvalue as and .

     

Hadron interactions in lattice QCD

Progress on the potential method, recently proposed to investigate hadron interactions in lattice QCD, is reviewed. The strategy to extract the potential in lattice QCD is explained in detail. The method is applied to extract NN potentials, hyperon potentials and the meson–baryon potentials. A theoretical investigation is made to understand the origin of the repulsive core using the operator product expansion. Some recent extensions of the method are also discussed.     

Elementary submodels in infinite combinatorics

The use of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams’s theorem on cycle decomposition of graphs, and finally we improve a decomposition theorem of Laviolette concerning bond-faithful decompositions of graphs.     

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.     

Sequential optimizing strategy in multi-dimensional bounded forecasting games

We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001) [10]. By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of a continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.     

秦九韶“正负开方术”是二阶收敛的

研究秦九韶"正负开方术"的收敛速度.采用分析算法的几何意义、使用这一算法求解具体的高次方程以及数学证明的方法.发现秦九韶法与牛顿切线法类似.秦九韶正负开方术是二阶收敛的.     

Solution of certain weakly singular integral equations of the first kind with indefinite parameters

In this paper we study a two-dimensional weakly singular integral equation of the first kind with logarithmic kernel. We construct a pair of spaces of the desired elements and the right-hand sides, where we prove the correctness of the problem under consideration and obtain inversion formulas for the integral operator.     

Bayesian factor analysis with fat-tailed factors and its exact marginal likelihood

The traditional Bayesian factor analysis method is extended. In contrast to the case for previous studies, the matrix variate t-distribution is utilized to provide a prior density on the latent factors. This is a natural extension of the traditional model and yields many advantages. The crucial issue is the selection of the number of factors. The marginal likelihood, constructed by asymptotic and computational approaches, is generally utilized for this problem. However, both theoretical and computational problems have arisen.In this paper, the exact marginal likelihood is derived. It enables us to evaluate posterior model probabilities without inducing the above problems. Monte Carlo experiments were conducted to examine the performance of the proposed Bayesian factor analysis modelling methodology. The simulation results show that the proposed methodology performs well.     

Bartholdi zeta functions of some graphs

We give a decomposition formula for the Bartholdi zeta function of a graph G which is partitioned into some irregular coverings. As a corollary, we obtain a decomposition formula for the Bartholdi zeta function of G which is partitioned into some regular coverings.