The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or uniform grids, the convergence behavior of the proposed algorithms is studied and a collection of numerical results is given.

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. he blow-up behaviors of DVIEs with non-vanishing delay vary with different nitial functions and the length of the lag, while DVIEs with pantograph delay wn the same blow-up behavior of VIEs. Some examples and applications to elay differential equations illustrate this influence.

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L^∞-norm and the weighted L²-norm. Numerical examples are presented to complement the theoretical convergence results.

We establish a general oracle inequality for regularized risk minimizers with strongly mixing observations, and apply this inequality to support vector machine (SVM) type algorithms. The obtained main results extend the previous known results for independent and identically distributed samples to the case of exponentially strongly mixing observations.

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L²-Poincaré inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.

Based on the structure of the rank-1 matrix and the different unfolding ways of the tensor, we present two types of structured tensors which contain the rank-1 tensors as special cases. We study some properties of the ranks and the best rank-γ approximations of the structured tensors. By using the upper-semicontinuity of the matrix rank, we show that for the structured tensors, there always exist the best rank-γ approximations. This can help one to better understand the sequential unfolding singular value decomposition (SVD) method for tensors proposed by J. Salmi et al. [IEEE Trans Signal Process, 2009, 57(12): 4719–4733] and offer a generalized way of low rank approximations of tensors. Moreover, we apply the structured tensors to estimate the upper and lower bounds of the best rank-1 approximations of the 3rd-order and 4th-order tensors, and to distinguish the well written and non-well written digits.

We study the functional limits of continuous-time random walks (CTRWs) with tails under certain conditions. We find that the scaled CTRWs with tails converge weakly to an α-stable Lévy process in D([0, 1]) with M₁-topology but the corresponding scaled CTRWs converge weakly to the same limit in D([0, 1]) with J₁-topology.

In this paper, we study Whittaker modules over the loop Virasoro algebra relative to some total order. We give a description of all Whittaker vectors for the universal Whittaker modules. We also show that any universal Whittaker module admits a unique simple quotient modules except for a special case.

A new family of finite-dimensional simple modular Lie superalgebra M is constructed based on results of Y. Z. Zhang and Q. C. Zhang [J. Algebra, 2009, 321: 3601–3619]. The simplicity and generators of M are discussed and the derivation superalgebra of M is characterized. Furthermore, the invariance of the nonnatural filtration of M is determined by the method of minimal dimension of image spaces.

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.

In 2007, Peng introduced It? integral with respect to G-Brownian motion and the related It?’s formula in G-expectation space. Motivated by the properties of multiple Wiener integral obtained by It? in 1951, we introduce multiple G-It? integral in G-expectation space, and investigate how to calculate it. Furthermore, We establish a relationship between Hermite polynomials and multiple G-It? integrals.