for square-free integers m, and we have obtained that the special value L(Em, 1) of L-function for Em is described in terms of positive definite ternary quadratic forms. It is an analogue result of Tunnell’s [Tun] for the classical congruent number problem. In this paper, we use a result of Gauss on ternary quadratic forms to give more useful criterion for non-2π/3and non-2π/3-congruentness. So...

This paper is devoted to prove the existence of the optimal interval where the Green’s function is negative definite. The left and right endpoints of the interval are found. Then, a new principle of comparison of a third-order differential equation is established. As an application of our results, the solvability of a non-conjugate boundary value problem is discussed.

Semi-definite rank minimization problems model a wide range of applications in both signal processing and machine learning fields. This class of problem is NP-hard in general. In this paper, we propose a proximal Alternating Direction Method (ADM) for the well-known semi-definite rank regularized minimization problem. Specifically, we first reformulate this NP-hard problem as an equivalent bico...

Recently the asymptotic convergence of some commonly used decomposition methods for support vector machines has been established. However, their local convergence rates are still unknown. In this paper, under the assumptions that the kernel matrix is positive definite and the problem is non-degenerate, we prove the linear convergence of a popular decomposition method.

We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: on-line learning with a simple square loss, and finding a symmetric positi...

We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: min{cond( [ A BH B X ] ) : [ A BH B X ] positive definite}, where A is an n × n Hermitian positive definite matrix, B a p × n matrix and X is a free p× p Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solut...

We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the w...

abstractsince heubner's (1985) pioneering study, there have been many studies on (mis) use/ non-use of articles by l2 learners from article-less and article languages the present study investigated how persian l2 learners of english produce and interpret english definite descriptions and demonstrative descriptions it was assumed that definite and demonstrative descriptions share the same centra...

Elliptic interface problems with multi-domains and triple junction points have wide applications in engineering and science. However, the corner singularity makes it a challenging problem for most existing methods. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve the elliptic interface...