Array-oriented programming languages such as APL [1] and J [2] pay special attention to manipulating array structures: rank-one vectors (sequences of values), rank-two matrices (which can be seen as rectangular sequences of sequences), rank-three cuboids (sequences of sequences of sequences), rank-zero scalars, and so on. One appealing consequence of this unification is the prospect of rank pol...

Abstract. This is the third and final installment in our series of papers applying the method of Atkin and Swinnerton-Dyer to deduce formulas for rank differences. The study of rank differences was initiated by Atkin and Swinnerton-Dyer in their proof of Dyson’s conjectures concerning Ramanujan’s congruences for the partition function. Since then, other types of rank differences for statistics ...

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently...

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.

Matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. However, the problem is in general NP-hard, and it is computationally hard to solve directly in practice. In this paper, we provide a new kind of approximation functions for the rank of matrix, and the corresponding approximation problems can be used to approximate ...

The rank function rank(·) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(·), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization p...

Abstract Affine matrix rank minimization problem aims to find a low-rank or approximately low-rank matrix that satisfies a given linear system. It is well known that this problem is combinatorial and NP-hard in general. Therefore, it is important to choose the suitable substitution for this matrix rank minimization problem. In this paper, a continuous promoting low rank non-convex fraction func...