In this note we prove that if K is a compact set of m×n matrices containing an isolated point X with no rank-one connection into the convex hull of K \ {X}, then the rank-one convex hull separates as K = ( K \ {X} )rc ∪ {X}. This is an extension of a result of P. Pedregal, which holds for 2× 2 matrices.

A Galois correspondence for finitely generated field extensions k/h is presented in the case characteristic h = p ^ 0. A field extension k/h is Galois if it is modular and h is separably algebraically closed in k. Galois groups are the direct limit of groups of higher derivations having rank a power of p. Galois groups are characterized in terms of abelian iterative generating sets in a manner ...

A partition of a non-negative integer n is a non-increasing sequence of positive integers whose sum is n. For example, The 5 partitions of 4 are 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. To explain Ramanujan’s famous three partition congruences, the partition statistics, the rank (resp. the crank) were introduced by Dyson [8] (resp. Andrews and Garvan [3]). Atkin and Garvan [5] initiated t...

A large variety of machine learning models which aim at vector quantization have been proposed. However, only very preliminary rigorous mathematical analysis concerning their learning behavior such as convergence speed, robustness with respect to initialization, etc. exists. In this paper, we use the theory of on-line learning for an exact mathematical description of the training dynamics of Ve...

Several attempts have been made to modify the quasi-Newton condition in order obtain rapid convergence with complete properties (symmetric and positive definite) of inverse Hessian matrix (second derivative objective function). There are many unconstrained optimization methods that do not generate definiteness matrix. One those is symmetric rank 1( H-version) update (SR1 update), where this sat...

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an `-tensor. The tensor product of s and t is a (k + `)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general...

where we use tick marks to denote the updated versions of D(t) and s(t) after receiving the new training data, and D(t) 1 2 is the matrix square-root of D(t). The updates to A consist of adding or subtracting an outer-product of a matrix of size (d × k)-by-d, which implies that each update has rank at most d. If we have already computed the eigenvalue decomposition of the old A, we can compute ...