let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

While tensor factorizations have become increasingly popular for learning on various forms of structured data, only very few theoretical results exist on the generalization abilities of these methods. Here, we discuss the tensor product as a principled way to represent structured data in vector spaces for machine learning tasks. To derive generalization error bounds for tensor factorizations, w...

PLUS factorizations, or customizable triangular factorizations, of nonsingular matrices have found applications in source coding and computer graphics. However, there are still some open problems. In this paper, we present a new necessary condition and a sufficient condition for the existence of generic PLUS factorizations. © 2004 Elsevier Inc. All rights reserved.

Some classes of I-factorizations of complete graphs are known. They are GK2n , AK2n , W K 2n and their variations, and automorphism-free I-factorizations. In this paper, for any positive integer t, we construct new I-factorizations NtK 2n which are defined for all 2n with 2n ~ 6t. They also have some variations.

We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional data which is a sequence naturally induced by coloring, we define matrix fac...

Methods to compute 1–factorizations of a complete graphs of even order are presented. For complete graphs where the number of vertices is a power of 2, we propose several new methods to construct 1–factorizations. Our methods are different from methods that make use of algebraic concepts such as Steiner triple systems, starters and all other existing methods. We also show that certain complete ...

A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is bound λ(a) on the lengths such factorizations. The factorization property one most basic finiteness properties in study non-unique Every commutative noetherian domain factorizations, but it open whether result holds noncommutative setting. We provide sufficient conditions for pr...

Abstract We consider bivariate polynomials over the skew field of quaternions, where indeterminates commute with all coefficients and each other. analyze existence univariate factorizations, that is, factorizations linear factors. A necessary condition for is factorization norm polynomial into a product polynomials. This however, not sufficient. Our central result states exist after multiplicat...