2 Integral Domains 12 2.1 Factorization in Integral Domains . . . . . . . . . . . . . . . . 12 2.2 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fermat’s Two Squares Theorem . . . . . . . . . . . . . . . . . 17 2.5 Maximal Ideals and Prime Ideals . . . . . . . . . . . . . . . . 20 2.6 Unique Fact...

Pietsch Factorization and Grothendieck Factorization are the two landmark theorems in modern functional analysis. They were first introduced to the numerical linear algebra community by the work of Joel A. Tropp in the column subset selection problem, which seeks to extract from a matrix a column submatrix that has lower spectral norm. Despite their broad application in functional analysis, the...

We describe mechanisms to emulate multiplicative properties of prime numbers by connecting them to computationally simpler representations involving bijections from natural numbers to sequences and multisets of natural numbers. Through a generic framework, we model interesting properties of a canonical representation of factorization and its simpler equivalents. Among their shared properties, w...

‎the number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving gaussian binomial coefficients is presented‎.

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, ...

nonnegative matrix factorization (nmf) is a common method in data mining that have been used in different applications as a dimension reduction, classification or clustering method. methods in alternating least square (als) approach usually used to solve this non-convex minimization problem.  at each step of als algorithms two convex least square problems should be solved, which causes high com...

this paper presents a modified digital image watermarking method based on nonnegative matrix factorization. firstly, host image is factorized to the product of three nonnegative matrices. then, the centric matrix is transferred to discrete cosine transform domain. watermark is embedded in low frequency band of this matrix and next, the reverse of the transform is computed. finally, watermarked ...

We present a distributed stochastic gradient descent algorithm for performing low-rank matrix factorization on streaming data. Low-rank matrix factorization is often used as a technique for collaborative filtering. As opposed to recent algorithms that perform matrix factorization in parallel on a batch of training examples [4], our algorithm operates on a stream of incoming examples. We experim...

Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not coobrantly generated. We also present a weak factorization system on the category of posets which is not coobrantly generated. No s...

The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) “reflects” factorization systems. In particular, a generalization of the well-known Cassidy-HébertKelly factorization theorem is given. The problem of relat...