Related papers: Normalization of Rings
We obtain two new algorithms for partial fraction decompositions; the first is over algebraically closed fields, and the second is over general fields. These algorithms takes $O(M^2)$ time, where $M$ is the degree of the denominator of the…
The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of…
This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis…
In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \leq 5$. Further we give some connections…
A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a…
Considering finite extensions K[A] \subseteq K[B] of positive affine semigroup rings over a field K we have developed in [1] an algorithm to decompose K[B] as a direct sum of monomial ideals in K[A]. By computing the regularity of…
We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…
Iterative regularization is a classic idea in regularization theory, that has recently become popular in machine learning. On the one hand, it allows to design efficient algorithms controlling at the same time numerical and statistical…
This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial…
A modified Gauss's algorithm for solving a system of linear equations in an integral ring is proposed, as well as an appropriate algorithm for calculating the elements of the adjoint matrix.
This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from…
An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a…
Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank…
This paper introduces the theory and hardware implementation of two new algorithms for computing a single component of the discrete Fourier transform. In terms of multiplicative complexity, both algorithms are more efficient, in general,…
In this note, we show that every Noetherian graded ring with an affine degree zero part is affine. As a result, a Noetherian graded Hopf algebra whose degree zero component is a commutative or a cocommutative Hopf subalgebra is affine.…
The ring of finite ad\`eles $\Af$ of the rational numbers $\Q$ is obtained in this article as a completion of $\Q$ with respect to a certain non--Archimedean metric. This ultrametric allows to represent any finite ad\`ele as a series…
Given a reduced affine algebra A over a perfect field K, we present parallel algorithms to compute the normalization \bar{A} of A. Our starting point is the algorithm of Greuel, Laplagne, and Seelisch, which is an improvement of de Jong's…
A method for computing singular or nearly singular integrals on closed surfaces was presented by J. T. Beale, W. Ying, and J. R. Wilson [Comm. Comput. Phys. 20 (2016), 733--753, arXiv:1508.00265] and applied to single and double layer…
The concept of integral as an inverse to that of derivation was already introduced for rings and recently also for lattices. Since semirings generalize both rings and bounded distributive lattices, it is natural to investigate integration…
Given a non-negative integer $n$, we establish a formula for the number of finite magmas on a set with cardinality $n$ up to isomorphism. We then generalize the method to operations with arbitrary finite arity, which yields a corrected…