GREATEST COMMON DIVISORS OF EUCLIDEAN DOMAIN MATRICES

2017 ◽  
Vol 101 (4) ◽  
pp. 855-872
Author(s):  
Nur Erawaty ◽  
Mawardi Bahri ◽  
Loecky Haryanto ◽  
Amir Kamal Amir
Keyword(s):  
Euclidean Domain ◽  
Greatest Common Divisors

scholarly journals Semi-Supervised Classification via Hypergraph Convolutional Extreme Learning Machine

2021 ◽  
Vol 11 (9) ◽  
pp. 3867
Author(s):  
Zhewei Liu ◽  
Zijia Zhang ◽  
Yaoming Cai ◽  
Yilin Miao ◽  
Zhikun Chen
Keyword(s):  
Extreme Learning Machine ◽  
Supervised Classification ◽  
Structured Data ◽  
Model Complex ◽  
Euclidean Domain ◽  
Noise Data ◽  
Data Points ◽  
Learning Machine ◽  
Random Hypergraph ◽  
Special Case

Extreme Learning Machine (ELM) is characterized by simplicity, generalization ability, and computational efficiency. However, previous ELMs fail to consider the inherent high-order relationship among data points, resulting in being powerless on structured data and poor robustness on noise data. This paper presents a novel semi-supervised ELM, termed Hypergraph Convolutional ELM (HGCELM), based on using hypergraph convolution to extend ELM into the non-Euclidean domain. The method inherits all the advantages from ELM, and consists of a random hypergraph convolutional layer followed by a hypergraph convolutional regression layer, enabling it to model complex intraclass variations. We show that the traditional ELM is a special case of the HGCELM model in the regular Euclidean domain. Extensive experimental results show that HGCELM remarkably outperforms eight competitive methods on 26 classification benchmarks.

scholarly journals Weighted greatest common divisors and weighted heights

2020 ◽  
Vol 213 ◽  
pp. 319-346
Author(s):  
L. Beshaj ◽  
J. Gutierrez ◽  
T. Shaska
Keyword(s):  
Greatest Common Divisors

Some Interesting Mathematics about Prime Factorization

1975 ◽  
Vol 68 (1) ◽  
pp. 73-74
Author(s):  
Joe Tom Rodgers
Keyword(s):  
Prime Factorization ◽  
Factoring Polynomials ◽  
Greatest Common Divisors ◽  
School Careers

Throughout their school careers, students are taught how to find the prime factorization of numbers. Most students readily learn the technique and find the process interesting. Most teachers then teach the ways in which prime factorizations can aid in factoring polynomials and finding greatest common divisors as well as least common multiples.

scholarly journals Operator Product Expansion, Heavy Quarks, QCD Duality and Its Violations

1997 ◽  
Vol 12 (11) ◽  
pp. 2075-2133 ◽  
Author(s):  
B. Chibisov ◽  
R. D. Dikeman ◽  
M. Shifman ◽  
N. G. Uraltsev
Keyword(s):  
Operator Product Expansion ◽  
Background Field ◽  
Decay Rates ◽  
Operator Product ◽  
Theoretical Treatment ◽  
Heavy Flavor ◽  
Wide Range ◽  
Euclidean Domain ◽  
Inclusive Processes ◽  
Hadron Duality

The quark(gluon)–hadron duality constitutes a basis for the theoretical treatment of a wide range of inclusive processes — from hadronic τ decays and Re+e- to semileptonic and nonleptonic decay rates of heavy flavor hadrons. A theoretical analysis of these processes is carried out by using the operator product expansion in the Euclidean domain, with subsequent analytic continuation to the Minkowski domain. We formulate the notion of the quark(gluon)–hadron duality in quantitative terms, then classify various contributions leading to violations of duality. A prominent role in the violations of duality seems to belong to the so-called exponential terms which, conceptually, may represent the (truncated) tail of the power series. A qualitative model, relying on an instanton background field, is developed, allowing one to get an estimate of the exponential terms. We then discuss a number of applications, mostly from heavy quark physics.

Homomorphisms of two-dimensional linear groups over Laurent polynomial rings and Gaussian domains

1991 ◽  
Vol 109 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Yu Chen
Keyword(s):  
Commutative Ring ◽  
Linear Group ◽  
Unit Group ◽  
Commutative Rings ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Linear Groups ◽  
Two Dimensional ◽  
Euclidean Domain ◽  
Invertible Matrices

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

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