scholarly journals ON THE PRIME DECOMPOSITION OF INTEGERS OF THE FORM zn-yn/z-y

2019 ◽  
Vol 44 (1) ◽  
pp. 41-61
Author(s):  
Rachid Marsli
Keyword(s):  
Prime Decomposition

scholarly journals Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

scholarly journals Unique prime decomposition results for factors coming from wreath product groups

2013 ◽  
Vol 265 (1) ◽  
pp. 221-232 ◽  
Author(s):  
J. Owen Sizemore ◽  
Adam Winchester
Keyword(s):  
Wreath Product ◽  
Prime Decomposition

Connections: Prime decomposition using tools

2010 ◽  
Vol 17 (4) ◽  
pp. 256-259
Author(s):  
Terri Kurz ◽  
Jorge Garcia
Keyword(s):  
Building Blocks ◽  
Prime Numbers ◽  
Prime Decomposition ◽  
Square Roots ◽  
Important Concept

Factoring numbers through multiplication using primes is an important concept for students to understand. Those in grades 3–5 should be able to find equivalent representations of the same number by de composing and composing numbers (NCTM 2000). One aspect of this essential skill is prime decomposition, which relates to such mathematical topics as simplifying, divisibility, square roots, and fractions. Prime numbers are crucial building blocks for understanding numbers and their multiplicative relationships (Zazkis and Liljedahl 2004).

scholarly journals Thinning a Triangulation of a Bayesian Network or Undirected Graph to Create a Minimal Triangulation

2017 ◽  
Vol 25 (03) ◽  
pp. 1750014
Author(s):  
Edmund Jones ◽  
Vanessa Didelez
Keyword(s):  
Bayesian Network ◽  
Graphical Model ◽  
Undirected Graph ◽  
Computer Experiment ◽  
R Package ◽  
The Other ◽  
Prime Decomposition ◽  
Original Graph ◽  
Minimal Triangulation

In one procedure for finding the maximal prime decomposition of a Bayesian network or undirected graphical model, the first step is to create a minimal triangulation of the network, and a common and straightforward way to do this is to create a triangulation that is not necessarily minimal and then thin this triangulation by removing excess edges. We show that the algorithm for thinning proposed in several previous publications is incorrect. A different version of this algorithm is available in the R package gRbase, but its correctness has not previously been proved. We prove that this version is correct and provide a simpler version, also with a proof. We compare the speed of the two corrected algorithms in three ways and find that asymptotically their speeds are the same, neither algorithm is consistently faster than the other, and in a computer experiment the algorithm used by gRbase is faster when the original graph is large, dense, and undirected, but usually slightly slower when it is directed.

Sign in / Sign up

Export Citation Format

Share Document