scholarly journals Minimal presentations of shifted numerical monoids

2018 ◽  
Vol 28 (01) ◽  
pp. 53-68 ◽  
Author(s):  
Rebecca Conaway ◽  
Felix Gotti ◽  
Jesse Horton ◽  
Christopher O’Neill ◽  
Roberto Pelayo ◽  
...  
Keyword(s):  
Recent Result ◽  
Commutative Algebra ◽  
Shift Parameter ◽  
Numerical Monoid ◽  
Factorization Theory ◽  
The Family ◽  
Combinatorial Commutative Algebra ◽  
Minimal Presentations

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid [Formula: see text], consider the family of “shifted” monoids [Formula: see text] obtained by adding [Formula: see text] to each generator of [Formula: see text]. In this paper, we examine minimal relations among the generators of [Formula: see text] when [Formula: see text] is sufficiently large, culminating in a description that is periodic in the shift parameter [Formula: see text]. We explore several applications to computation and factorization theory, and improve a recent result of Thanh Vu from combinatorial commutative algebra.

scholarly journals ON THE FAMILY OF DIOPHANTINE TRIPLES {k− 1,k+ 1, 16k3− 4k}

2007 ◽  
Vol 49 (2) ◽  
pp. 333-344 ◽  
Author(s):  
YANN BUGEAUD ◽  
ANDREJ DUJELLA ◽  
MAURICE MIGNOTTE
Keyword(s):  
Positive Integer ◽  
Recent Result ◽  
The Family ◽  
Diophantine Triples ◽  
Image Position ◽  
Diophantine Quadruples

AbstractIt is proven that ifk≥ 2 is an integer anddis a positive integer such that the product of any two distinct elements of the setincreased by 1 is a perfect square, thend= 4kord= 64k5−48k3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k− 1,k+ 1,c,d} are regular.

Combinatorial Commutative Algebra

2004 ◽  
pp. 1703-1746
Author(s):  
Irena Peeva ◽  
Volkmar Welker
Keyword(s):  
Commutative Algebra ◽  
Combinatorial Commutative Algebra

scholarly journals The computation of factorization invariants for affine semigroups

2019 ◽  
Vol 18 (01) ◽  
pp. 1950019 ◽  
Author(s):  
Pedro A. García-Sánchez ◽  
Christopher O’Neill ◽  
Gautam Webb
Keyword(s):  
Commutative Algebra ◽  
Improved Method ◽  
Dynamic Algorithm ◽  
Delta Set ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Combinatorial Commutative Algebra ◽  
Theoretical Results ◽  
New Algorithms ◽  
Standard Techniques

We present several new algorithms for computing factorization invariant values over affine semigroups. In particular, we give (i) the first known algorithm to compute the delta set of any affine semigroup, (ii) an improved method of computing the tame degree of an affine semigroup, and (iii) a dynamic algorithm to compute catenary degrees of affine semigroup elements. Our algorithms rely on theoretical results from combinatorial commutative algebra involving Gröbner bases, Hilbert bases, and other standard techniques. Implementation in the computer algebra system GAP is discussed.

scholarly journals The relevance of Freiman’s theorem for combinatorial commutative algebra

2018 ◽  
Vol 291 (3-4) ◽  
pp. 999-1014 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Guangjun Zhu
Keyword(s):  
Commutative Algebra ◽  
Combinatorial Commutative Algebra ◽  
Freiman’S Theorem

scholarly journals Split Lie–Rinehart algebras

2020 ◽  
pp. 2150164
Author(s):  
Helena Albuquerque ◽  
Elisabete Barreiro ◽  
A. J. Calderón ◽  
José M. Sánchez
Keyword(s):  
Lie Algebras ◽  
Commutative Algebra ◽  
Natural Extension ◽  
Nonzero Ideal ◽  
Direct Sums ◽  
Mild Conditions ◽  
The Family ◽  
The One

We introduce the class of split Lie–Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if [Formula: see text] is a tight split Lie–Rinehart algebra over an associative and commutative algebra [Formula: see text] then [Formula: see text] and [Formula: see text] decompose as the orthogonal direct sums [Formula: see text] and [Formula: see text], where any [Formula: see text] is a nonzero ideal of [Formula: see text], any [Formula: see text] is a nonzero ideal of [Formula: see text], and both decompositions satisfy that for any [Formula: see text], there exists a unique [Formula: see text] such that [Formula: see text]. Furthermore, any [Formula: see text] is a split Lie–Rinehart algebra over [Formula: see text]. Also, under mild conditions, it is shown that the above decompositions of [Formula: see text] and [Formula: see text] are by means of the family of their, respective, simple ideals.

The General Form of γ-Family of Quantum Relative Entropies

2011 ◽  
Vol 18 (02) ◽  
pp. 191-221
Author(s):  
Ryszard Paweł Kostecki
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Recent Result ◽  
Quantum Information Theory ◽  
Fenchel Duality ◽  
Commutative Case ◽  
Infinite Dimensional ◽  
Entropy Functionals ◽  
The Family

We use the Falcone–Takesaki non-commutative flow of weights and the resulting theory of non-commutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and classical relative entropies of Zhu–Rohwer, and belong to an intersection of families of Petz relative entropies with Bregman relative entropies. For the purpose of this task, we generalise the notion of Bregman entropy to the infinite-dimensional non-commutative case using the Legendre–Fenchel duality. In addition, we use the Falcone–Takesaki duality to extend the duality between coarse-grainings and Markov maps to the infinite-dimensional non-commutative case. Following the recent result of Amari for the Zhu–Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre–Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.

scholarly journals A fixed point theorem for a family of nonexpansive mappings

1976 ◽  
Vol 15 (1) ◽  
pp. 111-116 ◽  
Author(s):  
V.M. Sehgal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Vector Space ◽  
Recent Result ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Bounded Subset ◽  
The Family ◽  
Commutative Family ◽  
Locally Convex

Let E be a separated, locally convex topological vector space and F a commutative family of nonexpansive mappings defined on a quasi-complete convex (not necessarily bounded) subset X of E. In this paper, it is proved that if one of the mappings in F is condensing with a bounded range then the family F has a common fixed point in X. This result improves several well-known results and supplements a recent result of E. Tarafdar (Bull. Austral. Math. Soc. 13 (1975), 241–254) for such mappings.

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