scholarly journals Generalized Strongly Increasing Semigroups

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1370
Author(s):  
E. R. García Barroso ◽  
J. I. García-García ◽  
A. Vigneron-Tenorio
Keyword(s):  
Frobenius Number ◽  
Numerical Semigroups ◽  
New Class ◽  
Frobenius Numbers ◽  
Positive Integers

In this work, we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and other families of semigroups and we explicitly give their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied.

scholarly journals The genus, Frobenius number and pseudo-Frobenius numbers of numerical semigroups of type 2

2016 ◽  
Vol 146 (5) ◽  
pp. 1081-1090
Author(s):  
Aureliano M. Robles-Pérez ◽  
José Carlos Rosales
Keyword(s):  
Numerical Semigroup ◽  
Sufficient Conditions ◽  
Frobenius Number ◽  
The Other ◽  
Numerical Semigroups ◽  
Frobenius Numbers ◽  
The One ◽  
Necessary And Sufficient ◽  
Positive Integers

We study some questions on numerical semigroups of type 2. On the one hand, we investigate the relation between the genus and the Frobenius number. On the other hand, for two fixed positive integers g1, g2, we give necessary and sufficient conditions in order to have a numerical semigroup S such that {g1, g2} is the set of its pseudo-Frobenius numbers and, moreover, we explicitly build families of such numerical semigroups.

The Frobenius problem for a class of numerical semigroups

2017 ◽  
Vol 13 (05) ◽  
pp. 1335-1347 ◽  
Author(s):  
Ze Gu ◽  
Xilin Tang
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Embedding Dimension ◽  
Frobenius Problem ◽  
Positive Integers

Let [Formula: see text] be two positive integers such that [Formula: see text] and [Formula: see text] the numerical semigroup generated by [Formula: see text]. Then [Formula: see text] is the Thabit numerical semigroup introduced by J. C. Rosales, M. B. Branco and D. Torrão. In this paper, we give formulas for computing the Frobenius number, the genus and the embedding dimension of [Formula: see text].

On the numerical semigroup generated by {bn+1+i+bn+i−1b−1∣i∈N}$\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$

2020 ◽  
Vol 30 (4) ◽  
pp. 257-264
Author(s):  
Ze Gu
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Embedding Dimension ◽  
Order Relations ◽  
Positive Integers

AbstractLet b, n be two positive integers such that b ≥ 2, and S(b, n) be the numerical semigroup generated by $\begin{array}{} \{b^{n+1+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\} \end{array}$. Applying two order relations, we give formulas for computing the embedding dimension, the Frobenius number, the type and the genus of S(b, n).

COUNTING NUMERICAL SEMIGROUPS WITH SHORT GENERATING FUNCTIONS

2011 ◽  
Vol 21 (07) ◽  
pp. 1217-1235 ◽  
Author(s):  
VÍCTOR BLANCO ◽  
PEDRO A. GARCÍA-SÁNCHEZ ◽  
JUSTO PUERTO
Keyword(s):  
Generating Function ◽  
Polynomial Time ◽  
Generating Functions ◽  
Time Complexity ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Polynomial Time Complexity ◽  
Theoretical Results

This paper presents a new methodology to compute the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.

scholarly journals Minimal genus of a multiple and Frobenius number of a quotient of a numerical semigroup

2015 ◽  
Vol 25 (06) ◽  
pp. 1043-1053 ◽  
Author(s):  
Francesco Strazzanti
Keyword(s):  
Positive Integer ◽  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Minimal Genus

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is [Formula: see text], where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

scholarly journals Almost symmetric numerical semigroups with given Frobenius number and type

2019 ◽  
Vol 18 (11) ◽  
pp. 1950217
Author(s):  
M. B. Branco ◽  
I. Ojeda ◽  
J. C. Rosales
Keyword(s):  
Frobenius Number ◽  
Numerical Semigroups ◽  
Symmetric Numerical Semigroups

We give two algorithmic procedures to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number and type, and the whole set of almost symmetric numerical semigroups with fixed Frobenius number. Our algorithms allow to compute the whole set of almost symmetric numerical semigroups with fixed Frobenius number with similar or even higher efficiency that the known ones. They have been implemented in the GAP [The GAP Group, GAP — Groups, Algorithms and Programming, Version 4.8.6; 2016, https://www.gap-system.org ] package NumericalSgps [M. Delgado and P. A. García-Sánchez and J. Morais, “numericalsgps”: A GAP package on numerical semigroups, https://github.com/gap-packages/numericalsgps ].

scholarly journals Probability of ruin in discrete insurance risk model with dependent Pareto claims

2019 ◽  
Vol 7 (1) ◽  
pp. 215-233
Author(s):  
Corina D. Constantinescu ◽  
Tomasz J. Kozubowski ◽  
Haoyu H. Qian
Keyword(s):  
Power Law ◽  
Pareto Distribution ◽  
Risk Model ◽  
Probability Distributions ◽  
Insurance Risk ◽  
Probability Of Ruin ◽  
New Class ◽  
Compound Binomial ◽  
Compound Binomial Risk Model ◽  
Positive Integers

AbstractWe present basic properties and discuss potential insurance applications of a new class of probability distributions on positive integers with power law tails. The distributions in this class are zero-inflated discrete counterparts of the Pareto distribution. In particular, we obtain the probability of ruin in the compound binomial risk model where the claims are zero-inflated discrete Pareto distributed and correlated by mixture.

scholarly journals Parametrizing Arf numerical semigroups

2017 ◽  
Vol 16 (11) ◽  
pp. 1750209 ◽  
Author(s):  
P. A. García-Sánchez ◽  
B. A. Heredia ◽  
H. İ. Karakaş ◽  
J. C. Rosales
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Arf Numerical Semigroup

We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to 7.

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