scholarly journals Infinite sequences of almost symmetric non-Weierstrass numerical semigroups

2021 ◽  
Author(s):  
Jiryo Komeda
Keyword(s):  
Positive Integer ◽  
Infinite Number ◽  
Negative Integer ◽  
Numerical Semigroups ◽  
Infinite Sequences

AbstractLet u be any positive integer. We construct infinite sequences of almost symmetric non-Weierstrass numerical semigroups whose conductors are the genera double minus $$2u-1$$ 2 u - 1 . Moreover, let v be any non-negative integer. We give an infinite number of non-Weierstrass numerical semigroups whose conductors are the genera double minus 2v.

scholarly journals Reachability Relations and the Structure of Transitive Digraphs

10.37236/115 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Norbert Seifter ◽  
Vladimir I. Trofimov
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Equivalence Relations ◽  
Basic Relation ◽  
Factor Graphs ◽  
Locally Finite ◽  
General Properties

In this paper we investigate reachability relations on the vertices of digraphs. If $W$ is a walk in a digraph $D$, then the height of $W$ is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed opposite to their orientation. Two vertices $u,v\in V(D)$ are $R_{a,b}$-related if there exists a walk of height $0$ between $u$ and $v$ such that the height of every subwalk of $W$, starting at $u$, is contained in the interval $[a,b]$, where $a$ ia a non-positive integer or $a=-\infty$ and $b$ is a non-negative integer or $b=\infty$. Of course the relations $R_{a,b}$ are equivalence relations on $V(D)$. Factorising digraphs by $R_{a,\infty}$ and $R_{-\infty,b}$, respectively, we can only obtain a few different digraphs. Depending upon these factor graphs with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ it is possible to define five different "basic relation-properties" for $R_{-\infty,b}$ and $R_{a,\infty}$, respectively. Besides proving general properties of the relations $R_{a,b}$, we investigate the question which of the "basic relation-properties" with respect to $R_{-\infty,b}$ and $R_{a,\infty}$ can occur simultaneously in locally finite connected transitive digraphs. Furthermore we investigate these properties for some particular subclasses of locally finite connected transitive digraphs such as Cayley digraphs, digraphs with one, with two or with infinitely many ends, digraphs containing or not containing certain directed subtrees, and highly arc transitive digraphs.

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.

Multidimensional right-shift processes

1971 ◽  
Vol 3 (02) ◽  
pp. 200-201 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Positive Integer ◽  
Random Variable ◽  
Negative Integer

Let v be a positive integer and for each k = 1, · · ·, v let m k and N k be a positive and a non-negative integer, respectively. Denote by S'N k ,m k the set of (m k + 1) -tuples r k = (r k ,m k , · · ·, r k,1, r k,0) having non-negative components summing to N k , and by X k (t) = (X k,m k (t), · · ·, X k,1(t), X k ,0(t)) an (m k + 1)-tuple random variable taking on values only from the set S′ N k ,m k .

A physical model for teaching multiplication of integers

1968 ◽  
Vol 15 (6) ◽  
pp. 525-528
Author(s):  
Warren H. Hill
Keyword(s):  
Positive Integer ◽  
Physical Model ◽  
Number Line ◽  
Negative Integer ◽  
Physical Models ◽  
Addition And Subtraction ◽  
Positive Integers ◽  
Partial Success

One of the pedagogical pitfalls involved in teaching the multiplication of integers is the scarcity of physical models that can be used to illustrate their product. This problem becomes even more acute if a phyiscal model that makes use of a number line is desired. If the students have previously encountered the operations of addition and subtraction of integers using a number line, then the desirability of developing the operation of multiplication in a similar setting is obvious. Many of the more common physical models that can be used to represent the product of positive and negative integers are unsuitable for interpretation on a number line. Partial success is possible in the cases of the product of two positive integers and the product of a positive and a negative integer. But, how does one illustrate that the product of two negative integers is equal to a positive integer?

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

The “unknotting number” associated with other local moves than the crossing-change

2018 ◽  
Vol 27 (10) ◽  
pp. 1850051
Author(s):  
Eiji Ogasa
Keyword(s):  
Positive Integer ◽  
Negative Integer ◽  
Positive Integers

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let [Formula: see text] be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on [Formula: see text]-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is [Formula: see text]. For any positive integer [Formula: see text], there is a 1-knot whose “unknotting number” associated with the pass-move is [Formula: see text] and whose ordinary unknotting number is [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers. For the [Formula: see text]-move on [Formula: see text]-knots, which is a local move on knots, we have the following: Let [Formula: see text] be a non-negative integer. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one time. The “unknotting number” associated with the [Formula: see text]-move is unbounded. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one. The “unknotting number” associated with the [Formula: see text]-move is unbounded. We prove the following: For any positive integer [Formula: see text] and any positive integer [Formula: see text], there is a [Formula: see text]-knot which is changed into the unknot by [Formula: see text] times of the twist-move not by [Formula: see text] times.

Asymptotic behaviour of the fundamental solution to ∂ u /∂ t = — ( — ∆) m u

1993 ◽  
Vol 441 (1912) ◽  
pp. 423-432 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Infinite Number ◽  
Fourier Integral ◽  
Number Of Zeros

An asymptotic expansion is derived for the Fourier integral f ^ ( x ) = 1 ( 2 π ) n / 2 ∫ R n exp ( − | q | 2 m + i x ⋅ q ) d q , x ε R n as | x | →∞, where m is a positive integer. From this, it is deduced that the fundamental solution to the ‘heat’ equation ∂ u / ∂ t = − ( − Δ ) m u has an infinite number of zeros tending to infinity.

Multidimensional right-shift processes

1971 ◽  
Vol 3 (2) ◽  
pp. 200-201 ◽  
Author(s):  
Norman C. Severo
Keyword(s):  
Positive Integer ◽  
Random Variable ◽  
Negative Integer

Let v be a positive integer and for each k = 1, · · ·, v let mk and Nk be a positive and a non-negative integer, respectively. Denote by S'Nk,mk the set of (mk + 1) -tuples rk = (rk,mk, · · ·, rk,1, rk,0) having non-negative components summing to Nk, and by Xk(t) = (Xk,mk(t), · · ·, Xk,1(t), Xk,0(t)) an (mk + 1)-tuple random variable taking on values only from the set S′Nk,mk.

scholarly journals On the Diophantine Equation 8^x+n^y=z^2

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Integer Solution ◽  
Negative Integer

Let n be an positive integer with n = 10(mod15). In this paper, we prove that (1,0,3) is unique non negative integer solution (x,y,z) of the Diophantine equation 8^x+n^y=z^2 where x y, and z are non-negativeintegers.

On Positive Integer Solutions of the Equation xy + yz + xz = n

1996 ◽  
Vol 39 (2) ◽  
pp. 199-202 ◽  
Author(s):  
Al-Zaid Hassan ◽  
B. Brindza ◽  
Á. Pintér
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Class Number ◽  
Quadratic Forms ◽  
Negative Integer ◽  
Number Of Solutions ◽  
Integer Solutions

AbstractAs it had been recognized by Liouville, Hermite, Mordell and others, the number of non-negative integer solutions of the equation in the title is strongly related to the class number of quadratic forms with discriminant —n. The purpose of this note is to point out a deeper relation which makes it possible to derive a reasonable upper bound for the number of solutions.

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