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.

scholarly journals ON HYBRID SEQUENCES BUILT FROM NIEDERREITER–HALTON SEQUENCES AND KRONECKER SEQUENCES

2011 ◽  
Vol 84 (2) ◽  
pp. 238-254 ◽  
Author(s):  
ROSWITHA HOFER ◽  
PETER KRITZER
Keyword(s):  
Uniform Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Future Research ◽  
Short Summary ◽  
The One ◽  
Halton Sequences ◽  
Necessary And Sufficient

AbstractWe discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

Answerability

2018 ◽  
Author(s):  
Natalie Stoljar
Keyword(s):  
Moral Responsibility ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Alternative Approach ◽  
The One ◽  
Necessary And Sufficient

This chapter defends externalist or “constitutively relational” conceptions of autonomy through an examination of an alternative approach developed by Andrea Westlund. Westlund develops her approach in response to what has been called the “agency dilemma.” On the one hand, constraining external circumstances seem to undermine autonomy; on the other, the claim that people are nonautonomous because of their circumstances seems to erase their agency and disrespect their evaluative commitments. This chapter distinguishes the necessary and sufficient conditions of several interrelated aspects of agency: autonomy, authentic agential perspective, and moral responsibility. I argue that whereas answerability may be sufficient for moral responsibility, it is not sufficient for autonomy. Objections to externalist conceptions of autonomy, including the agency dilemma, wrongly assume that denying autonomy implies erasing agency. Once it is recognized that autonomy does not always overlap with authentic agential perspective or moral responsibility, the objections lose their force.

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.

Paternalism and the Justification of Compulsory Education

1983 ◽  
Vol 27 (2) ◽  
pp. 137-150 ◽  
Author(s):  
Peter Hobson
Keyword(s):  
Sufficient Conditions ◽  
Compulsory Education ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Welfare Of The Child ◽  
The One ◽  
Necessary And Sufficient

This paper examines at a philosophical level the justification for making education compulsory, concentrating on those arguments based on the welfare of the child. These arguments rest on the principle of paternalism which, as a form of interference with a person's liberty, is distinguished from such interferences based on concern for the welfare of society on the one hand and with upholding a moral or religious principle on the other. After noting the dangers involved in the use of paternalism, possible ways of determining when it is justified are examined prior to presenting what this author holds to be the two necessary and sufficient conditions for justified paternalism. After some discussion and amplification of these conditions, they are applied to the case of compulsory education. It is found that they are satisfied in this case but only with regard to education up to a certain level. The paper ends by attempting to answer some possible objections to the argument presented.

The Changing Temper of American Philosophy

1972 ◽  
Vol 34 (4) ◽  
pp. 129-137
Author(s):  
C. F. Delaney
Keyword(s):  
Present State ◽  
Cultural Revolution ◽  
Major Part ◽  
Historical Context ◽  
Sufficient Conditions ◽  
The Other ◽  
The Third ◽  
American Philosophy ◽  
The One ◽  
Necessary And Sufficient

When the question of a “new age” is put to a philosopher there are two quite different ways in which he can define the issue and accordingly respond. On the one hand, he might construe his task as that of a clarifier of some general claim about cultural revolutions. From this perspective he would set about the task of analyzing the concept of a cultural revolution in terms of some loose analogue of necessary and sufficient conditions. These criteria having been laid down, he could then make suggestions as to whether or not those conditions obtain which would justify the claim that we are in the midst of such a cultural revolution. On the other hand, he might construe his charge more specifically as that of assessing the present state of his own field to see if something like a shift in perspective is manifest in this narrower domain. In this paper I am going to take the latter tack. I will, first, briefly survey the contemporary scene in philosophy to illustrate the “changing temper” I am to talk about; secondly, locate these phenomena in a broader historical context; and, thirdly, try to get at the reasons underlying the changes on which I am focusing. The major part of the paper will be devoted to the third point.

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].

Does Global Justice Require More than Just Global Institutions?

2016 ◽  
Vol 3 (1) ◽  
pp. 19-31
Author(s):  
Kok-Chor Tan
Keyword(s):  
Global Justice ◽  
The Other ◽  
Global Institutions ◽  
Institutional Approach ◽  
The One ◽  
Necessary And Sufficient ◽  
Strong Reading

The ‘institutional approach’ to justice holds that persons’ responsibility of justice is primarily to support, maintain, and comply with the rules of just institutions. Within the rules of just institutions, so long as their actions do not undermine these background institutions, individuals have no further responsibilities of justice. But what does the institutional approach say in the non-ideal context where just institutions are absent, such as in the global case? One reading of the institutional approach, in this case, is that our duties are primarily to create just institutions, and that when we are doing our part in this respect, we may legitimately pursue other personal or associational ends. This ‘strong’ reading of our institutional duty takes it to be both a necessary and sufficient duty of justice of individuals that they do their part to establish just arrangements. But how plausible is this? On the one hand this requirement seems overly inflexible; on the other it seems overly lax. I clarify the motivation and context of this reading of the institutional duty, and suggest that it need not be as implausible as it seems.

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).

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Equipartitioning and balancing points of polygons

Pythagoras ◽  
2010 ◽  
Vol 0 (71) ◽  
Author(s):  
Shunmugam Pillay ◽  
Poobhalan Pillay
Keyword(s):  
General Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Sufficient Condition ◽  
Centre Of Mass ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The centre of mass G of a triangle has the property that the rays to the vertices from G sweep out triangles having equal areas. We show that such points, termed equipartitioning points in this paper, need not exist in other polygons. A necessary and sufficient condition for a quadrilateral to have an equipartitioning point is that one of its diagonals bisects the other. The general theorem, namely, necessary and sufficient conditions for equipartitioning points for arbitrary polygons to exist, is also stated and proved. When this happens, they are in general, distinct from the centre of mass. In parallelograms, and only in them, do the two points coincide.

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