A Network-Flow Feasibility Theorem and Combinatorial Applications

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.

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On γ-Sets in Rings

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i1.3873 ◽

2021 ◽

Vol 14 (1) ◽

pp. 314-326

Author(s):

Eva Jenny C. Sigasig ◽

Cristoper John S. Rosero ◽

Michael Jr. Patula Baldado

Keyword(s):

Lower Bound ◽

Division Ring ◽

Upper Bound ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Minimum Cardinality ◽

Division Rings ◽

Necessary And Sufficient

Let R be a ring with identity 1R. A subset J of R is called a γ-set if for every a ∈ R\J,there exist b, c ∈ J such that a+b = 0 and ac = 1R = ca. A γ-set of minimum cardinality is called a minimum γ-set. In this study, we identified some elements of R that are necessarily in a γ-sets, and we presented a method of constructing a new γ-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique γ-set; an upper bound for the total number of minimum γ-sets in a division ring; a lower bound for the total number of minimum γ-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial γ-set; necessary and sufficient conditions for an image of a γ-set to be a γ-set also; necessary and sufficient conditions for a ring to have a trivial γ-set; and, necessary and sufficient conditions for the families of γ-sets of two division rings to be isomorphic.

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A strict upper bound for size multipartite Ramsey numbers of paths versus stars

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2017.1.2.2 ◽

2017 ◽

Vol 1 (2) ◽

pp. 9

Author(s):

Chula Jayawardene

Keyword(s):

Natural Number ◽

Complete Graph ◽

Upper Bound ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Infinite Class ◽

Ramsey Numbers ◽

Necessary And Sufficient

Let $P_n$ represent the path of size $n$. Let $K_{1,m-1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a copy of $P_n$ in the first color or a copy of $S_{m+1}$ in the second color. The size Ramsey multipartite number $m_j(P_n, S_{m+1})$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (P_n,S_{m+1})$. Given $j,n,m$ if $s=\left\lceil \dfrac{n+m-1-k}{j-1} \right\rceil$, in this paper, we show that the size Ramsey numbers $m_j(P_n,S_{m+1})$ is bounded above by $s$ for $k=\left\lceil \dfrac{n-1}{j} \right\rceil$. Given $j\ge 3$ and $s$, we will obtain an infinite class $(n,m)$ that achieves this upper bound $s$. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.

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On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

Mathematica Slovaca ◽

10.1515/ms-2021-0059 ◽

2021 ◽

Vol 71 (6) ◽

pp. 1375-1400

Author(s):

Feyzi Başar ◽

Hadi Roopaei

Keyword(s):

Lower Bound ◽

Sequence Space ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Matrix Transformations ◽

Infinite Matrix ◽

The Matrix ◽

Alpha Beta ◽

Necessary And Sufficient ◽

Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

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The Direct Decomposition of l-algebras into Products of Subdirectly Irreducible Factors

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700003463 ◽

2003 ◽

Vol 75 (1) ◽

pp. 41-56 ◽

Cited By ~ 2

Author(s):

Sándor Radeleczki

Keyword(s):

Direct Product ◽

Present Author ◽

Sufficient Conditions ◽

Direct Decomposition ◽

Necessary And Sufficient Conditions ◽

Subdirectly Irreducible ◽

Bounded Lattice ◽

Structure Theorems ◽

Necessary And Sufficient

AbstractGeneralizing earlier results of Katriňák, El-Assar and the present author we prove new structure theorems for l-algebras. We obtain necessary and sufficient conditions for the decomposition of an arbitrary bounded lattice into a direct product of (finitely) subdirectly irreducible lattices.

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Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

Mathematics ◽

10.3390/math10020247 ◽

2022 ◽

Vol 10 (2) ◽

pp. 247

Author(s):

Kai An Sim ◽

Kok Bin Wong

Keyword(s):

Positive Integer ◽

Arithmetic Progression ◽

Upper Bound ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Arithmetic Progressions ◽

Minimum Number ◽

Necessary And Sufficient

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

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UPPER BOUNDS IN THE OHTSUKI–RILEY–SAKUMA PARTIAL ORDER ON 2-BRIDGE KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512500848 ◽

2012 ◽

Vol 21 (09) ◽

pp. 1250084 ◽

Cited By ~ 3

Author(s):

SCOTT M. GARRABRANT ◽

JIM HOSTE ◽

PATRICK D. SHANAHAN

Keyword(s):

Partial Order ◽

Upper Bound ◽

Continued Fractions ◽

Sufficient Conditions ◽

Upper Bounds ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K1 we characterize all other 2-bridge knots K2 such that {K1, K2} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.

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On size multipartite Ramsey numbers for stars

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2019.3.2.4 ◽

2020 ◽

Vol 3 (2) ◽

pp. 109

Author(s):

Anie Lusiani ◽

Edy Tri Baskoro ◽

Suhadi Wido Saputro

Keyword(s):

Lower Bound ◽

Disjoint Union ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Ramsey Number ◽

Complete Graphs ◽

Ramsey Numbers ◽

Simple Graphs ◽

Multipartite Graphs ◽

Necessary And Sufficient

Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also determine the lower bound of m3(kK1,m, C3), where kK1,m is a disjoint union of k copies of a star K1,m and C3 is a cycle of order 3.

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On the Borel-Cantelli Problem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-054-2 ◽

1970 ◽

Vol 13 (2) ◽

pp. 273-275 ◽

Cited By ~ 5

Author(s):

Jonathan Shuster

Keyword(s):

Lower Bound ◽

Probability Space ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient

Let (Ω, , P) be a probability space, and A1, A2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.In this paper, necessary and sufficient conditions will be given for infinitely many events Ak to occur, with probability 1. A lower bound for the probability that only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions that only finitely many Ak occur, with probability 1, are given.

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Robustness of Interval Monge Matrices in Fuzzy Algebra

Mathematics ◽

10.3390/math8040652 ◽

2020 ◽

Vol 8 (4) ◽

pp. 652

Author(s):

Máté Hireš ◽

Monika Molnárová ◽

Peter Drotár

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Sufficient Conditions ◽

Interval Matrix ◽

Monge Matrix ◽

Fuzzy Algebra ◽

Inexact Data ◽

Necessary And Sufficient ◽

Equivalent Conditions ◽

Extremal Algebra

Max–min algebra (called also fuzzy algebra) is an extremal algebra with operations maximum and minimum. In this paper, we study the robustness of Monge matrices with inexact data over max–min algebra. A matrix with inexact data (also called interval matrix) is a set of matrices given by a lower bound matrix and an upper bound matrix. An interval Monge matrix is the set of all Monge matrices from an interval matrix with Monge lower and upper bound matrices. There are two possibilities to define the robustness of an interval matrix. First, the possible robustness, if there is at least one robust matrix. Second, universal robustness, if all matrices are robust in the considered set of matrices. We found necessary and sufficient conditions for universal robustness in cases when the lower bound matrix is trivial. Moreover, we proved necessary conditions for possible robustness and equivalent conditions for universal robustness in cases where the lower bound matrix is non-trivial.

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The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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