scholarly journals On γ-Sets in Rings

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.

A Network-Flow Feasibility Theorem and Combinatorial Applications

1959 ◽  
Vol 11 ◽  
pp. 440-451 ◽  
Author(s):  
D. R. Fulkerson
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Network Flow ◽  
Present Author ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Private Communication ◽  
Circulatory Flow ◽  
Necessary And Sufficient ◽  
Circulation Theorem

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.

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

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

<p>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.</p>

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

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.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

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

Mathematics ◽  
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.

Isomorphisms of Simple Rings

1963 ◽  
Vol 15 ◽  
pp. 467-470
Author(s):  
Ti Yen
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Minimum Condition ◽  
Compatible Pair ◽  
Simple Ring ◽  
Common Extension ◽  
Necessary And Sufficient

Let A be a simple ring with minimum condition, and B1, B2, and C be regular subrings of A such that Bi > C, i = 1, 2. A pair of isomorphisms σi of Bi into A such that σi|C is the identity, and that Biσi are regular subrings of A (i = 1, 2), is called compatible if σ1|B1 ∩ B2 = σ2|B1 ∩ B2. Here σ|X means the restriction of σ to X. Bialynicki-Birula has proved some necessary and sufficient conditions that every compatible pair (σ1, σ2) has a common extention to an automorphism σ of A (1 ). When A is a division ring, he shows that the linear disjointness of the division subrings B1 and B2 is necessary and almost sufficient for the existence of a common extension of any compatible pair.

Right Bol Quasi-Fields

1969 ◽  
Vol 21 ◽  
pp. 1409-1420
Author(s):  
Michael J. Kallaher
Keyword(s):  
Division Ring ◽  
Near Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
The Right ◽  
Necessary And Sufficient ◽  
Bol Loops

We shall consider quasi-fields which satisfy the multiplicative Identity1.1(1.1) will be called the right Bol law and a quasi-field satisfying it will be called a right Bol quasi-held. Moufang quasi-fields, i.e., those satisfying the Moufang identity1.2were studied in (5). Quasi-fields satisfying the left Bol identity1.3were studied by Burn (3) and the author (6). Such quasi-fields are called Bol quasi-fields.Our investigation will parallel the investigations in (5; 6). In § 2 we derive necessary and sufficient conditions for a right Bol quasi-field to be an alternative division ring and also criteria for it to be a near-field. With this information we derive in §§ 3 and 4 new characterizations of Moufang planes similar to those in (5; 6).Loops satisfying (1.1) have been studied by Robinson (10). He calls such loops Bol loops.

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

2012 ◽  
Vol 21 (09) ◽  
pp. 1250084 ◽  
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.

scholarly journals On size multipartite Ramsey numbers for stars

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

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

On the Borel-Cantelli Problem

1970 ◽  
Vol 13 (2) ◽  
pp. 273-275 ◽  
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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