On a Theorem of Rav Concerning Egyptian Fractions

1975 ◽  
Vol 18 (1) ◽  
pp. 155-156 ◽  
Author(s):  
William A. Webb
Keyword(s):  
Positive Integer ◽  
Elementary Proof ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Egyptian Fractions ◽  
Image Position ◽  
Complete Bibliography ◽  
Necessary And Sufficient

Problems involving Egyptian fractions (rationals whose numerator is 1 and whose denominator is a positive integer) have been extensively studied. (See [1] for a more complete bibliography). Some of the most interesting questions, many still unsolved, concern the solvability ofwhere k is fixed.In [2] Rav proved necessary and sufficient conditions for the solvabilty of the above equation, as a consequence of some other theorems which are rather complicated in their proofs. In this note we give a short, elementary proof of this theorem, and at the same time generalize it slightly.

scholarly journals Characterizations and Identities for Isosceles Triangular Numbers

2021 ◽  
Vol 14 (2) ◽  
pp. 380-395
Author(s):  
Jiramate Punpim ◽  
Somphong Jitman
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Triangular Numbers ◽  
Triangular Number ◽  
Recursive Formulas ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Triangular numbers have been of interest and continuously studied due to their beautiful representations, nice properties, and various links with other figurate numbers. For positive integers n and l, the nth l-isosceles triangular number is a generalization of triangular numbers defined to be the arithmetic sum of the formT(n, l) = 1 + (1 + l) + (1 + 2l) + · · · + (1 + (n − 1)l).In this paper, we focus on characterizations and identities for isosceles triangular numbers as well as their links with other figurate numbers. Recursive formulas for constructions of isosceles triangular numbers are given together with necessary and sufficient conditions for a positive integer to be a sum of isosceles triangular  numbers. Various identities for isosceles triangular numbers are established. Results on triangular numbers can be viewed as a special case.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

scholarly journals On Generalizations of phi-2-Absorbing Primary Submodules

2018 ◽  
Vol 11 (1) ◽  
pp. 35
Author(s):  
Pairote Yiarayong ◽  
Manoj Siripitukdet
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Necessary And Sufficient

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or  $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.

scholarly journals Eventually Pointed Principally Ordered Regular Semigroups

2020 ◽  
Vol 24 (2) ◽  
pp. 139
Author(s):  
G.A. Pinto
Keyword(s):  
Positive Integer ◽  
Regular Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Completely Simple

An ordered regular semigroup, , is said to be principally ordered if for every  there exists . A principally ordered regular semigroup is pointed if for every element,  we have . Here we investigate those principally ordered regular semigroups that are eventually pointed in the sense that for all  there exists a positive integer, , such that . Necessary and sufficient conditions for an eventually pointed principally ordered regular semigroup to be naturally ordered and to be completely simple are obtained. We describe the subalgebra of  generated by a pair of comparable idempotents  and such that . 

On the Hausdorff and Trigonometric Moment Problems

1961 ◽  
Vol 13 ◽  
pp. 454-461
Author(s):  
P. G. Rooney
Keyword(s):  
Moment Problem ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

Coefficient Regions for Univalent Trinomials

1980 ◽  
Vol 32 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Waniurski
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].THEOREM A (Dieudonné criterion). The polynomial1is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial2does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the functionis regular in the unit disk.

Traces of Matrices of Zeros and Ones

1960 ◽  
Vol 12 ◽  
pp. 463-476 ◽  
Author(s):  
H. J. Ryser
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Critical Survey ◽  
Combinatorial Properties ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Sum Vector

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class1.1consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

On Sums of Progressions of Positive Integers

1970 ◽  
Vol 54 (388) ◽  
pp. 113-115
Author(s):  
R. L. Goodstein
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Common ◽  
Necessary And Sufficient ◽  
Positive Integers

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

Unique Extension and Product Measures

1967 ◽  
Vol 19 ◽  
pp. 757-763 ◽  
Author(s):  
Norman Y. Luther
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Product Measure ◽  
Unique Extension ◽  
Product Measures ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Finite Measures

Following (2) we say that a measure μ on a ring is semifinite ifClearly every σ-finite measure is semifinite, but the converse fails.In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

The displacement problem for elastic crystals

1989 ◽  
Vol 113 (1-2) ◽  
pp. 159-180 ◽  
Author(s):  
I. Fonseca ◽  
L. Tartar
Keyword(s):  
Boundary Conditions ◽  
Convex Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Displacement Boundary ◽  
Displacement Problem ◽  
Body Forces ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Elastic Crystals

SynopsisIn this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the typewhere h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.

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