Distributions of ballot problem random variables

1991 ◽  
Vol 23 (3) ◽  
pp. 586-597 ◽  
Author(s):  
Chern-Ching Chao ◽  
Norman C. Severo
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Positive Real Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Positive Real ◽  
Discrete Uniform Distribution ◽  
Ballot Problem ◽  
Discrete Uniform ◽  
Necessary And Sufficient

Suppose that in a ballot candidateAscoresavotes and candidateBscoresbvotes, and that all the possible voting records are equally probable. Corresponding to the firstrvotes, letαrandβrbe the numbers of votes registered forAandB, respectively. Let p be an arbitrary positive real number. Denote byδ(a, b, p)[δ*(a,b,ρ)] the number of values ofrfor which the inequality,r =1, ···,a+b, holds. Heretofore the probability distributions of δand δ* have been derived for only a restricted set of values ofa, b, andρ, although, as pointed out here, they are obtainable for all values of (a,b,ρ) by using a result of Takács (1964). In this paper we present a derivation of the distribution ofδ[δ*] whose development, for any (a, b, ρ), leads to both necessary and sufficient conditions forδ[δ*] to have a discrete uniform distribution.

Distributions of ballot problem random variables

1991 ◽  
Vol 23 (03) ◽  
pp. 586-597
Author(s):  
Chern-Ching Chao ◽  
Norman C. Severo
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Positive Real Number ◽  
Probability Distributions ◽  
Sufficient Conditions ◽  
Positive Real ◽  
Discrete Uniform Distribution ◽  
Ballot Problem ◽  
Discrete Uniform ◽  
Necessary And Sufficient

Suppose that in a ballot candidateAscoresavotes and candidateBscoresbvotes, and that all the possible voting records are equally probable. Corresponding to the firstrvotes, letαrandβrbe the numbers of votes registered forAandB, respectively. Let p be an arbitrary positive real number. Denote byδ(a, b, p)[δ*(a,b,ρ)] the number of values ofrfor which the inequality,r =1, ···,a+b, holds. Heretofore the probability distributions of δand δ* have been derived for only a restricted set of values ofa, b, andρ, although, as pointed out here, they are obtainable for all values of (a,b,ρ) by using a result of Takács (1964). In this paper we present a derivation of the distribution ofδ[δ*] whose development, for any (a, b, ρ), leads to both necessary and sufficient conditions forδ[δ*] to have a discrete uniform distribution.

scholarly journals Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients

2015 ◽  
Vol 3 (1) ◽  
pp. 12
Author(s):  
A. Murugesan ◽  
P. Sowmiya
Keyword(s):  
Real Number ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Difference Equation ◽  
Positive Real ◽  
Neutral Delay ◽  
First Order ◽  
Positive Roots ◽  
Necessary And Sufficient ◽  
Delay Difference

In this paper, we establish the necessary and sufficient conditions for oscillation of the following first order neutral delay difference equation <br />\begin{equation*} \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \quad\Delta[x(n)+px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0, \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}<br />where \(\tau\) and \(\sigma\) are positive integers, \(p\neq 0\) is a real number and \(q\) is a positive real number. We proved that every solution of (*) oscillates if and only if its characteristic equation<br />\begin{equation*}\quad \quad \quad \quad\quad \quad \quad \quad\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\lambda-1)(1+p\lambda^{-\tau})+q\lambda^{-\sigma}=0\quad \quad \quad \quad \quad \quad \quad \quad {(**)} \end{equation*}<br />has no positive roots.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

Ultradiversities and their spherical completeness

2020 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
Gholamreza H. Mehrabani ◽  
Kourosh Nourouzi
Keyword(s):  
Real Number ◽  
Finite Subset ◽  
Positive Real Number ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Ultrametric Spaces ◽  
Positive Real

AbstractDiversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

Ballots, queues and random graphs

1989 ◽  
Vol 26 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Real Number ◽  
Random Graph ◽  
Random Graphs ◽  
Limit Distribution ◽  
Positive Real Number ◽  
Positive Real ◽  
Ballot Theorem

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

Sign in / Sign up

Export Citation Format

Share Document