Arbitrary Real Numbers Research Articles

Qualitative Properties for a Fourth Order Rational Difference Equation

In this paper we deal with some properties of the solutions of the recursive sequence $$x_{n+1}=ax_{n-1}+\frac{bx_{n-1}x_{n-3}}{cx_{n-1}+dx_{n-3}},\quad n=0,1,\ldots,$$ where the initial conditions x −3, x −2, x −1, x 0 are arbitrary positive real numbers and a, b, c, d are constants. Also, we give the form of the solution of some special cases of this equation.

Dynamics of a rational difference equation

The authors investigate the global behavior of the solutions of the difference equation $$ x_{n + 1} = \frac{{ax_{n - l} x_{n - k} }} {{bx_{n - p} + cx_{n - q} }}, n = 0, 1, \cdots , $$ where the initial conditions x−r, x−r+1, x−r+2, …, x0 are arbitrary positive real numbers, r = max{l, k, p, q} is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.

Asymptotic Behavior of Equilibrium Point for a Class of Nonlinear Difference Equation

We study the asymptotic behavior of the solutions for the following nonlinear difference equation where the initial conditions are arbitrary nonnegative real numbers, are nonnegative integers, , and are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.

COMPUTER ALGEBRA REEXAMINATION OF THE SCALED PARTICLE THEORY FOR HARD-SPHERE AND LENNARD-JONES FLUIDS

In the present work, an extension of the scaled particle theory (ESPT) for fluid using computer algebra is developed to obtain an equation of state (EOS), for Lennard-Jones fluid. A suitable functional form for surface tension S(r,d,∊) is assumed with intermolecular separation r as a variable, given below: [Formula: see text] where m is arbitrary real number, and d and ∊ are related to physical property such as average or suitable molecular diameter and the binding energy of the molecule respectively. It is found that, for hard sphere fluid ∊ = 0, the above assumption when introduced in scaled particle theory (SPT) frame and choosing arbitrary real number, m = 1/3, the corresponding EOS is in good agreement with the computer simulation of molecular dynamics (MD) result. Furthermore, for the value of m = -1 it gives a Percus–Yevick (pressure), and for the value of m = 1, it corresponds Percus–Yevick (compressibility) EOS.

The Asymptotic Stability of xn+1-a2xn-1+bxn-k= 0

Abstract. We give the necessary and sufficient conditions for the asymptotic stability ofthe linear delay difference equation x n+1 − a 2 x n−1 + bx n−k = 0, n = 0,1,···, where aand b are arbitrary real numbers and k is a positive integer greater than 1. The obtainedconditions are given in terms of parameters a and b of difference equations. The methodof proof is based on arithematic of complex numbers as well as properties of analytic func-tions. 1. IntroductionIn [2] S.A. Kuruklis gave the necessary and sufficient conditions for the asymp-totic stability of the difference equation(A) x n+1 −ax n +bx n−k = 0, n = 0,1,··· ,where a and b are arbitrary real and k is a positive integer. The following is themain result obtained in [2]:Theorem A. Let a be a nonzero real, b an arbitrary real, and k a positive integergreater than 1. Equation (A) is asymptotically stable if and only if |a| < k+1k , and|a|−1 < b <a 2 −2|a|cosφ+1 12 for k : odd,|b−a| < 1 and |b| <a 2 −2|a|cosφ+1 12 for k : even,where φ is the solution in0,

Qualitative behavior of a rational recursive sequence

In this paper we study the behavior of the recursive sequence x n + 1 = a x n + b x n x n − 1 c x n + d x n − 1 , n = 0 , 1 , ... , where the initial conditions x −1, x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

On summing to arbitrary real numbers

Ein interessantes zahlentheoretisches Problem ist die Frage nach der Rationalität des Werts einer konvergenten Reihe reeller Zahlen. An diese Fragestellung anknüpfend nennen wir mit P. Erdős eine Folge \{a_{n}\}_{n=1}^{\infty} reeller Zahlen irrational, falls die Menge E=\{\sum_{n=1}^{\infty}1/(a_{n}c_{n})\mid c_{n}\in \mathbb{N}\} keine rationale Zahl enthält. In der vorliegenden Arbeit beweisen die Autoren für den Fall, dass die Reihe \sum_ {n=1}^{\infty}1/a_{n} bedingt konvergent ist, dass die Menge E jeweils die gesamte reelle Zahlengerade ausschöpft.

On the Difference Equation

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

A generalization of Dirichlet approximation theorem for the affine actions on real line

In this paper, we obtain strong density results for the orbits of real numbers under the action of the semigroup generated by the affine transformations T 0 ( x ) = x / a and T 1 ( x ) = b x + 1 , where a , b > 1 . These density results are formulated as generalizations of the Dirichlet approximation theorem and improve the results of Bergelson, Misiurewicz, and Senti. We show that for any x , u > 0 there are infinitely many elements γ in the semigroup generated by T 0 and T 1 such that | γ ( x ) − u | < C ( t 1 / | γ | − 1 ) , where C and t are constants independent of γ, and | γ | is the length of γ as a word in the semigroup. Finally, we discuss the problem of approximating an arbitrary real number by the ratios of prime numbers and the ratios of logarithms of prime numbers.

Construction of σ-orthogonal polynomials and gaussian quadrature formulas

Let dα be a measure on R and let σ = (m1, m2,...,mn), where mk ≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ωn(x) := (x − x1)(x − x2)...(x − xn) with x1 ≤ x2 ≤ ... ≤ xn is said to be the nth σ-orthogonal polynomial with respect to dα if the vector of zeros (x1, x2, ..., xn)T is a solution of the extremal problem $${\int_R {{\prod\limits_{k = 1}^n {{\left| {x - x_{k} } \right|}^{{m_{k} }} d\alpha {\left( x \right)} = {\mathop {\min }\limits_{y_{1} \leqslant y_{2} \leqslant ... \leqslant y_{n} } }} }} }\;{\int_R {{\prod\limits_{k = 1}^n {{\left| {x - y_{k} } \right|}^{{m_{k} }} d\alpha {\left( x \right)}.} }} }$$ In this paper the existence, uniqueness, characterizations, and continuity with respect to σ of a σ-orthogonal polynomial under a more mild assumption are established. An efficient iterative method based on solving the system of normal equations for constructing a σ-orthogonal polynomial is presented when all the mk are arbitrary real numbers no less than 2. A simple method to calculate the Cotes numbers of the corresponding generalized Gaussian quadrature formula when all the mk are positive integers no less than 2 is provided. Finally, some numerical examples are also given.

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d v denote the degree of the vertex v in G. The zeroth-order general Randic index of a graph is defined as $$R_{\alpha}^0(G)=\sum_{v\in V(G)}{d_{v}}^{\alpha}$$ where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randic index $$R_{\alpha}^0(G)$$ of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for $$R_{\alpha}^0(G)$$ depending on α in different intervals.

Vibrations of extensible beams: Unilateral problem

The present work is dedicated to study a unilateral problem relating to the operator L ˆ u ( x , t ) = ∂ 2 u ∂ t 2 − [ a ˆ ( t ) + b ˆ ( t ) ∫ α ( t ) β ( t ) ( ∂ u ∂ x ) 2 d x ] ∂ 2 u ∂ x 2 + q ∂ 4 u ∂ x 4 , which models small transverse deflections u ( x , t ) of an extensible beam with moving ends. Without restriction on the initial configuration u 0 and considering the initial velocity u 1 with a bounded gradient, we succeed to prove that, given T an arbitrary positive real number, there exists a unique solution for the unilateral problem defined for all t ∈ [ 0 , T ] .

Differences in dynamics between discrete strategies and continuous strategies in a multi-player game with a linear payoff structure

A deductive analysis concerning replicator dynamics proved that a continuous strategy game (in which a player chooses an arbitrary real number between [0, 1] as a cooperative fraction) has the same equilibrium as a discrete strategy game (in which a player chooses only C or D), which has the same linear payoff structure as a continuous strategy game. The deduction is shown for two-player and multi-player games.

Non-compact Littlewood–Paley theory for non-doubling measures

We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on R with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg. 0. Introduction. In this note we generalize a slightly non-standard stopping time argument from the usual Euclidean setting on R, with Lebesgue measure, to one in which the underlying measure is not assumed to satisfy any doubling condition. The original motivation for this work came from certain weighted inequalities proved by the author and Richard Wheeden in [WhWi]. They looked for conditions on weights v and non-negative measures μ which ensured that (∫ R + |∇u|q dμ )1/q ≤ (∫ Rd |f |p v dx )1/p (0.1) would hold for all f in some reasonable test class. Here we are assuming that p and q lie strictly between 1 and ∞, and that u is the harmonic (Poisson) extension of f into R + = R × (0,∞). The approach they used was to consider a dual form of (0.1): (∫ Rd |T (g)|p σ dx )1/p′ ≤ (∫ R + |g(t, y)|q dμ )1/q′ . (0.2) Here p′ and q′ are the dual indices to p and q, σ = v1−p ′ , and g is an arbitrary bounded, measurable, compactly-supported function mapping R + → R. The operator T is a certain “balayage”-like object, whose precise definition need not concern us here. The inequality (0.2) turned out, after some juggling, to follow from inequalities like this:  ∫ Rd | ∑ Q λQφ(Q)| ′ σ dx   1/p′ ≤ C  ∑ Q |λQ|qwQ   1/q′ . The summation is indexed over the dyadic cubes Q ⊂ R; the λQ’s are arbitrary real numbers, of which all but finitely many are assumed to be 0, and the wQ’s are certain positive numbers whose precise definition need not concern us. The functions φ(Q) are, if you will pardon the misnomer, non-compactly supported wavelets. This means: each φ(Q) is a bump function centered around Q, with size and smoothness decaying at a nice rate AMS Subject Classfication (2000): 42B25.

On the Rational Recursive Sequencexn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i)

The main objective of this paper is to study the boundedness character, the periodic character, the convergence, and the global stability of the positive solutions of the difference equationxn+1=(A+∑i=0kαixn−i)/(B+∑i=0kβixn−i), n=0,1,2,…,whereA, B, αi, βiand the initial conditionsx−k,...,x−1,x0are arbitrary positive real numbers, whilekis a positive integer number.

On the recursive sequence [formula omitted

We study the global stability, the boundedness character and the periodic nature of the positive solutions of the difference equation y n + 1 = α + y n - 1 β + y n + y n - 1 y n , n = 0 , 1 , 2 , … , where α, β ∈ (0, ∞), α ≠ β, β ≠ 2 and the initial conditions y −1, y 0 are arbitrary positive real numbers.

The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay

This paper is devoted to build the existence-and-uniqueness theorem of solutions to stochastic functional differential equations with infinite delay (short for ISFDEs) at phase space BC ( ( − ∞ , 0 ] ; R d ) . Under the uniform Lipschitz condition, the linear growth condition is weaked to obtain the moment estimate of the solution for ISFDEs. Furthermore, the existence-and-uniqueness theorem of the solution for ISFDEs is derived, and the estimate for the error between approximate solution and accurate solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is also valid for ISFDEs on [ t 0 , T ] . Moreover, the existence-and-uniqueness theorem still holds on interval [ t 0 , ∞ ) , where t 0 ∈ R is an arbitrary real number.

Self-similar solutions to a coagulation equation with multiplicative kernel

Existence of self-similar solutions to the Oort–Hulst–Safronov coagulation equation with multiplicative coagulation kernel is established. These solutions are given by s ( t ) − τ ψ τ ( y / s ( t ) ) for ( t , y ) ∈ ( 0 , T ) × ( 0 , ∞ ) , where T is some arbitrary positive real number, s ( t ) = ( ( 3 − τ ) ( T − t ) ) − 1 / ( 3 − τ ) and the parameter τ ranges in a given interval [ τ c , 3 ) . In addition, the second moment of these self-similar solutions blows up at time T . As for the profile ψ τ , it belongs to L 1 ( 0 , ∞ ; y 2 d y ) for each τ ∈ [ τ c , 3 ) but its behaviour for small and large y varies with the parameter τ .

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d v denote the degree of the vertex v in G. The zeroth-order general Randic index of a graph is defined as R α 0 (G) = ∑ v∈V(G) d α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randic index R α 0 (G) among all unicycle graphs G of order n. We give a clear picture for R α 0 (G) of unicycle graphs according to real number α in different intervals.

On the Difference equation xn+1=axn−bxn/(cxn−dxn−1)

We investigate some qualitative behavior of the solutions of the difference equation xn+ = ax n - bx n /(cx x - dxn-1), n = 0,1,..., where the initial conditions x-1, x0 are arbitrary real numbers and a, b, c, d are positive constants.