New results for oscillation of delay difference equations

In this paper, we consider the delay difference equation x_n+1 − x_n + p_nx_n−k = 0, n = 0, 1, 2, …, where p_n is a sequence of nonnegative real numbers and k is a positive integer. Some new results for the oscillation of this equation are obtained. Our theorems improve all known results in the literature.     

Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients

In this paper, we consider the following higher-order neutral delay difference equations with positive and negative coefficients: Δ^m(x_n + cx_n−k) + p_nx_n−r − q_nx_n−l = 0, n≥n₀, where c ϵ R, m ⩾ 1, k ⩾ 1, r, l ⩾ 0 are integers, and {p_n}^∞_n=n0 and {q_n}_n=n0^∞ are sequences of nonnegative real numbers. We obtain the global results (with respect to c) which are some sufficient conditions for the existences of nonoscillatory solutions.     

Algorithms forn-th root approximation

Fromf(x)=x ⁿ ?r and a polynomialQ _p (y)=∑ _i=0 ^p a _i y ⁱ , we consider Newton's method to solveF _p (x)=Q _p (f(x))=0. We obtain convergent iterative methods of orderp+1 to findr ^1/n for arbitraryp.     

Global attractivity for a nonlinear difference equation with variable delay

In this paper, we give sufficient conditions under which every solution of the nonlinear difference equation with variable delay x(n + 1) − x(n) + p_nf(x(g(n))) = 0, n = 0, 1, 2, … tends to zero as n → ∞. Here, p_n is a nonnegative sequence, f : R → R is a continuous function with xf(x) > 0 if x ≠ 0, and g : N → Z is nondecreasing and satisfies g(n) ≤ n for n ≥ 0 and lim_n→∞ g(n) = ∞.     

Expansion of Self-Similar Functions in the Faber–Schauder System

Let Ω = A^N be a space of right-sided infinite sequences drawn from a finite alphabet A = {0,1}, N = {1,2,…}. Let ρ(x, y)Σ_k=1^∞|x _k ? y _k |2^?k be a metric on Ω = AN, and μ the Bernoulli measure on Ω with probabilities p₀, p₁ > 0, p₀ + p₁ = 1. Denote by B(x,ω) an open ball of radius r centered at ω. The main result of this paper \(\mu (B(\omega ,r))r + \sum\nolimits_{n = 0}^\infty {\sum\nolimits_{j = 0}^{{2^n} - 1} {{\mu _{n,j}}} } (\omega )\tau ({2^n}r - j)\), where τ(x) = 2min {x,1 ? x}, 0 ≤ x ≤ 1, (τ(x) = 0, if x < 0 or x > 1 ), \({\mu _{n,j}}(\omega ) = (1 - {p_{{\omega _{n + 1}}}})\prod _{k = 1}^n{p_{{\omega _k}}} \oplus {j_k}\), \(j = {j_1}{2^{n - 1}} + {j_2}{2^{n - 2}} + ... + {j_n}\). The family of functions 1, x, τ(2 ⁿ r ? j), j = 0,1,…, 2 ⁿ ? 1, n = 0,1,…, is the Faber–Schauder system for the space C([0,1]) of continuous functions on [0, 1]. We also obtain the Faber–Schauder expansion for Lebesgue’s singular function, Cezaro curves, and Koch–Peano curves. Article is published in the author’s wording.     

Eine stets quadratisch konvergente Modifikation des Steffensen-Verfahrens

It is shown that the following modification of the Steffensen procedurex _n+1=x _n ?k _s (x _n )f(x _n ) (f[x _n ,x _n ?f(x _n )])^?1 (n=0,1,...) withk _s (x)=(1?z _s (x))^?1,z _s (x)=f(x) 2f[x?f(x),x,x+f(x)]×(f[x,x?f(x)])^?2 is quadratically convergent to the root of the equation \(f(x) = (x - \bar x)^p g(x) = 0(p > 0,g(\bar x) \ne 0)\) . Furthermore \(\mathop {\lim }\limits_{n \to \infty } k_s (x_n ) = p\) holds.     

Controllability of general nonlinear systems

Consider the nonlinear system $$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t)), x(0) = x_0 \in M}$$ whereM is aC ^∞ realn-dimensional manifold,f, g ₁,?.,g _m areC ^∞ vector fields onM, andu ₁ ,..,u _m are real-valued controls. Ifm=n?1 andf, g ₁ ,?,g _m are linearly independent, then the system is called a hypersurface system, and necessary and sufficient conditions for controllability are known. For a generalm, 1 ≤m ≤n?1, and arbitraryC ^∞ vector fields,f, g ₁ ,?,g _m , assume that the Lie algebra generated byf, g ₁ ,?,g _m and by taking successive Lie brackets of these vector fields is a vector bundle with constant fiber (vector space) dimensionp onM. By Chow's Theorem there exists a maximalC ^∞ realp-dimensional submanifoldS ofM containingx ₀ with the generated bundle as its tangent bundle. It is known that the reachable set fromx ₀ must contain an open set inS. The largest open subsetU ofS which is reachable fromx ₀ is called the region of reachability fromx ₀. IfO is an open subset ofS which is reachable fromx ₀,S we find necessary conditions and sufficient conditions on the boundary ofO inS so thatO = U. Best results are obtained when it is assumed that the Lie algebra generated byg ₁,?,g _m and their Lie brackets is a vector bundle onM.     

A quadratic-time algorithm for smoothing interval functions

In many real-life applications, physical considerations lead to the necessity to consider the smoothest of all signals that is consistent with the measurement results. Usually, the corresponding optimization problem is solved in statistical context. In this paper, we propose a quadratic-time algorithm for smoothing aninterval function. This algorithm, givenn+1 intervals x₀, ..., x _n with 0 ∈ x₀ and 0 ∈ x _n , returns the vectorx ₀, ...,x _n for whichx ₀=x ₀=0,x _i ∈ x _i , and Σ(x _i+1?x _i )² → min.     

Erzeugende Funktionen bei Spline-Interpolation mit äquidistanten Knoten

Using generating functions we obtain in the case ofn+1 equidistant data points a method for the calculation of the interpolating spline functions(x) of degree 2k+1 with boundary conditionss ^(κ) (x₀)=y ₀ ^(κ) ,s ^(κ) (x _n )=y _n ^(κ) , κ=1(1)k, which only needs the inversion of a matrix of orderk. The applicability of our method in the case of general boundary conditions is also mentioned.     

Quadrature rules for Prandtl's integral equation

In this paper we construct an interpolatory quadrature formula of the type $$\mathop {\rlap{--} \smallint }\limits_{ - 1}^1 \frac{{f'(x)}}{{y - x}}dx \approx \sum\limits_{i = 1}^n {w_{ni} (y)f(x_{ni} )} ,$$ wheref(x)=(1?x)^α(1+x)^β f _o(x), α, β>0, and {x _ni} are then zeros of then-th degree Chebyshev polynomial of the first kind,T _n (x). We also give a convergence result and examine the behavior of the quantity \( \sum\limits_{i = 1}^n {|w_{ni} (y)|} \) asn→∞.     

A note on the average depth of tries

LetA _n be the average root-to-leaf distance in a binary trie formed by the binary fractional expansions ofn independent random variablesX ₁,...,X _n with common densityf on [0, 1). We show thateither E(A _n )=∞ for alln≥2or \(\mathop {\lim }\limits_n E(A_n )/\log _2 n = 1\) depending on whether ∫f ² (x)dx = ∞ or ∫f ² (x)dx < ∞.     

How to Recognize Zero

An elementary point is a point in complexnspace, which is an isolated, nonsingular solution ofnequations innvariables, each equation being either of the formp = 0, wherepis a polynomial in [x₁,…,x_n], or of the formx_j − e^x_i = 0. An elementary number is the polynomial image of an elementary point. In this article a semi algorithm is given to decide whether or not a given elementary number is zero. It is proved that this semi algorithm is an algorithm, i.e. that it always terminates, unless it is given a problem containing a counterexample to Schanuel’s conjecture.     

Valutazione dell'errore per una formula di quadratura alla Tchebycheff di tipo chiuso

In this paper we study the remainder term of a quadrature formula of the form $$\int\limits_{ - 1}^1 {f(x)dx = A_n \left[ {f( - 1) + f(1)} \right] + C_n \sum\limits_{i = 1}^n {f(x_{n,i} ) + R_n \left[ f \right],} } $$ , withx _x,i ∈^-1,1, andR _n [f]=0 whenf(x) is a polynomial of degree ≤n+3 ifn is even, or ≤n+2 ifn is odd. Such a formula exists only forn=1(1)11. It is shown that, iff(x)∈ ^C(h+1) [-1,1], (h=n+3 orn+2), thenR _n [f]=f ^h+1 (τ)·± _n . The values α _n are given.     

Monotone Einschließung von Inversen positiver Elemente durch Verfahren vom Schulz-Typ

The conditional iterationx _n+1 =sup (x _n ,x _n +x _n (e?ax _n )),y _n?1 =inf (y _n ,x _n +y _n (e?ax _n )) generating sequences (x _n ) and (y _n ) is considered in partially ordered spaces. Under certain conditions it is shown, that the inversea ^?1 of a positive elementa≧0 is monotonously enclosed in the sensex _n ≦x _n+1 ≦a ^?1 ≦y _n+1 ≦y _n and that (x _n ) and (y _n ) converge toa ^?1 quadratically.     

Multiplicity of positive solutions of a class of nonlinear fractional differential equations

This paper is concerned with the nonlinear fractional differential equation L(D)u=f(x,u), u(0)=0, 0<x<1,where L(D) = Dsn − an−1Dsn−1 − … − a1Ds11 < s2 < … < sn < 1, and aj > 0, j = 1,2,…, n − 1. Some results are obtained for the existence, nonexistence, and multiplicity of positive solutions of the above equation by using Krasnoselskii's fixed-point theorem in a cone. In particular, it is proved that the above equation has N positive solutions under suitable conditions, where N is an arbitrary positive integer.     

Modifikationen des Pollard-Algorithmus

The factorization algorithm of Pollard generates a sequence in ? _n by $$x_0 : = 2;x_{i + 1} : = x_i^2 - 1(\bmod n),i = 1,2,3,...$$ wheren denotes the integer to be factored. The algorithm finds an factorp ofn within \(0\left( {\sqrt p } \right)\) macrosteps (=multiplications/divisions in ? _n ) on average. An empirical analysis of the Pollard algorithm using modified sequences $$x_{i + 1} = b \cdot x_i^\alpha + c(\bmod n),i = 1,2,...$$ withx ₀,b,c,α∈? and α≥2 shows, that a factorp ofn under the assumption gcd (α,p-1)≠1 now is found within $$0\left( {\sqrt {\frac{p}{{ged(\alpha ,p - 1}}} } \right)$$ macrosteps on average.     

Characterization of normal forms possessing inverse in the λ-β-η-calculus

The aim of this paper is to generalize a result given by Curry and Feys, who have shown that the only regular combinators possessing inverse in the λ-β-η-calculus are the permutators, whose definition is p=λzλx₁…λx_n(zx_i1…x_in) for n?0 where i₁,…, i_r is a permutation of 1,…, n. Here we extend this characterization to the set of normal forms, showing that the only normal forms possessing inverse in the λ-βη-calculus are the “hereditarily finite permutators” (h.f.p.), whose recursive definition is: if n?0, P_j (1?j?n) are h.f.p. and i₁,…,i_n is a permutation of 1,…, n, then the normal form of P = λzλx₁…λx_n(z(P₁x_i1))… (P_nin) is an h.f.p.     

On s-uniform property of compressing sequences derived from primitive sequences modulo odd prime powers

Let Z/(p^e) be the integer residue ring modulo p^e with p an odd prime and e ≥ 2. We consider the suniform property of compressing sequences derived from primitive sequences over Z/(p^e). We give necessary and sufficient conditions for two compressing sequences to be s-uniform with α provided that the compressing map is of the form ?(x₀, x₁,...,x_e?1) = g(x_e?1) + η(x₀, x₁,..., x_e?2), where g(x_e?1) is a permutation polynomial over Z/(p) and η is an (e ? 1)-variable polynomial over Z/(p).     

Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t ? 1), where p, q > 0, q = 1 ? p, p ≠ q. The variables M _n = ∫₀¹t ⁿ dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log₂x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.