On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators J_n,αf for functions f measurable and locally bounded on the interval [0, ∞) with α ? 1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of J_n,αf(x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann-Butzer-Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

On multivariate Baskakov operator

In this paper, we study multivariate Baskakov operator Bn,d(f,x). We first show that the operator can retain some properties of the original function f, such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator Bn,d(f,x) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K-functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.     

Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: Dαu(t)+f(t,u(t),Dμu(t))=0, u(0)=u(1)=0, where 1<α<2, 0<μ?α−1, Dα is the standard Riemann-Liouville fractional derivative, f is a positive Carathéodory function and f(t,x,y) is singular at x=0. By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.     

A fourth-order finite difference method based on uniform mesh for singular two-point boundary-value problems

We present a new fourth-order finite difference method based on uniform mesh for the (weakly) singular two-point boundary value problem: (x^αy′)′ = f(x, y), 0 < x ⩽ 1, y(0) = A, y(1) = B, 0 < α < 1. Our method provides O(h⁴)-convergent approximations for all α ∈ (0, 1); for α = 0 it reduces to the well-known fourth-order method of Numerov for $\text{y″ = ƒ(x, y)}$.     

A stability analysis for a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u), where ? ∞ < x < + ∞ and 0 < t < + ∞. Under suitable hypotheses pertaining to f, we exhibit a class of initial data φ(x), ? ∞ < x < + ∞, for which the corresponding solutions u(x, t) approach zero as t → + ∞. This convergence is uniform with respect to x on any compact subinterval of the real axis.     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

Sur les entiers dont la somme des chiffres est moyenne

The goal of this paper is to study sets of integers with an average sum of digits. More precisely, let g be a fixed integer, s(n) be the sum of the digits of n in basis g. Let f:N→N such that, in any interval [g^ν,g^ν+1[, f(n) is constant and near from (g-1)ν/2. We give an asymptotic for the number of integers n<x such that s(n)=f(n) and we prove that for every irrational α the sequence (αn) is equidistributed mod 1, for n satisfying s(n)=f(n).     

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

The extended gamma class and applications

The gamma class Γ _α (g) consists of positive and measurable functions that satisfy f(x+yg(x))/f(x)→exp(αy). In most cases, the auxiliary function g is Beurling varying, i.e. g(x)/x→0 and g∈Γ₀(g). Taking h=logf, we find that h∈EΓ _α (g,1), where EΓ _α (g,a) is the class of ultimately positive and measurable functions that satisfy (f(x+yg(x))?f(x))/a(x)→αy. In this paper, we discuss local uniform convergence for functions in the classes Γ _α (g) and EΓ _α (g,a). From this we obtain several representation theorems. We also prove some higher order relations for functions in the classes Γ _α (g) and EΓ _α (g,a). Some applications conclude the paper.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

Heat flow and Hardy inequality in complete Riemannian manifolds with singular initial conditions

Upper bounds are obtained for the heat content of an open set D with singular initial condition f on a complete Riemannian manifold, provided (i) the Dirichlet-Laplace-Beltrami operator satisfies a strong Hardy inequality, and (ii) f satisfies an integrability condition. Precise asymptotic results for the heat content are obtained for an open bounded and connected set D in Euclidean space with C² boundary, and with initial condition f(x)=δ(x)^−α,0<α<2, where δ(x) is the distance from x to the boundary of D.     

Vanishing exponential integrability for functions whose gradients belong to L(log(e+L))

If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−u_B|^n/(n−1), u_B=average of u over B, tends to 1 as the radius of B shrinks to zero—for quasi almost all center points. This refines a result of N. Trudinger (1967). We prove here a similar result for the class of gradients in Lⁿ(log(e+L))^α, 0?α?n−1. The results depend on a capacitary strong-type inequality for these spaces.     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.     

Basic Fourier series on a q-linear grid: Convergence theorems

For 0<q<1 define the symmetric q-linear operator acting on a suitable function f(x) by δf(x)=f(q1/2x)−f(q−1/2x). The q-linear initial value problem , f(0)=1, has two entire functions Cq(z) and Sq(z) as linearly independent solutions, which are orthogonal on a discrete set. Sufficient conditions for pointwise convergence and for uniform convergence of the corresponding Fourier expansion are given.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.