A note on approximation properties of q-Durrmeyer operators

In this paper, the approximation properties of q-Durrmeyer operators D_n,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process lim_n→∞D_n,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators D_n,q.     

Convergence of Modified Lagrange Interpolation toLp-Functions Based on the Zeros of Orthonormal Polynomials with Freud Weights

We consider the “Freud weight”W²_Q(x)=exp(−Q(x)). let 1<p<∞, and letL*_n(f) be a modified Lagrange interpolation polynomial to a measurable functionf∈{f; ess sup_x∈ |f(x)| W_Q(x)(1+|x|)^α<∞},α>0. Then we have lim_n→∞ ∫^∞_−∞ [|f(x)−L*_n(f; x)| W_Q(x)(1+|x|)^−Δ]^p dx=0, whereΔis a constant depending onpandα.     

Second order estimates for large solutions of elliptic equations

This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in R^N, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

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.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f○T:Rn×Q→Q with f○T(x,ω)=f(T(x)ω) where Q is a compact (Hausdorff topological) space, f∈C(Q) and T(x):Q→Q, x∈Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω∈Q the realization f(T(x)ω) belongs to an algebra with mean value, that is, an algebra of functions in BUC(Rn) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571-582, English transl. in Math. Notes 33 (1983) 294-300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of Rn provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in Rn with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

An example concerning the zero set of the Jacobian

Let f∈W1,1(Ω,Rn) be a homeomorphism of finite distortion K. It is known that if K1/(n−1)∈L¹(Ω), then the Jacobian Jf of f is positive almost everywhere in Ω. We will show that this integrability assumption on K is sharp in any Orlicz-scale: if α is increasing function (satisfying minor technical assumptions) such that limt→∞α(t)=∞, then there exists f such that K1/(n−1)/α(K)∈L¹(Ω) and Jf vanishes in a set of positive measure.     

Recurrent points and non-wandering points of graph maps

Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Asymptotic number of isometric generalized Fibonacci cubes

For a binary word f, let Q_d(f) be the subgraph of the d-dimensional cube Q_d induced on the set of all words that do not contain f as a factor. Let G_n be the set of words f of length n that are good in the sense that Q_d(f) is isometric in Q_d for all d. It is proved that lim_n→∞|G_n|/2ⁿ exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good.     

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u₀, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u₀, where a>0 is a parameter and u₀ is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen.