In this paper, we introduce the concepts of invariant convergence, lacunary invariant statistical convergence of sequences of fuzzy numbers and lacunary strongly invariant convergence of sequences of fuzzy numbers. We give some relations related to these concepts.

Let f be a 2π periodic function in L[0, 2π] and ∑∞ k=−∞ f̂(nk)e inkx be its Fourier series with ‘small’ gaps nk+1 − nk ≥ q ≥ 1. Here we have obtained sufficiency conditions for the absolute convergence of such series if f is of ∧ BV (p) locally. We have also obtained a beautiful interconnection between lacunary and non-lacunary Fourier series.

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

Three novel polyoxometalates (POMs) containing [Ni6(μ3-OH)3](9-) and [P2W15O56](12-) units were first made, showing the first hexa-nuclearity transition-metal substituted POMs (TMSPs) based on monomeric lacunary Dawson fragments, which further indicates that the hydrothermal technique can offer an effective way for making new TMSPs through lacunary POM fragments incorporated with high-nuclear T...

In this paper we obtain a universal lower bound on the 2-adic order of lacunary sums of binomial coefficients. By means of necessary and sufficient conditions, we determine the set of values for which the bound is achieved and show the periodicity of the set. We prove a congruential identity for the corresponding generating function. Our approach gives an alternative and transparent proof for s...

(1) On page 529: The statement “we will denote the set of all double lacunary sequences by Nθr,s ” should be removed. (2) On page 529: In Definition 2.4, the statement “for every ε > 0” should be removed. (3) On page 530: In the proof of (C), for all K should be replaced with for all k and l. (4) The references “Li, J. Lacunary statistical convergence and inclusion properties between lacunary m...

This paper presents introduce some new notions, f -asymptotically equivalent of multiple L, strong f -asymptotically equivalent of multiple L, and strong f -asymptotically lacunary equivalent of multiple L which is a natural combination of the definition for asymptotically equivalent, Statistically limit, Lacunary sequence, and Modulus function. We study some connections between the asymptotica...

A lacunary sequence is an increasing integer sequence θ = {kr } such that kr − kr−1 → ∞ as r → ∞. A sequence x is called sθ-convergent to L provided that for each ε > 0, limr (1/(kr −kr−1)){the number of kr−1 < k ≤ kr : |xk−L| ≥ ε} = 0. In this paper, we study the general description of inclusion between two arbitrary lacunary sequences convergent.

In this paper we introduce a new concept for lacunary strong P-convergent with respect to an Orlicz function and examine some properties of the resulting sequence space of fuzzy numbers. We also show that if a sequence is lacunary strong Pconvergence with respect to an Orlicz function then it is , ( ) r s S F θ -convergent.