Readjustment of the Paper [J.Kaur and S.S.Bhatia,Integrability and L^1-Convergence of Double Cosine Trigonometric Series,Anal.Theory Appl.,27(1)(2011),pp.32–39.]

In this paper, we show that new modified double cosine trigonometric sums introduced in [1] are inappropriate, the class of double sequences Jintroduced there is unusable for such sums and consequently the results obtained in it are completely incorrect. We here introduce appropriate modified double cosine trigonometric sums making the class Jusable considering a particular double cosine trigonometric series.

INTEGRABILITY AND L^1-CONVERGENCE OF DOUBLE COSINE TRIGONOMETRIC SERIES

We study here Ll-convergence of new modified double cosine trigonometric sum and obtain a new necessary and sufficient condition for Ll-cOnvergence of double cosine trigonometric series. Also, the results obtained by Moricz[1],[2] are particular cases of ours.

L^(2)-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING

In this paper,we are concerned with the asymptotic behavior of L^(∞) weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping-m/(1+t)^(λ).As λ∈(0,l/7],we prove tht the L^(∞) weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation(GPME)in L^(2)(R).As λ∈(1/7,1),we prove that the L^(∞) weak-entropy solution converges to an expansion around the nonlinear diffusion wave in L^(2)(R),which is the best asymptotic profile.The proof is based on intensive entropy analysis and an energy method.

Is there regional convergence between Morocco and its OECD partner countries in terms of well-being?

Morocco wants its 12 regions to play the role as the main lever of its public policies to initiate harmonized spatial multidimensional development. In the context of this goal and Morocco’s openness over the past two decades to bilateral and multilateral cooperation in an effort toward regional integration, this article studies the convergence of 389 regions in 36 countries(Morocco and 35 of its partner member countries in the Organization for Economic Co-operation and Development(OECD)) between 2000 and 2019 in terms of well-being. To this end, we considered the territorial dimension of β-convergence models for well-being and its four domains(economic, social, environmental, and governance). Then, we adapted the absolute β-convergence model by taking into account the existence of spatial heterogeneity according to five specifications of spatial models. Thus, apart from environmental domain, we found that β-convergence of regions is significant for well-being and three of its domains(economic, social, and governance). These convergences are made by a spatially autocorrelated error model(SEM). However, the speed and period of convergence are relatively low for social domain, partly explaining the very exacerbated tensions at the territorial level. The fastest convergence was achieved in governance domain, followed by economic domain. This suggests that emerging countries must pay particular attention to national public action in favor of social cohesion at the territorial level. The lack of convergence in environmental domain calls for common actions for all countries at the supranational level to protect the commons at the territorial level.

A relation between Г-convergence of functionals and their associated gradient flows

Giorgi conjectured in 1979 that if a sequence of functionals converges in the sense of P-convergence to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. It is shown that this conjecture holds true for a rather wide kind of functionals.

Γ-CONVERGENCE OF INTEGRAL FUNCTIONALS DEPENDING ON VECTOR-VALUED FUNCTIONS OVER PARABOLIC DOMAINS

This paper studies Γ-convergence for a sequence of parabolic functionals, where the integrand f is nonconvex, and periodic on the first variable. The author obtains the representation formula of the Γ-limit. The results in this paper support a conclusion which relates Γ-convergence of parabolic functionals to the associated gradiend flows and confirms one of De Giorgi’s conjectures partially.

THE ASYMPTOTIC PROPERTIES OF SUPERCRITICAL BISEXUAL GALTON-WATSON BRANCHING PROCESSES WITH IMMIGRATION OF MATING UNITS

In this article the supercritical bisexual Galton-Watson branching processes with the immigration of mating units is considered. A necessary condition for the almost sure convergence, and a sufficient condition for the L^1 convergence are given for the process with the suitably normed condition.

SOME NEW CONDITIONS IN FOURIER ANALYSIS AND APPLICATIONS

We investigate some conditions in Fourier analysis as a generalization of monotonicity condition. Especially, we give some applications of GBVS and sequences satisfy the condition ΔB1 and ΔB2.

АГ-Convergence Result for Thin Curved Films Bonded to a Fixed Substrate with a Noninterpenetration Constraint

The behavior of a thin curved hyperelastic film bonded to a fixed substrate is described by an energy composed of a nonlinearly hyperelastic energy term and a debonding interracial energy term. The author computes the Г-limit of this energy under a noninterpenetration constraint that prohibits penetration of the film into the substrate without excluding contact between them.

ASYMPTOTIC ANALYSIS OF SHELLS VIA г-CONVERGENCE

We give a new justification of the linear membrane and flexural shell models. We prove that the sequence of scaled energy functionals associated with the scaled problem r-converges to the energy functional associated with a two-dimensional model. This two-dimensional model is a membrane or flexural one, depending on the geometric and kinematic conditions. Then, a classical argument allows to give a new proof of the convergence theorems recently obtained by P.G. Ciarlet, V. Lods and B. Miara.

Application of <i>αδ</i>-Closed Sets

In this paper, we introduce the notion of αδ-US spaces. Also we study the concepts of αδ-convergence, sequentially αδ-compactness, sequentially αδ-continunity and sequentially αδ-sub-continuity and derive some of their properties.

HOMOGENIZATION PROBLEMS OF PARABOLIC MINIMA

We study the r-convergence of a class of 'parabolic' integral functionals for which the integrand f(z/,u/,Du) is allowed to satisfy a non-coercive assumption. Our results, restricted to the 'elliptic' case, generalize the main results of paper [7-8] and give an answer to an open problem in them.

Dynamic changes and convergence of China's regional green productivity:A dynamic spatial econometric analysis

Low-carbon economic development is at the heart of the post-pandemic green recovery scheme worldwide.It requires economic recovery without compromising on the environment,implying a critical role that green productivity plays in achieving the carbon neutrality goal.Green productivity measures the quality of economic growth with consideration for energy consumption and environmental pollution.This study employs the slacks-based measure directional distance function(SBM-DDF)approach and the Malmquist-Luenberger(ML)index to calculate green productivity and its components of 30 provinces in China between 2001 and 2018.Using a spatial panel data model,we empirically analyzed the conditionalβ-convergence of China's green productivity.We found that overall,since 2001,China's green productivity has demonstrated a continuous upward trend.When taking into account spatial factors,China's green productivity demonstrates a significant conditionalβ-convergence.In terms of regional effects,the results indicate that the green productivity of the eastern and western regions demonstrates club convergence,implying a more balanced green economic development.Moreover,the convergence rate of China's green productivity increases with the addition of environmental regulation variable,and so the corresponding convergence time decreases.It indicates that environmental regulations help to facilitate the convergence of China's green productivity,narrowing the gap between the regional green economic development.The findings provide guideline for achieving a low-carbon development and carbon neutrality from a regional green productivity perspective.

Global Behavior of a Class of Three-Dimensional Competitive Systems

In this paper the global behavior of a class of three-dimensional competitive systems is investigated. The existence of semi-stable periodic orbits is proved. A generalization of the results of [1-4] is obtained.

A copula-based approximation to Markov chains

The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation.In this paper,we propose an approximation method for Markov chains by applying the copula theory.For this purpose,we first discuss the checkerboard copula-based Markov chain,which is the Markov chain generated by the family of checkerboard copulas.This Markov chain has some appealing properties,such as self-similarity in copulas and having explicit forms of finite-dimensional distributions.Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains,and the error bounds of the approximate distributions are provided.Employing the checkerboard copula-based approximation method,we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains.This condition allows copulas of Markov chains to be asymmetric.Finally,by applying the approximation method,analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain,and numerical results are listed to show the approximation errors.

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

We consider anR^(d)-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_(n)(z)(z∈C^(d))be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^(α)-convergence of W_(n)(z)forα>1,as well as its uniform convergence region.

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets.The continuous model considers a free energy comprised of three components:the Ericksen’s energy for liquid crystals,the Cahn-Hilliard energy representing the interfacial energy of the droplet,and an anisotropic weak anchoring energy that enforces a condition such that the director field is aligned perpendicular to the interface of the droplet.Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics.We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy,as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part.Next,we present a discrete gradient flow method and show that it is monotone energy decreasing.Furthermore,we show that global discrete energy minimizersΓ-converge to global minimizers of the continuous energy.We conclude with numerical experiments illustrating different gradient flow dynamics,including droplet coales-cence and break-up.

A TAYLOR APPROXIMATION METHOD OF STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS

In this paper, we formulate a general framework of an analytic approximation of the solutions to some It o stochastic integro-differential equations defined on sub-intervals of an arbitrary part of the time-interval [0,1] and connected at successive division points. The integral parts and coefficients of the equation are approximated by their Taylor polynomial which series up to arbitrary derivatives. The suggested approximations converge to the initial solution in the Lp-th norm with some order.

＂Convergence＂ or ＂Divergence＇？ --Rethinking Regional Integration of the Past Two Decades

The European Union （EU）, North American Free Trade Agreement （NAFTA） and Association of Southeast Asian Nations （ASEAN） are the three biggest regional economic cooperation organizations in the world. What roles have these three organizations played in both domestic development and the evolving global trend of regional integration？ This paper investigates the inequality among these three organizations for regional economic cooperation based on cross-national parallel data from thirty-nine countries over the period 1989-2008. By using the Theil index decomposition, this paper finds that the interregional disparity is the main source of inequality. This paper also finds that intraregional disparity rose significantly from the mid-1990s. ASEAN contributes an equalizing force to the change, while NAFTA contributes a disequalizing force. From the empirical tests based on the Barro non-linear growth regression model, our results show that the whole sample and some subregional samples （ASEAN and EU） support the convergence hypothesis. ASEAN＇s convergence speed is the fastest, which testifies to the fact that the convergence speed of the transition path is faster than the long-term path.