Weak Type Estimates for Intrinsic Square Functions on Weighted Morrey Spaces

In this paper, we will obtain the weak type estimates of intrinsic square func- tions including the Lusin area integral, Littlewood-Paley g-function and g^-function on the weighted Morrey spaces L^1,k （w） for 0〈k〈 1 and w ∈ A1.

BOUNDEDNESS OF STEIN'S SQUARE FUNCTIONS ASSOCIATED TO OPERATORS ON HARDY SPACES

Let （X, d,μ） be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure μ. Let L be a second order non-negative self-adjoint operator on L^2（X）. Assume that the semigroup e^-tL generated by L satisfies the Davies-Gaffney estimates. Also, assume that L satisfies Plancherel type estimate. Under these conditions, we show that Stein＇s square function Gδ（L） arising from Bochner-Riesz means associated to L is bounded from the Hardy spaces HL^p（X） to L^p（X） for all 0 〈 p ≤ 1.

Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H x(R n)the associated Hardy-type space.In this article,we first establish the finite atomic characterization of H x(R n).As an application,we prove that the dual space of H x(Rn)is the Campanato space associated with X.For any given a∈(0,1]and s∈Z+,using the atomic and the Littlewood—Paley function characterizations of H x(Rn),we also establish its 5-order intrinsic square function characterizations,respectively,in terms of the intrinsic Lusin-area function S a,s,the intrinsic g-function g a,s,and the intrinsic g*λ-function g*λ,a,s,whereλcoincides with the best known range.

On the bilinear square Fourier multiplier operators and related multilinear square functions

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm（f1, f2）（x） =（∫0^∞︱∫（Rn）^2）e^2πix·（ξ1＋ξ2））m（tξ1, tξ2）f1（ξ1）f2（ξ2）dξ1dξ2︱^2dt/t）^1/2.Let s be an integer with s ∈ [n ＋ 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏i^2=1ω^i^p/p） and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p（νω） if p0 〈 p1, p2 〈 ∞ with 1/p = 1/p1 ＋ 1/p2. Moreover,if p0 〉 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L^p1（ω1） × L^p2（ω2） to L^p,∞（νω）. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.

Intrinsic Square Function Characterizations of Several Hardy-Type Spaces-a Survey

In this article,the authors give a survey about the recent developments of intrinsic square function characterizations and their applications on several Hardy-type spaces,including(weak)Musielak-Orlicz Hardy spaces,variable(weak)Hardy spaces,and Hardy spaces associated with ball quasi-Banach function spaces.The authors also present some open problems.

Hardy space estimates for bi-parameter Littlewood-Paley square functions

Suppose that g(f)are bi-parameter Littlewood-Paley square functions which were introduced by H.Martikainen.It is known that the L^2(R^n×R^m)boundedness and the H1(R^n×R^m)-L1(R^n×R^m)boundedness of g(f)have been proved by H.Martikainen and by Z.Li and Q.Xue,respectively.In this paper,we apply the vector-valued theory,the atomic decomposition of product Hardy spaces,and Journe’s covering lemma to show that g(f)are bounded from H^p(R^n×R^m)to Lp(R^n×R^m)with p smaller than 1.

Regional Logistics Demand Forecast Based on Least Square and Radial Basis Function

Regional logistics demand forecast is the basis for government departments to make logistics planning and logistics related policies.It has the characteristics of a small amount of data and being nonlinear,so the traditional prediction method can not guarantee the accuracy of prediction.Taking Xiamen City as an example,this paper selects the primary industry,the secondary industry,the tertiary industry,the total amount of investment in fixed assets,total import and export volume,per capita consumption expenditure,and the total retail sales of social consumer goods as the influencing factors,and uses a combining model least square and radial basis function(LS-RBF)neural network to analyze the related data from years 2000 to 2019,so as to predict the logistics demand from years 2020 to 2024.The model can well fit the training data,and the experimental results obtained from the comparison between the predicted value and the actual value in 2019 show that the error rate is very small.Therefore,the prediction results are reasonable and reliable.This method has high prediction accuracy,and it is suitable for irregular regional logistics demand forecast.

AREA FUNCTIONS ON HARDY SPACES ASSOCIATED TO SCHRDINGER OPERATORS

In this paper, the author gives a characterization of atomic Hardy spaces associated to Schrodinger operators by using area functions, and hence gets the dual spaces of atomic Hardy spaces.

AREA INTEGRAL FUNCTIONS FOR SECTORIAL OPERATORS ON L^p SPACES

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^∞ functional calculus of seetorial operators on LP-spaces hold true when the square functions are replaced by the area integral functions.

WEAK BOUNDS FOR ROUGH SQUARE AND MAXIMAL OPERATORS

With Ω ∈ L (log + L)(S n-1 ) and suitable h ∈ L γ (R 1)(1<γ≤2),the weak type (1,1) of the square function g(f)(x) =k｜ψ k*f｜ 2 12(x) and the maximal operator M ψ(f)(x) = sup k｜ψ k｜*｜f｜(x) where ψ(x)=｜x｜ -n Ω(x)h(｜x｜),ψ k(x)=ψ 2 k (x), are studied in this paper.As a corollary,the weak bounds of M Ω(f) proved by Christ in 1988 are given and the previous weak type results for M ψ(f)(x) are improved.In addition,the weighted weak type (1,1) estimates of the Littlewood Paley function g ψ(f) with power weights is also proved.

Least squares weighted twin support vector machines with local information

A least squares version of the recently proposed weighted twin support vector machine with local information(WLTSVM) for binary classification is formulated. This formulation leads to an extremely simple and fast algorithm, called least squares weighted twin support vector machine with local information(LSWLTSVM), for generating binary classifiers based on two non-parallel hyperplanes. Two modified primal problems of WLTSVM are attempted to solve, instead of two dual problems usually solved. The solution of the two modified problems reduces to solving just two systems of linear equations as opposed to solving two quadratic programming problems along with two systems of linear equations in WLTSVM. Moreover, two extra modifications were proposed in LSWLTSVM to improve the generalization capability. One is that a hot kernel function, not the simple-minded definition in WLTSVM, is used to define the weight matrix of adjacency graph, which ensures that the underlying similarity information between any pair of data points in the same class can be fully reflected. The other is that the weight for each point in the contrary class is considered in constructing equality constraints, which makes LSWLTSVM less sensitive to noise points than WLTSVM. Experimental results indicate that LSWLTSVM has comparable classification accuracy to that of WLTSVM but with remarkably less computational time.

ON THE CONCAVE φ-INEQUALITIES FOR NONNEGATIVE SUBMARTINGALES

Let 1,2 be nonnegative nondecreasing functions, and 1 be concave. Theauthors prove the equivalence of the following two conditions:(i) E1(Mf) < cE2(Zo+A) for every nonnegative submartingale f = (fn)n>o with it'sDoob's Decomposition: f= Z + A, where Z is a martingale in L1 and A is a nonnegativeincrasing and predictable process.(ii) There exists positive constants c, to such that > to.When 1 =2 the condition (ii) above is equivalent to the classical condition p < 1. Asa consequence, for a concave function ,if and only if E1(Mf) < cE2(Zo+A)for every nonnegative submartingale f.

THE LAW OF THE ITERATED LOGARITHM FOR WAVELET SERIES

We prove a law of iterated logarithm for wavelet series:holds almost everywhere on {x ∈ RN; S(f) = ∞}.

A computational model of the human glucose-insulin regulatory system

Objective：A computational model of insulin secretion and glucose metabolism for assisting the diagnosis of diabetes mellitus in clinical research is introduced.The proposed method for the estimation of parameters for a system of ordinary differential equations（ODEs）that represent the time course of plasma glucose and insulin concentrations during glucose tolerance test（GTT）in physiological studies is presented.The aim of this study was to explore how to interpret those laboratory glucose and insulin data as well as enhance the Ackerman mathematical model.Methods：Parameters estimation for a system of ODEs was performed by minimizing the sum of squared residuals（SSR）function,which quantifies the difference between theoretical model predictions and GTT＇s experimental observations.Our proposed perturbation search and multiple-shooting methods were applied during the estimating process.Results：Based on the Ackerman＇s published data,we estimated the key parameters by applying R-based iterative computer programs.As a result,the theoretically simulated curves perfectly matched the experimental data points.Our model showed that the estimated parameters,computed frequency and period values,were proven a good indicator of diabetes.Conclusion：The present paper introduces a computational algorithm to biomedical problems,particularly to endocrinology and metabolism fields,which involves two coupled differential equations with four parameters describing the glucose-insulin regulatory system that Ackerman proposed earlier.The enhanced approach may provide clinicians in endocrinology and metabolism field insight into the transition nature of human metabolic mechanism from normal to impaired glucose tolerance.

The improved element-free Galerkin method forthree-dimensional wave equation

The paper presents the improved element-free Galerkin （IEFG） method for three-dimensional wave propa- gation. The improved moving least-squares （IMLS） approx- imation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares （MLS） approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a dis- cretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scal- ing parameter, number of nodes and the time step length are considered for the convergence study.

A data-derived soft-sensor method for monitoring effluent total phosphorus

The effluent total phosphorus(ETP) is an important parameter to evaluate the performance of wastewater treatment process(WWTP). In this study, a novel method, using a data-derived soft-sensor method, is proposed to obtain the reliable values of ETP online. First, a partial least square(PLS) method is introduced to select the related secondary variables of ETP based on the experimental data. Second, a radial basis function neural network(RBFNN) is developed to identify the relationship between the related secondary variables and ETP. This RBFNN easily optimizes the model parameters to improve the generalization ability of the soft-sensor. Finally, a monitoring system, based on the above PLS and RBFNN, named PLS-RBFNN-based soft-sensor system, is developed and tested in a real WWTP. Experimental results show that the proposed monitoring system can obtain the values of ETP online and own better predicting performance than some existing methods.

The Impact of the Earth’s Movement through the Space on Measuring the Velocity of Light

Goal of this experiment is basically measuring the velocity of light. As usual we will measure two-way velocity of light (from A to B and back). In contrast to the similar experiments we will not assume that speeds of light from A to B and from B to A are equal. To achieve this we will take into account Earth’s movement through the space, rotation around its axis and apply “least squares method for cosine function”, which will be explained in Section 9. Assuming that direction East-West is already known, one clock, a source of light and a mirror, is all equipment we need for this experiment.

Three-Dimensional Meshfree Analysis of Interlocking Concrete Blocks for Step Seawall Structure

This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional(3D)complex geometry structures,which are the interlocking concrete blocks of step seawall.The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain.The 3D moving least square(MLS)approximation is applied to build the interpolation functions of unknowns.The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results.The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method(FEM)using linear shape functions on tetrahedral elements and the well-known commercial software,ANSYS.The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM.The studied meshfree method is effective in the analysis of static responses of complex geometry structures.The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes.

ONTHEPROPERTIESOFZCIRCULANTMATRICES

Some properties of Z circulant matrices are given.

A numerical approach based on the meshless collocation method in elastodynamics

In this paper, a collocation technique with the modified equilibrium on line method （ELM） for imposition of Neumann （natural） boundary conditions is presented for solving the two-dimensional problems of linear elastic body vibrations. In the modified ELM, equilibrium over the lines on the natural boundary is satisfied as Neumann boundary condition equations. In other words, the natural boundary conditions are satisfied naturally by using the weak formulation. The performance of the modified version of the ELM is studied for collocation methods based on two different ways to construct meshless shape functions： moving least squares approximation and radial basis point interpolation. Numerical examples of two-dimensional free and forced vibration analyses show that by using the modified ELM, more stable and accurate results would be obtained in comparison with the direct collocation method.