The Dual of the Two-Variable Exponent Amalgam Spaces (Lq(),lp())(Ω)

Wiener amalgam spaces are a class of function spaces where the function’s local and global behavior can be easily distinguished. These spaces are ex-tensively used in Harmonic analysis that originated in the work of Wiener. In this paper: we first introduce a two-variable exponent amalgam space (Lq(),lp())(Ω). Secondly, we investigate some basic properties of these spaces, and finally, we study their dual.

Multilinear Calder′on Zygmund operators on variableexponent Morrey spaces over domains

The boundedness of multilinear singular integrals of Calder′on-Zygmund type onproduct of variable exponent Lebesgue spaces over both bounded and unbounded domains areobtained. Further more, the boundedness for this type multilinear operators on product ofvariable exponent Morrey spaces over domains is shown in the paper.

Parameterized Littlewood-Paley Operators and Their Commutators on Lebesgue Spaces with Variable Exponent

In this paper, by applying the technique of the sharp maximal function and the equivalent representation of the norm in the Lebesgue spaces with variable exponent, the boundedness of the parameterized Litflewood-Paley operators, including the parameterized Lusin area integrals and the parameterized Littlewood-Paley gλ^＊- functions, is established on the Lebesgue spaces with variable exponent. Furthermore, the boundedness of their commutators generated respectively by BMO functions and Lipschitz functions are also obtained.

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

In this article, by extending classical Dellacherie＇s theorem on stochastic se- quences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis in- equality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier＇s inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

THE BOUNDEDNESS FOR A CLASS OF ROUGH FRACTIONAL INTEGRAL OPERATORS ON VARIABLE EXPONENT LEBESGUE SPACES

In this paper, we will discuss the behavior of a class of rough fractional integral operators on variable exponent Lebesgue spaces,and establish their boundedness from Lp1 （＇） （Rn） to Lp2（） （Rn）.

Boundedness for Commutators of Calder o′n-Zygmund Operator on Morrey Spaces with Variable Exponent

Our aim in the present paper is to prove the boundedness of commutators on Morrey spaces with variable exponent. In order to obtain the result, we clarify a relation between variable exponent and BMO norms.

DOOB’S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

In this paper we deal with the martingales in variable Lebesgue space over a probability space.We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space.The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob’s maximal operators in martingale Lebesgue space with a variable exponent.In particular,we present two kinds of weak-type Doob’s maximal inequalities and some necessary and sufficient conditions for strong-type Doob’s maximal inequalities.Finally,we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.

Variable dimensional state space based global path planning for mobile robot

A variable dimensional state space（VDSS） has been proposed to improve the re-planning time when the robotic systems operate in large unknown environments.VDSS is constructed by uniforming lattice state space and grid state space.In VDSS,the lattice state space is only used to construct search space in the local area which is a small circle area near the robot,and grid state space elsewhere.We have tested VDSS with up to 80 indoor and outdoor maps in simulation and on segbot robot platform.Through the simulation and segbot robot experiments,it shows that exploring on VDSS is significantly faster than exploring on lattice state space by Anytime Dynamic A＊（AD＊） planner and VDSS is feasible to be used on robotic systems.

ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

Let L:=-△+V be the Schrodinger operator on R^(n)with n≥3,where V is a non-negative potential satisfying△^(-1)(V)∈L^(∞)(R^(n)).Let w be an L-harmonic function,determined by V,satisfying that there exists a positive constantδsuch that,for any x∈Rn,0<δ≤w(x)≤1.Assume that p(·):R^(n)→(0,1]is a variable exponent satisfying the globally log-Hölder continuous condition.In this article,the authors show that the mappings HL^(p)(·))(R^(n))■f■wf∈H^(p)(·)(R^(n))and HL^(p(·))(R^(n))■f■(-△)^(1/2)L^(-1/2)(f)∈H^(p(·))(R^(n))are isomorphisms between the variable Hardy spaces HL^(p(·))(R^(n)),associated with L,and the variable Hardy spaces H^(p(·))(R^(n)).

Experimental Study on Interference Effect of Three Risers Arranged in Tandem with Variable Spacing Ratios

Taking the three-riser group arranged in tandem as the research subject,an experimental study was carried out on the risers arranged in tandem.The purpose is to explore the sensitivity of the dynamic response of each riser to spacing ratio and reveal the physical mechanism of riser groups under the interference effect.The spacing ratios of the adj acent risers are 4.0,5.0,6.0,and 8.0.At the spacing between the risers of 4.0D,the strong feedback effect increases the cross-flow(CF) displacement amplitude of the upstream riser.The shielding effect is the key factor affecting the interference effect on the midstream and downstream risers.At low reduced velocities,the shielding area initially appears,the displacement amplitude of the midstream and downstream risers varies greatly,the vibration of the two risers is still dominated by the first-order mode,and the transition between adjacent vibration modes is restrained.The multi-frequency superposition phenomenon is very significant at high reduced velocities.The most sensitive interference spacing under the test conditions is obtained.Due to the separation of the incoming flow and the double shielding effect of the upstream and midstream risers,the regular vortex-induced vibration in the wake area of the downstream riser is broken,and the vibration in the two directions is weakened.In general,the interference effect is more significant for the CF vibration of the three-riser groups than the in-line(IL) vibration.

On the Strong Law of Large Numbers for Non-Independent B-Valued Random Variables

This paper investigates some conditions which imply the strong laws of large numbers for Banach space valued random variable sequences. Some generalizations of the Marcinkiewicz-Zygmund theorem and the Hoffmann-J?rgensen and Pisier theorem are obtained. Key words strong law of large numbers - Banach space valued random variable sequence - p-smoothable Banach space CLC number O 211.4 - O 211.6 Foundation item: Supported by the National Natural Science Foundation of China (10071058)Biography: Gan Shi-xin (1939-), male, Professor, research direction: martingale theory, probability limiting theory and Banach space geometry theory.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

Blood-based magnetohydrodynamic Casson hybrid nanofluid flow on convectively heated bi-directional porous stretching sheet with variable porosity and slip constraints

Fluid flow through porous spaces with variable porosity has wide-range applications,notably in biomedical and thermal engineering,where it plays a vital role in comprehending blood flow dynamics within cardiovascular systems,heat transfer and thermal management systems improve efficiency using porous materials with variable porosity.Keeping these important applications in view,in current study blood-based hybrid nanofluid flow has considered on a convectively heated sheet.The sheet exhibits the properties of a porous medium with variable porosity and extends in both the x and y directions.Blood has used as base fluid in which the nanoparticles of Cu and Cu O have been mixed.Thermal radiation,space-dependent,and thermal-dependent heat sources have been incorporated into the energy equation,while magnetic effects have been integrated into the momentum equations.Dimensionless variables have employed to transform the modeled equations into dimensionless form and facilitating their solution using bvp4c approach.It has concluded in this study that,both the primary and secondary velocities augmented with upsurge in variable porous factor and declined with escalation in stretching ratio,Casson,magnetic,and slip factors along x-and y-axes.Thermal distribution has grown up with upsurge in Casson factor,magnetic factor,thermal Biot number,and thermal/space-dependent heat sources while has retarded with growth in variable porous and stretching ratio factors.The findings of this investigation have been compared with the existing literature,revealing a strong agreement among present and established results that ensured the validation of the model and method used in this work.

Attitude tracking control for variable structure near space vehicles based on switched nonlinear systems

An adaptive robust attitude tracking control law based on switched nonlinear systems is presented for a variable structure near space vehicle （VSNSV） in the presence of uncertainties and disturbances. The adaptive fuzzy systems are employed for approximating unknown functions in the flight dynamic model and their parameters are updated online. To improve the flight robust performance, robust controllers with adaptive gains are designed to compensate for the approximation errors and thus they have less design conservation. Moreover, a systematic procedure is developed for the synthesis of adaptive fuzzy dynamic surface control （DSC） approach. According to the common Lyapunov function theory, it is proved that all signals of the closed-loop system are uniformly ultimately bounded by the continuous controller. The simulation results demonstrate the effectiveness and robustness of the proposed control scheme.

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.

EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p(t)-LAPLACIAN

We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.

Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-H¨older continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

A Poincaré Inequality in a Sobolev Space with a Variable Exponent

Let Ω be a domain in RN. It is shown that a generalized Poincaré inequality holds in cones contained in the Sobolev space Wl,P（ ）（Ω）, where p（.） ： Ω → [1, ∞[ is a variable exponent. This inequality is itself a corollary to a more general result about equivalent norms over such cones. The approach in this paper avoids the difficulty arising from the possible lack of density of the space ;D（Ω） in the space {v ∈ Wl,P（ ）（Ω）; tr v = 0 on δΩ}. Two applications are also discussed.

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.

Global Well-posedness of Generalized Magnetohydrodynamics Equations in Variable Exponent Fourier–Besov–Morrey Spaces

A generalized incompressable magnetohydrodynamics system is considered in this paper.Furthermore, results of global well-posednenss are established with the aid of Littlewood–Paley decomposition and Fourier localization method in mentioned system with small initial condition in the variable exponent Fourier–Besov–Morrey spaces. Moreover, the Gevrey class regularity of the solution is also achieved in this paper.