Variable Spaces Research Articles

Component Analysis and Accuracy Improvement of Linear Power Flow Equation Based on Legendre Polynomial Expansion

The linearization of the power flow equation is widely used in power industry. Existing linearization approaches are generally regarded as nonlinear state variable space transformation, where the original power flow equation can be approximately reformulated as the linear forms in transformed state variable spaces. However, the optimality of linear power flow equations cannot be theoretically compared in different spaces and is only verified based on case studies. In this paper, based on Legendre polynomial expansion, power flow equations are uniformly transformed into a single state variable space, where dimensions correspond to certain orders of Legendre expansion. Through component analysis of power flow equations on each dimension, we find that the accuracy of the linear power flow equation can be improved by formulating an appropriate state variable space transformation to reduce Legendre component loss. Based on this, a linearization method minimizing Legendre component loss is formulated. Compared with the regular case-based linearization, the proposed method does not depend on empirical data, and can theoretically guide the selection of the linear power flow equation within the specified operating bound. The accuracy performance is verified in OPF calculation based on numerous IEEE and Polish test systems.

On Sobolev theorem for higher commutators of fractional integrals in grand variable Herz spaces

The higher order commutators of fractional integral operator Iζ(⋅),Vm of variable order is shown to be bounded from the grand variable Herz spaces K̇p(⋅)a(⋅),u,θ(Rn) into the weighted space K̇ρ,q(⋅)a(⋅),u,θ(Rn), where ρ=(1+|z1|)−λ and 1q(z)=1p(z)−β(z)n when p(z) is not necessarily constant at infinity.

Boundedness of Commutators of Integral Operators of Fractional Type and $$M_{\alpha , L^{r}\log L}$$ Maximal Operator in Variable Lebesgue Spaces

Boundedness of Commutators of Integral Operators of Fractional Type and $$M_{\alpha , L^{r}\log L}$$ Maximal Operator in Variable Lebesgue Spaces

Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator

ABSTRACT The Dirichlet boundary value problem for elliptic equations involving the (p(x), q(x))-Laplacian operator with a reaction term depending on the gradient and on two real parameters is considered in this paper. Using the topological degree theory for a class of demicontinuous operators of generalized (S +) and the theory of the variable exponent Sobolev spaces, we prove the existence of at least one weak solution of such problem.

Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces

AbstractThe main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

Some results concerning Riesz–Bessel transforms of high order

In the present study, we introduce the sharp function related to the Laplace–Bessel differential operator and investigate its properties on the variable Lebesgue spaces. Moreover, we obtain that Riesz–Bessel transforms of high order are bounded on variable Lebesgue spaces .

Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

AbstractLet be a complete probability space and let be two variable exponents. In this paper, we investigate several new variable Hardy–Lorentz martingale spaces associated with the variable Lorentz spaces , which are defined by nonincreasing rearrangement functions. To be precise, we first formulate the atomic decomposition theorems for these Hardy–Lorentz martingale spaces, and then establish martingale inequalities among them with the help of the boundedness of a σ‐sublinear operator. Furthermore, we prove the boundedness of fractional integrals on variable Hardy–Lorentz martingale spaces .

A Cyber-Physical monitoring and diagnosis scheme of energy consumption in Plant-Wide chemical processes

In the large-scale plant-wide chemical process, energy monitoring and diagnosis have a great impact on energy management and sustainable development. Most monitoring and diagnosis methods focused on the construction of the global model with process data, regardless of the meaningful process knowledge and in-depth analysis. Due to the multi-dimensional, correlative, and uncertain characteristics of the collected industrial data, it is laborious to obtain accurate and reliable energy monitoring and diagnosis results. To address these problems, a novel cyber-physical energy monitoring and diagnosis scheme is proposed in this paper. This scheme constructs a cyber-physical model based on process knowledge and data to describe the cause-effect relationships among the energy variables. For energy monitoring, a distributed monitoring approach is developed for energy state estimation based on the constructed monitoring statistics in both variable and residual spaces. For energy diagnosis, a faulty contribution degree index is further addressed and the corresponding root cause location strategy is discussed. The effectiveness and practicality of the proposed scheme are demonstrated via a numerical simulation example and practical ethylene oxide production. Two preset simulated faults and one practical process fault are used and all the abnormal states and root variables are monitored and diagnosed by the proposed scheme. The energy monitoring and diagnosis results provide great support for energy management and development in large-scale chemical plants.

Variable exponent Sobolev spaces and regularity of domains-II

We provide necessary conditions on Euclidean domains for inclusions W1,p(·)(Ω)↪Lq(·)(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p(\\cdot )}(\\Omega ) \\hookrightarrow L^{q(\\cdot )}(\\Omega ) $$\\end{document} of variable exponent Sobolev spaces. The conditions on the exponent p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot ) $$\\end{document} are log-Hölder and log-log-Hölder continuity, while those on the domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Omega $$\\end{document} are the measure and the log measure density conditions. Restrictions on the exponents q(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q(\\cdot ) $$\\end{document} and p(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p(\\cdot )$$\\end{document} appearing in Górka et al. (J. Geom. Anal. 310: 7304-7319, 2021) are relaxed, improving the results obtained in that work.

Local grand variable exponent Lebesgue spaces

We introduce local grand variable exponent Lebesgue spaces, where the variable exponent Lebesgue space is “aggrandized” only at a given closed set F of measure zero. It is shown that, for different aggrandizers with positive Matuszewska–Orlicz indices, the corresponding local grand variable exponent Lebesgue spaces coincide. We show that the maximal operator, singular operators, and maximal singular operators are bounded in such spaces. Lastly, an application to a Dirichlet problem for the Poisson equation, where F may be chosen as the boundary of the domain, is provided within the framework of such local grand spaces.

Weighted estimates for square functions associated with operators

AbstractLet L be a non‐negative self‐adjoint operator on . Suppose that the kernels of the analytic semigroup satisfy the upper bound related to a critical function ρ but without any assumptions of smooth conditions on spacial variables. In this paper, we consider the weighted inequalities for square functions associated with L, which include the vertical square functions, the conical square functions and the Littlewood–Paley g‐functions. A new bump condition related to the critical function is given for the two‐weighted boundedness of square functions associated with L. Besides, we also prove the weighted inequalities for square functions associated with L on weighted variable Lebesgue spaces with new classes of weights considered in [5]. As applications, our results can be applied to magnetic Schrödinger operator, Laguerre operators.

Existence of Steady Solutions for a Model for Micropolar Electrorheological Fluid Flows with Not Globally log-Hölder Continuous Shear Exponent

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function {hat{p}} that is log -Hölder continuous and an electric field textbf{E} for that vert textbf{E}vert ^2 is bounded and smooth. Note that these conditions do not imply that the variable shear exponent p={hat{p}}circ vert textbf{E}vert ^2 is globally log -Hölder continuous.

Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces

AbstractLet be a space of homogeneous type in the sense of Coifman and Weiss, and let be a ball quasi‐Banach function space on , which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space , associated with , via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of , which proves to be a ball Campanato‐type function space associated with . All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of , the authors further prove that can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space.

On the interpolation constants for variable Lebesgue spaces

AbstractFor and variable exponents and with values in [1, ∞], let the variable exponents be defined by The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space to the variable Lebesgue space for , then where C is an interpolation constant independent of T. We consider two different modulars and generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that and , as well as, lead to sufficient conditions for and . We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that , are Lipschitz continuous and bounded away from one and infinity (in this case, ).

ON PROPERTIES OF SPACES WITH MARTINGALE MIXED NORM WITH SUMMABILITY EXPONENTS CLOSE TO ONE

For spaces of random variables with martingale mixed norm the theorem of A. Pelczynski on the absence of an unconditional basis in the space $$L_1(0,1)$$ is generalized and (in terms of summability exponents) the cases are studied when these spaces coincide with the space $$L_1$$ . 2020 Mathematics subject classification: Primary 60G46; Secondary 60G42, 46E30.

Dual Spaces of Anisotropic Variable Hardy–Lorentz Spaces and Their Applications

Let A be a general expansive matrix on $$\mathbb {R}^n$$ and $$H_A^{p(\cdot ),q}(\mathbb R^n)$$ be the anisotropic variable Hardy–Lorentz space associated with A, where $$p(\cdot ):\ \mathbb R^n\rightarrow (0,\infty ]$$ denotes a variable exponent function satisfying the globally log-Hölder continuous condition and $$q\in (0,\infty )$$ . In this article, the authors give the appropriate dual space of $$H_A^{p(\cdot ),q}(\mathbb {R}^n)$$ with full range $$p(\cdot )$$ by introducing some anisotropic variable $$\eta $$ -type Campanato spaces. As an application, the Carleson measure characterizations of these $$\eta $$ -type Campanato spaces are established. To achieve these, the authors first deduce several equivalent characterizations of the anisotropic variable $$\eta $$ -type Campanato space, and then introduce the anisotropic variable tent-Lorentz spaces and establish their atomic decomposition. All these results are new even for the isotropic Hardy–Lorentz space $$H^{p,q}(\mathbb {R}^n)$$ and the isotropic variable Hardy–Lorentz space $$H^{p(\cdot ),q}(\mathbb {R}^n)$$ .

Scenography and the effectiveness of meaning in the theatrical space

The elements of theater formation that fall within the spatial experience of the scenography of the show, which the directors work in in the imaginary theater, are important and have an aesthetic, intellectual and cognitive dimension, working to highlight reality in an aesthetic image surrounding space and space. And its relationship to the distinct, multiple and variable spaces above the stage, to produce theatrical signals and endless meanings through the possibility of infinite reconfiguration of the theater's space and its public and private space through the distribution of a group of blocks within the scenic image.I dealt with in the first chapter (the methodological framework), which includes the research problem identified by the researcher, the importance of the research and the aim of the research, then the researcher determined the limits of the research and defined the terms and then defined them procedurally. The researcher also presented in the second chapter (the theoretical framework) two main topics: "Knowing scenography historically", the second topic is "the meaning of scenography in theatrical performance", and in the third topic "the aesthetics of scenography in the theatrical performance system", and then reviewed the most prominent indicators that resulted from the theoretical framework. The third chapter deals with the research procedures, and determines the research sample, the play "Red Carpet", scripted and directed by Dr. Jabbar Judy. And then in the fourth chapter, the results and conclusions were extracted, and recommendations and suggestions were made, in order to finally fix the references and sources of the research

Weak Type Estimate of Singular Integral Operators on Variable Weak Herz-Type Hardy Spaces

This paper is concerned with the boundedness properties of singular integral operators on variable weak Herz spaces and variable weak Herz-type Hardy spaces. Allowing our parameters to vary from point to point will raise extra difficulties, which, in general, are overcome by imposing regularity assumptions on these exponents, either at the origin or at infinity. Our results cover the classical results on weak Herz-type Hardy spaces with fixed exponents.

Existence result for a Neumann boundary value problem governed by a class of p ( x )-Laplacian-like equation

In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in Ω , ∂ u ∂ η = 0 on ∂ Ω , where Δ p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ, μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ), η is the outer unit normal to ∂ Ω and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.

Characterization of Lipschitz Space via the Commutators of Fractional Maximal Functions on Variable Lebesgue Spaces

We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol b belongs a variable Lipschitz space of pointwise type if and only if b belongs to a variable Lipschitz space of integral type under certain assumptions.