13,429 results on '"Compact space"'
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2. Some classes of topological spaces related to zero-sets
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F. Golrizkhatami and Ali Taherifar
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zero-set ,almost p-space ,compact space ,z-embedded subset ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - Abstract
An almost P-space is a topological space in which every zero-set is regular-closed. We introduce a large class of spaces, C-almost P-space (briefly CAP-space), consisting of those spaces in which the closure of the interior of every zero-set is a zero-set. In this paper we study CAP-spaces. It is proved that if X is a dense and Z#-embedded subspace of a space T, then T is CAP if and only if X is a CAP and CRZ-extended in T (i.e, for each regular-closed zero-set Z in X, clTZ is a zero-set in T). In 6P.5 of [8] it was shown that a closed countable union of zero-sets need not be a zero-set. We call X a CZ-space whenever the closure of any countable union of zero-sets is a zero-set. This class of spaces contains the class of P-spaces, perfectly normal spaces, and is contained in the cozero complemented spaces and CAP-spaces. In this paper we study topological properties of CZ (resp. cozero complemented)-space and other classes of topological spaces near to them. Some algebraic and topological equivalent conditions of CZ (resp. cozero complemented)-space are characterized. Examples are provided to illustrate and delimit our results.
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- 2022
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3. AMENDMENT TO "LINDELÖF WITH RESPECT TO AN IDEAL" [NEW ZEALAND J. MATH. 42, 115-120, 2012.
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HOQUE, JIARUL and MODAK, SHYAMAPADA
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- *
MATHEMATICS , *COMPACT spaces (Topology) - Abstract
We give a counterexample in this amendment to show that there is an error in consideration of the statement "if f: X → Y and J is an ideal on Y, then f-1(J) = ff-1(J): J ℇ J- is an ideal on X" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein. [ABSTRACT FROM AUTHOR]
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- 2023
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4. Efficient closed-form estimation of large spatial autoregressions
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Abhimanyu Gupta
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FOS: Computer and information sciences ,91B72, 62P20 ,Economics and Econometrics ,Applied Mathematics ,Econometrics (econ.EM) ,Function (mathematics) ,Parameter space ,Newton's method in optimization ,Least squares ,Methodology (stat.ME) ,FOS: Economics and business ,Compact space ,Autoregressive model ,Sample size determination ,Applied mathematics ,Statistics - Methodology ,Economics - Econometrics ,Mathematics ,Central limit theorem - Abstract
Newton-step approximations to pseudo maximum likelihood estimates of spatial autoregressive models with a large number of parameters are examined, in the sense that the parameter space grows slowly as a function of sample size. These have the same asymptotic efficiency properties as maximum likelihood under Gaussianity but are of closed form. Hence they are computationally simple and free from compactness assumptions, thereby avoiding two notorious pitfalls of implicitly defined estimates of large spatial autoregressions. For an initial least squares estimate, the Newton step can also lead to weaker regularity conditions for a central limit theorem than those extant in the literature. A simulation study demonstrates excellent finite sample gains from Newton iterations, especially in large multiparameter models for which grid search is costly. A small empirical illustration shows improvements in estimation precision with real data., 36 pages
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- 2023
5. Testing Linear-Invariant Properties
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Yufei Zhao and Jonathan Tidor
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FOS: Computer and information sciences ,Property testing ,Discrete mathematics ,Conjecture ,General Computer Science ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,01 natural sciences ,Prime (order theory) ,Computer Science - Computational Complexity ,Compact space ,Integer ,010201 computation theory & mathematics ,Bounded function ,FOS: Mathematics ,Mathematics - Combinatorics ,Degree of a polynomial ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Abstract
Fix a prime $p$ and a positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there exists an oblivious tester for this property with one-sided error and constant query complexity. Furthermore, a property is proximity oblivious-testable (PO-testable) if the test is also independent of the proximity parameter $\epsilon$. It is known that a number of natural properties such as linearity and being a low degree polynomial are PO-testable. These properties are examples of linear-invariant properties, meaning that they are preserved under linear automorphisms of the domain. Following work of Kaufman and Sudan, the study of linear-invariant properties has been an important problem in arithmetic property testing. A central conjecture in this field, proposed by Bhattacharyya, Grigorescu, and Shapira, is that a linear-invariant property is testable if and only if it is semi subspace-hereditary. We prove two results, the first resolves this conjecture and the second classifies PO-testable properties. (1) A linear-invariant property is testable if and only if it is semi subspace-hereditary. (2) A linear-invariant property is PO-testable if and only if it is locally characterized. Our innovations are two-fold. We give a more powerful version of the compactness argument first introduced by Alon and Shapira. This relies on a new strong arithmetic regularity lemma in which one mixes different levels of Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer, Hatami, Hatami, and Lovett by removing the bounded complexity restriction in their work. Our second innovation is a novel recoloring technique called patching. This Ramsey-theoretic technique is critical for working in the linear-invariant setting and allows us to remove the translation-invariant restriction present in previous work., Comment: 40 pages; updated with significantly improved main result
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- 2022
6. Extrapolation of compactness on weighted spaces: Bilinear operators
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Stefanos Lappas, Tuomas Hytönen, Tuomas Hytönen / Principal Investigator, and Department of Mathematics and Statistics
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Pure mathematics ,General Mathematics ,COMMUTATORS ,Mathematics::Classical Analysis and ODEs ,Extrapolation ,Bilinear interpolation ,NORM INEQUALITIES ,47B38 (Primary), 42B20, 42B35, 46B70, 47H60 ,Space (mathematics) ,Multilinear Muckenhoupt weights ,01 natural sciences ,Rubio de Francia extrapolation ,Compact operators ,111 Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Lp space ,Mathematics ,Calderon-Zygmund operators ,Fractional integral operators ,010102 general mathematics ,Muckenhoupt weights ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Range (mathematics) ,Compact space ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Fourier multipliers ,INTEGRAL-OPERATORS - Abstract
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages
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- 2022
7. Strong solutions of forward–backward stochastic differential equations with measurable coefficients
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Peng Luo, Ludovic Tangpi, and Olivier Menoukeu-Pamen
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Statistics and Probability ,Strong solutions ,Sobolev space ,Stochastic differential equation ,Compact space ,Applied Mathematics ,Modeling and Simulation ,Bounded function ,Applied mathematics ,Differentiable function ,Malliavin calculus ,Mathematics ,Variable (mathematics) - Abstract
This paper investigates solvability of fully coupled systems of forward–backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.
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- 2022
8. On the descriptive complexity of Salem sets
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Alberto Marcone and Manlio Valenti
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Algebra and Number Theory ,Degree (graph theory) ,Dimension (graph theory) ,Hausdorff space ,Dynamical Systems (math.DS) ,Mathematics - Logic ,Descriptive complexity theory ,Ambient space ,Combinatorics ,Compact space ,FOS: Mathematics ,03E15 28A75 28A78 03D32 ,Family of sets ,Mathematics - Dynamical Systems ,Logic (math.LO) ,Mathematics ,Descriptive set theory - Abstract
In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol{\Pi}^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $\boldsymbol{\Pi}^0_3$-complete when we endow the hyperspace of all closed subsets of $\mathbb{R}^d$ with the Fell topology. A similar result holds also for the Vietoris topology., Comment: Extended Lemma 3.1, fixed Lemma 5.3 and improved the presentation of the results. To appear in Fundamenta Mathematicae
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- 2022
9. Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation
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Jiaqi Yang, Changchun Liu, and Ming Mei
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Degenerate diffusion ,Applied Mathematics ,Mathematical analysis ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,010101 applied mathematics ,Compact space ,Rate of convergence ,0103 physical sciences ,Initial value problem ,Development (differential geometry) ,0101 mathematics ,Diffusion (business) ,Degeneracy (mathematics) ,Analysis ,Mathematics - Abstract
This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.
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- 2022
10. Elongation, flatness and compactness indices to characterise particle form
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Vasileios Angelidakis, Stefano Utili, and S Nadimi
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Range (mathematics) ,Compact space ,General Chemical Engineering ,Particle classification ,Flatness (systems theory) ,Particle ,Biological system ,Mathematics - Abstract
A century after the first attempts of Wentworth to characterise the shape of cobbles, our understanding of particle morphology is still expanding. A plethora of shape indices has been proposed in the literature to characterise the morphology of individual particles. This study aims to shed light on the merits and limitations of the indices currently used to characterise particle elongation, flatness and compactness, adopting a unified classification framework. Second, new indices are proposed to address the identified shortcomings. Third, a new particle classification system derived from the proposed indices is illustrated. It is shown the new system overcomes the misclassification of a range of particles that are incorrectly classified as bladed in the Zingg system.
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- 2022
11. Generalized Polynomial Complementarity Problems over a Polyhedral Cone
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Guo-ji Tang, Tong-tong Shang, and Jing Yang
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Polynomial (hyperelastic model) ,Combinatorics ,Control and Optimization ,Compact space ,Cone (topology) ,Applied Mathematics ,Complementarity (molecular biology) ,Theory of computation ,Solution set ,Tensor ,Extension (predicate logic) ,Management Science and Operations Research ,Mathematics - Abstract
The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.
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- 2021
12. Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary
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Monica Musso, Seunghyeok Kim, and Juncheng Wei
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Mathematics - Differential Geometry ,Positive mass theorem ,Boundary (topology) ,Conformal map ,01 natural sciences ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,Mean curvature ,Compactness ,010308 nuclear & particles physics ,Applied Mathematics ,Second fundamental form ,010102 general mathematics ,Mathematical analysis ,Scalar (physics) ,Yamabe problem ,16. Peace & justice ,Compact space ,Differential Geometry (math.DG) ,Boundary Yamabe problem ,Mathematics::Differential Geometry ,Blow-up analysis ,Analysis ,Linear equation ,Analysis of PDEs (math.AP) - Abstract
We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the $C^2$-compactness for all $4$-manifolds (which may be non-umbilic). For the $5$-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the $C^2$-compactness for all $5$-manifolds. Finally, we show that the $C^2$-compactness on $6$-manifolds is true if the trace-free second fundamental form on the boundary never vanishes., Comment: 29 pages, This version treats general 5-manifolds as well, Comments are welcome
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- 2021
13. Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of
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P. V. Paramonov
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Pure mathematics ,Algebra and Number Theory ,Compact space ,Approximations of π ,Order (group theory) ,IMG ,computer.file_format ,computer ,Mathematics - Abstract
Criteria for the uniform approximation of functions by solutions of second-order strongly elliptic equations on compact subsets of are obtained using the method of reduction to similar problems in , which were previously investigated by Mazalov. A number of metric properties of the capacities used are established. Bibliography: 16 titles.
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- 2021
14. A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology
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Philipp Getto, Gergely Röst, and István Balázs
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0303 health sciences ,Differential equation ,Applied Mathematics ,Ode ,State (functional analysis) ,Lipschitz continuity ,01 natural sciences ,Domain (mathematical analysis) ,Cell biology ,010101 applied mathematics ,03 medical and health sciences ,Compact space ,State space ,0101 mathematics ,Invariant (mathematics) ,Analysis ,030304 developmental biology ,Mathematics - Abstract
We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.
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- 2021
15. Boundedness and compactness of commutators associated with Lipschitz functions
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Dongyong Yang, Jianxun He, Huoxiong Wu, and Weichao Guo
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Applied Mathematics ,Commutator (electric) ,Type (model theory) ,Lipschitz continuity ,Space (mathematics) ,Omega ,law.invention ,Combinatorics ,Compact space ,Mathematics - Classical Analysis and ODEs ,law ,Iterated function ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Lp space ,Analysis ,Mathematics - Abstract
Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators., Comment: arXiv admin note: text overlap with arXiv:1712.08292
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- 2021
16. Second-Order Optimality Conditions for Infinite-Dimensional Quadratic Programs
- Author
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Duong Thi Viet An
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Control and Optimization ,Compact space ,Quadratic equation ,Series (mathematics) ,Orthogonality ,Applied Mathematics ,Euclidean geometry ,Theory of computation ,Banach space ,Applied mathematics ,Quadratic programming ,Management Science and Operations Research ,Mathematics - Abstract
Second-order necessary and sufficient optimality conditions for local solutions and locally unique solutions of generalized quadratic programming problems in Banach spaces are established in this paper. Since the decomposition procedures using orthogonality relations in Euclidean spaces and the compactness of finite-dimensional unit spheres, which worked well for finite-dimensional quadratic programs, cannot be applied to the Banach space setting, a series of new constructions and arguments are proposed. These results give a comprehensive extension of the corresponding theorems on finite-dimensional quadratic programs.
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- 2021
17. Dependence on parameters for nonlinear equations—Abstract principles and applications
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Marek Galewski, Michał Bełdziński, and Igor Kossowski
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Dirichlet problem ,General Mathematics ,General Engineering ,Boundary (topology) ,Dirichlet distribution ,Action (physics) ,Nonlinear system ,symbols.namesake ,Compact space ,Euler's formula ,symbols ,Applied mathematics ,Browder–Minty theorem ,Mathematics - Abstract
We provide parameter dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals is also investigated. Applications towards the dependence on parameters for both potential and non-potenial nonlinear Dirichlet boundary problems are given.
- Published
- 2021
18. Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences, without free disposal, and without compact consumption sets
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Nicholas C. Yannelis and Konrad Podczeck
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Consumption (economics) ,Economics and Econometrics ,Transitive relation ,Existence theorem ,Monotonic function ,Exchange economy ,Compact space ,Bounded function ,Completeness (order theory) ,Existence of Walrasian equilibrium ,Mathematical economics ,Continuous inclusion property ,Mathematics ,Public finance - Abstract
We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.
- Published
- 2021
19. Weighted iterated radial composition operators from logarithmic Bloch spaces to weighted‐type spaces on the unit ball
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Stevo Stević and Zhi-jie Jiang
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Unit sphere ,Pure mathematics ,Compact space ,Logarithm ,Iterated function ,General Mathematics ,General Engineering ,Composition (combinatorics) ,Type (model theory) ,Mathematics - Published
- 2021
20. Existence and Compactness of Conformal Metrics on the Plane with Unbounded and Sign-Changing Gaussian Curvature
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Chiara Bernardini
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Plane (geometry) ,General Mathematics ,Conformal map ,Sign changing ,symbols.namesake ,Mathematics - Analysis of PDEs ,Compact space ,FOS: Mathematics ,Gaussian curvature ,symbols ,35J60, 35J15 ,Total curvature ,Analysis of PDEs (math.AP) ,Mathematics ,Bar (unit) ,Mathematical physics - Abstract
We show that the prescribed Gaussian curvature equation in $\mathbb {R}^{2}$ $$ -\varDelta u= (1-|x|^{p}) e^{2u}, $$ has solutions with prescribed total curvature equal to ${{\varLambda }}:={\int \limits }_{\mathbb {R}^{2}}(1-|x|^{p})e^{2u}dx\in \mathbb {R}$ , if and only if $$ p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le{\varLambda}
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- 2021
21. Maximal abelian subalgebras of Banach algebras
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H. G. Dales, W. Żelazko, and H. L. Pham
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Banach function algebra ,Pure mathematics ,Arbitrarily large ,Cardinality ,Compact space ,General Mathematics ,Banach algebra ,Subalgebra ,Abelian group ,Commutative property ,Mathematics - Abstract
Let (Formula presented.) be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras (Formula presented.) such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.). For example, we shall prove that, in the case where (Formula presented.) is an infinite-dimensional, unital Banach function algebra, (Formula presented.) is a maximal abelian subalgebra in infinitely-many closed subalgebras of (Formula presented.) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples (Formula presented.) that are local algebras. We shall also give examples of uniform algebras of the form (Formula presented.), where (Formula presented.) is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras (Formula presented.) such that each (Formula presented.) contains (Formula presented.) as a closed subalgebra and is such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.).
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- 2021
22. Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form
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Utpal Manna and Akash Ashirbad Panda
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Compact space ,Mathematics::Probability ,Weak convergence ,Representation theorem ,Applied Mathematics ,Applied mathematics ,Canonical form ,Large deviations theory ,Martingale (probability theory) ,Navier–Stokes equations ,Noise (electronics) ,Analysis ,Mathematics - Abstract
We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Levy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.
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- 2021
23. Compact embedding theorems and a Lions' type Lemma for fractional Orlicz–Sobolev spaces
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Sabri Bahrouni, J.C. de Albuquerque, Marcos L. M. Carvalho, and Edcarlos D. Silva
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Sobolev space ,Mathematics::Functional Analysis ,Lemma (mathematics) ,Pure mathematics ,Compact space ,Applied Mathematics ,Bounded function ,Embedding ,Type (model theory) ,Space (mathematics) ,Nehari manifold ,Analysis ,Mathematics - Abstract
In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' “vanishing” Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrodinger equations, taking into account unbounded or bounded potentials.
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- 2021
24. On fractional and nonlocal parabolic mean field games in the whole space
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Espen R. Jakobsen and Olav Ersland
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Laplace transform ,Applied Mathematics ,Mathematical analysis ,Space (mathematics) ,Lévy process ,Parabolic partial differential equation ,35Q89, 47G20, 35A01, 35A09, 35Q84, 49L12, 45K05, 35S10, 35K61, 35K08 ,Moment (mathematics) ,Mathematics - Analysis of PDEs ,Compact space ,FOS: Mathematics ,Uniqueness ,Analysis ,Heat kernel ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump L\'evy processes with some $\sigma$-stable like behaviour. Included are $\sigma$-stable processes and fractional Laplace diffusion operators $(-\Delta)^{\frac{\sigma}2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $\sigma\in(1,2)$. We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs uses pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques requires no moment assumptions and uses a weaker topology than Wasserstein., Comment: Major update. A much larger class of anomalous diffusions, local coupling results are now complete, problems are posed in the whole space and not the compact torus. New equicontinuity, $L^1$, and $L^\infty$-results for the Fokker-planck equation are needed and proved by using pure PDE arguments. We now work in a more general metrics than Wasserstein
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- 2021
25. Expansive properties of induced dynamical systems
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Daniel Jardón and Iván Sánchez
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0209 industrial biotechnology ,Pure mathematics ,Dynamical systems theory ,Logic ,02 engineering and technology ,Metric space ,020901 industrial engineering & automation ,Compact space ,Artificial Intelligence ,Metric (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Locally compact space ,Dynamical system (definition) ,Expansive ,Mathematics - Abstract
For a given metric space X, we consider the set of all normal fuzzy sets on X, denoted by F ( X ) . In this work, we study expanding, positively expansive and weakly positively expansive dynamical systems ( X , f ) and how they are reflected in the dynamical system ( F ( X ) , f ˆ ) , where f ˆ is the Zadeh's extension of f and F ( X ) has one of the following metrics: the levelwise metric, the endograph metric, the sendograph metric and the Skorokhod metric. We mainly show that if we consider the following conditions: (i) ( X , f ) is positively expansive (resp. expanding); (ii) ( K ( X ) , f ‾ ) is positively expansive (resp. expanding); (iii) ( F ∞ ( X ) , f ˆ ) is positively expansive (resp. expanding); (iv) ( F 0 ( X ) , f ˆ ) is positively expansive (resp. expanding). Then (iv)⇒ (iii) ⇔ (ii) ⇒ (i). For expanding dynamical systems, we present a compact metric space and a locally compact metric space to show that (i) ⇏ (ii) and (iii) ⇏ (iv), respectively. For positively expansive dynamical systems, there is a compact metric space satisfying that (i) ⇏ (ii), but we don't know if (iii) ⇒ (iv).
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- 2021
26. Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent
- Author
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Chao Zhang, Yuzhou Fang, and Vicentiu Radulescu
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Nonlinear system ,Viscosity ,Double phase ,Compact space ,Variable exponent ,General Mathematics ,Degenerate energy levels ,Type (model theory) ,Scaling ,Mathematical physics ,Mathematics - Abstract
We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,\alpha}$ regularity of viscosity solutions to such equations.
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- 2021
27. New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces
- Author
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Stéphanie Nivoche and Oscar F. Bandtlow
- Subjects
Conjecture ,Degree (graph theory) ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Banach space ,01 natural sciences ,Measure (mathematics) ,Upper and lower bounds ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics - Spectral Theory ,Combinatorics ,Compact space ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,41A46 (Primary) 32A36, 32U20, 32W20, 35P15 (Secondary) ,Complex Variables (math.CV) ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics - Abstract
Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $\epsilon$-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$ and $K$ the asymptotics \begin{equation*} \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation*} where $C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of $K$ and $D$ by certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by Nivoche in 2004 thus settling the asymptotics above at the same time. We shall give a new proof of the asymptotics above for $D$ strictly hyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta's Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman-Weil formula together with an exhaustion procedure by special holomorphic polyhedra., Comment: 34 pages; strengthened result: compact $K$ now only assumed to be non-pluripolar
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- 2021
28. Hardy Factorization in Terms of Multilinear CalderÓN–Zygmund Operators using Morrey Spaces
- Author
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Nguyen Anh Dao and Brett D. Wick
- Subjects
Mathematics::Functional Analysis ,Multilinear map ,Pure mathematics ,Functional analysis ,Constructive proof ,Mathematics::Classical Analysis and ODEs ,Commutator (electric) ,Characterization (mathematics) ,Potential theory ,law.invention ,Compact space ,Factorization ,law ,Analysis ,Mathematics - Abstract
In this paper, we provide a constructive proof of $\mathbf {H}^{1}(\mathbb {R}^{n})$ factorization in terms of multilinear Calderon–Zygmund operators in Morrey spaces. As a direct application, we obtain a characterization of functions in $\text {BMO}(\mathbb {R}^{n})$ via commutators of multilinear Calderon–Zygmund operators. Furthermore, we prove a Morrey compactness characterization of [b, T]l, the commutator in the l-th entry.
- Published
- 2021
29. Quasi-linear functionals on locally compact spaces
- Author
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Svetlana V. Butler
- Subjects
Pure mathematics ,Applied Mathematics ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,46E27, 46G99, 28A25 (Primery ) 28C15 (Secondary) ,Mathematics (miscellaneous) ,Compact space ,Bounded function ,FOS: Mathematics ,Bijection ,Quasi linear ,Locally compact space ,Representation (mathematics) ,Mathematical Physics ,Mathematics - Abstract
This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested in quasi-linear functionals on locally compact non-compact spaces or on compact spaces. We study signed and positive quasi-linear functionals paying close attention to singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on $C_c(X)$ and for bounded quasi-linear functionals on $C_0(X)$ on a locally compact space, and for quasi-linear functionals on $C(X)$ on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also "isometric" when topological measures are finite. Finally, we further study properties of quasi-linear functionals and give an explicit example of a quasi-linear functional., Comment: 30 pages
- Published
- 2021
30. Principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type
- Author
-
Shigui Ruan and Hao Kang
- Subjects
Maximum principle ,Monotone polygon ,Operator (computer programming) ,Spectral theory ,Compact space ,Exponential growth ,Semigroup ,General Mathematics ,Mathematical analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119 , 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.
- Published
- 2021
31. Fixed accuracy estimation of parameters in a threshold autoregressive model
- Author
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Sergey E. Vorobeychikov and Victor Konev
- Subjects
Statistics and Probability ,Estimation ,оценка метода наименьших квадратов ,Compact space ,Autoregressive model ,Ergodicity ,Range (statistics) ,Applied mathematics ,пороговые авторегрессионные модели ,Ellipsoid ,Least squares ,Mathematics ,Parametric statistics - Abstract
For parameters in a threshold autoregressive process, the paper proposes a sequential modification of the least squares estimates with a specific stopping rule for collecting the data for each parameter. In the case of normal residuals, these estimates are exactly normally distributed in a wide range of unknown parameters. On the base of these estimates, a fixed-size confidence ellipsoid covering true values of parameters with prescribed probability is constructed. In the i.i.d. case with unspecified error distributions, the sequential estimates are asymptotically normally distributed uniformly in parameters belonging to any compact set in the ergodicity parametric region. Small-sample behavior of the estimates is studied via simulation data.
- Published
- 2021
32. On weighted compactness of commutator of semi-group maximal function and fractional integrals associated to Schrödinger operators
- Author
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Shifen Wang and Qingying Xue
- Subjects
Combinatorics ,Compact space ,Operator (computer programming) ,Closure (mathematics) ,law ,Group (mathematics) ,General Mathematics ,Commutator (electric) ,Maximal function ,Space (mathematics) ,Compact operator ,law.invention ,Mathematics - Abstract
Let $$\mathcal {T}^*$$ and $$\mathcal {I}_\alpha $$ be the semi-group maximal function and fractional integrals associated to the Schrodinger operator $$-\Delta +V(x)$$ , respectively, with V satisfying an appropriate reverse Holder inequality. In this paper, we show that the commutator of $$\mathcal {T}^*$$ is a compact operator on $$L^p(w)$$ for $$1
- Published
- 2021
33. On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative
- Author
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Mustapha Atraoui and Mohamed Bouaouid
- Subjects
Fractional differential equations ,Measure of noncompactness ,Algebra and Number Theory ,Functional analysis ,Applied Mathematics ,Banach space ,Fixed-point theorem ,Order (ring theory) ,Conformable fractional derivative ,Lipschitz continuity ,Fractional calculus ,Combinatorics ,Compact space ,Cosine family of linear operators ,QA1-939 ,Infinitesimal generator ,Nonlocal conditions ,Analysis ,Mathematics - Abstract
In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.
- Published
- 2021
34. Weak and strong semigroups in structural acoustic Kirchhoff-Boussinesq interactions with boundary feedback
- Author
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J.H. Rodrigues and Irena Lasiecka
- Subjects
Nonlinear system ,Compact space ,Semigroup ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Order (ring theory) ,Acoustic wave ,Space (mathematics) ,Analysis ,Energy (signal processing) ,Mathematics - Abstract
We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.
- Published
- 2021
35. Irreducible decomposition for Markov processes
- Author
-
Kazuhiro Kuwae
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,Markov process ,Absolute continuity ,Measure (mathematics) ,symbols.namesake ,Compact space ,Modeling and Simulation ,symbols ,Ergodic theory ,Irreducibility ,Invariant (mathematics) ,Resolvent ,Mathematics - Abstract
We prove an irreducible decomposition for Markov processes associated with quasi-regular symmetric Dirichlet forms or local semi-Dirichlet forms under the absolute continuity condition of transition probability with respect to the underlying measure. We do not assume the conservativeness nor the existence of invariant measures for the processes. As applications, we establish a concrete expression for Chacon–Ornstein type ratio ergodic theorem for such Markov processes and show a compactness of semi-groups under the Green-tightness of measures in the framework of symmetric resolvent strong Feller processes without irreducibility.
- Published
- 2021
36. On the controllability and stabilization of the Benjamin equation on a periodic domain
- Author
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F. Vielma Leal and Mahendra Panthee
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Domain (mathematical analysis) ,Exponential function ,010101 applied mathematics ,Controllability ,Arbitrarily large ,Compact space ,Exponential growth ,Exponential stability ,0101 mathematics ,Mathematical Physics ,Analysis ,Mathematics - Abstract
The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T . We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in H p s ( T ) , with s ≥ 0 . The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32] .
- Published
- 2021
37. Compactness on Fuzzy Soft r-Minimal Spaces
- Author
-
Islam M. Taha
- Subjects
Compact space ,Computational Theory and Mathematics ,Artificial Intelligence ,Logic ,Signal Processing ,Topology ,Fuzzy logic ,Computer Science Applications ,Mathematics - Published
- 2021
38. Difference of quaternionic weighted composition operators on slice regular Fock spaces
- Author
-
Yu-Xia Liang and J. Wang
- Subjects
Numerical Analysis ,Pure mathematics ,Generalization ,Applied Mathematics ,Nuclear Theory ,Operator theory ,Composition (combinatorics) ,Fock space ,Computational Mathematics ,Compact space ,Physics::Atomic Physics ,Mathematics::Differential Geometry ,Complex plane ,Analysis ,Mathematics - Abstract
Quaternionic Fock space is a useful generalization of the Fock space in the complex plane, which plays an important role in quantum mechanics. In view of quaternionic operator theory this topic att...
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- 2021
39. Fractional elliptic systems with critical nonlinearities
- Author
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Olímpio H. Miyagaki, Mousomi Bhakta, Patrizia Pucci, and Souptik Chakraborty
- Subjects
ground state solution ,positive solutions ,35R11, 35A15, 35B33, 35J60 ,min max method ,energy estimate ,Elliptic systems ,Palais Smale decomposition ,Applied Mathematics ,uniqueness ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Combinatorics ,Mathematics - Analysis of PDEs ,Compact space ,FOS: Mathematics ,Uniqueness ,Nonlocal system, uniqueness, ground state solution, Palais Smale decomposition, energy estimate, positive solutions, min max method ,Nonlocal system ,Ground state ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N}, &(-\Delta)^s v = \frac{\beta}{2_s^*}|v|^{\beta-2}v|u|^{\alpha}+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, v >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $\alpha,\,\beta>1$, $\alpha+\beta=2N/(N-2s)$, and $f,\, g$ are nonnegative functionals in the dual space of $\dot{H}^s(\mathbb{R}^{N})$. When $f=0=g$, we show that the ground state solution of the above system is {\it unique}. On the other hand, when $f$ and $g$ are nontrivial nonnegative functionals with ker$(f)$=ker$(g)$, then we establish the existence of at least two different positive solutions of the above system provided that $\|f\|_{(\dot{H}^s)'}$ and $\|g\|_{(\dot{H}^s)'}$ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system., Comment: 27 pages
- Published
- 2021
40. A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable
- Author
-
Vladimir V. Tkachuk
- Subjects
Pure mathematics ,Compact space ,General Mathematics ,Diagonal ,Countable set ,Complement (complexity) ,Mathematics - Abstract
A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.
- Published
- 2021
41. Generalized Reich–Ćirić–Rus-Type and Kannan-Type Contractions in Cone <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <mi>b</mi> </math>-Metric Spaces over Banach Algebras
- Author
-
Yan Han and Shaoyuan Xu
- Subjects
Pure mathematics ,Article Subject ,General Mathematics ,MathematicsofComputing_GENERAL ,Fixed-point theorem ,Type (model theory) ,Fixed point ,Lipschitz continuity ,Metric space ,Compact space ,Cone (topology) ,QA1-939 ,Computer Science::Programming Languages ,Uniqueness ,Mathematics - Abstract
In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone b -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of X . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone b -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.
- Published
- 2021
42. Variational Principles for Maximization Problems with Lower-semicontinuous Goal Functions
- Author
-
P.S. Kenderov, M. Ivanov, and Julian P. Revalski
- Subjects
Statistics and Probability ,Numerical Analysis ,Applied Mathematics ,Function (mathematics) ,Maximization ,Topological space ,Space (mathematics) ,Complete metric space ,Combinatorics ,Compact space ,Bounded function ,Geometry and Topology ,Equivalence (measure theory) ,Analysis ,Mathematics - Abstract
Let X be a completely regular topological space and f a real-valued bounded from above lower semicontinuous function in it. Let C(X) be the space of all bounded continuous real-valued functions in X endowed with the usual sup-norm. We show that the following two properties are equivalent: X is α-favourable (in the sense of the Banach-Mazur game); The set of functions h in C(X) for which f + h attains its supremum in X contains a dense and Gδ-subset of the space C(X). In particular, property (b) has place if X is a compact space or, more generally, if X is homeomorphic to a dense Gδ subset of a compact space.We show also the equivalence of the following stronger properties: X contains some dense completely metrizable subset; the set of functions h in C(X) for which f + h has strong maximum in X contains a dense and Gδ-subset of the space C(X). If X is a complete metric space and f is bounded, then the set of functions h from C(X) for which f + h has both strong maximum and strong minimum in X contains a dense Gδ-subset of C(X).
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- 2021
43. Approximation of functions and all derivatives on compact sets
- Author
-
Sotiris Armeniakos, Vassili Nestoridis, and Giorgos Kotsovolis
- Subjects
Pure mathematics ,Compact space ,General Mathematics ,Uniform convergence ,Open set ,Holomorphic function ,Order (ring theory) ,Rational function ,Type (model theory) ,Minimax approximation algorithm ,Mathematics - Abstract
In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.
- Published
- 2021
44. Wave Asymptotics for Waveguides and Manifolds with Infinite Cylindrical Ends
- Author
-
Tanya Christiansen and Kiril Datchev
- Subjects
Compact space ,General Mathematics ,Mathematical analysis ,Boundary (topology) ,Scattering theory ,Eigenfunction ,Asymptotic expansion ,Space (mathematics) ,Wave equation ,Mathematics ,Resolvent - Abstract
We describe wave decay rates associated to embedded resonances and spectral thresholds for waveguides and manifolds with infinite cylindrical ends. We show that if the cut-off resolvent is polynomially bounded at high energies, as is the case in certain favorable geometries, then there is an associated asymptotic expansion, up to a $O(t^{-k_0})$ remainder, of solutions of the wave equation on compact sets as $t \to \infty $. In the most general such case we have $k_0=1$, and under an additional assumption on the infinite ends we have $k_0 = \infty $. If we localize the solutions to the wave equation in frequency as well as in space, then our results hold for quite general waveguides and manifolds with infinite cylindrical ends. To treat problems with and without boundary in a unified way, we introduce a black box framework analogous to the Euclidean one of Sjöstrand and Zworski. We study the resolvent, generalized eigenfunctions, spectral measure, and spectral thresholds in this framework, providing a new approach to some mostly well-known results in the scattering theory of manifolds with cylindrical ends.
- Published
- 2021
45. Solvability and Optimal Control of Nonautonomous Fractional Dynamical Systems of Neutral-Type with Nonlocal Conditions
- Author
-
Madhukant Sharma
- Subjects
Pure mathematics ,Dynamical systems theory ,Semigroup ,General Mathematics ,Banach space ,General Physics and Astronomy ,General Chemistry ,Optimal control ,Fractional calculus ,Operator (computer programming) ,Compact space ,General Earth and Planetary Sciences ,Contraction principle ,General Agricultural and Biological Sciences ,Mathematics - Abstract
This paper discusses the solvability and existence of optimal controls of nonautonomous fractional dynamical systems of neutral-type in a general Banach space X along with nonlocal condition. We employ the Banach contraction principle, the classical semigroup theory, and the techniques of fractional calculus to establish the main results. The distinguish features of the presented work are that we establish the main results without imposing the compactness condition on semigroup, the continuity of linear operators $$-A(t)$$ and the strong continuity of operator D. At the end, an example is considered to demonstrate the developed results.
- Published
- 2021
46. Nonlinear perturbations of a periodic fractional Laplacian with supercritical growth
- Author
-
Giovany M. Figueiredo, Ricardo Ruviaro, and Sandra I. Moreira
- Subjects
Truncation ,Applied Mathematics ,Nonlinear perturbations ,Supercritical fluid ,Schrödinger equation ,Nonlinear system ,symbols.namesake ,Variational method ,Compact space ,symbols ,Fractional Laplacian ,Analysis ,Mathematics ,Mathematical physics - Abstract
Our main goal is to explore the existence of positive solutions for a class of nonlinear fractional Schrödinger equation involving supercritical growth given by $$ (- \Delta)^{\alpha} u + V(x)u=p(u),\quad x\in \mathbb{R^N},\ N \geq 1. $$ We analyze two types of problems, with $V$ being periodic and asymptotically periodic; for this we use a variational method, a truncation argument and a concentration compactness principle.
- Published
- 2021
47. The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth
- Author
-
Pawan Kumar Mishra, David G. Costa, and João Marcos do Ó
- Subjects
Continuous function (set theory) ,Applied Mathematics ,Type (model theory) ,Lambda ,Domain (mathematical analysis) ,35B33, 35J65, 35Q55 ,Combinatorics ,Mathematics - Analysis of PDEs ,Compact space ,FOS: Mathematics ,Nehari manifold ,Analysis ,Inclusion map ,Analysis of PDEs (math.AP) ,Sign (mathematics) ,Mathematics - Abstract
In this paper we study the following class of nonlocal problem involving Caffarelli-Kohn-Nirenberg type critical growth $$ L(u)-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\quad \text{in } \mathbb R^N, $$% where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1< q< 2< 4< p=2N/[N+2(b-a)-2]$, $0\leq a< b< a+1< N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div} \big(|x|^{-2a}\nabla u\big) $$ and the function $M\colon \mathbb R^+_0\to\mathbb R^+_0$ is exactly the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta> 0$. The above problem has a double lack of compactness, firstly because of the non-compactness of Caffarelli-Kohn-Nirenberg embedding and secondly due to the non-compactness of the inclusion map $$u\mapsto \int_{\mathbb R^N}h(x)|x|^{-2(a+1)}|u|^2dx,$$ as the domain of the problem in consideration is unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, we prove the existence of at least two positive solutions for suitable choices of parameters $\lambda$ and $\mu$.
- Published
- 2021
48. On the compactness of classes of the solutions of the Dirichlet problem
- Author
-
Evgeny Sevost'yanov and Oleksandr Petrovych Dovhopiatyi
- Subjects
Statistics and Probability ,Dirichlet problem ,Normalization (statistics) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Type (model theory) ,Beltrami equation ,Domain (mathematical analysis) ,Dirichlet distribution ,symbols.namesake ,Compact space ,symbols ,Mathematics - Abstract
Some theorems concerning the compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation and the characteristics of which are compactly supported and satisfy certain constraints of the theoretical-set type, have been proved. As a consequence, we obtained results on the compact classes of solutions of the corresponding Dirichlet problems considered in a certain Jordan domain.
- Published
- 2021
49. Genericity of historic behavior for maps and flows
- Author
-
Paulo Varandas, Maria Carvalho, and Faculdade de Ciências
- Subjects
Transitive relation ,Pure mathematics ,Matemática ,Mathematics::Dynamical Systems ,Continuous map ,Applied Mathematics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,Residual ,Set (abstract data type) ,Compact space ,FOS: Mathematics ,Ergodic theory ,Homoclinic orbit ,Mathematics - Dynamical Systems ,Mathematics ,Mathematical Physics ,Probability measure - Abstract
We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Ma\~n\'e and Shub examples of robustly transitive diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications., Comment: 14 pages, revised and improved version of previous preprint "Minimality and irregular sets"
- Published
- 2021
50. New contribution in fixed point theory via an auxiliary function with an application
- Author
-
Driss El Moutawakil, Amine Jaid, and Youssef Touail
- Subjects
Pure mathematics ,Metric space ,Compact space ,Differential equation ,Applied Mathematics ,General Mathematics ,Bounded function ,Fixed-point theorem ,Uniqueness ,Auxiliary function ,Convexity ,Mathematics - Abstract
In this paper, a new class of nonexpansive mappings is defined and some fixed point theorems for such newly type are proved in the setting of bounded metric spaces without using neither the compactness nor the so-called uniform convexity. Our theorems generalize and improve many known results in the fixed point theory. Furthermore, we apply the main results to show the existence and uniqueness of a solution for a differential equation.
- Published
- 2021
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