34 results
Search Results
2. Some weighted Hardy and Rellich inequalities on the Heisenberg group
- Author
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Jingbo Dou and Lin Xi
- Subjects
Pure mathematics ,Class (set theory) ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Mathematics::Spectral Theory ,01 natural sciences ,010101 applied mathematics ,Heisenberg group ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper, we establish some weighted Hardy and Rellich inequalities and discuss its best constants on the Heisenberg group. Moreover, we also present a class of higher-order weighted Hardy–Rellich inequalities with the remainder term.
- Published
- 2021
3. Time periodic traveling waves in a three-component non-autonomous and reaction-diffusion epidemic model
- Author
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Jiangbo Zhou, Zaili Zhen, Lixin Tian, and Jingdong Wei
- Subjects
Time periodic ,Component (thermodynamics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Astrophysics::Instrumentation and Methods for Astrophysics ,Fixed-point theorem ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,Infection rate ,010101 applied mathematics ,Reaction–diffusion system ,Traveling wave ,Computer Science::General Literature ,0101 mathematics ,Epidemic model ,Mathematics - Abstract
In this paper, we propose a non-autonomous and diffusive SIR epidemic model based on the fact that the infection rate, the removal rate and the death rate often vary in time. The explicit formulas of the basic reproduction number [Formula: see text] and the minimum wave speed [Formula: see text] are derived. Applying upper-lower solution method and Schauder’s fixed point theorem, we show that when [Formula: see text], [Formula: see text] and the diffusion rates satisfy a certain condition, a time periodic traveling wave solution exists in the model. By the method of contradiction analysis and the comparison arguments together with the properties of the spreading speed of an associated subsystem, we prove that when [Formula: see text] and [Formula: see text] or [Formula: see text] and [Formula: see text], the model possesses no time periodic traveling wave solutions.
- Published
- 2021
4. The complex green operator with Sobolev estimates up to a finite order
- Author
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Bingyuan Liu and Andrew Raich
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Sobolev space ,Hypersurface ,Computer Science::General Literature ,Order (group theory) ,CR manifold ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to explore the geometry of a smooth CR manifold of hypersurface type and its relationship to the higher regularity properties of the complex Green operator on [Formula: see text]-forms in the [Formula: see text]-Sobolev space [Formula: see text] for a fixed [Formula: see text] and [Formula: see text].
- Published
- 2020
5. A non-local expanding flow of convex closed curves in the plane
- Author
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Ke Shi
- Subjects
Plane (geometry) ,General Mathematics ,010102 general mathematics ,Convex curve ,Mathematical analysis ,Regular polygon ,Non local ,01 natural sciences ,010101 applied mathematics ,Perimeter ,Flow (mathematics) ,Euclidean geometry ,0101 mathematics ,Mathematics - Abstract
This paper presents a new non-local expanding flow for convex closed curves in the Euclidean plane which increases both the perimeter of the evolving curves and the enclosed area. But the flow expands the evolving curves to a finite circle smoothly if they do not develop singularity during the evolving process. In addition, it is shown that an additional assumption about the initial curve will ensure that the flow exists on the time interval [Formula: see text]. Meanwhile, a numerical experiment reveals that this flow may blow up for some initial convex curves.
- Published
- 2020
6. Regularity properties of nonlinear abstract Schrödinger equations and applications
- Author
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Veli B. Shakhmurov
- Subjects
Function space ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Banach space ,Type (model theory) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,Harmonic analysis ,Nonlinear system ,symbols.namesake ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator [Formula: see text] defined in a Banach space [Formula: see text], in which by choosing [Formula: see text] and [Formula: see text] we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.
- Published
- 2020
7. Conformal vector fields on Finsler manifolds
- Author
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Qiaoling Xia
- Subjects
Pure mathematics ,Conformal vector field ,General Mathematics ,010102 general mathematics ,Conformal map ,Characterization (mathematics) ,01 natural sciences ,010101 applied mathematics ,Metric (mathematics) ,Vector field ,Mathematics::Differential Geometry ,Finsler manifold ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper, we give an equivalent characterization of conformal vector fields on a Finsler manifold [Formula: see text], whose metric [Formula: see text] is defined by a Riemannian metric [Formula: see text] and a 1-form [Formula: see text]. This characterization contains all related results in [Z. Shen and Q. Xia, On conformal vector fields on Randers manifolds, Sci. China Math. 55(9) (2012) 1869–1882; Z. Shen and M. Yuan, Conformal vector fields on some Finsler manifolds, Sci. China Math. 59(1) (2016) 107–114; X. Cheng, Y. Li and T. Li, The conformal vector fields on Kropina manifolds, Diff. Geom. Appl. 56 (2018) 344–354] as special cases. Further, we determine conformal fields on some Finsler manifolds [Formula: see text] when [Formula: see text] is of constant sectional curvature and [Formula: see text] is a conformal 1-form with respect to [Formula: see text].
- Published
- 2020
8. On the Ricci–Bourguignon flow
- Author
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Pak Tung Ho
- Subjects
010101 applied mathematics ,Flow (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Geometric flow ,Mathematics::Differential Geometry ,Soliton ,0101 mathematics ,Constant (mathematics) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the Ricci–Bourguignon flow of all locally homogenous geometries on closed three-dimensional manifolds. We also consider the evolution of the Yamabe constant under the Ricci–Bourguignon flow. Finally, we prove some results for the Bach-flat shrinking gradient soliton to the Ricci–Bourguignon flow.
- Published
- 2020
9. Group analysis to the time fractional nonlinear wave equation
- Author
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Xiao-Jun Yang, Muhammad Iqbal, Jian-Gen Liu, and Yi-Ying Feng
- Subjects
010101 applied mathematics ,Waves and shallow water ,Conservation law ,Nonlinear phenomena ,Group analysis ,Nonlinear wave equation ,General Mathematics ,0103 physical sciences ,Mathematical analysis ,0101 mathematics ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Abstract
In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis scheme. First, the symmetry can be obtained make uses of the group analysis to the time fractional nonlinear wave equation. Based on the above-found symmetry, this equation was able to reduce into an ordinary differential equation of fractional order. As a result, some new invariant solutions were also constructed for this considered equation. Second, the scaling transformation was also obtained by introducing new independent and dependent variables. Finally, the conservation laws were also found to satisfy the time fractional nonlinear wave equation with the help of the Ibragimov theorem. These novel results show unique nonlinear phenomena.
- Published
- 2020
10. Cheng–Shen conjecture in Finsler geometry
- Author
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Behzad Najafi and Mona Atashafrouz
- Subjects
Pure mathematics ,Class (set theory) ,Closed manifold ,Conjecture ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Metric (mathematics) ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Finsler manifold ,0101 mathematics ,Mathematics - Abstract
The well-known Cheng–Shen conjecture says that every [Formula: see text]-quadratic Randers metric on a closed manifold is a Berwald metric. The class of [Formula: see text]-quadratic Randers metrics contains the class of generalized Douglas–Weyl Randers metrics. In this paper, we give a classification of left-invariant Randers metrics of generalized Douglas–Weyl type on three-dimensional Lie groups. Based on our classification theorem, we find a counter-example for the Cheng–Shen conjecture.
- Published
- 2020
11. Cyclic parallel hypersurfaces in complex Grassmannians of rank 2
- Author
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Young Jin Suh and Hyunjin Lee
- Subjects
010101 applied mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Rank (graph theory) ,Mathematics::Differential Geometry ,0101 mathematics ,Object (computer science) ,01 natural sciences ,Hermitian matrix ,Mathematics - Abstract
The object of the paper is to study cyclic parallel hypersurfaces in complex (hyperbolic) two-plane Grassmannians which have a remarkable geometric structure as Hermitian symmetric spaces of rank 2. First, we prove that if the Reeb vector field belongs to the orthogonal complement of the maximal quaternionic subbundle, then the shape operator of a cyclic parallel hypersurface in complex hyperbolic two-plane Grassmannians is Reeb parallel. By using this fact, we classify all cyclic parallel hypersurfaces in complex hyperbolic two-plane Grassmannians with non-vanishing geodesic Reeb flow. Next, we give a non-existence theorem for cyclic Hopf hypersurfaces in complex two-plane Grassmannians.
- Published
- 2019
12. Complex symmetric weighted composition operators on Dirichlet spaces and Hardy spaces in the unit ball
- Author
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Ze-Hua Zhou, Xiao-He Hu, and Zi-Cong Yang
- Subjects
Unit sphere ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Hardy space ,Composition (combinatorics) ,47B33, 47B15, 47B38, 32A35, 32A37 ,01 natural sciences ,Dirichlet space ,Dirichlet distribution ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,symbols.namesake ,FOS: Mathematics ,symbols ,Computer Science::General Literature ,0101 mathematics ,Self-adjoint operator ,Mathematics - Abstract
In this paper, we investigate when weighted composition operators acting on Dirichlet spaces $\mathcal{D}(\mathbb{B}_{N})$ are complex symmetric with respect to some special conjugations, and provide some characterizations of Hermitian weighted composition operators on $\mathcal{D}(\mathbb{B}_{N})$. Furthermore, we give a sufficient and necessary condition for $J$-symmetric weighted composition operators on Hardy spaces $H^2(\mathbb{B}_{N})$ to be unitary or Hermitian, then some new examples of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$ are obtained. We also discuss the normality of complex symmetric weighted composition operators on $H^2(\mathbb{B}_{N})$., 18 pages
- Published
- 2019
13. Higher-order geometric flow of hypersurfaces in a Riemannian manifold
- Author
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Zonglin Jia and Youde Wang
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Geometric flow ,Riemannian manifold ,01 natural sciences ,010101 applied mathematics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,Order (group theory) ,0101 mathematics ,Balanced flow ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this paper, we consider the high-order geometric flows of a compact submanifolds [Formula: see text] in a complete Riemannian manifold [Formula: see text] with [Formula: see text], which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on [Formula: see text]. Precisely, we show that if [Formula: see text] is strictly larger than the integer part of [Formula: see text] and [Formula: see text] is an immersion for all [Formula: see text] and if [Formula: see text] is bounded by a constant which relies on the injectivity radius [Formula: see text] and sectional curvature [Formula: see text] of [Formula: see text], then [Formula: see text] must be [Formula: see text].
- Published
- 2019
14. Constructions of nearly holomorphic Siegel modular forms of degree two
- Author
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Shuji Horinaga
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Modular form ,Holomorphic function ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Eisenstein series ,symbols ,0101 mathematics ,Representation (mathematics) ,Siegel modular form ,Mathematics - Abstract
Pitale, Saha and Schmidt studied the representation theoretic aspects of nearly holomorphic modular forms. By their theory, we obtain a classification of [Formula: see text]-modules which occur in the space of nearly holomorphic modular forms. In this paper, we give two constructions of nearly holomorphic Siegel modular forms of degree [Formula: see text] which generate reducible indecomposable modules. One construction is given by the Rankin–Cohen bracket of Shimura’s Eisenstein series and the other by Klingen Eisenstein series.
- Published
- 2019
15. Gap theorems for compact almost Ricci-harmonic solitons
- Author
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Abimbola Abolarinwa
- Subjects
Generalization ,General Mathematics ,010102 general mathematics ,Harmonic (mathematics) ,01 natural sciences ,010101 applied mathematics ,Theoretical physics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,0101 mathematics ,Focus (optics) ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
Almost Ricci-harmonic solitons are generalization of Ricci-harmonic solitons, almost Ricci solitons and harmonic-Einstein metrics. The main focus of this paper is to establish necessary and sufficient conditions for a gradient shrinking almost Ricci-harmonic soliton on a compact domain to be almost harmonic-Einstein.
- Published
- 2019
16. The isoperimetric problem in the 2-dimensional Finsler space forms with k = 0
- Author
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Linfeng zhou
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Space form ,0101 mathematics ,Isoperimetric inequality ,Space (mathematics) ,01 natural sciences ,Mathematics - Abstract
In this paper, the isoperimetric problem in the 2-dimensional Finsler space form [Formula: see text] with [Formula: see text] by using the Busemann–Hausdorff area is investigated. We prove that the circle centered the origin achieves the local maximum area of the isoperimetric problem.
- Published
- 2019
17. Heisenberg–Pauli–Weyl inequality for connected nilpotent Lie groups
- Author
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Kais Smaoui
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Lie group ,Center (group theory) ,01 natural sciences ,Representation theory ,010101 applied mathematics ,symbols.namesake ,Nilpotent ,Pauli exclusion principle ,Fourier transform ,symbols ,0101 mathematics ,Mathematics::Representation Theory ,media_common ,Mathematics - Abstract
The purpose of this paper is to formulate and prove an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality for connected nilpotent Lie groups with noncompact center. Representation theory and a localized Plancherel formula play an important role in the proof.
- Published
- 2018
18. Curvature bound for a curve flow with a prescribed rate of change in enclosed area
- Author
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Jianbo Fang, Yunlong Yang, and Shengliang Pan
- Subjects
010101 applied mathematics ,Thesaurus (information retrieval) ,Flow (mathematics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,Curvature ,01 natural sciences ,Mathematics - Abstract
This paper deals with the curvature bound for a nonlocal curve flow with a prescribed rate of change in enclosed area via Andrews–Bryan’s distance comparison. As a by-product, a partial answer to a conjecture given by Dallaston and McCue is obtained and the [Formula: see text] convergence of the curvature for the nonlocal flow is achieved.
- Published
- 2018
19. Reverse Bonnesen-style inequalities on surfaces of constant curvature
- Author
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Wenxue Xu and Baocheng Zhu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,Surface (topology) ,01 natural sciences ,Measure (mathematics) ,Upper and lower bounds ,010101 applied mathematics ,Constant curvature ,0101 mathematics ,Convex domain ,Isoperimetric inequality ,Mathematics - Abstract
This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface [Formula: see text] of constant curvature [Formula: see text] by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in [Formula: see text] and Bottema’s result in the Euclidean plane [Formula: see text].
- Published
- 2018
20. On the system of p-Laplacian equations with critical growth
- Author
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Junfang Zhao, Jiaquan Liu, and Xiangqing Liu
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Truncation method ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Bounded function ,p-Laplacian ,Computer Science::General Literature ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].
- Published
- 2018
21. Holomorphic correspondences related to finitely generated rational semigroups
- Author
-
Gautam Bharali and Shrihari Sridharan
- Subjects
Pure mathematics ,Distribution (number theory) ,Mathematics - Complex Variables ,Mathematics::Operator Algebras ,Semigroup ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Dynamical Systems (math.DS) ,Fixed point ,01 natural sciences ,Measure (mathematics) ,Julia set ,010101 applied mathematics ,Hausdorff dimension ,FOS: Mathematics ,Invariant measure ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,0101 mathematics ,37F05, 37F10 (Primary), 30G30 (Secondary) ,Mathematics - Abstract
In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyse the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup., Comment: 20 pages; minor revisions, Theorem 1.6 rephrased more simply; final version to be published in Internat. J. Math
- Published
- 2017
22. ITERATED FUNCTION SYSTEMS, REPRESENTATIONS, AND HILBERT SPACE
- Author
-
Palle E. T. Jorgensen
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,media_common.quotation_subject ,42C40 ,42A16 ,43A65 ,42A65 ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,Iterated function system ,Attractor ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Operator Algebras (math.OA) ,Representation (mathematics) ,media_common ,Mathematics ,Mathematics::Operator Algebras ,010102 general mathematics ,Mathematics - Operator Algebras ,Hilbert space ,Construct (python library) ,Infinity ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,symbols - Abstract
This paper studies a general class of Iterated Function Systems (IFS). No contractivity assumptions are made, other than the existence of some compact attractor. The possibility of escape to infinity is considered. Our present approach is based on Hilbert space, and the theory of representations of the Cuntz algebras O_n, n=2,3,.... While the more traditional approaches to IFS's start with some equilibrium measure, ours doesn't. Rather, we construct a Hilbert space directly from a given IFS; and our construction uses instead families of measures. Starting with a fixed IFS S_n, with n branches, we prove existence of an associated representation of O_n, and we show that the representation is universal in a certain sense. We further prove a theorem about a direct correspondence between a given system S_n, and an associated sub-representation of the universal representation of O_n., 22 pages, 3 figures containing 7 EPS graphics; LaTeX2e ("elsart" document class); v2 reflects change in Comments only
- Published
- 2004
23. Uniqueness results for algebraic and holomorphic curves into ℙn(ℂ)
- Author
-
Min Ru and Gul Ugur
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,Nevanlinna theory ,010101 applied mathematics ,Development (differential geometry) ,Uniqueness ,Algebraic curve ,0101 mathematics ,Algebraic number ,Value (mathematics) ,Mathematics - Abstract
This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into [Formula: see text].
- Published
- 2017
24. Translating solitons foliated by spheres
- Author
-
Daehwan Kim and Juncheol Pyo
- Subjects
Mean curvature flow ,Euclidean space ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010101 applied mathematics ,Hypersurface ,Hyperplane ,Foliation (geology) ,SPHERES ,Soliton ,0101 mathematics ,Surface of revolution ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
In this paper, we consider translating solitons in [Formula: see text] which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.
- Published
- 2017
25. 3-Manifold invariants derived from the intersecting kernels
- Author
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Jie Wu, Fengchun Lei, and Fengling Li
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,010101 applied mathematics ,Algebra ,Computer Science::General Literature ,Homomorphism ,0101 mathematics ,Algebraic number ,Quotient group ,Heegaard splitting ,3-manifold ,Mathematics - Abstract
The intersecting kernel of a Heegaard splitting [Formula: see text] for a compact orientable 3-manifold [Formula: see text] is the subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is the homomorphism induced by the inclusion [Formula: see text], [Formula: see text]. In the paper, we obtain some invariants of 3-manifolds [Formula: see text] from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.
- Published
- 2016
26. Measure expansive symplectic diffeomorphisms and Hamiltonian systems
- Author
-
Junmi Park and Manseob Lee
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Riemannian manifold ,Symplectic representation ,01 natural sciences ,010101 applied mathematics ,Computer Science::General Literature ,Diffeomorphism ,Anosov diffeomorphism ,0101 mathematics ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic geometry ,Mathematics ,Symplectic manifold - Abstract
Let [Formula: see text] be a [Formula: see text]-dimensional ([Formula: see text]), compact smooth Riemannian manifold endowed with a symplectic form [Formula: see text]. In this paper, we show that, if a symplectic diffeomorphism [Formula: see text] is [Formula: see text]-robustly measure expansive, then it is Anosov and a [Formula: see text] generic measure expansive symplectic diffeomorphism [Formula: see text] is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system [Formula: see text] is robustly measure expansive, then [Formula: see text] is Anosov.
- Published
- 2016
27. A direct blowing-up and rescaling argument on nonlocal elliptic equations
- Author
-
Yan Li, Wenxiong Chen, and Congming Li
- Subjects
Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Blowing up ,010101 applied mathematics ,Elliptic operator ,Nonlinear system ,Operator (computer programming) ,Argument ,A priori and a posteriori ,Extension method ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray–Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.
- Published
- 2016
28. Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation
- Author
-
Li Ma
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,symbols ,0101 mathematics ,Fractional Laplacian ,Ginzburg landau ,Mathematical physics ,Mathematics - Abstract
In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.
- Published
- 2016
29. Equations of complex Monge–Ampère type for arbitrary measures and applications
- Author
-
Le Mau Hai, Trieu Van Dung, and Tang Van Long
- Subjects
010101 applied mathematics ,Pure mathematics ,Class (set theory) ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,Type (model theory) ,Ampere ,01 natural sciences ,Mathematics - Abstract
In this paper, we prove the existence of weak solutions of equations of complex Monge–Ampère type for arbitrary measures, in particular, measures carried by pluripolar sets. As an application of the obtained result, we show the existence of weak solutions of equations of complex Monge–Ampère type in the class [Formula: see text] if there exist locally subsolutions.
- Published
- 2016
30. Non-uniform dependence on initial data for the generalized Degasperis–Procesi equation on the line
- Author
-
Yanggeng Fu, Zanping Yu, and Jianhe Shen
- Subjects
010101 applied mathematics ,Sobolev space ,Cauchy problem ,Solution map ,Uniform continuity ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Line (geometry) ,0101 mathematics ,Degasperis–Procesi equation ,01 natural sciences ,Mathematics - Abstract
In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces [Formula: see text] for [Formula: see text]. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the [Formula: see text] norm.
- Published
- 2016
31. Complex symmetry of weighted composition operators in several variables
- Author
-
Maofa Wang and Xingxing Yao
- Subjects
Pure mathematics ,Composition operator ,Spectral radius ,General Mathematics ,media_common.quotation_subject ,General function ,Open problem ,010102 general mathematics ,Operator theory ,01 natural sciences ,010101 applied mathematics ,Algebra ,Compact space ,0101 mathematics ,Equivalence (formal languages) ,Normality ,Mathematics ,media_common - Abstract
In this paper, we investigate analytic symbols [Formula: see text] and [Formula: see text] when the weighted composition operator [Formula: see text] is complex symmetric on general function space [Formula: see text]. As applications, we characterize completely the compactness, normality and isometry of complex symmetric weighted composition operators. Especially, we show that the equivalence of compactness and Hilbert–Schmidtness, and the existence of non-normal complex symmetric operators for such operators, which answers one open problem raised by Noor in [On an example of a complex symmetric composition operators on [Formula: see text], J. Funct. Anal. 269 (2015) 1899–1901] for higher dimensional case.
- Published
- 2016
32. Green matrices and continuity of the weak solutions for the elliptic systems with lower order terms
- Author
-
Jinping Zhuge and Zhenqiu Zhang
- Subjects
Pointwise ,Pure mathematics ,Elliptic systems ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Hölder condition ,Lower order ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Bounded function ,Schauder estimates ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the second-order elliptic systems with lower order terms coefficients belonging to Kato–Stummel type classes in a bounded domain [Formula: see text], where [Formula: see text]. We establish the existence, global pointwise estimates and [Formula: see text] estimates for Green matrices. Based on these estimates, we are able to show the continuity of weak solutions for the general elliptic systems by using method of potentials. We also obtain the optimal Schauder estimates for weak solutions.
- Published
- 2016
33. Growth property at infinity of the maximum modulus with respect to the Schrödinger operator
- Author
-
Jinjin Huang
- Subjects
Property (philosophy) ,Generalization ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Modulus ,Infinity ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Operator (computer programming) ,Cone (topology) ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics ,media_common - Abstract
In this paper, we consider the growth property at infinity of the maximum modulus with respect to the Schrödinger operator in a cone, which supplement Phragmén–Lindelöf theorems for subfunctions obtained by Qiao and Pan (Generalization of the Phragmén–Lindelöf theorems for subfunctions, Internat. J. Math. 24 (2013) Article ID: 1350062).
- Published
- 2016
34. On Siegel paramodular forms of degree 2 with small levels
- Author
-
Hiroki Aoki
- Subjects
010101 applied mathematics ,Algebra ,Lift (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,Graded ring ,0101 mathematics ,Differential operator ,01 natural sciences ,Mathematics ,Siegel modular form - Abstract
In this paper, we show that the graded ring of Siegel paramodular forms of degree [Formula: see text] with level [Formula: see text] has a very simple unified structure, taking with character. All are generated by six modular forms. The first five are obtained by a kind of Maass lift. The last one is obtained by a kind of Rankin–Cohen–Ibukiyama differential operator from the first five. This result is similar to the case of the graded ring of Siegel modular forms of degree [Formula: see text] with respect to [Formula: see text].
- Published
- 2016
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