2,018 results
Search Results
102. A spectral projection method for transmission eigenvalues
- Author
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Jiguang Sun, Fang Zeng, and Liwei Xu
- Subjects
Discretization ,General Mathematics ,Mathematical analysis ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Matrix (mathematics) ,Inverse scattering problem ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,35P30, 65M38, 31A10 ,Complex plane ,Boundary element method ,Eigendecomposition of a matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples., Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematics
- Published
- 2016
103. Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space
- Author
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Shuichi Kawashima and Yoshihiro Ueda
- Subjects
Isentropic process ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Perturbation (astronomy) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Exponential stability ,Euler's formula ,symbols ,A priori and a posteriori ,Uniqueness ,0101 mathematics ,Crucial point ,Linear stability ,Mathematics - Abstract
In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R3. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.
- Published
- 2016
104. A theorem of Piccard’s type in abelian Polish groups
- Author
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Eliza Jabłońska
- Subjects
010101 applied mathematics ,Discrete mathematics ,Meagre set ,General Mathematics ,010102 general mathematics ,Neighbourhood (graph theory) ,0101 mathematics ,Abelian group ,01 natural sciences ,Mathematics - Abstract
In the paper we prove a theorem of Piccard’s type which generalizes [9, Theorem 2]. More precisely, we show that in an abelian Polish group X the set \(\left\{ {\left( {{x_{1, \ldots ,\;}}{x_N}} \right) \in \;{X^N}\;:\;A\; \cap \;\bigcap\limits_{i = 1}^N {\left( {A + {x_i}} \right)} \;is\;not\;Haar\;meager\;in\;X} \right\}\) is a neighbourhood of 0 for every N ∈ N and every Borel non-Haar meager set A ⊂ X. The paper refers to the paper [3].
- Published
- 2016
105. Hölder Regularity of Grobman–Hartman Theorem for Dynamic Equations on Measure Chains
- Author
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Yonghui Xia, Kit Ian Kou, Lijun Chen, and Donal O'Regan
- Subjects
Discrete mathematics ,Work (thermodynamics) ,General Mathematics ,Existential quantification ,010102 general mathematics ,Linear system ,Hölder condition ,Function (mathematics) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Nonlinear system ,Transformation (function) ,0101 mathematics ,Mathematics - Abstract
It has been proven that there exists a one-to-one correspondence H(t, x) between solutions of the linear system and the nonlinear system in the previous work. However, there is no paper considering the Holder regularity of the transformation H(t, x) in the literature. This paper fills the gap. We establish a strict proof of the Holder regularity of the transformation H(t, x). We show that the conjugating function H(t, x) in the generalized Hartman–Grobman theorem is always Holder continuous.
- Published
- 2016
106. Approximate solution of integral equations with multidimensional convolution operators on large sets with piecewise smooth boundaries
- Author
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A. V. Kozak and D. I. Khanin
- Subjects
General Mathematics ,Mathematical analysis ,Boundary (topology) ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Integral equation ,Circular convolution ,Convolution ,010101 applied mathematics ,Kernel (image processing) ,Piecewise ,0101 mathematics ,Convolution theorem ,Mathematics - Abstract
This paper is devoted to the solution of multidimensional integral convolution-type equations on the m-dimensional Euclidian space’s subset with a piecewise boundary. The kernel of the convolution operator is assumed to belong to the \(L_1\)-space, and the right side and the required solution are assumed to belong to the \(L_p\)-space. At the points of the set in question, which are far from the boundary, the solution is sought for with the help of a convolution operator in the entire space or a convolution operator on a torus; near smooth parts of the boundary it is sought for with the help of convolution operators in half-spaces. At the remaining points, the solution is sought for numerically. The paper provides the estimates of arising errors, which prove the method is efficient for large sets with gently sloping boundaries. The work is essentially based on the local method of investigation of the projection method applicability developed by I. B. Symonenko and A. V. Kozak.
- Published
- 2016
107. Generalized Sasakian Space Forms and Riemannian Manifolds of Quasi Constant Sectional Curvature
- Author
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Tee-How Loo and Avik De
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Space form ,01 natural sciences ,010101 applied mathematics ,Differential Geometry (math.DG) ,Complex space ,Ricci-flat manifold ,Minkowski space ,FOS: Mathematics ,Hermitian manifold ,Mathematics::Differential Geometry ,Sectional curvature ,0101 mathematics ,Mathematics::Symplectic Geometry ,Real line ,Scalar curvature ,Mathematics - Abstract
In this paper, we show that a generalized Sasakian space form of dimension >3 is either of constant sectional curvature, or a canal hypersurface in Euclidean or Minkowski spaces, or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold, or locally a warped product of a real line and a generalized complex space form, or an $${\alpha}$$ -Sasakian space form, or it is of five dimension and admits an $${\alpha}$$ -Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension >5 is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension >3 is also given in this paper.
- Published
- 2016
108. Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation
- Author
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H. El Boujaoui, L. Maniar, and H. Bouslous
- Subjects
Discretization ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Finite difference method ,Finite difference ,Boundary (topology) ,Wave equation ,01 natural sciences ,010101 applied mathematics ,Observability ,0101 mathematics ,Exponential decay ,Fourier series ,Mathematics - Abstract
In this paper, we consider a finite difference fully discretization for the $$ 1-D $$ wave equation with a boundary feedback and we analyze the exponential decay of the energy of solutions. First, we prove that a uniform boundary observability has a positive answer for a conservative system when the mesh size tends to zero using Fourier series technique. Then, we prove that the decay rate of the energy is uniform with respect to the mesh size. Finally, we describe some numerical experiments developed in Matlab to illustrate the results proved in this paper.
- Published
- 2016
109. Sine functions on hypergroups
- Author
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László Székelyhidi and Żywilla Fechner
- Subjects
Mathematics::Functional Analysis ,Polynomial ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Mathematics::Classical Analysis and ODEs ,20N20, 43A62, 39B99 ,01 natural sciences ,Mathematics - Functional Analysis ,010101 applied mathematics ,Mathematics::Quantum Algebra ,Homomorphism ,Sine ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.
- Published
- 2016
110. On log local Cartier transform of higher level in characteristic p
- Author
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Sachio Ohkawa
- Subjects
Discrete mathematics ,Smooth morphism ,General Mathematics ,Modulo ,010102 general mathematics ,Scalar (mathematics) ,13N10, 16H05, 16S32 ,Differential operator ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics - Algebraic Geometry ,Azumaya algebra ,FOS: Mathematics ,Higgs boson ,Sheaf ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In our previous paper, given an integral log smooth morphism $X\to S$ of fine log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf of log differential operators of higher level and constructed a splitting module of it under an existence of a certain lifting modulo $p^{2}$. In this paper, under a certain liftability assumption which is stronger than our previous paper, we construct another splitting module of our Azumaya algebra over a scalar extension, which is smaller than our previous paper. As an application, we construct an equivalence, which we call the log local Cartier transform of higher level, between certain $\cal D$-modules and certain Higgs modules. We also discuss about the compatibility of the log Frobenius descent and the log local Cartier transform and the relation between the splitting module constructed in this paper and that constructed in the previous paper. Our result can be considered as a generalization of the result of Ogus-Vologodsky, Gros-Le Stum-Quir��s to the case of log schemes and that of Schepler to the case of higher level.
- Published
- 2016
111. Peano’s 1886 existence theorem on first-order scalar differential equations: a review
- Author
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Sonia Mazzucchi and Gabriele H. Greco
- Subjects
Pure mathematics ,Differential equation ,General Mathematics ,010102 general mathematics ,Existence theorem ,Mathematical proof ,01 natural sciences ,010101 applied mathematics ,Algebra ,Peano curve ,Peano axioms ,Ordinary differential equation ,Initial value problem ,0101 mathematics ,Peano existence theorem ,Mathematics - Abstract
In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem \(y^\prime =f(x,y)\), \(y(a)=b\), under the assumption of the continuity of the function f. The present paper gives a detailed description of Peano’s original statements and proofs, filling gaps, clarifying obscure points and avoiding ambiguous use of mathematical symbols. Peano’s 1886 work is compared with later papers of Peano himself as well as of Mie (Math Ann 43:553–568, 1893), Osgood (Monatsh Math Phys 9:331–345, 1898) and Perron (Math. Ann. 76:471–484, 1915).
- Published
- 2016
112. The incompressible limit in $$L^p$$ L p type critical spaces
- Author
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Raphaël Danchin and Lingbing He
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Torus ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,010101 applied mathematics ,Bounded function ,Barotropic fluid ,Besov space ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
This paper aims at justifying the low Mach number convergence to the incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $L^2$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $L^p$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $\dot B^{d/p-1}_{p,r}\cap\dot B^{-1}_{\infty,1}$ for some suitable $(p,r)\in[2,4]\times[1,+\infty].$ We still require $L^2$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.
- Published
- 2016
113. Blow-up of Solutions to Quasilinear Parabolic Equations with Singular Absorption and a Positive Initial Energy
- Author
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Bin Guo and Wenjie Gao
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,Geodetic datum ,01 natural sciences ,Upper and lower bounds ,Parabolic partial differential equation ,010101 applied mathematics ,Calculus ,Applied mathematics ,Absorption (logic) ,0101 mathematics ,Finite time ,Energy (signal processing) ,Mathematics - Abstract
In this paper, the authors prove that the solution blows up in a finite time for a larger class of initial data, namely positive initial energy. The results generalize and improve that of Giacomoni et al. (J Math Anal Appl 410:607–624, 2014). Furthermore, a lower bound estimate about blow-up time is also established. Finally, two examples are given in the paper to show the existence of the initial datum satisfying the conditions in the main theorem.
- Published
- 2015
114. Asymptotics and Attractors for Quasilinear Parabolic-Hyperbolic Systems Governing the Motions of Heavily Burdened Deformable Bodies
- Author
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Stuart S. Antman and Suleyman Ulusoy
- Subjects
General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Motion (geometry) ,Exact differential equation ,Type (model theory) ,Rigid body ,Inertia ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Ordinary differential equation ,0103 physical sciences ,Attractor ,010307 mathematical physics ,0101 mathematics ,media_common ,Mathematics - Abstract
This paper surveys recent nonlinear analyses of the motion of deformable solids to which are attached heavy rigid bodies. The deformable solids are described by geometrically exact equations of nonlinear viscoelasticity (of strain-rate type), which form quasilinear parabolic-hyperbolic systems. The main emphasis of this paper is on the treatment of problems in which the ratios of the inertias of the deformable body are small with respect to those of the rigid body. Some of these problems possess rigorous asymptotic expansions in a small inertia parameter with the leading terms of the reduced problems governed not by traditional ordinary differential equations for the rigid body, but by equations with memory. Some of these problems admit attractors with the dimensions of the attractors small when the inertia parameters are small. This paper describes a number of open problems.
- Published
- 2015
115. On a question of Hof, Knill and Simon on palindromic substitutive systems
- Author
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Tero Harju, Jetro Vesti, and Luca Q. Zamboni
- Subjects
Class (set theory) ,General Mathematics ,Palindrome ,Substitution (algebra) ,0102 computer and information sciences ,01 natural sciences ,Morphic word ,Omega ,010101 applied mathematics ,Combinatorics ,Morphism ,010201 computation theory & mathematics ,0101 mathematics ,Computer Science::Formal Languages and Automata Theory ,Word (group theory) ,Mathematics - Abstract
In a 1995 paper, Hof et al. obtain a sufficient combinatorial criterion on the subshift \(\Omega \) of the potential of a discrete Schrodinger operator which guarantees purely singular continuous spectrum on a generic subset of \(\Omega .\) In part, this condition requires that the subshift \(\Omega \) be palindromic, i.e., contains an infinite number of distinct palindromic factors. In the same paper, they introduce the class P of morphisms \(f:A^*\rightarrow B^*\) of the form \(a\mapsto pq_a\) with p and \(q_a\) palindromes, and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form \(a\mapsto q_ap.\) In this paper we give a partial affirmative answer to the question of Hof et al.: we show that every rich primitive substitutive subshift is generated by at most two morphisms each of which is conjugate to a morphism of class P. More precisely, we show that every rich (or almost rich in the sense of finite defect) primitive morphic word \(y\in B^\omega \) is of the form \(y=f(x)\) where \(f:A^*\rightarrow B^*\) is conjugate to a morphism of class P, and where x is a rich word fixed by a primitive substitution \(g:A^*\rightarrow A^*\) conjugate to one in class P.
- Published
- 2015
116. Approximate controllability of a non-autonomous differential equation
- Author
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Indira Mishra and Madhukant Sharma
- Subjects
010101 applied mathematics ,Controllability ,Schauder fixed point theorem ,Functional differential equation ,General Mathematics ,0103 physical sciences ,Applied mathematics ,0101 mathematics ,010301 acoustics ,01 natural sciences ,Resolvent ,Mathematics - Abstract
In this paper, we establish the approximate controllability results for a non-autonomous functional differential equation using the theory of linear evolution system, Schauder fixed point theorem, and by making use of resolvent operators. The results obtained in this paper, improve the existing ones in this direction, to a considerable extent. An example is also given to illustrate the abstract results.
- Published
- 2018
117. Second Hankel Determinant for a Certain Subclass of Bi-univalent Functions
- Author
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T. Janani, Gangadharan Mrugusundaramoorthy, and Nizami Mustafa
- Subjects
010101 applied mathematics ,Class (set theory) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Upper and lower bounds ,Unit disk ,Subclass ,Mathematics - Abstract
In this paper, we introduce a subclass of analytic and bi-univalent functions in the open unit disk. Here, we give upper bound estimates for the second Hankel determinant of the functions that belong to this class. Some interesting applications and conclusions of the results obtained in this paper are also discussed.
- Published
- 2018
118. Two-dimensional strain gradient damage modeling: a variational approach
- Author
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Emilio Barchiesi, Anil Misra, and Luca Placidi
- Subjects
Karush–Kuhn–Tucker conditions ,Deformation (mechanics) ,Applied Mathematics ,General Mathematics ,Linear elasticity ,Mathematical analysis ,Isotropy ,General Physics and Astronomy ,02 engineering and technology ,Dissipation ,01 natural sciences ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Damage mechanics ,0101 mathematics ,Galerkin method ,Energy functional ,Mathematics - Abstract
In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.
- Published
- 2018
119. Demicompactness Results for Strongly Continuous Semigroups, Generators and Resolvents
- Author
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Asrar Elleuch, Hedi Benkhaled, and Aref Jeribi
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,Semigroup ,Generator (category theory) ,General Mathematics ,010102 general mathematics ,Hilbert space ,Banach space ,01 natural sciences ,Bounded operator ,010101 applied mathematics ,symbols.namesake ,Product (mathematics) ,Bounded function ,symbols ,0101 mathematics ,Mathematics ,Resolvent - Abstract
Let $$(U(t))_ {t\ge 0}$$ be a strongly continuous semigroup of bounded linear operators on a Banach space X and B be a bounded operator on X. In this paper, we develop some aspects of the theory of semigroup for which U(t)B (respectively, BU(t), BU(t)B) is demicompact for some (respectively, every) $$t>0$$ . In addition, we study the demicompactness of similar, subspace and product semigroups. We also investigate the demicompactness of the resolvent. We close this paper by giving some conditions guaranteeing the demicompactness of a generator of a strongly continuous semigroup in a Hilbert space.
- Published
- 2018
120. On Ground-State Homoclinic Orbits of a Class of Superquadratic Damped Vibration Systems
- Author
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Mohsen Timoumi
- Subjects
Continuous function (set theory) ,Antisymmetric relation ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Matrix (mathematics) ,Bounded function ,Symmetric matrix ,Homoclinic orbit ,0101 mathematics ,Constant (mathematics) ,Ground state ,Mathematics - Abstract
In this paper, we are interested in the following damped vibration system: where B is an antisymmetric $$N\times N$$ constant matrix, $$q:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ is a continuous function, $$L(t)\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})$$ is a symmetric matrix, and $$W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})$$ are neither autonomous nor periodic in t. The novelty of this paper is that, supposing that $$Q(t)=\int ^{t}_{0}q(s)\mathrm{d}s$$ is bounded from below and L(t) is coercive unnecessarily uniformly positively definite for all $$t\in {\mathbb {R}}$$ , we establish the existence of ground-state homoclinic solutions for (1) when the potential W(t, x) satisfies a kind of superquadratic conditions due to Ding and Luan for Schr $${\ddot{o}}$$ dinger equation. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. Some recent results in the literature are generalized and significantly improved.
- Published
- 2018
121. Sparsity Regularization of the Diffusion Coefficient Identification Problem: Well-Posedness and Convergence Rates
- Author
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Pham Quy Muoi
- Subjects
Mathematical optimization ,General Mathematics ,Regularization perspectives on support vector machines ,010103 numerical & computational mathematics ,Backus–Gilbert method ,01 natural sciences ,Regularization (mathematics) ,Convexity ,010101 applied mathematics ,Parameter identification problem ,Rate of convergence ,Convex optimization ,0101 mathematics ,Energy functional ,Mathematics - Abstract
In this paper, we investigate sparsity regularization for the diffusion coefficient identification problem. Here, the regularization method is incorporated with the energy functional approach. The advantages of our approach are to deal with convex minimization problems. Therefore, the well-posedness of the problem is obtained without requiring regularity property of the parameter. The convexity of regularized problems also allows use the fast algorithms developed recently. Furthermore, the convergence rates of the method are obtained under a simple source condition. The main results of this paper are the well-posedness and convergence rates of sparsity regularization. We also obtain some new results of the continuity and the differentiability of related operators.
- Published
- 2015
122. Conformal metrics on the unit ball with prescribed mean curvature
- Author
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Xingwang Xu and Hong Zhang
- Subjects
Mean curvature flow ,Mean curvature ,General Mathematics ,Prescribed scalar curvature problem ,010102 general mathematics ,Mathematical analysis ,Center of curvature ,Curvature ,01 natural sciences ,010101 applied mathematics ,Total curvature ,Mathematics::Differential Geometry ,Sectional curvature ,0101 mathematics ,Mathematics ,Scalar curvature - Abstract
This paper focuses on the study of the prescribed mean curvature problem on the unit ball. If the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small, then the existence of positive solutions of conformal mean curvature equation has been known. The purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method.
- Published
- 2015
123. Interval Oscillation Criteria for Second-Order Damped Differential Equations with Mixed Nonlinearities
- Author
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Zhenlai Han, Dian-Wu Yang, Meirong Xu, and Yibing Sun
- Subjects
Mathematics::Commutative Algebra ,Oscillation ,Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Order (ring theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Interval (graph theory) ,0101 mathematics ,Quotient ,Mathematics - Abstract
In this paper, we consider the interval oscillation criteria for second-order damped differential equations with mixed nonlinearities $$\begin{aligned} \left( r(t)(x'(t))^\gamma \right) '+p(t)(x'(t))^\gamma +\sum ^n_{i=0}q_i(t)\left| x(g_i(t))\right| ^{\alpha _i}\text {sgn}\ x(g_i(t))=e(t), \end{aligned}$$ where $$\gamma $$ is a quotient of odd positive integers, $$\alpha _0=\gamma , \alpha _i>0, i=1,\ 2,\ldots ,n$$ with $$r,\ p,\ e$$ , and $$q_i\in C([t_0,\infty ),\mathbb {R}), r(t)>0, g_i:\ \mathbb {R}\rightarrow \mathbb {R}$$ are nondecreasing continuous functions on $$\mathbb {R}$$ and $$\lim _{t\rightarrow \infty }g_i(t)=\infty , i=0,\ 1,\ 2,\ldots ,n.$$ Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.
- Published
- 2015
124. The Derivative-Free Double Newton Step Methods for Solving System of Nonlinear Equations
- Author
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Na Huang, Changfeng Ma, and Ya-Jun Xie
- Subjects
Iterative method ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Derivative ,01 natural sciences ,Newton's method in optimization ,Local convergence ,010101 applied mathematics ,Nonlinear system ,Calculus ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose two classes of derivative-free Newton-like methods for solving system of nonlinear equations based on double Newton step. We also give the local convergence analysis of the iterative methods. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.
- Published
- 2015
125. Duality and Integral Transform of a Class of Analytic Functions
- Author
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Sushma Gupta, Sukhjit Singh, and Sarika Verma
- Subjects
010101 applied mathematics ,Combinatorics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Integral transform ,01 natural sciences ,Unit disk ,Analytic function ,Mathematics - Abstract
For \(\alpha , \gamma \ge 0\) and \(\beta 0, \, \, \, z\in E \end{aligned}$$ for some \(\phi \in {\mathbb {R}}\). For \(f\in {{\mathcal {W}}_{\beta }(\alpha , \gamma )}\), we consider the integral transform $$\begin{aligned} V_{\lambda }(f)(z):=\int _{0}^{1}\lambda (t)\frac{f(tz)}{t}\mathrm{d}t, \end{aligned}$$ where \(\lambda \) is a non-negative real-valued integrable function satisfying the condition \(\int _{0}^{1}\lambda (t)\mathrm{d}t=1\). In a very recent paper, Ali et al. (J Math Anal Appl 385:808–822, 2012) discussed the starlikeness of the integral transform \(V_{\lambda }(f)\) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). The aim of present paper is to find conditions on \(\lambda (t)\) such that \(V_{\lambda }(f)\) is starlike of order \(\delta \) (\(0\le \delta \le 1/2\)) when \(f\in {{\mathcal {W}}}_{\beta }(\alpha , \gamma )\). As applications, we study various choices of \(\lambda (t)\), related to classical integral transforms.
- Published
- 2015
126. A Numerical Method to Solve Higher-Order Fractional Differential Equations
- Author
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Ricardo Almeida and Nuno R. O. Bastos
- Subjects
Fractional differential equations ,Approximation formulas ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Fractional calculus ,Order (ring theory) ,Numerical Analysis (math.NA) ,01 natural sciences ,Fractional operator ,010101 applied mathematics ,FOS: Mathematics ,Numerical methods ,Mathematics - Numerical Analysis ,0101 mathematics ,Fractional differential ,Mathematics - Abstract
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method., Comment: This is a preprint of a paper whose final and definite form will be published in Mediterr. J. Math
- Published
- 2015
127. Global existence and finite time blow-up for a class of thin-film equation
- Author
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Zhihua Dong and Jun Zhou
- Subjects
Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,01 natural sciences ,Upper and lower bounds ,010101 applied mathematics ,Nonlinear system ,Thin-film equation ,0101 mathematics ,Finite time ,Constant (mathematics) ,Mathematics - Abstract
This paper deals with a class of thin-film equation, which was considered in Li et al. (Nonlinear Anal Theory Methods Appl 147:96–109, 2016), where the case of lower initial energy ( $$J(u_0)\le d$$ and d is a positive constant) was discussed, and the conditions on global existence or blow-up are given. We extend the results of this paper on two aspects: Firstly, we consider the upper and lower bounds of blow-up time and asymptotic behavior when $$J(u_0)d$$ .
- Published
- 2017
128. Connections on the Total Space of a Holomorphic Lie Algebroid
- Author
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Alexandru Ionescu and Gheorghe Munteanu
- Subjects
Lie algebroid ,Tangent bundle ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Connection (principal bundle) ,Holomorphic function ,Rank (differential topology) ,01 natural sciences ,Manifold ,010101 applied mathematics ,Section (fiber bundle) ,Algebra ,0101 mathematics ,Complex manifold ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The main purpose of the paper is the study of the total space of a holomorphic Lie algebroid E. The paper is structured in three parts. In the first section, we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section presents two approaches on the study of the geometry of the complex manifold E. The first part contains the study of the tangent bundle $$T_{\mathbb {C}}E=T'E\oplus T''E$$ and its link, via the tangent anchor map, with the complexified tangent bundle $$T_{\mathbb {C}}(T'M)=T'(T'M)\oplus T''(T'M)$$ . A holomorphic Lie algebroid structure is emphasized on $$T'E$$ . A special study is made for integral curves of a spray on $$T'E$$ . Theorem 2.8 gives the coefficients of a spray, called canonical, obtained from a complex Lagrangian on $$T'E$$ . In the second part of section two, we study the holomorphic prolongation $$\mathcal {T}'E$$ of the Lie algebroid E. In the third section, we study how a complex Lagrange (Finsler) structure on $$T'M$$ induces a Lagrangian structure on E. Three particular cases are analysed by the rank of the anchor map, the dimensions of manifold M, and those of the fibres. We obtain the correspondent on E of the Chern–Lagrange nonlinear connection from $$T'M$$ .
- Published
- 2017
129. Global regularity for a 3D Boussinesq model without thermal diffusion
- Author
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Zhuan Ye
- Subjects
010101 applied mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Compressibility ,General Physics and Astronomy ,0101 mathematics ,Boussinesq approximation (water waves) ,Thermal diffusivity ,01 natural sciences ,Heat kernel ,Mathematics - Abstract
In this paper, we consider a modified three-dimensional incompressible Boussinesq model. The model considered in this paper has viscosity in the velocity equations, but no diffusivity in the temperature equation. To bypass the difficulty caused by the absence of thermal diffusion, we make use of the maximal $$L_t^{p}L_x^{q}$$ regularity for the heat kernel to establish the global regularity result.
- Published
- 2017
130. Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well
- Author
-
Juntao Sun, Yuanze Wu, and Tsung-fang Wu
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Lambda ,Poisson distribution ,01 natural sciences ,Omega ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .
- Published
- 2017
131. Sum Relations of Multiple Zeta Star Values with Even Arguments
- Author
-
Chan-Liang Chung and Kwang-Wu Chen
- Subjects
010101 applied mathematics ,Combinatorics ,Identity (mathematics) ,Arithmetic zeta function ,Particular values of Riemann zeta function ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Star (graph theory) ,01 natural sciences ,Bernoulli number ,Prime zeta function ,Mathematics - Abstract
The purpose of this paper is the presentation of an identity which is closely related to the sum relation involving multiple zeta star values with even arguments. Let $$E^{\star }(m,n,k)$$ be the sum of all multiple zeta star values of depth k and weight mn with arguments multiples of $$m\ge 2$$ . In this paper, we give two formulas for $$E^{\star }(2s,n,k)$$ for $$s=1,2,3$$ and in particular, by comparing the two we obtain a Bernoulli numbers identity. There are corresponding results included in a special kind of alternating multiple zeta values.
- Published
- 2017
132. Some Theorems on Cauchy Problem for Nonlinear Fractional Differential Equations with Positive Constant Coefficient
- Author
-
H. T. Dinde and Shivaji Ramchandra Tate
- Subjects
Cauchy problem ,Constant coefficients ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fixed-point theorem ,Interval (mathematics) ,Type (model theory) ,Lambda ,01 natural sciences ,010101 applied mathematics ,Nonlinear fractional differential equations ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider a Cauchy problem for nonlinear fractional differential equation with constant coefficient $$\lambda >0$$ of the type: $${^c}{D}^{\alpha }x(t)=\lambda x(t)+f(t,x(t))$$ with $$x(0)=x_{0}.$$ The aim of this paper is to investigate the existence and interval of existence of solutions, uniqueness, continuous dependence of solutions on initial conditions, estimates on solutions and continuous dependence on parameters and functions involved in the equations. Finally, one illustrative example is given to demonstrate the theoretical results.
- Published
- 2017
133. Topology of Uniform Convergence and $$\varvec{m}$$ m -Topology on $$\varvec{C(X)}$$ C ( X )
- Author
-
Ľubica Holá and Branislav Novotný
- Subjects
010101 applied mathematics ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Metric (mathematics) ,Lower limit topology ,0101 mathematics ,Topological space ,Topology ,01 natural sciences ,Topology of uniform convergence ,Topology (chemistry) ,Mathematics - Abstract
In this paper, we prove that the density of the topology of uniform convergence d(C(X)) is equal to the density of the m-topology. For general topological spaces X, the density d(C(X)) is not known. In this paper we investigate the cardinal invariants of the m-topology on C(X), which is a classical topology. We prove that it behaves like a metric one in the sense that many cardinal invariants, including weight, cellularity and density, are equal for this topology. In fact, they are all equal to d(C(X)).
- Published
- 2017
134. Pullback attractors of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows
- Author
-
Bo You and Fang Li
- Subjects
Forcing (recursion theory) ,Field (physics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Order (ring theory) ,Pullback attractor ,01 natural sciences ,Fractal dimension ,Upper and lower bounds ,010101 applied mathematics ,Liquid crystal ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper is concerned with the long-time behaviour of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows introduced in Lin and Liu (Commun Pure Appl Math, 48:501–537, 1995) with a non-autonomous forcing bulk term and order parameter field boundary conditions. In this paper, we prove the existence of pullback attractors and estimate the upper bound of its fractal dimension under some suitable assumptions.
- Published
- 2016
135. Global smooth flows for compressible Navier–Stokes–Maxwell equations
- Author
-
Jiang Xu and Hongmei Cao
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Maxwell's equations ,Compressibility ,symbols ,Dissipative system ,Initial value problem ,Navier stokes ,0101 mathematics ,Mathematics - Abstract
Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.
- Published
- 2016
136. Asymptotic behavior of boundary blow-up solutions to elliptic equations
- Author
-
Shuibo Huang
- Subjects
Mean curvature ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Degenerate energy levels ,General Physics and Astronomy ,Boundary (topology) ,Function (mathematics) ,Infinity ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Asymptotic formula ,0101 mathematics ,Power function ,Mathematics ,media_common - Abstract
This paper is concerned with the asymptotic behavior on $${\partial\Omega}$$ of boundary blow-up solutions to semilinear elliptic equations $$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$ where b(x) is a nonnegative function on $${\Omega}$$ and may vanish on $${\partial\Omega}$$ at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.
- Published
- 2016
137. Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4
- Author
-
Chunhua Jin
- Subjects
Time periodic ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Space dimension ,General Physics and Astronomy ,01 natural sciences ,010101 applied mathematics ,Fixed-point iteration ,Compressibility ,Uniqueness ,0101 mathematics ,Compressible navier stokes equations ,Mathematics - Abstract
This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).
- Published
- 2016
138. Near-Best Univariate Spline Discrete Quasi-Interpolants on Nonuniform Partitions
- Author
-
Paul Sablonnière, Driss Sbibih, D. Barrera, M. J. Ibáñez, Departamento de Matematica Aplicada (E-GRAN-AM), Universidad de Granada = University of Granada (UGR), Departamento de Matemática Aplicada [Granada], Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université de Mathématiques et Informatique (MRC-UMIS), Université Mohammed Premier [Oujda], Universidad de Granada (UGR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
- Subjects
65D07 ,General Mathematics ,spline quasi-interpolants ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Uniform norm ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Linear combination ,Computer Science::Databases ,Mathematics ,65D17 ,Mathematical analysis ,Univariate ,Computer Science::Software Engineering ,Numerical Analysis (math.NA) ,spline approximation ,010101 applied mathematics ,Computational Mathematics ,Spline (mathematics) ,Norm (mathematics) ,Bounded function ,41A15 ,41A35 ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis ,Free parameter - Abstract
The univariate spline quasi-interpolants (QIs) studied in this paper are approximation operators using B-spline expansions with coefficients that are linear combinations of discrete values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs that have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound for the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of some families of near-best discrete quasi-interpolants (dQIs) of approximation order 3.
- Published
- 2008
139. Best approximation mappings in Hilbert spaces
- Author
-
Xianfu Wang, Heinz H. Bauschke, and Hui Ouyang
- Subjects
Primary 90C25, 41A50, 65B99, Secondary 46B04, 41A65 ,Sequence ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,Hilbert space ,Convex set ,02 engineering and technology ,Fixed point ,Quantitative Biology::Genomics ,01 natural sciences ,Linear subspace ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Rate of convergence ,Optimization and Control (math.OC) ,FOS: Mathematics ,symbols ,Affine space ,Affine transformation ,0101 mathematics ,Mathematics - Optimization and Control ,Software ,Mathematics - Abstract
The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces., 31 pages and 2 figures
- Published
- 2021
140. A Note on the Uniqueness of Certain Types of Differential-Difference Polynomials
- Author
-
S. Saha and S. Majumder
- Subjects
010101 applied mathematics ,Pure mathematics ,Difference polynomials ,Open problem ,General Mathematics ,010102 general mathematics ,Uniqueness ,0101 mathematics ,01 natural sciences ,Differential (mathematics) ,Mathematics ,Meromorphic function - Abstract
UDC 517.9 We study the uniqueness problems of certain types of differential-difference polynomials sharing a small function.In this paper, we not only solve the open problem occurred in [A. Banerjee, S. Majumder, On the uniqueness of certain types of differential-difference polynomials, Anal. Math., 43, № 3, 415-444 (2017)], but also present our main results in a more generalized way.
- Published
- 2021
141. Spectra and ergodic properties of multiplication and convolution operators on the space $${\mathcal S}({\mathbb R})$$
- Author
-
Claudio Mele, Angela A. Albanese, Albanese, A. A., and Mele, C.
- Subjects
General Mathematics ,Star (game theory) ,010102 general mathematics ,Rapidly decreasing functions, Multiplication operator, Convolution operator, Spectra, Power bounded operator, Mean ergodic operator ,Space (mathematics) ,01 natural sciences ,Spectral line ,Convolution ,010101 applied mathematics ,Combinatorics ,Multiplication (music) ,Schwartz space ,Bounded function ,Ergodic theory ,0101 mathematics ,Mathematics - Abstract
In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ S ( R ) of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ M h : S ( R ) → S ( R ) , $$f \mapsto h f $$ f ↦ h f , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ C T : S ( R ) → S ( R ) , $$f\mapsto T\star f$$ f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.
- Published
- 2021
142. Existence, uniqueness and stability of fractional impulsive functional differential inclusions
- Author
-
J. Vanterler da C. Sousa and Kishor D. Kucche
- Subjects
Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fixed-point theorem ,01 natural sciences ,Stability (probability) ,Fractional calculus ,010101 applied mathematics ,Computational Theory and Mathematics ,Differential inclusion ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the $$\psi $$ -Liouville–Caputo fractional derivative, through fixed point theorem, Arzela–Ascoli theorem and multivalued analysis theory.
- Published
- 2021
143. On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE
- Author
-
Birzhan Ayanbayev and Nikos Katzourakis
- Subjects
Minimisation (psychology) ,General Mathematics ,010102 general mathematics ,Inverse ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Parameter identification problem ,Nonlinear system ,symbols.namesake ,Elliptic curve ,Mathematics - Analysis of PDEs ,Operator (computer programming) ,Compact space ,FOS: Mathematics ,symbols ,Applied mathematics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals., 14 pages. arXiv admin note: text overlap with arXiv:1811.02845
- Published
- 2021
144. The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets
- Author
-
Abdul Aziz Shahid, Waqas Nazeer, and Krzysztof Gdawiec
- Subjects
Discrete mathematics ,Degree (graph theory) ,Picard–Mann iteration ,General Mathematics ,010102 general mathematics ,Mann iteration ,Julia set ,Fixed-point theorem ,escape criterion ,Mandelbrot set ,01 natural sciences ,Convexity ,010101 applied mathematics ,Fractal ,0101 mathematics ,Complex polynomial ,Mathematics - Abstract
In recent years, researchers have studied the use of different iteration processes from fixed point theory in the generation of complex fractals. For instance, the Mann, Ishikawa, Noor, Jungck–Mann and Jungck–Ishikawa iterations have been used. In this paper, we study the use of the Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets. We prove the escape criterion for the $$(k+1)$$ ( k + 1 ) st degree complex polynomial. Moreover, we present some graphical and numerical examples regarding Mandelbrot and Julia sets generated using the proposed iteration.
- Published
- 2021
145. The $$\partial \overline \partial $$-Bochner Formulas for Holomorphic Mappings between Hermitian Manifolds and Their Applications
- Author
-
Kai Tang
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Schwarz lemma ,General Mathematics ,010102 general mathematics ,Holomorphic function ,General Physics and Astronomy ,Type (model theory) ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Hermitian matrix ,010101 applied mathematics ,Hermitian manifold ,Mathematics::Differential Geometry ,0101 mathematics ,Degeneracy (mathematics) ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics - Abstract
In this paper, we derive some $$\partial \overline \partial $$ -Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) l-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results [5, 6] proved recently by L. Ni on Kahler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.
- Published
- 2021
146. Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer
- Author
-
Emmanuel Fouotsa, Hervé Talé Kalachi, and Franck Rivel Kamwa Djomou
- Subjects
Discrete mathematics ,Ring (mathematics) ,Rank (linear algebra) ,General Mathematics ,Modulo ,010102 general mathematics ,01 natural sciences ,Ring of integers ,010101 applied mathematics ,Finite field ,Integer ,Principal ideal ,0101 mathematics ,Prime power ,Mathematics - Abstract
Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring $$\mathbb {Z}_{m}$$ Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder.
- Published
- 2021
147. Numerical Approximation of Poisson Problems in Long Domains
- Author
-
Stefan A. Sauter, Alexander Veit, Wolfgang Hackbusch, Michel Chipot, University of Zurich, and Hackbusch, Wolfgang
- Subjects
Asymptotic analysis ,Discretization ,General Mathematics ,340 Law ,610 Medicine & health ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Cartesian product ,Differential operator ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,10123 Institute of Mathematics ,symbols.namesake ,510 Mathematics ,Exact solutions in general relativity ,Tensor (intrinsic definition) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Poisson's equation ,2600 General Mathematics ,Mathematics - Abstract
In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.
- Published
- 2021
148. Successive approximations for random coupled Hilfer fractional differential systems
- Author
-
Mouffak Benchohra, Fatima Si Bachir, Saïd Abbas, and Maamar Benbachir
- Subjects
010101 applied mathematics ,Section (fiber bundle) ,General Mathematics ,010102 general mathematics ,Convergence (routing) ,Applied mathematics ,Shaping ,Uniqueness ,0101 mathematics ,Fractional differential ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.
- Published
- 2021
149. Exact Convergence Rates for Particle Distributions in a Non-Lattice Branching Random Walk
- Author
-
Zhi-Qiang Gao
- Subjects
Discrete mathematics ,Characteristic function (probability theory) ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Motion (geometry) ,Random walk ,01 natural sciences ,010101 applied mathematics ,Counting measure ,Branching random walk ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Branching process - Abstract
Consider a discrete-time supercritical branching random walk, which is a branching process combined with random spatial motion of particles, the number of the descendants tending to infinity with positive probability. Let $$ Z_n(\cdot ) $$ be the counting measure which counts the number of particles of generation n in a given set. Revesz (1994) studied the convergence rates in the central and local limit theorems for $$ Z_n(\cdot ) $$ in some special cases, and then, the topic was further developed in various cases. In this paper, we give the exact convergence rates of the central and local limit theorems for $$ Z_n(\cdot ) $$ in the case the underlying motion is governed by a general non-lattice random walk on $${\mathbb {R}}^d$$ with the characteristic function of the motion law satisfying the weak Cramer condition.
- Published
- 2021
150. Besov Estimates for Weak Solutions of the Parabolic p-Laplacian Equations
- Author
-
Fengping Yao and Rumeng Ma
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,p-Laplacian ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we obtain the local regularity estimates in Besov spaces of weak solutions for the following parabolic p-Laplacian equations: $$\begin{aligned} u_{t}-\text {div} ~\! a \left( Du, x,t \right) =\text {div}~ {\mathbf {F}} \end{aligned}$$ under some proper assumptions on the functions a and $${\mathbf {F}}$$ . Moreover, we would like to point out that our results improve the known results for such equations.
- Published
- 2021
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