203 results
Search Results
2. Analysis of fractional COVID-19 epidemic model under Caputo operator
- Author
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Rahat Zarin, Amir Khan, Abdullahi Yusuf, Sayed Abdel‐Khalek, Mustafa Inc, and Mühendislik ve Doğa Bilimleri Fakültesi
- Subjects
Lyapunov function ,Special Issue Papers ,Coronavirus disease 2019 (COVID-19) ,General Mathematics ,Crossover ,General Engineering ,Regular polygon ,Fixed-point theorem ,Stability (probability) ,Numerical Simulations ,34d45 ,symbols.namesake ,Operator (computer programming) ,Sensitivity Analysis ,Stability Analysis ,Special Issue Paper ,Epidemic Model ,symbols ,Applied mathematics ,Uniqueness ,Sensitivity (control systems) ,26a33 ,Epidemic model ,Mathematics - Abstract
The article deals with the analysis of the fractional COVID‐19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease‐free equilibrium using the method of Castillo‐Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first‐order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.
- Published
- 2021
3. On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
- Author
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Thomas Y. Hou, De Huang, and Jiajie Chen
- Subjects
symbols.namesake ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,FOS: Mathematics ,symbols ,Finite time ,Analysis of PDEs (math.AP) ,Mathematics ,Euler equations - Abstract
We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version
- Published
- 2021
4. On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure
- Author
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Zhouping Xin and Beixiang Fang
- Subjects
35A01, 35A02, 35B20, 35B35, 35B65, 35J56, 35L65, 35L67, 35M30, 35M32, 35Q31, 35R35, 76L05, 76N10 ,Shock (fluid dynamics) ,Astrophysics::High Energy Astrophysical Phenomena ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Nozzle ,Mathematical analysis ,Boundary (topology) ,Euler system ,01 natural sciences ,Physics::Fluid Dynamics ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Free boundary problem ,Euler's formula ,symbols ,Boundary value problem ,0101 mathematics ,Transonic ,Astrophysics::Galaxy Astrophysics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages
- Published
- 2020
5. Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory
- Author
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Jonas Hirsch, Camillo De Lellis, Salvatore Stuvard, and Andrea Marchese
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Pure mathematics ,multiple valued functions, Dirichlet integral, regularity theory, area minimizing currents mod(p), minimal surfaces, linearization ,Generalization ,General Mathematics ,Dimension (graph theory) ,area minimizing currents mod(p) ,linearization ,minimal surfaces ,Dirichlet integral ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,0101 mathematics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Codimension ,regularity theory ,symbols ,multiple valued functions ,Analysis of PDEs (math.AP) - Abstract
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math
- Published
- 2020
6. For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering
- Author
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David Lafontaine, Euan A. Spence, and Jared Wunsch
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Helmholtz equation ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,symbols.namesake ,Helmholtz free energy ,Frequency domain ,symbols ,Scattering theory ,0101 mathematics ,Laplace operator ,Mathematics ,Resolvent - Abstract
It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.
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- 2020
7. A data assimilation process for linear ill-posed problems
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X.-M. Yang and Z.-L. Deng
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Well-posed problem ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,Bayesian probability ,Posterior probability ,General Engineering ,Markov chain Monte Carlo ,Inverse problem ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Data assimilation ,symbols ,Applied mathematics ,Ensemble Kalman filter ,0101 mathematics ,Randomness ,Mathematics - Abstract
In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
8. The asymptotic distribution of symbols on diagonals of random weighted staircase tableaux
- Author
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Amanda Lohss
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Conjecture ,Distribution (number theory) ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Diagonal ,Asymptotic distribution ,0102 computer and information sciences ,Asymmetric simple exclusion process ,Poisson distribution ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Connection (mathematics) ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Probability ,Software ,Mathematics - Abstract
Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016
- Published
- 2016
9. On existence of solutions of differential-difference equations
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Hai-chou Li
- Subjects
Independent equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,Euler equations ,010101 applied mathematics ,Stochastic partial differential equation ,Examples of differential equations ,Theory of equations ,symbols.namesake ,Simultaneous equations ,symbols ,Applied mathematics ,0101 mathematics ,C0-semigroup ,Differential algebraic equation ,Mathematics - Abstract
This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.
- Published
- 2015
10. On global stability of an HIV pathogenesis model with cure rate
- Author
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Yoshiaki Muroya and Yoichi Enatsu
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Lyapunov function ,Mathematical optimization ,General Mathematics ,General Engineering ,Human immunodeficiency virus (HIV) ,medicine.disease_cause ,Stability (probability) ,Upper and lower bounds ,Pathogenesis ,symbols.namesake ,Monotone polygon ,Stability theory ,medicine ,symbols ,Applied mathematics ,Logistic function ,Mathematics - Abstract
In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.
- Published
- 2014
11. Realisability conditions for second-order marginals of biphased media
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Raphaël Lachièze-Rey, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), and Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS )
- Subjects
realisability ,General Mathematics ,Gaussian ,random set ,Upper and lower bounds ,symbols.namesake ,Level set ,covariogram ,MSC 60D05 ,Statistics ,Applied mathematics ,Order (group theory) ,Variogram ,Mathematics ,biphased media ,Applied Mathematics ,Covariance ,Computer Graphics and Computer-Aided Design ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,If and only if ,covariance ,symbols ,marginal problems ,Constant (mathematics) ,[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] ,Software - Abstract
16 pages; International audience; This paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper.
- Published
- 2014
12. Multifrequency NLS Scaling for a Model Equation of Gravity-Capillary Waves
- Author
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Nader Masmoudi and Kenji Nakanishi
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Sequence ,Capillary wave ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Nonlinear system ,symbols.namesake ,symbols ,Limit (mathematics) ,Scaling ,Nonlinear Schrödinger equation ,Schrödinger's cat ,Mathematics - Abstract
This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrodinger (NLS) equation as well as other related systems starting from a model coming from the gravity-capillary water wave system in the long-wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrodinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence. © 2012 Wiley Periodicals, Inc.
- Published
- 2013
13. Paley-Wiener theorems and uncertainty principles for the windowed linear canonical transform
- Author
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Rui-Hui Xu, Yan‐Hui Zhang, and Kit Ian Kou
- Subjects
Uncertainty principle ,Paley–Wiener theorem ,General Mathematics ,Mathematical analysis ,Poisson summation formula ,General Engineering ,Sampling (statistics) ,Inverse ,Fractional Fourier transform ,symbols.namesake ,symbols ,Applied mathematics ,Series expansion ,Mathematics ,Interpolation - Abstract
In a recent paper, the authors have introduced the windowed linear canonical transform and shown its good properties together with some applications such as Poisson summation formulas, sampling interpolation, and series expansion. In this paper, we prove the Paley–Wiener theorems and the uncertainty principles for the (inverse) windowed linear canonical transform. They are new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.
- Published
- 2012
14. Random graphs containing few disjoint excluded minors
- Author
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Colin McDiarmid and Valentas Kurauskas
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Discrete mathematics ,Clique-sum ,Applied Mathematics ,General Mathematics ,Robertson–Seymour theorem ,Computer Graphics and Computer-Aided Design ,1-planar graph ,Planar graph ,Combinatorics ,symbols.namesake ,Pathwidth ,Graph power ,symbols ,Cograph ,Software ,Forbidden graph characterization ,Mathematics - Abstract
The Erdos-Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a 'blocking' set B of at most f(k) vertices such that the graph G - B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor-closed class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} of graphs, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, there is a set B of at most g(k) vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763-775), we showed that, amongst all graphs on vertex set [n] = {1,...,n} which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor-closed graph class \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} with 2-connected excluded minors, as long as \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} does not contain all fans (here a 'fan' is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on [n] which contain at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, all but an exponentially small proportion contain a set B of k vertices such that G - B is in \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}. (This is not the case when \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document} contains all fans.) For a random graph R sampled uniformly from the graphs on [n] with at most k disjoint excluded minors for \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}${\mathcal A}$\end{document}, we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc.
- Published
- 2012
15. On the zeros of Dirichlet -functions
- Author
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Raouf Ouni, Kamel Mazhouda, and Sami Omar
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Mathematical society ,General Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Riemann hypothesis ,symbols.namesake ,Number theory ,Computational Theory and Mathematics ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)
- Published
- 2011
16. On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: Pure radiation case
- Author
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Xin Zhou, Alexander Tovbis, and Stephanos Venakides
- Subjects
Applied Mathematics ,General Mathematics ,Semiclassical physics ,Schrödinger equation ,Nonlinear system ,symbols.namesake ,Quantum mechanics ,Inverse scattering problem ,symbols ,Initial value problem ,Limit (mathematics) ,Soliton ,Nonlinear Schrödinger equation ,Mathematical physics ,Mathematics - Abstract
In a previous paper [13] we calculated the leading-order term q0(x,t ,e )of the solution ofq(x,t ,e ), the focusing nonlinear (cubic) Schrodinger (NLS) equation in the semiclassical limit (e → 0) for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. The aim of the present paper is to calculate the long-term behavior of the semiclassical solution q(x,t ,e )in the pure radiation case. As before, our main tool is the Riemann-Hilbert problem (RHP) formulation of the inverse scattering problem and the corresponding system of “moment and integral conditions,” known also as a system of “modulation equations.” c � 2006 Wiley Periodicals, Inc.
- Published
- 2006
17. A lower-epiperimetric inequality for area-minimizing surfaces
- Author
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Tristan Rivière
- Subjects
Hölder's inequality ,Loomis–Whitney inequality ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Poincaré inequality ,Inequality of arithmetic and geometric means ,Minkowski inequality ,symbols.namesake ,Chebyshev's inequality ,symbols ,Isoperimetric inequality ,Cauchy–Schwarz inequality ,Mathematics - Abstract
The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k-density of area of an area-minimizing integral k-cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper-epiperimetric inequality. A direct consequence of this upper-epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper-epiperimetric inequality was proven by B. White in [5] for area-minimizing 2-cycles in ℝn. In the present paper we introduce the notion of a lower-epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k-density of area of an area-minimizing integral k-cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower-epiperimetric inequality for area-minimizing 2-cycles in ℝn. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2-rectifiable cycles, which plays a crucial role in the proof of the regularity of 1-1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.
- Published
- 2004
18. Pseudoholomorphic strips in symplectizations II: Fredholm theory and transversality
- Author
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Casim Abbas
- Subjects
Transversality ,Applied Mathematics ,General Mathematics ,Contact geometry ,Fredholm operator ,Mathematical analysis ,STRIPS ,Submanifold ,Fredholm theory ,law.invention ,symbols.namesake ,law ,symbols ,Mathematics::Differential Geometry ,Boundary value problem ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic geometry - Abstract
This paper is part of a larger program, the investigation of the chord problem in three dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds L0; L1 as boundary conditions. The submanifolds L0 and L1 do not intersect transversally. In this paper we will develop a nonlinear Fredholm theory that guarantees the existence of a family of embedded pseudoholomorphic strips near a given one with suitable properties. c 2003 Wiley Periodicals, Inc.
- Published
- 2003
19. Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces
- Author
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Chiun-Chuan Chen and Chang-Shou Lin
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Riemann surface ,Sobolev space ,Combinatorics ,Elliptic curve ,symbols.namesake ,Integer ,Quantum mechanics ,Euler characteristic ,symbols ,Uniform boundedness ,Compact Riemann surface ,Mathematics - Abstract
In this paper, we consider a sequence of multibubble solutions u k of the equation (0.1) Δ 0 u k + ρ k (he uk /f M he u k d μ -1)=0 in M, where h is a C 2,β positive function in a compact Riemann surface M, and ρ k is a constant satisfying lim k→+ ∞ ρ k = 8mπ for some positive integer m ≥ 1. We prove among other things that ρ k - 8mπ = 2/m m Σ/j=1h -1 (p k,j )(Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ))λ k,j e -λk,j + O(e -λk,j ), where p k,j are centers of the bubbles of u k and λ k,j are the local maxima of u k after adding a constant. This yields a uniform bound of solutions as ρ k converges to 8mπ from below provided that Δ 0 log h(p k,j ) + 8mπ - 2K(p k,j ) > 0. It generalizes a previous result, due to Ding, Jost, Li, and Wang [18] and Nolasco and Tarantello [31], which says that any sequence of minimizers u k is uniformly bounded if ρ k 0 for any maximum point p of the sum of 2 log h and the regular part of the Green function, where K is the Gaussian curvature of M. The analytic work of this paper is the first step toward computing the topological degree of (0.1), which was initiated by Li [24].
- Published
- 2002
20. Central Limit Theorem for Linear Eigenvalue Statistics of <scp>Non‐Hermitian</scp> Random Matrices
- Author
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Dominik Schröder, László Erdős, and Giorgio Cipolloni
- Subjects
Independent and identically distributed random variables ,Applied Mathematics ,General Mathematics ,Gaussian ,Hermitian matrix ,symbols.namesake ,Matrix (mathematics) ,Distribution (mathematics) ,Statistics ,symbols ,Random matrix ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics - Abstract
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Virag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices $X$ that are presented in the companion paper [Cipolloni, Erdős, Schroder 2019].
- Published
- 2021
21. A free boundary problem for a quasi-linear degenerate elliptic equation: Regular reflection of weak shocks
- Author
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Eun Heui Kim, Sunčica Čanić, and Barbara Lee Keyfitz
- Subjects
Conservation law ,Applied Mathematics ,General Mathematics ,Weak solution ,Mathematical analysis ,Regular solution ,symbols.namesake ,Reflection (mathematics) ,Riemann problem ,Shock position ,Free boundary problem ,symbols ,Boundary value problem ,Mathematics - Abstract
We prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance (UTSD) model for shock reflection by a wedge. In weak regular reflection, the state immediately behind the reflected shock is supersonic and constant. The flow becomes subsonic further downstream; the equation in self-similar coordinates is degenerate at the sonic line. The reflected shock becomes transonic and begins to curve there; its position is the solution to a free boundary problem for the degenerate equation. Using the Rankine-Hugoniot conditions along the reflected shock, we derive an evolution equation for the transonic shock, and an oblique derivative boundary condition at the unknown shock position. By regularizing the degenerate problem, we construct uniform bounds; we apply local compactness arguments to extract a limit that solves the problem. The solution is smooth in the interior and continuous up to the degenerate boundary. This work completes a stage in our program to construct self-similar solutions of two-dimensional Riemann problems. In a series of papers, we developed techniques for solving the degenerate elliptic equations that arise in self-similar reductions of hyperbolic conservation laws. In other papers, especially in joint work with Gary Lieberman, we developed techniques for solving free boundary problems of the type that arise from Rankine-Hugoniot relations. For the first time, in this paper, we combine these approaches and show that they are compatible. Although our construction is limited to a finite part of the unbounded subsonic region, it suggests that this approach has the potential to solve a variety of problems in weak shock reflection, including Mach and von Neumann reflection in the UTSD equation, and the analogous problems for the unsteady full potential equation. © 2002 John Wiley & Sons, Inc.
- Published
- 2001
22. Hamilton cycles containing randomly selected edges in random regular graphs
- Author
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Nicholas C. Wormald and Robert W. Robinson
- Subjects
Discrete mathematics ,Distribution (number theory) ,Applied Mathematics ,General Mathematics ,Contiguity ,Computer Graphics and Computer-Aided Design ,Hamiltonian path ,Combinatorics ,symbols.namesake ,Random regular graph ,symbols ,Cubic graph ,Probability distribution ,Almost surely ,Hamiltonian (quantum mechanics) ,Software ,Mathematics - Abstract
In previous papers the authors showed that almost all d-regular graphs for d≤3 are hamiltonian. In the present paper this result is generalized so that a set of j oriented root edges have been randomly specified for the cycle to contain. The Hamilton cycle must be orientable to agree with all of the orientations on the j root edges. It is shown that the requisite Hamilton cycle almost surely exists if and the limiting probability distribution at the threshold is determined when d=3. It is a corollary (in view of results elsewhere) that almost all claw-free cubic graphs are hamiltonian. There is a variation in which an additional cyclic ordering on the root edges is imposed which must also agree with their ordering on the Hamilton cycle. In this case, the required Hamilton cycle almost surely exists if j=o(n2/5). The method of analysis is small subgraph conditioning. This gives results on contiguity and the distribution of the number of Hamilton cycles which imply the facts above. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 128–147, 2001
- Published
- 2001
23. Fast convergence of the Glauber dynamics for sampling independent sets
- Author
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Eric Vigoda and Michael Luby
- Subjects
Applied Mathematics ,General Mathematics ,Sampling (statistics) ,Markov chain Monte Carlo ,Hardness of approximation ,Computer Graphics and Computer-Aided Design ,Combinatorics ,symbols.namesake ,Distribution (mathematics) ,Independent set ,Convergence (routing) ,symbols ,Constant (mathematics) ,Glauber ,Software ,Mathematics - Abstract
We consider the problem of sampling independent sets of a graph with maximum degree δ. The weight of each independent set is expressed in terms of a fixed positive parameter λ≤2/(δ−2), where the weight of an independent set σ is λ|σ|. The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. We show fast convergence (in O(n log n) time) of this dynamics. This paper gives the more interesting proof for triangle-free graphs. The proof for arbitrary graphs is given in a companion paper (E. Vigoda, Technical Report TR-99-003, International Computer Institute, Berkeley, CA, 1998). We also prove complementary hardness of approximation results, which show that it is hard to sample from this distribution when λ>c/δ for a constant c≤0. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 229–241, 1999
- Published
- 1999
24. Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces
- Author
-
Duy Minh Nhieu and Nicola Garofalo
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Poincaré inequality ,Carnot group ,Caccioppoli set ,Differential operator ,Sobolev inequality ,symbols.namesake ,symbols ,Interpolation space ,Isoperimetric inequality ,Mathematics ,Sobolev spaces for planar domains - Abstract
After Hormander's fundamental paper on hypoellipticity [54], the study of partial differential equations arising from families of noncommuting vector fields has developed significantly. In this paper we study some basic functional and geometric properties of general families of vector fields that include the Hormander type as a special case. Similar to their classical counterparts, such properties play an important role in the analysis of the relevant differential operators (both linear and nonlinear). To motivate our results, we recall some classical inequalities. Let E C R" be a Caccioppoli set (a measurable set having a locally finite perimeter); then one has the isoperimetric inequality
- Published
- 1996
25. Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise
- Author
-
Jean Daniel Mukam and Antoine Tambue
- Subjects
General Mathematics ,Numerical analysis ,finite element method ,General Engineering ,White noise ,Exponential integrator ,VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 ,Noise (electronics) ,Finite element method ,strong convergence ,Stochastic partial differential equation ,Galerkin projection method ,Nonlinear system ,symbols.namesake ,Wiener process ,symbols ,Applied mathematics ,stochastic convection–reaction–diffusion equations ,additive noise ,exponential integrators ,Mathematics - Abstract
In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided
- Published
- 2021
26. Quantitative Estimates for Regular Lagrangian Flows with<scp>BV</scp>Vector Fields
- Author
-
Quoc-Hung Nguyen
- Subjects
Combinatorics ,010104 statistics & probability ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Star (game theory) ,010102 general mathematics ,symbols ,Vector field ,0101 mathematics ,01 natural sciences ,Lagrangian ,Mathematics - Abstract
This paper is devoted to the study of flows associated to non-smooth vector fields. We prove the well-posedness of regular Lagrangian flows associated to vector fields $\mathbf{B}=(\mathbf{B}^1,...,\mathbf{B}^d)\in L^1(\mathbb{R}_+;L^1(\mathbb{R}^d)+L^\infty(\mathbb{R}^d))$ satisfying $ \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i*b_j,$ $b_j\in L^1(\mathbb{R}_+,BV(\mathbb{R}^d))$ and $\operatorname{div}(\mathbf{B})\in L^1(\mathbb{R}_+;L^\infty(\mathbb{R}^d))$ for $d,m\geq 2$, where $(\mathbf{K}_j^i)_{i,j}$ are singular kernels in $\mathbb{R}^d$. Moreover, we also show that there exist an autonomous vector-field $\mathbf{B}\in L^1(\mathbb{R}^2)+L^\infty(\mathbb{R}^2)$ and singular kernels $(\mathbf{K}_j^i)_{i,j}$, singular Radon measures $\mu_{ijk}$ in $\mathbb{R}^2$ satisfying $\partial_{x_k} \mathbf{B}^i=\sum_{j=1}^{m}\mathbf{K}_j^i\star\mu_{ijk}$ in distributional sense for some $m\geq 2$ and for $k,i=1,2$ such that regular Lagrangian flows associated to vector field $\mathbf{B}$ are not unique.
- Published
- 2021
27. Maximal full subspaces in random projective spaces-thresholds and Poisson approximation
- Author
-
Wojciech Kordecki
- Subjects
Random graph ,Discrete mathematics ,Generalization ,Applied Mathematics ,General Mathematics ,Poisson distribution ,Mathematical proof ,Computer Graphics and Computer-Aided Design ,Linear subspace ,Matroid ,Combinatorics ,symbols.namesake ,symbols ,Rank (graph theory) ,Projective test ,Software ,Mathematics - Abstract
Let Gn, p denote a random graph on n vertices. It is an interesting problem when small cliques arise and what distributions of the number of small cliques may occur. Matroids are natural generalization of graphs; therefore, we can try to investigate maximal flats of a small rank in random matroids. The most studied and most interesting seem to be “random projective geometries” introduced by Kelly and Oxley. Many of the theorems in our paper are based on the results published in a long paper of Barbour, Janson, Karoski, and Ruciski. However, proofs for projective geometries generally are more complicated. © 1995 Wiley Periodicals, Inc.
- Published
- 1995
28. A necessary and sufficient condition for the nirenberg problem
- Author
-
Congming Li and Wenxiong Chen
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Conformal map ,Function (mathematics) ,Symmetric function ,symbols.namesake ,Monotone polygon ,Metric (mathematics) ,Gaussian curvature ,symbols ,Counterexample ,Scalar curvature ,Mathematics - Abstract
We seek metrics conformal to the standard ones on Sn having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S2, for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.
- Published
- 1995
29. Integral equations for the generalized stokes operator: Applications to high reynolds number flows
- Author
-
Yves Achdou
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Reynolds number ,Summation equation ,Integral equation ,symbols.namesake ,Stokes' law ,Reynolds operator ,symbols ,Reynolds transport theorem ,Reynolds-averaged Navier–Stokes equations ,Stokes operator ,Mathematics - Abstract
This paper deals with a boundary integral equation for the generalized Stokes problem and its approximation by simpler integral equations when the Reynolds number tends to infinity. The two-dimensional case has been treated in [1]. This paper addresses the three-dimensional case. © 1994 John Wiley & Sons, Inc.
- Published
- 1994
30. Finite dimensional lie-poisson approximations to vlasov-poisson equations
- Author
-
Clint Scovel and Alan Weinstein
- Subjects
Applied Mathematics ,General Mathematics ,Discrete Poisson equation ,Mathematical analysis ,Lie group ,Poisson bracket ,symbols.namesake ,Uniqueness theorem for Poisson's equation ,Poisson manifold ,Lie algebra ,symbols ,First class constraint ,Mathematics ,Poisson algebra - Abstract
Many of the basic equations of conservative continuum mechanics (Euler, Vlasov-Poisson, Vlasov-Maxwell, MHD, etc.) are Hamiltonian systems with respect to Lie-Poisson brackets on dual spaces of infinite dimensional Lie algebras. The development of Lie-Poisson integrators for finite dimensional Lie-Poisson systems has shown that they are superior in the numerical simulations of these systems, especially with regard to long term phenomena. This paper shows how to truncate one of these systems, the Vlasov-Poisson equation of plasma physics, to a finite dimensional Lie-Poisson system. This requires replacing the functions on single-particle phase space, with their Poisson bracket Lie algebra structure, by a finite dimensional Lie algebra. Replacing the densities by their moments of order up to k about a fixed reference point corresponds to replacing the functions by their Taylor expansions up to order k. Unfortunately, these truncated Taylor expansions do not form a Lie algebra, since the functions which vanish through order k do not form an ideal under Poisson bracket. Geometrically, this corresponds to the fact that canonical transformations which fix the reference point do not form a normal subgroup. Introducing the location of a reference point in phase space as an extra variable and truncating with respect to this moving point turns out to decouple the “location” from “shape” coordinates of a lump of density in phase space, as far as the Poisson bracket is concerned. One can then replace the shape coordinates by a finite number of moments. The central result of the paper is a construction of the decoupling map described above in the general context of the decomposition of a Lie group as a product of subgroups. The main theorem is first proved by the general theory of Poisson reduction, then by explicit calculation, and lastly by showing that the Poisson isomorphism follows from the lift of a natural groupoid isomorphism. The groupoid aspect of the theory also provides natural Poisson maps, useful in the application of Ruth-type integration techniques, which do not seem easily derivable from the general theory of Poisson reduction. © 1994 John Wiley & Sons, Inc.
- Published
- 1994
31. Fast Computation of Orthogonal Systems with a <scp>Skew‐Symmetric</scp> Differentiation Matrix
- Author
-
Arieh Iserles and Marcus Webb
- Subjects
Tridiagonal matrix ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Matrix (mathematics) ,symbols.namesake ,symbols ,Skew-symmetric matrix ,Jacobi polynomials ,0101 mathematics ,Spectral method ,Mathematics ,Sine and cosine transforms ,Variable (mathematics) - Abstract
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skew-symmetric, tridiagonal and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first $N$ coefficients {of the expansion} can be computed to high accuracy in $\mathcal{O}(N\log_2N)$ operations. We consider two settings, one approximating a function $f$ directly in $(-\infty,\infty)$ and the other approximating $[f(x)+f(-x)]/2$ and $[f(x)-f(-x)]/2$ separately in $[0,\infty)$. In each setting we prove that there is a single family, parametrised by $\alpha,\beta > -1$, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where $\alpha, \beta= \pm 1/2$ are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.
- Published
- 2021
32. Minimal-energy splines: I. Plane curves with angle constraints
- Author
-
Weimin Han and Emery D. Jou
- Subjects
Box spline ,Plane curve ,General Mathematics ,General Engineering ,Geometry ,Mathematics::Numerical Analysis ,symbols.namesake ,Spline (mathematics) ,Smoothing spline ,Computer Science::Graphics ,Lagrange multiplier ,symbols ,Applied mathematics ,Spline interpolation ,Thin plate spline ,Mathematics - Abstract
This is the first in a series of papers on minimal-energy splines. The paper is devoted to plane minimal-energy splines with angle constraints. We first consider minimal-energy spline segments, then general minimal-energy spline curves. We formulate problems for minimal-energy spline segments and curves, prove the existence of solutions, justify the Lagrange multiplier rules, and obtain some nice properties (e.g., the infinite smoothness). Finally, we report our computational experience on minimal-energy splines.
- Published
- 1990
33. Analysis of the backward-euler/langevin method for molecular dynamics
- Author
-
Charles S. Peskin
- Subjects
Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Integral equation ,Backward Euler method ,Hamiltonian system ,Euler equations ,Langevin equation ,symbols.namesake ,Stochastic differential equation ,Classical mechanics ,symbols ,Brownian dynamics ,Mathematics - Abstract
This paper develops the theory of a recently introduced computational method for molecular dynamics. The method in question uses the backward-Euler method to solve the classical Langevin equations of a molecular system. Parameters are chosen to produce a cutoff frequency ωc, which may be set equal to kT/h to simulate quantum-mechanical effects. In the present paper, an ensemble of identical Hamiltonian systems modeled by the backward-Euler/Langevin method is considered, an integral equation for the equilibrium phase-space density is derived, and an asymptotic analysis of that integral equation in the limit Δt 0 is performed. The result of this asymptotic analysis is a second-order partial differential equation for the equilibrium phase-space density expressed as a function of the constants of the motion. This equation is solved in two special cases: a system of coupled harmonic oscillators and a diatomic molecule with a stiff bond.
- Published
- 1990
34. Generalized approximate boundary synchronization for a coupled system of wave equations
- Author
-
Yanyan Wang
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Boundary (topology) ,State (functional analysis) ,Kalman filter ,Wave equation ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Matrix (mathematics) ,symbols.namesake ,Synchronization (computer science) ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.
- Published
- 2020
35. A probabilistic approach to a non‐local quadratic form and its connection to the Neumann boundary condition problem
- Author
-
Zoran Vondraček
- Subjects
Dirichlet-to-Neumann operator, Hunt process, non-local normal derivative, non-local quadratic form ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Probabilistic logic ,Markov process ,Mathematics::Spectral Theory ,Directional derivative ,60J75, 31C25, 47G20, 60J45, 60J50 ,01 natural sciences ,Connection (mathematics) ,Interpretation (model theory) ,010101 applied mathematics ,symbols.namesake ,Operator (computer programming) ,Quadratic form ,FOS: Mathematics ,Neumann boundary condition ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov process: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neuman problem. We also study the Dirichlet-to-Neumann operator for non-local operators., Comment: 21 pages
- Published
- 2020
36. Transition Threshold for the <scp>3D</scp> Couette Flow in Sobolev Space
- Author
-
Zhifei Zhang and Dongyi Wei
- Subjects
Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Reynolds number ,01 natural sciences ,Physics::Fluid Dynamics ,Sobolev space ,010104 statistics & probability ,symbols.namesake ,symbols ,0101 mathematics ,Couette flow ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.
- Published
- 2020
37. Extending the choice of starting points for Newton's method
- Author
-
Argyros, Ioannis Konstantinos, Ezquerro, José Antonio, Hernández-Verón, Miguel Ángel, Kim, Young Ik, Magreñán, Ángel Alberto, and 0000-0002-6991-5706
- Subjects
General Mathematics ,010102 general mathematics ,General Engineering ,Lipschitz continuity ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Order (group theory) ,Applied mathematics ,Center (algebra and category theory) ,0101 mathematics ,Newton's method ,Second derivative ,Mathematics - Abstract
In this paper, we propose a center Lipschitz condition for the second derivative together with the use of restricted domains in order to improve the starting points for Newton's method when compared with previous results. Moreover, we present some numerical examples validating the theoretical results.
- Published
- 2019
38. The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus
- Author
-
Hamed Taghavian
- Subjects
Laplace inversion ,Laplace transform ,General Mathematics ,Composite number ,General Engineering ,Fractional calculus ,symbols.namesake ,Mittag-Leffler function ,symbols ,Partition (number theory) ,Applied mathematics ,Polynomial series ,Laplace transform inversion ,Mathematics - Abstract
This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact soluti ...
- Published
- 2019
39. Effective numerical evaluation of the double Hilbert transform
- Author
-
Min Ku, Xiaoyun Sun, Ieng Tak Leong, and Pei Dang
- Subjects
Pointwise ,General Mathematics ,010102 general mathematics ,General Engineering ,01 natural sciences ,010101 applied mathematics ,Periodic function ,Quadratic formula ,symbols.namesake ,symbols ,Applied mathematics ,Nyström method ,Hilbert transform ,0101 mathematics ,Remainder ,Energy (signal processing) ,Mathematics ,Trigonometric interpolation - Abstract
In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.
- Published
- 2020
40. Description of relations between regularity coefficients of time-varying linear systems
- Author
-
Michal Niezabitowski, Adam Czornik, and Aliaksei Vaidzelevich
- Subjects
Lyapunov function ,General Mathematics ,010102 general mathematics ,Linear system ,Lyapunov exponent ,01 natural sciences ,Nonlinear Sciences::Chaotic Dynamics ,010101 applied mathematics ,symbols.namesake ,symbols ,Applied mathematics ,0101 mathematics ,Characteristic exponent ,Mathematics - Abstract
In this paper we consider regularity coefficients of a discrete linear time‐varying systems. The main result presents a complete description of the relations between Lyapunov, Perron and Grobman regularity coefficients of mutually adjoint systems.
- Published
- 2018
41. Space‐time fractional Dirichlet problems
- Author
-
Boris Baeumer, Mark M. Meerschaert, and Tomasz Luks
- Subjects
Dirichlet problem ,Subordinator ,General Mathematics ,Open problem ,010102 general mathematics ,Markov process ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Bounded function ,Time derivative ,symbols ,Applied mathematics ,Infinitesimal generator ,0101 mathematics ,Mathematics - Abstract
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time‐changed by an inverse stable subordinator whose index equals the order of the fractional time derivative. Some applications are given, to demonstrate how to specify a well‐posed Dirichlet problem for space‐time fractional diffusions in one or several variables. This solves an open problem in numerical analysis.
- Published
- 2018
42. Perfect matchings and Hamiltonian cycles in the preferential attachment model
- Author
-
Paweł Prałat, Xavier Pérez-Giménez, Benjamin Reiniger, and Alan Frieze
- Subjects
Applied Mathematics ,General Mathematics ,Existential quantification ,0102 computer and information sciences ,Preferential attachment ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Hamiltonian path ,Vertex (geometry) ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Almost surely ,Combinatorics (math.CO) ,Hamiltonian (quantum mechanics) ,Software ,Mathematics - Abstract
In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with $m$ random vertices selected with probabilities proportional to their current degrees. (Constant $m$ is the only parameter of the model.) We prove that if $m \ge 1{,}260$, then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that $m \ge 29{,}500$. One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are "older" (i.e. are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting---sometimes called uniform attachment---in which vertices are added one by one and each vertex connects to $m$ older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version., Comment: 29 pages
- Published
- 2018
43. A new combinatorial representation of the additive coalescent
- Author
-
Jean-François Marckert, Minmin Wang, Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Université Pierre et Marie Curie - Paris 6 (UPMC), Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires] (CONICET), ANR-14-CE25-0014,GRAAL,GRaphes et Arbres ALéatoires(2014), Marckert, Jean-François, and Appel à projets générique - GRaphes et Arbres ALéatoires - - GRAAL2014 - ANR-14-CE25-0014 - Appel à projets générique - VALID
- Subjects
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,parking ,General Mathematics ,68R05 Key Words: additive coalescent ,Markov process ,0102 computer and information sciences ,01 natural sciences ,increasing trees ,Coalescent theory ,Combinatorics ,symbols.namesake ,60J25 ,60F05 ,Representation (mathematics) ,ComputingMilieux_MISCELLANEOUS ,construction Mathematics Subject Classification (2000) 60C05 ,Mathematics ,Block (data storage) ,Discrete mathematics ,Applied Mathematics ,Probabilistic logic ,Cayley trees ,Computer Graphics and Computer-Aided Design ,Tree (graph theory) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,random walks on trees ,60K35 ,010201 computation theory & mathematics ,symbols ,Node (circuits) ,Variety (universal algebra) ,Software - Abstract
The standard additive coalescent starting with n particles is a Markov process which owns several combinatorial representations, one by Pitman as a process of coalescent forests, and one by Chassaing & Louchard as the block sizes in a parking scheme. In the coalescent forest representation, some edges are added successively between a random node and a random root. In this paper, we investigate an alternative construction by adding edges between the roots. This construction induces the same process at the level of cluster sizes, but allows one to make numerous connections with some combinatorial and probabilistic models that were not known to be connected with additive coalescent. The variety of the combinatorial objects involved here – size biased percolation, parking scheme in a tree, increasing trees, random cuts of trees – justifies our interests in this Acknowledgement : The research has been supported by ANR-14-CE25-0014 (ANR GRAAL).
- Published
- 2018
44. Boundary Layers in Periodic Homogenization of Neumann Problems
- Author
-
Zhongwei Shen and Jinping Zhuge
- Subjects
35B27, 75Q05 ,Elliptic systems ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,16. Peace & justice ,01 natural sciences ,Homogenization (chemistry) ,Dirichlet distribution ,010101 applied mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Rate of convergence ,Boundary data ,FOS: Mathematics ,symbols ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in $L^2$ in dimension three or higher. Sharp regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to obtain a higher-order convergence rate for Neumann problems with non-oscillating data., Comment: 54 pages; minor revision of the first version
- Published
- 2018
45. Dynamics of a ratio-dependent stage-structured predator-prey model with delay
- Author
-
Yongli Song, Tao Yin, and Hongying Shu
- Subjects
Hopf bifurcation ,Steady state ,General Mathematics ,Dynamics (mechanics) ,General Engineering ,Structure (category theory) ,01 natural sciences ,Stability (probability) ,Instability ,010305 fluids & plasmas ,010101 applied mathematics ,symbols.namesake ,Control theory ,0103 physical sciences ,symbols ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,0101 mathematics ,Reduction (mathematics) ,Center manifold ,Mathematics - Abstract
In this paper, we investigate the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator. This predator-prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang.[26] We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.
- Published
- 2017
46. A new computational method for solving two-dimensional Stratonovich Volterra integral equation
- Author
-
Elham Hadadian and Farshid Mirzaee
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Linear system ,General Engineering ,01 natural sciences ,Volterra integral equation ,010104 statistics & probability ,Algebraic equation ,symbols.namesake ,Operational matrix ,Error analysis ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
This paper presents a method for computing numerical solutions of two-dimensional Stratonovich Volterra integral equations using one-dimensional modification of hat functions and two-dimensional modification of hat functions. The problem is transformed to a linear system of algebraic equations using the operational matrix associated with one-dimensional modification of hat functions and two-dimensional modification of hat functions. The error analysis of the method is given. The method is computationally attractive, and applications are demonstrated by a numerical example. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
47. A delayed prey-predator model with Crowley-Martin-type functional response including prey refuge
- Author
-
Balram Dubey, Atasi Patra Maiti, and Jai Tushar
- Subjects
Hopf bifurcation ,General Mathematics ,010102 general mathematics ,General Engineering ,Functional response ,Type (model theory) ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,Predation ,symbols.namesake ,Control theory ,0103 physical sciences ,symbols ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Prey predator ,0101 mathematics ,Predator ,Bifurcation ,Mathematics - Abstract
In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin-type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
48. Global well-posedness of the 2D Euler-Boussinesq system with stratification effects
- Author
-
Zhouyu Li
- Subjects
General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,General Engineering ,01 natural sciences ,Stratification (mathematics) ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Euler's formula ,symbols ,Applied mathematics ,Initial value problem ,0101 mathematics ,Well posedness ,Mathematics - Abstract
This paper is concerned with the Cauchy problem of the two-dimensional Euler–Boussinesq system with stratification effects. We obtain the global existence of a unique solution to this system without assumptions of small initial data in Sobolev spaces. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
49. Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response
- Author
-
Wenzhen Gan, Canrong Tian, and Peng Zhu
- Subjects
Hopf bifurcation ,General Mathematics ,General Engineering ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,010101 applied mathematics ,symbols.namesake ,Control theory ,Stability theory ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,Positive equilibrium ,Center manifold ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
50. Uncertainty principle for measurable sets and signal recovery in quaternion domains
- Author
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Yan Yang, Kit Ian Kou, and Cuiming Zou
- Subjects
Signal processing ,Hypercomplex number ,Uncertainty principle ,General Mathematics ,010102 general mathematics ,General Engineering ,020206 networking & telecommunications ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Harmonic analysis ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Applied mathematics ,0101 mathematics ,Quaternion ,Mathematics - Abstract
The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time-limiting data. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2017
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