58 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
- Subjects
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
- Subjects
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.
- Published
- 2020
7. The asymptotic distribution of symbols on diagonals of random weighted staircase tableaux
- Author
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Amanda Lohss
- Subjects
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
8. Realisability conditions for second-order marginals of biphased media
- Author
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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
9. 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
10. 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
11. 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
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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
12. Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise
- Author
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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
13. Quantitative Estimates for Regular Lagrangian Flows with<scp>BV</scp>Vector Fields
- Author
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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
14. Fast Computation of Orthogonal Systems with a <scp>Skew‐Symmetric</scp> Differentiation Matrix
- Author
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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
15. Generalized approximate boundary synchronization for a coupled system of wave equations
- Author
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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
16. A probabilistic approach to a non‐local quadratic form and its connection to the Neumann boundary condition problem
- Author
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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
17. Transition Threshold for the <scp>3D</scp> Couette Flow in Sobolev Space
- Author
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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
18. Space‐time fractional Dirichlet problems
- Author
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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
19. Perfect matchings and Hamiltonian cycles in the preferential attachment model
- Author
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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
20. A new combinatorial representation of the additive coalescent
- Author
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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
21. 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
22. 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
23. Gluing Eguchi-Hanson Metrics and a Question of Page
- Author
-
Simon Brendle and Nikolaos Kapouleas
- Subjects
Mathematics - Differential Geometry ,0209 industrial biotechnology ,Riemann curvature tensor ,Pure mathematics ,General Mathematics ,02 engineering and technology ,01 natural sciences ,K3 surface ,High Energy Physics::Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,020901 industrial engineering & automation ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Orbifold ,Ricci curvature ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Ricci flow ,Mathematics::Geometric Topology ,Orientation (vector space) ,Differential Geometry (math.DG) ,symbols ,Gravitational singularity ,Mathematics::Differential Geometry ,Symmetry (geometry) ,Analysis of PDEs (math.AP) - Abstract
In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat K\"ahler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics which are neither K\"ahler nor self-dual. In this paper, we focus on a configuration of maximal symmetry involving $8$ small Eguchi-Hanson manifolds of each orientation which are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a non-vanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of $(-t)^{\frac{1}{2}}$, while the maximum of the Ricci curvature converges to $0$., Comment: to appear in Comm Pure Appl Math
- Published
- 2016
24. Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces
- Author
-
Zhifei Zhang, Dongyi Wei, and Weiren Zhao
- Subjects
Class (set theory) ,Scattering ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Euler equations ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,Mathematics - Analysis of PDEs ,Monotone polygon ,Inviscid flow ,FOS: Mathematics ,symbols ,0101 mathematics ,Shear flow ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we prove the decay estimates of the velocity and $H^1$ scattering for the 2D linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960., 56 pages
- Published
- 2016
25. A collocation method of lines for two-sided space-fractional advection-diffusion equations with variable coefficients
- Author
-
Hassan Khosravian-Arab, Delfim F. M. Torres, Mohammed K. Almoaeet, and Mostafa Shamsi
- Subjects
General Mathematics ,Basis function ,01 natural sciences ,Domain (mathematical analysis) ,symbols.namesake ,Mathematics - Analysis of PDEs ,Space‐fractional advection‐diffusion equations ,Collocation method ,35R11, 65M20, 65M70 ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Diffusion (business) ,Mathematics ,Variable (mathematics) ,Spectral collocation method ,010102 general mathematics ,Method of lines ,General Engineering ,Numerical Analysis (math.NA) ,Fractional partial differential equations ,Fractional calculus ,Left and right Riemann‐Liouville fractional derivatives ,010101 applied mathematics ,Jacobi polynomials ,symbols ,Analysis of PDEs (math.AP) - Abstract
We present the Method Of Lines (MOL), which is based on the spectral collocation method, to solve space-fractional advection-diffusion equations (SFADEs) on a finite domain with variable coefficients. We focus on the cases in which the SFADEs consist of both left- and right-sided fractional derivatives. To do so, we begin by introducing a new set of basis functions with some interesting features. The MOL, together with the spectral collocation method based on the new basis functions, are successfully applied to the SFADEs. Finally, four numerical examples, including benchmark problems and a problem with discontinuous advection and diffusion coefficients, are provided to illustrate the efficiency and exponentially accuracy of the proposed method., This is a preprint of a paper whose final and definite form is with 'Math. Methods Appl. Sci.' Submitted 07-July-2018; Revised 21-Jan-2019; Accepted 24-02-2019
- Published
- 2019
26. Counting Hamilton cycles in sparse random directed graphs
- Author
-
Matthew Kwan, Asaf Ferber, and Benny Sudakov
- Subjects
Random graph ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Directed graph ,Binary logarithm ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Omega ,Hamiltonian path ,Vertex (geometry) ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Directed Graph ,Hamilton cycle ,0101 mathematics ,Software ,Mathematics - Abstract
Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles.
- Published
- 2018
27. Fuzzy transformations and extremality of Gibbs measures for the potts model on a Cayley tree
- Author
-
Christof Külske and Utkir Abdulloevich Rozikov
- Subjects
Discrete mathematics ,Markov chain ,Explicit formulae ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Interval (mathematics) ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Set (abstract data type) ,010104 statistics & probability ,symbols.namesake ,symbols ,Order (group theory) ,Tree (set theory) ,0101 mathematics ,Gibbs measure ,Software ,Mathematics ,Potts model - Abstract
We continue our study of the full set of translation-invariant splitting Gibbs measures TISGMs, translation-invariant tree-indexed Markov chains for the q-state Potts model on a Cayley tree. In our previous work Kulske et al., J Stat Phys 156 2014, 189-200 we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is 2q-1. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is non-extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least 2q-1+q extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 636-678, 2017
- Published
- 2016
28. An h p -adaptive Newton-Galerkin finite element procedure for semilinear boundary value problems
- Author
-
Thomas P. Wihler, Jens Markus Melenk, and Mario Amrein
- Subjects
Discretization ,General Mathematics ,General Engineering ,hp-FEM ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,symbols.namesake ,Robustness (computer science) ,symbols ,Applied mathematics ,A priori and a posteriori ,Boundary value problem ,0101 mathematics ,Galerkin method ,Newton's method ,Mathematics - Abstract
In this paper, we develop an hp-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an hp-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hp-adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
29. A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems
- Author
-
Hegagi Mohamed Ali, Sílvio M. A. Gama, and Fernando Lobo Pereira
- Subjects
Class (set theory) ,Generalization ,General Mathematics ,010102 general mathematics ,General Engineering ,Context (language use) ,Function (mathematics) ,Optimal control ,01 natural sciences ,symbols.namesake ,Nonlinear system ,Simple (abstract algebra) ,Mittag-Leffler function ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,010301 acoustics ,Mathematics - Abstract
In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.
- Published
- 2016
30. Four random permutations conjugated by an adversary generateSnwith high probability
- Author
-
Yuval Peres, Robin Pemantle, and Igor Rivin
- Subjects
Transitive relation ,Conjecture ,Intersection (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Sumset ,0102 computer and information sciences ,Poisson distribution ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,Symmetric group ,symbols ,0101 mathematics ,Element (category theory) ,Software ,Mathematics - Abstract
We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group Sn generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of An or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015
- Published
- 2015
31. Large Deviations from a Stationary Measure for a Class of Dissipative PDEs with Random Kicks
- Author
-
Armen Shirikyan, Claude-Alain Pillet, Vahagn Nersesyan, Vojkan Jaksic, Department of Mathematics and Statistics [Montréal], McGill University = Université McGill [Montréal, Canada], Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), CPT - E5 Physique statistique et systèmes complexes, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Fédération de Recherche des Unités de MAthématiques de Marseille (FRUMAM), Avignon Université (AU)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), ANR-11-BS01-0017,EMAQS,Estimation et manipulation à l'échelle Quantique(2011), ANR-11-BS01-0015,STOSYMAP,Systèmes stochastiques en mathématiques et physique mathématique(2011), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011), and Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Avignon Université (AU)-Université de Toulon (UTLN)
- Subjects
General Mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Markov process ,Measure (mathematics) ,large deviations ,symbols.namesake ,Mathematics - Analysis of PDEs ,35Q30, 76D05, 60B12, 60F10 ,Integer ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Applied mathematics ,Navier–Stokes system ,Mathematical Physics ,Ginzburg–Landau equation ,Mathematics ,Markov chain ,Applied Mathematics ,Probability (math.PR) ,Mathematical analysis ,Mathematical Physics (math-ph) ,Coupling (probability) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Bounded function ,dissipative PDE’s ,occupation measures ,symbols ,Dissipative system ,Large deviations theory ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
We study a class of dissipative PDE's perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer's large deviation criterion, a Lyapunov-Schmidt type reduction, and an abstract result on large-time asymptotic for generalised Markov semigroups., Comment: 42 pages
- Published
- 2015
32. Piecewise Legendre spectral-collocation method for Volterra integro-differential equations
- Author
-
Yanping Chen and Zhendong Gu
- Subjects
Polynomial ,Mathematical optimization ,Legendre wavelet ,General Mathematics ,Legendre's equation ,Volterra integral equation ,symbols.namesake ,Associated Legendre polynomials ,Computational Theory and Mathematics ,Collocation method ,symbols ,Piecewise ,Applied mathematics ,Legendre polynomials ,Mathematics - Abstract
Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in$h^{m}N^{-m}$-version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.
- Published
- 2015
33. Nonlinear Steepest Descent and Numerical Solution of Riemann-Hilbert Problems
- Author
-
Thomas Trogdon and Sheehan Olver
- Subjects
Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Stationary point ,symbols.namesake ,Nonlinear system ,Arbitrarily large ,Riemann hypothesis ,Hilbert's problems ,symbols ,Applied mathematics ,Gradient descent ,Mathematics - Abstract
The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for Riemann-Hilbert problems, the resulting numerical methods have been shown, in practice, to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann-Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painleve II equation and the modified Korteweg–de Vries equation to explicitly demonstrate the practical validity of the theory. © 2014 Wiley Periodicals, Inc.
- Published
- 2013
34. Towards a Mathematical Theory of Super-resolution
- Author
-
Carlos Fernandez-Granda and Emmanuel J. Candès
- Subjects
FOS: Computer and information sciences ,Computer Science - Information Theory ,Information Theory (cs.IT) ,Applied Mathematics ,General Mathematics ,Scale (descriptive set theory) ,Numerical Analysis (math.NA) ,Mathematical theory ,Noise ,Discontinuity (linguistics) ,symbols.namesake ,Fourier transform ,Dimension (vector space) ,Convex optimization ,FOS: Mathematics ,symbols ,Point (geometry) ,Mathematics - Numerical Analysis ,Algorithm ,Mathematics - Abstract
This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary., Comment: 48 pages, 12 figures
- Published
- 2013
35. Convergence of frozen Gaussian approximation for high-frequency wave propagation
- Author
-
Xu Yang and Jianfeng Lu
- Subjects
Asymptotic analysis ,Frequency wave ,Wave propagation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Hyperbolic systems ,Gaussian random field ,Gaussian approximation ,symbols.namesake ,Convergence (routing) ,Gaussian function ,symbols ,Mathematics - Abstract
The frozen Gaussian approximation provides a highly efficient computational method for high-frequency wave propagation. The derivation of the method is based on asymptotic analysis. In this paper, for general linear strictly hyperbolic systems, we establish the rigorous convergence result for frozen Gaussian approximation. As a byproduct, higher-order frozen Gaussian approximation is developed. © 2011 Wiley Periodicals, Inc.
- Published
- 2011
36. On the Hamiltonian structure of Hirota-Kimura discretization of the Euler top
- Author
-
Matteo Petrera and Yuri B. Suris
- Subjects
symbols.namesake ,Integrable system ,Discretization ,Hamiltonian structure ,General Mathematics ,Euler's formula ,symbols ,Elliptic function ,Applied mathematics ,Hamiltonian (quantum mechanics) ,Mathematics - Abstract
This paper deals with a remarkable integrable discretization of the so (3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi-Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville-Arnold sense (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
- Published
- 2010
37. Ginzburg-Landau vortex dynamics driven by an applied boundary current
- Author
-
Ian Tice
- Subjects
Current (mathematics) ,Applied Mathematics ,General Mathematics ,Boundary (topology) ,Vorticity ,Upper and lower bounds ,Vortex ,symbols.namesake ,Mathematics - Analysis of PDEs ,Classical mechanics ,Bounded function ,FOS: Mathematics ,symbols ,Relaxation (physics) ,Lorentz force ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain $\Omega \subset \Rn{2}$, subject to an electrical current applied on the boundary. The dynamics with an applied current are non-dissipative, but via the identification of a special structure in an interaction energy, we are able to derive a precise upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit $\ep \to 0$. We first consider the original time scale, in which the vortices do not move and the solutions undergo a "phase relaxation." Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current., Comment: 45 pages; v2: generalized BCs, streamlined presentation
- Published
- 2010
38. Tensor products and correlation estimates with applications to nonlinear Schrödinger equations
- Author
-
Nikolaos Tzirakis, Manoussos G. Grillakis, and James E. Colliander
- Subjects
Conservation law ,Vector operator ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Schrödinger equation ,symbols.namesake ,Nonlinear system ,Tensor product ,symbols ,Initial value problem ,Direct proof ,Nonlinear Schrödinger equation ,Mathematics - Abstract
We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schrodinger equations and derive a direct proof of Nakanishi's H 1 scattering result for every L 2 -supercritical nonlinearity. We also prove scattering below the energy space for a certain class of L 2 -supercritical equations. In this paper we obtain new 1 a priori estimates for solutions of the nonlinear Schrodinger equation in one and two dimension. We also provide a systematic way to obtain the known interaction a priori estimates for dimensions higher than three. These estimates are monotonicity formulae that take advantage of the conservation of the momentum of the equation. Due to the pioneering work (19), estimates of this type are referred to as Morawetz estimates in the literature. We then apply these estimates to study the global behavior of solutions to the nonlinear Schrodinger equation. To be more precise we want to study the global-in-time behavior of solutions to the following initial value problem
- Published
- 2009
39. Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations
- Author
-
Urszula Foryś
- Subjects
Differential equation ,General Mathematics ,Mathematical analysis ,General Engineering ,symbols.namesake ,Ordinary differential equation ,Dirichlet boundary condition ,Reaction–diffusion system ,Neumann boundary condition ,symbols ,Applied mathematics ,Boundary value problem ,Numerical stability ,Mathematics ,Variable (mathematics) - Abstract
In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre-malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.
- Published
- 2009
40. On the stabilizing effect of convection in three-dimensional incompressible flows
- Author
-
Zhen-L Lei and Thomas Y. Hou
- Subjects
Convection ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Rotational symmetry ,Mechanics ,Euler equations ,Term (time) ,Physics::Fluid Dynamics ,symbols.namesake ,Nonlinear system ,Classical mechanics ,Singularity ,symbols ,Euler's formula ,Compressibility ,Mathematics - Abstract
We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three-dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three-dimensional Euler or Navier-Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier-Stokes equations. We will present numerical evidence that seems to support that the three-dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three-dimensional model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.
- Published
- 2009
41. Geometry and analysis of spin equations
- Author
-
Tyler J. Jarvis, Yongbin Ruan, and Huijun Fan
- Subjects
Spin geometry ,Polynomial ,Applied Mathematics ,General Mathematics ,Riemann surface ,Degenerate energy levels ,Geometry ,Moduli space ,symbols.namesake ,Compactness theorem ,symbols ,Synthetic geometry ,Spin-½ ,Mathematics - Abstract
We introduce W-spin structures on a Riemann surface Σ and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W-spin equations when W = W(x1,... , xt ) is a non- degenerate quasi-homogeneous polynomial with fractional degrees (or weights) qi < 1/2 for all i. In particular, the compactness theorem holds for the superpotentials E6, E7, E8, or An−1, Dn+1 for n ≥ 3.
- Published
- 2008
42. On sparse reconstruction from Fourier and Gaussian measurements
- Author
-
Mark Rudelson and Roman Vershynin
- Subjects
Applied Mathematics ,General Mathematics ,Gaussian ,Banach space ,Sparse approximation ,Restricted isometry property ,Combinatorics ,symbols.namesake ,Fourier transform ,Probability theory ,Convex optimization ,symbols ,Probability distribution ,Mathematics - Abstract
This paper improves upon best-known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r-sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size k(r, n) = O(r log(n)·log 2 (r) log(r logn)) = O(r log 4 n ). A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n/r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short.
- Published
- 2008
43. Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds
- Author
-
Jeremie Szeftel, Jean-Marc Delort, Benoît Grébert, and Dario Bambusi
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Cauchy distribution ,01 natural sciences ,Manifold ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Distribution function ,symbols ,Mathematics::Differential Geometry ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Klein–Gordon equation ,Laplace operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.
- Published
- 2007
44. TheN-soliton of the focusing nonlinear Schrödinger equation forN large
- Author
-
Gregory Lyng and Peter D. Miller
- Subjects
Asymptotic analysis ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Schrödinger equation ,symbols.namesake ,Riemann problem ,Inverse scattering problem ,symbols ,Initial value problem ,Soliton ,Caustic (optics) ,Nonlinear Schrödinger equation ,Mathematics - Abstract
We present a detailed analysis of the solution of the focusing nonlinear Schrodinger equation with initial condition (x,0) = N sech(x) in the limit N ! 1. We begin by presenting new and more accurate numerical reconstructions of the N-soliton by inverse scattering (numerical linear algebra) for N = 5, 10, 20, and 40. We then recast the inverse-scattering problem as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this problem in the large-N limit. For those (x,t) where results have been obtained by other authors, we improve the error estimates from O(N 1/3 ) to O(N 1 ). We also analyze the Fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation. We then study the N-soliton for values of (x,t) where analysis has not been carried out before, and we uncover new phenomena. The main discovery of this paper is the mathematical mechanism for a secondary caustic (phase transition), which turns out to differ from the mechanism that generates the primary caustic. The mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum for the N-soliton, and more significantly, cannot be recovered from an analysis of an ostensibly similar Riemann-Hilbert problem in the conditions of which a certain formal continuum limit is taken on an ad hoc basis.
- Published
- 2007
45. A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations
- Author
-
Gautam Iyer and Peter Constantin
- Subjects
Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Burgers' equation ,Euler equations ,Physics::Fluid Dynamics ,Nonlinear system ,symbols.namesake ,Wiener process ,Inviscid flow ,symbols ,Applied mathematics ,Vector field ,Representation (mathematics) ,Navier–Stokes equations ,Mathematics - Abstract
In this paper we derive a probabilistic representation of the deterministic three-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for the Euler equations of ideal fluids is used to recover the velocity field. This method admits a self-contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic-type equations, including viscous Burgers equations and Lagrangian-averaged Navier-Stokes alpha models. © 2007 Wiley Periodicals, Inc.
- Published
- 2007
46. Hamiltonian ODEs in the Wasserstein space of probability measures
- Author
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Wilfrid Gangbo, Luigi Ambrosio, Ambrosio, Luigi, and Gangbo, W.
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Applied Mathematics ,General Mathematics ,Mathematical analysis ,Subderivative ,Symplectic matrix ,Combinatorics ,symbols.namesake ,Metric space ,Phase space ,Norm (mathematics) ,symbols ,Tangent vector ,Hamiltonian (quantum mechanics) ,Mathematics ,Probability measure - Abstract
In this paper we consider a Hamiltonian H on P_2(R^{2d} ), the set of probability measures with finite quadratic moments on the phase space R^{2d} = R^d × R^d , which is a metric space when endowed with the Wasserstein distance W_2. We study the initial value problem dμ_t/dt+∇·(J_dv_tμ_t ) = 0, where J_d is the canonical symplectic matrix, μ_0 is prescribed, and v_t is a tangent vector to P_2(R^{2d}) at μ_t , belonging to ∂H(μ_t ), the subdifferential of H at μ_t . Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ_0 is absolutely continuous. It ensures that μ_t remains absolutely continuous and v_t = ∇H(μ_t ) is the element of minimal norm in ∂H(μt ). The second method handles any initial measure μ_0. If we further assume that H is λ-convex, proper, and lowersemicontinuous on P_2(R^{2d} ), we prove that the Hamiltonian is preserved along any solution of our evolutive system, namely H(μ_t ) = H(μ_0).
- Published
- 2007
47. PDEs for the Gaussian ensemble with external source and the Pearcey distribution
- Author
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Pierre van Moerbeke and Mark Adler
- Subjects
Applied Mathematics ,General Mathematics ,Gaussian ,Mathematical analysis ,Hermitian matrix ,symbols.namesake ,Quartic function ,Diagonal matrix ,symbols ,Probability distribution ,Random matrix ,Eigenvalues and eigenvectors ,Brownian motion ,Mathematics - Abstract
The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues a. As a first result, the probability that the eigenvalues of the ensemble belong to an interval E satisfies a fourth-order PDE with quartic nonlinearity; the variables are the eigenvalue a and the boundary of E. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap, i.e., when the support of the equilibrium measure for large-size random matrices has a gap that can be made to close. The Gaussian Hermitian random matrix ensemble with external source, described above, has this feature. The Pearcey distribution is shown to satisfy a fourth-order PDE with cubic nonlinearity. This also gives the PDE for the transition probability of the Pearcey process, a limiting process associated with nonintersecting Brownian motions on R. (C) 2006 Wiley Periodicals, Inc.
- Published
- 2007
48. On the Floer homology of cotangent bundles
- Author
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Matthias Schwarz and Alberto Abbondandolo
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Pseudoholomorphic curve ,Mathematics::Geometric Topology ,Legendre transformation ,Algebra ,symbols.namesake ,Morse homology ,Floer homology ,Loop space ,symbols ,Cotangent bundle ,Mathematics::Symplectic Geometry ,Mathematics ,Singular homology ,Symplectic geometry - Abstract
This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.
- Published
- 2005
49. Layer solutions in a half-space for boundary reactions
- Author
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Xavier Cabré and J. Solà-Morales
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Phase transition ,Pure mathematics ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Half-space ,Maxima and minima ,symbols.namesake ,Monotone polygon ,Bounded function ,symbols ,Uniqueness ,Hamiltonian (quantum mechanics) ,Mathematics - Abstract
R n−1 are denoted by y = (y1, . . . , yn−1). Our main goal is to study bounded solutions of (1.1) that are monotone increasing, say from −1 to 1, in one of the y-variables. We call them layer solutions of (1.1), and we study their existence, uniqueness, symmetry, and variational properties, as well as their asymptotic behavior. The interest in such increasing solutions comes from some models of boundary phase transitions. When the nonlinearity f is given by f (u) = sin(cu) for some constant c, problem (1.1) in a half-plane is called the Peierls-Nabarro problem, and it appears as a model of dislocations in crystals (see [21, 36]). The Peierls-Nabarro problem is also central to the analysis of boundary vortices in the paper [28], which studies a model for soft thin films in micromagnetism recently derived by Kohn and Slastikov [26] (see also [27]). Our main result, Theorem 1.2, characterizes the nonlinearities f for which there exists a layer solution of (1.1) in dimension n = 2. We prove that the necessary and sufficient condition is that the potential G (defined by G ′ = − f ) has two, and only two, absolute minima in the interval [−1, 1], located at ±1. Under the additional hypothesis G ′′(±1) > 0, we also establish the uniqueness of a layer solution up to translations in the y-variable. The proofs of both the necessity and the sufficiency of the condition on G for existence use new ingredients, which we develop in this article. A first one is a nonlocal estimate, as well as a conserved or Hamiltonian quantity, satisfied by every layer solution in dimension 2 (see Theorem 1.3). The estimate can be seen as
- Published
- 2005
50. Delta and singular delta locus for one-dimensional systems of conservation laws
- Author
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Marko Nedeljkov
- Subjects
Shock wave ,Conservation law ,Variables ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,General Engineering ,Lipschitz continuity ,symbols.namesake ,Riemann hypothesis ,Riemann problem ,symbols ,Applied mathematics ,Locus (mathematics) ,Solution concept ,Mathematics ,media_common - Abstract
This work gives a condition for existence of singular and delta shock wave solutions to Riemann problem for 2×2 systems of conservation laws. For a fixed left-hand side value of Riemann data, the condition obtained in the paper describes a set of possible right-hand side values. The procedure is similar to the standard one of finding the Hugoniot locus. Fluxes of the considered systems are globally Lipschitz with respect to one of the dependent variables. The association in a Colombeau-type algebra is used as a solution concept. Copyright © 2004 John Wiley &Sons, Ltd.
- Published
- 2004
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