Let H be a separable Hilbert space and {en : n ∈ Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if 〈T ej, ei〉 = c2i− j, where cn is the nth Fourier coefficient of a bounded Lebesgue measurable function φ on the unit circle T = {z ∈ C : |z| = 1}. It has been shown [9], with some assumption on the smoothness and the zeros of φ, that T ∗ is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. These results, together with the theory of shifts (e.g., in [11]), allows us to identify all bounded operators on H commuting with such T .
设H是一个可分离的Hilbert空间，{en: n∈Z}是H中的一个标准正交基，如果< T ej, ei > = c2i−j，则有界算子T称为斜Toeplitz算子，其中cn是单位圆T = {Z∈C: | Z | = 1}上有界Lebesgue可测函数φ的第n个傅立叶系数。已经证明[9]，在对φ的光滑性和零的某些假设下，T *类似于一个移位的常数倍，或者类似于一个移位与一个秩一酉的直和的常数倍，具有无穷倍性。这些结果，连同移位理论(例如，在[11]中)，使我们能够识别H与这样的T交换上的所有有界算子。
{"title":"Operators That Commute with Slant Toeplitz Operators","authors":"Mark Ho, M. Wong","doi":"10.1093/AMRX/ABN003","DOIUrl":"https://doi.org/10.1093/AMRX/ABN003","url":null,"abstract":"Let H be a separable Hilbert space and {en : n ∈ Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if 〈T ej, ei〉 = c2i− j, where cn is the nth Fourier coefficient of a bounded Lebesgue measurable function φ on the unit circle T = {z ∈ C : |z| = 1}. It has been shown [9], with some assumption on the smoothness and the zeros of φ, that T ∗ is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. These results, together with the theory of shifts (e.g., in [11]), allows us to identify all bounded operators on H commuting with such T .","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75345859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
which was first derived in (Ulusoy, Nonlinearity 20 (2007): 685–712). We prove results on the regularity of non-negative solutions. In Ulusoy, an entropy dissipation–entropy estimate was provided for the p = 3 and n = 2 case using the energy functional Kq := ∫ hx hq dx. Here, we extend our calculations to include various other p and n values. After establishing some results on the support properties of solutions, we finally complete the analysis of the long-time behavior of non-negative weak solutions.
{"title":"On a New Family of Degenerate Parabolic Equations","authors":"S. Ulusoy","doi":"10.1093/AMRX/ABM010","DOIUrl":"https://doi.org/10.1093/AMRX/ABM010","url":null,"abstract":"which was first derived in (Ulusoy, Nonlinearity 20 (2007): 685–712). We prove results on the regularity of non-negative solutions. In Ulusoy, an entropy dissipation–entropy estimate was provided for the p = 3 and n = 2 case using the energy functional Kq := ∫ hx hq dx. Here, we extend our calculations to include various other p and n values. After establishing some results on the support properties of solutions, we finally complete the analysis of the long-time behavior of non-negative weak solutions.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73890176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Convergence of Leray-type Solutions of Inhomogeneous MHD Systems","authors":"M. Ghazel, J. Benameur","doi":"10.1093/AMRX/ABM011","DOIUrl":"https://doi.org/10.1093/AMRX/ABM011","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"151 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73712763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, order conditions for coefficients of a class of stochastic Runge–Kutta (SRK) methods with strong global order 1, which applied for solving Ito stochastic differential equations (SDEs) with a single noise process, are presented. In particular, explicit twostage and three-stage SRK methods of this class with minimum principal error constants are constructed. Numerical results with two test problems of our methods, the Ito method and Milstein method will be compared.
{"title":"Strong Runge–Kutta Methods With order one for Numerical Solution of Itô Stochastic Differential Equations","authors":"A. Soheili, M.Namjoo","doi":"10.1093/AMRX/ABM003","DOIUrl":"https://doi.org/10.1093/AMRX/ABM003","url":null,"abstract":"In this paper, order conditions for coefficients of a class of stochastic Runge–Kutta (SRK) methods with strong global order 1, which applied for solving Ito stochastic differential equations (SDEs) with a single noise process, are presented. In particular, explicit twostage and three-stage SRK methods of this class with minimum principal error constants are constructed. Numerical results with two test problems of our methods, the Ito method and Milstein method will be compared.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89108567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Boulakia, M. Fernández, Jean-Frédéric Gerbeau, N. Zemzemi
We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure.
{"title":"A Coupled System of PDEs and ODEs Arising in Electrocardiograms Modeling","authors":"M. Boulakia, M. Fernández, Jean-Frédéric Gerbeau, N. Zemzemi","doi":"10.1093/AMRX/ABN002","DOIUrl":"https://doi.org/10.1093/AMRX/ABN002","url":null,"abstract":"We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"75 1","pages":"24"},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84525045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a recent work, Blanc, Le Bris and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators. Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation. In the present article, we provide another proof of their results using the formalism of integral functionals. We also extend the analysis to cover the case of quasiconvex integrands.
在最近的工作中，Blanc, Le Bris和Lions引入了随机微分同胚的概念以及线性和单调椭圆算子的随机均匀化理论的一个变体。他们的证明依赖于遍历定理和对相关校正方程的分析。在这篇文章中，我们用积分泛函的形式证明了他们的结果。我们也将分析扩展到拟凸积分的情况。
{"title":"Stochastic diffeomorphisms and homogenization of multiple integrals","authors":"A. Gloria","doi":"10.1093/AMRX/ABN001","DOIUrl":"https://doi.org/10.1093/AMRX/ABN001","url":null,"abstract":"In a recent work, Blanc, Le Bris and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators. Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation. In the present article, we provide another proof of their results using the formalism of integral functionals. We also extend the analysis to cover the case of quasiconvex integrands.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84549973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Meghan McIntyre, Edward L. Rowe, M. Shearer, J. Gray, A. Thornton
A nonlinear first-order partial differential equation in two space variables and time describes the process of kinetic sieving in an avalanche, in which larger particles tend to rise to the surface while smaller particles descend, quickly leading to completely segregated layers. The interface between layers is a shock wave satisfying its own nonlinear equation. When the interface becomes vertical, it loses stability, and develops a mixing zone. The mixing zone is described explicitly under idealized initial conditions, and verified with numerical simulation. The problem and its solution are similar to twodimensional Riemann problems for scalar first-order conservation laws; the difference here is that the equation is not scale-invariant, due to shear in the avalanche, an essential ingredient of kinetic sieving.
{"title":"Evolution of a Mixing Zone in Granular Avalanches","authors":"Meghan McIntyre, Edward L. Rowe, M. Shearer, J. Gray, A. Thornton","doi":"10.1093/AMRX/ABM008","DOIUrl":"https://doi.org/10.1093/AMRX/ABM008","url":null,"abstract":"A nonlinear first-order partial differential equation in two space variables and time describes the process of kinetic sieving in an avalanche, in which larger particles tend to rise to the surface while smaller particles descend, quickly leading to completely segregated layers. The interface between layers is a shock wave satisfying its own nonlinear equation. When the interface becomes vertical, it loses stability, and develops a mixing zone. The mixing zone is described explicitly under idealized initial conditions, and verified with numerical simulation. The problem and its solution are similar to twodimensional Riemann problems for scalar first-order conservation laws; the difference here is that the equation is not scale-invariant, due to shear in the avalanche, an essential ingredient of kinetic sieving.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84852628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For the 3d cubic nonlinear Schrodinger (NLS) equation, which has critical (scaling) norms L 3 and u H 1/2 , we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up solutions, and prove a result on the concentration of the L 3 norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numer- ical experiments that consist of a specific bump profile with maximum at the origin and focus toward the origin at rate ∼ (T − t) 1/2 , where T > 0 is the blow-up time. For the other possibility, we propose the existence of "contracting sphere blow-up solutions", i.e. those that concentrate on a sphere of radius ∼ (T −t) 1/3 , but focus towards this sphere at a faster rate ∼ (T − t) 2/3 . These conjectured solutions are analyzed through heuristic arguments and shown (at this level of precision) to be consistent with all conservation laws of the equation.
{"title":"On Blow-up Solutions to the 3D Cubic Nonlinear Schrödinger Equation","authors":"J. Holmer, S. Roudenko","doi":"10.1093/AMRX/ABM004","DOIUrl":"https://doi.org/10.1093/AMRX/ABM004","url":null,"abstract":"For the 3d cubic nonlinear Schrodinger (NLS) equation, which has critical (scaling) norms L 3 and u H 1/2 , we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up solutions, and prove a result on the concentration of the L 3 norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numer- ical experiments that consist of a specific bump profile with maximum at the origin and focus toward the origin at rate ∼ (T − t) 1/2 , where T > 0 is the blow-up time. For the other possibility, we propose the existence of \"contracting sphere blow-up solutions\", i.e. those that concentrate on a sphere of radius ∼ (T −t) 1/3 , but focus towards this sphere at a faster rate ∼ (T − t) 2/3 . These conjectured solutions are analyzed through heuristic arguments and shown (at this level of precision) to be consistent with all conservation laws of the equation.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81980896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we introduce a novel modified synthetic evaluation method (M-TOPSIS) based on the concept of original TOPSIS and calculate the distance between the alternatives and ‘optimized ideal reference point’ in the D D-plane. It could avoid rank reversals and solve the problem on evaluation failure that often occurs in original TOPSIS, so we believe that the mechanism of M-TOPSIS is more reasonable. Furthermore, the MTOPSIS method is simple in both concept and calculation procedures.
{"title":"Comparative Analysis of a Novel M-TOPSIS Method and TOPSIS","authors":"L. Ren, Yanqiong Zhang, Yiren Wang, Zhen-qiu Sun","doi":"10.1093/AMRX/ABM005","DOIUrl":"https://doi.org/10.1093/AMRX/ABM005","url":null,"abstract":"In this study, we introduce a novel modified synthetic evaluation method (M-TOPSIS) based on the concept of original TOPSIS and calculate the distance between the alternatives and ‘optimized ideal reference point’ in the D D-plane. It could avoid rank reversals and solve the problem on evaluation failure that often occurs in original TOPSIS, so we believe that the mechanism of M-TOPSIS is more reasonable. Furthermore, the MTOPSIS method is simple in both concept and calculation procedures.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2010-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86357337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of known perturbations. The feedback law is determined by solving a Linear-Quadratic optimal control problem. The observation is the laminar-to-turbulent transition location linearized about its stationary position, the control is a suction velocity through a small slot in the plate, the state equation is the linearized Crocco equation about its stationary solution. This article is the continuation of [7] where we have studied the corresponding Linear-Quadratic control problem in the absence of perturbations. The solution to the algebraic Riccati equation determined in [7], together with the solution of an evolution equation taking into account the nonhomogeneous perturbations in the model, are used to define the feedback control law.
{"title":"Feedback Stabilization of a Boundary Layer Equation. Part 2: Nonhomogeneous State Equations and Numerical Simulations","authors":"J. Buchot, J. Raymond","doi":"10.1093/AMRX/ABP007","DOIUrl":"https://doi.org/10.1093/AMRX/ABP007","url":null,"abstract":"We study the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of known perturbations. The feedback law is determined by solving a Linear-Quadratic optimal control problem. The observation is the laminar-to-turbulent transition location linearized about its stationary position, the control is a suction velocity through a small slot in the plate, the state equation is the linearized Crocco equation about its stationary solution. This article is the continuation of [7] where we have studied the corresponding Linear-Quadratic control problem in the absence of perturbations. The solution to the algebraic Riccati equation determined in [7], together with the solution of an evolution equation taking into account the nonhomogeneous perturbations in the model, are used to define the feedback control law.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"22 1","pages":"87-122"},"PeriodicalIF":0.0,"publicationDate":"2010-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87146202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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