A $C^2$ function on $mathbb{R}^n$ is called strictly $(n-1)$-convex if the sum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, we establish a global $C^2$ estimates to the Monge-Amp`ere equation for strictly $(n-1)$-convex functions with Neumann condition. By the method of continuity, we prove an existence theorem for strictly $(n-1)$-convex solutions of the Neumann problems.
{"title":"The Monge-Ampère Equation for Strictly (n−1)-convex Functions with Neumann Condition","authors":"B. Deng","doi":"10.4208/jms.v53n1.20.04","DOIUrl":"https://doi.org/10.4208/jms.v53n1.20.04","url":null,"abstract":"A $C^2$ function on $mathbb{R}^n$ is called strictly $(n-1)$-convex if the sum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, we establish a global $C^2$ estimates to the Monge-Amp`ere equation for strictly $(n-1)$-convex functions with Neumann condition. By the method of continuity, we prove an existence theorem for strictly $(n-1)$-convex solutions of the Neumann problems.","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44537851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a system of parabolic PDEs with measure data modelling a problem of the time resolved diffuse optical tomography with a fluorescence term and Robin boundary conditions. We focus on the direct problem where the quantity of interest is the density of photons in the diffusion equations and which constitutes a major step to solve the inverse problem of identifiability and reconstruction of diffusion, absorption and concentration of the fluorescent markers. We study the problem under a variational form and its discretization with finite element method and we give some numerical simulation results for verification purpose as well as simulations with real data from a tomograph. AMS subject classifications: 65N21, 65J22, 78A46
{"title":"The Direct Robin Boundary Value Parabolic System of Time-Resolved Diffuse Optical Tomography with Fluorescence","authors":"Zakaria Belhachmi sci","doi":"10.4208/jms.v52n3.19.07","DOIUrl":"https://doi.org/10.4208/jms.v52n3.19.07","url":null,"abstract":"We consider a system of parabolic PDEs with measure data modelling a problem of the time resolved diffuse optical tomography with a fluorescence term and Robin boundary conditions. We focus on the direct problem where the quantity of interest is the density of photons in the diffusion equations and which constitutes a major step to solve the inverse problem of identifiability and reconstruction of diffusion, absorption and concentration of the fluorescent markers. We study the problem under a variational form and its discretization with finite element method and we give some numerical simulation results for verification purpose as well as simulations with real data from a tomograph. AMS subject classifications: 65N21, 65J22, 78A46","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45855959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Flow inside a lid-driven cavity (LDC) is studied here to elucidate bifurcation sequences of the flow at super-critical Reynolds numbers (Recr1) with the help of analyzing the time series at most energetic points in the flow domain. The implication of Recr1 in the context of direct simulation of Navier-Stokes equation is presented here for LDC, with or without explicit excitation inside the LDC. This is aided further by performing detailed enstrophy-based proper orthogonal decomposition (POD) of the flow field. The flow has been computed by an accurate numerical method for two different uniform grids. POD of results of these two grids help us understand the receptivity aspects of the flow field, which give rise to the computed bifurcation sequences by understanding the similarity and differences of these two sets of computations. We show that POD modes help one understand the primary and secondary instabilities noted during the bifurcation sequences. AMS subject classifications: 65M12, 65M15, 65M60, 76D05, 76F20, 76F65
{"title":"POD Applied to Numerical Study of Unsteady Flow Inside Lid-driven Cavity","authors":"Lucas Lestandi","doi":"10.4208/JMS.V51N2.18.03","DOIUrl":"https://doi.org/10.4208/JMS.V51N2.18.03","url":null,"abstract":"Flow inside a lid-driven cavity (LDC) is studied here to elucidate bifurcation sequences of the flow at super-critical Reynolds numbers (Recr1) with the help of analyzing the time series at most energetic points in the flow domain. The implication of Recr1 in the context of direct simulation of Navier-Stokes equation is presented here for LDC, with or without explicit excitation inside the LDC. This is aided further by performing detailed enstrophy-based proper orthogonal decomposition (POD) of the flow field. The flow has been computed by an accurate numerical method for two different uniform grids. POD of results of these two grids help us understand the receptivity aspects of the flow field, which give rise to the computed bifurcation sequences by understanding the similarity and differences of these two sets of computations. We show that POD modes help one understand the primary and secondary instabilities noted during the bifurcation sequences. AMS subject classifications: 65M12, 65M15, 65M60, 76D05, 76F20, 76F65","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41575680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose numerical schemes for stochastic differential equations driven by white noise and colored noise, respectively. For this purpose, we first discretize the white noise and colored noise, and give their regularity estimates. Then we use spectral element methods to solve the corresponding stochastic differential equations numerically. The approximation errors are derived, and the numerical results demonstrate high accuracy of the proposed schemes. AMS subject classifications: 65M70, 65L60, 41A10, 60H35
{"title":"Spectral Element Methods for Stochastic Differential Equations with Additive Noise","authors":"Chao Zhang","doi":"10.4208/jms.v51n1.18.05","DOIUrl":"https://doi.org/10.4208/jms.v51n1.18.05","url":null,"abstract":"In this paper, we propose numerical schemes for stochastic differential equations driven by white noise and colored noise, respectively. For this purpose, we first discretize the white noise and colored noise, and give their regularity estimates. Then we use spectral element methods to solve the corresponding stochastic differential equations numerically. The approximation errors are derived, and the numerical results demonstrate high accuracy of the proposed schemes. AMS subject classifications: 65M70, 65L60, 41A10, 60H35","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45728762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the cubature points based triangular spectral element method and provide accuracy results for elliptic problems in non polygonal domains using various isoparametric mappings. The capabilities of the method are here again clearly confirmed.
{"title":"Cubature Points Based Triangular Spectral Elements: an Accuracy Study","authors":"R. Pasquetti, F. Rapetti","doi":"10.4208/JMS.V51N1.18.02","DOIUrl":"https://doi.org/10.4208/JMS.V51N1.18.02","url":null,"abstract":"We investigate the cubature points based triangular spectral element method and provide accuracy results for elliptic problems in non polygonal domains using various isoparametric mappings. The capabilities of the method are here again clearly confirmed.","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43838310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution. AMS subject classifications: 35B40, 35B41, 35K51, 80A22, 80A20, 35Q53, 45K05, 35K55, 35G30, 92D50
{"title":"Attractors for a Caginalp Phase-field Model with Singular Potential","authors":"Alain Miranville and Charbel Wehbe","doi":"10.4208/JMS.V51N4.18.01","DOIUrl":"https://doi.org/10.4208/JMS.V51N4.18.01","url":null,"abstract":"We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution. AMS subject classifications: 35B40, 35B41, 35K51, 80A22, 80A20, 35Q53, 45K05, 35K55, 35G30, 92D50","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42790418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is well known that the approximation of eigenvalues and associated eigenfunctions of a linear operator under constraint is a difficult problem. One of the difficulties is to propose methods of approximation which satisfy in a stable and accurate way the eigenvalues equations, the constraint one and the boundary conditions. Using any non-stable method leads to the presence of non-physical eigenvalues: a multiple zero one called spurious modes and non-zero one called pollution modes. One way to eliminate these two families is to favor the constraint equations by satisfying it exactly and to verify the equations of the eigenvalues equations in weak ways. To illustrate our contribution in this field we consider in this paper the case of Stokes operator. We describe several methods that produce the correct number of eigenvalues. We numerically prove how these methods are adequate to correctly solve the 2D Stokes eigenvalue problem. AMS subject classifications: 76D07, 65N35, 34L16
{"title":"Numerical Assessment of a Class of High Order Stokes Spectrum Solver","authors":"E. Ahusborde","doi":"10.4208/jms.v51n1.18.01","DOIUrl":"https://doi.org/10.4208/jms.v51n1.18.01","url":null,"abstract":"It is well known that the approximation of eigenvalues and associated eigenfunctions of a linear operator under constraint is a difficult problem. One of the difficulties is to propose methods of approximation which satisfy in a stable and accurate way the eigenvalues equations, the constraint one and the boundary conditions. Using any non-stable method leads to the presence of non-physical eigenvalues: a multiple zero one called spurious modes and non-zero one called pollution modes. One way to eliminate these two families is to favor the constraint equations by satisfying it exactly and to verify the equations of the eigenvalues equations in weak ways. To illustrate our contribution in this field we consider in this paper the case of Stokes operator. We describe several methods that produce the correct number of eigenvalues. We numerically prove how these methods are adequate to correctly solve the 2D Stokes eigenvalue problem. AMS subject classifications: 76D07, 65N35, 34L16","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48926580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we consider a reaction-diffusion model for the spreading of farmers in Europe, which was occupied by hunter-gatherers; this process is known as the Neolithic agricultural revolution. The spreading of farmers is modelled by a nonlinear porous medium type diffusion equation which coincides with the singular limit of another model for the dispersal of farmers as a small parameter tends to zero. From the ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We show the existence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity. AMS subject classifications: 35K57, 35Q92, 92D40
{"title":"Large Time Behaviour of the Solution of a Nonlinear Diffusion Problem in Anthropology","authors":"J. Eliaš","doi":"10.4208/JMS.V51N3.18.04","DOIUrl":"https://doi.org/10.4208/JMS.V51N3.18.04","url":null,"abstract":"In this article we consider a reaction-diffusion model for the spreading of farmers in Europe, which was occupied by hunter-gatherers; this process is known as the Neolithic agricultural revolution. The spreading of farmers is modelled by a nonlinear porous medium type diffusion equation which coincides with the singular limit of another model for the dispersal of farmers as a small parameter tends to zero. From the ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We show the existence and uniqueness of a global in time solution and study its asymptotic behaviour as time tends to infinity. AMS subject classifications: 35K57, 35Q92, 92D40","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48147231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide a new information about the observation of the initial data problem. It is actually exponentially ill-posed. AMS subject classifications: MASC 65N20, 65F22
{"title":"Ill-posedness of Inverse Diffusion Problems by Jacobi's Theta Transform","authors":"F. B. Belgacem","doi":"10.4208/JMS.V51N2.18.01","DOIUrl":"https://doi.org/10.4208/JMS.V51N2.18.01","url":null,"abstract":"The subject is the ill-posedness degree of some inverse problems for the transient heat conduction. We focus on three of them: the completion of missing boundary data, the identification of the trajectory of a pointwise source and the recovery of the initial state. In all of these problems, the observations provide over-specified boundary data, commonly called Cauchy boundary conditions. Notice that the third problem is central for the controllability by a boundary control of the temperature. Presumably, they are all severely ill-posed, a relevant indicator on their instabilities, as formalized by G. Wahba. We revisit these issues under a new light and with different mathematical tools to provide detailed and complete proofs for these results. Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out to be a powerful tool to realize our objectives. In particular, based on the Laptev work [Matematicheskie Zametki 16, 741-750 (1974)], we provide a new information about the observation of the initial data problem. It is actually exponentially ill-posed. AMS subject classifications: MASC 65N20, 65F22","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47749689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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