There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

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Geodesic active contours

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

Viscosity solutions of fully nonlinear second-order elliptic partial differential equations

On considere l'existence des solutions d'equations aux derivees partielles non lineaires scalaires d'ordre 1: F(x, u, Du)=0 dans Ω, ou Ω est un sous-ensemble ouvert de R N , F:Ω×R×R N →R est continue, u:Ω→R est l'inconnue

Perron’s method for Hamilton-Jacobi equations

We prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second-order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.-L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions.

On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's

The solution m the Skorokhoci Problem defines a deieiminisiic mapping of paths that has been found to be useful in several areas of application. Typical uses of the mapping are construction and analysis of deterministic and stochastic processes that are constrained to remain in a given fixed set, such as stochastic differential equations with reflection and stochastic approximation schemes for problems with constraints In this paper we focus on the case where the set is a convex polyhedron and where the directions along which the constraint mechanism is applied arc possibly oblique and multivalued at corner points. Our goal is to characterize as completely as possible those situations in which the solution mapping is Lipschitz continuous. Our approach is geometric in nature, and shows that the Lipschitz continuity holds when a certain convex set, defined in terms of the normal directions to the faces of the polyhedron and the directions of the constraint mechanism, can be shown to exist. All previous inst...

Hitoshi Ishii

Papers

User’s guide to viscosity solutions of second order partial differential equations

Viscosity solutions of fully nonlinear second-order elliptic partial differential equations

Perron’s method for Hamilton-Jacobi equations

On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's

On lipschitz continuity of the solution mapping to the skorokhod problem, with applications