On Space Time Quasiconcave Solutions Of The Heat Equation
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| Author | : Chuanqiang Chen |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 94 |
| Release | : 2019-06-10 |
| Genre | : Mathematics |
| ISBN | : 1470435241 |
In this paper the authors first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, they obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain their ideas and for completeness, the authors also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.
| Author | : Chuanqiang Chen |
| Publisher | : |
| Total Pages | : |
| Release | : 2019 |
| Genre | : |
| ISBN | : 9781470452438 |
| Author | : Massimiliano Berti |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 184 |
| Release | : 2020-04-03 |
| Genre | : Education |
| ISBN | : 1470440695 |
The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.
| Author | : Luigi Ambrosio |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 134 |
| Release | : 2020-02-13 |
| Genre | : Education |
| ISBN | : 1470439131 |
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.
| Author | : Tvrtko Tadić |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 184 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 147043685X |
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure— so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
| Author | : Oskari Ajanki |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 146 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 1470436833 |
The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z∈H, including the vicinity of the singularities.
| Author | : Charles Collot |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 110 |
| Release | : 2019-09-05 |
| Genre | : Mathematics |
| ISBN | : 1470436264 |
The authors consider the energy super critical semilinear heat equation The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
| Author | : Chen Wan |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 102 |
| Release | : 2019-12-02 |
| Genre | : Education |
| ISBN | : 1470436868 |
Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.
| Author | : Carles Broto |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 176 |
| Release | : 2020-02-13 |
| Genre | : Education |
| ISBN | : 1470437724 |
For a finite group G of Lie type and a prime p, the authors compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of BG∧p in terms of Out(G).
| Author | : Mircea Mustaţă |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 92 |
| Release | : 2020-02-13 |
| Genre | : Education |
| ISBN | : 1470437813 |
The authors use methods from birational geometry to study the Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. They analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.
