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Register a new userIntegral representations of isotropic semi-classical functions and applications
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In cite{GUW} we introduced a class of semi-classical functions of isotropic type, starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the oscillatory functions of Lagrangian type that have played major role in semi-classical and micro-local analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual clean-intersection hypothesis). The simplest examples of isotropic states are the coherent states, a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrodinger in the 1920s. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary ref{cor:altWeyl}).
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General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the classical special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on authors habilitation thesis [Spi7] containing a more detailed account of the subject.
We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.
We show how many classes of partial differential systems with local and nonlocal nonlinearities are linearisable. By this we mean that solutions can be generated by solving a corresponding linear partial differential system together with a linear Fredholm integral equation. The flows of such nonlinear systems are examples of linear flows on Fredholm Stiefel manifolds that can be projected onto Fredholm Grassmann manifolds or of further projections onto natural subspaces thereof. Detailed expositions of such flows are provided for the Korteweg de Vries and nonlinear Schrodinger equations, as well as Smoluchowskis coagulation and related equations in the constant frequency kernel case. We then consider Smoluchowskis equation in the additive and multiplicative frequency kernel cases which correspond to the inviscid Burgers equation. The solution flow in these cases prompts us to introduce a new class of flows we call ``graph flows. These generalise flows on a Grassmann manifold from sets of graphs of linear maps to sets of graphs of nonlinear maps. We include a detailed discussion of directions in which these flows can be generalised to include many other partial differential systems with local and nonlocal nonlinearities.
In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.
We present few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.
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