Theory of extreme correlations using canonical Fermions and path integrals

The  t$t$–J$J$  model is studied using a novel and rigorous mapping of the Gutzwiller projected electrons, in terms of canonical electrons. The mapping has considerable similarity to the Dyson–Maleev transformation relating spin operators to canonical Bosons. This representation gives rise to a non Hermitian quantum theory, characterized by minimal redundancies. A path integral representation of the canonical theory is given. Using it, the salient results of the extremely correlated Fermi liquid (ECFL) theory, including the previously found Schwinger equations of motion, are easily rederived. Further, a transparent physical interpretation of the previously introduced auxiliary Greens function and the ‘caparison factor’, is obtained.     

Cardano's waves

The classical formula of Cardano for a cubic equation is deduced as a solution of an initial-value problem for a differential equation. A connection is noted with the wave equation.     

A variational formulation approach to a generalized coupled inhomogeneous Emden–Fowler system

We study a generalized coupled inhomogeneous Emden–Fowler system from the Lagrangian formulation standpoint. A special case of this system was considered in the literature, and necessary and sufficient conditions for the existence of multiple positive solutions were obtained. Here we perform preliminary Noether classification of the generalized system by the direct method. We obtain nine cases for which the system has Noether point symmetries. First integrals are then obtained for the cases which admit Noether point symmetries.     

Some Recent Developments in the Theory and Applications of a Class of Finite Fourier Transformations

In this talk we shall investigate various properties of a class of finite Fourier transformations (that is, Fourier transformations on finite intervals), not as Fourier coefficients, but as functions of a continuous variable. Some of these potentially useful properties of finite Fourier exponential, finite Fourier sine, and finite Fourier cosine transformations will then be applied to several families of special functions including (for example) Bessel functions, parabolic cylinder functions, and Chebyshev and Legendre (or spherical) polynomials.     

Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation

We consider a periodic pseudodifferential operator H = (? Δ) ^l  + A (l > 0) in R ^d which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l ? 1. Under these assumptions, we prove that the spectrum of H contains a half-line.     

Formation of Singularities and Self-Similar Vortex Motion Under the Localized Induction Approximation†

Abstract

The dynamical behavior of an isolated vortex filament in three dimensions within the localized induction approximation (LIA) is investigated. It is shown the existence of a uniparametric family of smooth solutions of LIA that generates a corner in finite time. An explicit formula of the solution is provided in terms of the curvature of the initial regular configuration. The behavior of the uniparametric family of solutions with respect to the free parameter is also considered.

     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.