We study the bosonic super Liouville system which is a statistical transmuta-tion of super Liouville system. Lax pair for the bosonic super Liouville system is constructed using prolongation method, ensuring the Lax integrability, and the solution to the equations of motion is also considered via Leznov-Saveliev analysis.

We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection we generalize several results for Sturm–Liouville equations on time scales which have been obtained by various authors in the past.

The quantum analogs of the derivatives with respect to coordinates qk and momenta pk are commutators with operators Pk and Qk. We consider quantum analogs of fractional Riemann-Liouville and Liouville derivatives. To obtain the quantum analogs of fractional Riemann-Liouville derivatives, which are defined on a finite interval of the real axis, we use a representation of these derivatives for an...

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http: //www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an ...

We give an algorithm that computes an absolutely normal Liouville number. 1. The main result The set of Liouville numbers is {x ∈ R \Q : ∀k ∈ N, ∃q ∈ N, q > 1 and ||qx|| < q−k} where ||x|| = min{|x−m| : m ∈ Z} is the distance of a real number x to the nearest integer and other notation is as usual. Liouville’s constant, ∑ k≥1 10 −k!, is the standard example of a Liouville number. Though uncount...

In [LWZ], we established Liouville-type theorems and decay estimates for solutions of a class of high order elliptic equations and systems without the boundedness assumptions on the solutions. In this paper, we continue our work in [LWZ] to investigate the role of boundedness assumption in proving Liouville-type theorems for fully nonlinear equations. We remove the boundedness assumption of sol...

A general program to show quantum-classical correspondence for bound conservative integrable and chaotic systems is described. The method is applied to integrable systems and the nature of the approach to the classical limit, the cancellation of essential singularities, is demonstrated. The application to chaotic systems requires an understanding of classical Liouville eigenfunctions and a Liou...

The Liouville-Green [or Wentzel-Kramers-Brillouin (WKB)] approximation for the two-dimensional cochlear mechanics problem disagrees with the finite-difference solution in the region after the response peak. This disagreement has left doubts about the validity of the Liouville-Green approximation, and has never been satisfactorily explained. In this paper, it is shown that the Liouville-Green ap...

We present our recent studies on the dynamics of boundary N = 2 Liouville theory. We use the representation theory of N = 2 superconformal algebra and the method of modular bootstrap to derive three classes of boundary states of the N = 2 Liouville theory. Class 1 and 2 branes are analogues of ZZ and FZZT branes of N = 0, 1 Liouville theory while class 3 branes come from U(1) degrees of freedom...

We review recent developments (up to January 2004) of the Liouville field theory and its matrix model dual. This review consists of three parts. In the part I, we review the bosonic Liouville theory. After briefly reviewing the necessary background, we discuss the bulk structure constants (the DOZZ formula) and the boundary states (the FZZT brane and the ZZ brane). Various applications are also...