Related papers: Path Integral Solubility of a General Two-Dimensio…
In this article we study existence of pathwise stochastic integrals with respect to a general class of $n$-dimensional Gaussian processes and a wide class of adapted integrands. More precisely, we study integrands which are functions that…
The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and…
The classical approach to visualizing a flow, in terms of its streamlines, motivates a topological/soft-analytic argument for constrained variational equations. In its full generality, that argument provides an explicit formula for…
The effective transport properties of heterogeneous nanoscale materials and structures are affected by several geometrical and physical factors. Among them the presence of imperfect interfaces plays a central role being often at the origin…
A critical analysis of the conformal approach to the theory of 2d turbulence is delivered. It is shown, in particular, that conformal minimal models cannot give a general turbulent solution, which should provide for constant fluxes of all…
The irreducible alternative superbimodules are studied. The complete classification is obtained for even bimodules of arbitrary dimension and for finite-dimensional irreducible superbimodules over an algebraically closed field.
New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…
In this thesis, we investigate various integral submodels and generalize them. In part I, we study the submodel of the nonlinear $\mathbf{C}P^1$-model and the related submodels in $(1+2)$ dimensions. In part II, we construct integrable…
In this contribution I summarize the achievements of separation of variables in integrable quantum systems from the point of view of path integrals. This includes the free motion on homogeneous spaces, and motion subject to a potential…
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
We construct a quantum theory of free scalar field in 1+1 dimensions based on a `Generalized Uncertainty Principle'. Both canonical and path integral formalism are employed. Higher dimensional extension is easily performed in the path…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
The paper presents shortly the geometric approach to the problem of a general quantization formalism, both physically meaningful and mathematically consistent.
Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimen\-sional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the…
These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum…
Continuum and discrete approaches to 2d gravity coupled to $c<1$ matter are reviewed.
We demonstrate that the necessary condition for $SO(N) \times SO(N)$ duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation,…
The dual-frame formalism leads to an approach to extend numerical relativity simulations in generalized harmonic gauge (GHG) all the way to null infinity. A major setback is that without care, even simple choices of initial data give rise…
In these talks, I discuss a few selected topics in integrable models that are of interest from various points of view. Some open questions are also described.