Related papers: 4-Dimensional Power Geometry and its Application t…
We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on…
Apparently, all partial differential equations that describe physical phenomena in space-time can be cast into a universal quasilinear, first-order form. In this paper, we do two things. First, we describe some broad features of systems of…
This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the…
The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size may be expressed in terms of a solution of the fifth Painleve transcendent. The generating function of a certain discontinuous linear…
As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE's for a class of special interest associated with the adjoint representation of $\mathfrak{sl}(3)$.…
In this article, we will define the Orlicz space and the Orlicz-Sobolev space, and develop their topological properties. We will also examine their applications to partial differential equations (PDEs), with an emphasis on the use of…
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…
The paper is about a Painlev\'e III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects…
Choosing the appropriate geometry in which to express the equations of fundamental physics can have a determinant effect on the simplicity of those equations and on the way they are perceived. The point of departure in this paper is the…
For a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector. The dimension is given in terms of the Euler quadratic form associated with a partially ordered set,…
Analytical solutions of variable coefficient nonlinear Schr\"odinger equations having four-dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional…
The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…
A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.
Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
We consider first order symmetry operators for the equations of motion of differential $p$-form fields in general $D$-dimensional background geometry of any signature for both massless and massive cases. For $p=1$ and $p=2$ we give the…
We develop a procedure to implement the method of quadric ansatz to a class of second order partial differential equations (PDEs), which includes the four-dimensional K\"ahler-Einstein equation with symmetry and the one-sided type-D…
Determining cosmological field equations represents a still very debated matter and implies a wide discussion around different theoretical proposals. A suitable conceptual scheme could be represented by gravity models that naturally…
We show that the particle states of Maxwell's theory, in $D$ dimensions, can be represented in an infinite number of ways by using different gauge fields. Using this result we formulate the dynamics in terms of an infinite set of duality…
Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e…
We propose a general method to construct symmetric tensor polynomials in the D-dimensional Euclidean space which are orthonormal under a general weight. The D-dimensional Hermite polynomials are a particular case of the present ones for the…