Related papers: Quantum extended crystal PDE's
We generalize our geometric theory on extended crystal PDE's and their stability, to the category $\mathfrak{Q}_S$ of quantum supermanifolds. By using algebraic topologic techniques, obstructions to the existence of global quantum smooth…
In this paper we show that between PDE's and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE's can be considered extensions of crystallographic subgroups. In this respect we can…
Following the previous works on the A. Pr\'astaro's formulation of algebraic topology of quantum (super) PDE's, it is proved that a canonical Heyting algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE. This is…
In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times $t=r\in\mathbb{R}$ are often called PDE entire or eternal. For example, consider the quadratic parabolic PDE \begin{equation*} \label{*}…
This paper proposes a framework to assess the stability of an ordinary differential equation which is coupled to a 1D-partial differential equation (PDE). The stability theorem is based on a new result on Integral Quadratic Constraints…
We are looking for the universal covering algebra for all symmetries of a given pde, using the sine-Gordon equation as a typical example for a non-evolution equation. For non-evolution equations, Estabrook-Wahlquist prolongation structures…
Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A geometrical reformulation of the stochastic…
Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential…
We demonstrate that higher-order electric susceptibilities in crystals can be enhanced and understood through nontrivial topological invariants and quantum geometry, using one-dimensional $\pi$-conjugated chains as representative model…
The existence and multiplicity of solutions to a quasilinear, elliptic partial differential equation (PDE) with singular non-linearity is analyzed. The PDE is a recently derived variant of a canonical model used in the modeling of…
In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, {\em exotic $n$-d'Alembert PDE's} are considered. These are $n$-d'Alembert PDE's, $(d'A)_n$, admitting Cauchy manifolds $N\subset (d'A)_n$…
In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, are considered {\em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or…
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
A number of characteristics of integrable nonlinear partial differential equations (PDE's) for classical fields are reviewed, such as Backlund transformations, Lax pairs, and infinite sequences of conservation laws. An algebraic approach to…
After a short introduction on the theory of homogeneous algebras we describe the application of this theory to the analysis of the cubic Yang-Mills algebra, the quadratic self-duality algebras, their "super" versions as well as to some…
The new method of solving quantum mechanical problems is proposed. The finite, i.e. cut off, Hilbert space is algebraically implemented in the computer code with states represented by lists of variable length. Complete numerical solution of…
Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied…
The Yang-Baxter equation and it's various forms have applications in many fields, including statistical mechanics, knot theory, and quantum information. Unitary solutions of the braided Yang-Baxter equation are of particular interest as…
The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a…