Related papers: Extended Crystal PDE's
When mathematical/computational problems reach infinity, extending analysis and/or numerical computation beyond it becomes a notorious challenge. We suggest that, upon suitable singular transformations (that can in principle be…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular…
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{*}…
We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a destriction functor and apply it to some well-known biset functors. The obstruction groups for this theory are reduced cohomology…
In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite…
Crystallographic tilings of the Euclidean space $\mathbb{E}^n$ are defined as simple tilings whose group of isometric automorphisms is crystallographic. To classify crystallographic tilings by their automorphism groups it is necessary to…
Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups.…
While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial…
The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a…
Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The…
Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its…
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…
Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness for different finite element schemes on a simplified version of the…
We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of the…
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. There are two different approaches for the construction of stochastic solutions:…
In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and…
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a p-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal…
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the…
We investigate a PDE-constrained optimization problem, with an intuitive interpretation in terms of the design of robust membranes made out of an arbitrary number of different materials. We prove existence and uniqueness of solutions for…