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We present a method of determining a Lax representation for similarity reductions of autonomous and non-autonomous partial difference equations. This method may be used to obtain Lax representations that are general enough to provide the…
We prove that the boundary distance map of a smooth compact Finsler manifold with smooth boundary determines its topological and differentiable structures. We construct the optimal fiberwise open subset of its tangent bundle and show that…
The Painleve test is very useful to construct not only the Laurent series solutions of systems of nonlinear ordinary differential equations but also the elliptic and trigonometric ones. The standard methods for constructing the elliptic…
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated…
We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of…
In this paper, an eigenvalue mapping-based discretization method is applied to discretize the generalized super-twisting algorithm. The existing eigenvalue mapping is extended to the complex domain which greatly enlarges the range of…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
Given a fibration of simply connected CW complexes of finite type, we study the evaluation subgroup of the fibre inclusion as an invariant of fibre-homotopy type. For spherical fibrations, we show the evaluation subgroup may be expressed as…
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…
The leaves of the Painlev{\'e} foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections. Therefore they are the fibers of the Riemann-Hilbert correspondence that sends each…
We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich…
We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more specifically, of graph classes with bounded tree-independence number. In [Dallard, Milani\v{c}, and…
In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or…
In this paper, we study the use of outer metrics, in particular Sobolev-type metrics on the diffeomorphism group in the context of PDE-constrained shape optimization. Leveraging the structure of the diffeomorphism group we analyze the…
We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models, in which the Hamiltonian is gradually simplified along a parallel…
Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
We construct a rational homotopy pullback decomposition for variants of the classifying space of the group of homeomorphisms for a large class of manifolds. This has various applications, including a rational section of the stabilisation…
This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of…