Related papers: The Hasse Principle for Geometric Variational Prob…
In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimisation, inference and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric…
Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all…
We investigate the Hasse principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Over purely transcendental field extensions of fields that…
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations containing forms of differing degree but also multiple forms…
We develop a language for describing the relationship among observations, mathematical models, and the underlying principles from which they are derived. Using Information Geometry, we consider geometric properties of statistical models for…
The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Recently, a variational principle has been derived from Einstein-Hilbert and a matter Lagrangian for the spherically symmetric system of a dust shell and a black hole. The so-called physical region of the phase space, which contains all…
We combine the parameterization method for invariant manifolds with the finite element method for elliptic PDEs,to obtain a new computational framework for high order approximation of invariant manifolds attached to unstable equilibrium…
After establishing the uniqueness of the continuation of local Cauchy data for harmonic maps between two Riemannian manifolds M and N, we prove (i) a reflection principle for a smooth minimal submanifold Y of a Riemannian manifold M that…
We study local-global principles for semi-integral points on orbifold pairs of Markoff type. In particular, we analyse when these orbifold pairs satisfy weak weak approximation, weak approximation and strong approximation off a finite set…
Numerical homogenization, i.e. the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a…
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…
We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
This article makes no claim to originality, other than, perhaps, the simple statement here called the {\it Abstract Maximum Principle}. Actually, the whole contents are strongly based on some H. Sussmann's and coauthors' papers, in which,…
Huygens principle (HP) is the cornerstone of wave optics, its mathematical model is a boundary value problem of wave equation. The solutions of this mathematical model should be partial derivative u sub n independent and satisfy the form of…
We present an algorithm for solving binary classification problems when the dataset is not fully representative of the problem being solved, and obtaining more data is not possible. It relies on a trained model with loose accuracy…
Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article…
According to the holographic principle, the information content assigned to a gravitational region is processed by its lower dimensional boundary. As an example setup compatible with this principle, the AdS/CFT correspondence relies on the…