Related papers: A theory of regularity structures
We formalise the self-referential definition of physical laws using monotone operators on a lattice of theories, resolving the pathologies of naive set-theoretic formulations. By invoking Tarski fixed point theorem, we identify physical…
For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic…
We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model…
We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach was studied by S. Mandelbrojt and becomes useful and necessary when working with…
These lecture notes for a graduate class present the regularization theory for linear and nonlinear ill-posed operator equations in Hilbert spaces. Covered are the general framework of regularization methods and their analysis via spectral…
We compute structure functions in the Hamiltonian formalism on a momentum lattice using a physically motivated regularisation that links the maximal parton number to the lattice size. We show for the $\phi ^4 _{3+1}$ theory that our method…
This paper contains an answer to the question of existence of regularities of the so called \textit{random in a broad sense} mass phenomena, asked by A. N. Kolmogorov in \cite{Kolmogorov}. It turns out that some family of finitely-additive…
Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems…
We study how maximal regularity estimates with respect to the continuous functions improve automatically in cases where the spatial norm is fundamentally different from the supremum norm. More precisely, we invoke properties such as weak…
Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable f (t) as a function of the time variable in some interval 0 < t < T, we investigate how to…
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical…
We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid…
This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing,…
We consider regularity for solutions of a class of de Rham's functional equations. Under some smoothness conditions of functions consisting the equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our results are…
We introduce an asymmetric operator of generalised translation, define the generalised modulus of smoothness by its means, and obtain the direct and inverse theorems in approximation theory for it.
We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(\tau, v^{\eta})|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the…
We introduce a class of rings using which we define the concept of skew regularity for quaternion-valued functions over quaternions. It is shown that the notion of skew regularity coincides with the concept of slice regularity over…
A new formulation of quantum mechanics is proposed based on a new principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a…
The relation between renormalization and short distance singular divergencies in quantum field theory is studied. As a consequence a finite theory is presented. It is shown that these divergencies are originated by the multiplication of…
The forced soliton equation is the starting point for semiclassical computations with solitons away from the small momentum transfer regime. This paper develops necessary analytical and numerical tools for analyzing solutions to the forced…