Related papers: Displacements
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
The problem of advancing coordinatization of mathematics is considered. The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation…
There is a substantial curricular overlap between calculus and physics, yet introductory physics students often struggle to connect the two. We introduce a quantity-based framing of the Fundamental Theorem of Calculus (FTC) to help unify…
Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…
A construction of integration, function calculus, and exterior calculus is made, allowing for integration of unital magma valued functions against (compactified) unital magma valued measures over arbitrary topological spaces. The Riemann…
This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce…
In this paper, starting from the microscopic dynamics of isolated dislocations, we explain how to derive formally mean field models for the dynamics of dislocation densities. Essentially these models are tranport equations, coupled with the…
The Reynolds Transport Theorem, colloquially known as 'differentiation under the integral sign', is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and…
The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work…
The basics of focused transport as applied to solar energetic particles are reviewed, paying special attention to areas of common misconception. The micro-physics of charged particles interacting with slab turbulence are investigated to…
Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i)…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…
Integrating with respect to functions which are constant on intervals whose bounds are discontinuity points (of those functions) is frequent in many branches of Mathematics, specially in stochastic processes. For such functions and alike…
This article aims to explain essential elements of perturbation theory and their conceptual underpinnings. It is not meant as a summary of popular perturbation methods, though some illustrative examples are given to underline the main…
We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise…
We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right…
In this work we state a Theorem on number theory and apply it to solve some ordinary and partial differential equations.
A new method for derivation of the equation of motion from the field equation is proposed. The problem of embedding the singularities in a field satisfying the field equations is discussed.