Related papers: Difference Problems and Differential Problems
The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to…
We describe explicit algorithms for factoring q-difference operators and solving q-difference equations. These are well known results, presented in a "concrete" form. ----- Nous decrivons des algorithmes explicites pour la factorisation…
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
We create a sequence version of calculus. First, we define equivalence, some fundamental operations, differential, and integral for sequences. Then, we propose sequence versions of identity function, power function, exponential function,…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.
We give a survey of the foundations of statistical queries and their many applications to other areas. We introduce the model, give the main definitions, and we explore the fundamental theory statistical queries and how how it connects to…
This is a kind of introduction to some basic topics in analysis, some of which would be covered in standard graduate courses, and some not. However, an important difference is that not much in the way of prerequisites are needed, beyond…
In this article, we explore a series of elementary yet insightful results involving integrals related to Gaussian sums. Using techniques rooted in classical calculus, we derive several identities and evaluate nontrivial definite integrals…
We sharpen a recent observation by Tim Maudlin: differential calculus is a natural language for physics only if additional structure, like the definition of a Hodge dual or a metric, is given; but the discrete version of this calculus…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
The Locker Problem is frequently used in introducing some topics in elementary number theory like divisors and multiples. It appears in many curricula ranging from elementary, secondary and up to tertiary level. In this paper, I will…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.
This paper establishes calculus upon two physical facts: (1) any average velocity is always between two instantaneous velocities, and (2) the motion of an object is determined once its velocity has been determined. It directly defines…
We present in this article a general approach (in the form of recommendations and guidelines) for tackling Diophantine equation problems (whether single equations or systems of simultaneous equations). The article should be useful in…
First year calculus is often taught in a way that is very burdensome to the student. Students have to memorize a diversity of processes for essentially performing the same task. However, many calculus processes can be simplified and…
This paper shows how partial differential problems can be solved thanks to cellular computing and an adaptation of the Least Squares Finite Elements Method. As cellular computing can be implemented on distributed parallel architectures,…
The main objective of this paper is to study the existence of solutions to some basic fractional difference equations. The tools employed are Krasnosel'skii fixed point theorem which guarantee at least two positive solutions.
The ability to categorize problems based upon underlying principles, rather than contexts, is considered a hallmark of expertise in physics problem solving. With inspiration from a classic study by Chi, Feltovich, and Glaser, we compared…