Automatic differentiation—the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately—dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learning. Despite this, practitioners in a variety of fields, including machine learnin...

although there are many excellent ways presenting the principle of the classical calculus, the novel presentations probably leads most naturally to the development of the non-newtonian calculus. the important point to note is that the non-newtonian calculus is a self-contained system independent of any other system of calculus. since this self-contained work is intended for a wide audience, inc...

Abstract Differential linear logic was introduced as a syntactic proof-theoretic approach to the analysis of differential calculus. Differential categories were subsequently introduce to provide a categorical model theory for differential linear logic. Differential categories used two different approaches for defining differentiation abstractly: a deriving transformation and a coderiliction. Wh...

It is shown how the techniques of automatic differentiation can be viewed in a broader context as an application of analysis on a nonarchimedean field. The rings used in automatic differentiation can be ordered in a natural way and form finite dimensional real algebras which contain infinitesimals. Some of these algebras can be extended to become a Cauchy-complete real-closed nonarchimedean fie...

Differential calculus is ubiquitous in digital movie production. We give a novel presentation of automatic differentiation, a method for computing derivatives of functions, that is not well known within the graphics community and describe some applications of this method. In particular we describe the implementation of a photogrammetric reconstruction tool used on the post-production of Matrix ...

These are lecture notes for the class Stochastic Calculus offered at the Courant Institute in the Fall Semester of 2012. It is a graduate level class. Students should have a solid background in probability and linear algebra. The topic selection is guided in part by the needs of our MS program in Mathematics in Finance. But it is not focused entirely on the Black Scholes theory of derivative pr...

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of noninteger orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for futu...

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account general form non-locality in kernels differential and integral operators. Self-consistency involves proving generalizations all fundamental theorems for generalized In the FVC from power-law nonlocality space, we use (GFC) Luchko approach, which was published 2021. This...

Automatic differentiation is a semantic transformation that applies the rules of differential calculus to source code. It thus transforms a computer program that computes a mathematical function into a program that computes the function and its derivatives. Derivatives play an important role in a wide variety of scientific computing applications, including numerical optimization, solution of no...