Related papers: A Generalization of Calculus for Use with Continuo…
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…
In this work we derive a functional equation in terms of the Hurwitz-Lerch zeta function along with definite integrals in terms of the incomplete gamma and Hurwitz-Lerch zeta functions. The method used in these derivations is contour…
We present a variant of the calculus of deductive systems developed in (Lambek 1972, 1974), and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of…
We present some lambda calculus with explicit substitutions and named variables. The characteristic feature of this calculus is as follows: renaming of bound variables when performing substitutions is done using special reductions and may…
We review the cumulant decomposition (a way of decomposing the expectation of a product of random variables (e.g. $\mathbb{E}[XYZ]$) into a sum of terms corresponding to partitions of these variables.) and the Wick decomposition (a way of…
A \emph{double extrema form} of the calculus of variations is put forward in which only the smallest one of the finite differences is physically meaningful to represent the variational derivatives defined on the discrete points. The most…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety…
We study an untyped lambda calculus with quantum data and classical control. This work stems from previous proposals by Selinger and Valiron and by Van Tonder. We focus on syntax and expressiveness, rather than (denotational) semantics. We…
We establish an integral representation for Popoviciu's convex functions of $d$ variables. This representation serves as a~foundation for deriving several functional inequalities, analogous to those well-known for usual convex functions.…
Objective of this paper is to introduce a new type of calculus which will be called G-Calculus based on non-Newtonian calculus introduced by Grossman and Katz \cite{GrossmanKatz}. The basic difference between geometric calculus defined by…
The smooth function reconstruction needs to use derivatives. In 2010, we used the gradually varied derivatives to successfully constructed smooth surfaces for real data. We also briefly explained why the gradually varied derivatives are…
Computing the rate-distortion function for continuous sources is commonly regarded as a standard continuous optimization problem. When numerically addressing this problem, a typical approach involves discretizing the source space and…
We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete,…
We introduce the structural resource lambda-calculus, a new formalism in which strongly normalizing terms of the lambda-calculus can naturally be represented, and at the same time any type derivation can be internally rewritten to its…
We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and integral on time scales are proved. Linear conformable differential equations with…
We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra ${\cal A}$. Such a derivation, introduced by Bresar in 1991, is given by a linear mapping $u: {\cal A} \rightarrow {\cal A}$…
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…
We prove dynamic inequalities of majorisation type for functions on time scales. The results are obtained using the notion of Riemann-Stieltjes delta integral and give a generalization of [App. Math. Let. 22 (2009), no. 3, 416--421] to time…
Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is…