Abstract The categorical models of differential linear logic (LL) are additive categories and those the lambda-calculus left-additive because Leibniz rule which requires summation two expressions. This means that, as far LL concerned, these feature finite nondeterminism indeed languages essentially non-deterministic. We introduce a framework for differentiation does not require additivity is co...

Milner’s π calculus features a clear dichotomy between replicable and non-replicable resources, very much in the spirit of Linear Logic (LL). Analyzing Milner’s encoding of the lazy λ-calculus in the π-calculus, Boudol introduced the λ-calculus with resources [1, 2] where functions can be applied to bags made of replicable and non-replicable arguments. This refinement of the syntax required to ...

We study the relationship between informatic and classical differentiation on the real line. The former arises when considering the interval domain over the reals equipped with the Lebesgue measurement. We show that informatic differentiation is a strict generalization of its classical counterpart, and wonder if it can provide a springboard towards extending techniques and results from calculus...

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...

We study the relationship between informatic and classical differentiation on the real line. The former arises when considering the interval domain over the reals equipped with the Lebesgue measurement. We show that informatic differentiation is a strict generalization of its classical counterpart, and wonder if it can provide a springboard towards extending techniques and results from calculus...

Calculus at the secondary school level has traditionally represented the peak of school mathematics in Australia, and has been available only to the most capable students. Until recently, many calculus curricula have focused on developing standard techniques, such as those concerned with differentiation and integration, with an emphasis on symbolic procedures for carrying these out in a range o...

We present early work on a PVS implementation of a model of simple control as signal flow graphs to enable formal verification of input/output behaviour of the control system. As has been shown by Rutten, Signal flow graphs can be described using Escardó’s coinductive stream calculus, which includes a definition of differentiation for streams over the real numbers and the use of differential eq...

1.2. The idea of quotient algebras was found, e.g. in [19], and the explicit descriptions, e.g. [3], based on Silva-differentiability on bornological vector spaces [2] produce a major impact in the field. The simplification [4] abandoned later in [5] indicates that a suitable framework of infinite dimensional differential calculus is definitely required. Our quick response with compact equicont...

Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki’s and Matiyasevich’s identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun.