The rôle of categorical structures in infinitesimal calculus

Abstract

The development of mathematics stands as one of the most important achievements of humanity, and the development of the calculus, differential calculus and integral calculus is one of the most important achievements in mathematics. Differential calculus is about finding the slope of a tangent to the graph of a function, or equivalently, differential calculus is about finding the rate of change of one quantity with respect to another quantity. On the other hand, integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The category theory is a mathematical approach to the study of algebraic structure that has become an important tool in theoretical computing science, particularly for semantics-based research. The notion of a limit in category theory generalizes various types of universal constructions that occur in diverse areas of mathematics. In our paper we illustrate how to represent some parts of infinitesimal calculus in categorical structures.

Matrix methods in evaluation of integrals

“…n in Apéry's proof of the irrationality of ζ (3) (see [15]) stimulated something of a vogue for the infinite series involving the reciprocals of central binomial coefficients starting in the late 1970s (see [16,17]), and some of this spilled over into works on the finite sums. In this connection, it is helpful to look at Sprugnoli [3], who gives the additional references to older papers (though he missed Parker [1]: the above formula (20), that appears at the end of Parker's note, is the Sprugnoli's Theorem 4.1 -which is also the Gould's equation (2.9)).…”

“…Moreover, it should be emphasized that the calculation method is used in this paper in terms of basic consideration. The theoretical foundations indicates the remarkable similarity, or even the full compliance, with the consideration carried out in paper [20]. It refers to the relations between the categorical structures and infinitesimal calculus.…”

“…During the review process of this paper, we discovered, by using Parker's formula (20), the following convolution type identity (see [18,19]), which holds for each n ∈ N:…”

An image encryption algorithm with a plaintext‐related quantisation scheme

“…Index denotes the previously scanned row, the first row () uses the bottom row () as previously scanned. The symbol is a substitution for all image pixels or sequence elements in a row and operators and denote operations of binary AND and binary XOR (eXclusive OR) [27]. The constant of 255 in second equation is used for computing negation of .…”

“…The symbol ∶ is a substitution for all image pixels or sequence elements in a row and operators ∧ and ⊕ denote operations of binary AND and binary XOR (eXclusive OR) [27]. The constant of 255 in second equation is used for computing negation of P ′ (l − 1, ∶).…”