Related papers: Integrations on rings
This article exemplifies a novel approach to the teaching of introductory differential calculus using the modern notion of ``infinitesimal'' as opposed to the traditional approach using the notion of ``limit''. I illustrate the power of the…
The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…
Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for…
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
In this paper, we show that a partitioned formula \phi is dependent if and only if \phi has uniform definability of types over finite partial order indiscernibles. This generalizes our result from a previous paper [1]. We show this by…
The main purpose of this paper is to develop new algorithms for computing invariant rings in a general setting. This includes invariants of nonreductive groups but also of groups acting on algebras over certain rings. In particular, we…
In this paper we define a type of generalized Riemann-Lebesgue (decomposition) integral for non-negative real functions with respect to two non-additive set functions. For this integral we present some classical properties.
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
In this paper we define a family of continuous functions of an arbitrary number of variables, and prove that they all satisfy a generalization of one of the classical functional equations of the inverse tangent function.
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition,…
This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings.
We introduce more general concepts of Riemann-Liouville fractional integral and derivative on time scales, of a function with respect to another function. Sufficient conditions for existence and uniqueness of solution to an initial value…
Integration is the final key step when turning an infinitesimal argument into a result applicable to quantities of finite size. Conceptually, it is about combining infinitesimal contributions to a finite whole. We make a first step towards…
The Hirsch function of a given continuous function is a new function depending on a parameter. It exists provided some assumptions are satisfied. If this parameter takes the value one, we obtain the well-known h-index. We prove some…
We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure…
The present paper studies structure of the ring of integer-valued entire functions. We characterize certain classes of prime and maximal ideals and investigate some of their properties.
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…
It is well known that over an infinite field the ring of symmetric functions in a finite number of variables is isomorphic to the one of polynomial functions on matrices that are invariants by the action of conjugation by general linear…
The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized…