相关论文: Iterated Differential Forms III: Integral Calculus
We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms.
I adapt a recently introduced method for integrating over the unitary group (S. Aubert and C.S. Lam, J.Math.Phys. 44, 6112-6131 (2003)) to the orthogonal group. I derive explicit formulas for a number of one, two and three-vector integrals,…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Several notions of multiplicativity are introduced for forms of degree $d\geq 3$ over a field of characteristic 0 or greater than d. Examples of multiplicative and strongly multiplicative forms of higher degree are given. Conditions…
We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.
A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first order differential calculus on such a…
A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
A Compact Introduction to Fractional Calculus is presented including basic definitions, fractional differential equations and special functions.
We study the algebra $\mathcal{I}^{QM}$ of iterated integrals of quasimodular forms for $\operatorname{SL}_2(\mathbb{Z})$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration.…
We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in…
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…
In the present paper we extend the concepts of multiplicative de- rivative and integral to complex-valued functions of complex variable. Some drawbacks, arising with these concepts in the real case, are explained satis- factorily.…
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type…
In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over…
This talk reviews Feynman integrals, which are associated to elliptic curves. The talk will give an introduction into the mathematics behind them, covering the topics of elliptic curves, elliptic integrals, modular forms and the moduli…
We construct integral forms for the universal enveloping algebras of certain twisted multiloop algebras and explicit integral bases for these integral forms.
In this paper complexes of generalized Verma modules over the infinite-dimensional exceptional Lie superalgebras $E (3,8)$ and $E(5,10)$ are constructed and studied.
In this article we would like to consider some approaches to non-integer integro-differentiations and its implementation in computer algebra system Wolfram Mathematics.
Formal transformations somehow resembling the usual derivative are surprisingly common in computer science, with two notable examples being derivatives of regular expressions and derivatives of types. A newcomer to this list is the…