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Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some…

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.

We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…

We propose an explicit and practical algorithm for computing Galois conjugates and irreducible polynomials for special values of modular functions evaluated at CM points associated with imaginary quadratic orders. Our approach builds upon…

This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising…

One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to…

This thesis determines some of the implications of non-universal and emergent universal statistics on arithmetic correlations and fluctuations of arithmetic functions, in particular correlations amongst prime numbers and the variance of the…

We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral…

In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional…

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

This is a research announcement on an alternative definition of the Casson invariants by means of virtual counting of the moduli space of irreducible representations of the fundamental group into $\SU(2)$. Along the way, by using derived…

This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to "count" an infinite number of points. We begin with the classical case of…

Functional methods can be applied to the quantum effective action to efficiently determine counterterms and matching conditions for effective field theories. We extend the toolbox to two-loop order and beyond and show how to evaluate the…

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…

This paper describes some validated numerics aspects of Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions and Hasse-Weil L-functions.

An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…

We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…