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This work continues the development of an intensional approach to computability initiated in previous work, in which programs and computations, rather than functions, constitute the primary objects of study. In this setting, models of…
The concept of input-independent computational time for chemistry-based analog computers was introduced in Anderson-Joshi (2025), where it was shown that arithmetic operations can be computed in a fixed time independent of the input values.…
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain a few new criteria equivalent to the Riemann hypothesis. Here, the same theorem is…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We…
A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets, and uniformly continuous maps is presented. In work of Berger it was shown how to extract certified algorithms working with the signed…
We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined…
We present a new way of organizing the few mathematical statements which form introduction to Calculus: the epsilon-delta characterization of the limit is now d e r i v e d from four simple, intuitive and frequently used statements, which…
Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of…
The decision time of an infinite time algorithm is the supremum of its halting times over all real inputs. The decision time of a set of reals is the least decision time of an algorithm that decides the set; semidecision times of…
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent…
In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, "most" preimages of almost all points are concentrated in a…
We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field.
The fractional order system, which is described by the fractional order derivative and integral, has been studied in many engineering areas. Recently, the concept of fractional order has been generalized to the distributed order concept,…
The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming…
We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…
The well-known Turing machine is an example of a theoretical digital computer, and it was the logical basis of constructing real electronic computers. In the present paper we propose an alternative, namely, by formalising arithmetic…
We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as co-inductive types. Four aspects are studied: the…