Related papers: New modular multiplication and division algorithms…
We study two important operations on polynomials defined over complete discrete valuation fields: Euclidean division and factorization. In particular, we design a simple and efficient algorithm for computing slope factorizations, based on…
We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this…
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among…
Motivated by cryptographic applications, we investigate two machine learning approaches to modular multiplication: namely circular regression and a sequence-to-sequence transformer model. The limited success of both methods demonstrated in…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings and…
By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such…
Gosper developed an algorithm for performing arithmetic operations on continued fractions (CFs), getting a CF as the result. Straightforward implementation of the algorithm leads to infinite loops on some inputs. Here we present a modified…
This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this…
In this paper we present a method to derive Eulerian continued fractions arising from a sequence of integrals. As examples, through a new derivation, we reproduce classical continued fraction expansions for the natural logarithm, the…
This paper describes several new improvements of modular arithmetic and how to exploit them in order to gain more efficient implementations of commonly used algorithms, especially in cryptographic applications. We further present a new…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
A formula for calculating Extensions of (mainly integral) Polynomial Functors is established, based upon projective resolutions. Sample computations are performed, which, in particular, exhibit a surprising non-trivial extension of Divided…
In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions. Ultradistributions permit us to unify the notion of integral and derivative in one only…
We formulate a thermodynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi-Vall\'ee. We introduce the modular partitions of continued fractions and observe that the statistics for modular…
The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve,…