Related papers: Series misdemeanors
From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries…
Algebraic power series are formal power series which satisfy a univariate polynomial equation over the polynomial ring in n variables. This relation determines the series only up to conjugacy. Via the Artin-Mazur theorem and the implicit…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
The problem is to evaluate a polynomial in several variables and its gradient at a power series truncated to some finite degree with multiple double precision arithmetic. To compensate for the cost overhead of multiple double precision and…
This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of…
In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface.…
Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or…
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
Probabilistic circuits (PCs) offer a promising avenue to perform embedded reasoning under uncertainty. They support efficient and exact computation of various probabilistic inference tasks by design. Hence, hardware-efficient computation of…
The divisibility of truncated binomial series by their exponent n is analyzed. Divisibility is shown to depends on the divisibility characteristics of the integers constituting the binomials. Series division by the highest possible powers…
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
Reversible computing models settings in which all processes can be reversed. Applications include low-power computing, quantum computing, and robotics. It is unclear how to represent side-effects in this setting, because conventional…
We transformed the generalized exponential power series to another functional form suitable for further analysis. By applying the Cauchy-Euler differential operator in the form of an exponential operator, the series became a sum of…
In this article, we provide partition-theoretic interpretations for some new truncated pentagonal number theorem and identities of Gauss. Also, we deduce few inequalities for some partition functions.
We provide algorithms computing power series solutions of a large class of differential or $q$-differential equations or systems. Their number of arithmetic operations grows linearly with the precision, up to logarithmic terms.
Power series are introduced that are simultaneously convergent for all real and p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and…
In this paper we study sequences, series, power series and uniform convergence in the $\mathcal{A}$-Calculus. Here $\mathcal{A}$ denotes an associative unital real algebra. We say a function is $\mathcal{A}$-differentiable if it is real…
Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice that…
Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $\tau\geq 2$ and any non-empty proper…
Pseudoinverses are ubiquitous tools for handling over- and under-determined systems of equations. For computational efficiency, sparse pseudoinverses are desirable. Recently, sparse left and right pseudoinverses were introduced, using…