arXiv subjects

Dominic C. Milioto

Publications and source records attributed to Dominic C. Milioto.

7 recordsLinked to original sources

arXiv2023

A numeric study of power expansions around singular points of algebraic functions, their radii of convergence, and accuracy profiles

An efficient method of computing power expansions of algebraic functions is the method of Kung and Traub and is based on exact arithmetic. This paper shows a numeric approach is both feasible and accurate while also introducing a performance improvement to Kung and Traub's method based on the ramification extent of the expansions. A new method is then described for computing radii of convergence using a series comparison test. Series accuracies are then fitted to a simple log-linear function in their domain of convergence and found to have low variance. Algebraic functions up to degree 50 were analyzed and timed. A consequence of this work provided a simple method of computing the Riemann surface genus and was used as a cycle check-sum. Mathematica ver. 13.2 was used to acquire and analyze the data on a 4.0 GHz quad-core desktop computer.

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arXiv2022

Determining radii of convergence of fractional power expansions around singular points of algebraic functions

The purpose of this paper is to introduce the branching geometry of algebraic functions around singular points and to describe a simple method of determining radii of convergence of their power expansions in terms of those singular points. Branching geometries are categorized into six types. Then a method is presented to determine radii of convergence of branch power expansions using analytic continuation and the identification of convergence-limiting singular points. Test cases exhibiting a variety of branching morphologies are analyzed, and convergence results obtained through analytic continuation are checked against Root Tests of the associated power series. All Root Tests agreed well with the results obtained by analytic continuation. Mathematica ver. 12.3 was used to implement the numeric algorithms.

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arXiv2020

A systematic approach to computing and indexing the fixed points of an iterated exponential

This paper describes a systematic method of numerically computing and indexing fixed points of $z^{z^w}$ for fixed $z$ or equivalently, the roots of $T_2(w;z)=w-z^{z^w}$. The roots are computed using a modified version of fixed-point iteration and indexed by integer triplets $\{n,m,p\}$ which associate a root to a unique branch of $T_2$. This naming convention is proposed sufficient to enumerate all roots of the function with $(n,m)$ enumerated by $\mathbb{Z}^2$. However, branches near the origin can have multiple roots. These cases are identified by the third parameter $p$. This work was done with rational or symbolic values of $z$ enabling arbitrary precision arithmetic. A selection of roots up to order $\{10^{12},10^{12},p\}$ with $|z|\leq 10^{12}$ was used as test cases. Results were accurate to the precision used in the computations, generally between $30$ and $100$ digits. Mathematica ver. $12$ was used to implement the algorithms.

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arXiv2019

On the Branching Geometry of Algebraic Functions

This paper describes an algorithm for determining the branching geometry of algebraic functions. The graphs of these complex-valued functions have a complicated interweaving structure that can be described by analytic branches separated by singular points. Power expansions for the branches in discs centered at a point can be computed using the Newton Polygon method, and expansions around annular regions centered at the origin computed using a version of Laurent's Theorem applied to algebraic functions. However, neither of the methods enable a determination of the region of convergence of the power series. In this paper, a method using analytic continuation is used to determine the domain of analyticity for the branches, and the Root Test used to numerically check the results.

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arXiv2013

An algorithm for determining the radii of convergence of algebraic power series

This paper describes an algorithm for determining radii of convergence of power expansions for algebraic functions and the testing done to check it. Since the current methods for computing these series are iterative, standard methods for computing radii of convergence cannot in general, be used. However, relying on geometric properties of algebraic functions, convergence radii of these series can be determined precisely.

math.AG↗

arXiv2010

A method for locating where the real part of the Riemann zeta function becomes negative for its real argument greater than one

This paper describes a search algorithm to locate values of t where the real part of the Riemann zeta function, zeta(sigma+it), is negative for sigma>1. The run-time to execute the search is much less than a brute-force approach and relies on certain symmetries of congruence equations related to the zeta function.

math.CA↗

arXiv2009

The Analytic Expression for Riemann's Prime Counting Function via the Residue Theorem

In his paper "On the Number of Primes Less Than a Given Magnitude", Bernhard Riemann introduced a prime counting function F(x) which counts the number of primes under x. Riemann obtained an analytic expression for F(x) by evaluating an inverse Laplace Transform. His method involved advanced techniques of analysis. However, this transform can be evaluated using the Residue Theorem when an appropriate branch of log(zeta) is defined. In this paper, a method for constructing a holomorphic branch of log(zeta) extending to the left half-plane is described along with it's geometry surrounding the logarithmic branch points. Using this information, an integral representation of F(x) is formulated in terms of this branch of log(zeta) which is then evaluated. The results are shown equal to Riemann's expression.

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