arXiv subjects

Steven Finch

Publications and source records attributed to Steven Finch.

At least 55 records · Page 3Linked to original sources

arXiv2020

Variance of Longest Run Duration in a Random Bitstring

We continue an earlier study, starting with unconstrained $n$-bitstrings, focusing now less on average behavior and more on uncertainty. The interplay between $\bullet$ longest runs of 0s and of 1s, when bitstrings are multus $\bullet$ longest runs of 0s and bitsums (# of 1s), when bitstrings are solus $\\$ is examined. While negative correlations approach zero as $n \rightarrow \infty$ in the former (for clumped 1s), the limit is evidently nonzero in the latter (for separated 1s). Similar analysis is possible when both 0s and 1s are clumped (bimultus), and when 0s are clumped but 1s are separated (persolus). Our methods are experimentally-based.

math.CO↗

arXiv2020

Cantor-solus and Cantor-multus Distributions

The Cantor distribution is obtained from bitstrings; the Cantor-solus distribution (a new name) admits only strings without adjacent 1 bits. We review moments and order statistics associated with these. The Cantor-multus distribution is introduced -- which instead admits only strings without isolated 1 bits -- and more complicated formulas emerge.

math.CO↗

arXiv2020

Recursive PGFs for BSTs and DSTs

We review fundamentals underlying binary search trees and digital search trees, with (atypical) emphasis on recursive formulas for associated probability generating functions. Other topics include higher moments of BST search costs and combinatorics for a certain finite-key analog of DSTs.

cs.DS↗

arXiv2019

Resolving Conflicts and Electing Leaders

We review distributed algorithms for transmitting data ($n$ real numbers) under a broadcast communication model, as well as for maximum finding and for sorting. Our interest is in the basics of recursive formulas and corresponding asymptotics as ${n\to\infty}$. The emphasis is on concrete examples rather than general theory.

cs.DC↗

arXiv2019

A translation of Zalgaller's "The shortest space curve of unit width" (1994)

This is an English translation of V. A. Zalgaller's article "On a problem of the shortest space curve of unit width" that appeared in $Matematicheskaya\ Fizika,\ Analiz,\ Geometriya$ v. 1 (1994) n. 3--4, 454--461. We refer interested readers to Ghomi (2018) for up-to-date discussion; the curve $L_{3}$ of length 3.9215... in Zalgaller (1994) still appears to be shortest, whereas the closed curve $L_{5}$ is provably not of unit width. I am thankful to Natalya Pluzhnikov for her dedicated work and to the B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine for permission to post this translation on the arXiv.

math.HO↗

arXiv2019

A translation of Henri Joris' "Le chasseur perdu dans la forêt" (1980)

This is an English translation of Henri Joris' article "Le chasseur perdu dans la forêt (Un problème de géométrie plane)" that appeared in $Elemente\ der\ Mathematik$ v. 35 (1980) n. 1, 1--14. Given a point $P$ and a line $L$ in the plane, what is the shortest search path to find $L$, given its distance but not its direction from $P$? The shortest search path was described by Isbell (1957), but a complete and detailed proof was not published until Joris (1980). I am thankful to Natalya Pluzhnikov for her dedicated work and to the Swiss Mathematical Society for permission to post this translation on the arXiv.

math.HO↗

arXiv2019

Number of Sign Changes: Segment of AR(1)

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider $n$ contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{n}$). Let its mean be zero and its lag-one serial correlation be $ρ$, which satisfies $|ρ| < 1$. Rice (1945) proved that $(n-1) \arccos(ρ)/π$ is the expected number of sign changes. A corresponding formula for higher-order moments was proposed by Nyberg, Lizana & Ambjörnsson (2018), based on an independent interval approximation. We focus on the variance only, for small $n$, and see a promising fit between theory and model.

math.HO↗

arXiv2019

Moments of Maximum: Segment of AR(1)

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider five contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{5}$). Let $M$ denote the maximum of these. Let $ρ$ be the lag-one serial correlation, which satisfies $|ρ| < 1$. For what value of $ρ$ is $\mathbb{E}(M)$ maximized? How does $\mathbb{V}(M)$ behave for increasing $ρ$? Answers to these questions lie in Afonja (1972), suitably decoded.

math.HO↗

arXiv2019

M/M/$c$ Queues and the Poisson Clumping Heuristic

In continuous time, customers arrive at random. Each waits until one of $c$ servers is available; each thereafter departs at random. The distribution of maximum line length of idle customers was studied over 25 years ago. We revisit two good approximations of this, employing a discrete Gumbel formulation and detailed graphics to describe simulation outcomes.

math.HO↗

arXiv2019

Geo/Geo/2 Queues and the Poisson Clumping Heuristic

In discrete time, customers arrive at random. Each waits until one of two servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. In the context of an emergency room (for medical treatment), the virtue of one fast doctor over two slow doctors is explored. Via limiting argument to continuous time, we study likewise the M/M/2 queue.

math.HO↗

arXiv2019

Traffic Light Queues and the Poisson Clumping Heuristic

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics for $2 \leq \ell \leq 3$.

math.PR↗

arXiv2018

Conjectures about Traffic Light Queues

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. The maximum line length of idle cars is fully understood for $\ell = 1$, but only partially for $2 \leq \ell \leq 3$.

math.HO↗

arXiv2017

Idempotents and Nilpotents Modulo n

We study asymptotic properties of periods and transient phases associated with modular power sequences. The latter are simple; the former are vaguely related to the reciprocal sum of square-free integer kernels.

math.NT↗

arXiv2016

Random Cyclic Quadrilaterals

The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively correlated, any two sides are independent. In contrast, for a uniform cyclic quadrilateral, any two sides are negatively correlated, whereas any two adjacent angles are uncorrelated yet dependent.

math.HO↗

arXiv2016

Ptolemy Constants as Described by Eccentricity

Let J denote a simple closed curve in the plane. Let points a, b, c, d \in J occur in this order when traversing J in a counterclockwise direction. Define p(a,b,c,d) to be the ratio of ab*cd+ad*bc to ac*bd, where zw denotes distance between z and w. Define P(J) to be the supremum of p over all such points. Harmaala & Klén [1] provided bounds on P(J) when J is an ellipse or rectangle of eccentricity ε. We nonrigorously give formulas for P(J) here, in the hope that someone else can fill gaps in our reasoning.

math.MG↗

arXiv2016

Quartic and Octic Characters Modulo n

The average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n->infty. The same is true for primitive cubic characters. It is therefore surprising that, as n->infty, the average number of primitive quartic characters of modulus n grows with ln(n), and that the average number of primitive octic characters of modulus n grows with ln(n)^2. Leading coefficients in the asymptotic expressions are also computed.

math.NT↗

arXiv2016

Squares and Cubes Modulo n

We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.

math.NT↗

arXiv2016

Capturing, Ordering and Gaussianity in 2D

We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.

math.HO↗