arXiv subjects

Steven Finch

Publications and source records attributed to Steven Finch.

At least 37 records · Page 2Linked to original sources

arXiv2024

What do sin$(x)$ and arcsinh$(x)$ have in Common?

N. G. de Bruijn (1958) studied the asymptotic expansion of iterates of sin$(x)$ with $0 < x \leq π/2$. Bencherif & Robin (1994) generalized this result to increasing analytic functions $f(x)$ with an attractive fixed point at 0 and $x > 0$ suitably small. Mavecha & Laohakosol (2013) formulated an algorithm for explicitly deriving required parameters. We review their method, testing it initally on the logistic function $\ell(x)$, a certain radical function $r(x)$, and later on several transcendental functions. Along the way, we show how $\ell(x)$ and $r(x)$ are kindred functions; the same is also true for sin$(x)$ and arcsinh$(x)$.

math.CA↗

arXiv2024

An Exceptional Convolutional Recurrence

A quadratic recurrence of Faltung type, arising via ancestral path lengths of random binary trees, turns out to be related to the Painlevé I differential equation.

math.CO↗

arXiv2024

Multisum Sets

Complete infinite multisum sets are eventually linear. After 30 years of sitting in a file cabinet, the proof (thanks to James H. Schmerl) is brought from darkness into light.

math.CO↗

arXiv2024

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.

math.HO↗

arXiv2024

On the Deepest Cycle of a Random Mapping

Let $\mathcal{T}_n$ be the set of all mappings $T:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$. The corresponding graph of $T$ is a union of disjoint connected unicyclic components. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random (i.e., with probability $n^{-n}$). The cycle of $T$ contained within its largest component is callled the deepest one. For any $T\in\mathcal{T}_n$, let $ν_n=ν_n(T)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of $ν_n/\sqrt{n}$ and find the limits of its expectation and variance as $n\to\infty$. For $n$ large enough, we also show that nearly $55\%$ of all cyclic vertices of a random mapping $T\in\mathcal{T}_n$ lie in the deepest cycle and that a vertex from the longest cycle of $T$ does not belong to its largest component with approximate probability $0.075$.

math.CO↗

arXiv2023

Stochastic Reservoir Calculations

Prabhu (1958) obtained the stationary distribution of storage level $Z_{t}$ in a reservoir of finite volume $v$, given an inflow $X_{t}$ and an outflow $Y_{t}$. Time $t$ is assumed to be discrete, $X_{t} \sim$ Gamma$(p,μ)$ are independent and $p$ is a positive integer. The mean inflow is $p/μ$; the target outflow is $m$ (constant). We attempt to clarify intricate details, often omitted in the literature, by working through several examples. Of special interest are the probabilities of depletion ($Z_{t}=0$) and spillage ($Z_{t}=v$). For prescribed {$v,p,μ$}, what value of $m$ minimizes both of these?

math.PR↗

arXiv2022

D/M/1 Queue: Policies and Control

Equilibrium G/M/1-FIFO waiting times are exponentially distributed, as first proved by Smith (1953). For other client-sorting policies, such generality is not feasible. Assume that interarrival times are constant. Symbolics for the D/M/1-LIFO density are completely known; numerics for D/M/1-SIRO arise via an unpublished recursion due to Burke (1967). Consider a weighted sum of two costs, one from keeping clients waiting for treatment and the other from having the server idle. With this in mind, what is the optimal interarrival time and how does this depend on the choice of policy?

math.PR↗

arXiv2022

M/G/1-FIFO Queue with Uniform Service Times

An exact formula for the equilibrium M/U/1 waiting time density is now effectively known. What began as a numeric exploration became a symbolic banquet. Inverse Laplace transforms provided breadcrumbs in the trail; delay differential equations subsequently gave clear-cut precision. We also remark on tail probability asymptotics and on queue lengths.

math.PR↗

arXiv2022

M/D/1 Queues with LIFO and SIRO Policies

While symbolics for the equilibrium M/D/1-LIFO waiting time density are completely known, corresponding numerics for M/D/1-SIRO are derived from recursions due to Burke (1959). Implementing an inverse Laplace transform-based approach for the latter remains unworkable.

math.PR↗

arXiv2022

How Long Might We Wait at Random?

In discrete time, customers arrive at random. Each waits until one of three servers is available; each thereafter departs at random. We seek the distribution of maximum line length of idle customers. Algebraic expressions obtained for the two-server scenario do not appear feasible here. We also review well-known distributional results for maximum wait time associated with an M/M/1 queue and speculate about their generalization.

math.HO↗

arXiv2022

Components and Cycles of Random Mappings

Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -- that exactly one component exists -- are also examined. For instance, the mean length of the longest cycle is $(0.7824...)\sqrt n$ in general, but for the special case, it is $(0.7978...)\sqrt n$, a difference of less than $2\%$.

math.CO↗

arXiv2022

Second Best, Third Worst, Fourth in Line

We investigate decomposable combinatorial labeled structures more fully, focusing on the exp-log class of type a=1 or 1/2. For instance, the modal length of the second longest cycle in a random n-permutation is (0.2350...)n, whereas the modal length of the second smallest component in a random n-mapping is 2 (conjecturally, given n>=434). As in earlier work, our approach is to establish how well existing theory matches experimental data and to raise open questions.

math.CO↗

arXiv2022

Joint Probabilities within Random Permutations

A celebrated analogy between prime factorizations of integers and cycle decompositions of permutations is explored here. Asymptotic formulas characterizing semismooth numbers (possessing at most several large factors) carry over to random permutations. We offer a survey of practical methods for computing relevant probabilities of a bivariate or trivariate flavor.

math.CO↗

arXiv2022

Random Gaussian Tetrahedra

Given independent normally distributed points A,B,C,D in Euclidean 3-space, let Q denote the plane determined by A,B,C and D^ denote the orthogonal projection of D onto Q. The probability that the tetrahedron ABCD is acute remains intractable. We make some small progress in resolving this issue. Let Gamma denote the convex cone in Q containing all linear combinations A+r*(B-A)+s*(C-A) for nonnegative r, s. We compute the probability that D^ falls in (B+C)-Gamma to be 0.681..., but the probability that D^ falls in Gamma to be 0.683.... The intersection of these two cones is a parallelogram in Q twice the area of the triangle ABC. Among other issues, we mention the distribution of random solid angles and sums of these.

math.PR↗

arXiv2022

Permute, Graph, Map, Derange

We study decomposable combinatorial labeled structures in the exp-log class, specifically, two examples of type a=1 and two examples of type a=1/2. Our approach is to establish how well existing theory matches experimental data. For instance, the median length of the longest cycle in a random n-permutation is (0.6065...)*n, whereas the median length of the largest component in a random n-mapping is (0.7864...)*n. Unsolved problems are highlighted, in the hope that someone else might address these someday.

math.CO↗

arXiv2022

Rounds, Color, Parity, Squares

This is a sequel to our paper "Permute, Graph, Map, Derange", involving decomposable combinatorial labeled structures in the exp-log class of type a=1/2, 1, 3/2, 2. As before, our approach is to establish how well existing theory matches experimental data and to raise open questions.

math.CO↗

arXiv2021

Traffic lights, clumping and QBDs

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 algebraically for $2\leq\ell\leq3$ and numerically for $\ell\geq4$.

math.PR↗

arXiv2020

Covariance within Random Integer Compositions

Fix a positive integer $N$. Select an additive composition $ξ$ of $N$ uniformly out of $2^{N-1}$ possibilities. The interplay between the number of parts in $ξ$ and the maximum part in $ξ$ is our focus. It is not surprising that correlations $ρ(N)$ between these quantities are negative; we earlier gave inconclusive evidence that $\lim_{N \to \infty} ρ(N)$ is strictly less than zero. A proof of this result would imply asymptotic dependence. We now retract our presumption in such an unforeseen outcome. Similar experimental findings apply when $ξ$ is a 1-free composition, i.e., possessing only parts $\geq 2$.

math.CO↗