arXiv subjects

Steven R. Finch

Publications and source records attributed to Steven R. Finch.

At least 19 recordsLinked to original sources

arXiv2022

Correlation between Angle and Side

Let alpha be an arbitrary angle in a random spherical triangle Delta and a be the side opposite alpha. (The sphere has radius 1; vertices of Delta are independent and uniform.) If some other side is constrained to be pi/2, then E(alpha*a)=3.05.... If instead some other angle is fixed at pi/2, then E(alpha*a)=2.87.... In our study of the latter scenario, both Apery's constant and Catalan's constant emerge. We also review Miles' 1971 proof that E(alpha*a)=pi^2/2-2 when no constraints are in place.

math.PR↗

arXiv2019

0-Pierced Triangles within a Poisson Overlay

Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is 0-pierced if no member of the line process intersects any side of T. Our starting point is Ambartzumian's 1982 joint density for angles of T; our exposition is elementary and raises several unanswered questions.

math.HO↗

arXiv2018

Four Vignettes on Apparent Size

Problems in optimization and geometric probability are discussed, all connected with angles subtended at an observer's eye by an object at a distance. Several of these remain unsolved.

math.HO↗

arXiv2018

The Maximum of an Asymmetric Simple Random Walk with Reflection

Consider the extreme value of a Bernoulli random walk on the one-dimensional integer lattice, with reflection at 0, over a finite discrete time interval. Only the asymmetric (biased) case is discussed. Asymptotic mean/variance results are given as the time interval length approaches infinity. We similarly solve an elementary traffic light problem from queueing theory.

math.HO↗

arXiv2018

Median Area for Broken Sticks

Breaking a line segment L in two places at random, the three pieces can be configured as a triangle T with probability 1/4. We determine both the PDF and CDF for area(T) in terms of elliptic integrals. In particular, if L has length 1, then the median area 0.031458... can be calculated to arbitrary precision. We also mention the analog involving cyclic quadrilaterals -- with corresponding probability 1/2 -- and ask some unanswered questions.

math.HO↗

arXiv2018

How Far Might We Walk at Random?

This elementary treatment first summarizes extreme values of a Bernoulli random walk on the one-dimensional integer lattice over a finite discrete time interval. Both the symmetric (unbiased) and asymmetric (biased) cases are discussed. Asymptotic results are given as the time interval length approaches infinity. Focus then shifts to such walks reflected at the origin -- in both strong and weak senses -- and related unsolved problems are meticulously examined.

math.HO↗

arXiv2018

Maximum Queue Length for Traffic Light with Bernoulli Arrivals

Cars arrive at an intersection with a stoplight, which is either red or green. The cars all travel in the same direction, that is, we ignore cross-traffic & oncoming traffic. Assume that the intersection is initially empty. Assume that, at every second, there is a probability p that one new car will arrive at the light, and the outcome is independent of past & future. Let L>=1 be an integer. A red light lasts L seconds; likewise for green. If the light is red, no cars can leave the intersection. If the light is green, cars will leave the intersection at a rate of one per second. Over a time period of n seconds, determine the (random) maximum queue length M of cars at the intersection. What is the distribution of M, as a function of (p,L,n)? We answer this question for the special case L=1 and introduce a conjecture for L>1.

math.HO↗

arXiv2018

Three Random Intercepts of a Segment

We construct random triangles via uniform sampling of certain families of lines in the plane. Two examples are given. The word "uniform" turns out to be vague; two competing models are examined. Everything we write is well-known to experts. Which model is more appropriate? Our hope is to engage a larger audience in answering this question.

math.HO↗

arXiv2017

Triangles Formed via Poisson Nearest Neighbors

We start with certain joint densities (for sides and for angles) corresponding to pinned Poissonian triangles in the plane, then discuss analogous results for staked and anchored triangles.

math.MG↗

arXiv2016

Width Distributions for Convex Regular Polyhedra

The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi.

math.MG↗

arXiv2016

Mean Width of a Regular Simplex

The mean width is a measure on n-dimensional convex bodies. An integral formula for the mean width of a regular n-simplex appeared in the electrical engineering literature in 1997. As a consequence, expressions for the expected range of a sample of n+1 normally distributed variables, for n<=6, carry over to widths of regular n-simplices. As another consequence, precise asymptotics for the mean width become available as n->infty.

math.MG↗

arXiv2016

Mean Width of a Regular Cross-Polytope

The expected range of a sample of n+1 normally distributed variables is known to be related to the mean width of a regular n-simplex. We show that the expected maximum mu_n of a sample of n half-normally distributed variables is related to the mean width of a regular n-crosspolytope. Both of these relations have mean square counterparts. An expression for mu_5 is found and is believed to be new.

math.MG↗

arXiv2016

Convex Hull of Two Orthogonal Disks

Three configurations of two perpendicular disks in R^3 are examined, the first in which the disks share centers and the other two in which the disks touch at precisely one point. Volume, surface area and mean width calculations dominate the discussion. Integrated mean curvature also appears as an indirect way to compute mean width.

math.MG↗

arXiv2016

Lost at Sea

What is the path of minimum expected length for escaping a planar convex region Omega? We rigorously obtain best 2-segment and 3-segment solutions when Omega is an infinite strip, and numerically examine 2-segment solutions when Omega is a disk.

math.OC↗

arXiv2016

Searching for a Shoreline

Logarithmic spirals are conjectured to be optimal escape paths from a half plane ocean. Assuming this, we find the rate of increase for both min-max and min-mean interpretations of "optimal". For the one-dimensional analog, which we call logarithmic coils, our min-mean solution differs from a widely-cited published account.

math.OC↗

arXiv2016

The Logarithmic Spiral Conjecture

When searching for a planar line, if given no further information, one should adopt a logarithmic spiral strategy (although unproven).

math.OC↗

arXiv2015

Random Spherical Triangles

Let Delta be a random spherical triangle (meaning that vertices are independent and uniform on the unit sphere). A closed-form expression for the area density of Delta has been known since 1867; a complicated integral expression for the perimeter density was found in 1994. Does there exist a closed-form expression for the latter? We attempt to answer this question from several directions. An outcome of our work is the exact value of the perimeter density at the point pi.

math.PR↗

arXiv2015

Covering a Sphere with Four Random Circular Caps

Let p(w) denote the probability that four random circular caps of angular radius 70deg 84deg; no improvement on the inequality p(w)>=0 for w<84deg is yet feasible. A dual problem involving randomly inscribed well-centered tetrahedra is also examined.

math.PR↗