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Steven R. Finch

Publications and source records attributed to Steven R. Finch.

33 records · Page 2Linked to original sources
arXiv2014

Lognormality and Triangles of Unit Area

To generate a triangle of unit perimeter, break a stick of length 1 in two places at random, with the condition that triangle inequalities are satisfied. Is there a similarly natural method for generating triangles of unit area? Study of a product (rather than a sum) of random lengths is facilitated by closure of multiplication within the lognormal family of distributions. Our (necessarily incomplete) answers to the question each draw upon this property.

math.PR↗
arXiv2014

Uniform Triangles with Equality Constraints

The equality constraint a+b+c=1 for random triangle sides corresponds to breaking a stick in two places. An analog a^2+b^2+c^2=1 has a remarkable feature: the bivariate density for angles coincides with that for 3D Gaussian triangles. Interesting complications also arise for a+b=1 and for a^2+b^2=1, with the understanding that the angle gamma opposite side c is Uniform[0,pi]. Closed-form expressions for several side moments remain open.

math.PR↗
arXiv2014

Appell F1 and Conformal Mapping

This is the last of a trilogy of papers on triangle centers. A fairly obscure "conformal center of gravity" is computed for the class of all isosceles triangles. This calculation appears to be new. A byproduct is the logarithmic capacity or transfinite diameter of such, yielding results consistent with Haegi (1951).

math.HO↗
arXiv2014

Least Capacity Point of Triangles

Let D be a compact convex domain in the plane. Pólya & Szegö and, independently, Levi & Pan defined the point p in D that is "best insulated from the boundary C of D". We compute p in the case when C is an isosceles right triangle, revisiting exact results from the study of complex conformal mappings.

math.HO↗
arXiv2014

In Limbo: Three Triangle Centers

Yet more candidates are proposed for inclusion in the Encyclopedia of Triangle Centers. Our focus is entirely on simple calculations.

math.HO↗
arXiv2013

Mutually Equidistant Spheres that Intersect

The setting for this brief paper is R^3. Distance between two spheres is understood as distance delta between spherical centers. For instance, a Reuleaux tetrahedron T is the intersection of four unit balls satisfying delta=1 pairwise. Volume and surface area of T are already well-known; our humble contribution is to calculate the mean width of T.

math.MG↗
arXiv2012

Oblique Circular Cones and Cylinders

Surface area and mean width of a cylinder (the convex hull of two parallel disks) in R^3 are computed. It is more difficult to obtain analogous results for a cone (the convex hull of a disk D and a point p). Oblique formulas for mean width, as well as those for mean curvature, are new. Let L denote the unique diameter of D whose endpoints are equidistant from p. We conclude with a question involving the plane that bisects the cone and contains {p,L}, as p varies. What is the minimum ratio of the smaller measure to the larger?

math.MG↗
arXiv2012

Rank-3 Projections of a 4-Cube

The orthogonal projection of a 4-cube onto a uniform random 3-subspace in R^4 is a convex 3-polyhedron P with 14 vertices almost surely. Three numerical characteristics of P -- volume, surface area and mean width -- are studied. These quantities, along with the Euler characteristic, form a basis of the space of all additive continuous measures that are invariant under rigid motions in R^3. While computing statistics of {vl, ar, mw}, we encounter the generalized hypergeometric function, elliptic integrals and Catalan's constant. A new constant 7.1185587167... also arises and deserves further attention.

math.MG↗
arXiv2012

Planar Projections and Second Intrinsic Volume

Consider random shadows of a cube and of a regular tetrahedron. Area and perimeter of the former are positively dependent (with correlation 0.915...), whereas area and perimeter of the latter appear to be negatively dependent. This is only one result of many, all involving generalizations of mean width.

math.MG↗
arXiv2011

Three Random Tangents to a Circle

Among several things, we find the side density for random triangles circumscribing the unit circle and calculate that its median is 5.5482.... An analogous exact computation for perimeter density remains open.

math.PR↗
arXiv2008

CLT Variance Associated with Baxendale's SDE

Simple analysis of the leftmost eigenvalue of Ince's equation (a boundary value problem with periodicity) resolves an open issue surrounding a stochastic Lyapunov exponent. Numerical verification is also provided.

math.CA↗
arXiv2007

Another Look at AR(1)

Given a stationary first-order autoregressive process X_t (with lag-one correlation rho satisfying |rho|<1), we examine the Central Limit Theorem for (1/n)*ln |X_1...X_n| and compute variances to high precision. Given a nonstationary process X_t (with |rho|>1), we examine instead (1/n)*ln|X_n| and study the distribution of ln|X_n|-n*ln|rho|.

math.DS↗
arXiv1999

Several Constants Arising in Statistical Mechanics

This is a brief survey of certain constants associated with random lattice models, including self-avoiding walks, polyominoes, the Lenz-Ising model, monomers and dimers, ice models, hard squares and hexagons, and percolation models.

math.CO↗
arXiv1997

Universality of finite-size corrections to the number of critical percolation clusters

Monte-Carlo simulations on a variety of 2d percolating systems at criticality suggest that the excess number of clusters in finite systems over the bulk value of nc is a universal quantity, dependent upon the system shape but independent of the lattice and percolation type. Values of nc are found to high accuracy, and for bond percolation confirm the theoretical predictions of Temperley and Lieb, and Baxter, Temperley, and Ashley, which we have evaluated explicitly in terms of simple algebraic numbers. Predictions for the fluctuations are also verified for the first time.

cond-mat.dis-nn↗