arXiv subjects

John H. Elton

Publications and source records attributed to John H. Elton.

6 recordsLinked to original sources

arXiv2024

Evolving the Euler rotation axis as a dynamical system, using the Euler vector and generalizations

Differential equations are derived which show how generalized Euler vector representations of the Euler rotation axis and angle for a rigid body evolve in time; the Euler vector is also known as a rotation vector or axis-angle vector. The solutions can exhibit interesting rotational features in this non-abstract, visualizable setting, including spinor-like behavior and quasiperiodicity. The equations are well-behaved at zero, reducing to the simple infinitesimal case there. One of them is equivalent to a known quaternion differential equation. The simple geometric derivation does not depend on Euler's rotation theorem, and yields a proof of Euler's theorem using only infinitesimal motions. With mild regularity conditions on the angular velocity function, there is a continuous evolution of the normalized axis and angle for all time. Dynamical systems properties are discussed, and numerical solutions are used to investigate them when the angular velocity is itself rotating, and the Euler vector trajectory traces out a torus-like shape, with a strobe plot that densely fills in a closed curve.

math-ph↗

arXiv2023

Exact Solution to the Chow-Robbins Game for almost all n, using the Catalan Triangle

The payoff in the Chow-Robbins coin-tossing game is the proportion of heads when you stop. Knowing when to stop to maximize expectation was addressed by Chow and Robbins(1965), who proved there exist integers ${k_n}$ such that it is optimal to stop when heads minus tails reaches this. Finding ${k_n}$ exactly was unsolved except for finitely many cases by computer. We show ${k_n} = \left\lceil {α\sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2ζ( - 1/2)} \right)\sqrt α}}{\sqrt π}{n^{ - 1/4}}} \right\rceil$ for almost all n, where $α$ is the Shepp-Walker constant.This comes from our estimate ${β_n} = α\sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2ζ( - 1/2)} \right)\sqrt α}}{\sqrt π}{n^{ - 1/4}} + O\left( {n^{ - 7/24}} \right)$ of real numbers defined by Dvoretzky(1967) for a more general Value function which is continuous in its first argument and easier to analyze. An $O({n^{ - 1/4}})$ dependence was conjectured by Christensen and Fischer(2022) from numerical evidence. Our proof uses moments involving Catalan and Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund(2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod's embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in a different way. We use them first for many more examples, but the new idea is to use them algebraically in the tree, with feedback to get a sharper Value estimate near the border, to settle almost all n.

math.PR↗

arXiv2022

Householder Meets Student

The Householder algorithm for the QR factorization of a tall thin n x p full-rank matrix X has the added bonus of producing a matrix M with orthonormal columns that are a basis for the orthocomplement of the column space of X. We give a simple formula for M-Transpose x when x is in that orthocomplement. The formula does not require computing M, it only requires the R factor of a QR factorization. This is used to get a remarkably simple computable concrete representation of independent "residuals" in classical linear regression. For Students problem, when p=1, if R(j)=Y(j)-Ybar are the usual (non-independent) residuals, W(j)=R(j+1) - R(1)/(sqrt(n)+1) gives n-1 i.i.d. mean-zero normal variables whose sum of squares is the same as that of the n residuals. Those properties of this formula can (in hindsight) easily be verified directly, yielding a new simple and concrete proof of Student's theorem. It also gives a simple way of generating n-1 exactly mean-zero i.i.d. samples from n samples with unknown mean. Yiping Cheng exhibited concrete linear combinations of the Y(j) with these properties, in the context of a constructive proof of Student's theorem, but that representation is not so simple. Analogous simple results are obtained for regression when there are more predictors, giving a very simple computable concrete formula for n-p i.i.d. independent residuals with the same sum of squares as that of the usual n non-independent residuals. A connection with Cochran's theorem is discussed.

math.ST↗

arXiv2011

Ham Sandwich with Mayo: A Stronger Conclusion to the Classical Ham Sandwich Theorem

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one point in each set. More generally, for n compactly-supported positive finite Borel measures in Euclidean n-space, there is always a hyperplane that bisects each of the measures and intersects the support of each measure. For example, at any given instant of time, there is one planet, one moon and one asteroid in our solar system and a single plane touching all three that exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal mass of the solar system.

math.MG↗

arXiv2009

Notes on Feige's gumball machines problem

We give a detailed proof, in the identically distributed case, of a conjecture of Feige about the maximum probability that the sum of n independent non-negative integer valued random variables, each of mean 1, exceeds n. The general case is reduced to two-point distributions.

math.PR↗

arXiv2009

Indefinite quadratic forms and the invariance of the interval in Special Relativity

A simple theorem on proportionality of indefinite real quadratic forms is proved, and is used to clarify the proof of the invariance of the interval in Special Relativity from Einstein's postulate on the universality of the speed of light; students are often rightfully confused by the incomplete or incorrect proofs given in many texts. The result is illuminated and generalized using Hilbert's Nullstellensatz, allowing one form to be a homogeneous polynomial which is not necessarily quadratic. Also a condition for simultaneous diagonalizabilityof semi-definite real quadratic functions is given.

math.GM↗