A mathematical analysis of fairness in shootouts

IMA Journal of Management Mathematics ◽

10.1093/imaman/dpaa023 ◽

2020 ◽

Author(s):

Roel Lambers ◽

Frits C R Spieksma

Keyword(s):

Sudden Death ◽

Mathematical Analysis ◽

Repetitive Sequence ◽

Equal Chance ◽

Explicit Expressions

Abstract A shootout is a popular mechanism to identify a winner of a match between two teams. It consists of rounds in which each team gets, sequentially, an opportunity to score a point. It has been shown empirically that shooting first or shooting second in a round has an impact on the scoring probability. This raises a fairness question: is it possible to specify a sequence such that identical teams have equal chance of winning? We show that, for a sudden death, no repetitive sequence can be fair. In addition, we show that the so-called Prohuet–Thue–Morse sequence is not fair. There is, however, an algorithm that outputs a fair sequence whenever one exists. We also analyze the popular best-of-$k$ shootouts and show that no fair sequence exists in this situation. In addition, we find explicit expressions for the degree of unfairness in a best-of-$k$ shootout; this allows sports administrators to asses the effect of the length of the shootout on the degree of unfairness.

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Deposition of Evaporating Droplets in a Decelerating Boundary Layer with Wall Suction: Mathematical Analysis

International Journal of Chemical Reactor Engineering ◽

10.2202/1542-6580.2399 ◽

2010 ◽

Vol 8 (1) ◽

Author(s):

Ro'ee Z Orland ◽

Rafael Tadmor ◽

David Katoshevski

Keyword(s):

Boundary Layer ◽

Liquid Phase ◽

Mathematical Analysis ◽

Free Stream ◽

Gas Velocity ◽

Lateral Direction ◽

Wall Suction ◽

Phase Concentration ◽

Spray Droplets ◽

Explicit Expressions

The problem of spray/droplets dynamics in a downstream decelerating boundary layer accompanied by evaporation and wall suction is treated analytically. A similarity approach is employed and explicit expressions are obtained for the distribution of the host-gas velocity and for the distribution of the liquid phase. Several options are considered in terms of evaporation, including the possibilities of constant evaporation in the lateral direction of the boundary layer and of evaporation dependent on proximity to the wall. The results reveal the possibility that, under certain conditions, the liquid phase concentration reaches an extremum inside the boundary layer domain, rather than at the wall or in the free stream.

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Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients

International Journal of Advanced Research in Mathematics ◽

10.18052/www.scipress.com/ijarm.5.35 ◽

2016 ◽

Vol 5 ◽

pp. 35-51 ◽

Cited By ~ 2

Author(s):

Victor Kozyakin

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Analysis ◽

Trigonometric Series ◽

Mathematical Analysis ◽

Tauberian Theorems ◽

Abel Transform ◽

Cosine Series ◽

Hardy Type ◽

Several Variables ◽

Explicit Expressions

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor.The approach developed in the paper is quite elementary and essentially algebraic. It does not rely on the classic machinery of the asymptotic analysis such as slowly varying functions, Tauberian theorems or the Abel transform. However, in our case, it allows obtaining explicit expressions for the asymptotics and extending to the general case of trigonometric series in several variables classical results of G. H. Hardy and other authors known for the case of one variable.

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Density waves in disk galaxies

Symposium - International Astronomical Union ◽

10.1017/s0074180900101135 ◽

1967 ◽

Vol 31 ◽

pp. 313-317 ◽

Cited By ~ 8

Author(s):

C. C. Lin ◽

F. H. Shu

Keyword(s):

Mathematical Analysis ◽

Spiral Structure ◽

Stellar System ◽

Collective Modes ◽

Disk Galaxies ◽

Spiral Pattern ◽

Density Waves ◽

The Galaxy ◽

Detailed Mathematical Analysis

Density waves in the nature of those proposed by B. Lindblad are described by detailed mathematical analysis of collective modes in a disk-like stellar system. The treatment is centered around a hypothesis of quasi-stationary spiral structure. We examine (a) the mechanism for the maintenance of this spiral pattern, and (b) its consequences on the observable features of the galaxy.

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Computing cell tractions from particle displacements on silicone rubber substrata

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100168542 ◽

1994 ◽

Vol 52 ◽

pp. 162-163

Author(s):

Tim Oliver ◽

Akira Ishihara ◽

Ken Jacobsen ◽

Micah Dembo

Keyword(s):

Plane Stress ◽

Linear Elasticity ◽

Silicone Rubber ◽

Mathematical Analysis ◽

Elastic Recovery ◽

Video Images ◽

Cell Traction Forces ◽

Latex Beads ◽

Cell Traction ◽

Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

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