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Len Bos

Publications and source records attributed to Len Bos.

At least 19 recordsLinked to original sources
arXiv2025

On the brachistochrone problem for cycling ascents

VAM ({\it velocit\`a ascensionale media}) is a measurement that quantifies a cyclist's climbing ability. We show that to minimize the time to attain a given height gain\, -- \,which is tantamount to maximizing VAM\, -- \,a cyclist should climb as steep a constant-grade hill as possible. Apart from the power-to-weight ratio, the limit of steepness is imposed by such factors as the efficiency of pedalling, which is related to feasible cadence, maintaining balance, preventing lifting of the front, and skidding of the rear, wheel. In an appendix, we discuss steepness constraints due to pedalling efficiency. The article itself is focused on consequences of the power available to the cyclist, which can be viewed as a necessary condition to examine other aspects of climbing strategy. We show that\, -- \,for given start and end points, and for any fixed average-power constraint\, -- \,the brachistochrone, which is the trajectory of minimum ascent time, is the straight line connecting these points, covered with a constant speed, which along such a line is equivalent to a constant power. This is in contrast to the classical solution of a descent brachistochrone under gravity, which is a cycloid along which the speed is not constant.

physics.pop-ph↗
arXiv2025

On minimizing cyclists' ascent times: Part II

We formulate an optimization of a bicycle ascent time under the constraints of the average, maximum, and minimum powers. In contrast to the first part of this study, we do not restrict the departure to flying starts with an initial speed determined by the model and its optimization. We allow for various initial speeds, from a standstill to a launched start. We accomplish this by generalizing the discontinuous piecewise constant speed model to a continuous piecewise linear speed model. Regardless of the initial speed, steepness or profile of the ascent the optimal strategy tends to a constant ground speed, in agreement with the conclusion of the previous, more restricted, formulation. This new formulation allows us to compare various initial-speed strategies and, hence, has a direct application to competitive cycling. Notably, in timetrials composed of flat and steep sections, it helps one decide whether or not to change bicycle, which requires stopping and restarting, from one that is more appropriate for flats to one that is more appropriate for uphills.

physics.class-ph↗
arXiv2024

Equations for the overlaps of a SIC

We give a holomorphic quartic polynomial in the overlap variables whose zeros on the torus are precisely the Weyl-Heisenberg SICs (symmetric informationally complete positive operator valued measures). By way of comparison, all the other known systems of equations that determine a Weyl-Heisenberg SIC involve variables and their complex conjugates. We also give a related interesting result about the powers of the projective Fourier transform of the group G = Z d x Z d .

cs.IT↗
arXiv2024

On minimizing cyclists' ascent times

We prove that, given an average power, the ascent time is minimized if a cyclist maintains a constant ground speed regardless of the slope. Herein, minimizing the time is equivalent to maximizing -- for a given uphill -- the corresponding mean ascent velocity (VAM: velocit\`a ascensionale media), which is a common training metric. We illustrate the proof with numerical examples, and show that, in general, maintaining a constant instantaneous power results in longer ascent times; both strategies result in the same time if the slope is constant. To remain within the athlete's capacity, we examine the effect of complementing the average-power constraint with a maximum-power constraint. Even with this additional constraint, the ascent time is the shortest with a modified constant-speed -- not constant-power -- strategy; as expected, both strategies result in the same time if the maximum and average powers are equal to one another. Given standard available information -- including level of fitness, quantified by the power output, and ascent profile -- our results allow to formulate reliable and convenient strategies of uphill timetrials.

physics.class-ph↗
arXiv2023

A Characterization of Optimal Prediction Measures via $\ell_1$ Minimization

Suppose that $K\subset\C$ is compact and that $z_0\in\C\backslash K$ is an external point. An optimal prediction measure for regression by polynomials of degree at most $n,$ is one for which the variance of the prediction at $z_0$ is as small as possible. Hoel and Levine (\cite{HL}) have considered the case of $K=[-1,1]$ and $z_0=x_0\in \R\backslash [-1,1],$ where they show that the support of the optimal measure is the $n+1$ extremme points of the Chebyshev polynomial $T_n(x)$ and characterizing the optimal weights in terms of absolute values of fundamental interpolating Lagrange polynomials. More recently, \cite{BLO} has given the equivalence of the optimal prediction problem with that of finding polynomials of extremal growth. They also study in detail the case of $K=[-1,1]$ and $z_0=ia\in i\R,$ purely imaginary. In this work we generalize the Hoel-Levine formula to the general case when the support of the optimal measure is a finite set and give a formula for the optimal weights in terms of a $\ell_1$ minimization problem.

math.ST↗
arXiv2023

On Waldron Interpolation on a Simplex in $\mathbb{R}^d$

We introduce explicit families of good interpolation points for interpolation on a triangle in $\mathbb{R}^2$ that may be used for either polynomial interpolation or a certain rational interpolation for which we give explicit formulas.

math.NA↗
arXiv2022

On Fekete Points for a Real Simplex

We survey what is known about Fekete points/optimal designs for a simplex in $\R^d.$ Several new results are included. The notion of Fej\'er exponenet for a set of interpolation points is introduced.

math.NA↗
arXiv2022

Modelling of a cyclist's power for time trials on a velodrome

We formulate a phenomenological model to study the power applied by a cyclist on a velodrome\, -- \,for individual timetrials\, -- \,taking into account the straights, circular arcs, connecting transition curves and banking. The dissipative forces we consider are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, power can be used to increase the kinetic and gravitational potential energy. Herein, to model a steady ride\, -- \,as expected for individual timetrials\, -- \,we assume a constant centre-of-mass speed, while allowing the cadence and power to vary during a lap. Hence, the kinetic energy is constant and the only mechanical energy whose change we need to consider is the increase of gravitational potential energy due to raising the centre of mass upon exiting each curve. The effect of dissipative forces is examined at each point of the lap; the effect of conservative forces is examined as an average. The latter is a small\, -- \,albeit not negligible\, -- \,part of the total power, and its inclusion within a model is a novelty presented herein. It increases the model's empirical adequacy. Following derivations and justifications of expressions that constitute this mathematical model, we present a numerical example. We show that the cadence and power vary slightly during a steady ride. In other words, a constant centre-of-mass speed entails nearly constant cadence and power, as expected for a steady ride and as supported by measurements. Also, we examine changes in the required power due to changes of various quantities, such as air density at a velodrome, laptime and several others, as well as the model sensitivity to input errors. Furthermore, we examine the effects on the required power of slight and gradual changes in speed, which are pertinent to individual time trials.

physics.class-ph↗
arXiv2021

Modelling of cyclist's power to overcome dissipative forces on a velodrome

We model the instantaneous power applied by a cyclist on a velodrome -- for individual pursuits and other individual time trials -- taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of empirical adequacy of the model.

physics.pop-ph↗
arXiv2020

On modelling bicycle power for velodromes: Part II Formulation for individual pursuits

We model the instantaneous power on a velodrome--as applied to individual pursuits and other individual time trials--taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of an empirical adequacy of the model. We also examine changes in the kinetic and potential energy.

physics.pop-ph↗
arXiv2020

On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations

The velocit\`a ascensionale media (VAM) is measurement that quantifies a cyclist's climbing ability. It depends on both the ground speed of a bicycle-cyclist system and the slope of an incline. To maximize the ascent speed, the solution to the brachistochrone problem determines that the optimal curve for the incline is a straight line, which is a hill with a constant slope. The maximum obtainable VAM value by a cyclist increases monotonically with the slope of the incline, but is limited by the maximum sustainable power. These properties -- which are theorems stemming from a standard mathematical model to account for the power required to propel a bicycle -- constitute a mathematical-physics background upon which various strategies for the VAM maximization can be examined in the context of the maximum sustainable power as a function of both the gear ratio and cadence. Recently established records provide an empirical support for these analytical results, which are based on theoretical considerations.

physics.pop-ph↗
arXiv2019

On Christoffel roots for nondetached slowness surfaces

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the $qP$ slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the $qP$ and $qSV$ slowness surfaces are not smooth.

physics.geo-ph↗
arXiv2019

On orthogonal transformations of Christoffel equations

The purpose of this paper is to prove the equivalence$-$under rotations of distinct terms$-$of different forms of a determinantal equation that appears in the studies of wave propagation in Hookean solids, in the context of the Christoffel equations. To do so, we prove a general proposition that is not limited to ${\mathbb R}^3$, nor is it limited to the elasticity tensor with its index symmetries. Furthermore, the proposition is valid for orthogonal transformations, not only for rotations. The sought equivalence is a corollary of that proposition.

physics.geo-ph↗
arXiv2018

On the Backus average of layers with randomly oriented elasticity tensors

As shown by Backus (1962), the average of a stack of isotropic layers results in a transversely isotropic medium. Herein, we consider a stack of layers consisting of a randomly oriented anisotropic elasticity tensor, which-one might expect-would result in an isotropic medium. However, we show-by means of a fundamental symmetry of the Backus average-that the corresponding Backus average is only transversely isotropic and not, in general, isotropic. In the process, we formulate, and use, a relationship between the Backus and Gazis et al. (1963) averages.

physics.geo-ph↗
arXiv2017

On commutativity and near commutativity of translational and rotational averages: Analytical proofs and numerical examinations

We show that, in general, the translational average over a spatial variable---discussed by Backus \cite{backus}, and referred to as the equivalent-medium average---and the rotational average over a symmetry group at a point---discussed by Gazis et al. \cite{gazis}, and referred to as the effective-medium average---do not commute. However, they do commute in special cases of particular symmetry classes, which correspond to special relations among the elasticity parameters. We also show that this noncommutativity is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the {\it square\/} of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology, where the weak-anisotropy assumption results in empirically adequate models.

physics.geo-ph↗
arXiv2017

Statistical and numerical considerations of Backus-average product approximation

In this paper, we examine the applicability of the approximation, $\overline{f\,g}\approx \overline f\,\overline g\,$, within Backus (1962) averaging. This approximation is a crucial step in the method proposed by Backus (1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions. Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions. Even though, according to the upper bound, Backus (1962) averaging might not appear as a viable approach, we show that$-$for cases representative of physical scenarios modelled by such an averaging$-$the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy. In particular, we examine a special case for which the approximation results in spurious values. However, such a case$-$though physically realizable$-$is not likely to appear in seismology, where Backus (1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (1962) averaging is also considered.

physics.geo-ph↗