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Stephen Montgomery-Smith

Publications and source records attributed to Stephen Montgomery-Smith.

At least 19 recordsLinked to original sources
arXiv2024

Using Lie derivatives with dual quaternions for parallel robots

We introduce the notion of the Lie derivative in the context of dual quaternions that represent rigid motions and twists. First we define the wrench in terms of dual quaternions. Then we show how the Lie derivative helps understand how actuators affect an end effector in parallel robots, and make it explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots. We also show how to use Lie derivatives with the Newton-Raphson Method to solve the forward kinematic problem for over constrained parallel actuators. Finally, we derive the equations of motion of the end effector in dual quaternion form, which include the effect of inertia from the actuators.

cs.RO↗
arXiv2023

Non-linear Instability of periodic orbits of suspensions of thin fibers in fluids

This paper is concerned with difficulties encountered by engineers when they attempt to predict the orientation of fibers in the creation of injection molded plastic parts. It is known that Jeffery's equation, which was designed to model a single fiber in an infinite fluid, breaks down very badly when applied, with no modifications, to this situation. In a previous paper, the author described how interactions between the fiber orientation and the viscosity of the suspension might cause instability, which could result in the simple predictions from Jeffery's equation being badly wrong. In this paper, we give some rigorous proofs of instability using Floquet Theory. We also give an example where the spectal radius of the monodromy operator is one, but there is still non-linear instability.

physics.flu-dyn↗
arXiv2022

Control of systems by parallel actuators

We describe a particular control method for a system controlled by several actuators with the same control constants. We show under certain assumptions that the control constants for the whole system can be obtained immediately from the control constants for a single actuator. This greatly simplifies the work in finding the control constants. Also, no gain scheduling is required. The authors have been unable to find any prior work in this direction, and so believe this is a rather new approach.

math.OC↗
arXiv2022

Functional calculus for dual quaternions

We give a formula for $f(\eta)$, where $f :\mathbb C \to \mathbb C$ is a continuously differentiable function satisfying $f(\bar z) = \overline{f(z)}$, and $\eta$ is a dual quaternion. Note this formula is straightforward or well known if $\eta$ is merely a dual number or a quaternion. If one is willing to prove the result only when $f$ is a polynomial, then the methods of this paper are elementary.

math.GM↗
arXiv2021

Perturbations of the coupled Jeffery-Stokes equations

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibers in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibers that might be responsible for the deviation from theory, can at least partially be modeled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data is slightly non-uniform, in that the probability distribution of the orientation has small spacial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibers to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory. This paper differs from the published version in that it contains a corrogendum at the end.

physics.flu-dyn↗
arXiv2021

The Fast Exact Closure for Jeffery's Equation with Diffusion

Jeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a `closure' that approximates the distribution function's fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when there are no fiber interactions, it exactly solves Jeffery's equation. Numerical examples for dense fiber suspensions are provided with both a Folgar-Tucker (1984) diffusion term and the recent anisotropic rotary diffusion term proposed by Phelps and Tucker (2009). Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations.

physics.flu-dyn↗
arXiv2021

Exact tensor closures for the three dimensional Jeffery's equation

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of \cite{cintra:95}, or equivalently, a natural closure in the sense of \cite{verleye:93}. The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fiber orientation was isotropic at some point in time. Finally, this paper presents the Fast Exact Closure (FEC), a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures, nor the elliptic integral computation.

physics.flu-dyn↗
arXiv2019

Shapley-like values without symmetry

Following the work of Lloyd Shapley on the Shapley value, and tangentially the work of Guillermo Owen, we offer an alternative non-probabilistic formulation of part of the work of Robert J. Weber in his 1978 paper "Probabilistic values for games." Specifically, we focus upon efficient but not symmetric allocations of value for cooperative games. We retain standard efficiency and linearity, and offer an alternative condition, "reasonableness," to replace the other usual axioms. In the pursuit of the result, we discover properties of the linear maps that describe the allocations. This culminates in a special class of games for which any other map that is "reasonable, efficient" can be written as a convex combination of members of this special class of allocations, via an application of the Krein-Milman theorem.

econ.TH↗
arXiv2016

Age-dependent Branching Processes and Applications to the Luria-Delbrück Experiment

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbrück that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution have spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by D. E. Lea, C. A. Coulson, and J. B. S. Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.

q-bio.PE↗
arXiv2016

Estimation of Mutation Rates from Fluctuation Experiments via Probability Generating Functions

This paper calculates probability distributions modeling the Luria-Delbrück experiment. We show that by thinking purely in terms of generating functions, and using a 'backwards in time' paradigm, that formulas describing various situations can be easily obtained. This includes a generating function for Haldane's probability distribution due to Ycart. We apply our formulas to both simulated and real data created by looking at yeast cells acquiring an immunization to the antibiotic canavanine. This paper is somewhat incomplete, having been last significantly modified in March 29, 2014. However the first author feels that this paper has some worthwhile ideas, and so is going to make this paper publicly available.

q-bio.PE↗
arXiv2014

The best constants for operator Lipschitz functions on Schatten classes

Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in B(H) is an operator such that the commutator [A, x] is contained in the Schatten class S_p. It is proved by the last two authors, that then also [f(A), x] is contained in S_p and there exists a constant C_p independent of x and f such that || [f(A), x] ||_p <= C_p || [A,x] ||_p. The main result of this paper is to give a sharp estimate for C_p in terms of p. Namely, we show that C_p ~ p^2/(p-1). In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of n self-adjoint operators, for which we prove the analogous result. The case described here in the abstract follows as a special case.

math.FA↗
arXiv2014

Hamiltonians representing equations of motion with damping due to friction

Suppose that $H(q,p)$ is a Hamiltonian on a manifold $M$, and $\tilde L(q,\dot q)$, the Rayleigh dissipation function, satisfies the same hypotheses as a Lagrangian on the manifold $M$. We provide a Hamiltonian framework that gives the equation $\dot q = \frac{\partial H}{\partial p}(q,p), \quad \dot p = - \frac{\partial H}{\partial q}(q,p) - \frac{\partial \tilde L}{\partial \dot q}(q,\dot q)$. The method is to embed $M$ into a larger framework where the motion drives a wave equation on the negative half line, where the energy in the wave represents heat being carried away from the motion. We obtain a version of Nöther's Theorem that is valid for dissipative systems. We also show that this framework fits the widely held view of how Hamiltonian dynamics can lead to the "arrow of time."

physics.class-ph↗
arXiv2013

Obtaining Laws of Thermodynamics for Ideal Gases using Elastic Collisions

The purpose of this note is to see to what extent ideal gas laws can be obtained from simple Newtonian mechanics, specifically elastic collisions. We present simple one-dimensional situations that seem to validate the laws. The first section describes a numerical simulation that demonstrates the second law of thermodynamics. The second section mathematically demonstrates the adiabatic law of expansion of ideal gases.

physics.class-ph↗
arXiv2010

Characterizing Complex Particle Morphologies Through Shape Matching: Descriptors, Applications, and Algorithms

Many standard structural quantities, such as order parameters and correlation functions, exist for common condensed matter systems, such as spherical and rod-like particles. However, these structural quantities are often insufficient for characterizing the unique and highly complex structures often encountered in the emerging field of nano and microscale self-assembly, or other disciplines involving complex structures such as computational biology. Computer science algorithms known as "shape matching" methods pose a unique solution to this problem by providing robust metrics for quantifying the similarity between pairs of arbitrarily complex structures. This pairwise matching operation, either implicitly or explicitly, lies at the heart of most standard structural characterization schemes for particle systems. By substituting more robust "shape descriptors" into these schemes we extend their applicability to structures formed from more complex building blocks. Here, we describe several structural characterization schemes and shape descriptors that can be used to obtain various types of structural information about particle systems. We demonstrate the application of shape matching algorithms to a variety of example problems, for topics including local and global structure identification and classification, automated phase diagram mapping, and the construction of spatial and temporal correlation functions. The methods are applicable to a wide range of systems, both simulated and experimental, provided particle positions are known or can be accurately imaged.

cond-mat.soft↗
arXiv2004

Conditions implying regularity of the three dimensional Navier-Stokes equation

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac like inequalities. As part of the our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam.

math.AP↗
arXiv2002

Power-bounded operators and related norm estimates

We consider whether L = limsup_{n to infty} n ||T^{n+1}-T^n|| < infty implies that the operator T is power bounded. We show that this is so if L<1/e, but it does not necessarily hold if L=1/e. As part of our methods, we improve a result of Esterle, showing that if sigma(T) = {1} and T != I, then liminf_{n to infty} n ||T^{n+1}-T^n|| >= 1/e. The constant 1/e is sharp. Finally we describe a way to create many generalizations of Esterle's result, and also give many conditions on an operator which imply that its norm is equal to its spectral radius.

math.FA↗