arXiv subjects

Christopher Cox

Publications and source records attributed to Christopher Cox.

At least 19 recordsLinked to original sources

arXiv2026

Nonholonomic billiards and bounded motion in cylinders

A widely used mathematical model for the bouncing motion of an ideally elastic ball -- referred to in previous work by the first two authors and collaborators as a {\em no-slip billiard} system -- exhibits some notable dynamical behavior that is not well-understood. For example, under certain initial conditions, the axial component of the position of the center of the ball moving inside a vertical solid cylinder under constant gravitational force does not accelerate downward as might be expected but remains bounded. There is not as yet, as far as we know, any analytical study of the bouncing ball dynamics, under gravity, in general cylinders (not necessarily having a circular cross-section) in $\mathbb{R}^3$. In this paper, we propose an approach by comparing the no-slip system with a smooth approximation of it that we call {\em nonholonomic billiards}. It consists of a $4$-dimensional ball rolling on the solid $3$-dimensional cylinder. We first review earlier work on no-slip billiards and their connection with nonholonomic (rolling) systems, explain how nonholonomic billiards approximate the no-slip kind (after work by the first two authors and B. Zhao), and illustrate the relationship with a few numerical case studies that demonstrate the utility of the soft (nonholonomic) system as a helpful tool for exploring the dynamics of no-slip billiard systems.

math.DS↗

arXiv2024

Photothermal Spectroscopy for Planetary Sciences: Mid-IR Absorption Made Easy

The understanding of the formation and evolution of the solar system still has many unanswered questions. Formation of solids in the solar system, mineral and organic mixing, and planetary body creation are all topics of interest to the community. Studying these phenomena is often performed through observations, remote sensing, and in-situ analysis, but there are limitations to the methods. Limitations such as IR diffraction limits, spatial resolution issues, and spectral resolution issues can prevent detection of organics, detection and identification of cellular structures, and the disentangling of granular mixtures. Optical-PhotoThermal InfraRed (O-PTIR) spectroscopy is a relatively new method of spectroscopy currently used in fields other than planetary sciences. O-PTIR is a non-destructive, highly repeatable, and fast form of measurement capable of reducing these limitations. Using a dual laser system with an IR source tuned to the mid-IR wavelength we performed laboratory O-PTIR measurements to compare O-PTIR data to existing IR absorption data and laboratory FTIR measurements for planetary materials. We do this for the purpose of introducing O-PTIR to the planetary science community. The technique featured here would serve to better measurements of planetary bodies during in-situ analysis. We find that, unlike other fields where O-PTIR produces almost one-to-one measurements with IR absorption measurements of the same material, granular materials relevant to planetary science do not. However, we do find that the materials compared were significantly close and O-PTIR was still capable of identifying materials relevant to planetary science.

astro-ph.EP↗

arXiv2022

Orthogonal realizations of random sign patterns and other applications of the SIPP

A sign pattern is an array with entries in $\{+,-,0\}$. A matrix $Q$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A.~Curtis and B.L.~Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228--259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $5\times n$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP.

math.CO↗

arXiv2022

Chaotic lensed billiards

Lensed billiards are an extension of the notion of billiard dynamical systems obtained by adding a potential function of the form $C1_{\mathcal{A}}$, where $C$ is a real valued constant and $1_{\mathcal{A}}$ is the indicator function of an open subset $\mathcal{A}$ of the billiard table whose boundaries (of $\mathcal{A}$ and the table) are piecewise smooth. Trajectories are polygonal lines that undergo either reflection or refraction at the boundary of $\mathcal{A}$ depending on the angle of incidence. After laying out some basic concepts and general facts, in particular reviewing the optical/mechanical analogy that motivates these billiard models, we explore how their dynamical properties depend on the potential parameter $C$ using a number of families of examples. In particular, we explore numerically the Lyapunov exponents for these parametric families and highlight the more salient common properties that distinguish them from standard billiard systems. We further justify some of these properties by characterizing lensed billiards in terms of switching dynamics between two open (standard) billiard subsystems and obtaining mean values associated to orbit sojourn in each subsystem.

nlin.CD↗

arXiv2022

The influence of physiological flow development on popular wall shear stress metrics in an idealized curved artery

We numerically investigate the influence of flow development on secondary flow patterns and subsequent wall shear stress distributions in a curved artery model, and we compute vascular metrics commonly used to assess variations in blood flow characteristics as it applies to arterial disease. We model a human artery with a simple, rigid 180-degree curved tube with circular cross-section and constant curvature, neglecting effects of taper, torsion and elasticity. High-fidelity numerical results are computed from an in-house discontinuous spectral element flow solver. The flow rate used in this study is physiological. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e. uniform). We observe and discuss differences in secondary flow patterns that emerge over the rapid acceleration and deceleration phases of the physiological waveform, and we directly connect the variation in intensity of these secondary flow patterns along the curvature to differences in the wall shear stress metrics for each entrance condition. Results indicate that decreased axial velocities under an undeveloped condition produce less intense secondary flow that, in turn, reduces both the oscillatory and multidirectional nature of the wall shear stress vector, and we link this effect to abnormalities in computed stress metrics. These results suggest potentially lower prevalence of disease in curvatures where entrance flow is rather undeveloped-a physiologically relevant result to further understand the influence of blood flow development on disease.

physics.flu-dyn↗

arXiv2022

Dynamics of the no-slip Galton board

The ideal Galton board and Lorentz gas billiard models have been studied numerically and analytically primarily in settings where friction and rotational velocity are neglected. We eliminate these simplifying assumptions and study the resulting dynamics of a more general model using no-slip collisions, in which particles rotate and may exchange linear and angular momentum at collisions while adhering to certain conservation laws. Using numerical experiments and phase portrait analysis we show that (in contrast to specular dispersing billiards) regularity persists when a small force is introduced while (consistent with specular billiards) under a stronger force new structure including invariant regions may arise. We also show analytically that with the introduction of an external force periodicity proliferates, with new types of periodic orbits not present in the no-force case.

math.DS↗

arXiv2021

The effect of entrance flow development on vortex formation and wall shear stress in a curved artery model

We numerically investigate the effect of entrance condition on the spatial and temporal evolution of multiple three-dimensional vortex pairs and wall shear stress distribution in a curved artery model. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e. uniform). We discuss the connection along the axial direction between regions of organized vorticity observed at various cross-sections of the model and compare results between the different entrance conditions. We model a human artery with a simple, rigid $180^\circ$ curved pipe with circular cross-section and constant curvature, neglecting effects of taper, torsion and elasticity. Numerical results are computed from a discontinuous high-order spectral element flow solver. The flow rate used in this study is physiological. We observe differences in secondary flow patterns, especially during the deceleration phase of the physiological waveform where multiple vortical structures of both Dean-type and Lyne-type coexist. We highlight the effect of the entrance condition on the formation of these structures and subsequent appearance of abnormal inner wall shear stresses - a potentially significant correlation since cardiovascular disease is known to progress along the inner wall of curved arteries under varying degrees of flow development.

physics.flu-dyn↗

arXiv2021

Accumulation points of the edit distance function

Given a hereditary property $\mathcal H$ of graphs and some $p\in[0,1]$, the edit distance function $\operatorname{ed}_{\mathcal H}(p)$ is (asymptotically) the maximum proportion of "edits" (edge-additions plus edge-deletions) necessary to transform any graph of density $p$ into a member of $\mathcal H$. For any fixed $p\in[0,1]$, $\operatorname{ed}_{\mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $p\in[0,1]$ for which infinitely many CRGs are required to compute $\operatorname{ed}_{\mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $\operatorname{ed}_{\mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points.

math.CO↗

arXiv2021

On the Number of Equilibria Balancing Newtonian Point Masses with a Central Force

We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular restricted $n$-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by $n+1$, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in $n$. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as ``ring configurations'', consisting of $n-1$ equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as $5n-5$ equilibria. We verify analytically that the ring configuration has at least $5n-5$ equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell's problem from electrostatics and the image counting problem from gravitational lensing.

math-ph↗

arXiv2021

The maximum number of 10- and 12-cycles in a planar graph

For a fixed planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle, the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this by answering the following question for $m\in\{5,6\}$, which is interesting in its own right: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ independent samples from $\mu$ form an $m$-cycle?

math.CO↗

arXiv2021

Counting paths, cycles and blow-ups in planar graphs

For a planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_7)\sim{4\over 27}n^4$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_6)\sim(n/3)^3$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_8)\sim(n/4)^4$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,K_4\{1\})\sim(n/6)^6$, where $K_4\{1\}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_{2m+1})$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_{2m})$ for $m\geq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.

math.CO↗

arXiv2020

User-friendly automatic transcription of low-resource languages: Plugging ESPnet into Elpis

This paper reports on progress integrating the speech recognition toolkit ESPnet into Elpis, a web front-end originally designed to provide access to the Kaldi automatic speech recognition toolkit. The goal of this work is to make end-to-end speech recognition models available to language workers via a user-friendly graphical interface. Encouraging results are reported on (i) development of an ESPnet recipe for use in Elpis, with preliminary results on data sets previously used for training acoustic models with the Persephone toolkit along with a new data set that had not previously been used in speech recognition, and (ii) incorporating ESPnet into Elpis along with UI enhancements and a CUDA-supported Dockerfile.

cs.CL↗

arXiv2020

Accuracy, Stability, and Performance Comparison between the Spectral Difference and Flux Reconstruction Schemes

We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.

physics.comp-ph↗

arXiv2019

Periodic words, common subsequences and frogs

Let $W^{(n)}$ be the $n$-letter word obtained by repeating a fixed word $W$, and let $R_n$ be a random $n$-letter word over the same alphabet. We show several results about the length of the longest common subsequence (LCS) between $W^{(n)}$ and $R_n$; in particular, we show that its expectation is $\gamma_W n-O(\sqrt{n})$ for an efficiently-computable constant $\gamma_W$. This is done by relating the problem to a new interacting particle system, which we dub "frog dynamics". In this system, the particles (`frogs') hop over one another in the order given by their labels. Stripped of the labeling, the frog dynamics reduces to a variant of the PushTASEP. In the special case when all symbols of $W$ are distinct, we obtain an explicit formula for the constant $\gamma_W$ and a closed-form expression for the stationary distribution of the associated frog dynamics. In addition, we propose new conjectures about the asymptotic of the LCS of a pair of random words. These conjectures are informed by computer experiments using a new heuristic algorithm to compute the LCS. Through our computations, we found periodic words that are more random-like than a random word, as measured by the LCS.

math.PR↗

arXiv2019

Restricted online Ramsey numbers of matchings and trees

Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or a large tree. We show that in any $t$-coloring of $E(K_n)$, Builder can locate a monochromatic matching on at least ${n-t+1\over t+1}$ edges by revealing at most $O(n\log t)$ edges. We show also that in any $3$-coloring of $E(K_n)$, Builder can locate a monochromatic tree on at least $n/2$ vertices by revealing at most $5n$ edges.

math.CO↗

arXiv2018

Regolith behavior under asteroid-level gravity conditions: low-velocity impact experiments

The dusty regolith covering the surfaces of asteroids and planetary satellites differs in size, shape, and composition from terrestrial soil particles and is subject to very different environmental conditions. Experimental studies of the response of planetary regolith in the relevant environmental conditions are thus necessary to facilitate future Solar System exploration activities. We combined the results and provided new data analysis elements for a series of impact experiments into simulated planetary regolith in low-gravity conditions using two experimental setups: the Physics of Regolith Impacts in Microgravity Experiment (PRIME) and the COLLisions Into Dust Experiment (COLLIDE). Results of these experimental campaigns found that there is a significant change in the regolith behavior with the gravity environment. In a 10-2g environment (Lunar g levels), only embedding of the impactor was observed and ejecta production was produced for most impacts at > 20 cm/s. Once at microgravity levels (<10-4g), the lowest impact energies also produced impactor rebound. In these microgravity conditions, ejecta started to be produced for impacts at > 10 cm/s. The measured ejecta speeds were lower than the ones measured at reduced-gravity levels, but the ejected masses were higher. The mean ejecta velocity shows a power-law dependence on the impact energy with an index of ~0.7. When projectile rebound occurred, we observed that its coefficients of restitution on the bed of regolith simulant decrease by a factor of 10 with increasing impact speeds from ~5 cm/s up to 100 cm/s. We could also observe an increased cohesion between the JSC-1 grains compared to the quartz sand targets.

astro-ph.EP↗

arXiv2018

Rolling and no-slip bouncing in cylinders

The purpose of this paper is to compare a classical non-holonomic system---a sphere rolling against the inner surface of a vertical cylinder under gravity---and a class of discrete dynamical systems known as no-slip billiards in similar configurations. A well-known notable feature of the non-holonomic system is that the rolling sphere does not fall; its height function is bounded and oscillates harmonically up and down. The central issue of the present work is whether similar bounded behavior can be observed in the no-slip billiard counterpart. Our main results are as follows: for circular cylinders in dimension $3$, the no-slip billiard has the bounded orbits property, and very closely approximates rolling motion, for a class of initial conditions which we call transversal rolling impact. When this condition does not hold, trajectories undergo vertical oscillations superimposed to an overall downward acceleration. Considering cylinders with different cross-section shapes, we show that no-slip billiards between two parallel hyperplanes in Euclidean space of arbitrary dimension are always bounded even under a constant force parallel to the plates; for general cylinders, when the orbit of the transverse system (a concept that depends on a factorization of the motion into transversal and longitudinal components) has period two---a very common occurrence in planar no-slip billiards---the motion in the longitudinal direction, under no forces, is generically not bounded. This is shown using a formula for a longitudinal linear drift that we prove in arbitrary dimensions. While the systems for which we can prove the existence of bounded orbits have relatively simple transverse dynamics, we also briefly explore numerically a no-slip billiard system, namely the stadium cylinder billiard, that can exhibit chaotic transversal dynamics.

math.DS↗

arXiv2018

Nearly orthogonal vectors and small antipodal spherical codes

How can $d+k$ vectors in $\mathbb{R}^d$ be arranged so that they are as close to orthogonal as possible? In particular, define $\theta(d,k):=\min_X\max_{x\neq y\in X}|\langle x,y\rangle|$ where the minimum is taken over all collections of $d+k$ unit vectors $X\subseteq\mathbb{R}^d$. In this paper, we focus on the case where $k$ is fixed and $d\to\infty$. In establishing bounds on $\theta(d,k)$, we find an intimate connection to the existence of systems of ${k+1\choose 2}$ equiangular lines in $\mathbb{R}^k$. Using this connection, we are able to pin down $\theta(d,k)$ whenever $k\in\{1,2,3,7,23\}$ and establish asymptotics for general $k$. The main tool is an upper bound on $\mathbb{E}_{x,y\sim\mu}|\langle x,y\rangle|$ whenever $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, which may be of independent interest. Our results translate naturally to the analogous question in $\mathbb{C}^d$. In this case, the question relates to the existence of systems of $k^2$ equiangular lines in $\mathbb{C}^k$, also known as SIC-POVM in physics literature.

math.CO↗