arXiv subjects

Christopher Cox

Publications and source records attributed to Christopher Cox.

31 records · Page 2Linked to original sources

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↗

arXiv2017

Ergodicity in umbrella billiards

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is $0$. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to non-ergodic lemon and moon billiards, small increases in the new parameter transform elliptic $2$-periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.

math.DS↗

arXiv2016

Stability of periodic orbits in no-slip billiards

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, no-slip. This leads to the study of no-slip billiards--planar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutkin) and linear stability of periodic orbits for a Sinai-type billiard (Wojtkowski) for no-slip billiards. We show among other facts that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than $π$, no-slip billiard dynamics will always contain elliptic period-$2$ orbits; (ii) polygonal no-slip billiards always admit small invariant open sets and thus cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the no-slip version of a Sinai billiard must contain linearly stable periodic orbits of period $2$ and, more generally, we provide a curvature threshold at which a commonly occurring period-$2$ orbit shifts from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in a class of polygonal billiards.

math.DS↗

arXiv2016

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.

math.CO↗

arXiv2015

(4,2)-choosability of planar graphs with forbidden structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at most $s$ common colors in their lists. Every planar graph is $(4,1)$-choosable, but there exist planar graphs that are not $(4,3)$-choosable. It is an open question whether planar graphs are always $(4,2)$-choosable. A chorded $\ell$-cycle is an $\ell$-cycle with one additional edge. We demonstrate for each $\ell \in \{5,6,7\}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $\ell$-cycles.

math.CO↗

arXiv2015

Counting prime juggling patterns

Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length $n$ is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the state graph). For the case of $b=2$ balls we give an expression for the number of prime juggling patterns of length $n$ by establishing a connection with partitions of $n$ into distinct parts. From this we show the number of two-ball prime juggling patterns of length $n$ is $(γ-o(1))2^n$ where $γ=1.32963879259...$. For larger $b$ we show there are at least $b^{n-1}$ prime cycles of length $n$.

math.CO↗

arXiv2015

A generalization of Eulerian numbers via rook placements

We consider a generalization of Eulerian numbers which count the number of placements of $cn$ "rooks" on an $n\times n$ board where there are exactly $c$ rooks in each row and each column, and exactly $k$ rooks below the main diagonal. The standard Eulerian numbers correspond to the case $c=1$. We show that for any $c$ the resulting numbers are symmetric and give generating functions of these numbers for small values of $k$.

math.CO↗

arXiv2015

Chvátal-type results for degree sequence Ramsey numbers

A sequence of nonnegative integers $π=(d_1,d_2,...,d_n)$ is graphic if there is a (simple) graph $G$ of order $n$ having degree sequence $π$. In this case, $G$ is said to realize or be a realization of $π$. Given a graph $H$, a graphic sequence $π$ is potentially $H$-graphic if there is some realization of $π$ that contains $H$ as a subgraph. In this paper, we consider a degree sequence analogue to classical graph Ramsey numbers. For graphs $H_1$ and $H_2$, the potential-Ramsey number $r_{pot}(H_1,H_2)$ is the minimum integer $N$ such that for any $N$-term graphic sequence $π$, either $π$ is potentially $H_1$-graphic or the complementary sequence $\overlineπ=(N-1-d_N,\dots, N-1-d_1)$ is potentially $H_2$-graphic. We prove that if $s\ge 2$ is an integer and $T_t$ is a tree of order $t> 7(s-2)$, then $$r_{pot}(K_s, T_t) = t+s-2.$$ This result, which is best possible up to the bound on $t$, is a degree sequence analogue to a classical 1977 result of Chvátal on the graph Ramsey number of trees vs. cliques. To obtain this theorem, we prove a sharp condition that ensures an arbitrary graph packs with a forest, which is likely to be of independent interest.

math.CO↗

arXiv2015

Differential Geometry of Rigid Bodies Collisions and Non-standard Billiards

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires boundary conditions that specify how orbits should be continued after collisions. A boundary condition is the assignment of a collision map at each tangent space on the boundary of $M$ that gives the post-collision state of the system as a function of the pre-collision state. Our main result is a complete description of the space of linear collision maps satisfying energy and (linear and angular) momentum conservation, time reversibility, and the natural requirement that impulse forces only act at the point of contact of the colliding bodies. These assumptions can be stated in geometric language by making explicit a family of vector subbundles of the tangent bundle to the boundary of $M$: the diagonal, non-slipping, and impulse subbundles. Collision maps at a boundary configuration are shown to be the isometric involutions that restrict to the identity on the non-slipping subspace. The space of such maps is naturally identified with the union of Grassmannians of $k$-dimensional subspaces of $\mathbb{R}^{n-1}$, $0\leq k\leq n-1$, each subspace specifying the directions of contact roughness. We then consider non-standard billiard systems, defined by fixing the position of one of the bodies and allowing boundary conditions different from specular reflection. We also make a few observations of a dynamical nature for simple examples of non-standard billiards and provide a sufficient condition for the billiard map on the space of boundary states to preserve the canonical (Liouville) measure on constant energy hypersurfaces.

math.DS↗

arXiv2014

Ordered Ramsey numbers of loose paths and matchings

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height one less than the maximum degree. We also extend theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers, arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to provide upper bounds on the ordered Ramsey number of $k$-uniform matchings under certain orderings.

math.CO↗

arXiv2014

Classification with Sparse Overlapping Groups

Classification with a sparsity constraint on the solution plays a central role in many high dimensional machine learning applications. In some cases, the features can be grouped together so that entire subsets of features can be selected or not selected. In many applications, however, this can be too restrictive. In this paper, we are interested in a less restrictive form of structured sparse feature selection: we assume that while features can be grouped according to some notion of similarity, not all features in a group need be selected for the task at hand. When the groups are comprised of disjoint sets of features, this is sometimes referred to as the "sparse group" lasso, and it allows for working with a richer class of models than traditional group lasso methods. Our framework generalizes conventional sparse group lasso further by allowing for overlapping groups, an additional flexiblity needed in many applications and one that presents further challenges. The main contribution of this paper is a new procedure called Sparse Overlapping Group (SOG) lasso, a convex optimization program that automatically selects similar features for classification in high dimensions. We establish model selection error bounds for SOGlasso classification problems under a fairly general setting. In particular, the error bounds are the first such results for classification using the sparse group lasso. Furthermore, the general SOGlasso bound specializes to results for the lasso and the group lasso, some known and some new. The SOGlasso is motivated by multi-subject fMRI studies in which functional activity is classified using brain voxels as features, source localization problems in Magnetoencephalography (MEG), and analyzing gene activation patterns in microarray data analysis. Experiments with real and synthetic data demonstrate the advantages of SOGlasso compared to the lasso and group lasso.

cs.LG↗

arXiv2013

Sparse Overlapping Sets Lasso for Multitask Learning and its Application to fMRI Analysis

Multitask learning can be effective when features useful in one task are also useful for other tasks, and the group lasso is a standard method for selecting a common subset of features. In this paper, we are interested in a less restrictive form of multitask learning, wherein (1) the available features can be organized into subsets according to a notion of similarity and (2) features useful in one task are similar, but not necessarily identical, to the features best suited for other tasks. The main contribution of this paper is a new procedure called Sparse Overlapping Sets (SOS) lasso, a convex optimization that automatically selects similar features for related learning tasks. Error bounds are derived for SOSlasso and its consistency is established for squared error loss. In particular, SOSlasso is motivated by multi- subject fMRI studies in which functional activity is classified using brain voxels as features. Experiments with real and synthetic data demonstrate the advantages of SOSlasso compared to the lasso and group lasso.

cs.LG↗