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A hyperbolic transformation of the torus is an example of a function that is Devaney chaotic; that is, it is topologically transitive and has dense periodic points. An irrational rotation of the torus, on the other hand, is not chaotic because it has no periodic points. We show that a hyperbolic transformation of the torus followed by a translation (an affine hyperbolic toral automorphism) has dense periodic points and maintains transitivity. As a consequence, affine toral automorphisms are chaotic, even when the translation is an irrational rotation.
We show that the rings of invariants for the three-dimensional modular representations of an elementary abelian -group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau (Transform. Groups 18 (2013), 1–22) for these representations.
Bootstrapping is a nonparametric statistical technique that can be used to estimate the sampling distribution of a statistic of interest. This paper focuses on implementation of bootstrapping in a new setting, where the data of interest are 3-dimensional rotations. Two measures of center, the mean rotation and spatial average, are considered, and bootstrap confidence regions for these measures are proposed. The developed techniques are then used in a materials science application, where precision is explored for measurements of crystal orientations obtained via electron backscatter diffraction.
We show that the 20-graph Heawood family, obtained by a combination of and moves on , is precisely the set of graphs of at most 21 edges that are minor-minimal with respect to the property “not -apex”. As a corollary, this gives a new proof that the 14 graphs obtained by moves on are the minor-minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven-graph Petersen family, obtained from , is the set of graphs of at most 17 edges that are minor-minimal with respect to the property “not apex”.
A partial differential equation (PDE)-based model combining the effects of surface electromigration and substrate wetting is developed for the analysis of the morphological instability of a monocrystalline metal film in a high temperature environment typical to operational conditions of microelectronic interconnects and nanoscale devices. The model accounts for the anisotropies of the atomic mobility and surface energy. The goal is to describe and understand the time-evolution of the shape of the film surface. The formulation of a nonlinear parabolic PDE problem for the height function of the film in the electric field is presented, followed by the results of the linear stability analysis of a planar surface. Computations of a fully nonlinear evolution equation are presented and discussed.
The size of the automorphism group of a compact Riemann surface of genus is bounded by . Curves with automorphism group of size equal to this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group . We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple.
Ochem, Rampersad, and Shallit gave various examples of infinite words avoiding what they called approximate repetitions. An approximate repetition is a factor of the form , where and are close to being identical. In their work, they measured the similarity of and using either the Hamming distance or the edit distance. In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence. Our principal technique is the so-called “entropy compression” method, which has its origins in Moser and Tardos’s algorithmic version of the Lovász local lemma.
A simple and connected -vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, including cycle pendant stars, cycle chains, prisms, and generalized books.
The Jones polynomial for knots and links was a breakthrough discovery in the early 1980s. Since then, it’s been generalized in many ways; in particular, by considering knots and links which live in thickened surfaces and by allowing arcs between punctures or marked points on the boundary of the surface. One such generalization was recently introduced by Roger and Yang and has connections with hyperbolic geometry. We provide generators and relations for Roger and Yang’s Kauffman bracket arc algebra of the torus with one puncture and the sphere with three or fewer punctures.
Tessellate the plane into rows of hexagons. Consider a subset of rows of these hexagons, each row containing hexagons, forming a rhombus-shaped chessboard of spaces. Two kings placed on the board are said to “attack” each other if their spaces share a side or corner. Placing kings in alternating spaces of every other row results in an arrangement where no two of the kings are attacking each other. According to our specific distance metric, is in fact the largest number of kings that can be placed on such a board with no two kings attacking one another, for a maximum “density” of . We consider a generalization of this maximum density problem, instead requiring that no king attacks more than other kings for . For instance when the density is at most . For each we give constructive lower bounds on the density, and use systems of inequalities and discharging arguments to yield upper bounds, where the bounds match in most cases.
The gonality of a graph is a discrete analogue of the similarly named geometric invariant of algebraic curves. Motivated by recent progress in Brill–Noether theory for graphs, we study the gonality of random graphs. In particular, we show that the gonality of a random graph is asymptotic to the number of vertices.