Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex

Orbits Around Black Holes in Triaxial Nuclei

We discuss the properties of orbits within the influence sphere of a supermassive black hole (BH), in the case that the surrounding star cluster is nonaxisymmetric. There are four major orbit families; one of these, the pyramid orbits, have the interesting property that they can approach arbitrarily closely to the BH. We derive the orbit-averaged equations of motion and show that in the limit of weak triaxiality, the pyramid orbits are integrable: the motion consists of a two-dimensional libration of the major axis of the orbit about the short axis of the triaxial figure, with eccentricity varying as a function of the two orientation angles, and reaching unity at the corners. Because pyramid orbits occupy the lowest angular momentum regions of phase space, they compete with collisional loss cone repopulation and with resonant relaxation in supplying matter to BHs. General relativistic advance of the periapse dominates the precession for sufficiently eccentric orbits, and we show that relativity imposes an upper limit to the eccentricity: roughly the value at which the relativistic precession time is equal to the time for torques to change the angular momentum. We argue that this upper limit to the eccentricity should apply also to evolution driven by resonant relaxation, with potentially important consequences for the rate of extreme-mass-ratio inspirals in low-luminosity galaxies. In giant galaxies, we show that capture of stars on pyramid orbits can dominate the feeding of BHs, at least until such a time as the pyramid orbits are depleted; however this time can be of order a Hubble time.Comment: 20 pages, 15 figure

Boundedness in languages of infinite words

We define a new class of languages of $\omega$-words, strictly extending $\omega$-regular languages. One way to present this new class is by a type of regular expressions. The new expressions are an extension of $\omega$-regular expressions where two new variants of the Kleene star $L^*$ are added: $L^B$ and $L^S$. These new exponents are used to say that parts of the input word have bounded size, and that parts of the input can have arbitrarily large sizes, respectively. For instance, the expression $(a^Bb)^\omega$ represents the language of infinite words over the letters $a,b$ where there is a common bound on the number of consecutive letters $a$. The expression $(a^Sb)^\omega$ represents a similar language, but this time the distance between consecutive $b$'s is required to tend toward the infinite. We develop a theory for these languages, with a focus on decidability and closure. We define an equivalent automaton model, extending B\"uchi automata. The main technical result is a complementation lemma that works for languages where only one type of exponent---either $L^B$ or $L^S$---is used. We use the closure and decidability results to obtain partial decidability results for the logic MSOLB, a logic obtained by extending monadic second-order logic with new quantifiers that speak about the size of sets

Velocity renormalization and Dirac cone multiplication in graphene superlattices with various barrier edge geometries

The electronic properties of one-dimensional graphene superlattices strongly depend on the atomic size and orientation of the 1D external periodic potential. Using a tight-binding approach, we show that the armchair and zigzag directions in these superlattices have a different impact on the renormalization of the anisotropic velocity of the charge carriers. For symmetric potential barriers, the velocity perpendicular to the barrier is modified for the armchair direction while remaining unchanged in the zigzag case. For asymmetric barriers, the initial symmetry between the forward and backward momentum with respect to the Dirac cone symmetry is broken for the velocity perpendicular (armchair case) or parallel (zigzag case) to the barriers. At last, Dirac cone multiplication at the charge neutrality point occurs only for the zigzag geometry. In contrast, band gaps appear in the electronic structure of the graphene superlattice with barrier in the armchair direction.Comment: 13 pages, 14 figure

Boxy/peanut/X bulges, barlenses and the thick part of galactic bars: What are they and how did they form?

Bars have a complex three-dimensional shape. In particular their inner part is vertically much thicker than the parts further out. Viewed edge-on, the thick part of the bar is what is commonly known as a boxy-, peanut- or X- bulge and viewed face-on it is referred to as a barlens. These components are due to disc and bar instabilities and are composed of disc material. I review here their formation, evolution and dynamics, using simulations, orbital structure theory and comparisons to observations.Comment: 21 pages, 7 figures, invited review to appear in "Galactic Bulges", E. Laurikainen, R. Peletier, D. Gadotti, (eds.), Springe

The star height of reset-free events and strictly locally testable events

An algorithm is presented for determining the star height of reset-free events and strictly locally testable events

The Crabb Site (41TT650), a Prehistoric Caddo Site on Tankersley Creek, Titus County, Texas

In this article, we discuss the archaeological findings at the Crabb site (411T650), a prehistoric Caddo settlement on an upland remnant/knoll in the Tankersley Creek floodplain in Titus County, Texas. Tankersley Creek is one of the principal tributaries of Big Cypress Creek, and it flows south past the Crabb site to merge with the larger creek a few miles downstream. This part of Titus County is in the Pineywoods, an area with abundant moisture and a warm, moderate climate in modem times, with an overstory vegetation of pines and a variety of hardwoods (principally several species of oaks and hickory). Along Tankersley Creek itself, the vegetation would have been dominated by more mesic hardwoods that tolerated occasional inundation from seasonal floods and standing water in the floodplain, such as water oaks and sweetgum trees, bushes, vines, and shrubs. The Tankersley Creek valley is well-known for its archaeological resources, particularly those of prehistoric Caddo age, and numerous prehistoric Caddo sites have been identified and recorded in the valley in the last 20 years. The Crabb site is one of several dozen prehistoric Caddo sites known in the creek valley, and one where limited investigations over the last 10 years have recovered interesting archaeological information about an intensive Caddo settlement that was apparently occupied at least 600-700 years ago, if not longer. Additional archaeological work is planned at the Crabb site should time and money become available to conduct more work at this significant site, but we believe it is important for future Caddo archaeological researchers that we make available now the archaeological information that has been obtained to date from the site

The Stressors of the Military Lifestyle and the Services Available to Help Military Families

This quantitative, descriptive study focuses on the effectiveness of the various services available for military families from the perspective of Rhode Island military residents. A review of the literature outlines the major stressors of the military lifestyle. The services available to help military families deal with these stressors are analyzed in the study. Five Rhode Island military families participated in this study. Results revealed overall satisfaction with the resources available through the military itself, dissatisfaction with services available through other areas, such as schools, churches, or organizations, the necessity of aid during the reintegration experience, the minimal usage of the services available to military families, the need for aid when going through a problem with a child, and effectiveness of the services that are used. Implications from this study include the need for future research on the potential reasons as to why these services are used so infrequently and for such a limited amount of time. More research also needs to be done about what military families need specifically during the reintegration period, especially since the War on Terror is coming to an end and many service members will be returning home soon. This study proved the effectiveness and necessity for services to remain in action to help the service members and their families that sacrifice so much for the good of the United States