Ideal Structure of the Kauffman and Related Monoids

The generators of the Temperley-Lieb algebra generate a monoid with an appealing geometric representation. It has been much studied, notably by Louis Kauffman. Borisavljevi?, Do?en, and Petri? gave a complete proof of its abstract presentation by generators and relations, and suggested the name “Kauffman monoid”. We bring the theory of semigroups to the study of a certain finite homomorphic image of the Kauffman monoid. We show the homomorphic image (the Jones monoid) to be a combinatorial and regular *-semigroup with linearly ordered ideals. The Kauffman monoid is explicitly described in terms of the Jones monoid and a purely combinatorial numerical function. We use this to describe the ideal structure of the Kauffman monoid and two other of its homomorphic images.     

On the representation of long-range potentials by a [1, 0] Padé approximant

The utility of the [1, 0] Padé approximant in the determination of long-range interaction constants is investigated using the recently calculated Rydberg-Klein-Rees potentials for the B³Π_Ou ⁺ states of I₂, Br₂ and Cl₂. The conditions under which the [1, 0] Padé approximant is a valid representation of the long-range potential outside the electron-overlap region are studied in detail, using as model systems the B states of I₂, Br₂ and Cl₂, and the X¹Σg⁺, B¹Σu⁺ and B′¹Σu⁺ states of H₂. In general, the validity of the representation depends on the nature of the potential, and is governed by

if the long-range potential can be adequately represented by a maximum number of three terms (n, m, m1) in the inverse-power expansion.     

Design and fabrication of antireflective GaN subwavelength grating structures using periodic silica sphere monolayer array patterning

We designed and fabricated gallium nitride (GaN) subwavelength grating (SWG) structures on GaN/sapphire via patterning using the periodic silica sphere monolayer array as an etch mask and a subsequent dry etching for efficient antireflection coatings. Theoretical optimization of GaN SWG structures was performed in terms of their geometrical parameters by the rigorous coupled-wave analysis simulation using a theoretical structural model. The bullet-like parabola-shaped SWGs with a large height-to-diameter ratio (R _H/D) yielded good broadband and wide-angle antireflective surface properties. Considering the R _H/D, the GaN SWG structure using 320-nm silica spheres theoretically and experimentally exhibited the most efficient antireflection property because it provided a linearly graded effective refractive index profile with relatively long relaxation length. For various geometries of the fabricated GaN SWGs on GaN/sapphire, the calculated reflectance results showed a similar tendency with the experimental results.     

Fast Inference for the Latent Space Network Model Using a Case-Control Approximate Likelihood

Multilevel modeling is a popular statistical technique for analyzing data in hierarchical format, and thus naturally fits within a distributed database framework. We consider the computational aspects of multilevel modeling across distributed databases. In addition, we consider these aspects under a generalization of the multilevel model where the distributed groups (or databases) are allowed to specify different models at both level-1 (individual) and level-2 (group). For a variety of scenarios, we develop the distributed computation algorithm for two-step least squares (LS) estimators and also for iterative MLE estimators of the parameters of interest; in particular, we determine the required data structure at each computing site, the necessary information (original data, cross-product matrices, coefficient vectors), and the order in which such information needs to be passed between sites. Finally, we discuss recursive updating, fault tolerance, and security issues.     

Generalized higher order spt-functions

We give a new generalization of the spt-function of G.E. Andrews, namely $\operatorname {Spt}_{j}(n)$ , and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function $\operatorname {spt}_{k}(n)$ , due to F.G. Garvan, to ${}_{j\!}\operatorname {spt}_{k}(n)$ , thus providing a two-fold generalization of $\operatorname {spt}(n)$ , and give its combinatorial interpretation. Lastly, we show how the positivity of _j spt _k (n) can be used to generalize Garvan’s inequality between rank and crank moments to the moments of j-rank and (j+1)-rank.     

Equations of state in the Brans–Dicke cosmology

We investigate the Brans–Dicke (BD) theory with the potential as cosmological model to explain the present accelerating universe. In this work, we consider the BD field as a perfect fluid with the energy density and pressure in the Jordan frame. Introducing the power-law potential and the interaction with the cold dark matter, we obtain the phantom divide which is confirmed by the native and effective equation of state. Also we can describe the metric f(R) gravity with an appropriate potential, which shows a future crossing of the phantom divide in viable f(R) gravity models when employing the native and effective equations of state.     

Radial Hetero[5]catenanes: Peripheral Isomer Sequences of the Interlocked Macrocycles

A pair of radial [5]catenanes, with either an isomeric cyclic ‐AABB‐ or ‐ABAB‐ type sequence of the interlocked β‐cyclodextrin (β‐CD) and cucurbit[6]uril (CB[6]) units, has been efficiently synthesized. Because of a marked difference in the binding strength and interlocking sequence of the peripheral macrocycles, interesting sequence‐dependent properties, characteristic of mechanically bonded macrocycles, were realized. Variable‐temperature ¹H NMR studies showed that the ‐ABAB‐ isomer has a more independent β‐CD dynamic, whereas the β‐CD motions in the ‐AABB‐ isomer are coupled. Dynamics of the pH‐insensitive β‐CD can also be further modulated upon base‐triggered mobilization of the CB[6]. These unique properties of the mechanical bond expressed in a sequence‐specific fashion and the transmission of the control on the macrocycle dynamics from one interlocked component to another, highlight the potential of similar complex hetero[n]catenanes in the design of advanced, multicomponent molecular machines.     

Classification of complete regularities for finite topological spaces

We show that there are exactly five different classes of complete regularity determined by finite topological spaces.