On a rudimentary model of the W(001) surface instability and reconstruction

A simple qualitative theory of the reconstruction of the W (001) surface based on the assumption of an important role of Shockley surface states from the vicinity of the Fermi energy in the electron-phonon interaction is presented. While translational symmetry chooses state coupling via the pseudo-Jahn-Teller effect, the local symmetry determines the magnitude of the corresponding matrix elements. It appears that the in-plane reconstruction modes M?₅ and X?₃ are more promising whereas the buckled modes seem unfavourable since they lead to an ionic superstructure on the surface.     

Design and synthesis of pyrankacin: a pyranmycin class of broad-spectrum aminoglycoside antibiotic

[STRUCTURE: SEE TEXT] A novel broad-spectrum aminoglycoside antibiotic, pyrankacin, has been prepared. In addition to the synthetic innovation in dideoxygenation and regioselective Staudinger reduction, we have obtained prominent antibacterial activity against several clinically important pathogens in the course of this work.     

Synthesis of 3‐(2‐hydroxy‐1‐phenylethyl)‐ and 3‐(2‐hydroxy‐2‐phenylethyl)adenine,DNA adducts derived from styrene

3‐(2‐Hydroxy‐2‐phenylethyl)‐ and 3‐(2‐hydroxy‐1‐phenylethyl)adenine, DNA adducts derived from styrene, along with their 9‐substituted analogues were prepared by alkylation of 8‐bromoadenine with corresponding allyl‐protected bromohydrins followed by a new deallylation procedure using tetrakis(triphenylphosphine)palladium catalyzed reductive cleavage by poly(methylhydrosiloxane) in the presence of p‐toluenesulphonic acid. This novel procedure proved to be useful for purine derivatives, which were resistant to other deallylation protocols. Structure of positional isomers was assigned using 2D NMR experiments HMBC and HMQC.     

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Numerical Computations for the Schramm-Loewner Evolution

We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems.     

Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc.     

Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

Let CC _d,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let AC _d,k be the largest order in an Abelian Cayley graph for given d and k. We show that \({CC_{d,2} \geq \frac{13}{36} (d + 2)(d -4)}\) for any d= 6p?2 where p is a prime such that \({p \neq 13}\) , \({p \not\equiv 1}\) (mod 13), and \({AC_{d,3} \geq \frac{9}{128} (d + 3)^2(d - 5)}\) for d = 8q?3 where q is a prime power.     

A Sharp Threshold for a Modified Bootstrap Percolation with Recovery

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: ‘healthy’ or ‘infected’ and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.