Recent developments in algebraic combinatorics

We survey three recent developments in algebraic combinatorics. The first is the theory of cluster algebras and the Laurent phenomenon of Sergey Fomin and Andrei Zelevinsky. The second is the construction of toric Schur functions and their application to computing three-point Gromov-Witten invariants, by Alexander Postnikov. The third development is the construction of intersection cohomology for nonrational fans by Paul Bressler and Valery Lunts and its application by Kalle Karu to the torich-vector of a nonrational polytope. We also briefly discuss the “half hard Lefschetz theorem” of Ed Swartz and its application to matroid complexes. Partially supported by NSF grant #DMS-9988459.     

The Linear Rational Pseudospectral Method with Preassigned Poles

We present a linear rational pseudospectral (collocation) method with preassigned poles for solving boundary value problems. It consists in attaching poles to the trial polynomial so as to make it a rational interpolant. Its convergence is proved by transforming the problem into an associated boundary value problem. Numerical examples demonstrate that the rational pseudospectral method is often more efficient than the polynomial method.     

The interlace polynomial of a graph

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable “interlace polynomial” for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.     

Reagent concentration effects in the REM resin solid phase synthesis of tertiary amines

The use of reagent concentration has resulted in increased rates for all stages of the REM resin synthesis of tertiary amines. These increases in rate translate into faster reaction times, higher yields and lower reagent consumption. Of the methods examined, the most successful was the use of perfluorous solvents, either alone or with a small amount of organic co-solvent.     

Bundles, Handcuffs, and Local Freedom

We answer a question of J. Anderson's by producing infinitely many commensurability classes of fibered hyperbolic 3-manifolds whose fundamental groups contain subgroups that are locally free and not free. These manifolds are obtained by performing 0–surgery on a collection of knots with the same properties.     

Verifiable Distributed Oblivious Transfer and Mobile Agent Security

The mobile agent is a fundamental building block of the mobile computing paradigm. In mobile agent security, oblivious transfer (OT) from a trusted party can be used to protect the agent’s privacy and the hosts’ privacy. In this paper, we introduce a new cryptographic primitive called Verifiable Distributed Oblivious Transfer (VDOT), which allows us to replace a single trusted party with a group of threshold trusted servers. The design of VDOT uses a novel technique called consistency verification of encrypted secret shares. VDOT protects the privacy of both the sender and the receiver against malicious attacks of the servers. We also show the design of a system to apply VDOT to protect the privacy of mobile agents. Our design partitions an agent into the general portion and the security-sensitive portion. We also implement the key components of our system. As far as we know, this is the first effort to implement a system that protects the privacy of mobile agents. Our preliminary evaluation shows that protecting mobile agents not only is possible, but also can be implemented efficiently. This work was supported in part by the DoD University Research Initiative (URI) program administered by the Office of Naval Research under grant N00014-01-1-0795. Sheng Zhong was supported by ONR grant N00014-01-1-0795 and NSF grants ANI-0207399 and CCR-TC-0208972. Yang Richard Yang was supported in part by NSF grant ANI-0207399. A preliminary version of this paper was presented at the DialM-POMC Joint Workshop on Foundations of Mobile Computing in 2003. Sheng Zhong received his Ph.D. in computer science from Yale University in the year of 2004. He holds an assistant professor position at SUNY Buffalo and is currently on leave for postdoctoral research at the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS). His research interests, on the practical side, are security and incentives in data mining, databases, and wireless networks. On the theoretical side, he is interested in cryptography and game theory. Yang Richard Yang is an Assistant Professor of Computer Science at Yale University. His research interests include computer networks, mobile computing, wireless networking, sensor networks, and network security. He leads the LAboratory of Networked Systems (LANS) at Yale. His recent awards include a Schlumberger Fellowship and a CAREER Award from the National Science Foundation. He received his B.E. degree from Tsinghua University (1993), and his M.S. and Ph.D. degrees from the University of Texas at Austin (1998 and 2001).     

Small Exotic Molecules with Bosonic Constituents

We study the stability and the properties of the ground state of neutral systems containing up to four positively charged bosons and their antiparticles. Examples are the di-pionium molecule (π ⁺ π ⁻)², which is almost identical to the positronium molecule (e ⁺ e ⁻)², the tri-pionium (π ⁺ π ⁻)³, and the quadri-pionium (π ⁺ π ⁻)⁴ molecules. We briefly compare our results on the energy to those on the large-N limit of (π ⁺ π ⁻) ^N . We also show that the annihilation probability can be calculated accurately with simple wave functions when one uses the generalized Schwinger rule. Received September 5, 1994; accepted for publication October 15, 1994     

High quality gate dielectrics formed by rapid thermal chemical vapor deposition of silane and nitrous oxide

In the present study, we have performed electrical characterization of oxides deposited via rapid thermal chemical vapor deposition using SiH₄ and N₂O. We have investigated the effect of temperature, pressure, and SiH₄ to N₂O ratio on the electrical and material properties of as-deposited films. We have found that as-deposited oxides deposited at low temperatures, low pressures, and with a low silane to nitrous oxide ratio of ～0.5% give good material and electrical properties. The as-deposited films are stoichiometric in nature and have high deposition rates. As-deposited films had very low D_it values, high breakdown fields, and excellent subthreshold swing. The leakage currents and metal oxide semiconductor field effect transistor current drive, although lower than thermal oxides, were found to be quite acceptable. We have also investigated the thickness dependence of the films and found that as the film thickness is reduced below 50Å, the reliability improves for all oxides including the silicon-rich deposited oxides.     

Non-homogeneous equations of the thermoacoustics: Thermally forced radial thermo-elastic oscillations of the visco-thermal fluid sphere

The purpose of this paper is to solve the coupled non-homogeneous equations derived from the basic classical theory of the acoustic propagation in visco-thermal fluid. The 1D radial distribution of pressure and temperature inside a spherical object was found. It is supposed that both acoustic and thermal conditions on a surface of the object are known. The starting equations are considered in the time domain and their solution was found without initial conditions for steady harmonic motion.