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.     

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     

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.     

Thermotropic liquid-crystalline behaviour from mixtures of non-mesogenic molecules

A class of non-mesogenic molecules, which as mixtures exhibit monotropic nematic phases, is described.