A FORTRAN program for cluster enumeration

A simple FORTRAN program based primarily on the algorithm of Martin is presented for enumerating isolated connected clusters of up to a given specified number of particles on the square lattice. A brief explanation of the workings of the program is also provided to facilitate its use by those interested in this problem. To order 15, the program enumerates clusters at an average rate of approximately 63,000/sec when implemented on an IBM 370/168 with the extended optimizing compiler.Supported in part by grants from the ARO, NSF, and ONR.     

Nucleation and metastability in three-dimensional Ising models

We present Monte Carlo experiments on nucleation theory in the nearest-neighbor three-dimensional Ising model and in Ising models with long-range interactions. For the nearest-neighbor model, our results are compatible with the classical nucleation theory (CNT) for low temperatures, while for the long-range model a breakdown of the CNT was observed near the mean-field spinodal. A new droplet model and a zeroth-order theory of droplet growth are also presented.Supported in part by grants from ARO, ONR, and NSF.     

Progress in metric-affine gauge theories of gravity with local scale invariance

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection acquire the status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective low-energy theory.Based on a plenary talk given by one of us (EWM) at the 53rd annual meeting of the Deutsche Physikalische Gesellschaft in Bonn on March 14, 1989.Supported by the German-Israeli Foundation for Scientific Research and Development (GIF), Jerusalem and Munich.Supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, project He 528/12-1.Supported in part by DOE Grant DE-FG05-85-ER40200.     

The large time stability of sound waves

We demonstrate the existence of solutions to the full 3×3 system of compressible Euler equations in one space dimension, up to an arbitrary timeT>0, in the case when the initial data has arbitrarily large total variation, and sufficiently small supnorm. The result applies to periodic solutions of the Euler equations, a nonlinear model for sound wave propagation in gas dynamics. Our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem. This length scale plays no role in 2×2 systems, (or any system possessing a full set of Riemann coordinates), nor in the small total variation problem forn×n systems, the cases originally addressed by Glimm in 1965. Recent work by a number of authors has demonstrated that when the total variation is sufficiently large, solutions of 3×3 systems of conservation laws can in general blow up in finite time, (independent of the supnorm), due to amplifying instabilities created by the non-trivial Lie algebra of the vector fields that define the elementary waves. For the large total variation problem, there is an interaction between large scale effects that amplify and small scale effects that are stable, and we show that the length scale on which this interaction occurs isd. In the limitd, we recover Glimm's theorem, and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp.Supported in part by NSF Applied Mathematics grant numbers DMS-92-06631, DMS-95000694, in part by ONR, US Navy grant number N00014-94-1-0691, A Guggenheim fellowship, and by the Institute of Theoretical Dynamics, UC-Davis.Partially supported by DOE grant number DE-FG02-88ER25053 while at the Courant Institute, and by NSF grant number DMS-9201581 and DOE grant number DE-FG02-90ER25084.     

The GHS and other correlation inequalities for a class of even ferromagnets

We prove the GHS inequality for families of random variables which arise in certain ferromagnetic models of statistical mechanics and quantum field theory. These include spin –1/2 Ising models, ⁴ field theories, and other continuous spin models. The proofs are based on the properties of a classG ^– of probability measures which contains all measures of the form const exp(–V(x))dx, whereV is even and continuously differentiable anddV/dx is convex on [0, ). A new proof of the GKS inequalities using similar ideas is also given.Supported in part by National Science Foundation Grant MPS 71-02838 A 04.Supported by National Science Foundation Grant MPS 74-24696.Supported in part by National Science Foundation Grant MPS 74-04870.     

AnS matrix theory for classical nonlinear physics

The basic concepts appropriate for anS matrix theory for classical nonlinear physics are formulated here. These concepts are illustrated by a discussion of shock wave diffraction patterns. Other information concerning solutions of non-linear conservation laws is surveyed, so that a coherent picture of this theory can be seen. Within thisS matrix framework, a number of open problems as well as a few solved ones will be discussed.Dedicated to John A. Wheeler on the occasion of his 75th birthday.Supported in part by the National Science Foundation, grant DMS-831229.Supported in part by the Aplied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DE-AC02-76ER03077.Supported in part by the Army Research Office, grant DAAG29-83-K-0007.Work supported by the U.S. Department of Energy.     

The Classification of Non-Local Chiral CFT with <Emphasis Type="Italic">c</Emphasis> < 1

All non-local but relatively local irreducible extensions of Virasoro chiral CFTs with c < 1 are classified. The classification, which is a prerequisite for the classification of local c < 1 boundary CFTs on a two-dimensional half-space, turns out to be 1 to 1 with certain pairs of A-D-E graphs with distinguished vertices. Dedicated to Hans-Jürgen Borchers on the occasion of his 80th birthday Supported in part by JSPS. Supported in part by EU network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-00280. Supported in part by GNAMPA and MIUR.     

The baker's transformation is not embeddable

The baker's transformation is one of the earliest and simplest examples of a bijective mixing transformation. By determining its cycle structure, we show that this transformation is not embeddable in any flow, i.e., one-parameter semigroup, on the nonnegative rationals and,a fortiori, not emdeddable in any flow on the reals.Dedicated to the memory of Charles H. Randall, colleague and friend.Research supported in part by ONR Contract N-00014-87-K-0379.     

Statistical Mechanics of Complex Systems for Pattern Identification

This paper presents a statistical mechanics concept for identification of behavioral patterns in complex systems based on measurements (e.g., time series data) of macroscopically observable parameters and their operational characteristics. The tools of statistical mechanics, which provide a link between the microscopic (i.e., detailed) and macroscopic (i.e., aggregated) properties of a complex system are used to capture the emerging information and to identify the quasi-stationary evolution of behavioral patterns. The underlying theory is built upon thermodynamic formalism of symbol sequences in the setting of a generalized Ising model (GIM) of lattice-spin systems. In this context, transfer matrix analysis facilitates construction of pattern vectors from observed sequences. The proposed concept is experimentally validated on a richly instrumented laboratory apparatus that is operated under oscillating load for identification of evolving microstructural changes in polycrystalline alloys. This work has been supported in part by the U.S. Army Research Office (ARO) under Grant No. W911NF-07-1-0376, by NASA under Cooperative Agreement No. NNX07AK49A, and by the U.S. Office of Naval Research (ONR) under Grant No. N00014-08-1-0380. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsoring agencies.     

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.Research supported in part by the NSF, Contract No. DMS 89-05205Research supported in part by the ONR, Contract No. DOD-C-N-00014-88-K-0082Research supported in part by the DOE, Grant No. DE-FG02-88ER25065Research supported in part by the U.K. Science and Engineering Research Council     

Space-dependent dirac operators and effective quantum field theory for fermions

For the operator, wherem(x) can change sign, we develop a cluster expansion for computing the determinant and Green's functions. We use a local chiral transformation to relate the space-dependent case to the ordinary Dirac operator.Supported in part by National Science Foundation grant PHY/DMS 88-16214Supported in part by National Science Foundation grants DMS 90-08827 and DMS 88-580873Supported in part by National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

On Transonic Shocks in a Nozzle with Variable End Pressures

In the book, Courant and Friedrichs (Supersonic Flow and Shock Waves. New York: Interscience Publishers, 1948) described the following transonic shock phenomena in a de Laval nozzle: Given the appropriately large receiver pressure p _r , if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed. The position and the strength of the shock front are automatically adjusted so that the end pressure at the exit becomes p _r . When the end pressure p _r varies and lies in an appropriate scope, in general, it is expected that a curved transonic shock is still formed in a nozzle. In this paper, we solve this problem for the two-dimensional steady Euler system with a variable exit pressure in a nozzle whose divergent part is an angular sector. Both existence and uniqueness are established. Supported by the National Natural Science Foundation of China (No.10571082) and the National Basic Research Programm of China (No.2006CB805902). Supported in part by Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research Grants CUHK4028/04P, CUHK4040/06P and RGC Central Allocation Grant CA05/06.SC01.     

From Dynamical to Non-Dynamical Twists

We provide a construction which gives a twisting element for a universal enveloping algebra starting from a certain dynamical twist. This construction is a quantization of the analogous quasi-classical process given in [Karolinsky and Stolin, Lett. Math. Phys. 60 (2002), 257–274]. In particular, we reduce the computation of the twisting element for the classical r-matrix constructed from the Frobenius algebra , the maximal parabolic subalgebra of related to the simple root _n-1, to the computation of the universal dynamical twist for .*Supported in part by the Royal Swedish Academy of Sciences**Supported in part by RFFI grant 02-01-00085a and CRDF grant RM1-2334-MO-02Mathematical Subject Classifications (2000). 17B35, 17B10, 17B20.     

Localized excitation transport on substitutionally disordered lattices

Average-T-matrix and coherent medium theories are used to study the motion of localized excitations on Substitutionally disordered lattices. We derive equations which relate coherent medium results for bond and site averaging and show how these reduce to the two-body solution results of Gochanour, Andersen, and Fayer. Numerical results forP ₀(t), the probability of remaining at the origin for two-dimensional nearest-neighbor lattices are presented.Supported in part by the National Science Foundation. (CHE81-00407).     

Updating the effective range expansion of low-energyN \bar N-scattering

Low energyN¯N scattering is reanalyzed using the new data on annihilation cross-sections from CERN [1], the¯np-absorption measurement from BNL [2] and the annihilation width of the 2p state in antiprotonic hydrogen [3]. Imposing the condition that the imaginary part of the effective range is negative we find low energy scattering parameters which interpolate between the existing ITEP [4] and CERN/Heidelberg [5] values.Supported by the BMFT-grant 06 HD 756     

The smooth cohomology ofN=2 supersymmetric Landau-Ginzburg field theories

We compute the smooth cohomology (both unrestricted and compactly supported) of the supercharge of an ultraviolet cutoffN=2 supersymmetric Landau-Ginzburg field theory.Supported in part by the National Science Foundation under grant DMS-9206936.Supported in part by the Department of Energy under grant DE-FG02-88ER25065.     

Loschmidt's and Zermelo's paradoxes do not exist

A strict operational (i.e., informational) analysis of the meaning of preparing a system to realize the paradoxes of Loschmidt or Zermelo is made. Where reversal or recurrence are operationally realizable, no contradiction with the irreversible nature of macroscopic operations occurs. Paradox results either from neglecting irreversible phenomena in the means for preparing a reversed state, or from confusing elements or ensembles, which are meaningful in microstate language but meaningless operationally, with preparable macrostates, whoserepresentation in microstate language is an ensemble whose very definition is incompatible with that of any paradox-generating element or ensemble,Supported in part by NSF Grant No. GN 534.1.     

Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions

We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.Supported by NSF Grant MCS-8301364, ONR Contract N00014-81-K-0012 and a Fellowship from John S. Guggenheim Memorial Foundation     

Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point. Supported in part by the Swiss National Science Foundation. Supported in part by the Swiss National Science Foundation, the programme SPECT and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652).     

Parallel transport in the determinant line bundle: The zero index case

For a product family of invertible Weyl operators on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator onR×X.Supported in party by NSF Grants PHY8605978 and PHY-82-15249 and the Robert A. Welch FoundationSupposed in part by NSF Grant PHY-82-15249