Triviality of Hierarchical Ising Model¶in Four Dimensions

Existence of critical renormalization group trajectory for a hierarchical Ising model in 4 dimensions is shown. After 70 iterations of renormalization group transformations, the critical Ising model is mapped into a vicinity of the Gaussian fixed point. Convergence of the subsequent trajectory to the Gaussian fixed point is shown by power decay of the effective coupling constant. The analysis in the strong coupling regime is computer-aided and Newman's inequalities on truncated correlations are used to give mathematical rigor to the numerical bounds. In order to obtain a criterion for convergence to the Gaussian fixed point, characteristic functions and Newman's inequalities are systematically used. Received: 27 April 2000 / Accepted: 5 January 2001     

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schr?dinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation. Received: 26 February 2001 / Accepted: 28 May 2001     

Weak Solutions with Decreasing Energy¶of Incompressible Euler Equations

Weak solution of the Euler equations is an L²-vector field u(x, t), satisfying certain integral relations, which express incompressibility and the momentum balance. Our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations, as both viscosity and compressibility tend to zero; thus, we believe that weak solutions describe turbulent flows with very high Reynolds numbers. Every physically meaningful weak solution should have kinetic energy decreasing in time. But the existence of such weak solutions have been unclear, and should be proven. In this work an example of weak solution with decreasing energy is constructed. To do this, we use generalized flows (GF), introduced by Y. Brenier. GF is a sort of a random walk in the flow domain, such that the mean kinetic energy of particles is finite, and the particle density is constant. We construct a GF such that fluid particles collide and stick; this sticking is a sink of energy. The GF which we have constructed is a GF with local interaction; this means that there are no external forces. The second important property is that the particle velocity depends only on its current position and time; thus we have some velocity field, and we prove that this field is a weak solution with decreasing energy of the Euler equations. The GF is constructed as a limit of multiphase flows (MF) with the mass exchange between phases.     

Derivation of the Euler Equations¶from a Caricature of Coulomb Interaction

A caricature of collisionless plasma involving 2N particles of opposite charge is introduced. The N first particles are called "ions" and don't move. The N other particles are called "electrons". At each time, there is a one-to-one matching between electrons and ions and each pair is linked by a "spring" so that each electron oscillates with fixed frequency )^у. The essential point is that the matching between electrons and ions is updated at every discrete time n‰, n˸,1,2,..., so that the total potential energy of the system stays minimal. This leads to a non trivial interaction which turns out to be a caricature of Coulomb interaction. It is proven that, provided the N ions are equally spaced in a bounded domain D and ), ‰ and N^у tend to zero at appropriate rates, the electrons behave as the fluid parcels of an incompressible inviscid liquid moving inside D according to the Euler equations. Our proof relies on a result of P. Lax on the approximation of volume-preserving transformations by permutations.     

Certain Extensions¶of Vertex Operator Algebras of Affine Type

We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras A _k (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type. Received: 7 March 2000 / Accepted: 10 November 2000     

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway–Norton–Queen and to equivariant elliptic cohomology. Received: 7 January 1999 / Accepted: 14 March 2000     

A New Type of Non-Noether Adiabatic Invariants for Disturbed Lagrangian Systems： Adiabatic Invariants of Generalized Lutzky Type

For a Lagrangian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariant are presented in general infinitesimal transformation groups. On the basis of the invariance of disturbed Lagrangian systems under general infinitesimal transformations, the determining equations of Lie symmetries of the system are constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of adiabatic invariant, i.e. generalized Lutzky adiabatic invariants, of a disturbed Lagrangian system are obtained by investigating the perturbation of Lie symmetries t＇or a Lagrangian system with the action of small disturbance. Finally, an example is given to illustrate the application of the method and results.     

Self Duality Equations for Ginzburg–Landau¶and Seiberg–Witten Type Functionals¶with 6th Order Potentials

The abelian Chern–Simons–Higgs model of Hong-Kim-Pac and Jackiw–Weinberg leads to a Ginzburg–Landau type functional with a 6^th order potential on a compact Riemann surface. We derive the existence of two solutions with different asymptotic behavior as the coupling parameter tends to 0, for any number of prescribed vortices. We also introduce a Seiberg–Witten type functional with a 6^th order potential and again show the existence of two asymptotically different solutions on a compact K?hler surface. The analysis is based on maximum principle arguments and applies to a general class of scalar equations. Received: 13 October 1998 / Accepted: 21 October 2000     

A Statistical Approach to the Asymptotic Behavior of a Class of Generalized Nonlinear Schrödinger Equations

A statistical relaxation phenomenon is studied for a general class of dispersive wave equations of nonlinear Schrödinger-type which govern non-integrable, non-singular dynamics. In a bounded domain the solutions of these equations have been shown numerically to tend in the long-time limit toward a Gibbsian statistical equilibrium state consisting of a ground-state solitary wave on the large scales and Gaussian fluctuations on the small scales. The main result of the paper is a large deviation principle that expresses this concentration phenomenon precisely in the relevant continuum limit. The large deviation principle pertains to a process governed by a Gibbs ensemble that is canonical in energy and microcanonical in particle number. Some supporting Monte-Carlo simulations of these ensembles are also included to show the dependence of the concentration phenomenon on the properties of the dispersive wave equation, especially the high frequency growth of the dispersion relation. The large deviation principle for the process governed by the Gibbs ensemble is based on a large deviation principle for Gaussian processes, for which two independent proofs are given.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0202309)This research was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship from the National Science Foundation.This research was supported in part by grants from the Department of Energy (DE-FG02-99ER25376) and from the National Science Foundation (NSF-DMS-0207064).     

The Oscillator Representation and¶Groups of Heisenberg Type

We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms | of "Clifford type"-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G · \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K · \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a |: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra.     

Generalized Jordanian R-Matrices of Cremmer–Gervais Type

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang–Baxter equation, yielding a family of R-matrices which generalize to higher dimensions the Jordanian R-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa–Ueno, Yang–Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang–Baxter equation; via a vertex-IRF transformation from solutions to the dynamical Yang–Baxter equation.     

Twisted Traces of Quantum Intertwiners¶and Quantum Dynamical R-Matrices¶Corresponding to Generalized Belavin–Drinfeld Triples

We consider weighted traces of products of intertwining operators for quantum groups U _q (?), suitably twisted by a “generalized Belavin–Drinfeld triple”. We derive two commuting sets of difference equations – the (twisted) Macdonald–Ruijsenaars system and the (twisted) quantum Knizhnik–Zamolodchikov–Bernard (qKZB) system. These systems involve the nonstandard quantum R-matrices defined in a previous joint work with T. Schedler ([ESS]). When the generalized Belavin–Drinfeld triple comes from an automorphism of the Lie algebra ?, we also derive two additional sets of difference equations, the dual Macdonald–Ruijsenaars system and the \textit{dual} qKZB equations. Received: 20 March 2000 / Accepted: 11 December 2000     

Solutions of the Dirac–Fock Equations for Atoms¶and Molecules

The Dirac-Fock equations are the relativistic analogue of the well-known Hartree-Fock equations. They are used in computational chemistry, and yield results on the inner-shell electrons of heavy atoms that are in very good agreement with experimental data. By a variational method, we prove the existence of infinitely many solutions of the Dirac-Fock equations "without projector", for Coulomb systems of electrons in atoms, ions or molecules, with Z h 124, N h 41, N h Z. Here, Z is the sum of the nuclear charges in the molecule, N is the number of electrons.     

Non-Noether Conserved Quantity of Poincaré-Chetaev Equations of a Generalized Classical Mechanics

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincaré-Chetaev equations of a generalized classical mechanics under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced. Finally, an example is given to illustrate the application of the results.     

Construction of Generalized Synchronization for a Kind of Array Differential Equations and Applications

Chaos synchronization, as a special complex phenomenon, has been studied for about a decade. Only recently, have generalized chaotic synchronization phenomena been realized to be general in the real world and to have potential applications. We present two theorems for constructing a kind of array differential equations with generalized synchronization (GS) with respect to linear transformations. Two array differential equation systems with GS are introduced based on our theorems. Numerical simulations show that the two systems display periodic GS and chaotic GS, respectively.     

Reciprocal Transformations of Two Camassa–Holm Type Equations

The relation between the Camassa–Holm equation and the Olver–Rosenau–Qiao equation is obtained,and we connect a new Camassa–Holm type equation proposed by Qiao etc. with the first negative flow of the Kd V hierarchy by a reciprocal transformation.     

On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations

We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations in ℝ ⁿ (n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends Lions' existence theorem [15] to the case 1< γ <γ _n for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ _n = 3/2 for n= 2 and γ _n = 9/5 for n= 3. Dedicated to Professor Rolf Leis on the occasion of his 70th birthday Received: 17 January 2000 / Accepted: 3 July 2000     

Decomposition of Higher-Order Equations of Monge–Ampère Type

It is demonstrated that the fourth-order PDE