On the connection between the one-dimensionalS=1/2 Heisenberg chain and Haldane-Shastry model

Extra integrals of motion and the Lax representation are found for interacting spin systems with the HamiltonianH = (J/2) _{j, k=1,j k} ^N P(j – k) _{j k} , where one of the periods of the WeierstrassP function is equal toN. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues ofH corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.     

The Integrals of Motion for the Deformed W-Algebra $${W_{q,t}(\widehat{gl_N})}$$. II. Proof of the Commutation Relations

We explicitly construct two classes of infinitely many commutative operators in terms of the deformed W-algebra , and give proofs of the commutation relations of these operators. We call one of them local integrals of motion and the other nonlocal, since they can be regarded as elliptic deformations of local and nonlocal integrals of motion for the Virasoro algebra and the W ₃ algebra [1,2]. Dedicated to Professor Tetsuji Miwa on the occasion of the 60th birthday     

Power counting and regularization through loop integrations for multiple Feynman integrals in Minkowski space

A regularization procedure with a regularization parameter is developed which may be applied to multiple Feynman integrals in Minkowski space. The regularization is carried out inmomentum space and provides a rigorous method for studying Feynman integrals as multiple integrals in real variable theory. The regularized integrals are defined by changing the measure of integration _i dx _i to _i (1+x _i ² )^–/2 dx _i , >0, with a corresponding change defined inMinkowski space. We then develop a power counting convergence criterion for the absolute convergence of the integrals in terms of the parameter as a function of the so-called power asymptotic coefficients of Feynman integrands. An application to quantum electrodynamics is carried out.Work supported by the Department of National Defence Award under CRAD No. 3610-637:F4122.     

Coulomb pair density matrix I

The Coulomb pair density matrixG (r, r) for attractive and repulsive potentials is not only interesting for determining the two-particle effective potentials, but it is also essential in numerical studies of quantum systems. A high-temperature approximation is obtained for logG (r, r), in the form of simple integrals or series expansions; large-distance expansions are also given.     

Integrable Hénon-Heiles Hamiltonians: A Poisson algebra approach

The three integrable two-dimensional Hénon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl(2,R)⊕h₃ as their underlying Poisson symmetry algebra. In general, the procedure here introduced can be applied in order to obtain N-dimensional integrable generalizations of any 2D integrable potential of the form , and the formalism gives the explicit form of all the integrals of the motion. Further applications of this algebraic approach in different contexts are suggested.     

Contribution to the expression of particle movement in magnetic mirror systems in cyclic variables

The aim of this paper is to express the Hamiltonian function of a particle in a static, axially symmetric magnetic field in convenient variables. Assuming adiabaticity (the relative change of the magnetic field during one cyclotron revolution can be neglected), the Hamiltonian function is determined with cyclicity in two degrees of freedom.For the determination of the Hamiltonian function either the Carthesian, or the orthogonal curvilinear system was used as a starting coordinate system. The latter consists of a natural system of lines of force and equipotentials of the field. In both cases a Hamiltonian function of the formH=H(P₁ P ₂,P ₃,Q ₃) is obtained, where P_i are generalized impulses and Q₃ longitudinal coordinates.The form of the Hamiltonian function is very simple; it facilitates appreciably the integration of the equations of motion and provides simple expressions for the integrals of motion.The author wishes to express his thanks to Ing. J. Váa, the Director of the Institute, for his interest in the work and to Dr. M. Scidl for encouragement.     

Higher Spin Conserved Currents in Sp(2M) Symmetric Spacetime

An infinite set of higher spin conserved charges is found for the sp(2M) symmetric dynamical systems in M(M+ 1)/2-dimensional generalized spacetime _M. Since the dynamics in _M is equivalent to the conformal dynamics of infinite towers of fields in d-dimensional Minkowski spacetime with d = 3, 4, 6, 10, ... for M = 2, 4, 8, 16, ..., respectively, the constructed currents in _M generate infinite towers of (mostly new) higher spin conformal currents in Minkowski spacetime. The charges have a form of integrals of M-forms which are bilinear in the field variables and are closed as a consequence of the field equations. Conservation implies independence of a value of charge of a local variation of a M-dimensional integration surface _M analogous to Cauchy surface in the usual spacetime. The scalar conserved charge provides an invariant bilinear form on the space of solutions of the field equations that gives rise to a positive-definite norm on the space of quantum states.     

On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

Coulomb pair density matrix. II

The Coulomb pair density matrixG (r, r) for attractive and repulsive potentials and for all values of parameters is determined in the form of simple series or integrals. These results are useful in both theoretical and numerical studies.     

Evaluation of certain statistical sums

Summary Certain statistical ensembles,e.g. open chemical systems with randomly varying number of particles, are characterized by partition functions of the type ,n being a natural number anda _j ’s generalized temperatures. The state of the system is well defined if one knows the dependence ofa _j ’s on ensemble averages 〈n ^j 〉. For making the equations 〈n ^j 〉=〈n ^j 〉 (a ₁, ...,a _s) at least more accessible for numerical calculations a transformation of the partition function to a series of Fourier integrals is proposed. In the special case of the integrals can be calculated analytically transforming the statistical sum into a series of error functions.     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

On the gas-liquid phase transition

A new approach to the problem of the gas-liquid phase transition, based on the Mayer cluster expansion of the partition function, is proposed. It is shown that the necessary and sufficient condition for phase transition to occur is that there exist a temperatureT= T_c > 0 such that forT T _c, all theb _l (except perhaps a finite number of them) are positive, where theb _l, are the cluster integrals (as defined by Mayer) in the thermodynamic limit. Explicit expressions for the isotherms for gas-saturated vapor and liquid phases are given.     

On the euler equation: Bi-Hamiltonian structure and integrals in involution

We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the physical phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.     

Motion of vortex lines and mutual friction in superfluid4He nearT λ

The motion of a single vortex line in superfluid⁴He nearT is studied within modelF. The linear response of the vortex-line velocityv _L to a homogeneous counterflowv _s –v _n is calculated up to lowest order of renormalized perturbation theory. The critical temperature dependence is taken into account via the renormalizationgroup theory. Non-asymptotic critical effects are found to be important. The results are generalized to describe collective vortex motion and mutual friction in rotating superfluid⁴He. The phenomenological mutual-friction coefficientsB andB of Hall and Vinen are determined without adjustment of parameters. ForB quantitative agreement with experiments nearT is found whereas forB the agreement is only semiquantiative.     

Quantum Yang-Mills on the two-sphere

We obtain the quantum expectations of gauge-invariant functions of the connection on a principalG=SU(N) bundle overS ². We show that the spaceA/g _m of connections modulo gauge transformations which are the identity at one point is itself a principal bundle over G, based loops in the symmetry group. The fiber inA/g _m is an affine linear space. Quantum expectations are iterated path integrals first over this fiber then over G, each with respect to the push-forward toA/g _m of the measure s^-S(A) DA.S(A) denotes the Yang-Mills action onA. There is a global section ofA/g _m on which the first integral is a Gaussian. The resulting measure on G is the conditional Wiener measure. We explicitly compute the expectations of a special class of Wilson loops.     

Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.     

Exact calculation of one-dimensional irreducible integrals

One-dimensional irreducible integrals (_k) are computed in the form of Mayerf-function polynomials for a general interparticle potential. Obeisance to the exact specification of the irreducible integral definition produces regularities in the interaction of star graphs with the integration process. Tables of _k fork 5 and test solutions are presented.     

Dimension and entropy analysis of MEG time series from human α-rhythm

Magnetoencephalograms (MEG) from human brain were measured by means of a 37 channel SQUID magnetometer (KRENIKON). Correlation integrals were calculated from time series exhibiting strong -rhythm in order to give estimates of correlation dimension andK ₂ entropy. The results are discussed regarding the length and the stationarity of the data. It is shown that low spurious correlation dimensions andK ₂-entropies may easily be obtained as artefacts due to time correlations in phase space and data length. When time correlations are avoided and the length of time series is taken into account, estimates of correlation dimension andK ₂ entropy indicate no evidence of the existence of low dimensionality.     

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities ² and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions E. The functions E are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. ³ These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.     

Self and mutual admittance of slot antennas on a dielectric half-space

In this paper, an efficient implementation of the spectral domain moment technique is presented for computing the self and mutual coupling between slot antennas on a dielectric half-space. It is demonstrated that by the proper selection of the weighting functions in the method of moments, the analytic evaluation or simplification of the transverse moment integrals is enabled. This results into a significant reduction of the required computational labor. The method is then utilized in order to provide design data for the self and mutual admittances between two slot antennas on a dielectric substrate lens in the case of fused quartz ( _r =3.80), crystal quartz ( _r =4.53), silicon ( _r =11.9) and GaAs ( _r =12.8). The presented technique and associated results are useful when designing twin slot quasi-optical receivers, imaging arrays, phased arrays or power-combining arrays of slot elements at millimeter-wave frequencies.