critical phenomena and universal exponents in statistical physics on dyson s hierarchical model on etudie le comportement des etats de gibbs a la temperature critique ou a proximite on etudie en details le modele hierarchique de dyson et on presente les plus importants resultats

the geometric foundations of the integrability property of differential equations and physical systems ii mechanics on affinely connected manifolds and the work of kowalewski and painleve in the 1880 s sophie kowalewski proposed to study integrability of differential equations in terms of the analyticity properties of their solutions the model she studied the spinning top moving under gravity has an interesting geometric and algebraic structure which will be generalized the differential geometric properties of such models will be investigated in terms of the theory of affinely connected manifolds and lie theory

parameterization of the glottal area glottal flow and vocal fold contact area a new set of parameters is described for analysis and synthesis of glottal area vocal fold contact area and glottal volume flow parameters are all nondimensionalized and consist of an abduction quotient a shape quotient a phase quotient and a load quotient in addition to fundamental frequency and vibrational amplitude the parameters show promise in interpretation of electroglottographic photoglottographic and inverse filtered volume velocity waveforms in terms of the glottal configuration some comparisons between modeled and measured glottographic waveforms are made

periodic orbits and non integrability of henon heiles systems we apply the averaging theory of second order to study the periodic orbits for a generalized henon heiles system with two parameters which contains the classical henon heiles system two main results are shown the first result provides sufficient conditions on the two parameters of these generalized systems which guarantee that at any positive energy level the hamiltonian system has periodic orbits these periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy the second result shows that for the non integrable henon heiles systems in the sense of liouville arnol d which have the periodic orbits analytically found with averaging theory cannot exist any second first integral of class in particular for any second first integral of class we prove that the classical henon heiles system and many generalizations of it are not integrable in the sense of liouville arnol d moreover the tools we use for studying the periodic orbits and the non liouville arnol d integrability can be applied to hamiltonian systems with an arbitrary number of degrees of freedom

formant frequency estimation of high pitched vowels using weighted linear prediction all pole modeling is a widely used formant estimation method but its performance is known to deteriorate for high pitched voices in order to address this problem several all pole modeling methods robust to fundamental frequency have been proposed this study compares five such previously known methods and introduces a technique weighted linear prediction with attenuated main excitation wlp ame wlp ame utilizes temporally weighted linear prediction lp in which the square of the prediction error is multiplied by a given parametric weighting function the weighting downgrades the contribution of the main excitation of the vocal tract in optimizing the filter coefficients consequently the resulting all pole model is affected more by the characteristics of the vocal tract leading to less biased formant estimates by using synthetic vowels created with a physical modeling approach the results showed that wlp ame yields improved formant frequencies for high pitched sounds in comparison to the previously known methods e g relative error in the first formant of the vowel a decreased from 11 to 3 when conventional lp was replaced with wlp ame experiments conducted on natural vowels indicate that the formants detected by wlp ame changed in a more regular manner between repetitions of different pitch than those computed by conventional lp

the c neumann problem as a completely integrable system on an adjoint orbit it is shown by purely lie algebraic methods that the c neumann problem the motion of a material point on a sphere under the influence of a quadratic potential is a completely integrable system of euler poisson equations on a minimal dimensional orbit of a semidirect product of lie algebras 1 the c neumann problem the motion of a point on the sphere s 1 under the influence of a quadratic potential u x ax x x e r a diag al an is a completely integrable hamiltonian system for n 3 this has been shown by c neumann in 1859 12 and for arbitrary n by k uhlenbeck 16 r devaney 3 j moser 10 11 m adler and p van moerbeke 2 in this paper we show how this problem fits naturally in the framework of euler poisson equations 41 5 14 17 proving that the c neumann problem is a hamiltonian system on a minimaldimensional adjoint orbit in a semidirect product of lie algebras thus its complete integrability will follow entirely from lie algebraic considerations the equations of motion are i ax1 axx i 1 n 1 1 where the lagrange multiplier a ax x 11x112 is chosen such that x e sn i during the motion set x y and get the equivalent system to 1 1 xi yi yi a1xi ax x 11y112 xi llxll 1 x y 0 1 2 the following crucial remark that motivated the present investigation is due to k uhlenbeck and can be verified without any difficulties lemma 1 1 put x xixj p yixj xiyj system 1 2 is equivalent to x p x p x a lixl 1 x y o 1 3 remark that if one replaces x and a by x id n and a tr a id n respectively where id is the n x n identity matrix equations 1 3 remain unchanged from now on we shall assume that in 1 3 this change has been made so that x p a e sl n the next section gives a lie algebraic interpretation to these equations received by the editors march 12 1980 1980 mathematics subject classification primary 58f07 70h05 secondary 53c15 17b99

discrete all pole modeling a method for parametric modeling and spectral envelopes when only a discrete set of spectral points is given is introduced this method called discrete all pole dap modeling uses a discrete version of the itakura saito distortion measure as its error criterion one result is an autocorrelation matching condition that overcomes the limitations of linear prediction and produces better fitting spectral envelopes for spectra that are representable by a relatively small discrete set of values such as in voiced speech an iterative algorithm for dap modeling that is shown to converge to a unique global minimum is presented results of applying dap modeling to real and synthetic speech are also presented dap modeling is extended to allow frequency dependent weighting of the error measure so that spectral accuracy can be enhanced in certain frequency regions

motion of four dimensional rigid body around a fixed point an elementary approach i the goal of this letter is to give an elementary approach to the solution of euler frahm equations for the manakov four dimensional case for this we use the kotter approach and some results from the original papers by schottky weber and caspary we hope that such an approach will be useful for the solution of the problem of an n dimensional top

almost gibbsian description of the sign fields of sos fields an example is presented of a measure on a lattice system which has a measure zero set of points configurations where some conditional probability can be discontinuous but does not become a gibbs measure under decimation or other transformations we also discuss some related issues

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

algebraic geometry and stability for integrable systems abstract in 1970s a method was developed for integration of nonlinear equations by means of algebraic geometry starting from a lax representation with spectral parameter the algebro geometric method allows to solve the system explicitly in terms of theta functions of riemann surfaces however the explicit formulas obtained in this way fail to answer qualitative questions such as whether a given singular solution is stable or not in the present paper the problem of stability for equilibrium points is considered and it is shown that this problem can also be approached by means of algebraic geometry

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

the full geometry of kowalewski s top and 1 2 abelian surfaces revue sur les varietes abeliennes les automorphismes d une courbe elliptique surfaces abeliennes t 2 de polarisation 1 2 jacobiens hyperelliptiques comme recouvrements doubles de t 2 les tores t 2 comme doubles recouvrements des jacobiens hyperelli courbe de kowalewski et surfaces abeliennes de polarisation 2 4

generalized tops and moment maps to loop algebras using a combination of hamiltonian reduction under continuous and discrete symmetry groups the lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable hamiltonian systems in loop algebras

a lie theoretic galois theory for the spectral curves of an integrable system i in the study of integrable systems of ode s arising from a lax pair with a parameter the constants of the motion occur as spectral curves the specific curves depend upon the representation of the lie algebra in this paper a galois theory of spectral curves is given that classifies the spectral curves from an integrable system the spectral curves correspond to conjugacy classes of certain subgroups of the weyl group for the lie algebra the theory is illustrated with the periodic toda lattice

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

integrable hamiltonian systems on lie groups kowalewski type equations for the heavy top extend to a larger class of hamiltonian systems on lie groups this paper explains these extensions and along the way reveals further geometric significance of her work in the theory of elliptic curves specifically in this paper we shall be concerned with the solutions of the following differential system in six variables hi h2 h3 h1 h2 h3 dh 1 h2h d1 h2h3 h 23 ha2 dt c3 c2

comparing measurement errors for formants in synthetic and natural vowels the measurement of formant frequencies of vowels is among the most common measurements in speech studies but measurements are known to be biased by the particular fundamental frequency f0 exciting the formants approaches to reducing the errors were assessed in two experiments in the first synthetic vowels were constructed with five different first formant f1 values and nine different f0 values formant bandwidths and higher formant frequencies were constant input formant values were compared to manual measurements and automatic measures using the linear prediction coding burg algorithm linear prediction closed phase covariance the weighted linear prediction attenuated main excitation wlp ame algorithm alku pohjalainen vainio laukkanen and story 2013 j acoust soc am 134 2 1295 1313 spectra smoothed cepstrally and by averaging repeated discrete fourier transforms formants were also measured manually from pruned reassigned spectrograms rss fulop 2011 speech spectrum analysis springer berlin all but wlp ame and rs had large errors in the direction of the strongest harmonic the smallest errors occur with wlp ame and rs in the second experiment these methods were used on vowels in isolated words spoken by four speakers results for the natural speech show that f0 bias affects all automatic methods including wlp ame only the formants measured manually from rs appeared to be accurate in addition rs coped better with weaker formants and glottal fry

the kowalevski top its r matrix interpretation and bihamiltonian formulation the lax pair representation for the kowalevski top and for the kowalevski gyrostat are obtained directly via the r matrix method the construction of reyman and semenov tian shansky is then used to give a bihamiltonian description of the kowalevski gyrostat

the effects of measurement error and vowel selection on the locus equation measure of coarticulationa effects on the slope of introducing error in the f2 hz values in locus equations les and of using fewer than ten vowels were investigated for each of the initial consonants b d g 2000 simulated sets were generated using monte carlo techniques the sets were altered with 50 100 or 200 hz error being randomly applied to each f2 value within a set selected vowels were then removed from the sets and the effects on the slopes were measured results suggest that the le slopes are generally resistant to error and reduced number of vowels effects of adding 50 hz of random error to the f2 values in sets using eight or ten vowels were minimal yielding a mean absolute change in slope less than 0 07

mathematical properties of renormalization group transformations abstract renormalization group transformations of the type introduced into statistical physics by k g wilson have been widely used with great success for a variety of many body calculations the basic approach consists in integrating out certain degrees of freedom and using a new image hamiltonian to describe the statistical properties of those that remain the mathematical foundations of the procedure however remain somewhat obscure p a pearce and the author have pointed out that there are serious questions about the existence and properties of a thermodynamic limit for renormalization group transformations in particular certain real space transformations show peculiarities in this limit these suggest that there may be serious mathematical problems with renormalization group procedures despite their considerable practical sucess

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

remarks on kowalevski s top we present a review of some results on kowalevski s top kt in classical and quantum mechanics the following items are considered i the generalized kt gkt on the lie algebras so 4 e 3 and so 3 1 ii kowalevski s gyrostat on these algebras iii action of the gkt iv quantum counterparts of the kt v semiclassical quantization of the gkt vi generalization of the kt by chaplygin and goryachev at l 0 and its lax representation unsolved questions are also discussed

lax representation and poisson geometry of the kowalevski top we discuss the poisson structure underlying the two field kowalevski gyrostat and the kowalevski top we start from their lax structure and construct a suitable pencil of poisson brackets which endows these systems with the structure of bi hamiltonian completely integrable systems we study the casimir functions of such pencils and show how it is possible to frame the kowalevski systems within the so called gel fand zakharevich bi hamiltonian setting for integrable systems

general integrable problems of classical mechanics several classical problems of mechanics are shown to be integrable for the special systems of coupled rigid bodies introduced in this work and calledck central configurations it is proven that dynamics of an arbitraryck central configuration in a newtonian gravitational field with an arbitrary quadratic potential is integrable in the liouville sense and in the theta functions of riemann surfaces hidden symmetry of the inertial dynamics of these configurations is disclosed and reductions of the lagrange equations to the euler equations on lie coalgebras are obtained reductions and integrable cases of a heavyck central configuration rotation around a fixed point are indicated separation of rotations of a space station type orbiting system being ack central configuration of rigid bodies is proven this result leads to the possibility of the independent stabilization of rotations of the rigid bodies in such orbiting configurations

sim simultaneous inverse filtering and matching of a glottal flow model for acoustic speech signals a new method simultaneous inverse filtering and model matching sim is proposed that allows one to calculate voice source measures without any user interaction it is based on the discrete all pole modeling dap technique for inverse filtering if which is modified to include a model of the glottal flow as integral part lf model fant et al stl qpsr stockholm 4 1985 1 13 1986 as the correct lf parameters are initially unknown they are estimated in an iterative procedure using multi dimensional optimization techniques that are initialized according to the results of an exhaustive search the error criteria applied reflect how well the if is performed after the spectral contribution of the glottal flow has been removed the resulting optimal lf parameter constellation serves as the basis to calculate 11 voice source measures the performance was evaluated using synthesized signals and recordings of natural utterances for the synthesized signals the accuracy to reproduce the original parameters was high correlations exceeding 0 88 for measures where the starting point of the glottal cycle did not enter explicitly errors were smaller compared to conventional estimation methods where the measures were estimated from the if signal the analysis of natural utterances indicates that problems still exist with regard to robustness but that under advantageous conditions the open quotient the speed quotient the closing quotient the parabolic spectral parameter and the negative peak amplitude of the glottal flow derivative can indeed be determined automatically by the sim method

periodic motions of the kowalevski gyrostat in two constant fields the case of the gyrostat motion found by a g reyman and m a semenov tian shansky is known as the liouville integrable hamiltonian system with three degrees of freedom without symmetry groups we find the set of points at which the integral map has rank 1 this set consists of special periodic motions generating the singular points of bifurcation diagrams on iso energetic surfaces for such motions all phase variables are expressed as algebraic functions of one auxiliary variable satisfying the differential equation integrable in elliptic functions of time it is shown that the corresponding points in three dimensional space of the integral constants belong to the intersection of two sheets of the discriminant surface of the lax curve

measures for lattice systems abstract i review issues related with the presence or absence of gibbsianness in measures describing random fields in lattices after a brief exposition of the definition and properties of gibbs measures i discuss the phenomenon of non gibbsianness its examples characterization the proposed classification schemes and notions of generalized gibbsianness the review closes with comments on directions for further work

an amplitude quotient based method to analyze changes in the shape of the glottal pulse in the regulation of vocal intensity this study presents an approach to visualizing intensity regulation in speech the method expresses a voice sample in a two dimensional space using amplitude domain values extracted from the glottal flow estimated by inverse filtering the two dimensional presentation is obtained by expressing a time domain measure of the glottal pulse the amplitude quotient aq as a function of the negative peak amplitude of the flow derivative dpeak the regulation of vocal intensity was analyzed with the proposed method from voices varying from extremely soft to very loud with a spl range of approximately 55db when vocal intensity was increased the speech samples first showed a rapidly decreasing trend as expressed on the proposed aq dpeak graph when intensity was further raised the location of the samples converged toward a horizontal line the asymptote of a hypothetical hyperbola this behavior of the aq dpeak graph indicates that the intensity regulation strategy changes from laryngeal to respiratory mecha

non gibbsian limit for large block majority spin transformations we generalize a result of lebowitz and maes that projections of massless gaussian measures onto ising spin configurations are non gibbs measures this result provides the first evidence for the existence of singularities in majority spin transformations of critical models indeed under the assumption of the folk theorem that an average block spin transformation applied to a critical ising model in 5 or more dimensions converges to a gaussian fixed point we show that the limit of a sequence of majority spin transformations with increasing block size applied to a critical ising model is a measure that is not of gibbsian type

closed phase covariance analysis based on constrained linear prediction for glottal inverse filtering closed phase cp covariance analysis is a widely used glottal inverse filtering method based on the estimation of the vocal tract during the glottal cp since the length of the cp is typically short the vocal tract computation with linear prediction lp is vulnerable to the covariance frame position the present study proposes modification of the cp algorithm based on two issues first and most importantly the computation of the vocal tract model is changed from the one used in the conventional lp into a form where a constraint is imposed on the dc gain of the inverse filter in the filter optimization with this constraint lp analysis is more prone to give vocal tract models that are justified by the source filter theory that is they show complex conjugate roots in the formant regions rather than unrealistic resonances at low frequencies second the new cp method utilizes a minimum phase inverse filter the method was evaluated using synthetic vowels produced by physical modeling and natural speech the results show that the algorithm improves the performance of the cp type inverse filtering and its robustness with respect to the covariance frame position

lie point symmetries and first integrals the kowalevski top we show how the lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations the method of reduction increase of order developed by nucci j math phys 37 1772 1775 1996 is essential noether s theorem is neither necessary nor considered the most striking example we present is the relationship between lie group analysis and the famous first integral of the kowalevski top

synergistic modes of vocal tract articulation for american english vowels the purpose of this study was to investigate the spatial similarity of vocal tract shaping patterns across speakers and the similarity of their acoustic effects vocal tract area functions for 11 american english vowels were obtained from six speakers three female and three male using magnetic resonance imaging mri each speaker s set of area functions was then decomposed into mean area vectors and representative modes eigenvectors using principal components analysis pca three modes accounted for more than 90 of the variance in the original data sets for each speaker the general shapes of the first two modes were found to be highly correlated across all six speakers to demonstrate the acoustic effects of each mode both in isolation and combined a mapping between the mode scaling coefficients and f1 f2 pairs was generated for each speaker the mappings were unique for all six speakers in terms of the exact shape of the f1 f2 vowel space but the general effect of the modes was the same

a simple stochastic cluster dynamics rigorous results motivated by the swendsen wang algorithm for the ising model with no external magnetic field the authors consider a particle system on a discrete lattice evolving according to the following parallel algorithm at each time step connected clusters of particles are removed with probability 1 2 and at each empty site a particle is created with probability p due to the presence of arbitrarily large clusters the interaction can have an arbitrary range in the one dimensional case they solve the model completely by means of a novel path expansion in spacetime in the two dimensional case they show that due to strong time correlation the probability of having a large cluster of particles does not rapidly decay exponentially in the cardinality of the cluster even for p very small they then prove ergodicity for 1 p close to zero

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

technique for tuning vocal tract area functions based on acoustic sensitivity functions a technique for modifying vocal tract area functions is developed by using sum and difference combinations of acoustic sensitivity functions to perturb an initial vocal tract configuration first sensitivity functions e g fant and pauli proc speech comm sem 74 1975 are calculated for a given area function at its specific formant frequencies the sensitivity functions are then multiplied by scaling coefficients that are determined from the difference between a desired set of formant frequencies and those supported by the current area function the scaled sensitivity functions are then summed together to generate a perturbation of the area function this produces a new area function whose associated formant frequencies are closer to the desired values than the previous one this process is repeated iteratively until the coefficients are equal to zero or are below a threshold value

the kowalewski top a new lax representation the 2 2 monodromy matrices for the kowalewski top on the lie algebras e 3 so 4 and so 3 1 are presented the corresponding quadratic r matrix structure is the dynamical deformation of the standard r matrix algebras some tops and toda lattices related to the kowalewski top are discussed

exact solution of the goryachev chaplygin problem in quantum mechanics the quantum mechanical analogue of the classical integrable system originally founded by goryachev and chaplygin 1900 is considered in detail the problem is formulated in terms of the euclid e 3 group the euler poisson equations of motion and their integrals are derived the determination of the spectra of the integrals of motion is equivalent to construction of the special basis of representation of e 3 the eigenvalue problem admits separation in new variables which are closely connected with two boson creation annihilation operators they are the same as in the majorana representation of the lorentz group in the new variables the quantum mechanical equations of motion look similar to the classical ones the constants of motion are determined by the spectral problems for the two jacobi type tridiagonal infinite matrices some numerical results are given

new generalizations of all the known integrable problems in rigid body dynamics in the present paper we consider a quite general problem of motion of a rigid body under the action of an axially symmetric combination of potential and gyroscopic forces special versions of this problem are the classical problems of motion of a heavy body gyrostat or a body in a liquid another example is the problem of motion of a heavy magnetized and electrically charged body under the combined action of gravitational magnetic electric and lorentz forces we introduce a method based on the invariance of the equations of motion under certain transformations this method enables a systematic construction of several integrable problems generalizing all the known general for arbitrary initial conditions and conditional on a single level of the cyclic integral integrable problems as well as particular solutions five general integrable cases are constructed that generalize known ones by the inclusion of one to four additional free parameters fifteen conditional integrable cases generalizing all the known general and conditional cases and containing an additional arbitrary function are introduced the latter type of generalization can be applied to any one of a large collection of particular solutions known in many versions of the problem considered the method also enables the construction of the explicit solutions of the equations of motion in the generalized cases from those of the original cases detailed physical interpretation is given for some cases

systems of hess appel rot type we construct higher dimensional generalizations of the classical hess appel rot rigid body system we give a lax pair with a spectral parameter leading to an algebro geometric integration of this new class of systems which is closely related to the integration of the lagrange bitop performed by us recently and uses mumford relation for theta divisors of double unramified coverings based on the basic properties satisfied by such a class of systems related to bi poisson structure quasi homogeneity and conditions on the kowalevski exponents we suggest an axiomatic approach leading to what we call the class of systems of hess appel rot type

robustness of the non gibbsian property some examples we discuss some examples of measures on lattice systems which lack the property of being a gibbs measure in a rather strong sense

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

order and the ubiquitous occurrence of chaos abstract for a large class of ode s which includes the van der pol equation we determine analytically the asymptotic location of the singularities in the complex t plane by integrating these ode s numerically we show that if the singularities are dense which is the generic case the solution is chaotic in the sense of sensitive dependence on initial conditions in the exceptional case that the singularities are not dense the solution exhibits order taxis chaos is ubiquitous even for first order ode s in complex t

lax representation for the systems of s kowalevskaya type we describe a number of dynamical systems that are generalizations of the s kowalevskaya system and admit the lax representation

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

on integrable deformations of the spherical top the motion on the sphere s2 in potential v x1x2x3 2 3 is considered the lax representation and the linearization procedure for this two dimensional integrable system are discussed

simultaneous separation for the kowalevski and goryachev chaplygin gyrostats in the special case of zero square integral the kowalevski gyrostat and goryachev chaplygin gyrostat share a simple separation of variables originated from the 4 4 lax matrix

transformations of mechanical systems with cyclic coordinates and new integrable problems we show that the well known routhian procedure of ignoring cyclic coordinates is far more than a tool for obtaining equations of motion of reduced order in a lagrangian form a transformation is introduced that involves a number of arbitrary functions or additional parameters in the system while preserving the routhian equations of motion although these parameters invoke new physical effects in the transformed system the solution of the latter is always obtained in a simple way from that of the original system in particular from any integrable system with k cyclic degrees of freedom we obtain a family of systems integrable on a fixed level of the cyclic integrals and physically generalizing that system through the inclusion of k arbitrary functions that depend only on the noncyclic coordinates in many problems of physical interest a general integrable case can also be generalized to an integrable case for arbitrary initial conditions the method is applied to some problems of rigid body dynamics four new integrable problems are obtained as generalizations of known cases by including certain additional combinations of gravitational and electromagnetic forces the new cases are presented in an explicit way that enables direct verification of the constancy of the integrals using the equations of motion explicit time solution of the new cases is discussed physical interpretation is given for two cases

a fixed point equation for the high temperature phase of discrete lattice spin systems a fixed point equation on an infinite dimensional space is proposed as an alternative to the usual definition of the infinite volume limit in discrete lattice spin systems in the high temperature phase it is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities a theorem is then proved that says that if a certain finite volume condition is satisfied then this fixed point equation has a solution and the resulting free energy is analytic in the parameters in the hamiltonian for particular values of the temperature this finite volume condition may be checked with the help of a computer the two dimensional ising model is considered as a test case and it is shown that the finite volume condition is satisfied for 0 77 critical

on the inverse problem in statistical mechanics for quantum spin systems it is known that for a suitable space of potentials the equilibrium states arew dense in the set of all translation invariant states the problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential a necessary and sufficient condition for a state to be an equilibrium state for some potential is given in sect 3

the lagrange bitop on so 4 so 4 and geometry of the prym varieties a four dimensional integrable rigid body system is considered and it is shown that it represents two twisted three dimensional lagrange tops a polynomial lax representation which doesn t fit neither in dubrovin s nor in adler van moerbeke s picture is presented the algebro geometric integration procedure is based on deep facts from the geometry of the prym varieties of double coverings of hyperelliptic curves mumford s relation and mumford dalalyan theory the correspondence between all such coverings with prym varieties splitted as a sum of two varieties of the same dimension and the integrable hierarchy associated to the initial system is established

a new integrable case of the motion of the 4 dimensional rigid body a lax pair for a new family of integrable systems on so 4 is presented the construction makes use of a twisted loop algebra of theg 2 lie algebra we also describe a general scheme producing integrable cases of the generalized rigid body motion in an external field which have a lax representation with spectral parameter several other examples of multi dimensional tops are discussed

kowalevski top and genus 2 curves kowalevski s curve of genus 2 is related to two other curves arising from the solution of the kowalevski top by the method of spectral curves in the case when the angular momentum of the top is orthogonal to the gravity vector one is the bobenko reyman semenov tian shansky curve of genus 2 the other is the spectral curve of the kuznetsov tsiganov lax matrix of genus 3 the relations between the curves are given by correspondences that is multivalued maps inducing isogenies of the corresponding jacobian or prym varieties

reduction theory and the lagrange routh equations reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of euler jacobi lagrange hamilton routh poincare and others the modern vision of mechanics includes besides the traditional mechanics of particles and rigid bodies field theories such as electromagnetism fluid mechanics plasma physics solid mechanics as well as quantum mechanics and relativistic theories including gravity symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas to subtle symmetries underlying integrable systems reduction theory concerns the removal of symmetries and their associated conservation laws variational principles along with symplectic and poisson geometry provide fundamental tools for this endeavor reduction theory has been extremely useful in a wide variety of areas from a deeper understanding of many physical theories including new variational and poisson structures to stability theory integrable systems as well as geometric phases this paper surveys progress in selected topics in reduction theory especially those of the last few decades as well as presenting new results on non abelian routh reduction we develop the geometry of the associated lagrange routh equations in some detail the paper puts the new results in the general context of reduction theory and discusses some future directions

geodesic flow on so 4 and the intersection of quadrics this note studies the relationship between the complete integrability of geodesic flow on so 4 with a left invariant metric and the geometry of the intersection of four quadrics in p 6

poisson maps and integrable deformations of the kowalevski top we construct a poisson map between manifolds with linear poisson brackets corresponding to the lie algebras e 3 and so 4 using this map we establish a connection between the deformed kowalevski top on e 3 proposed by sokolov and the kowalevski top on so 4 the connection between these systems leads to the separation of variables for the deformed system on e 3 and yields the natural 5 5 lax pair for the kowalevski top on so 4

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

integrable euler equations on so 4 and their physical applications for the lie algebra so 4 and other six dimensional lie algebras we find some euler s equations which have an additional fourth order integral and are algebraically integrable in terms of elliptic functions in a one parameter set of orbits integrable euler s equations having an additional second order integral and generalizing steklov s case are presented equations for rotation of a rigid body havingn ellipsoid cavities filled with the ideal incompressible fluid being in a state of homogeneous vortex motion are derived it is shown that the obtained equations are euler s equations for the lie algebra of the groupgn 1 so 3 so 3 new physical applications of euler s equations on so 4 are discussed

a tower of genus two curves related to the kowalewski top several curves of genus 2 are known such that the equations of motion of the kowalewski top are linearized on their jacobians one can expect from tran scendental approaches via solutions of equations of motion in theta functions that their jacobians are isogeneous the paper focuses on two such curves kowalewski s and that of bobenko reyman semenov tian shansky the latter arising from the solution of the problem by the method of spectral curves an isogeny is established between the jacobians of these curves by purely algebraic means using richelot s transformation of a genus 2 curve it is shown that this isogeny respects the hamil tonian flows the two curves are completed into an infinite tower of genus 2 curves with isogeneous jacobians

integrable mechanical systems related to the harry dym hierarchy it is shown that all stationary and restricted flows of the harry dym hierarchy have a parametrization as a set of newton equations they are new classes of integrable mechanical systems properties of the harry dym hierarchy are used for deriving systematically spectral parameter dependent lax representations and bi hamiltonian formulation for some of these systems they yield complete sets of integrals in involution which render these systems to be completely integrable

euler equations on finite dimensional lie algebras arising in physical problems real physical problems are presented in which euler equations on lie algebras of arbitrarily high finite dimension arise a new integrable case of rotation of a magnetized rigid body in constant gravitational and magnetic fields is found it generalizes the kowalewski classical integrable case

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

geometrization and generalization of the kowalevski top a new view on the kowalevski top and the kowalevski integration procedure is presented for more than a century the kowalevski 1889 case has attracted full attention of a wide community as the highlight of the classical theory of integrable systems despite hundreds of papers on the subject the kowalevski integration is still understood as a magic recipe an unbelievable sequence of skillful tricks unexpected identities and smart changes of variables the novelty of our present approach is based on our four observations the first one is that the so called fundamental kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the kowalevski variables w x 1 x 2 as the pencil parameter and the darboux coordinates respectively the second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials all steps of the kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials the third observation connects the kowalevski integration and the pencil equation with the theory of multi valued groups the kowalevski change of variables is now recognized as an example of a two valued group operation and its action the final observation is surprising equivalence of the associativity of the two valued group operation and its action to the n 3 case of the great poncelet theorem for pencils of conics

on a trajectory isomorphism of the kowalevski gyrostat and the clebsch problem for the kowalevski gyrostat a change of variables similar to that for the kowalevski top is done we establish one to one correspondence between solutions of the kowalevski gyrostat and the clebsch system and demonstrate that kowalevski variables for the gyrostat practically coincide with elliptic coordinates on a sphere for the clebsch case

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

lax representation and new formulae for the goryachev chaplygin top a lax pair for the goryachev chaplygin top gct is presented the pair is obtained from that for the kowalewski top found by reyman and semenov tian shansky 1987 explicit formulae are constructed in terms of theta functions for the dynamical variables

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

kowalewski s top on the lie algebras o 4 e 3 and o 3 1 the authors extend the lax pair for kowalewski s top kt obtained by haine and horozov 1987 to the lie algebras o 4 and o 3 1 the results are expressed in terms of a solution of the neumann system they derive formulae for the action variables by analogy with the e 3 kt case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables

bifurcations of invariant manifolds in the generalized henon heiles system abstract the generalized henon heiles hamiltonian h 1 2 q dot 1 2 q dot 2 2 aq 1 2 bq 2 2 q 1 2 q 2 e 3 q 2 3 in the integrable case e 6 is studied from topological point of view the topology of the real level sets for all generic values of the constants of the motion and all generic bifurcations of the liouville tori and cylinders are described by making use of the reach algebraic structure of the problem

new integrable problem of classical mechanics complete integrability in liouville s sense is proven for rotation of an arbitrary rigid body with a fixed point in a newtonian field with an arbitrary homogeneous quadratic potential a consequences is the complete integrability of rotation of a rigid body with fixed center of mass in the field of arbitrary sufficiently remote objects in the second approximation explicit formulae are obtained expressing angular velocities of the rigid body in terms of functions for riemannian surfaces integrable cases are found for rotation of a rigid body in nonlinear newtonian potential fields