uniform convergence of the free energy of the classical heisenberg model to that of the gaussian model we show that the free energy of the classical heisenberg model converges to the free energy of the gaussian in the low temperature limit the limit is uniform as the field is taken to zero

tunstall adaptive coding and miscoding in the first part of this paper we tackle the case where a variable length to block tunstall code is used to encode the wrong source miscoding it turns out that exactly as happens in the case with huffman coding the asymptotic excess rate is given by the informational divergence between the probability distribution ruling the source and the probability distribution for which the tunstall code had been devised we also prove asymptotic equality between the individual rate of a codeword and the corresponding self information this allows us to bound the maximal and the minimal length in a tunstall tree in the second part of the paper we study the case where the probability distribution of the source changes in time if this happens it is necessary to frequently update the current code to ensure that the optimality conditions concerning its rate are met this coding procedure is known as adaptive coding we propose some schemes for adaptive tunstall coding based on the structure of the coding tree on the ordering property analogous to gallager s 1978 sibling property of the huffman code and on the tunstall regions analogous to the attraction regions due to longo and galasso 1982

a universal variable to fixed length source code based on lawrence s algorithm it is shown that the modified lawrence algorithm is universal over the class of binary memoryless sources and that the rate converges asymptotically optimally fast to the source entropy it is proven that no codes exist that have a better asymptotic performance the asymptotic bounds show that universal variable to fixed length codes can have a significantly lower redundancy than universal fixed to variable length codes with the same number of codewords

variable to fixed length codes for markov sources petry s efficient and optimal variable to fixed length source code for discrete memoryless sources was described by schalkwijk by extending this coding technique we are able to give an algorithm for markov sources that is easy to implement we can bound the loss of efficiency as a function of the code complexity and the mismatch between the source and the code rates arbitrarily close to the source entropy are shown to be achievable in this sense the codes introduced are optimal

construction of classical superintegrable systems with higher order integrals of motion from ladder operators we construct integrals of motion for multidimensional classical systems from ladder operators of one dimensional systems this method can be used to obtain new systems with higher order integrals we show how these integrals generate a polynomial poisson algebra we consider a one dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion we obtain also the polynomial algebra generated by these integrals we calculate numerically the trajectories and show that all bounded trajectories are closed

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

a new superintegrable hamiltonian we identify a new superintegrable hamiltonian in three degrees of freedom obtained as a reduction of pure keplerian motion in six dimensions the new hamiltonian is a generalization of the keplerian one and has the familiar 1 r potential with three barrier terms preventing the particle crossing the principal planes in three degrees of freedom there are five functionally independent integrals of motion and all bound classical trajectories are closed and strictly periodic the generalization of the laplace runge lenz vector is identified and shown to provide functionally independent isolating integrals they are quartic in the momenta and do not arise from separability of the hamilton jacobi equation a formulation of the system in action angle variables is presented

higher order superintegrability of a holt related potential in a recent paper post and winternitz 2011 j phys a math theor 44 162001 studied the properties of two dimensional euclidean potentials that are linear in one of the two cartesian variables in particular they proved the existence of a potential endowed with an integral of third order and an integral of fourth order in this paper we show that these results can be obtained in a more simple and direct way by noting that this potential is directly related to the holt potential it is proved that the existence of a potential with higher order superintegrability is a direct consequence of the integrability of the family of holt type potentials

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

a note on approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

a new approach to the higher order superintegrability of the tremblay turbiner winternitz system the higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized smorodinsky winternitz system the idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions here denoted by m and n this technique leads to a proof of the superintegrability of the tremblay turbiner winternitz system and to the explicit expression of the constants of motion a second family related with the first one of superintegrable systems is also studied

a low temperature expansion for classical n vector models iii a complete inductive description fluctuation integrals we give a complete inductive description of the effective densities generated by applications of the renormalization transformations to the densities of the classical n vector models described in the paper 1 and we formulate the main theorems of this sequence of papers this description includes now all the large field domains which we ve excluded from the discussions of 1 4 we apply the renormalization transformation to a density satisfying the inductive hypotheses we introduce large field small field partitions and we determine and analyze the fluctuation integrals in the general setting

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

reduction of superintegrable systems the anisotropic harmonic oscillator we introduce a 2n parametric family of maximally superintegrable systems in n dimensions obtained as a reduction of an anisotropic harmonic oscillator in a 2n dimensional configuration space these systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta they generalize known examples of superintegrable models in the euclidean plane

superintegrability of the calogero moser system constants of motion master symmetries and time dependent symmetries the classical n dimensional calogero moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion in the first part of the article some properties related with the existence of several families of constants of motion are analyzed in the second part the master symmetries and the time dependent symmetries of this system are studied

lattice systems with a continuous symmetry iii low temperature asymptotic expansion for the plane rotator model we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

an application of informational divergence to huffman codes a classification of all probability distributions over the finite alphabet of an information source is given where the classes are the sets of distributions sharing the same binary huffman code such a classification can be used in noiseless coding when the distribution of the finite memoryless source varies in time or becomes gradually known instead of applying the huffman algorithm to each new estimate of the probability distribution if a simple test based on the above classification is passed then the huffman code used previously is optimal also for the new distribution

polynomial constants of motion for calogero type systems in three dimensions we give an explicit and concise formula for higher degree polynomial first integrals of a family of calogero type hamiltonian systems in dimension three these first integrals together with the already known ones prove the maximal superintegrability of the systems

validity of the spin wave approximation for the free energy of the heisenberg ferromagnet we consider the quantum ferromagnetic heisenberg model in three dimensions for all spins s 1 2 we rigorously prove the validity of the spin wave approximation for the excitation spectrum at the level of the first non trivial contribution to the free energy at low temperatures our proof comes with explicit constructive upper and lower bounds on the error term it uses in an essential way the bosonic formulation of the model in terms of the holstein primakoff representation in this language the model describes interacting bosons with a hard core on site repulsion and a nearest neighbor attraction this attractive interaction makes the lower bound on the free energy particularly tricky the key idea there is to prove a differential inequality for the two particle density which is thereby shown to be smaller than the probability density of a suitably weighted two particle random process on the lattice

n 2 supersymmetric extension of the tremblay turbiner winternitz hamiltonians on a plane the family of tremblay turbiner winternitz hamiltonians hk on a plane corresponding to any positive real value of k is shown to admit an supersymmetric extension of the same kind as that introduced by freedman and mende for the calogero problem and based on an superalgebra the irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion xk of hk bases for them are explicitly constructed the ground state of each supersymmetrized hamiltonian is shown to belong to an atypical lowest weight state irreducible representation

a class of superintegrable systems of calogero type we show that the three body calogero model with inverse square potentials can be interpreted as a maximally superintegrable and multiseparable system in euclidean three space as such it is a special case of a family of systems involving one arbitrary function of one variable

on the superintegrability of calogero moser sutherland model the superintegrability of calogero moser sutherland hyperbolic model and nonsuperintegrability of trigonometric one are shown

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

superintegrable deformations of superintegrable systems quadratic superintegrability and higher order superintegrability the superintegrability of four hamiltonians hr hr r a b c d where hr are known hamiltonians and is a certain function defined on the configuration space and depended on a parameter is studied the new hamiltonians and the associated constants of motion jri i 1 2 3 are continous functions of the parameter the first part is concerned with separability and quadratic superintegrability the integrals of motion are quadratic in the momenta and the second part is devoted to the existence of higher order superintegrability the results obtained in the second part are related with the tremblay turbiner winternitz and the post winternitz systems

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

third order superintegrable systems separable in parabolic coordinates in this paper we investigate superintegrable systems which separate in parabolic coordinates and admit a third order integral of motion we give the corresponding determining equations and show that all such systems are multi separable and so admit two second order integrals the third order integral is their lie or poisson commutator we discuss how this situation is different from the cartesian and polar cases where new potentials were discovered which are not multi separable and which are expressed in terms of painleve transcendents or elliptic functions

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

on the composition of tunstall messages we study the composition of messages in an encoding tree for a tunstall code and more generally in a tree whose skewness is bounded for such trees a sort of law of large numbers holds true actually we provide a direct and converse coding theorem for variable length to block length source codes when a vanishing error probability is allowed

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

third order superintegrable systems separating in polar coordinates a complete classification of quantum and classical superintegrable systems in e2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum new quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth painleve transcendent or in terms of the weierstrass elliptic function

on asymptotic limits for the quantum heisenberg model the authors discuss various asymptotic limits of the classical and quantum heisenberg model they give a new proof that the thermodynamic free energy of the quantum model converges to the free energy of the classical model in the limit of large spins they also obtain gaussian and free bose gas limits for the classical and quantum models respectively

bi hamiltonian structure of the bi dimensional superintegrable nonlinear isotonic oscillator the higher order superintegrability of the two dimensional isotonic oscillator noncentral oscillator with inversely quadratic nonlinearities also known as caged anisotropic oscillator with rational ratio of frequencies is directly related with the existence of some complex functions with interesting poisson bracket properties first the properties of these functions are studied and then it is proved that these complex functions determine the existence of a bi hamiltonian complex structure in the second part several real symplectic structures are obtained and the properties of the recursion operators are studied

random walk representations of the heisenberg model we develop random walk representations for the spin s heisenberg ferromagnet with nearest neighbor interactions we show that the spin s heisenberg model is a diffusion with local times controlled by the spin s ising model as a consequence expectations for the heisenberg model conditioned on zero diffusion are shown to be ising expectations

a low temperature expansion for classical n vector models ii renormalization group equations this paper continues the analysis of the low temperature expansions for classicaln vector models started in 1 a main part of it is a derivation of renormalization group equations and a construction of their solutions to do this we have to introduce a fluctuation integral connected with a next renormalization transformation and to make its preliminary analysis the results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites or actions described inductively in 1

periodic orbits for an infinite family of classical superintegrable systems we show that all bounded trajectories in the two dimensional classical system with the potential are closed for all integer and rational values of k the period is and does not depend on k this agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable