superintegrable systems on the two dimensional sphere s2 and the hyperbolic plane h2 the existence of superintegrable systems with n 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two dimensional sphere s2 and on the hyperbolic plane h2 the approach used is based on enforcing the conditions for the existence of two independent integrals further than the energy this is done in a way which allows us to discuss at once the cases of the sphere s2 and the hyperbolical plane h2 by considering the curvature as a parameter different superintegrable potentials are obtained as the solutions of certain systems of two dependent second order partial differential equations the euclidean results are directly recovered for 0 and the superintegrable potentials on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent superintegrable potentials for the values 1 and 1 some new superintegrable potentials are f

energy decay for timoshenko systems of memory type abstract linear systems of timoshenko type equations for beams including a memory term are studied the exponential decay is proved for exponential kernels while polynomial kernels are shown to lead to a polynomial decay the optimality of the results is also investigated

contractions of lie algebras and separation of variables the n dimensional sphere inonu wigner contractions from the rotation group o n 1 to the euclidean group e n are used to relate the separation of variables in laplace beltrami operators on n dimensional spheres and euclidean spaces respectively in this article we consider all subgroup type coordinates corresponding to different chains of subgroups of o n 1 and e n respectively in particular the contractions relate the graphical formalism of trees on spheres to the clusters on euclidean spaces introduced in this article the contractions are considered analytically on several levels the vector fields realizing the lie algebras the complete sets of commuting operators characterizing separable coordinate systems the coordinate systems themselves and the separated eigenfunctions

asymptotic stability of abstract dissipative systems with infinite memory we consider in this paper the problem of asymptotic behavior of solutions to an abstract linear dissipative integrodifferential equation with infinite memory past history modeling linear viscoelasticity we show that the stability of the system holds for a much larger class of the convolution kernels than the one considered in the literature and we provide a relation between the decay rate of the solutions and the growth of the kernel at infinity some applications are also given

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

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

exact solvability of superintegrable benenti systems we establish quantum and classical exact solvability for two large classes of maximally superintegrable benenti systems in n dimensions with arbitrarily large n namely we solve the hamilton jacobi and schrodinger equations for the systems in question the results obtained are illustrated for a model with the cubic potential

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

integrable deformations of oscillator chains from quantum algebras a family of completely integrable nonlinear deformations of systems of n harmonic oscillators are constructed from the non standard quantum deformation of the sl 2 algebra explicit expressions for all the associated integrals of motion are given and the long range nature of the interactions introduced by the deformation is shown to be linked to the underlying co algebra structure separability and superintegrability properties of such systems are analysed and their connection with classical angular momentum chains is used to construct a non standard integrable deformation of the xxx hyperbolic gaudin system

the number of synaptic inputs and the synchrony of large sparse neuronal networks the prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons we study synchronization in models of large networks of spiking neurons with random sparse connectivity synchrony occurs only when the average number of synapses m that a cell receives is larger than a critical value mc below mc the system is in an asynchronous state in the limit of weak coupling assuming identical neurons we reduce the model to a system of phase oscillators that are coupled via an effective interaction in this framework we develop an approximate theory for sparse networks of identical neurons to estimate mc analytically from the fourier coefficients of our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it we apply this theory to compute mc for a model of inhibitory networks of integrate and fire i f neurons as a function of the intrinsic neuronal properties e g the refractory period tr the synaptic time constants and the strength of the external stimulus iext the number mc is found to be nonmonotonous with the strength of iext for tr 0 we estimate the minimum value of mc over all the parameters of the model to be 363 8 above mc the neurons tend to fire in smeared one cluster states at high firing rates and smeared two or more cluster states at low firing rates refractoriness decreases mc at intermediate and high firing rates these results are compared to numerical simulations we show numerically that systems with different sizes n behave in the same way provided the connectivity m is such that 1 meff 1 m 1 n remains constant when n varies this allows extrapolating the large n behavior of a network from numerical simulations of networks of relatively small sizes n 800 in our case we find that our theory predicts with remarkable accuracy the value of mc and the patterns of synchrony above mc provided the synaptic coupling is not too large we also study the strong coupling regime of inhibitory sparse networks all of our simulations demonstrate that increasing the coupling strength reduces the level of synchrony of the neuronal activity above a critical coupling strength the network activity is asynchronous we point out a fundamental limitation for the mechanisms of synchrony relying on inhibition alone if heterogeneities in the intrinsic properties of the neurons and spatial fluctuations in the external input are also taken into account

a maximally superintegrable system on an n dimensional space of nonconstant curvature abstract we present a novel hamiltonian system in n dimensions which admits the maximal number 2 n 1 of functionally independent quadratic first integrals this system turns out to be the first example of a maximally superintegrable hamiltonian on an n dimensional riemannian space of nonconstant curvature and it can be interpreted as the intrinsic smorodinsky winternitz system on such a space moreover we provide three different complete sets of integrals in involution and solve the equations of motion in closed form

conformal symmetries of spacetimes in this paper we give a unified and global new approach to the study of the conformal structure of the three classical riemannian spaces as well as of the six relativistic and non relativistic spacetimes minkowskian de sitter anti de sitter and both newton hooke and galilean we obtain general expressions within a cayley klein framework holding simultaneously for all these nine spaces whose cycles including geodesics and circles are explicitly characterized in a new way the corresponding cycle preserving symmetries which give rise to mobius like conformal lie algebras together with their differential realizations are then deduced without having to resort to solving the conformal killing equations we show that each set of three spaces with the same signature type and any curvature have isomorphic conformal algebras these are related through an apparently new conformal duality laplace and wave type differential equations with conformal algebra symmetry are finally constructed

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

general stability result for viscoelastic wave equations in this paper we consider a viscoelastic equation and establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity our result allows a wider class of relaxation functions and improves earlier results in the literature

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

a systematic construction of completely integrable hamiltonians from coalgebras a universal algorithm to construct n particle classical and quantum completely integrable hamiltonian systems from representations of coalgebras with casimir elements is presented in particular this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of hamiltonian systems with coalgebra symmetry in order to illustrate this general method the algebra and the oscillator algebra are used to derive new classical integrable systems including a generalization of gaudin calogero systems and oscillator chains quantum deformations are then used to obtain some explicit integrable deformations of the previous long range interacting systems and a non coboundary deformation of the 1 1 poincare algebra is shown to provide a new ruijsenaars schneider like hamiltonian

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

superintegrable systems on 3 dimensional curved spaces eisenhart formalism and separability the eisenhart geometric formalism which transforms an euclidean natural hamiltonian h t v into a geodesic hamiltonian t with one additional degree of freedom is applied to the four families of quadratically superintegrable systems with multiple separabilities in the euclidean plane first the separability and superintegrability of such four geodesic hamiltonians t r r a b c d in a three dimensional curved space are studied and then these four systems are modified with the addition of a potential u r leading to h r t r u r second we study the superintegrability of the four hamiltonians h r h r r where r is a certain position dependent mass that enjoys the same separability as the original system h r all the hamiltonians here studied describe superintegrable systems on non euclidean three dimensional manifolds with a broken spherical symmetry

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

integro differential equations of hyperbolic type with positive definite kernels abstract the main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in hilbert spaces for kernels that decay exponentially at infinity and possess strongly positive definite primitives the exponential stability of weak solutions is obtained in the energy norm it is also shown that this theory applies to several examples of kernels with possibly variable sign and to a problem in nonlinear viscoelasticity

maximal superintegrability on n dimensional curved spaces a unified algebraic construction of the classical smorodinsky winternitz systems on the nd sphere euclidean and hyperbolic spaces through the lie groups so n 1 iso n and so n 1 is presented firstly general expressions for the hamiltonian and its integrals of motion are given in a linear ambient space n 1 and secondly they are expressed in terms of two geodesic coordinate systems on the nd spaces themselves with an explicit dependence on the curvature as a parameter on the sphere the potential is interpreted as a superposition of n 1 oscillators furthermore each lie algebra generator provides an integral of motion and a set of 2n 1 functionally independent ones are explicitly given in this way the maximal superintegrability of the nd euclidean smorodinsky winternitz system is shown for any value of the curvature

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

the relationship between structural and functional connectivity graph theoretical analysis of an eeg neural mass model we investigated the relationship between structural network properties and both synchronization strength and functional characteristics in a combined neural mass and graph theoretical model of the electroencephalogram eeg thirty two neural mass models nmms each representing the lump activity of reasonably large groups of interacting excitatory and inhibitory neurons were reciprocally and excitatory coupled using random rewiring as described by watts and strogatz numerical analysis of the network revealed an abrupt transition towards a synchronized state as a function of increasing coupling strength alpha synchronization increased with increasing degree and decreasing regularity of the network parameters of the functional network showed a diverse dependency on structural connectivity normalized clustering coefficient gamma and path length lambda increased with increasing alpha for sufficiently large alpha however gamma decreased with increasing rewiring probability p while lambda increased hence a structured functional network exists despite the randomness of the underlying structural network that is patterns of functional connectivity are influenced by patterns of the corresponding structural level but do not necessarily agree with those

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

general decay of the solution energy in a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u t t u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove that for certain class of relaxation functions and certain initial data the rate of decay of energy is similar to that of g this result improves earlier ones in the literature in which only the exponential and polynomial decay rates are considered

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

asymptotic stability of semigroups associated with linear weak dissipative systems with memory we study the asymptotic behavior of the solutions of a class of linear dissipative integral differential equations we show in the abstract setting a necessary and sufficient condition to get an exponential decay of the solution in the case of the lack of exponential decay we find the polynomial rate of decay of the solution some examples are given

synchronization in the kuramoto model a dynamical gradient network approach we propose a dynamical gradient network approach to consider the synchronization in the kuramoto model our scheme to adaptively adjust couplings is based on the dynamical gradient networks where the number of links in each time interval is the same as the number of oscillators but the links in different time intervals are also different the gradient network in the n 1 th time interval is decided by the oscillator dynamics in the nth time interval according to the gradient network in the n 1 th time interval only one inlink s coupling for each oscillator is adjusted by a small incremental coupling during the transition to synchronization the intensities for all oscillators are identical direct numerical simulations fully verify that the synchronization in the kuramoto model is realized effectively even if there exist delayed couplings and external noise

optimal detection of functional connectivity from high dimensional eeg synchrony data computing phase locking values between eeg signals is a popular method for quantifying functional connectivity however this method involves large scale high resolution datasets which impose a serious multiple testing problem standard multiple testing methods fail to exploit the information from the complex dependence structure that varies across hypotheses in spectral temporal and spatial dimensions and result in a severe loss of power they tend to control the false positives at the cost of hiding true positives we introduce a new approach called optimal discovery procedure odp for identifying synchrony that is statistically significant odp maximizes the number of true positives for a given number of false positives and thus offers a theoretical optimum for detecting significant synchrony in a multiple testing situation we demonstrate the utility of this method with plv data obtained from a visual search study we also present simulation analysis to confirm the validity and relevance of using odp in comparison with the standard fdr method for given configurations of true synchrony we also compare the effectiveness of odp with our previously published investigation of hierarchical fdr method singh and phillips 2010

note on intrinsic decay rates for abstract wave equations with memory in this paper we consider a viscoelastic abstract wave equation with memory kernel satisfying the inequality g h g 0 s 0 where h s is a given continuous positive increasing and convex function such that h 0 0 we shall develop an intrinsic method based on the main idea introduced by lasiecka and tataru uniform boundary stabilization of semilinear wave equation with nonlinear boundary dissipation differential and integral equations 6 507 533 1993 for determining decay rates of the energy given in terms of the function h s this will be accomplished by expressing the decay rates as a solution to a given nonlinear dissipative ode we shall show that the obtained result while generalizing previous results obtained in the literature is also capable of proving optimal decay rates for polynomially decaying memory kernels h s sp and for the full range of admissible parameters p 1 2 while such result has been known for certain restrictive ranges of the parameters p 1 3 2

isochronic potentials and new family of superintegrable systems a general scheme for constructing superintegrable systems for separated hamiltonians in an arbitrary number of degrees of freedom is presented the resulting family contains previously known superintgrable systems with separated hamiltonians in cartesian coordinates at least however in general the models belonging to the family admit additional integrals which are nonpolynomial functions of momenta an application of the method for the construction of superintegrable models of liouville type is described

decay estimates for second order evolution equations with memory abstract this paper develops a unified method to derive decay estimates for general second order integro differential evolution equations with semilinear source terms depending on the properties of convolution kernels at infinity we show that the energy of a mild solution decays exponentially or polynomially as t our approach is based on integral inequalities and multiplier techniques these decay results can be applied to various partial differential equations we discuss three examples a semilinear viscoelastic wave equation a linear anisotropic elasticity model and a petrovsky type system

on harmonic oscillators on the two dimensional sphere s2 and the hyperbolic plane h2 ii two harmonic oscillators isotropic and nonisotropic 2 1 are studied on the two dimensional sphere s2 and the hyperbolic plane h2 both systems are integrable and super integrable with constants of motion quadratic in the momenta these properties are shown to derive from a complex factorization for the constants of motion which holds for arbitrary values of the curvature and the dynamics of the euclidean harmonic 1 1 and 2 1 oscillators is directly recovered for 0 the harmonic oscillators on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent potentials for the values 1 and 1 finally a particular potential is proposed for representing the general spherical hyperbolic n 1 anisotropic harmonic oscillator on a two dimensional manifold of constant curvature

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory abstract we are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation which describes the dynamics of many viscoelastic systems and covers the well known timoshenko system we present an optimal energy decay estimate under very weak conditions more specifically we prove that the energy of the system decays to zero at least at the rate of t 1 only under the condition that the related kernel functions are non increasing and integrable moreover we apply our abstract results to three classes of coupled systems of second order partial differential equations arising in viscoelasticity

second order superintegrable systems in conformally flat spaces iv the classical 3d stackel transform and 3d classification theory this article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces in the first part of the article we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different three dimensional spaces we show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is stackel equivalent to a system on a constant curvature space in the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates we find that there are eight families of these systems

coupled second order evolution equations with fading memory optimal energy decay rate abstract of concern is the cauchy problem for coupled second order semilinear evolution equations in a hilbert space with indirect memory damping we find an approach to obtain successfully an optimal rate of uniform decay for the system energy only under basic conditions on the memory kernels simultaneously the same rate is also obtained with less difficulty for the corresponding single memory dissipative second order evolution equations as can be seen our results essentially improve the previously related ones in the literature the abstract results are then applied to several concrete problems in the real world

hamiltonian systems admitting a runge lenz vector and an optimal extension of bertrand s theorem to curved manifolds bertrand s theorem asserts that any spherically symmetric natural hamiltonian system in euclidean 3 space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a kepler system in this paper we extend this classical result to curved spaces by proving that any hamiltonian on a spherically symmetric riemannian 3 manifold which satisfies the same conditions as in bertrand s theorem is superintegrable and given by an intrinsic oscillator or kepler system as a byproduct we obtain a wide panoply of new superintegrable hamiltonian systems the demonstration relies on perlick s classification of bertrand spacetimes and on the construction of a suitable globally defined generalization of the runge lenz vector

coupled second order semilinear evolution equations indirectly damped via memory effects abstract in this paper we study a system of coupled second order semilinear evolution equations in a hilbert space which is partially damped through memory effects a global existence and uniqueness theorem regarding the solutions to its cauchy problem is given following this we analyze stability of the system energy and obtain various decay rates corresponding to the integral kernel some of them being optimal moreover we apply our abstract theory to concrete systems including that of timoshenko type for which we essentially improve the previous related results