stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

on equivalence of two metrics in classical thermodynamics abstract the concepts of distance between states in equilibrium thermodynamics introduced by weinhold and ruppeiner were based on the stability conditions and theory of fluctuations respectively it has been shown that both these riemannian matrices are equivalent up to a multiplicative function they are connected by a contact transformation acting on the thermodynamic phase space the family of contact transformation produces a set of new metrics from some reference metric such transformations can be also considered as a practical method for evaluating partial derivatives of thermodynamic functions

instability in bacterial populations and the curvature tensor in the geometry associated with equilibrium thermodynamics the scalar curvature rs is a measure of the volume of correlation and therefore the singularities of rs indicates the system instabilities we explore the use of a similar approach to study instabilities in non equilibrium systems and we choose as a test example a colony of bacteria in this regard we follow the proposal made by obata et al of using the curvature tensor for studying system instabilities bacterial colonies are often found in nature in concentrated biofilms or other colony types which can grow into spectacular patterns visible under the microscope for instance it is known that a decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities this is opposed to the logistic law for birth and death that allows only a single uniform density to be stable although this homogeneous configuration is stable in the absence of bacterial interactions without logistic growth a density dependent swim speed v leads to phase separation via a spinodal instability thus we relate the singularities in the curvature tensor r to the spinodal instability that is the appearance of regions of different densities of bacteria

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

the periodic cauchy problem for a two component non isospectral cubic camassa holm system abstract in this paper we study the periodic cauchy problem for a two component non isospectral cubic camassa holm system which includes the fokas olver rosenau qiao forq or modified camassa holm mch equation and the two component mch system as two special cases the system is integrable in the sense of possessing a non isospectral lax pair with the spectrum depending on time t and admits multi peakon solutions in an explicit form furthermore we establish the local well posedness for the system in the besov space b 2 r s t with s 3 2 1 r where the key ingredients include the friedrichs regularization method the littlewood paley decomposition theory and the transport theory in besov spaces then we derive a precise blow up criteria which is dependent of the parameters t and t moreover by the intrinsic structure of the system we obtain a new blow up result for strong solutions with sufficient conditions on the initial data and parameters the entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves

gibbs entropy invariants of motion abstract invariants of motion are constructed departing from the gibbs entropy the method we applied also allows us to find the subspace of a real vector space to which the system is confined to during its temporal evolution we also find that the uncertainty principle is itself an invariant of motion for harmonic oscillator like hamiltonians

time dependent invariants of motion for complete sets of non commuting observables abstract using the generalized ehrenfest theorem the dynamics of the mean values of a complete set of non commuting observables csnco associated to a given hamiltonian is expressed we found refined time dependent invariants of motion tdim for the csnco and associated them with different lie algebras we also show that when the csnco are related through an antisymmetric matrix then the characteristic polynomial of the covariance matrix for the csnco is also a tdim examples for the heisemberg su 2 and so 2 groups are outlined

blow up phenomena for an integrable two component camassa holm system with cubic nonlinearity and peakon solutions abstract this paper is devoted to an integrable two component camassa holm system with cubic nonlinearity which includes the cubic camassa holm equation also called the fokas olver rosenau qiao equation as a special case the one peaked solitons peakons and two peakon solutions are described in an explicit formula then the local well posedness for the cauchy problem of the system is studied moreover we target at the precise blow up scenario for strong solutions to the system and establish a new blow up result with respect to the initial data

analytic solutions of the cauchy problem for two component shallow water systems this paper is mainly concerned with the cauchy problem for a modified two component camassa holm shallow water system with analytic initial data the analyticity of its solutions is proved in both variables globally in space and locally in time the obtained results can be also applied to the classical integrable two component camassa holm shallow water system and the b equation

riemannian geometry and stability of thermodynamical equilibrium systems a geometrical approach to statistical thermodynamics is proposed it is shown that any r parameter generalised gibbs distribution leads to a riemannian metric of parameter space the components of the metric tensor are represented by second moments of stochastic variables the scalar curvature r as a geometrical invariant is a function of the second and third moments so is strictly connected with fluctuations of the system in the case of a real gas r is positive and tends to infinity as the system approaches the critical point in the case of an ideal gas r 0 the obtained results and the results of the authors previous work suggests that for a wide class of models r tends to infinity near the critical point they treat r as a measure of the stability of the system they propose some sort of statistical principle only such models may be accepted for which r tends to infinity if the system is approaching the critical point it is shown that if this criterion is adopted for a class of models for which the scaling hypothesis holds then they obtain the new inequalities for the critical indices these inequalities are in good agreement with model calculations and experiment

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

reduced complexity hmm filtering with stochastic dominance bounds a convex optimization approach this paper uses stochastic dominance principles to construct upper and lower sample path bounds for hidden markov model hmm filters we consider an hmm consisting of an x state markov chain with transition matrix p by using convex optimization methods for nuclear norm minimization with copositive constraints we construct low rank stochastic matrices underline p and bar p so that the optimal filters using underline p bar p provably lower and upper bound with respect to a partially ordered set the true filtered distribution at each time instant since underline p and bar p are low rank say r the computational cost of evaluating the filtering bounds is o x r instead of o x 2 a monte carlo importance sampling filter is presented that exploits these upper and lower bounds to estimate the optimal posterior finally explicit bounds are given on the variational norm between the true posterior and the upper and lower bounds in terms of the dobrushin coefficient

a new integrable two component system with cubic nonlinearity in this paper a new integrable two component system mt m uxvx uv uvx uxv x nt n uxvx uv uvx uxv x where m u uxx and n v vxx is proposed our system is a generalized version of the integrable system mt m ux2 u2 x which was shown having cusped solution cuspon and w m shape soliton solutions by qiao j math phys 47 112701 2006 the new system is proven integrable not only in the sense of lax pair but also in the sense of geometry namely it describes pseudospherical surfaces accordingly infinitely many conservation laws are derived through recursion relations furthermore exact solutions such as cuspons and w m shape solitons are also obtained

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

convergence analysis of self adaptive equalizers a theoretical analysis of self adaptive equalization for data transmission is carried out starting from known convergence results for the corresponding trained adaptive filter the development relies on a suitable ergodicity model for the sequence of observations at the output of the transmission channel thanks to the boundedness of the decision function used for data recovery it can be proved that the algorithm is bounded strong convergence results can be reached when a perfect noiseless equalizer exists the algorithm will converge to it if the eye pattern is initially open otherwise convergence may take place towards certain other stationary points of the algorithm for which domains of attraction have been defined some of them will result in a poor error rate the case of a noisy channel exhibits limit points for the algorithm that differ from those of the classical trained algorithm the stronger the noise the greater the difference is one of the principal results of this study is the proof of the stability of the usual decision feedback algorithms once the learning period is over

information geometry of the spherical model motivated by the observation that geometrizing statistical mechanics offers an interesting alternative to more standard approaches we calculate the scaling behavior of the curvature mathcal r of the information geometry metric for the spherical model we find that mathcal r ensuremath sim ensuremath epsilon ensuremath 2 where ensuremath epsilon ensuremath beta c ensuremath ensuremath beta is the distance from criticality the discrepancy from the naively expected scaling mathcal r ensuremath sim ensuremath epsilon ensuremath 3 is explained and compared with that for the ising model on planar random graphs which shares the same critical exponents

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

blind adaptive multiuser detection the decorrelating detector and the linear minimum mean square error mmse detector are known to be effective strategies to counter the presence of multiuser interference in code division multiple access channels in particular those multiuser detectors provide optimum near far resistance when training data sequences are available the mmse multiuser detector can be implemented adaptively without knowledge of signature waveforms or received amplitudes this paper introduces an adaptive multiuser detector which converges for any initialization to the mmse detector without requiring training sequences this blind multiuser detector requires no more knowledge than does the conventional single user receiver the desired user s signature waveform and its timing the proposed blind multiuser detector is made robust with respect to imprecise knowledge of the received signature waveform of the user of interest

qualitative analysis for a new integrable two component camassa holm system with peakon and weak kink solutions this paper is devoted to a new integrable two component camassa holm system with peaked solitons peakons and weak kink solutions it is the first integrable system that admits weak kink and kink peakon interactional solutions in addition the new system includes both standard quadratic and cubic camassa holm equations as two special cases in the paper we first establish the local well posedness for the cauchy problem of the system and then derive a precise blow up scenario and a new blow up result for strong solutions to the system with both quadratic and cubic nonlinearity furthermore its peakon and weak kink solutions are discussed as well

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

dynamical behaviour of a statistical manifold associated with correlated walks a riemann space associated with a correlated walk cw model is investigated by a numerical method which is applicable to many other stochastic processes the behaviour of the riemann curvature is shown to have a close relation to the stability of the cw system the riemann space is spanned by two parameters representing jump probabilities of the cw model the cw manifold evolves through some characteristic eras it starts at a point and changes to a line and then to a homogeneous spherical surface of the scalar curvature r 1 2 the homogeneous space immediately transforms to an inhomogeneous space the inhomogeneity gradually grows a region of the inhomogeneous space violently oscillates in time and another region expands fast as a whole the curvature of the space decreases and soon becomes negative the oscillations have already started to fade away the negative curvature goes on decreasing and finally the space converges to a homogeneous saddle surface of r 1 such a behaviour of the curvature is shown to be well understood by the terms of stability and order parameter of stochastic processes in other words the inhomogeneity the oscillation and the decrease behaviour of the curvature are closely related to stability and orderliness of the correlated system

generalized metric phase space for quantum systems and the uncertainty principle abstract we demonstrate that when the gibbs entropy is an invariant of motion following information theory procedures it is possible to define a generalized metric phase space for the temporal evolution of the mean values of a given hamiltonian the metric is positive definite and this fact leads to a metric tensor k t whose properties are well defined working with these properties we shown that a the generalized uncertainty principle gup is always the summation over the principal minors of order 2 belonging to k t b several invariants of motion can be derived from the metric tensor and c particularly under certain conditions the gup itself is also a motion invariant

information geometry of boltzmann machines a boltzmann machine is a network of stochastic neurons the set of all the boltzmann machines with a fixed topology forms a geometric manifold of high dimension where modifiable synaptic weights of connections play the role of a coordinate system to specify networks a learning trajectory for example is a curve in this manifold it is important to study the geometry of the neural manifold rather than the behavior of a single network in order to know the capabilities and limitations of neural networks of a fixed topology using the new theory of information geometry a natural invariant riemannian metric and a dual pair of affine connections on the boltzmann neural network manifold are established the meaning of geometrical structures is elucidated from the stochastic and the statistical point of view this leads to a natural modification of the boltzmann machine learning rule

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

orbital stability of the train of peakons for an integrable modified camassa holm equation abstract in this paper we consider the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peakons and multi peakons using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm we prove that the sum of n sufficiently decoupled peakons is orbitally stable in the energy space

the cauchy problem and blow up phenomena of a new integrable two component camassa holm system abstract this paper considers the cauchy problem and blow up phenomena of a new integrable two component camassa holm system which is a natural extension of the fokas olver rosenau qiao equation firstly the local well posedness of the system in the critical besov space b 2 1 1 2 r b 2 1 1 2 r is investigated and it is shown that the data to solution mapping is holder continuous then a blow up criteria for the cauchy problem in the critical besov space is derived moreover with conditions on the initial data a new blow up criteria is obtained by virtue of the blow up criteria at hand and the conservative property of m and n along the characteristics finally a global existence result for the strong solution is established

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

thermodynamics of boson systems related to dunkl differential difference operators abstract we study the thermodynamics of systems based on a fock space representation inspired by the differential difference operators proposed in dunkl 1989 we calculate thermodynamic functions as the entropy and heat capacity and compare them with the standard boson case a calculation of the second virial coefficient and the scalar curvature in two and three dimensions show that these systems become repulsive within an interval of negative values of the reflection operator parameter in addition the stability of this system is examined as a function of

geometrical aspects of a generalized statistical mechanics abstract we discuss here the use of generalized forms of entropy taken as information measures to characterize phase transitions and critical behavior in thermodynamic systems our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models the thermodynamics of the system is the focus of attention in this geometric context

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

blow up phenomena and persistence properties for an integrable two component peakon system abstract in this paper we are concerned with an integrable two component peakon system which was proposed by xia qiao and zhou we present a precise blow up scenario and a new blow up result for strong solutions to the system moreover we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically respectively

stability and anyonic behavior of systems with m statistics abstract starting with the partition function z for systems with m statistics as proposed in chen et al 1996 1 we calculate from the metric g ln z the scalar curvature r in two and three dimensions our results exhibit the details of the anyonic behavior as a function of the fugacity z and the identical particle maximum occupancy number m we also compare the stability of systems for m 1 with the fermionic m 1 and bosonic m cases

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

information geometry of the ising model on planar random graphs it has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure particularly in the case where there are two such parameters such as the ising model with inverse temperature ensuremath beta and external field h in various two parameter calculable models the scalar curvature mathcal r of the information metric has been found to diverge at the phase transition point ensuremath beta c and a plausible scaling relation postulated mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath alpha ensuremath 2 for spin models the necessity of calculating in nonzero field has limited analytic consideration to one dimensional mean field and bethe lattice ising models in this paper we use the solution in field of the ising model on an ensemble of planar random graphs where ensuremath alpha ensuremath 1 ensuremath beta 1 2 ensuremath gamma 2 to evaluate the scaling behavior of the scalar curvature and find mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath 2 the apparent discrepancy is traced back to the effect of a negative ensuremath alpha

analysis of adaptive step size sa algorithms for parameter tracking we present proofs and data for adaptive step size algorithms for tracking time varying parameters when recursive stochastic approximation type algorithms are used a classical problem in adaptive control and communication theory concerns the tracking of the best fit of a given form when the statistics or the parameters change slowly a major and yet unresolved problem has been the choice of the step sizes in the tracking algorithm an algorithm for adapting the step size using the same system measurements which are used for the tracking was suggested by benveniste and various examples worked out by brossier the numerical results were very encouraging but proofs were lacking these proofs are supplied here together with supporting numerical data the proofs are based on recent results in stochastic approximation the adaptive step size technique works very well indeed much supporting analysis is presented particularly concerning the interpretation of certain stationary processes as stationary pathwise derivatives finite difference forms are also treated these are mathematically simpler and can be applied to a wide variety of systems even when the system is not well modeled the data shows that they work well

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

a q analogue of the supersymmetric oscillator and its q superalgebra a q analogue of the supersymmetric oscillator is constructed out of q boson and q fermion creation and annihilation operators for a fixed nb nf 2n where nb and nf are q boson and q fermion occupation numbers the irreducible representations of the q superalgebra generated by the q oscillator are 2n 1 dimensional particular cases when q is a root of or 1 are discussed a realization of the quantum group suq 2 is obtained using a pair of 1 fermion creation and annihilation operators

weak convergence and asymptotic properties of adaptive filters with constant gains the basic adaptive filtering algorithm x n 1 epsilon x n epsilon epsilon y n y n x n epsilon psi n is analyzed using the theory of weak convergence apart from some very special cases the analysis is hard when done for each fixed epsilon 0 but the weak convergence techniques are set up to provide much information for small epsilon the relevant facts from the theory are given define x epsilon cdot by x epsilon t x n epsilon on n epsilon n epsilon epsilon then weak distributional convergence of x epsilon cdot and of x epsilon cdot t epsilon is proved under very weak assumptions where t epsilon rightarrow infty as epsilon rightarrow 0 the normalized errors x n epsilon theta sqrt epsilon are analyzed where theta is a stable point for the mean algorithm the asymptotic properties of a projection algorithm are developed where the x n epsilon are truncated at each iteration if they fall outside of a given set

performance analysis of general tracking algorithms a general family of tracking algorithms for linear regression models is studied it includes the familiar least mean square gradient approach recursive least squares and kalman filter based estimators the exact expressions for the quality of the obtained estimates are complicated approximate and easy to use expressions for the covariance matrix of the parameter tracking error are developed these are applicable over the whole time interval including the transient and the approximation error can be explicitly calculated

continuous time tracking algorithms involving two time scale markov chains this work is concerned with least mean squares lms algorithms in continuous time for tracking a time varying parameter process a distinctive feature is that the true parameter process is changing at a fast pace driven by a finite state markov chain the states of the markov chain are divisible into a number of groups within each group the transitions take place rapidly among different groups the transitions are infrequent introducing a small parameter into the generator of the markov chain leads to a two time scale formulation the tracking objective is difficult to achieve nevertheless a limit result is derived yielding algorithms for limit systems moreover the rates of variation of the tracking error sequence are analyzed under simple conditions it is shown that a scaled sequence of the tracking errors converges weakly to a switching diffusion in addition a numerical example is provided and an adaptive step size algorithm developed

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

the camassa holm hierarchy n dimensional integrable systems and algebro geometric solution on a symplectic submanifold this paper shows that the camassa holm ch spectral problem yields two different integrable hierarchies of nonlinear evolution equations nlees one is of negative order ch hierachy while the other one is of positive order ch hierarchy the two ch hierarchies possess the zero curvature representations through solving a key matrix equation we see that the well known ch equation is included in the negative order ch hierarchy while the dym type equation is included in the positive order ch hierarchy furthermore under two constraint conditions between the potentials and the eigenfunctions the ch spectral problem is cast in 1 a new neumann like n dimensional system when it is restricted into a symplectic submanifold of 2n which is proven to be integrable by using the dirac poisson bracket and the r matrix process and 2 a new bargmann like n dimensional system when it is considered in the whole 2n which is proven to be integrable by using the standard poisson bracket and the r matrix process

riemann scalar curvature of ideal quantum gases obeying gentile s statistics the scalar curvature r of ideal quantum gases obeying gentile s statistics is investigated by the method of information geometrical theory the r value is specified by the fugacity and the maximum number p of particles in a state the lowest case p 1 corresponds to fermi dirac statistics and the unbounded case p to bose einstein statistics in contrast to r 0 for ideal classical gases obeying boltzmann statistics we find r 2 1 2 32 for p2 and r 2 1 2 32 for p 1 in 0 which is the classical limit this means that a quantum statistical character is left in r in the classical limit also a correlation between the sign of r and a quantum mechanical exchange effect is recognized for 0 and 1 furthermore we obtain results that support the instability interpretation of r proposed by janyszek and mrugala

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation