a deterministic analysis of stochastic approximation with randomized directions we study the convergence of two stochastic approximation algorithms with randomized directions the simultaneous perturbation stochastic approximation algorithm and the random direction kiefer wolfowitz algorithm we establish deterministic necessary and sufficient conditions on the random directions and noise sequences for both algorithms and these conditions demonstrate the effect of the random directions on the sample path behavior of the algorithms studied we discuss ideas for further research in the analysis and design of these algorithms

towards hilbert s tenth problem for rings of integers through iwasawa theory and heegner points for a positive proportion of primes p and q we prove that mathbb z is diophantine in the ring of integers of mathbb q root 3 of p sqrt q this provides a new and explicit infinite family of number fields k such that hilbert s tenth problem for o k is unsolvable our methods use iwasawa theory and congruences of heegner points in order to obtain suitable rank stability properties for elliptic curves

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

an asymptotic property of the camassa holm equation abstract asymptotic densities are used to study an asymptotic property of the dispersionless camassa holm equation an asymptotic density of a global solution is a weak limit of its scaled momentum density along a sequence of time increasing to infinity for a global solution with non negative compactly supported initial momentum density we show that if the asymptotic density is unique then it is a positive combination of dirac measures supported in a bounded interval in the non negative axis with zero as the only possible accumulation point in other words if the scaled momentum density does not oscillate as time goes to infinity the solution behaves as a combination of peakons moving to the right at different speeds in contrast to many investigations on the topic our approach is not spectral theoretic and hence is independent of the structure of the isospectral problem associated with the equation

long time behaviour for a class of low regularity solutions of the camassa holm equation in this paper we investigate the long time behaviour for a class of low regularity solutions of the camasssa holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

blow up solutions of the general b equationa in this short paper we consider the b equation a new precise blow up scenario is described and some new results are obtained for the blow up phenomena

variational derivation of a geophysical camassa holm type shallow water equation abstract we consider two dimensional one layer irrotational equatorial oceanic flows in the f plane approximation in the small amplitude shallow water regime we deduce by a variational method in the lagrangian formalism a camassa holm type equation

fredholm determinants and the camassa holm hierarchy the equation of camassa and holm 2 2 is an approximate description of long waves in shallow water

some new soliton like solutions and periodic wave solutions with loop or without loop to a generalized kdv equation abstract in this paper by using the integral bifurcation method we study a generalized kdv equation which was first derived by fokas from physical considerations via a methodology of fuchssteiner all kinds of soliton like or kink like wave solutions and periodic wave solutions with loop or without loop are obtained smooth compacton like periodic wave solution and non smooth periodic cusp wave solution are also obtained their dynamic properties are investigated and their profiles are given by mathematical software

global weak solutions for a two component camassa holm shallow water system abstract in this paper we prove the existence of global weak solution for an integrable two component camassa holm shallow water system provided the initial data satisfying some certain conditions

a robust information clustering algorithm we focus on the scenario of robust information clustering ric based on the minimax optimization of mutual information mi the minimization of mi leads to the standard mass constrained deterministic annealing clustering which is an empirical risk minimization algorithm the maximization of mi works out an upper bound of the empirical risk via the identification of outliers noisy data points furthermore we estimate the real risk vc bound and determine an optimal cluster number of the ric based on the structural risk minimization principle one of the main advantages of the minimax optimization of mi is that it is a nonparametric approach which identifies the outliers through the robust density estimate and forms a simple data clustering algorithm based on the square error of the euclidean distance

robustness of discrete time direct adaptive controllers the problem of preserving stability of discrete time adaptive controllers in spite of reduced order modeling and output disturbances is addressed in this paper conditions for global stability convergence of the tracking error with bounded signals are derived for a discrete time pole zero placement adaptive controller where the parameter estimator is modified in terms of normalized signals following an input output perpective the overall system is decomposed into two subsystems reflecting the parameter estimation and modeling errors respectively and its stability is studied using the sector stability and passivity theorems first the analysis is carried for the class of disturbances and reference inputs that are either decaying or can be exactly hulled by a linear controller of the chosen structure in this l 2 framework it is shown that the only substantive assumption to assure stability is the existence of a linear controller such that the closed loop transfer function verifies certain conicity conditions the convergence speed and alertness properties of various parameter adaptation algorithms regarding this condition are discussed the results are further extended to a broader class of l infty disturbances and reference inputs

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

elliptic curves retaining their rank in finite extensions and hilbert s tenth problem for rings of algebraic numbers using poonen s version of the weak vertical method we produce new examples of large and small rings of algebraic numbers including rings of integers where z and or the ring of integers of a subfield are existentially definable and or where the ring version of mazur s conjecture on the topology of rational points does not hold

on adaptive learning rate that guarantees convergence in feedforward networks this paper investigates new learning algorithms lf i and lf ii based on lyapunov function for the training of feedforward neural networks it is observed that such algorithms have interesting parallel with the popular backpropagation bp algorithm where the fixed learning rate is replaced by an adaptive learning rate computed using convergence theorem based on lyapunov stability theory lf ii a modified version of lf i has been introduced with an aim to avoid local minima this modification also helps in improving the convergence speed in some cases conditions for achieving global minimum for these kind of algorithms have been studied in detail the performances of the proposed algorithms are compared with bp algorithm and extended kalman filtering ekf on three bench mark function approximation problems xor 3 bit parity and 8 3 encoder the comparisons are made in terms of number of learning iterations and computational time required for convergence it is found that the proposed algorithms lf i and ii are much faster in convergence than other two algorithms to attain same accuracy finally the comparison is made on a complex two dimensional 2 d gabor function and effect of adaptive learning rate for faster convergence is verified in a nutshell the investigations made in this paper help us better understand the learning procedure of feedforward neural networks in terms of adaptive learning rate convergence speed and local minima

neural network based approximate output regulation of discrete time nonlinear systems the existing approaches to the discrete time nonlinear output regulation problem rely on the offline solution of a set of mixed nonlinear functional equations known as discrete regulator equations for complex nonlinear systems it is difficult to solve the discrete regulator equations even approximately moreover for systems with uncertainty these approaches cannot offer a reliable solution by combining the approximation capability of the feedforward neural networks nns with an online parameter optimization mechanism we develop an approach to solving the discrete nonlinear output regulation problem without solving the discrete regulator equations explicitly the approach of this paper can be viewed as a discrete counterpart of our previous paper on approximately solving the continuous time nonlinear output regulation problem

holder continuity for a modified camassa holm system this paper is concerned with the holder continuity for a modified camassa holm system we show that the date to solution for the modified camassa holm system is holder continuous from bounded set of sobolev spaces with exponent s 3 2 measured in a weaker sobolev norm with index r s on the line our result also holds for the periodic case

robust neural network tracking controller using simultaneous perturbation stochastic approximation this paper considers the design of robust neural network tracking controllers for nonlinear systems the neural network is used in the closed loop system to estimate the nonlinear system function we introduce the conic sector theory to establish a robust neural control system with guaranteed boundedness for both the input output i o signals and the weights of the neural network the neural network is trained by the simultaneous perturbation stochastic approximation spsa method instead of the standard backpropagation bp algorithm the proposed neural control system guarantees closed loop stability of the estimation system and a good tracking performance the performance improvement of the proposed system over existing systems can be quantified in terms of preventing weight shifts fast convergence and robustness against system disturbance

discrete time adaptive backstepping nonlinear control via high order neural networks this paper deals with adaptive tracking for discrete time multiple input multiple output mimo nonlinear systems in presence of bounded disturbances in this paper a high order neural network honn structure is used to approximate a control law designed by the backstepping technique applied to a block strict feedback form bsff this paper also includes the respective stability analysis on the basis of the lyapunov approach for the whole controlled system including the extended kalman filter ekf based nn learning algorithm applicability of the scheme is illustrated via simulation for a discrete time nonlinear model of an electric induction motor

blow up phenomena and global existence for a two component camassa holm system with an arbitrary smooth function abstract in the paper we mainly investigate blow up phenomena and global existence of strong solutions to a two component camassa holm systems with an arbitrary smooth function h for three types of smooth functions h by using a conservation law and the sign preserving property of strong solutions we obtain two new blow up results and a new global existence result for the two component system our obtained results generalize and cover the recent results in yan et al 2015 zhang and yin 2015 2016

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

learning and convergence analysis of neural type structured networks a class of feedforward neural networks structured networks has recently been introduced as a method for solving matrix algebra problems in an inherently parallel formulation a convergence analysis for the training of structured networks is presented since the learning techniques used in structured networks are also employed in the training of neural networks the issue of convergence is discussed not only from a numerical algebra perspective but also as a means of deriving insight into connectionist learning bounds on the learning rate are developed under which exponential convergence of the weights to their correct values is proved for a class of matrix algebra problems that includes linear equation solving matrix inversion and lyapunov equation solving for a special class of problems the orthogonalized back propagation algorithm an optimal recursive update law for minimizing a least squares cost functional is introduced it guarantees exact convergence in one epoch several learning issues are investigated

the integral bifurcation method and its application for solving a family of third order dispersive pdes abstract in this paper an improved method named the integral bifurcation method is introduced in order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations a family of third order dispersive partial differential equations which were given by a degasperis d holm and a hone are studied many integral bifurcations are obtained for different parameter conditions by using these integral bifurcations many travelling wave solutions such as loop soliton solutions solitary wave solutions cusp soliton solutions and periodic wave solutions are obtained in particular under the conditions c 1 0 c 2 c 3 1 a very peculiar periodic wave solution is obtained

local well posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude abstract we study the local well posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water we use an approach proposed by kato which is based on semigroup theory for quasi linear equations we also show that singularities for the model equation can occur only in the form of wave breaking in particular surging breakers

a note on the camassa holm equation abstract we consider a generalized camassa holm equation we improve the compactness argument of 5 6 7 17 establishing the existence of global weak solutions in h 1 without the use of an ole nik type estimate the potential interest of this improvement is linked to convergence analysis of numerical schemes for which it may not be easy to verify an ole nik type estimate

well posedness and blow up phenomena for a higher order shallow water equation this paper deals with the cauchy problem for a higher order shallow water equation yt auxy buyx 0 where y 2ku i x2 ku and k 2 the local well posedness of solutions for the cauchy problem in sobolev space hs r with s 7 2 is obtained under some assumptions the existence and uniqueness of the global solutions to the equation are shown and conditions that lead to the development of singularities in finite time for the solutions are also acquired finally the weak solution for the equation is considered

local in space blow up criteria for a class of nonlinear dispersive wave equations abstract this paper is concerned with blow up phenomena for the nonlinear dispersive wave equation on the real line u t u x x t f u x f u x x x g u f u 2 u x 2 x 0 that includes the camassa holm equation as well as the hyperelastic rod wave equation f u k u 2 2 and g u 3 k u 2 2 as special cases we establish some a local in space blow up criterion i e a criterion involving only the properties of the data u 0 in a neighborhood of a single point simplifying and precising earlier blow up criteria for this equation

well posedness and blow up solution for a new coupled camassa holm equations with peakons a new camassa holm system with two component admitting peakon solitons is proposed the local well posedness for the system is established a criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained and an existence result for a class of local weak solution is also given

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

stable neural network based adaptive control for sampled data nonlinear systems for a class of mimo sampled data nonlinear systems with unknown dynamic nonlinearities a stable neural network nn based adaptive control approach which is an integration of an nn approach and the adaptive implementation of the variable structure control with a sector is developed the sampled data nonlinear system is assumed to be controllable and its state vector is available for measurement the variable structure control with a sector serves two purposes one is to force the system state to be within the state region in which the nn s are used when the system goes out of neural control and the other is to provide an additional control until the system tracking error metric is controlled inside the sector within the network approximation region the proof of a complete stability and a tracking error convergence is given and the setting of the sector and the nn parameters is discussed it is demonstrated that the asymptotic error of the system can be made dependent only on inherent network approximation errors and the frequency range of unmodeled dynamics simulation studies of a two link manipulator show the effectiveness of the proposed control approach

on the cauchy problem for a two component b family system with high order nonlinearity abstract this paper is devoted to the cauchy problem of two component b family system with high order nonlinearity which includes the well known camassa holm equation degasperis procesi equation novikov equation and their two component extension systems as special cases firstly we establish the local well posedness of the system in the non homogeneous besov space by using the littlewood paley decomposition and the transport equation theory secondly we investigate the blow up of strong solutions to the system thirdly we study the gevrey regularity and analyticity of the solutions to the system in the gevrey sobolev spaces by using the generalized ovsyannikov theorem moreover we find a lower bound of the lifespan and prove the continuity of the data to solution mapping

global synchronization criteria of linearly coupled neural network systems with time varying coupling in this paper global synchronization of linearly coupled neural network nn systems with time varying coupling is investigated the dynamical behavior of the uncoupled system at each node is general which can be chaotic or others the coupling configuration is time varying i e the coupling matrix is not a constant matrix based on lyapunov function method and the specific property of householder transform some criteria for the global synchronization are obtained by these criteria one can verify whether the coupled system with time varying coupling is globally synchronized which is important and useful for both understanding and interpreting synchronization phenomena and designing coupling configuration finally two simulations are given to demonstrate the effectiveness of the theoretical results

on some wave breaking for the nonlinear integrable shallow water wave equations mathematicians in this paper based on the conservation laws and the blow up scenario some new blow up phenomena are derived for the camassa holm equation and degasperis procesi equation which have attracted much attention due to their structure c 2015 elsevier ltd all rights reserved

the local well posedness existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude abstract this paper deals with the cauchy problem for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime which was proposed by constantin and lannes 2009 20 applying the pseudoparabolic regularization technique the local well posedness of strong solutions in sobolev space h s r with s 3 2 is established via a limiting procedure moreover a sufficient condition for the existence of weak solutions of the equation in lower order sobolev space h s r with 1 s 3 2 is obtained

global solutions to a special case of the generalized weakly dissipative periodic two component camassa holm system abstract we establish global well posedness to the generalized weakly dissipative periodic two component camassa holm system in some special case

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

discrete time neural net controller for a class of nonlinear dynamical systems a family of two layer discrete time neural net nn controllers is presented for the control of a class of mnth order mimo dynamical system no initial learning phase is needed so that the control action is immediate in other words the neural network nn controller exhibits a learning while functioning feature instead of a learning then control feature a two layer nn is used which is linear in the tunable weights the structure of the neural net controller is derived using a filtered error approach it is indicated that delta rule based tuning when employed for closed loop control can yield unbounded nn weights if 1 the net cannot exactly reconstruct a certain required function or 2 there are bounded unknown disturbances acting on the dynamical system certainty equivalence is not used overcoming a major problem in discrete time adaptive control in this paper new online tuning algorithms for discrete time systems are derived which are similar to spl epsiv modification for the case of continuous time systems that include a modification to the learning rate parameter and a correction term to the standard delta rule

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

on permanent and breaking waves in hyperelastic rods and rings we prove that the only global strong solution of the periodic rod equation vanishing in at least one point t0 x0 r s1 is the identically zero solution such conclusion holds provided the physical parameter of the model related to the finger deformation tensor is outside some neighborhood of the origin and applies in particular for the camassa holm equation corresponding to 1 we also establish the analogue of this unique continuation result in the case of non periodic solutions defined on the whole real line with vanishing boundary conditions at infinity our analysis relies on the application of new local in space blowup criteria and involves the computation of several best constants in convolution estimates and weighted poincare inequalities

diophantine definability and decidability in extensions of degree 2 of totally real fields we investigate diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of q among other results we prove the following the big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields are shown to hold for all extensions of degree 2 of totally real number fields the definability and undecidability results for integral closures of small and big subrings of number fields in the infinite algebraic extensions of q previously shown by the author to hold for totally real fields are extended to a large class of extensions of degree 2 of totally real fields this class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes

exact controllability of scalar conservation laws with an additional control in the context of entropy solutions in this paper we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions with the boundary data and a source term depending only on the time as controls we provide sufficient conditions for a state to be reachable in arbitrary small time to do so we introduce a slightly modified wave front tracking algorithm

a convergent finite difference scheme for the camassa holm equation with general h 1 initial data we suggest a finite dfference scheme for the camassa holm equation that can handle general h 1 initial data the form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality we prove that the difference scheme converges strongly in h 1 towards a dissipative weak solution of the camassa holm equation

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

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

existence of weak solutions in lower order sobolev space for a camassa holm type equation a generalized camassa holm equation containing a nonlinear dissipative effect is investigated the existence of the weak solution of the equation in lower order sobolev space hs with is established by using the techniques of the pseudoparabolic regularization and some a priori estimates derived from the equation itself

well posedness and analytic solutions of a two component water wave equation abstract we first establish the local well posedness for the cauchy problem of a two component water waves system in a critical nonhomogeneous besov space then we derive a continuation criterion for strong solutions to the system finally we prove the existence of analytic solutions to the system

global conservative solutions of the generalized hyperelastic rod wave equation abstract we prove existence of global and conservative solutions of the cauchy problem for the nonlinear partial differential equation u t u x x t f u x f u x x x g u 1 2 f u u x 2 x 0 where f is strictly convex or concave and g is locally uniformly lipschitz this includes the camassa holm equation f u u 2 2 and g u u u 2 as well as the hyperelastic rod wave equation f u u 2 2 and g u 3 u 2 2 as special cases it is shown that the problem is well posed for initial data in h 1 r if one includes a radon measure that corresponds to the energy of the system with the initial data the solution is energy preserving stability is proved both with respect to initial data and the functions f and g the proof uses an equivalent reformulation of the equation in terms of lagrangian coordinates

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

conservative weak solutions of the periodic cauchy problem for μhs equation we prove the existence and uniqueness of conservative weak solutions of the periodic cauchy problem for an integrable evolution equation from mathematical physics our method is to first prove the existence and uniqueness of solutions for all time in lagrangian coordinates and construct the maps to carry this result to the eulerian coordinates

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

asymptotic variational wave equations we investigate the equation ut f u x x f u ux 2 2 where f u is a given smooth function typically f u u 2 2 or u 3 3 this equation models unidirectional and weakly nonlinear waves for the variational wave equation utt c u c u ux x 0 which models some liquid crystals with a natural sinusoidal c the equation itself is also the euler lagrange equation of a variational problem two natural classes of solutions can be associated with this equation a conservative solution will preserve its energy in time while a dissipative weak solution loses energy at the time when singularities appear conservative solutions are globally defined forward and backward in time and preserve interesting geometric features such as the hamiltonian structure on the other hand dissipative solutions appear to be more natural from the physical point of view we establish the well posedness of the cauchy problem within the class of conservative solutions for initial data having finite energy and assuming that the flux function f has lipschitz continuous second order derivative in the case where f is convex the cauchy problem is well posed also within the class of dissipative solutions however when f is not convex we show that the dissipative solutions do not depend continuously on the initial data

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

the study of global weak solutions for a generalized hyperelastic rod wave equation abstract the global weak solution to the cauchy problem for a generalized hyperelastic rod wave equation or the generalized camassa holm equation is investigated in the space c 0 r l 0 h 1 r under the assumption that the initial value u 0 x belongs to the space h 1 r the limit of the viscous approximation for the equation is used to establish the existence of the global weak solution the key elements in our analysis include a one sided super bound estimate and a space time higher norm estimate on the first order derivatives of the solution with respect to the space variable

non uniform continuity of periodic holm staley b family of equations abstract we consider a family of non evolutionary partial differential equations known as the holm staley b family which includes the integrable camassa holm and degasperis procesi equations we show that the solution map is not uniformly continuous the proof relies on a construction of smooth periodic traveling waves with small amplitude

the periodic b equation and euler equations on the circle in this note we show that the periodic b equation can only be realized as a euler equation on the lie group diff s1 of all smooth and orientation preserving diffeomorphisms on the circle if b 2 i e for the camassa holm equation in this case the inertia operator generating the metric on diff s1 is given by a 1 x2 in contrast the degasperis procesi equation for which b 3 is not a euler equation on diff s1 for any inertia operator our result generalizes a recent result of kolev some geometric investigations on the degasperis procesi shallow water equation wave motion 46 412 419 2009

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

on diophantine definability and decidability in some infinite totally real extensions of ℚ let m be a number field and w m a set of its non archimedean primes then let o m wm x m ord t x 0 t w m let p 1 p r be a finite set of prime numbers let f inf be the field generated by all the p j i th roots of unity as j and i 1 r let k inf be the largest totally real subfield of f inf then for any e 0 there exist a number field m k inf and a set w m of non archimedean primes of m such that w m has density greater than 1 e and z has a diophantine definition over the integral closure of o m w m in k inf

blow up zero α limit and the liouville type theorem for the euler poincare equations in this paper we study the euler poincare equations in mathbb r n we prove local existence of weak solutions in w 2 p mathbb r n p n and local existence of unique classical solutions in h k mathbb r n k n 2 3 as well as a blow up criterion for the zero dispersion equation 0 we prove a finite time blow up of the classical solution we also prove that as the dispersion parameter vanishes the weak solution converges to a solution of the zero dispersion equation with sharp rate as 0 provided that the limiting solution belongs to c 0 t h k mathbb r n with k n 2 3 for the stationary weak solutions of the euler poincare equations we prove a liouville type theorem namely for 0 any weak solution mathbf u in h 1 mathbb r n is u 0 for 0 any weak solution mathbf u in l 2 mathbb r n is u 0

model free control of nonlinear stochastic systems with discrete time measurements consider the problem of developing a controller for general nonlinear and stochastic systems where the equations governing the system are unknown using discrete time measurement this paper presents an approach for estimating a controller without building or assuming a model for the system such an approach has potential advantages in accommodating complex systems with possibly time varying dynamics the controller is constructed through use of a function approximator such as a neural network or polynomial this paper considers the use of the simultaneous perturbation stochastic approximation algorithm which requires only system measurements a convergence result for stochastic approximation algorithms with time varying objective functions and feedback is established it is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite difference gradient approximations