measuring time frequency information content using the renyi entropies the generalized entropies of renyi inspire new measures for estimating signal information and complexity in the time frequency plane when applied to a time frequency representation tfr from cohen s class or the affine class the renyi entropies conform closely to the notion of complexity that we use when visually inspecting time frequency images these measures possess several additional interesting and useful properties such as accounting and cross component and transformation invariances that make them natural for time frequency analysis this paper comprises a detailed study of the properties and several potential applications of the renyi entropies with emphasis on the mathematical foundations for quadratic tfrs in particular for the wigner distribution we establish that there exist signals for which the measures are not well defined

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

exact solutions for the non conservative elastic stability of non uniform columns abstract the exact characteristic equations for the non conservative elastic stability of general elastic restrained non uniform columns governed by a fourth order ordinary differential equation with arbitrarily varying polynomial coefficients are derived the results are compared with those in the existing literature in addition the influence of the boundary conditions taper ratio and variation of linearly distributed tangential forces on the non conservative elastic stability of two kinds of tapered columns subjected to linearly distributed tangential forces is investigated it is shown that the parameter e of the linearly distributed tangential forces g x g 0 1 ex l has great influence on the instability mechanism for the clamped clamped columns

characterizations of δ volterra lattice a symmetric orthogonal polynomials interpretation abstract in this paper we introduce the volterra lattice which is interpreted in terms of symmetric orthogonal polynomials it is shown that the measure of orthogonality associated with these systems of orthogonal polynomials evolves in t like 1 x 2 1 t x where is a given positive borel measure moreover the volterra lattice is related to the toda lattice from miura or backlund transformations the main ingredients are orthogonal polynomials which satisfy an appell condition with respect to the forward difference operator and the characterization of the point spectrum of a jacobian operator that satisfies a volterra equation lax type theorem we also provide an explicit example of solutions of volterra and toda lattices and connect this example with the results presented in the paper

divergence of a rotating shaft with an intermediate support and conservative axial loads abstract the dynamic behavior of a rotating shaft with an intermediate support and conservative axial loads is analyzed using euler beam theory and the assumed mode method the equations of motion are then transformed to the standard form of eigenvalue problems for determining the critical dimensionless rotational speeds and the critical axial loads corresponding to the divergence type instability of the shaft results of numerical simulations are presented for various combinations of support locations axial loads and rotational speeds

upper bounds on shannon and renyi entropies for central potentials the renyi and shannon entropies are information theoretic measures which have enabled to formulate the position momentum uncertainty principle in a much more adequate and stringent way than the variance based heisenberg like relation moreover they are closely related to various energetic density functionals of quantum systems here we derive upper bounds on these quantities in terms of the second order moment r2 for general central potentials this improves previous results of this type the proof uses the renyi maximization procedure with a covariance constraint due to costa et al in proceedings of the fourth international workshop on energy minimization methods in computer vision and pattern recognition emmcvpr edited by a rangarajan m a t figueiredo and j zerubia springer verlag lisbon 2003 lect notes comput sci 52 211 2003 the contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account finally the appl

extension of the module of invertible transformations classification of integrable systems we demonstrate that for the systems of equations which are invariant under a point group or possess conservation laws of the zeroth or first order a nontrivial extension of the module of invertible transformations is possible that simplifies greatly a classification of the integrable systems of equations here we present an exhaustive list and a classification of the second order systems of the formu t u xx f u v u x v x v t v xx g u v u x v x which possess the conservation laws of higher order the reduction group approach allows us to define the lax type representations for some new equations of our list

entropic properties of d dimensional rydberg systems abstract the fundamental information theoretic measures the renyi r p and tsallis t p entropies p 0 of the highly excited rydberg quantum states of the d dimensional d 1 hydrogenic systems which include the shannon entropy p 1 and the disequilibrium p 2 are analytically determined by use of the strong asymptotics of the laguerre orthogonal polynomials which control the wavefunctions of these states we first realize that these quantities are derived from the entropic moments of the quantum mechanical probability r densities associated to the rydberg hydrogenic wavefunctions n l r which are closely connected to the l p norms of the associated laguerre polynomials then we determine the n asymptotics of these norms in terms of the basic parameters of our system the dimensionality d the nuclear charge and the hyperquantum numbers n l of the state by use of recent techniques of approximation theory finally these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states

on quantizer distortion and the upper bound for exponential entropy a sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions this result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources it is known that for a large class of quantizers both zero memory and vector the rth power distortion as well as some other distortion criteria are bounded below by a constant depending on r multiplied by a certain integral of the source s probability density it is pointed out that this bound can be rewritten in terms of the exponential entropy the exponential entropy measures the quantitative extent or range of the source distribution this fact gives a physical interpretation of the indicated limits of quantizer performance further elucidated by the main result

elastic stability of non uniform columns abstract a simple and efficient method is proposed to investigate the elastic stability of three different tapered columns subjected to uniformly distributed follower forces the influences of the boundary conditions and taper ratio on critical buckling loads are investigated the critical buckling loads of columns of rectangular cross section with constant depth and linearly varied width t 1 constant width and linearly varied depth t 2 and double taper t 3 are investigated among the three different non uniform columns considered taper ratio has the greatest influence on the critical buckling load of column t 3 and the lowest influence on that of column t 1 the types of instability mechanisms for hinged hinged and cantilever non uniform columns are divergence and flutter respectively however for clamped hinged and clamped clamped non uniform columns the type of instability mechanism for column t 1 is divergence while that for columns t 2 and t 3 is divergence only when the taper ratio of the columns is greater than certain critical values and flutter for the rest value of taper ratio when the type of instability mechanism changes from divergence to flutter there is a finite jump for the critical buckling load the influence of taper ratio on the elastic stability of cantilever column t 3 is very sensitive for small values of the taper ratio and there also exist some discontinieties in the critical buckling loads of flutter instability for a hinged hinged non uniform column t 2 or t 3 with a rotational spring at the left end of the column when the taper ratio is less than the critical value the instability mechanism changes from divergence to flutter as the rotational spring constant is increased for a clamped elastically supported non uniform column when the taper ratio is greater than the critical value the instability mechanism changes from flutter to divergence as the translational spring constant is increased

explicit auto transformations of integrable chains a construction scheme for explicit auto transformations of integrable discrete differential equations chains is presented these transformations are rather convenient to obtain the exact solutions for chains as well as associated partial differential systems on the other hand they exemplify new integrable discrete mappings their group properties are also of great interest the scheme is illustrated by several examples of integrable systems which contains the nonlinear schrodinger system and the landau lifshits model

on some linear integral equations generating solutions of nonlinear partial differential equations abstract two types of linear inhomogeneous integral equations which yield solutions of a broad class of nonlinear evolution equations are investigated one type is characterized by a two fold integration with an arbitrary measure and contour over a complex variable k and thier complex conjugates whereas the other one has a two fold integration over one and the same contour the inhomogeneous term which may contain an arbitrary function of k makes it possible to define a matrix structure on the solutions of the integral equations the elements of these matrices are shown to obey a system of partial differential equations the special form of which depends on the choice of the dispersion relation occurring in the integral equations for special elements of the matrices closed partial differential equations can be derived such as e g the nonlinear schrodinger equation and the real and complex modified korteweg de vries and sine gordon equations the relations between the matrix elements are shown to lead to miura transformations between the various partial differential equations

elastic stability of euler columns with a continuous elastic restraint using variational iteration method in this paper analysis of elastic stability for continuously restrained euler columns is conducted by means of variational iteration method vim a uniform homogeneous column is assumed to be restrained along its length the restraint considered in this study is an elastic foundation model in engineering practice and it is of great interest to foundation engineers hence the variation of the critical buckling loads with the stiffness of elastic restraint is investigated using vim for the restrained columns with different end conditions obtaining analytical solutions for these types of problems is not a simple procedure since the equations of stability criteria are highly nonlinear this study presents the application of vim for obtaining exact solutions for continuously restrained euler columns the study proves that vim is a very efficient and promising approach in the elastic stability analysis of specified problems

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

dimensional regularization of renyi s statistical mechanics abstract we show that typical renyi s statistical mechanics quantifiers exhibit poles we are referring to the partition function z and the mean energy u renyi s entropy is characterized by a real parameter the poles emerge in a numerable set of rational numbers belonging to the line physical effects of these poles are studied by appeal to dimensional regularization as usual interesting effects are found as for instance gravitational ones in particular negative specific heats

on strong and weak backlund transformations the authors show on a set of examples containing the hierarchies of equations associated with the zakharov and shabat spectral problem 1972 the modified korteweg de vries burgers spectral problem and the chiral field equation that one can easily prove the strength or weakness of a backlund transformation for a whole hierarchy of equations by the use of the darboux matrix approach and thus with just the knowledge of half a backlund transformation

integrable discretization of time and its application on the fourier pseudospectral method to the korteweg de vries equation this paper presents a new integrable discretization of the korteweg de vries kdv equation different from other discrete analogues we discretize the variable time and obtain an integrable differential difference system this system has the original kdv equation as a standard limit when the step size tends to zero the main idea is based on hirota s bilinear method and backlund transformation and can be applied to other integrable systems by applying the fourier pseudospectral method to the space variable we derive a new numerical scheme for the kdv equation numerical results are found to agree with the exact solution and the first five conservation quantities are preserved quite well

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

a hierarchy of coupled korteweg de vries equations and the normalisation conditions of the hilbert riemann problem the authors discuss the normalisation conditions of the hilbert riemann problem which is of importance for the construction of the ladder of soliton solutions for a given spectral problem they provide the prescription for constructing the appropriate normalisation condition all this is applied to a new hierarchy of coupled korteweg de vries equations to find the soliton solution which displays some peculiar phenomena

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

the non autonomous ydkn equation and generalized symmetries of boll equations in this paper we study the integrability of a class of nonlinear non autonomous quad graph equations compatible around the cube introduced by boll in the framework of the generalized adler bobenko and suris abs classification we show that all these equations possess three point generalized symmetries which are subcases of either the yamilov discretization of the krichever novikov equation or of its non autonomous extension we also prove that all those symmetries are integrable as they pass the algebraic entropy test

functional representation of the ablowitz ladik hierarchy the ablowitz ladik hierarchy alh is considered in the framework of the inverse scattering approach after establishing the structure of solutions of the auxiliary linear problems the alh which has been originally introduced as an infinite system of differential difference equations is presented as a finite system of difference functional equations the representation obtained when rewritten in terms of hirota s bilinear formalism is used to demonstrate relations between the alh and some other integrable systems the kadomtsev petviashvili hierarchy in particular

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

some extensions of the uncertainty principle abstract we study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities extending results recently derived by bialynicki birula i bialynicki birula formulation of the uncertainty relations in terms of the renyi entropies physical review a 74 5 2006 052101 and zozor et al s zozor c vignat on classes of non gaussian asymptotic minimizers in entropic uncertainty principles physica a 375 2 2007 499 517 those inequalities can be considered as generalizations of the heisenberg uncertainty principle since they measure the mutual uncertainty of a wave function and its fourier transform through their associated renyi entropies with conjugated indices we consider here the general case where the entropic indices are not conjugated in both cases where the state space is discrete and continuous we discuss the existence of an uncertainty inequality depending on the location of the entropic indices and in the plane our results explain and extend a recent study by luis a luis quantum properties of exponential states physical review a 75 2007 052115 where states with quantum fluctuations below the gaussian case are discussed at the single point 2 2

integrable symplectic maps associated with the zs akns spectral problem the darboux transformations dts associated with the well known zs akns spectral problem are regarded as discrete spectral problems two kinds of integrable symplectic maps are derived from them through nonlinearization procedures as an application they are used to calculate the finite genus solutions of some new integrable lattice models constructed from these dts