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

quasi variational inequalities and ergodic impulse control in this paper we solve the general ergodic control problem for reflected diffusion processes and the associated quasi variational inequalities our method relies on some a priori estimates for the solutions of quasi variational inequalities where the discount factor i e the zero order term is set to 0

discrete time hybrid control in borel spaces average cost optimality criterion abstract this paper addresses an optimal hybrid control problem in discrete time with borel state and action spaces by hybrid we mean that the evolution of the state of the system may undergo deep changes according to structural modifications of the dynamic such modifications occur either by the position of the state or by means of the controller s actions the optimality criterion is of a long run ratio average or ratio ergodic type we provide the existence of optimal average policies for this hybrid control problem by analyzing an associated dynamic programming equation we also show that this problem can be translated into a standard or non hybrid optimal control problem with cost constraints besides we show that our model includes some special and important families of control problems such as those with an impulsive or switching mode finally to illustrate our results we provide an example on a pollution accumulation problem

quantification of topological changes of vorticity contours in two dimensional navier stokes flow a characterization of reconnection of vorticity contours is made by direct numerical simulations of the two dimensional navier stokes flow at a relatively low reynolds number we identify all the critical points of the vorticity field and classify them by solving an eigenvalue problem of its hessian matrix on the basis of critical point theory the numbers of hyperbolic saddles and elliptic minima and maxima points are confirmed to satisfy euler s index theorem numerically time evolution of these indices is studied for a simple initial condition generally speaking we have found that the indices are found to decrease in number with time this result is discussed in connection with related works on streamline topology in particular the relationship between stagnation points and the dissipation associated elementary procedures in physical space the merging of vortices are studied in detail for a number of snapshots a similar analysis is also done using the stream function

a general approach for obtaining joint representations in signal analysis i characteristic function operator method a general approach for obtaining joint representations for arbitrary physical quantities is presented the characteristic function operator method of moyal 1949 and ville 1960 and generalized by cohen 1966 1976 and scully and cohen 1987 for the time frequency case is developed for arbitrary variables

lagrangian coherent structures and mixing in two dimensional turbulence abstract we introduce a lagrangian definition for the boundaries of coherent structures in two dimensional turbulence the boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors such material lines are responsible for stretching and folding in the mixing of passive tracers we derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets the criterion provides a rigorous link between the lagrangian concept of hyperbolicity the okubo weiss criterion and vortex boundaries we apply the results to simulations of two dimensional barotropic turbulence

optimal switching for ordinary differential equations we consider the problem of controlling an ordinary differential equation subject to positive switching costs and show in particular that the value functions form the viscosity solution cf 6 7 of the dynamic programming quasi variational inequalities this interpretation allows for a rigorous application of various dynamic programming techniques

singular values of compact pseudodifferential operators abstract this paper investigates the asymptotic decay of the singular values of compact operators arising from the weyl correspondence the motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay the problem is approached by using a gabor frame expansion of the symbol to construct an approximating finite rank operator this establishes a variety of sufficient conditions for the associated operator to be in a particular schatten class in particular an improvement of a sufficient condition of daubechies for an operator to be trace class is obtained in addition a new development and improvement of the calderon vaillancourt theorem in the context of the weyl correspondence is given additional results of this type are then obtained by interpolation

positivity properties of phase plane distribution functions the aim of this paper is to compare the members of cohen s class of phase plane distributions with respect to positivity properties it is known that certain averages which are in a sense compatible with heisenberg s uncertainty principle of the wigner distribution over the phase plane yield non negative values for all states it is shown in this paper that the wigner distribution is unique in this respect among the members of cohen s class that have correct marginals or that satisfy moyal s formula for all states the subset of members of cohen s class not necessarily satisfying one of these two conditions with positivity properties comparable with those for the wigner distribution is shown to be rather small

integral bounds for radar ambiguity functions and wigner distributions an upper bound is proved for the lp norm of woodward s ambiguity function in radar signal analysis and of the wigner distribution in quantum mechanics when p 2 a lower bound is proved for 1 p 2 in addition a lower bound is proved for the entropy these bounds set limits to the sharpness of the peaking of the ambiguity function or wigner distribution the bounds are best possible and equality is achieved in the lp bounds if and only if the functions f and g that enter the definition are both gaussians

stagnation point von karman coefficient on the basis of various direct numerical simulations dns of turbulent channel flows the following picture is proposed i at a distance y from either wall the taylor microscale lambda is proportional to the average distance l s between stagnation points of the fluctuating velocity field i e lambda y b 1 l s y with b 1 constant for delta nu y or approximately equal to delta where the wall unit delta nu is defined as the ratio of kinematic viscosity nu to skin friction velocity u tau and delta is the channel s half width ii the number density n s of stagnation points varies with height according to n s c s delta nu 3 y 1 where y identical with y delta nu and c s is constant in the range delta nu y or approximately equal to delta iii in that same range the kinetic energy dissipation rate per unit mass equals 2 3 e u tau 3 kappa s y where e is the total kinetic energy per unit mass normalized by u tau 2 and kappa s b 1 2 c s is the stagnation point von karman coefficient iv in the limit of exceedingly large reynolds numbers re tau identical with delta delta nu large enough for the reynolds stress uv to equal u tau 2 in the range delta nu y delta and assuming that production of turbulent kinetic energy balances dissipation locally in that range and limit the mean velocity u normalized by u tau obeys d dy u approximately equal to 2 3 e kappa s y in that same range v it follows that the von karman coefficient kappa is a meaningful and well defined coefficient and the log law holds in turbulent channel pipe flows only if e is independent of y and re tau in that range in which case kappa approximately kappa s vi in support of d dy u approximately equal to 2 3 e kappa s y dns data of turbulent channel flows which include the highest currently available values of re tau are best fitted by e approximately equal to 2 3 b 4 y 2 15 and d dy u approximately equal to b 4 kappa s y 1 2 15 with b4 independent of y in delta nu y delta if the significant departure from uv approximately equal to u tau 2 at these re tau values is taken into account

an exact criterion for the stirring properties of nearly two dimensional turbulence abstract an exact criterion can be found for partitioning the fluid into regions with different dynamical properties from both the points of view of particle dispersion and tracer gradient evolution this criterion differs markedly both in its magnitude and spatial scales from the okubo weiss criterion which depends upon the differential geometry of the streamfunction field and coincides with the eigenvalues of the velocity gradient tensor the new criterion corresponds to the eigenvalues of the acceleration gradient tensor whose spatial distribution depends instead upon the topology of the pressure field this result holds for all flows for which a continuous momentum equation can be prescribed we provide numerical evidence for the quantitative importance of the time change of the strain rate components in the dispersion problem in freely decaying two dimensional turbulence

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

separability positivity and minimum uncertainty in time frequency energy distributions gaussian signals play a very special role in classical time frequency analysis because they are solutions of apparently unrelated problems such as minimum uncertainty and positivity and separability of wigner ville distributions we investigate here some of the logical connections which exist between these different features and we discuss some examples and counterexamples of their extension to more general joint distributions within cohen s class and the affine class

bilinear phase plane distribution functions and positivity there is a theorem of wigner that states that phase plane distribution functions involving the state bilinearly and having correct marginals must take negative values for certain states the purpose of this paper is to support the statement that these phase plane distribution functions are for hardly any state everywhere non negative in particular it is shown that for certain generalized wigner distribution functions there are no smooth states except the gaussians for the wigner distribution function itself whose distribution function is everywhere non negative this class of wigner type distribution functions contains the margenau hill distribution furthermore the argument used in the proof of wigner s theorem is augmented to show that under mild conditions one can find for any two states f g with non negative distribution functions a linear combination h of f and g whose distribution function takes negative values unless f and g are proportional

ergodic problem for optimal stochastic switching abstract we study the minimization of the long run average cost for a family of reflected diffusion processes when the control is to switch from one process to another one this leads to solving the ergodic problem for a system of quasi variational inequalities with neumann boundary conditions this problem is studied as the limit of the discounted case when the discount factor goes to zero

on the probability density interpretation of smoothed wigner functions it has been conjectured that averages of the wigner function over phase space volumes larger than those of minimum uncertainty are always positive this is true for gaussian averaging so that the husimi distribution is positive however the authors provide a specific counterexample for the averaging with a discontinuous hat function the analysis of the specific system of a one dimensional particle in a box also elucidates the respective advantages of the wigner and the husimi functions for the study of the semiclassical limit the falsification of the averaging conjecture is shown not to depend on the discontinuities of the hat function by considering the latter as the limit of a sequence of analytic functions

some properties of the smoothed wigner function abstract recently it has been proposed a modification of the wigner function which consists in smoothing it by convolution with a phase space gaussian function this smoothed wigner function is non negative if the gaussian parameters and satisfy the condition we analyze in this paper the predictions of this modified wigner function for the harmonic oscillator for anharmonic oscillator and finally for the hydrogen atom we find agreement with experiment in the linear case but for strongly nonlinear systems such as the hydrogen atom the results obtained are completely wrong

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

virtues and vices of quartic time frequency distributions we present results concerning three different types of quartic fourth order time frequency distributions tfds first we present new results on the previously introduced local ambiguity function and show that it provides more reliable estimates of instantaneous chirp rate than the wigner distribution second we introduce the class of quartic shift covariant time frequency distributions and investigate distributions that localize quadratic chirps finally we present a shift covariant distribution of time and chirp rate

on positivity of time frequency distributions this correspondence addresses the problem of how to regard the fundamental impossibility with time frequency energy distributions of cohen s class always to be nonnegative and at the same time to have correct marginal distributions it is shown that the wigner distribution is the only member of a large class of bilinear time frequency distributions that becomes nonnegative after smoothing in the time frequency plane by means of gaussian weight functions with bt product equal to unity

ergodicity of robust switching control and nonlinear system of quasi variational inequalities we analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities these equations are related to robust switching control problems introduced in 3 we prove that as time horizon goes to infinity resp discount factor goes to zero the long run average solution to the parabolic system resp the limiting discounted solution to the elliptic system is characterized by a solution of a nonlinear system of ergodic variational inequalities our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities

a class of affine wigner functions with extended covariance properties affine wigner functions are phase space representations based on the affine group in place of the usual weyl heisenberg group of quantum mechanics such representations are relevant to the time frequency analysis of real signals an interesting family is singled out by the requirement of covariance with respect to each solvable three parameter group containing the affine group explicit forms are given in each case and properties such as unitarity and localization are discussed some particular distributions are recovered

optimum smoothing of the wigner ville distribution a compromise is found between the different requirements that we would like to be fulfilled by a time frequency distribution namely positivity and obtention of a distribution close to the dirac one for the unimodular signal s t exp i phi t the fulfillment of the marginal conditions being of less interest in signal theory starting from the usual wigner ville distribution we define an optimum smoothing by minimizing the width of the different functions approximating the desired dirac distribution the smoothing is obtained by a convolution through a double gaussian of width t and such that t 1 2 two possibilities appear in the first one we do not introduce any correlation between t and in the convolution kernel and obtain a simple result in the second one extrapolating the frequency variation and still using a gaussian we obtain a better result although the smoothing process becomes more complex these results to be physically meaningful impose inequalities on the successive derivatives of which are equivalent to those used for the obtention of the classical limit for the corresponding quantum problem