the space and scale dependencies of the self similar structure of turbulence two new techniques for analysing the space and scale dependencies of the self similar structure of turbulence are introduced both methods are based on the wavelet transform but use different wavelets first the concept of a turbulence eddy is defined in terms of the mexican hat wavelet transform of the velocity signal in such a way that the eddy has a size and a location in space a new scaling exponent the eddy capacity d text e 0 leq d text e leq 1 is defined in terms of the zero crossings of this wavelet transform d text e is a measure of the space fillingness of eddies and of their scaling in real space whereas the hausdorff dimension d text h prime and the kolmogorov capacity d text k prime are scaling exponents that are sensitive to the scaling and space fillingness in wavenumber space d text e differs from both d text k prime and d text h prime in that it is particularly sensitive to phase correlations for any random phase signal d text e 1 whereas d text h prime and d text k prime depend on the energy spectrum for well defined spiral signals and for signals with a fractal intermittency in space d text e d text k prime secondly we propose a practical test for discriminating between h fractals and k fractals i e between a dense and a sparse distribution of singularities this method is based on successive averages of wavelet transforms and the morlet wavelet is used the technique is shown to be robust with respect to large amounts of phase noise both methods are applied to a one dimensional 1d turbulence velocity signal of very high reynolds number re lambda 2720 it is found that d text e 1 for the inertial scales larger than the taylor microscale lambda thus indicating some degree of phase scrambling of the 1d turbulence velocity signal at the larger inertial scales there is some indication of a spiral type structure in this signal at scales below lambda finally we measure the spatial fluctuation of wavelet energy comparing an experimental turbulence signal with random phase and spiral like signals the magnitude of spatial fluctuations is set by the scrambled phase part of the turbulence but the increase of energy fluctuation at small scales may be due to locally self similar small scale structures

random multifractals negative dimensions and the resulting limitations of the thermodynamic formalism the story described in this paper has started with the death or survival criterion which the author published in 1972 1974 and had obtained in 1968 while investigating kolmogorov s hypothesis that the turbulent dissipation epsilon dx in a box is log normally distributed using this criterion the present paper discusses the concrete significance of negative fractal dimensions they arise in those random multifractal measures for which the cramer function f alpha the spectrum of singularities satisfies f alpha 0 and concerns three aspects of such measures a the fine grained multifractal properties which are non random and the same for almost all realizations b the properties obtained by using the partition function formalism and c the typical coarse grained multifractal properties however the f alpha s in the range where f alpha 0 say nothing about the variability of coarse grained properties between samples a description of these fluctuations hence a fuller multifractal description of the measure is shown to be provided by the values of f alpha in the range where f alpha 0 we prefer to reserve the term thermodynamic for the fine grained and partition functional properties and to say that the coarse grained properties go beyond the thermodynamics i e are not macroscopic but mesoscopic

scalings of scalar structure functions in a velocity field with coherent vortical structures in planar turbulence modeled as an isotropic and homogeneous collection of two dimensional noninteracting compact vortices the structure functions s p r of a statistically stationary passive scalar field have the following scaling behavior in the limit where the peclet number pe s n r const in r l pe 1 3 for l pe 1 3 r l s p r r l pe 1 3 6 1 d for l pe 1 2 r l pe 1 3 where l is a large scale and d is the fractal codimension of the spiral scalar structures generated by the vortices d 2 3 note that l pe 1 2 is the scalar taylor microscale that stems naturally from our analytical treatment of the advection diffusion equation the essential ingredients of our theory are the locality of interscale transfer and lundgren s time average assumption a phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields k ln k 1 generalized power spectra for the advected scalar fields

on the uniqueness of lyapunov matrices for a time delay system abstract in this contribution we study lyapunov matrices for a time delay system the main contribution consists in the demonstration that the set of matrix equations that defines the matrices admit a unique solution if the spectrum of the time delay system does not contain a point s 0 such that s 0 also belongs to the spectrum

potential theory and analytic properties of a cantor set the authors consider the electrostatic potential due to a uniform distribution of charge on a cantor set mellin transforms are used to find expansions for the potential at a distance r from the end of the charge distribution it is found that the potential is a power law multiplied by a function periodic in ln r together with a power series in r a recursion relation for the moments of the distribution is used to reveal a similar structure in these moments finally fourier transform techniques are used to find explicit representations for the distribution and its moments

computing non fragile pi controllers for delay models of tcp aqm networks this article focuses on the stabilisation problem of fluid flow delay models of transmission control protocol active queue management tcp aqm networks by using a proportional integral pi controller as aqm strategy more precisely the complete set of pi controllers that exponentially stabilises the corresponding linear time delay system is derived using the particular geometric properties of this set of the controller parameters the issues of robustness to uncertainty in the network parameters and to perturbation in the controller coefficients are addressed then a methodology to compute a non fragile pi aqm controller is provided finally exponential estimates for the closed loop system solutions allowing to evaluate the performance of the corresponding pi controlled closed loop system are proposed by using a lyapunov krasovskii functional approach an illustrative example completes the presentation

wavelets and orthogonal polynomials based on harmonic oscillator eigenstates we present a new method for the construction of wavelets the main tool is the harmonic oscillator eigenstates with the help of their properties we are able to construct wavelets which have explicit expressions and have exponential decay in both time domain and frequency domain moreover we get a new family of polynomials which are orthogonal in a weighted squarely integrable function space

the kolmogorov spectrum and its oceanic cousins a review kolmogorov s theory of turbulence in an incompressible fluid of constant density at high reynolds numbers has provided a cornerstone for the interpretation of oceanic spectral measurements of turbulence the most convincing verification of the theory came from observations by grant stewart and moilliet under conditions that clearly satisfy the basic premises of the theory but subsequent measurements have explored the influence of ambient stratification and shear on both the energy and temperature spectra as the turbulence decays the larger scales of motion interact more weakly as do internal waves so that kolmogorov s cascade becomes disrupted it has long been known that the existence of a k 5 3 region in the spectrum does not require local isotropy and it is indicated that the success of kolmogorov scaling in collapsing measured spectra in the dissipation range does not require a continuing energy cascade from larger scales several questions remain unresolved particularly the reasons for the shape of temperature spectra that have been measured in turbulence generated by large scale internal waves in a tidal channel

fast chemical reaction and multiple scale concentration fields in singular vortices two species involved in a simple fast reaction tend to become segregated in patches composed of a single of these reactants these patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs fed by diffusion stirred by advection this boundary and the concentration fields within the patches may tend to present multiple scale characteristics based on this segregated state this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it when concentrations presenting initial self similar fluctuations are advected by a singular vortex first the two sources of singularity i e the self similar initial conditions and the singular vortex are considered isolatedly on the one hand self similar initial conditions are imposed to a diffusion reaction system for one and two dimensional cases on the other hand an imposed singular vortex advects initially on off concentration fields in combination with diffusion and reaction this problem is addressed analytically by characterizing the boundary by a box counting dimension and the concentration fields by a h older exponent and numerically by direct numerical simulations of the advection diffusion reaction equations second the way the two sources hang together shows that depending on the self similar properties of the initial concentration fields the vortex promotes the chemical activity close to its inner smoothed out core or close to the outer region where the boundary starts to spiral for all the considered situations the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws

analytic solution of a delay differential equation arising in cost functionals for systems with distributed delays the solvability of a delay differential equation arising in the construction of quadratic cost functionals i e lyapunov functionals for a linear time delay system with a constant and a distributed delay is investigated we present a delay free auxiliary ordinary differential equation system with algebraically coupled split boundary conditions which characterizes the solutions of the delay differential equation and is used for solution synthesis a spectral property of the time delay system yields a necessary and sufficient condition for existence and uniqueness of solutions to the auxiliary system equivalently the delay differential equation the result is a tractable analytic solution framework to the delay differential equation

lyapunov matrices for time delay systems the construction of complete type lyapunov krasovki i functionals for a linear time invariant delay system depends on so called delay lyapunov matrices which satisfy a matrix delay equation with additional boundary conditions we study existence and uniqueness issues for these delay lyapunov matrices 2006 elsevier b v all rights reserved

constrained multi project plannings problems a lagrangean decomposition approach abstract this paper describes a lagragean decomposition technique for solving multi project planning problems with resource constraints and alternative modes of performing each activity in the projects the decompsition can be useful in several ways from one side it provides bounds on the optimum so that the quality of approximate solutions can be evaluated furthermore in the context of branch and bound algorithms it can be used for more effective fathoming of the tree nodes finally in the modelling perspective the la grangean optimal multipliers can provide insights to project managers as prices for assigning the resources to different projects

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

formal improvements in the solution of the wavelet transformed poisson and diffusion equations hermitian wavelets relation to the laplace operator leads to a natural measure of the scale factor that emphasizes the largest component wave number for the poisson equation e g the pressure equation in navier stokes turbulence the wavelet transform of the solution at a given location and scale depends on the wavelet transform of the source field at the same location and at nearby and larger scales for the diffusion problem the hamiltonian formulation is simplified through a canonical transformation

on local isotropy of passive scalars in turbulent shear flows an assessment of local isotropy and universality in high reynolds number turbulent flows is presented the emphasis is on the behaviour of passive scalar fields advected by turbulence but a brief review of the relevant facts is given for the turbulent motion itself experiments suggest that local isotropy is not a natural concept for scalars in shear flows except perhaps at such extreme reynolds numbers that are of no practical relevance on earth yet some type of scaling exists even at moderate reynolds numbers the relation between these two observations is a theme of this paper

wavelet spectra compared to fourier spectra the relation between fourier spectra and spectra obtained from wavelet analysis is established small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments these results are related to regularity theorems in besov spaces for the analysis of infinitely regular signals a new wavelet with an infinite number of cancellations is proposed

mixing entrainment and fractal dimensions of surfaces in turbulent flows some basic thoughts are set down on the relation between the fractal dimension of various surfaces in turbulent flows and the practically important processes of mixing between two streams reacting or otherwise separated by a convoluted surface as well as of entrainment of irrotational flow by a turbulent stream an expression based on heuristic arguments is derived for the flux of transportable properties such as mass momentum and energy across surfaces and a prediction made on this basis for the fractal dimension of surfaces in fully turbulent flows is shown to be in essential agreement with measurements it is further shown that this prediction remains robust when corrected for the non uniform effects along the surface a related prediction concerning the dependence of mixing on the reynolds number and the fractal dimension of the surface is substantiated in the developing as well as the fully developed states by independent measurements of both the fractal dimension and the amount of mixing between reactants in a temporally evolving countercurrent shear flow

wavelet analysis of fractal boundaries part 2 multifractal analysis this second part deals with the global analysis of the boundary of domains omega rd we develop methods for determining the dimensions of the sets where the local behaviors introduced in part 1 occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

kolmogorov s contributions to the physical and geometrical understanding of small scale turbulence and recent developments this paper reviews how kolmogorov postulated for the first time the existence of a steady statistical state for small scale turbulence and its defining parameters of dissipation rate and kinematic viscosity thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes we present theoretical arguments and experimental evidence to indicate when the small scale motions might tend to a universal form paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the inertial range spectrum these may be particular forms of eddy or eigenstructure such as spiral vortices which may not be unique to turbulent flows also they tend to lead to the notable spiral contours of scalars in turbulence whose self similar structure enables the boxcounting technique to be used to measure the capacity dk of the contours themselves or of their intersections with lines dk although the capacity a term invented by kolmogorov and studied thoroughly by kolmogorov tikhomirov is like the exponent 2p of a spectrum in being a measure of the distribution of length scales d being related to 2p in the limit of very high reynolds numbers the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the reynolds number thus kolmogorov tikhomirov provide the basis for a more widely applicable measure of the self similar structure of turbulence finally we also review how kolmogorov s concept of the universal spatial structure of the small scales together with appropriate additional physical hypotheses enables other aspects of turbulence to be understood at these scales in particular the general forms of the temporal statistics such as the highfrequency inertial range spectra in eulerian and lagrangian frames of reference and the perturbations to the small scales caused by non isotropic non gaussian and inhomogeneous large scale motions 1 kolmogorov s papers review and comments a introduction in this review we join with the other contributors to this special publication in celebrating some of kolmogorov s great contributions to fluid mechanics and mathematics and showing in some small way how his genius has inspired further

oscillatory finite time singularities in finance population and rupture abstract we present a simple two dimensional dynamical system where two nonlinear terms exerting respectively positive feedback and reversal compete to create a singularity in finite time decorated by accelerating oscillations the power law singularity results from the increasing growth rate the oscillations result from the restoring mechanism as a function of the order of the nonlinearity of the growth rate and of the restoring term a rich variety of behavior is documented analytically and numerically the dynamical behavior is traced back fundamentally to the self similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin the interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log periodic oscillations with remarkable scaling properties three domains of applications are discussed 1 the stock market with a competition between nonlinear trend followers and nonlinear value investors 2 the world human population with a competition between a population dependent growth rate and a nonlinear dependence on a finite carrying capacity 3 the failure of a material subjected to a time varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing

from global scaling a la kolmogorov to local multifractal scaling in fully developed turbulence in the first part of the paper a modern presentation of scaling ideas is made it includes a reformulation of kolmogorov s 1941 theory bypassing the universality problem pointed out by landau and a presentation of the multifractal theory with emphasis on scaling rather than on cascades in the second part various historical aspects are discussed the importance of kolmogorov s rigorous derivation of the 4 5 l law for the third order structure function in his last 1941 turbulence paper is stressed this paper also contains evidence that he was aware of universality not being essential to the 1941 theory an inequality is established relating the exponents 2 p of the structure functions of order 2 p and the maximum velocity excursion it follows that models such as the obukhov kolmogorov 1962 log normal model in which 2 p does not increase monotonically are inconsistent with the basic physics of incompressible flow this result is independent of novikov s 1971 inequality in particular the proof presented here does not rely on the questionable relation proposed by obukhov and kolmogorov between instantaneous velocity increments and local averages of the dissipation

singularity spectrum of fractal signals from wavelet analysis exact results the multifractal formalism for singular measures is revisited using the wavelet transform for bernoulli invariant measures of some expanding markov maps the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima the generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures it is demonstrated that the hausdorff dimensiond h of the set of singularities of holder exponenth can be directly determined from the wavelet transform modulus maxima the singularity spectrum so obtained is shown to be not disturbed by the presence in the signal of a superimposed polynomial behavior of ordern provided one uses an analyzing wavelet that possesses at leastn n vanishing moments however it is shown that ac behavior generally induces a phase transition in thed h singularity spectrum that somewhat masks the weakest singularities this phase transition actually depends on the numbern of vanishing moments of the analyzing wavelet its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal these theoretical results are illustrated with numerical examples they are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional brownian motions and turbulent velocity signals

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

kolmogorov s two equation model of turbulence a n kolmogorov in 1941 proposed a method of predicting the behaviour of turbulent flows which is remarkably similar to the turbulence modelling techniques which have subsequently become commonplace the present article describes kolmogorov s method and discusses its similarities to and differences from the subsequently developed techniques a translation of the 1941 paper is provided as an appendix

wavelet analysis of fractal boundaries part 1 local exponents let be a domain of rd in part 1 of this paper we introduce new tools in order to analyse the local behavior of the boundary of classifica tions based on geometric accessibility conditions are introduced and compared they are related to analytic criteria based either on local lp regularity of the characteristic function 1 or on its wavelet coefficients part 2 deals with the global analysis of the boundary of we develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur these methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

maximal covering code multiplexing access telecommunication networks abstract we present a maximal covering model for the network design of code division multiplexing access cdma telecommunication system properties of the solution and the algorithm have been studied the model has been developed to solve actual problems they are extremely large so exact decomposition methods cannot be used to solve it a lagrangian decomposition heuristic has been defined different algorithm options have been computationally compared using random generated and actual data model and algorithm parameters have been adjusted and sensibility of the solution to the data has been considered in these tests

lyapunov matrices for a class of time delay systems with piecewise constant kernel abstractin this contribution we study time delay systems with distributed delay whose kernel is piecewise constant it is shown that the problem of finding a lyapunov matrix for this class of time delay systems can be reduced to the computation of solutions to an auxiliary system of linear differential equations with boundary conditions the problem of solving the auxiliary system is then reduced to the solution of a system of linear algebraic equations it is established that the auxiliary system with boundary conditions admits a unique solution if and only if there exists a unique lyapunov matrix

brief paper lyapunov matrices existence and uniqueness issues a necessary and sufficient condition for the existence and uniqueness of lyapunov matrices for neutral type single delay systems is presented

hamiltonian formulation for the diffusion equation a hamiltonian formulation is presented for the diffusion equation going beyond the conservation of property in diffusive transport conservation is shown to apply in the sense of classical hamiltonian dynamics provided the equation is transformed with hermitian wavelets the characteristic equations obtained previously for the wavelet transformed diffusion equation are the canonical equations corresponding to a time independent hamiltonian the configuration variables are defined by the canonical structure and its invariants while the momenta determine the evolution of the system irreversibility results from the finite time escape of trajectories initiating from the smallest scales and eroding increasingly larger scales however this scale dependent erosion of initial conditions does not necessarily imply memory loss

project scheduling with resource constraints a branch and bound approach abstract this paper describes a branch and bound algorithm for project scheduling with resource constraints the algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods four lower bounds are examined the first is a simple lower bound based on longest path computations the second and third bounds are derived from a relaxed integer programming formulation of the problem the second bound is based on the linear programming relaxation with the addition of cutting planes and the third bound is based on a lagrangean relaxation of the formulation this last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient though not polynomial algorithm is given the fourth bound is based on the disjunctive arcs used to model the problem as a graph we report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources