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Publications (301)
We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for sine-wave initial conditions. These so far undetected complex singularities are arranged in an eye shape centered around the origin in time. Interestingly, since the eye is squashed along the imaginary-time axis, complex-time singulariti...
We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for sine-wave initial conditions. These so far undetected complex singularities are arranged in an eye shape centered around the origin in time. Interestingly, since the eye is squashed along the imaginary time axis, complex-time singulariti...
We treat the incompressible, and axisymmetric Euler equations for a three-dimensional cylindrical domain with boundaries. The equations are solved by the novel Cauchy-Lagrange algorithm (CLA), which uses the time-analyticity of the Lagrangian trajectories of an incompressible Euler flow and computes the time-Taylor coefficients of the Lagrangian ma...
It is known that the gravitational collapse of cold dark matter leads to infinite-density caustics that seed the primordial dark-matter halos in the large-scale structure. The development of these caustics begins, generically, as an almost one-dimensional phenomenon with the formation of pancakes. Focusing on the one-dimensional case, we identify a...
Finite-dimensional, inviscid equations of hydrodynamics, obtained through a Fourier-Galerkin projection, thermalize with an energy equipartition. Hence, numerical solutions of such inviscid equations, which typically must be Galerkin-truncated, show a behavior at odds with the parent equation. An important consequence of this is an uncertainty in t...
Finite-dimensional, inviscid equations of hydrodynamics, such as the zero-viscosity, one-dimensional Burgers equation or the three-dimensional incompressible Euler equation, obtained through a Fourier-Galerkin projection, thermalise---mediated through structures known as tygers [Ray et al., Phys. Rev. E 84, 016301 (2011)]---with an energy equiparti...
Gravitational collapse of cold dark matter leads to infinite-density caustics that seed the primordial dark-matter halos in the large-scale structure. The development of these caustics begins, generically, as an almost one-dimensional phenomenon with the formation of pancakes. Focusing on the one-dimensional case, we identify a landscape of so far...
Regeneration of magnetic fields by dynamo effect in the presence of turbulence is of interest both for astro-geophysical applications and because it may produce unwanted effects in large scale fast breader reactors. In such machines magnetic Reynolds numbers R M of the order of 100 can be reached, possibly in excess of the critical value for dynamo...
The present paper is a companion to the paper by Villone and Rampf (2017) containing the translation of Hankel's 1861 prized manuscript (Preisschrift), titled "On the general theory of the motion of fluids" together with connected documents. Here we give a critical assessment of Hankel's work, which covers many important aspects of fluid dynamics c...
The present paper is a companion to the paper by Villone and Rampf (2017), titled "Hermann Hankel's On the general theory of motion of fluids, an essay including an English translation of the complete Preisschrift from 1861" together with connected documents. Here we give a critical assessment of Hankel's work, which covers many important aspects o...
Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions....
Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions....
The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries. For a good understanding, it is crucial to carry out, besides mathematical studies, high...
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of threedimensional (3D) ideal flow (Frisch and Zheligovsky 2014; Podvigina et al. 2016). Looking at such invariants with the modern tools of differential geometry and of geodesic flow on the space SDiff of volume-preserving transformations (Arnold 1966),...
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools of differential geometry and o...
The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or absence of boundaries. For a good understanding, it is crucial to carry out, besides mathematical studies, high...
We present theoretical and numerical results for the one-dimensional
stochastically forced Burgers equation decimated on a fractal Fourier set of
dimension $D$. We investigate the robustness of the energy transfer mechanism
and of the small-scale statistical fluctuations by changing $D$. We find that a
very small percentage of mode-reduction ($D \l...
We present theoretical and numerical results for the one-dimensional stochastically forced Burgers equation decimated on a fractal Fourier set of dimension $D$. We investigate the robustness of the energy transfer mechanism and of the small-scale statistical fluctuations by changing $D$. We find that a very small percentage of mode-reduction ($D \l...
A novel semi-Lagrangian method is introduced to solve numerically the Euler
equation for ideal incompressible flow in arbitrary space dimension. It
exploits the time-analyticity of fluid particle trajectories and requires, in
principle, only limited spatial smoothness of the initial data. Efficient
generation of high-order time-Taylor coefficients...
It is shown here that in a flat, cold dark matter (CDM)-dominated Universe with positive cosmological constant (Λ), modelled
in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent
Taylor series) until at least a finite time after decoupling. The time variable used for this statem...
Two prized papers, one by Augustin Cauchy in 1815, presented to the French
Academy and the other by Hermann Hankel in 1861, presented to G\"ottingen
University, contain major discoveries on vorticity dynamics whose impact is now
quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal
incompressible flow in terms of three invariants th...
It is known that the Eulerian and Lagrangian structures of fluid flow can be
drastically different; for example, ideal fluid flow can have a trivial
(static) Eulerian structure, while displaying chaotic streamlines. Here we show
that ideal flow with limited spatial smoothness (an initial vorticity that is
just a little better than continuous), neve...
An energy-spectrum bottleneck, a bump in the turbulence spectrum between the inertial and dissipation ranges, is shown to occur in the nonturbulent, one-dimensional, hyperviscous Burgers equation and found to be the Fourier-space signature of oscillations in the real-space velocity, which are explained by boundary-layer-expansion techniques. Pseudo...
It has been known for some time that a 3D incompressible Euler flow that has
initially a barely smooth velocity field nonetheless has Lagrangian fluid
particle trajectories that are analytic in time for at least a finite time (Ph.
Serfati C.R. Acad. Sci. S\'erie I 320, 175-180 (1995); A. Shnirelman
arXiv:1205.5837 (2012)). Here an elementary deriva...
It is well known that the overwhelming majority of both natural and man-made flows of fluids do not vary smoothly in space and time but fluctuate in a quite disordered manner, exhibiting sudden and irregular (but still continuous) space- and time-variations. Such irregular flows are called “turbulent”. A very large amount of information is required...
An energy-spectrum bottleneck, a bump in the turbulence spectrum between the inertial and dissipation ranges, is shown to occur in the non-turbulent, one-dimensional, hyperviscous Burgers equation and found to be the Fourier-space signature of oscillations in the real-space velocity, which are explained by boundary-layer-expansion techniques. Pseud...
Fractal decimation reduces the effective dimensionality of a flow by keeping
only a (randomly chosen) set of Fourier modes whose number in a ball of radius
$k$ is proportional to $k^D$ for large $k$. At the critical dimension D=4/3
there is an equilibrium Gibbs state with a $k^{-5/3}$ spectrum, as in [V. L'vov
{\it et al.}, Phys. Rev. Lett. {\bf 89...
Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extend...
Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this procedure works already for Reynolds numbers of the order of 100 (or even lower when combined with a suitable extend...
One of the simplest models used in studying the dynamics of large-scale
structure in cosmology, known as the Zeldovich approximation, is equivalent to
the three-dimensional inviscid Burgers equation for potential flow. For smooth
initial data and sufficiently short times it has the property that the mapping
of the positions of fluid particles at an...
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold K(G) exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. For large K(G) and smooth initi...
Robert Harry Kraichnan {(1928-2008)} was one of the leaders in the theory of turbulence for a span of about forty years (mid-fifties to mid-nineties). Among his many contributions, he is perhaps best known for his work on the inverse energy cascade (i.e. from small to large scales) for forced two-dimensional turbulence. This is a review of Kraichna...
We present three novel forms of the Monge–Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the dev...
We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with
space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose
Fourier symbol grows exponentially as e|k|/kd{{{\rm e}^{|k|/k_{\rm d}}}} at high wavenumbers |k|. Us...
It is shown that the solutions of inviscid hydrodynamical equations with
suppression of all spatial Fourier modes having wavenumbers in excess of a
threshold $\kg$ exhibit unexpected features. The study is carried out for both
the one-dimensional Burgers equation and the two-dimensional incompressible
Euler equation. At large $\kg$, for smooth init...
Extended Self-Similarity (ESS), a procedure that remarkably extends the range of scaling for structure functions in Navier--Stokes turbulence and thus allows improved determination of intermittency exponents, has never been fully explained. We show that ESS applies to Burgers turbulence at high Reynolds numbers and we give the theoretical explanati...
We present three novel forms of the Monge-Ampere equation, which is used,
e.g., in image processing and in reconstruction of mass transportation in the
primordial Universe. The central role in this paper is played by our Fourier
integral form, for which we establish positivity and sharp bound properties of
the kernels. This is the basis for the dev...
It was argued that application of hyperviscosity in turbulence simulation will lead to artifacts caused by partial thermalization --- the tendency to the physical behavior corresponding to the equilibrium of the Galerkin-truncated inviscid system [U. Frisch et al., Phys. Rev. Letts. in press; or, Arxiv:0803.4269]. We study the partial thermalizatio...
We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at high wavenumbers $|k|$. Using estimates i...
It is shown that the use of a high power alpha of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid conservative dynamics with a finite range of spatial Fourier modes. Those at large wave numbers thermalize, whereas modes at small wave numbers obey ordinary viscous dynamics [C. Cichowlas et...
The Euler equations of hydrodynamics, which appeared in their present form in the 1750s, did not emerge in the middle of a desert. We shall see in particular how the Bernoullis contributed much to the transmutation of hydrostatics into hydrodynamics, how d’Alembert was the first to describe fluid motion using partial differential equations and a ge...
We show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the early development of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the drag for a body of arbitrary shape; for this he exploited and significantly extended the existing ideas on decomp...
Physics Today, 61, p. 70, http://dx.doi.org./10.1063/1.2930746
It is shown that the use of a high power $\alpha$ of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid \textit{conservative} dynamics with a finite range of spatial Fourier modes. Those at large wavenumbers thermalize, whereas modes at small wavenumbers obey ordinary viscous dynamics [C. Ci...
This is an adapatation by U. Frisch of an English translation by Thomas E. Burton of Euler's memoir `Principes g\'en\'eraux du mouvement des fluides' (Euler, 1775b). Burton's translation appeared in Fluid Dynamics, 34} (1999) pp. 801-82, Springer and is here adapted by permission. A detailed presentation of Euler's published work can be found in Tr...
Physica D Nonlinear Phenomena, 237, pp. 1855-1869, http://dx.doi.org./10.1016/j.physd.2007.08.003
We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination of complex-space Lagrangian singularities. Lagrangian singularities are found to be closer to the real domain t...
We have built a three-dimensional 24-bit lattice gas algorithm with improved collision rules. Collisions are defined by a look-up table with 224 entries, fine-tuned to maximize the Reynolds number. External flow past a circular plate at Reynolds number around 190 has been simulated. The flow is found to evolve from axi-symmetric to fully 3D. Such s...
The 3D Navier-Stokes equations are obtained from two different lattice gas models. The first one has its sites on a cubic lattice and has particle speeds zero, one and √2. The second one is a 3D projection of a lattice gas implementation of the 4D Navier-Stokes equations, residing on a face-centred hypercubic lattice.
It is shown that the multifractal model of fully developed turbulence predicts a new form of universality for the energy spectrum E(k), which can be tested experimentally. Denoting by R the Reynolds number, log E/log R should be a universal function of log k/log R. This includes an intermediate dissipation range in which a continuous range of multi...
Recent work on intermittency for a passive scalar advected by a Gaussian white-in-time velocity field with a power-law energy
spectrum has led for the first time to a systematic understanding of how intermittency can come about and to testable predictions for anomalous exponents. Some of the key ideas
of such work are here presented in a way which...
Fully Developed Turbulence is concerned with the behaviour of turbulent flows (here assumed incompressible) as the Reynolds number R becomes very large. For such flows there is evidence of scaling (e.g. the -5/3 law for the energy spectrum). It is shown that the Kolmogorov 1941 theory is equivalent to an invariance principle: as R → ∞ all the invar...
It is shown analytically and by Monte
Carlo simulations that a passive scalar with finite diffusivity,
advected by a white-in-time velocity field with a power law spectrum
∝ k−1−ξ (0 < ξ < 2), has an inertial-diffusive range with
a spectrum ∝ k−3−ξ. This is the analog of the
Batchelor-Howells-Townsend (J. Fluid Mech., 5 (1959) 134)
phenomenological...
It is conjectured that for many equations of hydrodynamical type, including the three-dimensional Navier-Stokes equations, the Burgers equation and various models of turbulence, the use of hyperviscous dissipation with a high power alpha (dissipativity) of the Laplacian and suitable rescaling of the hyperviscosity becomes asymptotically equivalent...
After Kolmogorov's (1941) formulation of the energy-cascade picture, several theoretical models have been proposed to study the statistics of fully developed turbulence. Frisch et al. (1978) have recently presented the beta model on the basis of a discrete eddy picture. This model provides the intermittency correction to the energy spectrum in term...
The equations of fully nonlinear MHD turbulence are analyzed in the
homogeneous isotropic case with helicity included in both the velocity
and magnetic fields. A quasi-normal Markovianized eddy-damped
approximation is used which leads to a closed set of coupled equations
for the evolution of kinetic and magnetic energy spectra, the kinetic
helicity...
We are concerned with the global (in time) regularity properties of the Burgers MRCM equation, which arises in the theory
of turbulence (with α = 1)
\frac¶U¶t(t,x) = - \frac¶2 ¶x2 [U(t,0) - U(t,x)]2 - n( - \frac¶2 ¶x2 )a U(t,x)\frac{{\partial U}}{{\partial t}}(t,x) = - \frac{{\partial ^2 }}{{\partial x^2 }}[U(t,0) - U(t,x)]^2 - \nu ( - \frac{{\pa...
Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of P\'olya relates the large d istance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from t...
A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic régime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that t...
The problem of deterministic reconstruction of the past kinetic history of the Universe is shown to be reduced, within the
Zeldovich approximation, to solving a Monge-Ampere equation. A variational representation, due to Y. Brenier, is then employed
to devise a “Monge-Ampere-Kantorovich” numerical method of cosmological reconstruction. Results of t...
Without Abstract
Very interesting perspectives of massively parallel and interactive simulations for phenomena in fluid mechanics have appeared in 1985, thanks to a technique of a drastically new conception called lattice gas hydrodynamics. It has been developed by scientists from the observatory of Nice (M. Henon, J.P. Rivet and the author) together with scientist...
A new proof is given that a one-dimensional random walk starting from the origin, with independent steps having an even p.d.f. K(x), has a probability p
n
, of never visiting the negative half-space for the n first steps, which is universal, i.e. independent of K(x).
The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random...
A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that...
The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random...
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum approximately 1/k, where k is the wave number. From very high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot...
Using a very high precision spectral calculation applied to the incompressible and inviscid flow with initial condition , we find that the width δ(t) of its analyticity strip follows a ln(1/t) law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times [Frisch, U., Mat...
In his first 1941 paper Kolmogorov assumed that the velocity has increments which are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. The inconsistency is illustrated numerically...
A reconstruction method for recovering the initial conditions of the Universe starting from the present galaxy distribution is presented which guarantees uniqueness of solutions. We show how our method can be used to obtain the peculiar velocities of a large number of galaxies, hence trace galaxies orbits back in time and obtain the entire past dyn...
We study turbulence in the one-dimensional Burgers equation with a white-in-time, Gaussian random force that has a Fourier-space spectrum $\sim 1/k$, where $k$ is the wave number. From very-high-resolution numerical simulations, in the limit of vanishing viscosity, we find evidence for multiscaling of velocity structure functions which cannot be fa...
Available from http://taylorandfrancis.metapress.com/openurl.asp?genre=article&issn=0309-1929&volume=98&issue=2&spage=173
We show that the deterministic past history of the Universe can be uniquely reconstructed from knowledge of the present mass density field, the latter being inferred from the three-dimensional distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those sca...
Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for...
Using a very high precision spectral calculation applied to the incompressible and inviscid flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-sp...
It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities (f(α). This is a new way of characterizing multifractality in dynamical systems, so far applied only to mult...
A new method for reconstruction of the primordial density fluctuation
field is presented. Various previous approaches to this problem rendered
non-unique solutions. Here, it is demonstrated that the initial
positions of dark matter fluid elements, under the hypothesis that their
displacement is the gradient of a convex potential, can be reconstruct...
It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity nu and no force are of the form u=-2nu inverted Delta ln P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on t...
We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a fe...
The Monge-Kantorovich mass transportation problem dates back to work by Monge in 1781 on how to optimally move earth from one place to another, knowing only the initial and final landscapes, the cost being a prescribed function of the distance travelled by "molecules" of earth. We solve the cosmological reconstruction problem of mapping the present...
Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for...
Reconstructing the density fluctuations in the early Universe that evolved into the distribution of galaxies we see today is a challenge to modern cosmology. An accurate reconstruction would allow us to test cosmological models by simulating the evolution starting from the reconstructed primordial state and comparing it to observations. Several rec...
It is shown that the generalizations to more than one space dimension of the pole decomposition for the Burgers equation with finite viscosity and no force are of the form u = -2 viscosity grad log P, where the P's are explicitly known algebraic (or trigonometric) polynomials in the space variables with polynomial (or exponential) dependence on tim...
It is shown that non-helical (more precisely, parity-invariant) flows
capable of sustain- ing a large-scale dynamo by the negative magnetic
eddy diffusivity effect (Lanotte et al. 1999) are quite common. This
conclusion is based on numerical examination of a large number of
randomly selected flows. Furthermore, it is shown that, for parity-
invaria...
It is shown phenomenologically that the fractional derivative ξ = D
α
u of order α of a multifractal function has a power-law tail ∝\(\left| \xi \right|^{ - p_ \star}\) in its cumulative probability, for a suitable range of α's. The exponent is determined by the condition \(\zeta _{p_ \star } = {\alpha }p_ \star\), where ζ
p
is the exponent of the...
The asymptotic decay of passive scalar fields is solved analytically for the Kraichnan model, where the velocity has a short correlation time. At long times, two universality classes are found, both characterized by a distribution of the scalar-generally non-Gaussian-with global self-similar evolution in time. Analogous behavior is found numericall...
The unified classical path (UCP) theory is tested for a line model similar to the Anderson model, for which the true profile is known. The central intensity is found to be zero, unlike that of the true profile. It is also shown that the central intensity of Ly alpha predicted by the UCP theory is much lower than the true value (corrected for absorp...
Extending work of E, Khanin, Mazel and Sinai (1997 PRL 78:1904-1907) on the one-dimensional Burgers equation, we show that density pdf's have universal power-law tails with exponent -7/2. This behavior stems from singularities, other than shocks, whose nature is quite different in one and several dimensions. We briefly discuss the possibility of de...
Using multi-scale ideas from wavelet analysis, we extend singular-spectrum analysis (SSA) to the study of nonstationary time series, including the case where intermittency gives rise to the divergence of their variance. The wavelet transform resembles a local Fourier transform within a finite moving window whose width W , proportional to the major...
