# Benoit B. Mandelbrot's research while affiliated with Yale University and other places

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## Publications (183)

The familiar cascade measures are sequences of random positive measures obtained on (0,1) via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose pos- sible strong or weak limits are natural candidates for modeli...

In order to understand better the morphology and the asymptotic behavior in Diffusion Limited Aggregation (DLA), we studied a large numbers of very large off-lattice circular clusters. We inspected both dynamical and geometric asymptotic properties, namely the moments of the particle's sticking distances and the scaling behavior of the transverse g...

Cowles Foundation Discussion Paper, n° 1166/1997

Cowles Foundation Discussion Paper, n° 1165/1997

The familiar cascade measures are sequences of random positive measures obtained on [0, 1] via b-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modelin...

Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in th...

Applied blindly, the formula for the dimension of the intersection can give negative results. Extending Minkowski's definition of the dimension by � -neighborhoods to � -pseudo- neighborhoods, that is, replacing (A ∩ B)� with A� ∩ B� , we introduce the notion of negative dimensions through several examples of random fractal constructions.

The standard “Brownian” model of competitive markets asserts that the increments of price (or of its logarithm) are statistically independent and Gaussian, implying that price itself is a continuous function of time. This model arose in 1900, at an immediately high level of perfection, in the work of L. Bachelier. In many fields of science it becam...

Positive T-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval T = [0, 1] and replace random measures by random functions. We...

Positive $T$-martingales were developed as a general framework that
extends the positive measure-valued martingales and are meant to model
intermittent turbulence. We extend their scope by allowing the
martingale to take complex values. We focus on martingales constructed
on the interval $T=[0,1]$ and replace random measures by random
functions. We...

The partly random fractal sums of pulses (PFSP) are a family of random functions that depend on a kernel function K and at least two positive parameters C and δ. Given K, the construction of F(t; C, δ) consists in adding N affine versions of a pulse as follows. The pulse height Λ and its width W are random variables related by w/λ
δ = a constant. T...

In order to understand better the morphology and the asymptotic behavior in Diffusion-Limited Aggregation (DLA), we studied a large number of very large off-lattice circular clusters. We inspected both dynamical and geometric asymptotic properties via the scaling behavior of the transverse growth crosscuts, i.e. the one-dimensional cuts by circles....

The temporal development of patterns in diffusion-limited aggregation (DLA) growth in cylinder geometry is accompanied by various fluctuating quantities. We give experimental evidence that the fluctuations of the highest growth probability and those of the thickness of the interface and of the distance between the highest site and the average heigh...

The Earth's pole motion is characterized by an annual term, the Chandler wobble, and a strong secular motion. The annual is nearly periodic and the Chandler wobble is nearly a damped oscillation, but data are so limited that little about the secular term can be learned by conventional statistical analysis. Instead, we have used a new technique call...

The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of $\frac{1% }{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is...

Roughness is, among human sensations, just as fundamental as color or pitch, or as heaviness or hotness. But its study had
remained in a more primitive state, by far. The reason was that both geometry and science were first drawn to smooth shapes.
Thus, color and pitch came to be measured in cycles per seconds, that is, were reduced to sinusoids, i...

Various distinct aspects of the geometry of turbulence can be studied with the help of a wide family of shapes for which I have recently coined the neologism “fractals.” These shapes are loosely characterized as being violently convoluted and broken up, a feature denoted in Latin by the adjective “fractus,” Fractal geometry approaches the loose not...

This article has no abstract.

Obukhov, Kolmogoroff and Yaglom, and in effect (independently) the geologist deWijs, have argued that energy dissipation in intermittent turbulence is lognormally distributed. However, this hypothesis will be shown to be probably untenable: depending upon the precise formulation chosen, it is either unverifiable or inconsistent.The present paper pr...

Physical and geometrical properties are studied on self similar fractal lattices. Properties of spin systems are shown to depend on various topological factors, in addition to the fractal dimensionality. A (non random) fractal model is proposed for the backbone of the infinite cluster near percolation in d dimensions, and its properties agree with...

In this paper, the classical Lvy flights are generalized, their jumps being replaced by more involved pulses. This generates a wide family of selfaffine random functions. Their versatility makes them useful in modeling. Their structure throws new conceptual light on the difficult issue of global statistical dependence, especially in the case of pro...

This short paper advances and defends a strong statement concerning financial modeling. It argues that, even when the present fractal models become superseded, fractal tools are bound to remain central to finance. The reasons are that the main feature of price records is roughness and that the proper language of the theory of roughness in nature an...

Having been crafted to welcome a new scientific journal, this paper looks forward but requires no special prerequisite. The argument builds on a technical wrinkle (used earlier but explained here fully for the first time), namely, the author’s grid-bound variant of Brownian motion B(t). While B(t) itself is additive, this variant is a multiplicativ...

The bulk of this text consists in nonsystematic sketches of the current status of diverse very difficult questions in various mathematical sciences. All were triggered by actual fractal pictures generated by computer. In physics some of those questions outline a nascent “rational rugometry,” involving quantitative measures of roughness. Other quest...

Chapter foreword (2003). This chapter reproduces Chapter 19 of M 1982F, with a few additions clearly marked as such: Plate C5 of M 1982F (identical to its jacket) and a paragraph and a few illustrations from M 1985g that fit here better than they would elsewhere in this book.

THIS CHAPTER BEGAN AS A SINGLE PAGE TO ACKNOWLEDGE my indebtedness to three individuals: Mandelbrojt, Douady, and Hubbard. But it grew and — unavoidably — became increasingly autobiographical. It even extended the scope of the word “acknowledgment” by commenting about Bourbaki, my Nemesis.

This non-technical text serves two purposes.

THIS CHAPTER DESCRIBES THE CIRCUMSTANCES under which I had the privilege of discovering in 1980 the set that is the main topic of this book. As will be meticulously documented in Chapters C12 and C14, I actually saw this set in 1979 but bumbled and fumbled for about a year.

It is conjectured that the boundary of the ℳ-set of the complex map has a Hausdorff-Besicovich fractal dimension equal to 2.

DLA is nearly self-similar, but departures from simple self-similarity are unquestionable and quantifying their statistical nature has proven to be a daunting task. We show that DLA follows a surprising new scaling rule. It expresses that the screened region, in which the harmonic measure is tiny, increases more than proportionately as the cluster...

IN HISTORICAL SEQUENCE, Kleinian groups came before the Fatou-Julia theory, and I explored those topics in the same sequence. But in this book, I inverted history and have put forward my best foot, or at least the best known one. This introduction will also explain the mysterious initials IFS. As in Chapter C2, some names are first printed in bold...

THIS CHAPTER DRAWS FROM A CACHE OF LONG “LOST” illustrations that were prepared made in 1979 with the assistance of Mark R. Laff. They are published here for the first time, at long last, together with explanations and comments, describe in documented detail the story told in Chapter C12. Some are attractive, and most affect the basic distinction m...

HAVING A BROAD INTENDED READERSHIP, this chapter should begin by a very rough characterization. Mathematical analysis is “like calculus but far more advanced.” An exact definition does not exist. This is not surprising, because truly important notions are left undefined, even in mathematics! This is argued in an appendix to the preface of M 1999N a...

Foreword to this chapter and the appended figure (2003). The n
2 conjecture advanced in this chapter’s Section 2 was first proven in Guckenheimer & McGehee 1984. The two authors and I were participating in a special year on iteration that Lennart Carleson and Peter W. Jones convened during 1983–1984 at the Mittag-Leffler Institute in Djursholm (Swe...

Allow the very elementary transformation, from z to z2 + c to be repeated indefinitely. The process reveals geometric shapes—fractal attractors, fractal repellers and other fractal sets—that astonish by their number, their variety, and their beauty.

A “normalized radical” ℛ of the ℳ-set is defined as the shape that satisfies exactly all the self-similarity properties that hold approximately for the molecules of the ℳ-set of the quadratic map. Explicit constructions show that the complement of ℛ is a σ-lune, and prove that the ℛ-set does not self-overlap. The fractal dimension D of the boundary...

The author has recently shown that an important singular non-random measure defined in 1900 by Hermann Minkowski is multifractal and has the characteristic that a is almost surely infinite. Hence αmax = ∞ and its f(α) distribution has no descending right-side corresponding to a decreasing f(α). Its being left-sided creates many very interesting com...

"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in th...

The Julia set F* of the map z → ỹ(z) = z2-μ may be the boundary of an atom, of a molecule, or of a “devil’s polymer” in the z-plane. Denote the boundary of one of the atoms of F* by H. When μ ≠ 0 is the nucleus of a cardioid-shaped atom of the M-set, it is conjectured that the fractal dimension D of H is 1. Thus, H may be a be a rectifiable curve (...

To Chapter 18 of M 1982F, this chapter adds a related figure from Chapter 20. Changing from the unusual format of M 1982F to this book’s conventional format entailed a few nonlinear rearrangements and called for wording to be inserted for continuity or consistence. The last page of the original text summarized M 1983m{C18}; it was deleted. The art...

The terms “chaos” and “order in chaos” prove extremely valuable but elude definition. It remains important to single out instances when the progress to planar chaos can be followed in a detailed and objective fashion. This paper proposes to show that an excellent such example is provided by the iterates of a map for which z and λ are both complex....

WHEN A NEW MATHEMATICAL DEVELOPMENT IS EVALUATED, some purists allow the computer but are offended by strong physical motivation. This part is a new opportunity to discuss an example.

This work reports several observations concerning the dynamics of a continuous interpolate, forward and backward, of the quadratic map of the complex plane. In the difficult limit case |λ| = 1, the dynamics is known to have rich structures that depend on whether arg λ/2π is rational or a Siegel number. This paper establishes that these structures,...

THE THEORY OF ITERATION OF RATIONAL FUNCTIONS goes back to the mid-nineteenth century and perhaps even to Abel. During its first classic period, it indissolubly linked the names of Pierre Fatou and Gaston Julia. It also generated great controversy accompanied by hasty anecdotes and schematic or fanciful stories. The account in Alexander 1994 is exc...

For each complex μ, denote by F(μ) the largest bounded set in the complex plane that is invariant under the action of the map z → f(z) = z2-μ. M 1980n{C3} and M 1982F{FGN}, Chapter 19 {C4} reported various remarkable properties of the M0 set (the set of those values of the complex μ for which F(μ) contains domains) and of the closure ℳ of ℳ0. {P.S....

Within the M-set of the map z → λz(1-z), consider a sequence of points λm having a limit point λ. Denote the corresponding F* -sets by ℱ*(λm) and ℱ*(λ). In general, lim ℱ*(λm) = ℱ*(lim λm), but there is a very important exception. In some cases, the sets ℱ*(λm) do not converge to either a curve or a dust, but converge to a domain of the A -plane, p...

THE PATH OF SCIENTIFIC INVESTIGATION AND DISCOVERY is not necessarily logical, as history never tires of reminding both laymen and scientists. From many viewpoints, the complex quadratic map, reducible to either z→z2 + c, z → λ(z2-2), or z → λz(1-z), is the simplest of all nonlinear maps. Its global action was therefore the first to be studied care...

Chapter foreword concerning the illustrations, especially the “missing specks” of Figure 1 (2003). As described in Chapter C1, this paper boasts many “firsts” and was instrumental in reviving the theory of iteration. The many new observations it contains concern the set in the μ-plane for which A. Douady and J.H. Hubbard soon proposed the term “Man...

The mesofractal model is founded on the stable processes that date to Cauchy and Lévy. The unifractal model uses the fractional Brownian motions introduced by the author. By now, both are well-understood.

A nonnegative 1-periodic multifractal measure on is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. They are martingale structures, therefore converge. The criterion on W for nondegeneracy is provided. It di...

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. A new method...

Divergence of high moments and dimension of the carrier is the subtitle of Mandelbrot's 1974 seed paper on random multifractals. The key words divergence and dimension met very different fates. Dimension expanded into a multifractal formalism based on an exponent and a function f(). An excellent exposition in Halsey et al. 1986 helped this formalis...

New multiplicative and statistically self-similar measures A are defined on R as limits of measure-valued martingales. Those martingales are constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process defined in a strip of the plane. Several fundamental problems are solved, including the...

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial diffusion-limited aggregation is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D = 1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than M = 10(...

When suitably rescaled, the distribution of the angular gaps between branches of off-lattice radial DLA is shown to approach a size-independent limit. The power-law expected from an asymptotic fractal dimension D=1.71 arises only for very small angular gaps, which occur only for clusters significantly larger than one million particles. Intermediate...

The present paper is the first in a series of three closely related papers in which theinverse measureμ*(dt) of a given measure μ(dt) on [0, 1] is introduced. In the first case discussed in detail, μ and μ* are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not differentiable, and both are non-zero for e...

In a previous paper the authors introduced the inverse measure μ†of a probability measure μ on [0, 1]. It was argued that the respective multifractal spectra are linked by the “inversion formula”f†(α) = αf(1/α). Here, the statements of the previous paper are put into more mathematical terms and proofs are given for the inversion formula in the case...

The author proposed fractal scenarios for the distribution of galaxies in 1975 and 1977; they were expanded in 1982 and are
being developed further. The claim is that galaxies have a scale-invariant fractal distribution with a dimension well below3.
Fractality is the only input that is needed to account for the observed clustering combined with voi...

A new class of random multiplicative and statistically self-similar measures is defned on IR. It is the limit of measure-valued martingales constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process in a strip of the plane. Several fundamental problems are solved, including the non-degen...

This article concerns the fractal trees that are obtained recursively by symmetric binary branching. A trunk of length 1 divides into two branches of length r, each of which makes an angle θ > 0° with the linear extension of the trunk. Each branch then divides by the same rule. Some basic information on such trees is found in Chapter 16 of [FGN], o...

In diverse sciences that lack Hamiltonians, the analysis of complex systems is helped by the powerful tools provided by renormalization, fixed points and scaling. As one example, an intrinsic form of exact renormalizability was long used by the author in economics and related fields, most notably in finance. In 1962–3, its use led to a model of pri...

This paper defines and studies a general algorithm for constructing new families of fractals in Euclidean space. This algorithm involves a sequence of linear interscale transformations that proceed from large to small scales. The authors find that the fractals obtained in this fashion decompose in intrinsic fashion into linear combinations of a var...

In thin metallic films, semiconductors, nerve tissues, and many other media, the measured spectral density of noise is proportional to ƒD − 2, where ƒ is the frequency and D a constant 0 < D < 2. The energy of these “ƒD − 2 noises” behaves more “erratically” in time than is expected from functions subject to the Wiener-Khinchin spectral theory. Mor...

The purpose of this paper is twofold. From the viewpoint of engineering, it presents a model of certain random perturbations that appear to come in clusters, or bursts. This is achieved by introducing the concept of a “self-similar stochastic point process in continuous time.”KeywordsIEEE TransactionMarginal DistributionConditional DensityJoint Den...

For pt.II see ibid. vol.17, p.435 (1984). In the first two papers of this series the authors considered self-similar fractal lattices with a finite order of ramification R. In the present paper they study physical models defined on a family of fractals with R= infinity . In order to characterise the geometry of these systems, they need the connecti...

This book’s variety of purpose was played down when Chapter N1 put forward a modern synthesis centered around the concept of self-affine variability. This chapter, to the contrary, describes how the reprints classified into Parts II, III and IV fit together historically. It also examines conceptual connections and relevant historical events; all as...

The distribution of the rate of dissipation of free turbulence, as observed in the ocean and the atmosphere, seldom satisfies the homogeneity assumption of the classic Kolmogorov-Obukhov theory. Analogous “intermittency” (with “clustering” of the “active regions”) is also observed for the energy of various “1/ƒ noises,” for the spatial distribution...

Chapter N1 argued that Fƒ−B
noises with the same exponent B can take any of many very different forms. This brief chapter takes the next step and faces the challenge of going beyond the spectrum and discriminating between those various possibilities. Two very different questions come to mind.
Is it legitimate to view the measurements performed on a...

This brief chapter’s goal is to show that, next to obvious and deep differences and without actual intent or interaction, fruitful and sur prising parallels exist between my scientific work from 1956 to 1972 (as exemplified in M1997E and this book) and modern statistical physics.
The latter, to be denoted in this chapter as MSP, will be understood...

Simple multifractal measures are constructed by multiplying a periodically extended function with copies of itself. The frequencies of the copies form a geometric series, (1, b, b 2,…, b n ,…), where b is a real number larger than 1. This deterministic construction leads to measures that are similar to random multifractal measures, yet are easier t...

Obukhov, Kolmogorov and others argued that energy dissipation in intermittent turbulence is lognormally distributed. This hypothesis is shown to be untenable: depending upon the precise formulation chosen, it is either unverifiable or inconsistent. The paper proposes a variant of the generating model leading to the lognormal. This variant is consis...

Fractals constructed by recursive processes are introduced to model growth phenomena. These fractals are simultaneously directed and self-similar in analogy with patterns growing under diffusion-limited conditions. The multifractal nature of the harmonic measure associated with Laplacian interfaces is qualitatively interpreted using the models. Cal...

(This text was omitted in the original but inserted when distributing the original’s reprints). This paper introduces and investigates the new concept of “sporadically varying generalized random function.” A sporadic process attributes to the members of a family of functions a set of non-normalizable “probability” weights, in such a fashion that ev...

For pt.I see ibid., vol.16, p.1267 (1983). The authors construct and investigate a family of fractals which are generalisations of the Sierpinski gaskets (SGs) to all Euclidean dimensionalities. These fractal lattices have a finite order of ramification, and can be considered 'marginal' between one-dimensional and higher-dimensional geometries. Phy...

We study the Horton-Strahler ordering for random binary trees, which are statistically self-similar branching structures. Extending previously obtained results, we show that near the top of these trees, the expected bifurcation ratios tend strongly to the value 4. But at the root of the tree, the expected bifurcation ratio is less than 4, becoming...

Diffusion-limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords. To estimate the harmonic measure at the bottom of a fjord seems a prohibitive task, but the authors find that a new mathematical equality due to Beurling, Carleson and Jones makes it easy. They find that the harmonic measure at...

The iteration of random multiplications yields new random functions that are interesting theoretically and practically. For example, they represent intermittent turbulence, M 1974f{N15}, and the distribution of minerals. This paper also generalizes the stable random variables: Lévy’s criterion of invariance under non-random averaging is replaced by...

This paper proposes a new mathematical model to describe the occurrence of errors in data transmission on telephone lines. We claim that the distribution of inter-error intervals can be well approximated by a Pareto distribution of exponent less than one. It follows that the relative number of errors and the information-theoretical equivocation ten...

We study the branching structure of very large DLA clusters, of up to 100 million particles. The Horton-Strahler ordering of the branches in these clusters shows a relaxation towards a state with the stream numbers forming a geometric series. This behaviour is compared with those of several self-similar trees. It indicates that DLA clusters converg...

Finitely ramified fractal lattices show anomalous diffusion with (r2) varies as t2H. There is a hierarchy of transit times which are shown by Monte Carlo simulation to satisfy ( tau n) varies as alpha n for large n, where alpha =b1H/ and b is the base of the lattice. The lattice resistivity scales with size as rho varies as Rn where R is characteri...

Following his critical analysis of the random model of turbulence due to A, M. Yaglom, M 1974f{N14} and M1974c{N15} introduced his own model, which he calls “canonical.” It proceeds from a brick, that is subsequently divided into b, b
2, …, b
n
, … similar bricks; each brick of the n-th stage is divided into b equal bricks in the (n + l)-th stage....

Magnetic spin models and resistor networks are studied on certain self-similar fractal lattices, which are described as 'quasi-linear', because they share a significant property of the line: finite portions can be isolated from the rest by removal of two points (sites). In all cases, there is no long-range order at finite temperature. The transitio...

The authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries. The Laplacian potential around a linearly self-similar square Koch tree is studied in detail. The multiplicative nature of this potential, and the consequent multifractality of the ha...

Consider a trigonometric series that converges towards zero outside of a set E. Cantor posed the problem of determining whether it necessarily follows that this series also converges towards zero on E. When the answer is positive, E is called a set of unicity; when it is negative, a set of multiplicity. Basic references are Zygmund 1959, Chapter IX...

Multifractals and 1/ƒ noises arose independently, as will be seen in Chapter N2. Therefore, it is not surprising that they continue near-unanimously to be viewed as separate and unrelated concepts.

It is conjectured that turbulent dispersion in a closed vessel involves surfaces whose fractal dimension exceeds 2. The different singularities and quasi-singularities of the motion are carried by a hierarchy of sets whose dimensions are fractions. The quasi-singularities are viewed as being singularities of the Euler equations, after they have bee...

In the author’s fractal models, proposed in 1975, galaxies have a scale invariant fractal distribution of dimension well below 3. Fractality suffices to account for the observed clustering, and the presence of voids and walls. However, early examples of fractals presented too many holes (“lacunas”) to agree with the approximate isotropy of the sky....

The following facts are established: Critical Ising clusters in the plane are (as expected) self-similar. In their shape, they conform to the idea of the Present address: Departament de F'isica Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain y Present address: Computer Science Dept, Stanford University, Stanford CA 9430...

This paper presents the multifractal model of asset returns ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused...

The Multifractal Model of Asset Returns ("MMAR," see Mandelbrot, Fisher, and Calvet, 1997) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the loc...

This paper presents the first empirical investigation of the Multifractal Model of Asset Returns ("MMAR"). The MMAR, developed in Mandelbrot, Fisher, and Calvet (1997), is an alternative to ARCH-type representations for modelling temporal heterogeneity in financial returns. Typically, researchers introduce temporal heterogeneity through time-varyin...

## Citations

... Iteration is an extensive phenomenon in nature and human life. It causes many complexities such as bifurcations, chaos and fractals [12][13][14]. The computation of iteration of a general order or a higher order in the one-dimensional case is also a complicated work although efforts have been made to polynomials [1][2][3][4]21], quasi-polynomials [17,19], linear fractions [20] and rational fractions [11]. ...

... The Hurst exponent (H ) can be defined in terms of the serial correlation structure [10]. For each lag k of an exact fractal time series, ...

... Clearly, the p.d.f.s of θ * for both FIT and channel flow at all Re λ nearly coincide with that of the normal distribution (Obukhov 1962;Yeung & Pope 1989). It must be noted, however, that due to intermittency, the p.d.f. of pseudo-dissipation cannot be precisely log-normal (Orszag 1970;Mandelbrot 1999). To illustrate this further, the p.d.f.s are plotted in the log-linear scale in figure 13(b), which shows some dependence on Re λ and the type of forcing in the flow. ...

... where f l is a lower and f u an upper frequency limit of 1/f noise. For f l → 0 the integral diverges; Mandelbrot denoted this phenomenon, the infrared catastrophe [7]. In resistors or semiconductor materials, for example, 1/f noise has been observed down to a frequency of 10 −6 Hz [8]. ...

... On the technical side, first theoretical thoughts of fractals are said to reach back up to the 17th century. However, much of the today's popularity is attributable to Benot Mandelbrot and what he called the "heroic" nineteen-seventies (e.g., [67][68][69][70][71]73]). A bit earlier, with the introduction of rescaled range (R/S) analysis [47], Harold E. Hurst was first to measure a time-series (monofractal) persistence empirically. ...

... In reference [16] observations that correspond to z g ranging from 3.5 to 6 are given. In [17], z g of a wired channel is reported to be in the range of 3 to 10. The parameter z g is the only parameter comparable in all papers, since it directly corresponds to the slope of the CCDF in the double-logarithmic plot. ...

... This is shown in figure 2.1. What is most notable about this figure is perhaps that it clearly belongs to a class of functions called 'multi-fractal measures'[12,21]. The measure µ is known as the 'Minkowski measure'; its relationship to the Question Mark function is straight-forward, and may be stated as a formally: ...

Reference: On the Minkowski Measure

... [5] Por otro lado, Benoit Mandelbrot estudió el espacio de parámetros de polinomios cuadráticos en un artículo publicado en 1980 y despertó el interés global por el mismo. [19] El estudio matemático riguroso de este conjunto realmente comenzó con el trabajo de los matemáticos Adrien Douady y John H. Hubbard, quienes demostraron muchas de sus propiedades fundamentales y nombraron el conjunto en honor de Mandelbrot. Entre otras propiedades, probaron que es un conjunto conexo y formularon la conjetura MLC, que formula la creencia de que el conjunto de Mandelbrot es localmente conexo. ...

... For self-similar processes it is only defined at H with f (H ) = 1. The multifractal spectrum plays an important role in multifractal measures and it is defined as the dimension of sets with local Hölder exponent α (see Calvet et al. (1997) for details). ...

... In particular, one needs stationary multifractal measures that are natural and simple to define and simulate numerically. The heuristics and the pictures in [7] suggested that these goals could be fulfilled by the measures µ studied in this paper. In fact, as we show, mathematics defeated this hope. ...