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In what area, specifically, were your post graduation studies concentrated? Why did you choose this area? The topic of my Ph.D. thesis was conceived and written completely on my own, I chose a very unusual combination of linguistics, statistics (power-law distributions) and statistical physics. This was viewed as strange but I thought was far more interesting than the conventional topics that my teachers mentioned as possibilities. The thesis was very badly written, but much of its content has survived and is now viewed as "classical".

New multiplicative and statistically self-similar measures μ 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 non-degeneracy and the multifractal analysis of μ. On a bounded interval, the positive and negative moments of diverge under broad conditions.

the Brownian model, even the largest of N successive daily price increments contributes negligibly to the overall sample variance. The resulting 'absent' concentration justifies the role of variance in measuring Brownian volatility. Mandelbrot introduced in 1963 an alternative 'mesofractal model', in which the population variance is infinite. A significant proportion of the overall sample variance comes from an absolutely small number of large contributions, expressing a 'hard' form of concentration. To achieve a prescribed proportion of the overall measured variance, those 1900 and 1963 models require numbers of days of the order of N 1 and N 0 , respectively. This paper shows that an intermediate possibility exists: a new and very flexible 'soft' form of concentration is provided by the 'multifractal' model Mandelbrot introduced in 1997. The standard 'extreme values' theory applies to mesofractals but multifractals behave very differently. The single largest contribution to sample variance is asymptotically negligible; however, an arbitrarily high proportion of the overall variance is contributed by a number of days of the order of N D , where 0

This article describes a versatile family of functions that are increasingly roughened by successive interpolations. They reproduce, in the simplest way possible, the main features of financial prices: continually varying volatility, discontinuity or concentration, and the fact that many changes fall far outside the mildly behaving Brownian 'norm'. Being illuminating but distorted and incomplete, these constructions deserve to be called 'cartoons'. They address both the observed variation of financial prices and the generalized model the author introduced in 1997, namely, Brownian motion in multifractal time. Special cases of the same construction provide cartoons of the Bachelier model - the Wiener Brownian motion - or the two models the author proposed in the 1960s, namely, Levy stable and fractional Brownian motions. The cartoons are the embodiment of the author's 'principle of scaling in economics'. While rich in structure, they are unexpectedly parsimonious, easily computed, and easily compared to on...

The scaling properties of financial prices raise many questions. To provide background - appropriately so in the first issue of a new journal! - this paper, part I (sections 1 to 3), is largely a survey of the present form of some material that is well known yet repeatedly rediscovered. It originated in the author's work during the 1960s. Part II follows as sections 4 to 6, but can to a large extent be read separately. It is more technical and includes important material on multifractals and the 'star equation'; part of it appeared in 1974 but is little known or appreciated - for reasons that will be mentioned. Part II ends by showing the direct relevance to finance of a very recent improvement on the author's original (1974) theory of multifractals.

is a direct continuation of the preceding paper, with which it shares the front material and the numbering of the sections. A little repetition makes it possible to read this paper, part II, by itself. It describes the progression of the formalism from the financial model the author introduced in 1963, with independence and α< 2, to the financial model the author developed 1997, with multifractal dependence and 1

Individual investors and professional stock and currency traders know better than ever that prices quoted in any financial market often change with heart-stopping swiftness. Fortunes are made and lost in sudden bursts of activity when the market seems to speed up and the volatility soars. Last September, for instance, the stock for Alcatel, A French telecommunications equipment manufacturer, dropped about 40 percent one day and fell another 6 percent over the next few days. In a reversal, the stock shot up 10 percent on the fourth day.

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 price variation founded on the (Cauchy-Polya-Levy) stable distribution, with striking data collapse that accounted for observed large deviations from Gaussianity. In 1965, a different form of exact renormalization led to fractional Brownian motion, which neglected large deviations but accounted for long dependence and the resulting non periodic cyclic behavior. Finally, from a seed planted in 1972, exact renormalizability and scaling led to a model of price variation of which the M1963 and M1965 models are special examples. This broader model, fractional Brownian motion in multifractal time, accounts simultaneously for both large deviations and long dependence. These three steps are in loose parallelism with space, time and joint renormalization in statistical physics. This presentation surveys the old works and many new developments described in the author's 1997 books on fractals and scaling in finance.

It is often said that 20th-century 'modern' buildings are sterile, not built to human scale and, in fact, unnatural. The more I ponder such statements and their variants, the more elusive I find their meaning and the more I feel that a discussion of the logic and of the aesthetics of the notion of scale needs to be resumed. Clearly, a building's absolute or even relative height and number of stories are incidental and other aspects of scale are more important: I propose that it might be interesting to introduce into aesthetics the distinction between scalebound objects and scaling objects, a broad distinc tion that is proving increasingly useful in several scien tific contexts. One of my conclusions is that it is fruitful to call Mies van der Rohe's buildings scalebound?a term a physicist would use to describe a flawless crystal and the solar system?and to call the Paris Opera House a scaling building?the term scaling also being applic able to typical views of the Alps and to the visual characteristics of many other objects in nature, some of them visible (large or small) and others invisible to the naked eye. Before I elaborate on this dichotomy, I must emphasize it has limits. I realize that numerous exam ples both in the sciences and in the visual arts combine scalebound and scaling features. In addition, one must keep in mind the pitfalls of analogy; mine are unlikely to be entirely new and are certain to be found eventual ly unsatisfactory through counterexample and contra diction. In very rough outline (important features will be introduced gradually later), I propose the term scale bound to denote any object, whether in nature or one made by an engineer or an artist, for which characteris tic elements of scale, such as length and width, are few in number and each with a clearly distinct size. A spherical radome for sheltering a radar assembly is particularly scalebound, since its perceived scale' is determined by the smallest possible number of mea surements: a single intrinsic measure, its radius, and a single extrinsic one, the distance from which it is viewed; if a scalebound object is on the human scale, it will not, in general, be on the scale of a fly. Consider another good example: a Bauhaus style, glass cube type building. If its windows are identical rectangles delimited by metal grids, its only characteristic lengths are the height and width of the whole and of the windows, where the smaller is a harmonic (whole divisor) of the bigger fundamental one. Its basic struc ture is so spare that as the result of slight changes, such as interrupting the fa?ade by a few floors of different design or by changing the divisions between the win dows from thin metal to thick concrete, the overall structure is enormously enriched. Some Post-Bauhaus architects are experts at such enrichment. In any event, any scalebound object does have a scale of its own, and the nearest it can come to being on a human scale is when it has the same size as a human standing next to it.

In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and ∞.

Objective Although second-trimester and third-trimester reference curves for human fetal hand growth allow for identification of several genetic syndromes, little is known about first-trimester hand growth. We investigated first-trimester hand growth in euploid and aneuploid fetuses. Method Between 9 and 12 weeks' gestational age (GA), wrist width, hand width, hand length, and hand index were measured in three-dimensional (3D) ultrasound datasets of 112 euploid and 65 aneuploid pregnancies. We constructed reference curves for these measurements in euploid pregnancies and calculated z-scores for measurements in aneuploid pregnancies. Reproducibility was established in a subset of 20 datasets. Results While wrist width, hand width, and hand length increased with gestational age, hand index decreased. Intraobserver and interobserver intraclass correlation coefficient values were > 0.97. In trisomy 21 cases, wider wrists and hands were observed compared with euploid pregnancies (mean z-scores 1.06, SD 2.04, p< 0.001 and 1.16, SD 1.30, p< 0.001, respectively). Trisomy 18 cases showed narrower and shorter hands (mean z-scores -0.74, SD 1.20, p = 0.009 and -0.97, SD 0.86, p = 0.005, respectively). In trisomy 13 cases, no differences were observed. Conclusion Reference values are available for first-trimester studies on human hand development. First-trimester hand measurements in trisomies 21 and 18 differ significantly from those in euploid pregnancies and may be useful for early identification of abnormal development. (C) 2014 John Wiley & Sons, Ltd.

We study the Levy dusts on the line on two accounts: the fluctuations around the average power law that characterizes the mass-radius relation for self-similar fractals, and the statistics of the intervals between strides along the logarithmic axis (their tail distribution is related to the dust`s fractal dimension). The Levy dusts are suggested as a yardstick of neutral lacunarity, against which non-neutral lacunarity can be measured objectively. A notion of perceived dimension is introduced. We conclude with an application of the Mittag-Leffler statistics to a nonlinear electrical network. {copyright} {ital 1997} {ital The American Physical Society}

We showed in an earlier paper (1995a) that negatively correlated fractional Brownian motion (FBM) can be generated as a fractal sum of one kind of micropulses (FSM). That is, FBM of exponent H 1 2 is the limit (in the sense of finite-dimensional distributions) of a certain sequence of processes obtained as sums of rectangular pulses. We now show that more general pulses yield a wide range of FBMs: either negatively (as before) or positively ( H> 1 2 ) correlated. We begin with triangular (conical and semi-conical) pulses. To transform them into micropulses, the base angle is made to decrease to zero, while the number of pulses, determined by a Poisson random measure, is made to increase to infinity. Then we extend our results to more general pulse shapes.

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We begin with stochastic processes obtained as sums of “up-and-down” pulses with random moments of birth τ and random lifetime w determined by a Poisson random measure. When the pulse amplitude e → 0, while the pulse density δ increases to infinity, one obtains a process of “fractal sum of micropulses.” A CLT style argument shows convergence in the sense of finite dimensional distributions to a Gaussian process with negatively correlated increments. In the most interesting case the limit is fractional Brownian motion (FBM), a self-affine process with the scaling constant 0 1 2 . The construction is extended to the multidimensional FBM field as well as to micropulses of more complicated shape.

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 emerging picture corresponds qualitatively and quantitatively to the scenario of infinite drift that starts from the familiar five-armed shape for small sizes and proceeds through increasingly tight multi-armed shapes. The transverse crosscuts show quantitatively how the lacunarity of circular clusters becomes increasingly "compact" with size. Finally, we find the transverse-cut dimensions to be in agreement for clusters grown in circular and cylindrical geometry, suggesting that the question of universality is best addressed on the crosscut.

There are strong reasons to believe that the multifractal spectrum of DLA shows anomalies which have been termed left-sided. In order to show that this is compatible with strictly multiplicative structures Mandelbrot and co-workers introduced a one-parameter family of multifractal measures, invariant under infinitely many linear maps, on the real line. Under the assumption that the usual multifractional formalism holds, they showed that the multifractal spectrum of these measures is indeed left-sided, i.e., increasing over the whole @a range ]@a"m"i"n, ~[. Here, it is shown that the multifractal formalism for self-similar measures does indeed hold also in the infinite case, in particular that the singularity exponents @t(q) satisfy the usual equation @Sp^q"i @l^@t"i = 1 and that the spectrum f(@a) is the Legendre transform of @t(q).

The behavior of very large clusters of diffusion-limited aggregation (DLA) was investigated to help discriminate between the two geometric scenarios recently described by Mandelbrot: finite transient and infinite drift. Using 50 DLA clusters of 1 million particles, we follow the increase during growth of the maximum radius of the clusters and of various relative moments. One can distinguish two regions: an inactive completely grown core and an active growing region. In the growing region, scale factors were defined the moments of the atoms distances from the original "seed". They do not cross-over to the behavior characteristic of self-similarity for finite sizes and support the novel "drift" scenario that postulate an infinite continuing "transient". The moment's "misbehavior" may help understand the disagreement between previous estimates of the clusters' fractal dimension.

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 converge to topologically self-similar objects.

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 asymptotically a periodic function of log4 n.

This paper amplifies upon earlier theories and observations by the authors, advances a probabilistic model of rain fields, and exhibits realistic simulations, which the reader is advised to scan before continuing with the text. The model is also compared with other approaches. The theory of fractals has been in part motivated by the Hurst effect, which is an empirical observation in hydrology and climatology. A fractal is an unsmooth shape that is scaling, that is, where shape appears unchanged when examined by varying magnifications. The study of fractals is characterized by the prevalence of hyperbolically distributed random variables, for which Pr( U > u ) ∝ u a, where Pr is the probability that the value of the variable exceeds u , and a is a positive exponent. Lovejoy established the applicability of fractals in meteorology, by showing that cloud and rain areas project on the Earth along shapes whose boundaries are fractal curves, and that the temporal and spatial structure of rain is rife with hyperbolically distributed features. These observations, as amplified in this paper, set up the challenge of constructing fractal models with the observed properties. The models presented here belong to a very versatile family of random processes devised in Mandelbrot: fractal sums of (simple) pulses, or FSP processes. The simulations to be presented reveal that these processes involve a scaling heirarchy of “bands”, “fronts”, and “clusters”, as well as other complex shapes, none of which had been deliberately incorporated in the process. This very rich morphology and the related statistical distributions exemplify the power of simple fractal models to generate complex structures, and are in accord with the wide diversity of actual rainfall shapes. It is argued that this model already provides a useful context in which the basic statistical properties of the rainfield, including the relationship between the temporal and spatial structure, may be studied. DOI: 10.1111/j.1600-0870.1985.tb00423.x

Affiche - résumé. Session GC: Microbes in the Changing Environmenl: Global Climate Change and Soif under Human Impact; International audience; The spatial organization of soil microbial communities over large areas and the identification of environmental factors structuring their distribution have been liUle investigated. The overall objective of this study was to determine the spatial paUerning of microbial biomass in soils on wide extent and to rank the environmental fiUers most influencing this distribution, by using the French Soil Quality Monitoring Network. This network covers ail the French terrilory and soils were sampled at2,150 sites along a systematic grid sampling. The soil DNA extracted from ail these soils was expressed in terms of soil molecular microbial biomass and related to other soil and land use data over the French territory. This study provides the first extensive map of microbial biomass and reveals the heterogeneous and spatially structured distribution of this biomass on the scale of France. The main factors driving biomass distribution are the soil physico-chemical properties (texture, pH and totai organic carbon) and land use also. Intensive agricultural crops, especially monoculture and vineyards, exhibited the smallest biomass pools in solI. Interestingly, factors known to influence the large scale distribution of macro organisms, such as dimatic factors, were not identified as important drivers for microbial communities. Microbial abundance is spatially structured and dependent on local filters such as soil characteristics and land use but is relatively independent of global fiUers such as climatic factors or the presence of natural barriers. Our study confirms that the biogeography of microorganisms differs fundamentally from the biogeography of "macroorganisms" and that soil management can have significant large-scale effects.

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