Fractal Geometry: Mathematical Foundations and Applications.

3-Adic Cantor function on local fields and its p-adic derivative

Exponential demand for broadband applications has elicited renewed interest in characterizing noise in power line communication (PLC) networks for optimal error performance analysis. PLC is economical and ubiquitous, however due to multipath phenomena and nonstationary impulsive noise, the PLC channel is very harsh to signal transmission. Recent studies reveal that PLC noise is cyclostationary and modelling its characteristic should exploit the spectral and temporal redundancy. In addition, preliminary research shows that PLC noise exhibits self-similarity and can be predicted by fractal theory. In modelling noise employing fractal theory, the Hausdorff dimension (D) needs to be approximated. D is related to Hurst parameter (H), which is a measure of self-similarity. Hence, it is important to estimate H for accurate noise prediction using fractal theory. H parameter estimators assume stationarity in time series data. Therefore, preprocessing is required for the case of PLC noise before estimating H to make it stationary as periodicity and nonstatioanrity affects the H estimators. In this study, we remove the periodicity hidden in the PLC noise obtained from measurement campaign within several university offices, workshops and laboratories. Then long-range dependence is estimated from this pre-processed noise data by employing three estimation algorithms: R/S (rescaled adjusted range) method, aggregated variance method and absolute values of aggregated series method. Results from the said estimation methods though not the same, show that the Hurst parameters obtained from various noise measurements are greater than 0.5, implying presence of self-similarity with long-term memory. Results from this study will find valuable applications in PLC noise modelling using fractal and chaos theory and in error performance analysis in both narrowband and broadband PLC systems.

The energy of a mechanical system subjected to a random force with zero mean increases irreversibly and diverges with time in the absence of friction or dissipation. This random heating effect is usually encountered in phenomenological theories formulated in terms of stochastic differential equations, the epitome of which is the Langevin equation of Brownian motion. We discuss a simple discrete impulsive model that captures the essence of random heating and Brownian motion. The model may be regarded as a discrete analog of the Langevin equation, although it is developed ab initio. Its analysis requires only simple algebraic manipulations and elementary averaging concepts, but no stochastic differential equations (or even calculus). The irreversibility in the model is shown to be a consequence of a natural causal stochastic condition that is closely analogous to Boltzmann’s molecular chaos hypothesis in the kinetic theory of gases. The model provides a simple introduction to several ostensibly more advanced topics, including random heating, molecular chaos, irreversibility, Brownian motion, the Langevin equation, and fluctuation-dissipation theorems.

Micro food web networks on suspended sediment

This paper discusses to establish the characteristic model of particles' Brownian motion by fractal theory. Particles' Brownian motion is a kind of moving at random and particles' position oscillates with extreme irregularity. After having analyzed this process, authors present a fractal model to describe the physical process and prove theoretically the model is valid. The relationship between fractal characteristic information and particle size parameters is deduced. Combining image processing technique, it can provide a new method and technique of submicron particle sizing.

Many types of financial time series, most notably market returns, have been found to exhibit long-range memory as well as dramatic day-to-day swings that cannot be adequately represented by light-tailed distributions such as the normal distribution. In particular, this means that for such time series, the usual variance parameter (i.e., the sum of covariances at all time lags) is not defined because the covariance function does not converge to zero fast enough as the time lag increases. Moreover in such time series, often the tails of the marginal density converge to zero so slowly that higher-order marginal moments such as skewness and kurtosis fail to exist. Therefore conventional methods for analyzing simulation-generated time series cannot generally be applied to high-fidelity simulations of financial markets.Building on earlier work in fractal geometry and fractal time series, Mandelbrot et al. proposed the multifractal model of asset returns (MMAR) as an alternative to the ARCH models for analyzing time series exhibiting volatility clustering, long-range dependence, and heavy-tailed returns (Mandelbrot et al. 1997). They defined the multifractal spectrum as the renormalized probability density function of the Holder exponents observed in the time series; and they used the multifractal spectrum to measure the ability of MMAR to match the statistical properties of real data. In 2002 Kantelhardt et al. formulated multifractal detrended fluctuation analysis (MF-DFA), an algorithm for extracting the multifractal spectrum from a time series (Kantelhardt et al. 2002).Many economists have recently adopted agent-based simulation for modeling financial markets. Fads based on new products, shocks related to world events, scandals involving company leaders, or outright criminal activity can drastically change the processes and relationships governing a market. Already there is evidence that agent-based models yield more accurate approximations to the true or observed behavior in financial markets. Agent-based models exhibit emergent behaviors that have been linked to non-Gaussian interaction metrics and singularities in the time series they generate (Chan 2011). Farmer et al. constructed a agent-based model that is capable of exhibiting many of the statistical properties observed in real financial data simply due to the rules governing a double auction order book (Farmer et al. 2005).During the 20th century, economists relied heavily on Brownian motion to construct theoretical models of finance. The advantages of using Brownian motion stem from its foundation in the Gaussian distribution; and with advances in stochastic calculus and differential equations, it became a staple in models of empirical finance. The well-known Black-Scholes model for options pricing is a prime example of this application. Multifractal analysis has the ability to illuminate the underlying difference between a monofractal like Brownian motion and a considerably more diverse construction like the multifractal binomial measure. Thus, combining multifractal analysis with an agent-based model has the potential to provide insight into the underlying processes and behaviors of a dynamic economic system that is constantly changing and evolving.To analyze market-return time series exhibiting volatility clustering, long-range dependence, or heavy-tailed marginals, we exploit multifractal analysis and agent-based simulation. We develop a robust, automated software tool for extracting the multifractal spectrum of a time series based on MF-DFA. Guidelines are given for setting MF-DFA's parameters in practice. The software is tested on simulated data with closed-form monofractal and multifractal spectra as well as on observed data, and the results are analyzed. We also present a prototype agent-based financial market model based on the zero-intelligence model of Farmer et al. and analyze its output using MF-DFA. The figure below compares the multifractal spectrum of the output from the prototype model with that of the multifractal binomial measure. The ultimate objective is to expand this model to study the effects of microlevel agent behaviors on the macrolevel time series output as analyzed by MF-DFA. Finally we explore the potential for validating agent-based models using MF-DFA and thus being able to tune these models to the multifractal spectrum of empirical data.

Mathematics is often taught as an “absolute science” where there is a clear distinction between true and false or right and wrong, which should be uni- versally accepted by all professional mathematicians and every enlightened layman. This is true to a large extent, but there are important aspects of mathematics where agreement has been lacking and still is lacking. The development of mathematics in fact includes as fierce quarrels as any other science. In the beginning of the 20th century, the very foundations of mathematics were under intense discussion. In parallel, a split between “pure” and “applied” mathematics developed, which had never existed before. Traditionally, mathematicians were generalists combining theoretical mathematical work with applications of mathematics and even work in mechanics, physics and other disciplines. Leibniz, Lagrange, Gauss, Poincare and von Neumann all worked with concrete problems from mechanics, physics and a variety of applications, as well as with theoretical mathematical questions.

This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem.

Yolk/shell particles composed of a submicrometer-sized movable core and a silica shell are promising building blocks for novel optical colloidal crystals, because the locations of cores in the shell compartment can be reversibly changed by using external stimuli. Two dimensional arrays of yolk/shell particles incorporating movable cores were prepared by a self-assembly method. The movable cores of colloidal crystals in water could be observed with an optical microscope under application of external electric field. The motions of inner silica cores depended on the electric field strength and frequency and were categorized into three cases: (1) Random Brownian motion, (2) anisotropic motion of cores moving in a direction orthogonal to a field, and (3) suppressed motion fixed in the center of shell compartment. Random Brownian motion of cores was scarcely affected by field strength when a high frequency (in the MHz range) electric field was applied. On the other hand, an increase in field strength at low-frequency fields (kHz) transiently changed the core motion from (1) to (2) and a further increase in field strength changed it from (2) to (3). When the silica core was incorporated in a large void a stronger electric field was needed to suppress its motion than when it was in a small void. The high responsivity to electric fields in a low-frequency range indicated the importance of electric double layer (EDL) interaction between core and inner shell in controlling the core location in yolk/shell colloidal crystals. It was also shown that movable titania cores in yolk/shell particles required a low-frequency field with a high strength to change from the random to anisotropic motion. The result suggested that the electrostatic interaction between EDLs of the silica core and the inner silica wall could be stronger than that between EDLs of the titania core and the silica shell.

Hunting Strategy of Phages that Infect Bacteria

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, in other words to uniform motions around a circle - the epitome of a smooth shape. A study of roughness had necessarily to wait until specific mathematical tools had been discovered and, much later, suitably interpreted. Fractal geometry began when I reinterpreted the flight from nature that had led mathematicians to conceive of notions like the Holder exponent, the Cantor set, or the Hausdorff dimension. They boasted of these notions being ‘monstrous’ but in fact I turned them over into everyday tools of science. I also added further tools that - taken together - made roughness quantitatively measurable for the first time. Acquiring a quantitative measure is the step that moves a field into maturity. And this move instantly led to a striking conjecture. In 1984, ‘Nature’ published an article I wrote with D. A. Passoja and A. J. Paullay on metal fractures. We found that the traditional measures of their roughness range all over. To the contrary, their fractal roughness varies very little not only between samples but also between materials. Last time I checked the “universality” had been extended but not explained. The new intrinsic measure created a major intellectual mystery. The first major new tool that I added to those contributed by the likes of Holder, Cantor, and Hausdorff was multifractality, for both measures and functions. I was motivated by the urge to model the intermittence of turbulence but my first full paper (in 1972) also noted that the same techniques ought to apply to the intermittence in the variation of financial prices. An ancient adage claimed that the City of London is as unpredictable as the weather. I found unexpectedly quantitative truth to this adage by showing that both phenomena can be tackled with essentially the same tools. Roughness is everywhere therefore fractal geometry has little fear of running out of problems. This address will sketch the fractal geometry of roughness and explore some new developments relevant to this Congress.

Die Untersuchung der Ähnlichkeiten zwischen der mikroskopischen und makroskopischen
\nGrößenordnung ist Teil dieser Arbeit, die sich in der Forschungsfrage
\nwiderspiegelt: Was können wir über die mikroskopische Skala durch Untersuchung
\nder Makroskala erfahren? Untersuchungen der Umgebung, in der das
\nSelbst-assembly stattfindet, und das Selbst-assembly an sich helfen diese Frage zu
\nbeantworten.
\nWir imitierten die Mikroskala und identifizierten mehrere analoge Parameter.
\nAnstelle von Hitze verwenden wir Turbulenz, statt mikroskopisch kleine Partikel
\nkommen zentimetergroße Partikel zum Einsatz. Der Schwerkraft wurde durch
\neinen nach oben gerichteten Wasserstrom entgegengewirkt, da ihr Einfluss auf
\nmakroskopische Partikel beträchtlich ist. Auf mikroskopische Partikel hat sie dagegen
\nnur einen geringen Einfluss. Auf die gleiche Art und Weise hat Wärme einen
\ngroßen Einfluss auf die mikroskopische Skala, aber einen geringen Einfluss auf
\ndie makroskopische Partikel. Die Turbulenz erwies sich als geeignetes Analogon
\nfür Wärme und wurde wie im mikroskopischen Maßstab modelliert. Dabei kamen
\nthermodynamische Konzepte wie Brownsche Bewegung, Diffusion, Kinetik und
\ndie Einstein-Relation zum Einsatz. Diese Konzepte erwiesen sich auch im makroskopischen
\nMaßstab als geeignet.
\nDie Teilchengeschwindigkeit ist Maxwell-Boltzmann verteilt und die durchschnittliche
\nquadratische Verschiebung ist in Übereinstimmung mit einer begrenzten
\nZufallsverschiebung. Der Diffusionskoeffizient und die Geschwindigkeit sind
\nunabhängig von der Partikelgröße. Dies führt zu der Interpretation, dass die
\nBewegung einer einzelnen zentimetergroßen Kugel der Bewegung eines mikroskopischen
\nTeilchens a¨hnelt, indem sie eine Zufallsverschiebung und eine Brownsche
\nBewegung ausführt. Um mikro- oder nanoskopische Partikel sichtbar zu machen,
\nwerden häufig Rasterelektronen- oder Lichtmikroskopie verwendet. Anstelle von
\nMikroskopen verwendeten wir Videokameras, um die Experimente mit zentimetergroßen
\nPartikeln aufzuzeichnen. Eine Schwimmbadpumpe und asymmetrische Einströmung
\n(eines von vier Einlassventilen ist immer geschlossen) wurden eingesetzt,
\num Turbulenzen und asymmetrische Strömung zu erzeugen. Dieser asymmetrische
\nZufluss kann große makroskopische Wirbel verursachen, die die Eingangswärme
\nim Mikromaßstab darstellen. Im mikroskopischen Fall ist die Brownsche Bewegung
\n90
\nZusammenfassung 91
\nder Teilchen das Ergebnis derAusbreitung vonWärme, die von ihrer Quelle ausgeht,
\nwährend im makroskopischen Maßstab die von der Asymmetrie der Strömung herrührende
\nVermehrung die Brownsche Bewegung großer Teilchen verursacht. Trotz
\ndieser Ähnlichkeiten zwischen Wärme und Turbulenz variieren die Werte für die
\nPerturbationsenergie erheblich, je nachdem, ob sie über eine Einzelkugel-Diffusion
\n(Einstein-Beziehung) und Geschwindigkeit (kinetische Energie) oder über Wechselwirkungen
\nzwischen zwei magnetische Kugeln über die Entfernung bestimmt
\nwurden. Der letztere Fall ist um eine Größenordnung niedriger, etwa 6.5 μJ verglichen
\nmit 80 μJ. Dies legt nahe, dass die Wärme- oder Turbulenzenergie-Spektren
\nin Bezug auf ihreWirkung auf die Partikel unterschiedlich sein können. Es besteht
\neine gerichtete Abhängigkeit von Teilchengeschwindigkeit, Diffusion und Perturbationsenergie.
\nDie horizontalen Dimensionen sind ähnlich, aber die vertikale
\nKomponente zeigt eine stärkere Abhängigkeit in Bezug auf Strömungsasymmetrie
\nund Turbulenz. Der Richtungsabhängigkeit kann voraussichtlich über zukünftige
\ntechnische Anpassungen entgegengewirkt werden. Die gerichtete Abhängigkeit
\nkann auch als Temperaturgradient interpretiert werden.
\nDas Self-assembly wurde durch Strukturbildung mehrerer magnetischer Kugeln
\noder eines Dodekaheders aus zwölf heptagonalen magnetischen Plättchen durch
\nsystematische Variation von Turbulenz und Asymmetrie untersucht. Die multiplen
\nmagnetischen Kugeln bilden Linien und Ringe, und ihr Auftreten entsprach der
\nTheorie, jedoch wichen die absoluten Energien der Strukturen von der Theorie ab.
\nFür Experimente mit zunehmender Anzahl von Kugeln stellen vier Kugeln einen
\nÜbergang zwischen Linien und Ringen dar. Das System erwies sich als geeignet
\num die minimale Energiestruktur anzustreben, was wiederum zeigt, dass sich unser
\nmakroskopisches System ähnlich wie die Mikroskala verhält. Die Turbulenz wirkte
\nähnlich wie dieWärme, da fast nur einzelne Teilchen bei hoher Turbulenz beobachtet
\nwurden, während sich Linien und Ringe, bei abnehmender Turbulenz bildeten,
\nwas einem Phasenübergang zwischen einer Flüssigkeit und einem Feststoff oder
\neinem Gas und einer Flüssigkeit ähnelt.
\nDas Self-assembly von zwölf Zentimeter großen pentagonalen Plättchen
\nzeigte das gleiche Energieminimierungsverhalten wie die Kugelstrukturen. Ein
\nvollständiges Self-assembly des Dodekaeders wurde nicht erreicht. Vorherrschend
\nwaren intermediäre Strukturen mit maximalen Kontakten zwischen jedem Partikel
\n(Trimer und Tetramer etc.), was der minimalen Energiestruktur entspricht. Auch in
\ndiesem komplexeren Fall verhält sich das System ähnlich wie im Mikromaßstab.
\nDie beiden Beispiele des Self-assembly repräsentieren zum einen die Bildung
\neinfacher Strukturen (Ringe und Linien) und zum anderen einen komplexeren Fall
\ndes Self-assembly (ein hohles Dodekaeder). Das letztere Beispiel kann als Selfassembly
\neines geometrischen Konstrukts oder als Modell f ür das Selbst-assembly
\neines sphärischen Virus interpretiert werden. Dies unterstreicht das Potenzial des
\nmakroskopischen Self-assembly; es kann bei der Untersuchung von allgemein
\nweitgehend größenunabhängigen Problemen oder als analoge Darstellung bei der
\nUntersuchung von natürlich vorkommenden Phänomenen eingesetzt werden.

We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections into N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of to X. The same quantity also bounds the factor by which the Hausdorff dimension of X can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of X need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number d of a set X with box-counting and Hausdorff dimension d in the real Hilbert space 2 such that for all projections into N, no matter how large N is, the Hausdorff dimension of (X) is less than d (and in fact, is less than two, no matter how large d is).

The field of “Data Assimilation” has been driven by applications from the geosciences where complex mathematical models are interfaced with observational data in order to improve model forecasts. Mathematically, data assimilation is closely related to filtering and smoothing on the one hand and inverse problems and statistical inference on the other. Key challenges of data assimilation arise from the high-dimensionality of the underlying models, combined with systematic spatio-temporal model errors, pure model uncertainty quantification and relatively sparse observation networks. Advances in the field of data assimilation will require combination of a broad range of mathematical techniques from differential equations, statistics, machine learning, probability, scientific computing and mathematical modeling, together with insights from practitioners in the field. The workshop brought together a collection of scientists representing this broad spectrum of research strands.

Chapter 3 - Dimension Theory