This document discusses Mandelbrot's work on non-Gaussian stochastic processes and their implications for estimating risk and extremes. It contains the following key points:
1) Mandelbrot made three "giant leaps" in the 1960s-70s by proposing alpha-stable models to describe heavy-tailed fluctuations, applying self-similarity to model long-range dependence, and developing multifractal cascades.
2) If natural phenomena exhibit heavy tails and long memory, but models don't account for this, it can lead to underestimating risk and extremes. Mandelbrot's fractional stable motion combines heavy tails and long-range dependence.
3) Events that are rare according to short
This document discusses stochastic models of movement and search behavior. It begins by describing how Karl Pearson originally connected animal motion to random walks over a century ago. It then discusses how the Brownian motion and random walk paradigm dominated modeling of movement in biology. It introduces the continuous time random walk as an exemplar model and discusses how variations were developed to better capture biological data. The document notes that some data shows anomalous diffusion patterns that motivate considering non-Brownian models involving Lévy flights with heavy-tailed jump lengths or fractional time processes with heavy-tailed waiting times. It discusses how these can be combined and gives examples fitting these models to data. It concludes by discussing the Lévy foraging hypothesis proposing these models may confer evolutionary advantages.
Five Dimensional Bianchi Type-V Space-Time in f (R,T) Theory of GravitywIJERA Editor
We study the spatially homogeneous anisotropic Bianchi type-V universe in f(R,T) theory of gravity, where R is
the Ricci scalar and T is the trace of the energy-momentum tensor. We assume the variation law of mean
Hubble parameter and constant deceleration parameter to find two different five dimensional exact solutions of
the modified field equations. The first solution yields a singular model for n 0 while the second gives a nonsingular
model for n 0. The physical quantities are discussed for both models in future evolution of the
universe.
This document summarizes a research paper on chaotic group actions. It begins with an abstract discussing how group actions can be considered "chaotic" if they exhibit sensitive dependence on initial conditions and have a dense set of points with finite orbits. The paper then provides definitions and examples of chaotic group actions, discussing how the concept generalizes the definition of chaotic maps. It explores which groups can admit chaotic dynamics on topological spaces and which spaces admit chaotic group actions. Specifically, it shows a group has a chaotic action if and only if it is residually finite. It also constructs examples of chaotic actions and proves several theorems about when group actions are chaotic.
Workshop presented by
Helena Trigo Teixeira,José Orlando,Escola Maria Amália Vaz de Carvalho from Eskola Secundaria Maria Amalia Vaz de Carvalho, Lisbon,Portugal
First discovery of_a_magnetic_field_in_a_main_sequence_delta_scuti_star_the_k...Sérgio Sacani
Coralie Neiner do Laboratory for Space Studies and Astrophysics Instrumentation, LESIA (CNRS/Observatoire de Paris/UPMC/Université Paris Diderot) e Patricia Lampens (Royual OIbservatory of Belgium), descobriram a primeira estrela magnética do tipo delta Scuti, através de observações espectropolarimétricas, realizadas com o telescópio CFHT. As estrelas do tipo delta Scuti, são estrelas pulsantes, sendo que algumas delas mostram assinaturas atribuídas para um segundo tipo de pulsação. A descoberta mostra que isso é na verdade a assinatura de um campo magnético. Essa descoberta tem importantes implicações para o entendimento do interior das estrelas.
Dois tipos de estrelas pulsantes existem entre as estrelas com massa entre 1.5 e 2.5 vezes a massa do Sol: as estrelas do tipo delta Scuti e as estrelas do tipo gamma Dor. A teoria nos diz que as estrelas com temperatura entre 6900 e 7400 graus Kelvin podem ter ambos os tipos de pulsação. Essas são então chamadas de estrelas híbridas. Contudo, o satélite Kepler da NASA tem detectado um grande número de estrelas híbridas com temperaturas maiores ou menores do que esse limite pensado anteriormente. A existência dessas estrelas híbridas com temperaturas maiores é algo muito controverso, já que desafia o nosso entendimento sobre as estrelas pulsantes do tipo delta Scuti e gamma Dor.
Small scatter and_nearly_isothermal_mass_profiles_to_four_half_light_radii_fr...Sérgio Sacani
This document summarizes the results of a study analyzing the total mass density profiles of 14 early-type galaxies using two-dimensional stellar kinematic data out to large radii of 2-6 half-light radii. The study finds that the total density profiles are well described by a nearly-isothermal power law with density proportional to radius from 0.1 to at least 4 half-light radii. The average logarithmic slope is -2.19 with a small scatter of only 0.11. This places tight constraints on galaxy formation models and illustrates the power of extended two-dimensional stellar kinematic observations.
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Sp...Hontas Farmer
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Space with Experimental Predictions - http://meetings.aps.org/Meeting/APR19/Session/Z13.5
Invited Seminar presented at the VIA Forum Astroparticle Physics Forum COSMOVIA
21 March 2020
http://viavca.in2p3.fr/2010c_o_s_m_o_v_i_a__forum_sd24fsdf4zerfzef4ze5f4dsq34sdteerui45788789745rt7yr68t4y54865h45g4hfg56h45df4h86d48h48t7uertujirjtiorjhuiofgrdsqgxcvfghfg5h40yhuyir/viewtopic.php?f=73&t=3705&sid=c56cbf76f87536fc4c3ff216d9edaba2
Author: O.M. Lecian
Speaker: O.M. Lecian
Abstract: The LHAASO experiment is aimed at detecting highly-energetic particles of cosmological origin within a large
range of energies.
The sensitivity of the experimental apparatus can within the frameworks of statistical fluctuations of the
background.
Acceleration and lower-energy particles can be analyzed.
The anisotropy mass composition of cosmic rays can analytically described.
The LHAASO Experiment is also suited for detecting particles of cosmological origin originated from the breach
(and/or other kinds of modifications) of particle theories paradigms comprehending other symmetry groups.
Some physical implications of anisotropies can be looked for.
The study of anisotropy distribution for particles of cosmological origin as well as the anisotropies of their velocities
both in the case of a flat Minkowskian background as well as in the case of curved space-time can be investigated,
as far as the theoretical description of the cross-section is concerned, as well as for the theoretical expressions of
such quantities to be analyzed.
The case of a geometrical phase of particles can be schematized by means of a geometrical factor.
Particular solutions are found under suitable approximations.
A comparison with the study of ellipsoidal galaxies is achieved.
The case of particles with anisotropies in velocities falling off faster than dark matter (DM) is compared.
The study of possible anisotropies in the spatial distribution of cosmological particles can therefore be described
also deriving form the interaction of cosmic particles with the gravitational field, arising at quantum distances, at
the semiclassical level and at the classical scales, within the framework of the proper description of particles
anisotropies properties.
This document discusses stochastic models of movement and search behavior. It begins by describing how Karl Pearson originally connected animal motion to random walks over a century ago. It then discusses how the Brownian motion and random walk paradigm dominated modeling of movement in biology. It introduces the continuous time random walk as an exemplar model and discusses how variations were developed to better capture biological data. The document notes that some data shows anomalous diffusion patterns that motivate considering non-Brownian models involving Lévy flights with heavy-tailed jump lengths or fractional time processes with heavy-tailed waiting times. It discusses how these can be combined and gives examples fitting these models to data. It concludes by discussing the Lévy foraging hypothesis proposing these models may confer evolutionary advantages.
Five Dimensional Bianchi Type-V Space-Time in f (R,T) Theory of GravitywIJERA Editor
We study the spatially homogeneous anisotropic Bianchi type-V universe in f(R,T) theory of gravity, where R is
the Ricci scalar and T is the trace of the energy-momentum tensor. We assume the variation law of mean
Hubble parameter and constant deceleration parameter to find two different five dimensional exact solutions of
the modified field equations. The first solution yields a singular model for n 0 while the second gives a nonsingular
model for n 0. The physical quantities are discussed for both models in future evolution of the
universe.
This document summarizes a research paper on chaotic group actions. It begins with an abstract discussing how group actions can be considered "chaotic" if they exhibit sensitive dependence on initial conditions and have a dense set of points with finite orbits. The paper then provides definitions and examples of chaotic group actions, discussing how the concept generalizes the definition of chaotic maps. It explores which groups can admit chaotic dynamics on topological spaces and which spaces admit chaotic group actions. Specifically, it shows a group has a chaotic action if and only if it is residually finite. It also constructs examples of chaotic actions and proves several theorems about when group actions are chaotic.
Workshop presented by
Helena Trigo Teixeira,José Orlando,Escola Maria Amália Vaz de Carvalho from Eskola Secundaria Maria Amalia Vaz de Carvalho, Lisbon,Portugal
First discovery of_a_magnetic_field_in_a_main_sequence_delta_scuti_star_the_k...Sérgio Sacani
Coralie Neiner do Laboratory for Space Studies and Astrophysics Instrumentation, LESIA (CNRS/Observatoire de Paris/UPMC/Université Paris Diderot) e Patricia Lampens (Royual OIbservatory of Belgium), descobriram a primeira estrela magnética do tipo delta Scuti, através de observações espectropolarimétricas, realizadas com o telescópio CFHT. As estrelas do tipo delta Scuti, são estrelas pulsantes, sendo que algumas delas mostram assinaturas atribuídas para um segundo tipo de pulsação. A descoberta mostra que isso é na verdade a assinatura de um campo magnético. Essa descoberta tem importantes implicações para o entendimento do interior das estrelas.
Dois tipos de estrelas pulsantes existem entre as estrelas com massa entre 1.5 e 2.5 vezes a massa do Sol: as estrelas do tipo delta Scuti e as estrelas do tipo gamma Dor. A teoria nos diz que as estrelas com temperatura entre 6900 e 7400 graus Kelvin podem ter ambos os tipos de pulsação. Essas são então chamadas de estrelas híbridas. Contudo, o satélite Kepler da NASA tem detectado um grande número de estrelas híbridas com temperaturas maiores ou menores do que esse limite pensado anteriormente. A existência dessas estrelas híbridas com temperaturas maiores é algo muito controverso, já que desafia o nosso entendimento sobre as estrelas pulsantes do tipo delta Scuti e gamma Dor.
Small scatter and_nearly_isothermal_mass_profiles_to_four_half_light_radii_fr...Sérgio Sacani
This document summarizes the results of a study analyzing the total mass density profiles of 14 early-type galaxies using two-dimensional stellar kinematic data out to large radii of 2-6 half-light radii. The study finds that the total density profiles are well described by a nearly-isothermal power law with density proportional to radius from 0.1 to at least 4 half-light radii. The average logarithmic slope is -2.19 with a small scatter of only 0.11. This places tight constraints on galaxy formation models and illustrates the power of extended two-dimensional stellar kinematic observations.
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Sp...Hontas Farmer
Formal Unification of Gravity and Particle Physics in Lagrangian Euclidean Space with Experimental Predictions - http://meetings.aps.org/Meeting/APR19/Session/Z13.5
Invited Seminar presented at the VIA Forum Astroparticle Physics Forum COSMOVIA
21 March 2020
http://viavca.in2p3.fr/2010c_o_s_m_o_v_i_a__forum_sd24fsdf4zerfzef4ze5f4dsq34sdteerui45788789745rt7yr68t4y54865h45g4hfg56h45df4h86d48h48t7uertujirjtiorjhuiofgrdsqgxcvfghfg5h40yhuyir/viewtopic.php?f=73&t=3705&sid=c56cbf76f87536fc4c3ff216d9edaba2
Author: O.M. Lecian
Speaker: O.M. Lecian
Abstract: The LHAASO experiment is aimed at detecting highly-energetic particles of cosmological origin within a large
range of energies.
The sensitivity of the experimental apparatus can within the frameworks of statistical fluctuations of the
background.
Acceleration and lower-energy particles can be analyzed.
The anisotropy mass composition of cosmic rays can analytically described.
The LHAASO Experiment is also suited for detecting particles of cosmological origin originated from the breach
(and/or other kinds of modifications) of particle theories paradigms comprehending other symmetry groups.
Some physical implications of anisotropies can be looked for.
The study of anisotropy distribution for particles of cosmological origin as well as the anisotropies of their velocities
both in the case of a flat Minkowskian background as well as in the case of curved space-time can be investigated,
as far as the theoretical description of the cross-section is concerned, as well as for the theoretical expressions of
such quantities to be analyzed.
The case of a geometrical phase of particles can be schematized by means of a geometrical factor.
Particular solutions are found under suitable approximations.
A comparison with the study of ellipsoidal galaxies is achieved.
The case of particles with anisotropies in velocities falling off faster than dark matter (DM) is compared.
The study of possible anisotropies in the spatial distribution of cosmological particles can therefore be described
also deriving form the interaction of cosmic particles with the gravitational field, arising at quantum distances, at
the semiclassical level and at the classical scales, within the framework of the proper description of particles
anisotropies properties.
Venice 2012 Two topics in the history of complexity: Bunched Black Swans and ...Nick Watkins
This document discusses two topics in the history of complexity: "bunched black swans" and the "minds eye". It summarizes the work of Benoit Mandelbrot in developing self-similar models, such as fractional Brownian motion and fractional hyperbolic motion, to model long-range dependence and heavy-tailed distributions seen in natural phenomena like cotton prices and the Nile River. It discusses how Mandelbrot later developed multifractal models to address limitations with purely self-similar models, such as their inability to capture "volatility clustering". The document also discusses cognitive styles and individual differences in visual versus verbal thinking and their relevance to the history of mathematics, science, and future developments in information and communication
Cambridge 2014 Complexity, tails and trendsNick Watkins
This document discusses two types of complexity that can affect trend detection in time series data: long range dependence and heavy tails.
Long range dependence, if present in a system, implies the presence of low frequency "slow" fluctuations that can complicate trend detection. Heavy tails in a probability distribution are a source of "wild" fluctuations due to more frequent extreme events.
The document reviews several examples of long range dependence and heavy tails observed in real-world datasets like financial data and space weather data. Statistical models like linear fractional stable motion (LFSM) and autoregressive fractionally integrated moving average (ARFIMA) processes are discussed for modeling systems with both properties. Better statistical inference methods are also needed to distinguish true
Cyprus 2011 complexity extreme bursts and volatility bunching in solar terres...Nick Watkins
Nick Watkins presented research on complexity and volatility clustering in solar-terrestrial physics. The research began 15 years ago studying multiscale complexity in Earth's magnetosphere. Watkins' focus has been on self-similar and multifractal time series models. He highlighted work on temporal scaling of bursts above thresholds and volatility clustering, a multifractal feature also seen in some financial time series. Watkins advocated using the Kesten stochastic process as a "null" model for framing hypotheses about volatility bunching. The relevance extends beyond solar-terrestrial physics to complex systems modeling.
2014 Wroclaw Mandelbrot's Other Model of 1/f NoiseNick Watkins
1) Mandelbrot gave an invited talk discussing several of his models for 1/f noise processes, including fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) which introduced the concept of long-range dependence.
2) However, he later emphasized that reducing 1/f noise solely to the property of self-similarity was an oversimplification. His other models, like fractional hyperbolic motion and multiplicative multifractal cascades, exhibited different visual properties.
3) One lesser known class was his "dustborne" models from 1965-1967, which involved renewal processes with heavy-tailed waiting times. These exhibited properties like ergodicity breaking and time-dependent power spectra
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noiseNick Watkins
This document summarizes a talk given by Nick Watkins on distinguishing between the Hurst effect, 1/f spectra, and long range dependence (LRD) often conflated in time series analysis. It discusses how the Hurst effect refers to anomalous growth of the range over time, 1/f refers to a singularity in the Fourier spectrum, and LRD describes long memory seen in stationary processes with 1/f spectra. Mandelbrot originally showed the Hurst effect could be explained by fractional Gaussian noise exhibiting LRD. The document cautions against assuming stationarity and reviews alternative non-stationary models proposed by Mandelbrot to explain 1/f spectra without LRD.
Hyderabad 2010 Distributions of extreme bursts above thresholds in a fraction...Nick Watkins
The document summarizes simulations of burst sizes and durations in fractional Lévy motion processes with varying parameters α and H. The simulations show:
- For fractional Brownian motion (α=2), the predicted scaling relationships for burst length (β=2-H) and size (γ=2/(1+H)) agree reasonably well with simulation results.
- For heavier tailed processes with α=1.6, the predictions also show similar levels of agreement, though not perfect.
- By α=1.2, the very heavy tails cause a clear problem with the predictions not fitting the simulation results well.
The simulations thus provide a test of theoretical scaling relationships for burst statistics in fractional Lé
IMA 2013 Bunched Black Swans at the British Antarctic SurveyNick Watkins
- THE WHAT: The speaker will discuss "bunched black swans", which are long range correlated, wild events. Specifically, the speaker will discuss "bursts" where the time series exceeds a threshold.
- THE WHY: Scientists studying climate, space weather, and finance care about bursts. The speaker will provide examples of why this is relevant to the British Antarctic Survey (BAS) and beyond.
- THE HOW: The speaker will discuss how they have modeled one class of "routinely extreme" burst problems using an approach building on Mandelbrot's work, involving fractional stable models.
2015 Dresden Mandelbrot's other route to1 over fNick Watkins
Mandelbrot made important contributions to understanding the phenomenon of 1/f noise through several models between 1963-1972:
1) He proposed heavy-tailed distributions to explain wild fluctuations in cotton prices (1963) and the Hurst effect in Nile river levels.
2) This led to his fractional Gaussian noise (fGn) and fractional Brownian motion models which produce a 1/f spectral shape through long-range dependence or self-similarity (1965-1968). These stationary and non-stationary models were controversial due to their singular power spectra.
3) Less well known are his "conditionally stationary" models from 1963-1967 which achieved 1/f noise through long-tailed waiting times
2014 OU Mandelbrot's eyes and 1/f noiseNick Watkins
The document discusses Benoit Mandelbrot's work on modeling wild fluctuations and long-range dependence that he observed in various real-world datasets. It describes how his initial attempts to explain these phenomena with heavy-tailed distributions and fractional Brownian motion led him to abstract the property of self-similarity in the spectral domain. The document also notes that Mandelbrot warned against over-interpreting any single property and advocated using our eyes alongside formulas to fully understand complexity.
This document provides an overview of teaching kinetic theory and the behavior of gases. It outlines learning outcomes, common student misconceptions, teaching challenges, and evidence that supports the atomic model of matter. Examples are given to illustrate gas properties, laws, and the historical development of the kinetic theory of gases. Suggestions for demonstrations and simulations to teach these concepts in an engaging, hands-on manner are also provided.
AGU 2012 Bayesian analysis of non Gaussian LRD processesNick Watkins
Contributed talk at American Geophysical Union Fall Meetinfg, San Francisco, 2012.
Part of work now submitted to Bayesian Analysis, 2014, see eprint: http://arxiv.org/abs/1403.2940
This document discusses Huygens' principle as a universal model of propagation and transport. It begins by introducing Huygens' principle and noting some confusion in its interpretation in different fields like optics. The document then aims to clarify the principle and generalize it in a way that applies to virtually all propagation phenomena described by explicit linear differential or difference equations, including light, heat, matter waves, and Maxwell's equations. It discusses representing Huygens' principle using Green's functions and the Chapman-Kolmogorov equation. Finally, it outlines how the principle applies in various areas like mechanics, electromagnetism, discrete models, fractional Fourier transforms, and linguistics.
This document discusses fractal geometry and its applications in materials science. It begins by providing background on fractals and how they were discovered to describe natural patterns. Fractals have fractional dimensions and self-similar patterns across different scales. Non-linear dynamics and chaos theory are then introduced to study irregular patterns in nature. Specific fractal objects like the Cantor set and Koch curve are described. The document outlines how fractal analysis can be used to characterize microstructures, surfaces, cracks and particles in materials using techniques like box counting to determine fractal dimension. Finally, the role of image processing in materials science images for quantitative microstructure analysis is briefly discussed.
The korteweg dvries equation a survey of resultsHaziqah Hussin
This document provides a survey of results for the Korteweg-de Vries (KdV) equation, an important nonlinear partial differential equation that describes the evolution of small-amplitude dispersive waves. Some key results discussed include:
- The concept of solitons was introduced to describe solitary wave solutions of the KdV equation. Inverse scattering theory was also developed to exactly solve the initial value problem.
- The KdV equation has applications in fluid dynamics, plasma physics, anharmonic lattices, and other areas. It has a variety of interesting properties and solutions that have been studied extensively.
- Exact soliton solutions were derived and shown to correspond one-to-one with eigenvalues of the scattering
The document discusses the fluid-gravity correspondence, which provides a map between dynamical black hole solutions in asymptotically anti-de Sitter spacetimes and fluid flows in a strongly interacting boundary field theory. It reviews how the gravitational duals to fluid dynamics are constructed via a boundary derivative expansion. This allows extracting the hydrodynamic transport coefficients from the gravitational analysis and provides a way to algorithmically construct black hole geometries from fluid equations of motion.
This document provides an overview of cosmic microwave background radiation and its theoretical framework. It discusses how CMB was discovered and its importance as evidence for the Big Bang model. The document outlines the theoretical concepts behind CMB, including the standard cosmological model, Friedman equations, perturbations in spacetime metric, and using the Boltzmann equation to model changes in particle densities as the universe expands. It also provides background on key observational findings about CMB from missions like COBE, WMAP and Planck.
This dissertation examines the stability and nonlinear evolution of an idealized hurricane model. The author uses the quasi-geostrophic shallow water equations to model the hurricane as a simple axisymmetric annular vortex with a predefined potential vorticity distribution. Both single-layered and two-layered models are considered. For the single-layer case, linear stability analysis reveals barotropic and baroclinic instabilities that depend on parameters like the potential vorticity within the core and the Rossby deformation length. Nonlinear simulations then show how the annular structure breaks down in different ways depending on these parameters. For the two-layer case, linear stability is found to be identical to the single-layer case, and phase diagrams
This document discusses the connection between deterministic evolution over time and differential equations from philosophical, historical, and mathematical perspectives.
From a philosophical viewpoint, the author argues that deterministic motion can be associated with semigroups and is characterized by differential equations with time derivatives. Historically, the exponential function and semigroup theory emerged from efforts to solve linear differential equations. Mathematically, the document outlines the basic theory of uniformly, strongly, and σ(X,F)-continuous semigroups of linear operators and their generators.
- Handbook of Mathematical Fluid Dynamics.pdfAhmedElEuchi
The Boltzmann equation describes rarefied gas flows where the mean free path of gas molecules is not negligible compared to characteristic lengths. It provides an alternative to the Navier-Stokes equations in these regimes. The Boltzmann equation accounts for the molecular nature of gases and models intermolecular collisions. It has applications in aerospace engineering for high-altitude flight and in environmental problems involving particulate matter.
A colleague of yours has given you mathematical expressions for the f.pdfarjuntiwari586
The document analyzes and compares different mathematical expressions for calculating electromagnetic fields given time-dependent charge and current distributions. It summarizes a derivation showing that an expression given by Panofsky and Phillips (involving retarded potentials) can be transformed into a form that better highlights the transverse nature of radiation fields. It also clarifies why a term in the electric field expression vanishes in the static case.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
Venice 2012 Two topics in the history of complexity: Bunched Black Swans and ...Nick Watkins
This document discusses two topics in the history of complexity: "bunched black swans" and the "minds eye". It summarizes the work of Benoit Mandelbrot in developing self-similar models, such as fractional Brownian motion and fractional hyperbolic motion, to model long-range dependence and heavy-tailed distributions seen in natural phenomena like cotton prices and the Nile River. It discusses how Mandelbrot later developed multifractal models to address limitations with purely self-similar models, such as their inability to capture "volatility clustering". The document also discusses cognitive styles and individual differences in visual versus verbal thinking and their relevance to the history of mathematics, science, and future developments in information and communication
Cambridge 2014 Complexity, tails and trendsNick Watkins
This document discusses two types of complexity that can affect trend detection in time series data: long range dependence and heavy tails.
Long range dependence, if present in a system, implies the presence of low frequency "slow" fluctuations that can complicate trend detection. Heavy tails in a probability distribution are a source of "wild" fluctuations due to more frequent extreme events.
The document reviews several examples of long range dependence and heavy tails observed in real-world datasets like financial data and space weather data. Statistical models like linear fractional stable motion (LFSM) and autoregressive fractionally integrated moving average (ARFIMA) processes are discussed for modeling systems with both properties. Better statistical inference methods are also needed to distinguish true
Cyprus 2011 complexity extreme bursts and volatility bunching in solar terres...Nick Watkins
Nick Watkins presented research on complexity and volatility clustering in solar-terrestrial physics. The research began 15 years ago studying multiscale complexity in Earth's magnetosphere. Watkins' focus has been on self-similar and multifractal time series models. He highlighted work on temporal scaling of bursts above thresholds and volatility clustering, a multifractal feature also seen in some financial time series. Watkins advocated using the Kesten stochastic process as a "null" model for framing hypotheses about volatility bunching. The relevance extends beyond solar-terrestrial physics to complex systems modeling.
2014 Wroclaw Mandelbrot's Other Model of 1/f NoiseNick Watkins
1) Mandelbrot gave an invited talk discussing several of his models for 1/f noise processes, including fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) which introduced the concept of long-range dependence.
2) However, he later emphasized that reducing 1/f noise solely to the property of self-similarity was an oversimplification. His other models, like fractional hyperbolic motion and multiplicative multifractal cascades, exhibited different visual properties.
3) One lesser known class was his "dustborne" models from 1965-1967, which involved renewal processes with heavy-tailed waiting times. These exhibited properties like ergodicity breaking and time-dependent power spectra
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noiseNick Watkins
This document summarizes a talk given by Nick Watkins on distinguishing between the Hurst effect, 1/f spectra, and long range dependence (LRD) often conflated in time series analysis. It discusses how the Hurst effect refers to anomalous growth of the range over time, 1/f refers to a singularity in the Fourier spectrum, and LRD describes long memory seen in stationary processes with 1/f spectra. Mandelbrot originally showed the Hurst effect could be explained by fractional Gaussian noise exhibiting LRD. The document cautions against assuming stationarity and reviews alternative non-stationary models proposed by Mandelbrot to explain 1/f spectra without LRD.
Hyderabad 2010 Distributions of extreme bursts above thresholds in a fraction...Nick Watkins
The document summarizes simulations of burst sizes and durations in fractional Lévy motion processes with varying parameters α and H. The simulations show:
- For fractional Brownian motion (α=2), the predicted scaling relationships for burst length (β=2-H) and size (γ=2/(1+H)) agree reasonably well with simulation results.
- For heavier tailed processes with α=1.6, the predictions also show similar levels of agreement, though not perfect.
- By α=1.2, the very heavy tails cause a clear problem with the predictions not fitting the simulation results well.
The simulations thus provide a test of theoretical scaling relationships for burst statistics in fractional Lé
IMA 2013 Bunched Black Swans at the British Antarctic SurveyNick Watkins
- THE WHAT: The speaker will discuss "bunched black swans", which are long range correlated, wild events. Specifically, the speaker will discuss "bursts" where the time series exceeds a threshold.
- THE WHY: Scientists studying climate, space weather, and finance care about bursts. The speaker will provide examples of why this is relevant to the British Antarctic Survey (BAS) and beyond.
- THE HOW: The speaker will discuss how they have modeled one class of "routinely extreme" burst problems using an approach building on Mandelbrot's work, involving fractional stable models.
2015 Dresden Mandelbrot's other route to1 over fNick Watkins
Mandelbrot made important contributions to understanding the phenomenon of 1/f noise through several models between 1963-1972:
1) He proposed heavy-tailed distributions to explain wild fluctuations in cotton prices (1963) and the Hurst effect in Nile river levels.
2) This led to his fractional Gaussian noise (fGn) and fractional Brownian motion models which produce a 1/f spectral shape through long-range dependence or self-similarity (1965-1968). These stationary and non-stationary models were controversial due to their singular power spectra.
3) Less well known are his "conditionally stationary" models from 1963-1967 which achieved 1/f noise through long-tailed waiting times
2014 OU Mandelbrot's eyes and 1/f noiseNick Watkins
The document discusses Benoit Mandelbrot's work on modeling wild fluctuations and long-range dependence that he observed in various real-world datasets. It describes how his initial attempts to explain these phenomena with heavy-tailed distributions and fractional Brownian motion led him to abstract the property of self-similarity in the spectral domain. The document also notes that Mandelbrot warned against over-interpreting any single property and advocated using our eyes alongside formulas to fully understand complexity.
This document provides an overview of teaching kinetic theory and the behavior of gases. It outlines learning outcomes, common student misconceptions, teaching challenges, and evidence that supports the atomic model of matter. Examples are given to illustrate gas properties, laws, and the historical development of the kinetic theory of gases. Suggestions for demonstrations and simulations to teach these concepts in an engaging, hands-on manner are also provided.
AGU 2012 Bayesian analysis of non Gaussian LRD processesNick Watkins
Contributed talk at American Geophysical Union Fall Meetinfg, San Francisco, 2012.
Part of work now submitted to Bayesian Analysis, 2014, see eprint: http://arxiv.org/abs/1403.2940
This document discusses Huygens' principle as a universal model of propagation and transport. It begins by introducing Huygens' principle and noting some confusion in its interpretation in different fields like optics. The document then aims to clarify the principle and generalize it in a way that applies to virtually all propagation phenomena described by explicit linear differential or difference equations, including light, heat, matter waves, and Maxwell's equations. It discusses representing Huygens' principle using Green's functions and the Chapman-Kolmogorov equation. Finally, it outlines how the principle applies in various areas like mechanics, electromagnetism, discrete models, fractional Fourier transforms, and linguistics.
This document discusses fractal geometry and its applications in materials science. It begins by providing background on fractals and how they were discovered to describe natural patterns. Fractals have fractional dimensions and self-similar patterns across different scales. Non-linear dynamics and chaos theory are then introduced to study irregular patterns in nature. Specific fractal objects like the Cantor set and Koch curve are described. The document outlines how fractal analysis can be used to characterize microstructures, surfaces, cracks and particles in materials using techniques like box counting to determine fractal dimension. Finally, the role of image processing in materials science images for quantitative microstructure analysis is briefly discussed.
The korteweg dvries equation a survey of resultsHaziqah Hussin
This document provides a survey of results for the Korteweg-de Vries (KdV) equation, an important nonlinear partial differential equation that describes the evolution of small-amplitude dispersive waves. Some key results discussed include:
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LSE 2012 Extremes, Bursts and Mandelbrot's Eyes ... and Five Ways to Mis-estimate Risk
1. Extremes, bursts & Mandelbrot’s
eyes ... and five ways to mis-
estimate risk
Nick Watkins nww@bas.ac.uk
NERC British Antarctic Survey, Cambridge, UK
Visiting Fellow, Centre for the Analysis of Time Series, LSE
Associate Fellow, Department of Physics, University of Warwick
2. Thank many people: Tim Graves (Cambridge), Dan
Credgington (Now UCL) , Sam Rosenberg (Now Barclays
Capital), Christian Franzke (BAS), Bogdan Hnat (Warwick),
Sandra Chapman (Warwick), Nicola Longden (BAS),
Mervyn Freeman (BAS), Bobby Gramacy (Chicago), Dave
Stainforth and Lenny Smith (LSE), Jean Boulton, Ed
Bullmore, Bill Faw, Brian Levine, ...
Watkins et al, Space Sci. Rev., 121, 271-284 (2005)
Watkins et al, Phys. Rev. E 79, 041124 (2009a)
Watkins et al, Phys. Rev. Lett. , 103, 039501 (2009b)
Franzke et al, Phil Trans Roy Soc, (2012)
Watkins et al, in press AGU Hyderabad Chapman Conference Proceedings (2012)
4. Two themes interwoven
• The reasons why Mandelbrot was led to
study “non-classical” models that had features
like extremely fat tails (infinite variance) in
fluctuation amplitude, and extremely long
range memory (1/f power spectra) in time.
• Why, if such models in fact apply, but we don’t
use them, we would tend to underestimate
“risk”-used simply to mean P(fluctuation)
• Disclaimers: Not a professional historian or
philosopher of science, nor an economist. Led
to these questions from physical science.
5. One more acknowledgement
“They misunderestimated me ...”
... One of his "most memorable additions to the language, and an
incidentally expressive one: it may be that we rather needed a word for
'to underestimate by mistake’”. – Philip Hensher
6. 5 ways to misestimate risk
First 3 ( all "misunderestimation“, as they typically under-
estimate fluctuations), would be to use:
• short tailed pdfs if they should have been longer.
• short memory if you should instead have used lrd
• additive models if system is in fact multiplicative
Will just briefly note also the problem of :
• in multivariate models, using iid variables if instead should
have used coupled ones
And for balance, a fifth case, of "misoverestimation“:
e.g. generating heavy tails (~ 4 days) from spurious
measurements [Edwards, Philips, Watkins et al, Nature,
2007] although heavy tails (up to ~ 12 hours) may still be
buried in the data ... debate continues.
7. Why did an Antarctic scientist get
interested in complexity ? via
coupled solar
wind-ionosphereSolar wind
Magnetosphere
Ionosphere
Ultraviolet Imager –
NASA Polar
Instrumentation
Solar wind
Problem
8. and “Extremes”
• Now a “hot topic” across many areas of
science and policy.
• Term used both loosely (“black swans”) and
precisely (statistical Extreme Value Theory
(EVT), most mature for iid case).
• Today using it loosely, as “events which are
“bigger” than expected ...” which immediately
poses question of whether “size” here means
amplitude, duration, ...
9. “Extremes” in space weather
• Example: Riley,
Space Weather [2012]
Drew inference from
distribution of flare intensities,
CME speeds etc that large events
more common than was thought:
“suggest that the likelihood
of another Carrington
event occurring within the next
decade is ~ 12%”
10. Heavy tails & “Grey Swans”
Light tailed
Gaussian
Heavy
tailed
power law
Plot number of
events (#) versus
magnitude (x). In
red “normal” case,
a magnitude 25
event essentially
never happens.
In the blue heavy
tailed case, it
becomes a “1 in
2000” event.
“Extreme events
… [are ] the norm”
-John Prescott
This matters because it applies in many
natural and man-made situations
e.g. Gutenberg-Richter law,
insurer’s “80-20” rule of thumb
#
11. Burst idea
• Very general idea – inspired by energy release
measures used in “sandpile” models. My
interest grew from these and our application
of the burst idea to solar-terrestrial coupling
data (e.g. Freeman et al, GRL, 2000).
1
( ( ) )
i
i
t
I t
A Y t L dt
+
′ ′= −∫
12. Bursts in climate
• Rather than, e.g. an unexpectedly high
temperature, “extreme” might be a long
duration.
• Runs of hot days above a fixed threshold, e.g.
summer 1976 in UK, or summer 2003 in
France.
• Direct link to weather
derivatives [e.g. book
by Jewson]
13. log s
log
P(T)
log
P(τ)
logT
log τ
Poynting flux in solar wind plasma from NASA
Wind Spacecraft at Earth-Sun L1 point
Freeman, Watkins & Riley [PRE, 2000].
log
P(s) size
length
waiting time
“Fat tailed” burst pdfs seen in
solar wind data ...Data
... and ionospheric in currents (not shown).
14. Our initial guess (1997-98): …
23 May 2012 14
Does Bak et al’s SOC paradigm apply to
magnetospheric energy storage/release cycle ?
Lui et al, GRL, 2001
15. Bak et al’s aim was to unify fractals in space
with “1/f” noise in time directly, via a physical
mechanism:
Answering Kadanoff’s
question: [spacetime] ...”fractals: where’s the
physics” ? (often traduced, was plea, not a criticism)
16. A different way ?
Experience with SOC and complexity in space physics
[summarised in Freeman & Watkins, Science, 2002; Chapman
& Watkins, Space Science Reviews], and the difficulty of
uniquely attributing complex natural phenomena led us to
“back up” one step.
Got interested in applying the known models for non-Gaussian
and non iid random walks. Partly to try and see what physics
was embodied in any particular choice, partly for “calibration”
of the measurement tools. Link to risk and extremes. Such
models go beyond the CLT. They are not always general “laws”,
but they are mapping out a range of widely observed
“tendencies”. In learning about these we have become
interested in the history of Mandelbrot’s paradigmatic models
and their relatives.
17. Approaches to extremes
• Stochastic processes
• Dynamical systems [e.g. Franzke, submitted]
• Mixture of both
• Complex models like GCMs
• ...
I am concerned today with stochastics, but
clearly models that mix these properties are
of interest, for example Rypdal and Rypdal’s
stochastic model for SOC and developments.
18. “Textbook” stochastic models
• “White” noise
• Gaussian “short-tailed” distribution of
amplitudes
• Successive values independent
ACF is short-tailed
• When integrated leads to an additive random
walk model
1 2 3), ( ), (( ),X t X tX t …
1 1( )( ), XX tt τ< + >
1
() )( N
N i i
X tY t =
= ∑
19. 3 “giant leaps” madem beyond these 1963-74 by Mandelbrot.
All “well known” and yet process is instructive-recap
1. BBM observes heavy tailed fluctuations in 1963 in cotton
prices---proposes alpha-stable model , self-similarity idea
2. BBM hears about River Nile and “Hurst effect”. Initially (see
his Selecta) believes this will be explained by heavy tails,
but when he sees that fluctuations are ~ Gaussian
applies self-similarity [Comptes Rendus1965] in the form of a
long range dependent (LRD) model, roots of fractional Brownian motion. BBM’s
classic series of papers on fBm in mathematical & hydrological literature with
Van Ness and Wallis in 1968-1969. BBM unites them in a new self-similar model,
fractional hyperbolic motion, in 1969 paper with Wallis on robustness of R/S.
Combines 1 & 2 above (heavy tails & LRD).
3. BBM becomes dissatisfied with purely self-similar models, develops multifractal
cascade, initially in context of turbulence, JFM 1974.
Later applications include finance.
20. “Noah effect”- e.g. Lévy
flights where a < 2
increases tail fatness
a=1
e.g. Hnat et al, NPG [2004]
a=2
Levy flight model
Ionospheric
Data (AE index)
BBM observes heavy tailed fluctuations in 1963 in
cotton prices---proposes alpha-stable model ,
abstracts out self-similarity idea
21. • H is the selfsimilarity parameter. Relates a
walk time series to same series dilated by a
factor c.
Selfsimilar scaling
0 0( ) ( ) ( )Y t t Y t Y t∆ − = −
0 0=( ( )) ( )H
Y c t t c Y t t∆ − ∆ −
22. Droughts & Bunching
23 May 2012 22
Mandelbrot’s climate example: Pharoah’s dream 7 years of plenty
(green boxes) and 7 years of drought (red boxes). Now shuffle ...
23. Droughts & Bunching
23 May 2012 23
Mandelbrot’s climate example: Pharoah’s dream 7 years of plenty
(green boxes) and 7 years of drought (red boxes). Now shuffle ...
Point is that frequency distribution is same (c.f. Previous slide) but
that the two series represent very different hazards. Don’t even
need to come from heavy tails, e.g. a long run of very hot days ...
24. “Joseph effect”- e.g. fractional Brownian
(fBm) walk: steepness of log(psd) of Y(t)
with log(f) increases with memory
parameter d
d=-1/2
d=0
S(f) ~ f-2(1+d)
Fractional Brownian
walk model Y(t)
Mandelbrot heard about River Nile and
“Hurst effect”. Initially (see his Selecta)
believed this would be explained by heavy
tails.
When he saw that fluctuations are ~
Gaussian applied self-similarity [Comptes
Rendus1965] in the form of a long range
dependent (lrd) model for Y(t).
Related to the ordinary
Brownian random walk
But with long ranged memory,
a fractional Brownian
motion (fBm)
Mandelbrot’s classic series of papers
on fBm in mathematical &
hydrological literature with
Van Ness and Wallis in 1968-1969.
25. What if heavy tailed and LRD ?
• Mandelbrot & Wallis [1969] looked at this, proposed a version
of fractional Brownian motion Y(t) which substitutes heavy
tailed “hyperbolic” innovations for the Gaussian ones. First
difference of this was their fractional hyperbolic noise X(t)
• In such a model you not only get “grey swan” (heavy tail)
events, but they are “bunched” by the long range dependence
...
( )X t
( )X t
26. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007
27. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007
28. Financial Bunched Black Swans
“ “We were seeing things that were 25-standard deviation
moves, several days in a row,” said David Viniar,
Goldman’s CFO ... [describing catastrophic losses on their
flagship Global Alpha hedge fund]. “What we have to look
at more closely is the phenomenon of the crowded trade
overwhelming market
fundamentals”,
he said. “It makes
you reassess how
big the extreme
moves can be””.
--- FT, August 13th, 2007
29. H = d+1/α: allows
H “subdiffusive” (i.e. < ½) while
“superdiffusive” (i.e. <2).
R/S, DFA etc, measure d but not α (e.g. Franzke et al, Phil Trans
Roy Soc, 2012) , two series can share a value of H (or d, or α )
and be otherwise quite different c.f. Rypdal and Rypdal’s
critique of Scaffetta and West.
Memory kernel:
Joseph
α-stable jump:
Noah
An H-selfsimilar, stable successor to
Mandelbrot’s model
To combine effects 1 & 2 (heavy tails & LRD) we nowadays would use
e.g Linear Fractional Stable Motion or its derivative noise.
1 1
1
( ) ( ) ( ) ( )
H H
H H R
Y t C t s s dL sα α
α α α
− −−
, , + +
= − − −∫
30. Bursts in LFSM model
• We have begun to study how bursts, defined as integrated
area above thresholds, scale for the LFSM walk Y(t). [Watkins
et al, PRE, 2009] Scaling depends both on alpha and d, via H.
• Our study benefits from earlier work of Kearney and
Majumdar [J Phys A, 2005] on area defined by curve to its
first return (for Brownian motion started epsilon above a
threshold)
• and of Carbone and Stanley , PRE & Physica A on bursts
defined in fBm using a running average (similar to that used in
detrended fluctuation analysis (DFA)).
• We’ve used the scaling properties of LFSM
walk Y(t) to predict its burst distribution.
31. First passage-based burst
• Illustrate idea first for Brownian motion. Instead of set of all
threshold crossings we can use just the time at which a
Brownian motion returns to the level L that it exceeded at L
(i.e. the first passage time) to define a burst :
• We exploit the famous scaling behaviour of a random walk.
( )
f
i
t
FP t
A t dY t′ ′= ∫
ft
1/2
( ) ~Y t t
32. Relation of burst area to FPT
• Get burst area in terms of FPT
• and vice versa
2/3
~f FPt A
3/2
~FPA t
33. Then fold in standard result for
distribution of Brownian FPTs
• Note that expectation value here is infinite !
• Above can be combined with our previous result to give a
distribution for burst sizes in Brownian walk
3/2
( ) ~f fP t t−
4/3
( )P A A−
≈
34. Repeat for LFSM
• Instead of FPT we used level crossings to define bursts here
(1 )
~ H
I It A− +
2/(1 )
( ) H
P A A− +
=
Simulations [Watkins et al, PRE, 2009] confirm this works for fBm at least.
Though agreement less tight than seen by Carbone and Stanley, evidence that
DFA-style detrending indeed helps remove the nonstationary element of the walk ?
However, predicts an exponent of about -(2/1.4) i.e. roughly -4/3 for AE index.
Observations sufficiently different (more like -6/5) to motivate further work.
AE data
35. Natural examples include ice cores (e.g. Davidsen and Griffin, PRE , 2009),
and returns of ionospheric AE index (above), see also Consolini et al, PRL,
1996. Man-made example from which name volatility is taken is finance.
Effect not seen in fractional Levy models c.f. Rypdal & Rypdal, JGR 2010
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10
4
-600
-400
-200
0
200
400
600
increments,r
First differences of AE index January-June 1979
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lag
acf
AE data: acf of returns
-100 -80 -60 -40 -20 0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
lag
acf
AE data: acf of squared returns
First differenced AE data
ACF of diff. AE
ACF of (diff. AE) squared
Having introduced 3 models in 6 years, Why did BBM
remain dissatisfied ? Partly because his eyes told him ...
Effect that multifractals capture is “volatility clustering”
36. Meneveau
& Srinivasan
P-model
Multiplicative models:
BBM becomes dissatisfied with purely self-similar models, develops
multifractal cascade, initially in context of turbulence, JFM 1974.
Later applications include finance in late 1990s by BBM, Ghashgaie et al.
37. 2
( ) H
q qζ =
( ) (( ) ( )) ~q q
q t tS X Xτ τ τ=< + − >
Open: what do we expect bursts
to do in multifractals ?
For a monofractal
Instead we see a downward curvature of the zeta function at higher orders
in a multifractal, but high variability over ensembles at these high orders c.f.
Dudok de Wit, NPG. A line drawn through zeta plot would look like a
smaller H value ?
Intuitively should act to reduce size of a burst of a given duration ? Or make
P(A) plot steeper i.e. more negative exponent ? Now looking at this with
Martin Rypdal and Ola Lovsletten.
Some early indicative results from multifractal models and turbulence in
Bartolozzi et al; in Uritsky et al, 2010, and in Watkins et al, PRE, 2009.
38. Importance can be seen from simple thought experiment,
and the fact we don’t know the answer to it ...
If we could not only see and hear these talks, but also see the
pictures drawn (or not) in individual audience’s minds by them
---how much variety would we see ?
Many steps made over millions of years to allow human
beings to be convinced that we are communicating about
same thing. Language itself, cave painting [c.f. Herzog’s film
The Cave of Forgotten Dreams], maths, blackboards,
visualisation ….
Cognitive styles:
39. How can we (will we) make further leaps towards this collective
communication while discovering and appreciating why (it is
evolutionary advantage ?) that we don’t all see the same thing ?
Cognitive styles:
40. Dichotomies :
Solar wind
Magnetosphere
Ionosphere
I am fascinated by the possible ways in which Tulving’s episodic vs
semantic and James’ verbal/visual dichotomies may illuminate :
... the tension (metaphorical, and by Mandelbrot’s own account
sometimes literal !) between the very visual thinking of BBM and the
non-visual, formal proofs of e.g. the Bourbaki, of which his uncle,
Szolem Mandelbrojt was a founder member ...
And various reports about mathematical and scientific thought in the 19th
and 20th centuries.
41. “It's very strange that in high school I never knew, I never felt that I
had this very particular gift, but in that year in that special cramming
school it became more and more pronounced, and in fact in many
ways saved me. In the fourth week again I understood nothing, but
after five or six weeks of this game it became established that I
could spontaneously just listen to the problem and do one
geometric solution, then a second and a third. Whilst the professor
was checking whether they were the same, I would provide other
problems having the same structure. It went on. I didn't learn much
algebra. I just learned how better to think in pictures because I
knew how to do it. I would see them in my mind's eye, intersecting,
moving around, or not intersecting, having this and that property,
and could describe what I saw in my eye. Having described it, I
could write two or three lines of algebra, which is much easier if you
know the results than if you don't”
---Mandelbrot, at www.webofstories.com
Mandelbrot
42. Dirac
• “Her fundamental laws do not govern the
world as it appears in our mental picture in
any very direct way, but instead they control a
substratum of which we cannot form a mental
picture without introducing irrelevancies."
--- Preface to The Principles of Quantum
Mechanics [1930]
43. Euclid
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“In Euclid, you find some drawings but it is known
that most of them were added after Euclid, in
later editions. Most of the drawings in the original
are abstract drawings. You make some reasoning
about some proportions and you draw some
segments, but they are not intended to be
geometrical segments, just representations of
some abstract notions.”
The Continuing Silence of Bourbaki—An Interview with Pierre Cartier [Bourbaki
1955-83], June 18, 1997
by Marjorie Senechal in The Mathematical Intelligencer, № 1 (1998) · pp.22–28
44. Lagrange & Russell
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Also Lagrange proudly stated, in his textbook
on mechanics, "You will not find any drawing
in my book!“
The analytical spirit was part of the French tradition
and part of the German tradition. And I suppose it
was also due to the influence of people like Russell,
who claimed that they could prove everything
formally—that so-called geometrical intuition
was not reliable in proof.
Senechal , op cit
45. 19th Century Scientists
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“The earliest results of my inquiry amazed me. I had begun by
questioning friends in the scientific world, as they were the most likely
class of men to give accurate answers concerning this faculty of
visualizing, to which novelists and poets continually allude, which has left
an abiding mark on the vocabularies of every language, and which
supplies the material out of which dreams and the well-known
hallucinations of sick people are built.”
“To my astonishment, I found that, the great majority of the men of
science to whom I first applied protested that mental imagery way
unknown to them, and they looked on me as fanciful and fantastic in
supposing that the words 'mental imagery' really expressed what I
believed everybody supposed them to mean. They had no more notion of
its true nature than a color-blind man, ... has of the nature of color.” -
Francis Galton quoted in James, op cit
46. No mind’s eye ?
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“They [...] naturally enough supposed that those who affirmed they
possessed [visual images] were romancing. To illustrate their mental
attitude it will be sufficient to quote a few lines from the letter of one of my
correspondents, who writes:
"These questions presuppose assent to some sort of a proposition
regarding the "mind's eye," and the "images" which it sees. . . . This
points to some initial fallacy. . . . It is only by a figure of speech that I can
describe my recollection of a scene as a "mental image" which I can "see"
with my "mind's eye. " . . . I do not see it . . . any more than a man sees
the thousand lines of Sophocles which under due pressure he is ready to
repeat. The memory possesses it,' etc”.”
The Principles of Psychology, Volume 2.
47. Bourbaki
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“Again Bourbaki's abstractions and disdain for
visualization were part of a global fashion,
as illustrated by the abstract tendencies in the
music and the paintings of that period.“
Senechal op cit in The Mathematical Intelligencer, № 1 (1998) · pp.22–28
http://www.ega-math.narod.ru/Bbaki/Cartier.htm
48. Memory & the prospective
brain:
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Endel Tulving distinguished in 1970s between episodic and semantic
memory. Episodic means re-experiencing the past
(“remembering”) while semantic is about
the facts we store about it (“knowing”).
More recently fMRI studies c. 2005 onwards
[Schacter group Nature; Tulving group
PNAS] are beginning to
link centres of the brain used in episodic
memory to the simulation of future events.
Tulving sees ep mem and future prognosis
as linked “mental time travel” capability,
which he sees as an element of “autonoetic”
consciousness. Has made provocative
suggestion that this is unique to H. Sapiens
(i.e. our “killer app” compared to Neanderthals).
49. William James [1890]
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“A person whose visual imagination is strong finds it
hard to understand how those who are without the
faculty can think at all. Some people undoubtedly
have no visual images at all worthy of the name,
and instead of seeing their breakfast-table, they tell
you that they remember it or know what was on it.
This knowing and remembering takes place
undoubtedly by means of verbal images, ...”
- The Principles of Psychology, Volume 2.
50. Recap Themes
• Why do space and climate physicists care about extremes ? Several
approaches to extremes including stochastic.
• What might we lose either by failing to spot scaling and correlations when
present, or alternatively by inferring them when actually absent ? [“Five
ways to misestimate risk”, NERC-KTN PURE white paper in prep, 2012]
• Idea of selfsimilar extreme “bursts” from SOC. Can we predict statistics of
bursts from scaling? [Watkins et al, PRE; 2009; Hyderabad Chapman
Conference book, in press 2012 ]
• But how often is reality actually selfsimilar ? Why did Mandelbrot come to
embrace richer, multifractal models? [c.f. Rypdal & Rypdal, 2011].
Indications of how multifractality affects a time series’ properties
including bursts [Watkins et al, PRL, 2009 ].
• Open issues, next steps, collaboration ?
• And how does the kind of animal that we are enter the process of finding
new models ? Cognitive diversity.