ergodicity

The Encyclopedia of Mathematics (2002) defines ergodic theory as the “metric theory of dynamical systems. The branch of the theory of dynamical systems that studies systems with an invariant measure and related problems.” This modern definition implicitly identifies the birth of ergodic theory with proofs of the mean ergodic theorem by von Neumann (1932) and the pointwise ergodic theorem by Birkhoff (1931). These early proofs have had significant impact in a wide range of modern subjects. For example, the notions of invariant measure and metric transitivity used in the proofs are fundamental to the measure theoretic foundation of modern probability theory (Doob 1953; Mackey 1974). Building on a seminal contribution to probability theory (Kolmogorov 1933), in the years immediately following it was recognized that the ergodic theorems generalize the strong law of large numbers. Similarly, the equality of ensemble and time averages – the essence of the mean ergodic theorem – is necessary to the concept of a strictly stationary stochastic process. Ergodic theory is the basis for the modern study of random dynamical systems, e.g., Arnold (1988). In mathematics, ergodic theory connects measure theory with the theory of transformation groups. This connection is important in motivating the generalization of harmonic analysis from the real line to locally compact groups.
2

The Swallows

If you want your computer to write a novel, why not program it to simulate something like a text adventure?

I wouldn’t call it great literature, but of all the novels that came out of NaNoGenMo 2013, I think the most influential was Chris Pressey’s four-book series of The Swallows, The Swallows of Summer, Swallows and Sorrows, and Dial S for Swallows. It immediately generated a comprehensible plot, complete with a MacGuffin, and its structure suggested a number of ways the idea could be built upon–and I’ll be discussing those projects in the future.

Do peek at the Python source code if you’re interested in how it was constructed.

https://github.com/catseye/The-Swallows

4

CSA: The Ergodicity Exhibition

Developed from their Evolo skyscraper competition entries, Ergodicity, an exhibition hosted by Canterbury School of Architecture, presented thesis work from eleven Graduate Diploma students.

With over 70 percent of the worlds growing population soon to live within major cities, the exhibition reconsiders the effect of increasing densities. Projects developed their research and design to accommodate for a variety of topics affecting our urban areas today, including: population increase, the rising demand for resources, pollution, waste management, and the digital revolution.

The projects which were shown covered a wide range of locations and programmatic responses, but as a collective all questioned ‘what role can the Skyscraper play in improving our urban areas?’

Responses included approaches such as Tiny Tokyo by Carma Masson, a mixed-use community micro scraper based in the business district of central Tokyo. Tiny Tokyo re-evaluates the approach towards designing skyscrapers, using them as a tool for reviving local heritage and culture, whilst introducing relevance for the people they are designed for, rather than designing them as a corporate tool. 

The future of our history is a concept which has been explored within Luke Hill’s project titled Dis.Assemble. This project involves a complex network composed of 6 miles of disused rail systems buried deep beneath London’s streets which provides a subterranean industrial waste facility: its sole intention to ‘Dis.Assemble’ materials produced by the metropolis above.

Unused space has also been explored within Jake Mullery’s SYMCITY thesis, describing an architectural construct that occupies the ‘dead’ space between existing skyscrapers. 

A comedic thesis by Paul Sohi told the story of one man’s life growing and living in a world of 10 billion people, where 90% of society lives in urbanised cities. The comic explores what such a world would be like.

The launch night was attended by many and with special guest Peter Wynne Rees, chief planner for the City of London, the exhibition was an opportunity to showcase the work of students at the Canterbury School of Architecture ahead of the end of year summer show which starts on the 31st of May.

-Text+photography by Taylor Grindley

Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies. […] Why is the difference between ensemble and time averages of such importance? Well, basically, because when you assume the processes to be ergodic, ensemble and time averages are identical. Let me give an example even simpler than the one Peters gives:

Assume we have a market with an asset priced at 100€. Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be 100€ – because we here envision two parallel universes (markets) where the asset price falls in one universe (market) with 50% to 50 €, and in another universe (market) it goes up with 50% to 150€, giving an average of 100€ ((150+50)/2). The time average for this asset would be 75€ – because we here envision one universe (market) where the asset price first rises by 50% to 150€, and then falls by 50% to 75€ (0.5*150).

From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen. Assuming ergodicity there would have been no difference at all.

Is ergodicity at the root of all macroeconomic opinions?

Schools of macroeconomic thought differ widely in their policy preferences to achieve social optima. A broad chiasm exists between Keynesians and neoclassical economists with respect to monetary policy and fiscal policy preferences. While the following description is a summary, it will suffice to illustrate how different views on ergodicity explain the differences in these schools of thoughts.

Keynesians and allies believe that there are economic conjectures whereby monetary intervention can generate real growth (situations where the output gap is significant and inflation is below target for example). Neoclassical economists and their monetary allies believe that the gravity of market forces is so powerful that monetary surprises cannot yield real economic benefits.

On the monetary debate, neoclassical economists & monetarists believe that economies are ergodic as market forces ensure price adjustments that maintain the economy at potential at most times and thus any gains due to a monetary surprise today will be balanced by a price change that will annihilate those nominal gain. Keynesians and allies believe that a short-term gain will forever alter the development path of an economy, hence initial conditions matter. Depending on each perspective the economy either has a long run steady state or a path that can be altered at each short-term junction. While neoclassical economics believes in the ergodicity of economic systems, Keynesians and associates believe in path dependence.

With respect to the fiscal debate, neoclassical economists believe that changes in government expenditures cannot efficiently modulate economic activity and change potential output because agents’ behaviour is altered by the expectations of a balancing fiscal change in the future. Since the government must over time keep a reasonable balance, a tax cut that leads to a deficit heralds higher future taxes and leads agents to save the tax cut (Ricardian equivalence). Keynesians on the other hand feel that short-term stimuli may create a boost in the economy’s growth path whose value exceeds the amount of the stimulus. 

Who should we believe? Both schools of thought have a point. Unlike natural systems ergodicity does not apply always and everywhere with the same power. The challenge of wise economic management lies in the ability to recognize with a certain degree of certainty when a change in expected policy can yield positive results from those instances where a change in policy simply changes the timeframe of economic consequences.

The anti-black swan: oversignifying unlikely events and large deviations is as dangerous as undersignifying?

http://www.youtube.com/watch?v=f1vXAHGIpfc Time for a Change: Introducing irreversible time in economics Ole Peters

An exploration of the remarkable consequences of using Boltzmann’s 1870s probability theory and cutting-edge 20th Century mathematics in economic settings. An understanding of risk, market stability and economic inequality emerges.

The lecture presents two problems from economics: the leverage problem “by how much should an investment be leveraged”, and the St Petersburg paradox. Neither can be solved with the concepts of randomness prevalent in economics today. However, owing to 20th-century developments in mathematics these problems have complete formal solutions that agree with our intuition. The theme of risk will feature prominently, presented as a consequence of irreversible time.

Our conceptual understanding of randomness underwent a silent revolution in the late 19th century. Prior to this, formal treatments of randomness consisted of counting favourable instances in a suitable set of possibilities. But the development of statistical mechanics, beginning in the 1850s, forced a refinement of our concepts. Crucially, it was recognised that whether possibilities exist is often irrelevant – only what really materialises matters. This finds expression in a different role of time: different states of the universe can really be sampled over time, and not just as a set of hypothetical possibilities. We are then faced with the ergodicity problem: is an average taken over time in a single system identical to an average over a suitable set of hypothetical possibilities? For systems in equilibrium the answer is generally yes, for non-equilibrium systems no. Economic systems are usually not well described as equilibrium systems, and the novel techniques are appropriate. However, having used probabilistic descriptions since the 1650s economics retains its original concepts of randomness to the present day.

The solution of the leverage problem is well known to professional gamblers, under the name of the Kelly criterion, famously used by Ed Thorp to solve blackjack. The solution can be phrased in many different ways, in gambling typically in the language of information theory. Peters pointed out that this is an application of the ergodicity problem and has to do with our notion of time. This conceptual insight changes the appearance of Kelly’s work, Thorp’s work and that of many others. Their work - fiercely rejected by leading economists in the 1960s and 1970s - is not an oddity of a specific case of an unsolvable problem solved. Instead, it is a reflection of a deeply meaningful conceptual shift that allows the solution of a host of other problems.

The transcript and downloadable versions of the lecture are available from the Gresham College website:

valiantprincess asked:

punkcop + hurt/comfort but in this horror vein

“Do you feel it?” Beth asks, her heartbeat, calm, steady, but loud in her chest, in her ears, drowning out all sound, reminding her with every two-toned pulse that she’s
a-live
         a-live
                   a-live.

But Sarah takes her hand, warmth and running blood and lifelifelife thrumming beneath the skin, beneath the surface, just out of reach, out of sight, but still there, still  b  u  r  n  i  n  g  within her.

And she (Sarah) says,
                                                                                       “Of course I do.”

Horror Minifics!

academia.edu
Self Assembling Hyper-Stories and In-Between Places: The Shamanic Intent Behind the Hypertextual and Ergodic Architecture of Grant Morrison's Superhero Poetics in Final Crisis and Seven Soldiers of Victory
Grant Morrison has been writing comics since 1985. Over the years, his stories have evolved from being deconstructions of the superhero archetype to magical explorations of the medium itself and recently with mainstream superhero fiction. Starting
ebookspoint.us
Lectures on Ergodic Theory free ebook

Lectures on Ergodic Theory free ebook ,

Lectures on Ergodic Theory free ebook

<p>English | 1956 | ISBN: 0821841254 | PDF | 108 Pages | 1 mb<br /> <br /> This classic book is based on lectures given by the author at the University of Chicago in 1956. The topics covered include, in particular, recurrence, the ergodic theorems, and a general discussion of ergodicity and mixing properties. There is also a general discussion of the relation between conjugacy and equivalence. With minimal prerequisites of some analysis and measure theory, this work can be used for a one-semester course in ergodic theory or for self-study. Readership Graduate students and research mathematicians interested in number theory. Table of Contents Introduction Examples Recurrence Mean convergence Pointwise convergence Comments on the ergodic theorem Ergodicity Consequences of ergodicity Mixing Measure algebras Discrete spectrum Automorphisms of compact groups Generalized proper values Weak topology Weak approximation Uniform topology Uniform approximation Category Invariant measures Invariant measures: the solution Invariant measures: the problem Generalized ergodic theorems Unsolved problems References.</p> ,
ebookspoint.us
Dynamical Entropy in Operator Algebras free ebook

Dynamical Entropy in Operator Algebras free ebook ,

Dynamical Entropy in Operator Algebras free ebook

<p>2006 | 293 Pages | ISBN: 3540346708 | PDF | 2.4 MB<br /> <br /> The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. It is the only monograph on this topic. Although the authors assume a basic knowledge of operator algebras, they give precise definitions of the notions and in most cases complete proofs of the results which are used.</p> ,
arxiv.org
[1507.06520] Quantum ergodicity for quantum graphs without back-scattering

[ Authors ]
Matthew Brammall, Brian Winn
[ Abstract ]
We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.

arxiv.org
[1507.06348] Anosov C-systems and random number generators

[ Authors ]
George Savvidy
[ Abstract ]
We are developing further our earlier suggestion to use hyperbolic Anosov C-systems for the Monte-Carlo simulations in high energy particle physics. The hyperbolic dynamical systems have homogeneous instability of all trajectories and as such they have mixing of all orders, countable Lebesgue spectrum and positive Kolmogorov entropy. These extraordinary ergodic properties follow from the C-condition introduced by Anosov. The C-condition defines a rich class of dynamical systems which span an open set in the space of all dynamical systems. The important property of C-systems is that they have a countable set of everywhere dense periodic trajectories and that their density exponentially increases with entropy. Of special interest are C-systems that are defined on a high dimensional torus. The C-systems on a torus are perfect candidates to be used for Monte-Carlo simulations. Recently an efficient algorithm was found, which allows very fast generation of long trajectories of the C-systems. These trajectories have high quality statistical properties and we are suggesting to use them for the QCD lattice simulations and at high energy particle physics.