Probably more likely than probable – Reblog

This is a reblog from here Probably more likely than probable // Revolutions

What kind of probability are people talking about when they say something is “highly likely” or has “almost no chance”? The chart below, created by Reddit user zonination, visualizes the responses of 46 other Reddit users to “What probability would you assign to the phase: <phrase>” for various statements of probability. Each set of responses has been converted to a kernel destiny estimate and presented as a joyplot using R.

Probable

Somewhat surprisingly, the results from the Redditors hew quite closely to a similar study of 23 NATO intelligence officers in 2007. In that study, the officers — who were accustomed to reading intelligence reports with assertions of likelihood — were giving a similar task with the same descriptions of probability. The results, here presented as a dotplot, are quite similar.

CIA

For details on the analysis of the Redditors, including the data and R code behind the joyplot chart, check out the Github repository linked below.

Github (zonination): Perceptions of Probability and Numbers

A new “Mathematician’s Apology” – Reblog

In the two and a half years (or so) since I left academia for industry, I’ve worked with a number of math majors and math PhDs outside of academia and talked to a number of current grad students who were considering going into industry. As a result, my perspective on the role of the math research […]

via A new “Mathematician’s Apology” — Low Dimensional Topology

Bessel series for a constant

Fourier series express functions as a sum of sines and cosines of different frequencies. Bessel series are analogous, expressing functions as a sum of Bessel functions of different orders.

Fourier series arise naturally when working in rectangular coordinates. Bessel series arise naturally when working in polar coordinates.

The Fourier series for a constant is trivial. You can think of a constant as a cosine with frequency zero.

The Bessel series for a constant is not as simple, but more interesting. Here we have:

1=J_0(x)+2J_2(x)+2J_4(x)+2J_g(x)\cdots

Since J_{-n}=(-1)^n J_n(x) we can write the series above as the following infinite series:

1=\sum_{n=-\infty}^{\infty} J_{2n}(x)

Cool, right?

 

The Winton Gallery opens at the Science Museum

During the recent Christmas and New Year break I had the opportunity to visit the Science Museum (yes, again…). This time to see the newly opened Winton Gallery that housed the Mathematics exhibit in the museum. Not only is the exhibit about a subject matter close to my heart, but also the gallery was designed by Zaha Hadid Architects. I must admit, that the first I heard of this was in a recent visit to the IMAX at the Science Museum to see Rogue One… Anyway, I took some pictures that you can see in the photo gallery here, and I am also re-posting an entry that appeared in the London Mathematical Society newsletter Number 465 for January 2017.

Mathematics: The Winton Gallery opens at the Science Museum, London

On 8 December 2016 the Science Museum opened a pioneering new gallery that explores how mathematicians, their tools and ideas have helped shape the modern world over the last 400 years. Mathematics: The Winton Gallery places mathematics at the heart of all our lives, bringing  the subject to life through remarkable stories, artefacts and design.

More than 100 treasures from the Science Museum’s world-class science, technology, engineering and mathematics collections help tell powerful stories about how mathematical practice has shaped and been shaped by some of our most fundamental human concerns – including money, trade, travel, war, life and death.

From a beautiful 17th-century Islamic astrolabe that used ancient mathematical techniques to map the night sky to an early example of the famous Enigma machine, designed to resist even the most advanced mathematical techniques for codebreaking, each historical object has an important story to tell about how mathematics has shaped our world. Archive photography and lm helps capture these stories and digital exhibits alongside key objects introduce the wide range of people who made, used or were affected by each mathematical device.

Dramatically positioned at the centre of the gallery is the Handley Page ‘Gugnunc’ aircraft, built in 1929 for a competition to construct a safe aircraft. Ground-breaking aerodynamic research influenced the wing design of this experimental aircraft, helping transform public opinion about the safety of ying and securing the future of the aviation industry. This aeroplane highlights perfectly the central theme of the gallery about how mathematical practice is driven by, and in uences, real-world concerns and activities.

Mathematics also defines Zaha Hadid Architects’ design for the gallery. Inspired by the Handley Page aircraft, the gallery is laid out using principles of mathematics and physics. These principles also inform the three-dimensional curved surfaces representing the patterns of air ow that would have streamed around this aircraft.

Patrik Schumacher, Partner at Zaha Hadid Architects, recently noted that mathematics was part of Zaha Hadid’s life from a young age and was always the foundation of her architecture, describing the new mathematics gallery as ‘an important part of Zaha’s legacy in London’. Gallery curator David Rooney, who was respon- sible for the Science Museum’s recent award- winning Codebreaker: Alan Turing’s Life and Legacy exhibition, explained that the gallery tells ‘a rich cultural story of human endeavor that has helped transform the world’.

The mathematics gallery was made possible through an unprecedented donation from long-standing supporters of science, David and Claudia Harding. Additional support was also provided by Principal Sponsor Samsung, Major Sponsor MathWorks and a number of individual donors.

A lavishly illustrated new book, Mathematics: How It Shaped Our World, written by David Rooney and published by Scala Arts & Heritage Publishers, accompanies the new display. It expands the stories covered in the gallery and contains an absorbing series of newly commissioned essays by prominent historians and mathematicians including June Barrow-Green, Jim Bennett, Patricia Fara, Dame Celia Hoyles and Helen Wilson, with an afterword from Dame Zaha Hadid with Patrick Schumacher.

Nobel Prize in Physics 2016: Exotic States of Matter

Yesterday the 2016 Nobel Prize in Physics was announced. I immediately got a few tweets asking for more information about what these “exotic” states of matter were and explain more about them… Well in short the prize was awarded for the  theoretical discoveries that help scientists understand unusual properties of materials, such as superconductivity and superfluidity, that arise at low temperatures.

Physics Nobel 2016

The prize was awarded jointly to David J. Thouless of the University of Washington in Seattle, F. Duncan M. Haldane of Princeton University in New Jersey, and J. Michael Kosterlitz of Brown University in Rhode Island. The citation from the Swedish Academy reads: “for theoretical discoveries of topological phase transitions and topological phases of matter.”

“Topo…what?” – I hear you cry… well let us start at the beginning…

Thouless, Haldane and Kosterliz work in a field of physics known as Condensed Matter Physics and it is interested in the physical properties of “condensed” materials such as solids and liquids. You may not know it, but results from research in condensed matter physics have made it possible for you to save a lot of data in your computer’s hard drive: the discovery of giant magnetoresistance has made it possible.

The discoveries that the Nobel Committee are highlighting with the prize provide a better understanding of phases of matter such as superconductors, superfluids and thin magnetic films. The discoveries are now guiding the quest for next generation materials for electronics, quantum computing and more. They have developed mathematical models to describe the topological properties of materials in relation to other phenomena such as superconductivity, superfluidity and other peculiar magnetic properties.

Once again that word: “topology”…

So, we know that all matter is formed by atoms. Nonetheless matter can have different properties and appear in different forms, such as solid, liquid, superfluid, magnet, etc. These various forms of matter are often called states of matter or phases. According to condensed matter physics , the different properties of materials originate from the different ways in which the atoms are organised in the materials. Those different organizations of the atoms (or other particles) are formally called the orders in the materials. Topological order is a type of order in zero-temperature phase of matter (also known as quantum matter). In general, topology is the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures. In our case, we are talking about properties of matter that remain unchanged when the object is flattened or expanded.

Although, research originally focused on topological properties in 1-D and 2-D materials, researchers have discovered them in 3-D materials as well. These results are particularly important as they enable us to understanding “exotic” phenomena such as superconductivity, the property of matter that lets electrons travel through materials with zero resistance, and superfluidity, which lets fluids flow with zero loss of kinetic energy. Currently one of the most researched topics in the area is the study of topological insulators, superconductors and metals.

Here is a report from Physics Today about the Nobel Prize announcement:

Thouless, Haldane, and Kosterlitz share 2016 Nobel Prize in Physics

David Thouless, Duncan Haldane, and Michael Kosterlitz are to be awarded the 2016 Nobel Prize in Physics for their work on topological phases and phase transitions, the Royal Swedish Academy of Sciences announced on Tuesday. Thouless, of the University of Washington in Seattle, will receive half the 8 million Swedish krona (roughly $925 000) prize; Haldane, of Princeton University, and Kosterlitz, of Brown University, will split the other half.

This year’s laureates used the mathematical branch of topology to make revolutionary contributions to their field of condensed-matter physics. In 1972 Thouless and Kosterlitz identified a phase transition that opened up two-dimensional systems as a playground for observing superconductivity, superfluidity, and other exotic phenomena. A decade later Haldane showed that topology is important in considering the properties of 1D chains of magnetic atoms. Then in the 1980s Thouless and Haldane demonstrated that the unusual behavior exhibited in the quantum Hall effect can emerge without a magnetic field.

From early on it was clear that the laureates’ work would have important implications for condensed-matter theory. Today experimenters are studying 2D superconductors and topological insulators, which are insulating in the bulk yet channel spin-polarized currents on their surfaces without resistance (see Physics Today, January 2010, page 33). The research could lead to improved electronics, robust qubits for quantum computers, and even an improved understanding of the standard model of particle physics.

Vortices and the KT transition

When Thouless and Kosterlitz first collaborated in the early 1970s, the conventional wisdom was that thermal fluctuations in 2D materials precluded the emergence of ordered phases such as superconductivity. The researchers, then at the University of Birmingham in England, dismantled that argument by investigating the interactions within a 2D lattice.

Thouless and Kosterlitz considered an idealized array of spins that is cooled to nearly absolute zero. At first the system lacks enough thermal energy to create defects, which in the model take the form of localized swirling vortices. Raising the temperature spurs the development of tightly bound pairs of oppositely rotating vortices. The coherence of the entire system depends logarithmically on the separation between vortices. As the temperature rises further, more vortex pairs pop up, and the separation between partners grows.

The two scientists’ major insight came when they realized they could model the clockwise and counterclockwise vortices as positive and negative electric charges. The more pairs that form, the more interactions are disturbed by narrowly spaced vortices sitting between widely spaced ones. “Eventually, the whole thing will fly apart and you’ll get spontaneous ‘ionization,’ ” Thouless told Physics Today in 2006.

That analog to ionization, in which the coherence suddenly falls off in an exponential rather than logarithmic dependence with distance, is known as the Kosterlitz–Thouless (KT) transition. (The late Russian physicist Vadim Berezinskii made a similar observation in 1970, which led some researchers to add a “B” to the transition name, but the Nobel committee notes that Berezinskii did not theorize the existence of the transition at finite temperature.)

Unlike some other phase transitions, such as the onset of ferromagnetism, no symmetry is broken. The sudden shift between order and disorder also demonstrates that superconductivity could indeed subsist in the 2D realm at temperatures below that of the KT transition. Experimenters observed the KT transition in superfluid helium-4 in 1978 and in superconducting thin films in 1981. More recently, the transition was reproduced in a flattened cloud of ultracold rubidium atoms (see Physics Today, August 2006, page 17).

A topological answer for the quantum Hall effect

Thouless then turned his attention to the quantum foundations of conductors and insulators. In 1980 German physicist Klaus von Klitzing had applied a strong magnetic field to a thin conducting film sandwiched between semiconductors. The electrons traveling within the film separated into well-organized opposing lanes of traffic along the edges (see Physics Today, June 1981, page 17). Von Klitzing had discovered the quantum Hall effect, for which he would earn the Nobel five years later.

Crucially, von Klitzing found that adjusting the strength of the magnetic field changed the conductance of his thin film only in fixed steps; the conductance was always an integer multiple of a fixed value, e2/h. That discovery proved the key for Thouless to relate the quantum Hall effect to topology, which is also based on integer steps—objects are often distinguished from each other topologically by the number of holes or nodes they possess, which is always an integer. In 1983 Thouless proposed that the electrons in von Klitzing’s experiment had formed a topological quantum fluid; the electrons’ collective behavior in that fluid, as measured by conductance, must vary in steps.

Not only did Thouless’s work explain the integer nature of the quantum Hall effect, but it also pointed the way to reproducing the phenomenon’s exotic behavior under less extreme conditions. In 1988 Haldane proposed a means for electrons to form a topological quantum fluid in the absence of a magnetic field. Twenty-five years later, researchers reported such behavior in chromium-doped (Bi,Sb)2Te3, the first observation of what is known as the quantum anomalous Hall effect.

Exploring topological materials

Around 2005, physicists began exploring the possibility of realizing topological insulators, a large family of new topological phases of matter that would exhibit the best of multiple worlds: They would robustly conduct electricity on their edges or surfaces without a magnetic field and as a bonus would divide electron traffic into lanes determined by spin. Since then experimenters have identified topological insulators in two and three dimensions, which may lead to improved electronics. Other physicists have created topological insulators that conduct sound or light, rather than electrons, on their surfaces (see Physics Today, May 2014, page 68).

Haldane’s work in the 1980s on the fractional quantum Hall effect was among the theoretical building blocks for proposals to use topologically protected excitations to build a fault-tolerant quantum computer (see Physics Today, October 2005, page 21). And his 1982 paper on magnetic chains serves as the foundation for efforts to create topologically protected excitations that behave like Majorana fermions, which are their own antiparticle. The work could lead to robust qubits for preserving the coherence of quantum information and perhaps provide particle physicists with clues as to the properties of fundamental Majorana fermions, which may or may not exist in nature.

—Andrew Grant

 

I quite like this version of my name including the square root of pi

I recently got a newsletter from the ODI. Nothing unusual there, except for the fact that my name was spelled wrongly. It is clear that their mail merge does not know how to handle accented characters, but I must admit that I quite like this version of my name… I mean it includes the square root of \pi! How cool is that‽

I shall start using that version!

Pi_J_Name

Life lessons from differential equations

Ten life lessons from differential equations:

  1. Some problems simply have no solution.
  2. Some problems have no simple solution.
  3. Some problems have many solutions.
  4. Determining that a solution exists may be half the work of finding it.
  5. Solutions that work well locally may blow up when extended too far.
  6. Boundary conditions are the hard part.
  7. Something that starts out as a good solution may become a very bad solution.
  8. You can fool yourself by constructing a solution where one doesn’t exist.
  9. Expand your possibilities to find a solution, then reduce them to see how good the solution is.
  10. You can sometimes do what sounds impossible by reframing your problem.

n sweets in a bag, some are orange…

SweetsThe other day in the news there was a note about a particular question in one of the national curriculum exams… I thought it was a bit of an odd thing for a maths question to feature in the news and so I thought of having a look a the question. Here it is:

There are n sweets in a bag.

6 of the sweets are orange.

The rest of the sweets are yellow.

Hannah takes at random a sweet form the bag. She eats the sweet.

Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hanna eats two orange sweets is \frac{1}{3}.

a) Show that n^2-n-90=0

It sounds like an odd question, but after giving it a bit of thought it is actually quite straightforward; and I am glad they ask something that makes you think, rather than something that is purely a mechanical calculation.

So, let’s take a look: Hannah is taking sweets from the bag at random and without replacement (she eats the sweets after all). So we are told that there are 6 orange sweets, so at the beginning of the sweet-eating binge, the probability of picking an orange sweet is:

\displaystyle P(\text{1 orange sweet})=\frac{6}{n}.

Hannah eats the sweet, remember… so in the second go at the sweets, the probability of an orange sweet is now:

\displaystyle P(\text{2nd orange sweet})=\frac{5}{n-1}.

Now, they tell us that the probability of eating two orange sweets is \frac{1}{3}, so we have that:

\displaystyle \left( \frac{6}{n} \right)\left( \frac{5}{n-1} \right)=\frac{1}{3},

\displaystyle \frac{30}{n^2-n}=\frac{1}{3},

\displaystyle n^2-n=90,

which is the expression we were looking for. Furthermore, you can then solve this quadratic equation to find that the total number of sweets in the bag is 10.

The only thing we don’t know is if the sweets are just orange in colour, or also in flavour! We will have to ask Hannah!