To stay up to date you can also follow on Mastodon.

🔗 Statistics

In statistics, a Pólya urn model (also known as a Pólya urn scheme or simply as Pólya's urn), named after George Pólya, is a type of statistical model used as an idealized mental exercise framework, unifying many treatments.

In an urn model, objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. In the basic Pólya urn model, the urn contains x white and y black balls; one ball is drawn randomly from the urn and its color observed; it is then returned in the urn, and an additional ball of the same color is added to the urn, and the selection process is repeated. Questions of interest are the evolution of the urn population and the sequence of colors of the balls drawn out.

This endows the urn with a self-reinforcing property sometimes expressed as the rich get richer.

Note that in some sense, the Pólya urn model is the "opposite" of the model of sampling without replacement, where every time a particular value is observed, it is less likely to be observed again, whereas in a Pólya urn model, an observed value is more likely to be observed again. In both of these models, the act of measurement has an effect on the outcome of future measurements. (For comparison, when sampling with replacement, observation of a particular value has no effect on how likely it is to observe that value again.) In a Pólya urn model, successive acts of measurement over time have less and less effect on future measurements, whereas in sampling without replacement, the opposite is true: After a certain number of measurements of a particular value, that value will never be seen again.

One of the reasons for interest in this particular rather elaborate urn model (i.e. with duplication and then replacement of each ball drawn) is that it provides an example in which the count (initially x black and y white) of balls in the urn is not concealed, which is able to approximate the correct updating of subjective probabilities appropriate to a different case in which the original urn content is concealed while ordinary sampling with replacement is conducted (without the Pólya ball-duplication). Because of the simple "sampling with replacement" scheme in this second case, the urn content is now static, but this greater simplicity is compensated for by the assumption that the urn content is now unknown to an observer. A Bayesian analysis of the observer's uncertainty about the urn's initial content can be made, using a particular choice of (conjugate) prior distribution. Specifically, suppose that an observer knows that the urn contains only identical balls, each coloured either black or white, but he does not know the absolute number of balls present, nor the proportion that are of each colour. Suppose that he holds prior beliefs about these unknowns: for him the probability distribution of the urn content is well approximated by some prior distribution for the total number of balls in the urn, and a beta prior distribution with parameters (x,y) for the initial proportion of these which are black, this proportion being (for him) considered approximately independent of the total number. Then the process of outcomes of a succession of draws from the urn (with replacement but without the duplication) has approximately the same probability law as does the above Pólya scheme in which the actual urn content was not hidden from him. The approximation error here relates to the fact that an urn containing a known finite number m of balls of course cannot have an exactly beta-distributed unknown proportion of black balls, since the domain of possible values for that proportion are confined to being multiples of ${\displaystyle 1/m}$, rather than having the full freedom to assume any value in the continuous unit interval, as would an exactly beta distributed proportion. This slightly informal account is provided for reason of motivation, and can be made more mathematically precise.

This basic Pólya urn model has been enriched and generalized in many ways.

🔗 Aviation 🔗 Soviet Union 🔗 Military history 🔗 Military history/Military aviation 🔗 Military history/North American military history 🔗 Military history/United States military history 🔗 Aviation/Aviation accident 🔗 Military history/Cold War 🔗 Military history/Russian, Soviet and CIS military history 🔗 Belgium

On 4 July 1989, a pilotless MiG-23 jet fighter of the Soviet Air Forces crashed into a house in Kortrijk, Belgium, killing one person. The pilot had ejected over an hour earlier near Kołobrzeg, Poland, after experiencing technical problems, but the aircraft continued flying for around 900 km (600 mi) before running out of fuel and descending into the ground.

🔗 Linguistics 🔗 Energy 🔗 Linguistics/Philosophy of language

Long-term nuclear waste warning messages are intended to deter human intrusion at nuclear waste repositories in the far future, within or above the order of magnitude of 10,000 years. Nuclear semiotics is an interdisciplinary field of research, first done by the American Human Interference Task Force in 1981.

A 1993 report from Sandia National Laboratories recommended that such messages be constructed at several levels of complexity. They suggested that the sites should include foreboding physical features which would immediately convey to future visitors that the site was both man-made and dangerous, as well as providing pictographic information attempting to convey some details of the danger, and written explanations for those able to read it.

🔗 Mathematics

Bellard's formula is used to calculate the nth digit of π in base 16.

Bellard's formula was discovered by Fabrice Bellard in 1997. It is about 43% faster than the Bailey–Borwein–Plouffe formula (discovered in 1995). It has been used in PiHex, the now-completed distributed computing project.

One important application is verifying computations of all digits of pi performed by other means. Rather than having to compute all of the digits twice by two separate algorithms to ensure that a computation is correct, the final digits of a very long all-digits computation can be verified by the much faster Bellard's formula.

Formula:

${\displaystyle {\begin{aligned}\pi ={\frac {1}{2^{6}}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2^{10n}}}\,\left(-{\frac {2^{5}}{4n+1}}\right.&{}-{\frac {1}{4n+3}}+{\frac {2^{8}}{10n+1}}-{\frac {2^{6}}{10n+3}}\left.{}-{\frac {2^{2}}{10n+5}}-{\frac {2^{2}}{10n+7}}+{\frac {1}{10n+9}}\right)\end{aligned}}}$

🔗 Biography

William Angus Wallace (born 31 October 1948) is a Scottish orthopaedic surgeon. He is Professor of Orthopaedic and Accident Surgery at the Faculty of Medicine & Health Sciences of the University of Nottingham. He came to widespread public notice for a life-saving surgery he performed using improvised equipment on a British Airways flight in 1995, and for treating Wayne Rooney before the 2006 FIFA World Cup.

🔗 Military history 🔗 Military history/Military science, technology, and theory 🔗 Military history/Weaponry 🔗 Military history/Russian, Soviet and CIS military history 🔗 Military history/Post-Cold War

The Poseidon (Russian: Посейдон, "Poseidon", NATO reporting name Kanyon), previously known by Russian codename Status-6 (Russian: Статус-6), is an autonomous, nuclear-powered, and nuclear-armed unmanned underwater vehicle under development by Rubin Design Bureau, capable of delivering both conventional and nuclear payloads.

The Poseidon is one of the six new Russian strategic weapons announced by Russian President Vladimir Putin on 1 March 2018.

🔗 Business

Topgrading is a corporate hiring and interviewing methodology that is intended to identify preferred candidates for a particular position. In the methodology, prospective employees undergo a 12-step process that includes extensive interviews, the creation of detailed job scorecards, research into job history, coaching, and more. After being interviewed and reference-checked, job candidates are grouped into one of three categories: A Players, B Players, or C Players. A Players have the most potential for high performance in their role while B and C Players may require more work to be successful. The methodology has been used by major corporations and organizations like General Electric, Lincoln Financial, Honeywell, Barclays, and the American Heart Association.

🔗 Philosophy 🔗 Philosophy/Logic 🔗 Philosophy/Ancient philosophy 🔗 Philosophy/Epistemology

"I know that I know nothing" is a saying derived from Plato's account of the Greek philosopher Socrates. Socrates himself was never recorded as having said this phrase, and scholars generally agree that Socrates only ever asserted that he believed that he knew nothing, having never claimed that he knew that he knew nothing. It is also sometimes called the Socratic paradox, although this name is often instead used to refer to other seemingly paradoxical claims made by Socrates in Plato's dialogues (most notably, Socratic intellectualism and the Socratic fallacy).

This saying is also connected or conflated with the answer to a question Socrates (according to Xenophon) or Chaerephon (according to Plato) is said to have posed to the Pythia, the Oracle of Delphi, in which the oracle stated something to the effect of "Socrates is the wisest person in Athens." Socrates, believing the oracle but also completely convinced that he knew nothing, was said to have concluded that nobody knew anything, and that he was only wiser than others because he was the only person who recognized his own ignorance.

🔗 Physics

Certain systems can achieve negative thermodynamic temperature; that is, their temperature can be expressed as a negative quantity on the Kelvin or Rankine scales. This should be distinguished from temperatures expressed as negative numbers on non-thermodynamic Celsius or Fahrenheit scales, which are nevertheless higher than absolute zero.

The absolute temperature (Kelvin) scale can be understood loosely as a measure of average kinetic energy. Usually, system temperatures are positive. However, in particular isolated systems, the temperature defined in terms of Boltzmann's entropy can become negative.

The possibility of negative temperatures was first predicted by Lars Onsager in 1949, in his analysis of classical point vortices confined to a finite area. Confined point vortices are a system with bounded phase space as their canonical momenta are not independent degrees of freedom from their canonical position coordinates. Bounded phase space is the essential property that allows for negative temperatures, and such temperatures can occur in both classical and quantum systems. As shown by Onsager, a system with bounded phase space necessarily has a peak in the entropy as energy is increased. For energies exceeding the value where the peak occurs, the entropy decreases as energy increases, and high-energy states necessarily have negative Boltzmann temperature.

A system with a truly negative temperature on the Kelvin scale is hotter than any system with a positive temperature. If a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system. A standard example of such a system is population inversion in laser physics.

Temperature is loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature as the tradeoff between internal energy and entropy contained in the system, with "coldness", the reciprocal of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.

Thermodynamic systems with unbounded phase space cannot achieve negative temperatures: adding heat always increases their entropy. The possibility of a decrease in entropy as energy increases requires the system to "saturate" in entropy. This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however (see the examples below), have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease. The limited range of states accessible to a system with negative temperature means that negative temperature is associated with emergent ordering of the system at high energies. For example in Onsager's point-vortex analysis negative temperature is associated with the emergence of large-scale clusters of vortices. This spontaneous ordering in equilibrium statistical mechanics goes against common physical intuition that increased energy leads to increased disorder.

🔗 Transport 🔗 Trains

The Cosmopolitan Railway was a proposed global railroad network advocated by William Gilpin, formerly the first territorial governor of Colorado (1861–62), in his 1890 treatise Cosmopolitan Railway: Compacting and Fusing Together All the World's Continents. Gilpin named his capital city of Denver as the "railroad centre of the West".