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🔗 Mathematics 🔗 Germany 🔗 Military history 🔗 Statistics 🔗 Military history/Intelligence 🔗 Military history/World War II 🔗 Military history/German military history 🔗 Military history/European military history

In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose we have an unknown number of items which are sequentially numbered from 1 to N. We take a random sample of these items and observe their sequence numbers; the problem is to estimate N from these observed numbers.

The problem can be approached using either frequentist inference or Bayesian inference, leading to different results. Estimating the population maximum based on a single sample yields divergent results, whereas estimation based on multiple samples is a practical estimation question whose answer is simple (especially in the frequentist setting) but not obvious (especially in the Bayesian setting).

The problem is named after its historical application by Allied forces in World War II to the estimation of the monthly rate of German tank production from very few data. This exploited the manufacturing practice of assigning and attaching ascending sequences of serial numbers to tank components (chassis, gearbox, engine, wheels), with some of the tanks eventually being captured in battle by Allied forces.

🔗 Technology 🔗 Business 🔗 Electronics 🔗 Energy 🔗 Home Living

The Phoebus cartel existed to control the manufacture and sale of incandescent light bulbs. They appropriated market territories and fixed the useful life of such bulbs. Corporations based in Europe and America founded the cartel on January 15, 1925 in Geneva. They had intended the cartel to last for thirty years (1925 to 1955). The cartel ceased operations in 1939 owing to the outbreak of World War II. The cartel included manufacturers Osram, General Electric, Associated Electrical Industries, and Philips, among others.

The Phoebus cartel created a notable landmark in the history of the global economy because it engaged in large-scale planned obsolescence to generate repeated sales and maximize profit. It also reduced competition in the light bulb industry for almost fifteen years. Critics accused the cartel of preventing technological advances that would produce longer-lasting light bulbs. Phoebus based itself in Switzerland. The corporation named itself Phœbus S.A. Compagnie Industrielle pour le Développement de l'Éclairage (French for "Phoebus, Inc. Industrial Company for the Development of Lighting").

🔗 Astronomy 🔗 Sweden 🔗 Solar System 🔗 Astronomy/Solar System

The Sweden Solar System is the world's largest permanent scale model of the Solar System. The Sun is represented by the Ericsson Globe in Stockholm, the largest hemispherical building in the world. The inner planets can also be found in Stockholm but the outer planets are situated northward in other cities along the Baltic Sea. The system was started by Nils Brenning and Gösta Gahm and is on the scale of 1:20 million.

🔗 Technology 🔗 Lists 🔗 Futures studies 🔗 Invention

Emerging technologies are those technical innovations that represent progressive innovations within a field for competitive advantage.

🔗 Numbers 🔗 Law

An illegal number is a number that represents information which is illegal to possess, utter, propagate, or otherwise transmit in some legal jurisdiction. Any piece of digital information is representable as a number; consequently, if communicating a specific set of information is illegal in some way, then the number may be illegal as well.

🔗 Agriculture 🔗 England 🔗 Folklore

Telling the bees is a traditional custom of many European countries in which bees would be told of important events in their keeper's lives, such as births, marriages, or departures and returns in the household. If the custom was omitted or forgotten and the bees were not "put into mourning" then it was believed a penalty would be paid, such as the bees leaving their hive, stopping the production of honey, or dying. The custom is best known in England, but has also been recorded in Ireland, Wales, Germany, Netherlands, France, Switzerland, Bohemia, and the United States.

🔗 California 🔗 Business 🔗 Politics 🔗 Trains 🔗 Politics/American politics 🔗 Trains/Rapid transit 🔗 Buses 🔗 Trains/Streetcars

The notion of a General Motors streetcar conspiracy emerged after General Motors (GM) and other companies were convicted of monopolizing the sale of buses and supplies to National City Lines (NCL) and its subsidiaries. In the same case, the defendants were accused of conspiring to own or control transit systems, in violation of Section 1 of the Sherman Antitrust act. The suit created lingering suspicions that the defendants had in fact plotted to dismantle streetcar systems in many cities in the United States as an attempt to monopolize surface transportation.

Between 1938 and 1950, National City Lines and its subsidiaries, American City Lines and Pacific City Lines—with investment from GM, Firestone Tire, Standard Oil of California (through a subsidiary), Federal Engineering, Phillips Petroleum, and Mack Trucks—gained control of additional transit systems in about 25 cities. Systems included St. Louis, Baltimore, Los Angeles, and Oakland. NCL often converted streetcars to bus operations in that period, although electric traction was preserved or expanded in some locations. Other systems, such as San Diego's, were converted by outgrowths of the City Lines. Most of the companies involved were convicted in 1949 of conspiracy to monopolize interstate commerce in the sale of buses, fuel, and supplies to NCL subsidiaries, but were acquitted of conspiring to monopolize the transit industry.

The story as an urban legend has been written about by Martha Bianco, Scott Bottles, Sy Adler, Jonathan Richmond, and Robert Post. It has been explored several times in print, film, and other media, notably in Who Framed Roger Rabbit, Taken for a Ride, Internal Combustion, and The End of Suburbia.

Only a handful of U.S. cities, including San Francisco, New Orleans, Newark, Cleveland, Philadelphia, Pittsburgh, and Boston, have surviving legacy rail urban transport systems based on streetcars, although their systems are significantly smaller than they once were. Other cities are re-introducing streetcars. In some cases, the streetcars do not actually ride on the street. Boston had all of its downtown lines elevated or buried by the mid-1920s, and most of the surviving lines at grade operate on their own right of way. However, San Francisco's and Philadelphia's lines do have large portions of the route that ride on the street as well as using tunnels.

🔗 Mathematics 🔗 Festivals

Pi Day is an annual celebration of the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in the month/day format) since 3, 1, and 4 are the first three significant digits of π. In 2009, the United States House of Representatives supported the designation of Pi Day. UNESCO's 40th General Conference decided Pi Day as the International Day of Mathematics in November 2019.

Pi Approximation Day is observed on July 22 (22/7 in the day/month format), since the fraction ​²²⁄₇ is a common approximation of π, which is accurate to two decimal places and dates from Archimedes.

Two Pi Day, also known as Tau Day for the mathematical constant Tau, is observed on June 28 (6/28 in the month/day format).

🔗 Mathematics 🔗 Statistics

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

The graph to the right shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization from leading 1 digit to leading n digits.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants. Like other general principles about natural data—for example the fact that many data sets are well approximated by a normal distribution—there are illustrative examples and explanations that cover many of the cases where Benford's law applies, though there are many other cases where Benford's law applies that resist a simple explanation. It tends to be most accurate when values are distributed across multiple orders of magnitude, especially if the process generating the numbers is described by a power law (which are common in nature).

It is named after physicist Frank Benford, who stated it in 1938 in a paper titled "The Law of Anomalous Numbers", although it had been previously stated by Simon Newcomb in 1881.

🔗 United States