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🔗 Mathematics 🔗 Physics

A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties. This is like the effort required to cut a string and reattach the ends to form a different braid, as opposed to a ball (representing an ordinary quantum particle in four-dimensional spacetime) bumping into a wall. Alexei Kitaev proposed topological quantum computation in 1997. While the elements of a topological quantum computer originate in a purely mathematical realm, experiments in fractional quantum Hall systems indicate these elements may be created in the real world using semiconductors made of gallium arsenide at a temperature of near absolute zero and subjected to strong magnetic fields.

🔗 Biography 🔗 Mathematics 🔗 France 🔗 Women scientists 🔗 Biography/science and academia 🔗 Women's History 🔗 Mathematics/Mathematicians

Marie-Sophie Germain (French: [maʁi sɔfi ʒɛʁmɛ̃]; 1 April 1776 – 27 June 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Leonhard Euler, and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss (under the pseudonym of «Monsieur LeBlanc»). One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred. On 27 June 1831, she died from breast cancer. At the centenary of her life, a street and a girls’ school were named after her. The Academy of Sciences established the Sophie Germain Prize in her honor.

🔗 Technology 🔗 Computing 🔗 Mathematics

A slide rule is a hand-operated mechanical calculator consisting of slidable rulers for evaluating mathematical operations such as multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest analog computers.

Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler, the slide rule is not meant to be used for measuring length or drawing straight lines. Nor is it designed for addition or subtraction, which is usually performed using other methods, like using an abacus. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results.

English mathematician and clergyman Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. It made calculations faster and less error-prone than evaluating on paper. Before the advent of the scientific pocket calculator, it was the most commonly used calculation tool in science and engineering. The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through the 1950s and 1960s, even as desktop electronic computers were gradually introduced. But after the handheld scientific calculator was introduced in 1972 and became inexpensive in the mid-1970s, slide rules became largely obsolete, so most suppliers departed the business.

In the United States, the slide rule is colloquially called a slipstick.

🔗 Mathematics 🔗 Statistics

The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.

🔗 Mathematics 🔗 Statistics 🔗 Psychology

Accuracy and precision are two measures of observational error. Accuracy is how close a given set of measurements (observations or readings) are to their true value. Precision is how close the measurements are to each other.

The International Organization for Standardization (ISO) defines a related measure: trueness, "the closeness of agreement between the arithmetic mean of a large number of test results and the true or accepted reference value."

While precision is a description of random errors (a measure of statistical variability), accuracy has two different definitions:

1. More commonly, a description of systematic errors (a measure of statistical bias of a given measure of central tendency, such as the mean). In this definition of "accuracy", the concept is independent of "precision", so a particular set of data can be said to be accurate, precise, both, or neither. This concept corresponds to ISO's trueness.
2. A combination of both precision and trueness, accounting for the two types of observational error (random and systematic), so that high accuracy requires both high precision and high trueness. This usage corresponds to ISO's definition of accuracy (trueness and precision).

🔗 Mathematics 🔗 Ancient Near East 🔗 Archaeology

YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest known computational accuracy ... in the ancient world". The tablet is believed to be the work of a student in southern Mesopotamia from some time in the range from 1800–1600 BC, and was donated to the Yale Babylonian Collection by J. P. Morgan.

🔗 Mathematics

Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it apart from its Western counterpart. The most remarkable of those features is the fact that it is written from right to left following the normal direction of the Arabic script. Other differences include the replacement of the Greek and Latin alphabet letters for symbols with Arabic letters and the use of Arabic names for functions and relations.

🔗 Mathematics 🔗 Statistics 🔗 Linguistics 🔗 Linguistics/Applied Linguistics

Zipf's law (, not as in German) is an empirical law formulated using mathematical statistics that refers to the fact that many types of data studied in the physical and social sciences can be approximated with a Zipfian distribution, one of a family of related discrete power law probability distributions. Zipf distribution is related to the zeta distribution, but is not identical.

Zipf's law was originally formulated in terms of quantitative linguistics, stating that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc.: the rank-frequency distribution is an inverse relation. For example, in the Brown Corpus of American English text, the word the is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences (69,971 out of slightly over 1 million). True to Zipf's Law, the second-place word of accounts for slightly over 3.5% of words (36,411 occurrences), followed by and (28,852). Only 135 vocabulary items are needed to account for half the Brown Corpus.

The law is named after the American linguist George Kingsley Zipf (1902–1950), who popularized it and sought to explain it (Zipf 1935, 1949), though he did not claim to have originated it. The French stenographer Jean-Baptiste Estoup (1868–1950) appears to have noticed the regularity before Zipf. It was also noted in 1913 by German physicist Felix Auerbach (1856–1933).

🔗 Mathematics 🔗 Percussion

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory.

"Can One Hear the Shape of a Drum?" is the title of a 1966 article by Mark Kac in the American Mathematical Monthly which made the question famous, though this particular phrasing originates with Lipman Bers. Similar questions can be traced back all the way to Hermann Weyl . For his paper, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968.

The frequencies at which a drumhead can vibrate depend on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known; for example, whether a circle-shaped triangle can be recognized in this way. Kac admitted he did not know if it was possible for two different shapes to yield the same set of frequencies. The question of whether the frequencies determine the shape was finally answered in the negative in the early 1990s by Gordon, Webb and Wolpert.

🔗 Mathematics

In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blissard (1861) and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.