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https://en.wikipedia.org/wiki/Linking%20number
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In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all).
The linking number was introduced by Gauss in the form of the linking integral. It is an important object of study in knot theory, algebraic topology, and differential geometry, and has numerous applications in mathematics and science, including quantum mechanics, electromagnetism, and the study of DNA supercoiling.
Definition
Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions. This determines the linking number:
Each curve may pass through itself during this motion, but the two curves must remain separated throughout. This is formalized as regular homotopy, which further requires that each curve be an immersion, not just any map. However, this added condition does not change the definition of linking number (it does not matter if the curves are required to always be immersions or not), which is an example of an h-principle (homotopy-principle), meaning that geometry reduces
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https://en.wikipedia.org/wiki/Communication%20Theory%20of%20Secrecy%20Systems
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"Communication Theory of Secrecy Systems" is a paper published in 1949 by Claude Shannon discussing cryptography from the viewpoint of information theory. It is one of the foundational treatments (arguably the foundational treatment) of modern cryptography. It is also a proof that all theoretically unbreakable ciphers must have the same requirements as the one-time pad.
Shannon published an earlier version of this research in the formerly classified report A Mathematical Theory of Cryptography, Memorandum MM 45-110-02, Sept. 1, 1945, Bell Laboratories. This report also precedes the publication of his "A Mathematical Theory of Communication", which appeared in 1948.
See also
Confusion and diffusion
Product cipher
One-time pad
Unicity distance
References
Shannon, Claude. "Communication Theory of Secrecy Systems", Bell System Technical Journal, vol. 28(4), page 656–715, 1949.
Shannon, Claude. "A Mathematical Theory of Cryptography", Memorandum MM 45-110-02, Sept. 1, 1945, Bell Laboratories.
Notes
https://www.itsoc.org/about/shannon
External links
Online retyped copy of the paper
Scanned version of the published BSTJ paper
History of cryptography
Cryptography publications
1945 in science
1949 documents
1949 in science
1945 documents
Mathematics papers
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https://en.wikipedia.org/wiki/Nondeterminism
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Nondeterminism or nondeterministic may refer to:
Computer science
Nondeterministic programming
Nondeterministic algorithm
Nondeterministic model of computation
Nondeterministic finite automaton
Nondeterministic Turing machine
Indeterminacy in computation (disambiguation)
Other
Indeterminism (philosophy)
See also
Indeterminacy (disambiguation)
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https://en.wikipedia.org/wiki/Michael%20Clark%20%28British%20politician%29
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Dr Michael Clark (born 8 August 1935) is a Conservative Party politician in the United Kingdom.
Early life
He was educated at King Edward VI Grammar School, East Retford and King's College London, where he graduated with a BSc (1st class Hons) in Chemistry in 1956, and subsequently studied at the University of Minnesota on a Fulbright Scholarship, before completing a PhD in Chemistry at St John's College, Cambridge in 1960.
He worked for some years with ICI Plastics Division, initially as a Research Scientist, but subsequently as a Factory Manager. Later, he joined the motor industry, introducing plastics into the manufacture of cars and commercial vehicles. In 1969, he became a manufacturing consultant with the PA Consulting Group in London, and was a Trustee from 1994 to 2000.
Political career
Dr Clark held office at constituency level in Cambridgeshire between 1969 and 1983 being County Treasurer 1975–78, and Chairman 1980–83. He first stood, unsuccessfully, at Ilkeston in 1979, being defeated by the Labour incumbent Ray Fletcher. He served as Conservative Party Member of Parliament for Rochford from 1983 until 1997, then, with its abolition, for the new constituency of Rayleigh from 1997 until he stood down at the 2001 general election.
In Parliament he was active in all matters involving science and technology joining, in 1983, the Energy Select Committee, becoming Chairman 1989–92. With the demise of that Committee he was elected Chairman of the All-Party Group for
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https://en.wikipedia.org/wiki/Eric%20de%20Sturler
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Eric de Sturler (born 15 January 1966, Groningen) is a Professor of Mathematics at Virginia Tech in Blacksburg, Virginia. He is on the editorial board of Applied Numerical Mathematics and the Open Applied Mathematics Journal.
Prof. de Sturler completed his Ph.D. under the direction of Henk van der Vorst at Technische Universiteit Delft in 1994. His thesis is entitled Iterative Methods on Distributive Memory Computers. He was a second-place winner of the Leslie Fox Prize for Numerical Analysis in 1997.
His research focuses on preconditioned iterative methods for solving linear and nonlinear systems, with applications in computational physics, material science, and mathematical biology.
References
External links
Eric de Sturler's personal webpage
1966 births
Living people
Dutch mathematicians
De Sturler, Eric
De Sturler, Eric
Delft University of Technology alumni
Scientists from Groningen (city)
Dutch expatriates in the United States
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https://en.wikipedia.org/wiki/DikuMUD
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DikuMUD is a multiplayer text-based role-playing game, which is a type of multi-user domain (MUD). It was written in 1990 and 1991 by Sebastian Hammer, Tom Madsen, Katja Nyboe, Michael Seifert, and Hans Henrik Stærfeldt at DIKU (Datalogisk Institut Københavns Universitet)—the department of computer science at the University of Copenhagen in Copenhagen, Denmark.
Commonly referred to as simply "Diku", the game was greatly inspired by AberMUD, though Diku became one of the first multi-user games to become popular as a freely-available program for its gameplay and similarity to Dungeons & Dragons. The gameplay style of the great preponderance of DikuMUDs is hack and slash, which is seen proudly as emblematic of what DikuMUD stands for.
Diku's source code was first released in 1990.
Development and history
DikuMUD was created by the University of Copenhagen's Department of Computer Science among a group of student friends: Katja Nyboe, Tom Madsen, Hans Henrik Staerfeldt, Michael Seifert, and Sebastian Hammer. According to Richard Bartle, co-creator of the first MUD, DikuMUD's developers sought to create a better version of AberMUD. Unlike TinyMUD and LPMUD, which encouraged live changes to the virtual world, DikuMUD hard-coded its virtual world.
The making of DikuMUD was first announced on Usenet by Hans Henrik Stærfeldt March 27, 1990. At the time Madsen, Hammer, and Stærfeldt were the only developers, joined by Michael Seifert in June 1990. Stærfeldt stated that their inte
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https://en.wikipedia.org/wiki/GuRoo
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GuRoo is a humanoid robot developed at the Mobile Robotics Laboratory in the School of Information Technology and Electrical Engineering at the University of Queensland. The design of the GuRoo is based on the human form and it is kept as anthropomorphic as possible. GuRoo is completely autonomous. It is used for research in different areas including dynamic stability, human-robot interaction and machine learning. GuRoo competes in the annual RoboCup. The goal of this competition is to foster the development of robotics through an annual soccer competition. It is the dream of the RoboCup federation to develop a team of fully autonomous humanoid robots, to play against and beat the human team that wins the World Cup in the year 2050.
Specifications
Mechanical
GuRoo was designed with the proportions of a child of approximately six years of age. The robot is able to interface with typical human environments such as bench tops and door handles. The mechanical design began in 2001 as an undergraduate thesis project. SolidEdge a solid modeling package was used to draft all sections of the robot under construction, it took physical form one year later. The majority of the structure is made of 3mm aluminium plate and angle sections. The structure is heavily milled to reduce weight and improve airflow over the motors and power electronics.
Electro Mechanical
In an effort to mimic the human body, the GuRoo has been built with 23 degrees of freedom. The actuators chosen tended towa
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https://en.wikipedia.org/wiki/Leopold%20Ru%C5%BEi%C4%8Dka
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Leopold Ružička (; born Lavoslav Stjepan Ružička; 13 September 1887 – 26 September 1976) was a Croatian-Swiss scientist and joint winner of the 1939 Nobel Prize in Chemistry "for his work on polymethylenes and higher terpenes" "including the first chemical synthesis of male sex hormones." He worked most of his life in Switzerland, and received eight doctorates honoris causa in science, medicine, and law; seven prizes and medals; and twenty-four honorary memberships in chemical, biochemical, and other scientific societies.
Early life
Ružička was born in Vukovar (at the time in the Kingdom of Hungary, Austro-Hungarian Empire, today in Croatia). His family of craftsmen and farmers was mostly of Croat origin, with a Czech great-grandparent, Ružička, and a great-grandmother and a great-grandfather from Austria. He lost his father, Stjepan, at the age of four, and his mother, Amalija Sever, took him and his younger brother Stjepan, to live in Osijek.
Ružička attended the classics program secondary school in Osijek. He changed his original idea of becoming a priest and switched to studying technical disciplines. Chemistry was his choice, probably because he hoped to get a position at the newly opened sugar refinery built in Osijek.
Owing to the excessive hardship of everyday and political life, he left and chose the High Technical School in Karlsruhe in Germany. He was a good student in areas he liked and that he thought would be necessary and beneficial in the future, which was
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https://en.wikipedia.org/wiki/Free%20body%20diagram
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In physics and engineering, a free body diagram (FBD; also called a force diagram) is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems.
Purpose
Free body diagrams are used to visualize forces and moments applied to a body and to calculate reactions in mechanics problems. These diagrams are frequently used both to determine the loading of individual structural components and to calculate internal forces within a structure. They are used by most engineering disciplines from Biomechanics to Structural Engineering.
In the educational environment, a free body diagram is an important step in understanding certain topics, such as statics, dynamics and other forms of classical mechanics.
Features
A free body diagram is not a scaled drawing, it is a diagram. The symbols used in a free body diagram depends upon how a body is modeled.
Free body diagrams consist of:
A simplified version of the body (often a dot or a box)
Forces shown as straight arrows pointing in the direction they act on the body
Moments are shown as curves with an arrow head or a vector with two arrow heads pointing in the
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https://en.wikipedia.org/wiki/Wave%20packet
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In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Any signal of a limited width in time or space requires many frequency components around a center frequency within a bandwidth inversely proportional to that width; even a gaussian function is considered a wave packet because its Fourier transform is a "packet" of waves of frequencies clustered around a central frequency. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant (no dispersion) or it may change (dispersion) while propagating.
Historical background
Ideas related to wave packets – modulation, carrier waves, phase velocity, and group velocity – date from the mid 1800s. The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.
Erwin Schrödinger introduced the idea of wave packets just after publishing his famous wave equation. He solved his wave equation for a quantum harmonic oscillator, in
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https://en.wikipedia.org/wiki/Limit%20cycle
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In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Definition
We consider a two-dimensional dynamical system of the form
where
is a smooth function. A trajectory of this system is some smooth function with values in which satisfies this differential equation. Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some such that for all . An orbit is the image of a trajectory, a subset of . A closed orbit, or cycle, is the image of a closed trajectory. A limit cycle is a cycle which is the limit set of some other trajectory.
Properties
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
Given a limit cycle and a trajectory in its interior that approaches the limit cycle for time approaching , then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching . The correspond
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https://en.wikipedia.org/wiki/Golden%20number
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Golden number may mean:
Golden number (time), a number assigned to a calendar year denoting its place in a Metonic cycle
Golden ratio, an irrational mathematical constant with special properties in arts and mathematics
Fibonacci number, a sequence of numbers which converges on the golden ratio
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https://en.wikipedia.org/wiki/Orthogonal%20functions
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In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:
The functions and are orthogonal when this integral is zero, i.e. whenever . As with a basis of vectors in a finite-dimensional space, orthogonal functions can form an infinite basis for a function space. Conceptually, the above integral is the equivalent of a vector dot product; two vectors are mutually independent (orthogonal) if their dot-product is zero.
Suppose is a sequence of orthogonal functions of nonzero L2-norms . It follows that the sequence is of functions of L2-norm one, forming an orthonormal sequence. To have a defined L2-norm, the integral must be bounded, which restricts the functions to being square-integrable.
Trigonometric functions
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions and are orthogonal on the interval when and n and m are positive integers. For then
and the integral of the product of the two sine functions vanishes. Together with cosine functions, these orthogonal functions may be assembled into a trigonometric polynomial to approximate a given function on the interval with its Fourier series.
Polynomials
If one begins with the monomial sequence on the interval and applies the Gram–Schmidt pr
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https://en.wikipedia.org/wiki/Sharif%20University%20of%20Technology
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Sharif University of Technology (SUT; ) is a public research university in Tehran, Iran. It is widely considered as the nation's most prestigious and leading institution for science, technology, engineering, and mathematics (STEM) fields.
Established in 1966 under the reign of Mohammad Reza Shah Pahlavi, it was formerly named the Aryamehr University of Technology () and for a short period after the 1979 revolution, the university was called Tehran University of Technology. Following the revolution, the university was named after Majid Sharif Vaghefi.
Today, the university provides both undergraduate and graduate programs in 15 main departments. The student body consists of about 6,000 undergraduate students and 4,700 graduate students from all the 31 provinces of Iran. Funding for Sharif University is provided by the government and through private funding. Undergraduate admission to Sharif is limited to the top 800 of the 500,000 students who pass the national entrance examination administered annually by the Iranian Ministry of Science, Research and Technology.
In the 2013 Academic Ranking of World Universities Engineering/Technology and Computer Sciences rankings, SUT was ranked 5th in the Middle East. It is in the top 251–275 universities in the world and 37th in Asia in the 2014 Times Higher Education World University Rankings. In the 2014 Times Higher Education top 100 for newer universities (less than 50 years old), SUT ranked 1st in the Middle East, 6th in Asia, and
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https://en.wikipedia.org/wiki/Transport%20%28disambiguation%29
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Transport is the movement of people or goods from place to place.
Transport may also refer to:
Related terms
Especially in military contexts, a vehicle used to carry supplies or personnel, e.g. transport aircraft (disambiguation) or troopship
Transport industry
Penal transportation, also known as "sentence to transport"
Biology and medicine
Movement of molecules or ions across cell membranes, including active transport and passive transport; see also secretion
Movement of electrons in electron transport chains
Movement of blood and other bodily fluids in the circulatory system
Geology and earth science
Movement of products of erosion, e.g. by a river, prior to their deposition as a sedimentary rock
Physics and technology
Transport phenomena, in physics, mechanisms by which particles or quantities move from one place to another
In computer networking, the function of issuing and responding to service requests in transport layers and associated "transport protocols"
Transport (recording), a device that handles a storage medium and extracts or records the information from and to it
MPEG transport stream, a communications protocol for audio, video, and data
Transport (SAP), a process of moving some or all the modifications from one SAP installation to another
Transport layer, the fourth layer of the OSI model for networking
Other uses
Transport Canada, a department of the government of Canada
Transport (typeface), a typeface used on British, Italian, Span
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https://en.wikipedia.org/wiki/Parametrization%20%28geometry%29
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In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters".
Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.
For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed origin. If are the coordinates of the point, the movement is thus described by a parametric equation
where is the parameter and denotes the time. Such a parametric equation completely determines the curve, without the need of any interpretation of as time, and is thus called a parametric equation of the curve (this is sometimes abbreviated by saying that one has a parametric curve). One similarly gets the parametric equation of a surface by considering functions of two parameters and .
Non-uniquen
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https://en.wikipedia.org/wiki/Glossary%20of%20order%20theory
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This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
completeness properties of partial orders
distributivity laws of order theory
preservation properties of functions between posets.
In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by
A
Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric.
Adjoint. See Galois connection.
Alexandrov topology. For a preordered set P, any upper set O is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open.
Algebraic poset. A poset is algebraic if it has a base of compact elements.
Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation.
Approximates relation. See way-below relation.
Antisymmetric relation. A homogeneous relation R on a set X is antisymmetric, if x R y and y R x implies x = y, for a
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https://en.wikipedia.org/wiki/Tagged%20union
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In computer science, a tagged union, also called a variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixed, types. Only one of the types can be in use at any one time, and a tag field explicitly indicates which one is in use. It can be thought of as a type that has several "cases", each of which should be handled correctly when that type is manipulated. This is critical in defining recursive datatypes, in which some component of a value may have the same type as that value, for example in defining a type for representing trees, where it is necessary to distinguish multi-node subtrees and leaves. Like ordinary unions, tagged unions can save storage by overlapping storage areas for each type, since only one is in use at a time.
Description
Tagged unions are most important in functional programming languages such as ML and Haskell, where they are called datatypes (see algebraic data type) and the compiler is able to verify that all cases of a tagged union are always handled, avoiding many types of errors. Compile-time checked sum types are also extensively used in Rust, where they are called enum. They can, however, be constructed in nearly any programming language, and are much safer than untagged unions, often simply called unions, which are similar but do not explicitly track which member of a union is in use currently.
Tagged unions are often a
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https://en.wikipedia.org/wiki/Harold%20Urey
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Harold Clayton Urey ( ; April 29, 1893 – January 5, 1981) was an American physical chemist whose pioneering work on isotopes earned him the Nobel Prize in Chemistry in 1934 for the discovery of deuterium. He played a significant role in the development of the atom bomb, as well as contributing to theories on the development of organic life from non-living matter.
Born in Walkerton, Indiana, Urey studied thermodynamics under Gilbert N. Lewis at the University of California, Berkeley. After he received his PhD in 1923, he was awarded a fellowship by the American-Scandinavian Foundation to study at the Niels Bohr Institute in Copenhagen. He was a research associate at Johns Hopkins University before becoming an associate professor of chemistry at Columbia University. In 1931, he began work with the separation of isotopes that resulted in the discovery of deuterium.
During World War II, Urey turned his knowledge of isotope separation to the problem of uranium enrichment. He headed the group located at Columbia University that developed isotope separation using gaseous diffusion. The method was successfully developed, becoming the sole method used in the early post-war period. After the war, Urey became professor of chemistry at the Institute for Nuclear Studies, and later Ryerson professor of chemistry at the University of Chicago.
Urey speculated that the early terrestrial atmosphere was composed of ammonia, methane, and hydrogen. One of his Chicago graduate students was St
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https://en.wikipedia.org/wiki/Strangeness
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In particle physics, strangeness ("S") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a particle is defined as:
where n represents the number of strange quarks () and n represents the number of strange antiquarks (). Evaluation of strangeness production has become an important tool in search, discovery, observation and interpretation of quark–gluon plasma (QGP). Strangeness is an excited state of matter and its decay is governed by CKM mixing.
The terms strange and strangeness predate the discovery of the quark, and were adopted after its discovery in order to preserve the continuity of the phrase: strangeness of particles as −1 and anti-particles as +1, per the original definition. For all the quark flavour quantum numbers (strangeness, charm, topness and bottomness) the convention is that the flavour charge and the electric charge of a quark have the same sign. With this, any flavour carried by a charged meson has the same sign as its charge.
Conservation
Strangeness was introduced by Murray Gell-Mann, Abraham Pais, Tadao Nakano and Kazuhiko Nishijima to explain the fact that certain particles, such as the kaons or the hyperons and , were created easily in particle collisions, yet decayed much more slowly than expected for their large masses and large production cross sections. Noting that collisions seemed to always produc
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https://en.wikipedia.org/wiki/127%20%28number%29
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127 (one hundred [and] twenty-seven') is the natural number following 126 and preceding 128. It is also a prime number.
In mathematics
As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known Mersenne prime exponent for a Mersenne number, , which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years.
is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime.
Furthermore, 127 is equal to , and 7 is equal to , and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime.
There are a total of 127 prime numbers between 2,000 and 3,000.
127 is also a cuban prime of the form , . The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime. 127 is greater than the arithmetic mean of its two neighboring primes; thus, it is a strong prime.
127 is a centered hexagonal number.
It is the seventh Motzkin number.
127 is a palindromic prime in nonary and binary.
127 is the first Friedman prime in decimal. It is also the first nice Friedman number in decimal, since , as well as binary since .
127 is the sum of the sums of the divisors of the first twelve positive integers.
127 is the smallest prime that can be written as the sum of the first two or more odd primes: .
127 is the smallest odd
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https://en.wikipedia.org/wiki/Exact%20functor
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In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
Definitions
Let P and Q be abelian categories, and let be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever
is a short exact sequence in P then
is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→A→B→C→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".)
Further, we say that F is
left-exact if whenever 0→A→B→C→0 is exact then 0→F(A)→F(B)→F(C) is exact;
right-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C) is exact. This is distinct from the notion of a topological half-exact functor.
If G is a contravariant additive functor from P to Q, we similarly define G to be
exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A)→0 is exact;
left-exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A) is exact;
right-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A)→0 is exact;
half-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A) is exact.
It is not always necessary to start with an enti
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https://en.wikipedia.org/wiki/Integral%20test%20for%20convergence
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In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
Statement of the test
Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series
converges to a real number if and only if the improper integral
is finite. In particular, if the integral diverges, then the series diverges as well.
Remark
If the improper integral is finite, then the proof also gives the lower and upper bounds
for the infinite series.
Note that if the function is increasing, then the function is decreasing and the above theorem applies.
Proof
The proof basically uses the comparison test, comparing the term with the integral of over the intervals
and , respectively.
The monotonous function is continuous almost everywhere. To show this, let . For every , there exists by the density of a so that . Note that this set contains an open non-empty interval precisely if is discontinuous at . We can uniquely identify as the rational number that has the least index in an enumeration and satisfies the above property. Since is monotone, this defines an injective mapping and thus is countable. It follows that is continuous almost everywhere. This is sufficient for Riemann integrability.
Since is a monotone decreasing function, we know tha
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https://en.wikipedia.org/wiki/Truthmaker%20theory
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Truthmaker theory is "the branch of metaphysics that explores the relationships between what is true and what exists". The basic intuition behind truthmaker theory is that truth depends on being. For example, a perceptual experience of a green tree may be said to be true because there actually is a green tree. But if there was no tree there, it would be false. So the experience by itself does not ensure its truth or falsehood, it depends on something else. Expressed more generally, truthmaker theory is the thesis that "the truth of truthbearers depends on the existence of truthmakers". A perceptual experience is the truthbearer in the example above. Various representational entities, like beliefs, thoughts or assertions can act as truthbearers. Truthmaker theorists are divided about what type of entity plays the role of truthmaker; popular candidates include states of affairs and tropes.
Truthmaker maximalism is the thesis that every truth has a truthmaker. An alternative view is truthmaker atomism, the thesis that only atomic sentences have truthmakers. Truthmaker atomism remains true to the basic intuition that truth depends on being by holding that the truth of molecular sentences depends on the truth of atomic sentences, whose truth in turn depends on being. All non-maximalist positions accept that there are truthmaker gaps: truths without truthmakers. Opponents have tried to disprove truthmaker theory by showing that there are so-called deep truthmaker gaps: truthbearer
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https://en.wikipedia.org/wiki/Charles%20Friedel
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Charles Friedel (; 12 March 1832 – 20 April 1899) was a French chemist and mineralogist.
Life
A native of Strasbourg, France, he was a student of Louis Pasteur at the Sorbonne. In 1876, he became a professor of chemistry and mineralogy at the Sorbonne.
Friedel developed the Friedel-Crafts alkylation and acylation reactions with James Crafts in 1877, and attempted to make synthetic diamonds.
His son Georges Friedel (1865–1933) also became a renowned mineralogist.
Lineage
Friedel's wife's father was the engineer, Charles Combes. The Friedel family is a rich lineage of French scientists:
Georges Friedel (1865–1933), French crystallographer and mineralogist; son of Charles
Edmond Friedel (1895–1972), French Polytechnician and mining engineer, founder of BRGM, the French geological survey; son of Georges
Jacques Friedel (1921–2014), French physicist; son of Edmond
References
Further reading
External links
Charles Friedel
1832 births
1899 deaths
Scientists from Strasbourg
Academic staff of the University of Paris
19th-century French chemists
French mineralogists
Members of the French Academy of Sciences
Corresponding members of the Saint Petersburg Academy of Sciences
University of Paris alumni
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https://en.wikipedia.org/wiki/Michel%20Plancherel
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Michel Plancherel (16 January 1885 – 4 March 1967) was a Swiss mathematician. He was born in Bussy (Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribourg and then his doctoral degree in 1907 with a thesis written under the supervision of Mathias Lerch. Plancherel was a professor in Fribourg (1911), and from 1920 at ETH Zurich.
He worked in the areas of mathematical analysis, mathematical physics and algebra, and is known for the Plancherel theorem in harmonic analysis. He was an Invited Speaker of the ICM in 1924 at Toronto and in 1928 at Bologna.
He was married to Cécile Tercier, had nine children, and presided at the Mission Catholique Française in Zürich.
References
External links
Short biography, Department of mathematics, University of Fribourg
1885 births
1967 deaths
20th-century Swiss mathematicians
Swiss Roman Catholics
Academic staff of ETH Zurich
University of Fribourg alumni
Academic staff of the University of Fribourg
People from the canton of Fribourg
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https://en.wikipedia.org/wiki/Plancherel%20theorem
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In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if is a function on the real line, and is its frequency spectrum, then
A more precise formulation is that if a function is in both Lp spaces and , then its Fourier transform is in , and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to has a unique extension to a linear isometric map , sometimes called the Plancherel transform. This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions.
Plancherel's theorem remains valid as stated on n-dimensional Euclidean space . The theorem also holds more generally in locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of non-commutative harmonic analysis.
The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series.
Due to the polarization identity, one can also a
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https://en.wikipedia.org/wiki/List%20of%20important%20publications%20in%20computer%20science
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This is a list of important publications in computer science, organized by field. Some reasons why a particular publication might be regarded as important:
Topic creator – A publication that created a new topic
Breakthrough – A publication that changed scientific knowledge significantly
Influence – A publication which has significantly influenced the world or has had a massive impact on the teaching of computer science.
Artificial intelligence
Computing Machinery and Intelligence
Alan Turing
Mind, 59:433–460, 1950.
Online copy
Description: This paper discusses the various arguments on why a machine can not be intelligent and asserts that none of those arguments are convincing. The paper also suggested the Turing test, which it calls "The Imitation Game" as according to Turing it is pointless to ask whether or not a machine can think intelligently, and checking if it can act intelligently is sufficient.
A Proposal for the Dartmouth Summer Research Project on Artificial Intelligence
John McCarthy
Marvin Minsky
N. Rochester
C.E. Shannon
Online copy
Description: This summer research proposal inaugurated and defined the field. It contains the first use of the term artificial intelligence and this succinct description of the philosophical foundation of the field: "every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it." (See philosophy of AI) The proposal invited researchers to
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https://en.wikipedia.org/wiki/William%20Elford%20Leach
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William Elford Leach FRS (2 February 1791 – 25 August 1836) was an English zoologist and marine biologist.
Life and work
Elford Leach was born at Hoe Gate, Plymouth, the son of an attorney. At the age of twelve he began a medical apprenticeship at the Devonshire and Exeter Hospital, studying anatomy and chemistry. By this time he was already collecting marine animals from Plymouth Sound and along the Devon coast. At seventeen he began studying medicine at St Bartholomew's Hospital in London, finishing his training at the University of Edinburgh before graduating MD from the University of St Andrews (where he had never studied).
From 1813 Leach concentrated on his zoological interests and was employed as an 'Assistant Librarian' (what would later be called Assistant Keeper) in the Natural History Department of the British Museum, where he had responsibility for the zoological collections. Here he threw himself into the task of reorganising and modernising these collections, many of which had been neglected since Hans Sloane left them to the nation. In 1815, he published the first bibliography of entomology in Brewster's Edinburgh Encyclopedia (see Timeline of entomology – 1800–1850). He also worked and published on other invertebrates, amphibians, reptiles, mammals and birds. and was the naturalist who separated the centipedes and millipedes from the insects, giving them their own group, the Myriapoda. In his day he was the world's leading expert on the Crustacea and was
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https://en.wikipedia.org/wiki/Abstraction%20%28mathematics%29
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Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two of the most highly abstract areas of modern mathematics are category theory and model theory.
Description
Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic.
Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. For example, the first steps in the abstraction of geometry were historically made by the ancient Greeks, with Euclid's Elements being the earliest extant documentation of the axioms of plane geometry—though Proclus tells of an earlier axiomatisation by Hippocrates of Chios. In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Eu
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https://en.wikipedia.org/wiki/Analytical%20mechanics
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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.
By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation.
Analytical mechanics takes advantage of a system's constraints to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics a
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https://en.wikipedia.org/wiki/Nine%20lemma
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In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.
The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).
Linderholm (p. 201) offers a satirical view of the nine lemma:
"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines. Make faces. You have now proved:
(a) the Nine Lemma
(b) the Sixteen Lemma
(c) the Twenty-five Lemma..."
There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of ).
References
Homological algebra
Lemmas in category theory
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https://en.wikipedia.org/wiki/Archie%27s%20Weird%20Mysteries
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Archie's Weird Mysteries (French: Archie, mystères et compagnie) is an animated television series based on the characters by Archie Comics. The series premise revolves around a Riverdale High physics lab gone awry, making the town of Riverdale a "magnet" for B movie-style monsters. All the main characters solve strange mysteries in a format similar to both Scooby-Doo and The X-Files.
Produced by Les Studios Tex and DIC Productions, L.P., the series was initially shown mornings on the PAX network premiering on 2 October 1999, often with infomercials bookending the program. The series eventually premiered in France on M6 on 19 January 2000.
Voice cast and characters
Main
Andrew Rannells as Archie Andrews – a redheaded, freckled student at Riverdale High, and a reporter for the school newspaper. All of his news stories are centered around the surreal occurrences that take place within each episode and always end with "...in a little town called Riverdale." Despite his clumsiness and rash decision-making, he frequently attracts girls, human or otherwise.
America Young as Betty Cooper – a literal girl-next-door who is intelligent, comely, cute, level-headed, caring, and the head of the cheerleading squad. She is best friends with Veronica, despite them both competing for Archie's affection. She has bright blonde hair that's always worn in a ponytail.
Camille Schmidt as Veronica Lodge – a beautiful, rich, popular, and somewhat self-centered girl. She is best friends with Bett
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https://en.wikipedia.org/wiki/1740%20in%20science
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The year 1740 in science and technology involved some significant events.
Mathematics
Jean Paul de Gua de Malves publishes his work of analytic geometry, .
Metallurgy
Benjamin Huntsman develops the technique of crucible steel production at Handsworth, South Yorkshire, England.
Physics
Jacques-Barthélemy Micheli du Crest creates a spirit thermometer, making use of two fixed points, 0 for "Temperature of earth" based on a cave at Paris Observatory and 100 for the heat of boiling water.
Émilie du Châtelet publishes Institutions de Physique, including a demonstration that the energy of a moving object is proportional to the square of its velocity (Ek = mv²).
Louis Bertrand Castel publishes L'Optique des couleurs in Paris, including the observation that the colours of white light split by a prism depend on distance from the prism.
Technology
Henry Hindley of Yorkshire invents a device to cut the teeth of clock wheels.
Awards
Copley Medal: Alexander Stuart
Births
February 17 – Horace Bénédict de Saussure, Genevan pioneer of Alpine studies (died 1799)
March 28 (bapt.) – James Small, Scottish inventor (died 1793)
June 27 – John Latham, English physician and naturalist, "grandfather of Australian ornithology" (died 1837)
July 1 – Franz-Joseph Müller von Reichenstein, Austrian mineralogist and discoverer of tellurium (died 1825)
August 26 – Joseph Michel Montgolfier, French pioneer balloonist (died 1810)
September 29 – Thomas Percival, English reforming physician and
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https://en.wikipedia.org/wiki/Category%20of%20abelian%20groups
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In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object of Ab is the trivial group {0} which consists only of its neutral element.
The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
Ab is a full subcategory of Grp, the category of all groups. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism:
(f+g)(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y)
= f(x) + g(x) + f(y) + g(y) = (f+g)(x) + (f+g)(y)
The third equality requires the group to be abelian. This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category.
In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism f : A → B is the subgroup K of A defined by K = {x ∈ A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp:
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https://en.wikipedia.org/wiki/BGI%20Group
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BGI Group, formerly Beijing Genomics Institute, is a Chinese genomics company with headquarters in Yantian District, Shenzhen. The company was originally formed in 1999 as a genetics research center to participate in the Human Genome Project. It also sequences the genomes of other animals, plants and microorganisms.
BGI has transformed from a small research institute, notable for decoding the DNA of pandas and rice plants, into a diversified company active in animal cloning, health testing, and contract research. BGI's earlier research was continued by the Beijing Institute of Genomics, Chinese Academy of Sciences. BGI Research, the group's nonprofit division, works with the Institute of Genomics and operates the China National GeneBank under a contract with the Chinese government. BGI Genomics, a subsidiary, was listed on the Shenzhen Stock Exchange in 2017.
In 2021, details came to light about multiple controversies involving the BGI Group. These controversies include alleged collaboration with the People's Liberation Army (PLA) and use of genetic data from prenatal tests. BGI denied that it shares prenatal genetics data with the PLA.
History
Wang Jian, Yu Jun, Yang Huanming and Liu Siqi created BGI, originally named Beijing Genomics Institute, in September 1999, in Beijing, China as a non-governmental independent research institute in order to participate in the Human Genome Project as China's representative. After the project was completed, funding dried up, after whic
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https://en.wikipedia.org/wiki/Set
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Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics
Mathematics
Set (mathematics), a collection of elements
Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electronics and computing
Set (abstract data type), a data type in computer science that is a collection of unique values
Set (C++), a set implementation in the C++ Standard Library
Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems
Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks
Single-electron transistor, a device to amplify currents in nanoelectronics
Single-ended triode, a type of electronic amplifier
Set!, a programming syntax in the scheme programming language
Biology and psychology
Set (psychology), a set of expectations which shapes perception or thought
Set or sett, a badger's den
Set, a small tuber or bulb used instead of seed, especially:
Potato set
Onion set
SET (gene), gene for a human protein involved in apoptosis, transcription and nucleosome assembly
Single Embryo Transfer, used in in vitro fertilization
Physics and chemistry
A chemical change in an adhesive from unbonded to bonded
Set, to make/become solid; see Solidification
Stress–energy tensor, a physical quantity in the theory of fields
Single electron transfer
Other uses in science and technology
Saw set, the process of setting the t
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https://en.wikipedia.org/wiki/Positive%20linear%20functional
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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements that is it holds that
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.
When is a complex vector space, it is assumed that for all is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace and the partial order does not extend to all of in which case the positive elements of are the positive elements of by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
Sufficient conditions for continuity of all positive linear functionals
There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.
This includes all topological vector lattices that are sequentially complete.
Theorem Let be an Ordered topological vector
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https://en.wikipedia.org/wiki/Positive%20operator%20%28Hilbert%20space%29
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In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every , and , where is the domain of . Positive-semidefinite operators are denoted as . The operator is said to be positive-definite, and written , if for all .
In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism.
Cauchy–Schwarz inequality
If then
Indeed, let Applying Cauchy–Schwarz inequality to the inner product
as proves the claim.
It follows that If is defined everywhere, and then
On a complex Hilbert space, if A ≥ 0 then A is symmetric
Without loss of generality, let the inner product be anti-linear on the first argument and linear on the second. (If the reverse is true, then we work with instead). For the polarization identity
and the fact that for positive operators, show that so is symmetric.
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define to be an operator of rotation by an acute angle Then but so is not symmetric.
If A ≥ 0 and Dom A = , then A is self-adjoint and bounded
The symmetry of implies that and For to be self-adjoint, it is necessary that In our case, the equality of domains holds because so is indeed self-adjoint. The fact that is bounded now
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https://en.wikipedia.org/wiki/State%20%28functional%20analysis%29
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In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both . Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M .
In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number).
Jordan decomposition
States can be viewed as noncommutative generalizations of probability measures. By Gelfand representation, every commutative C*-algebra A is of the form C0(X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the are the evaluation functionals on X.
More generally, the GNS construction shows that every state is, after choosing a suitable representation, a vector state.
A bounded linear functional on a C*-algebra A is said to be self-adjoint if it is real-valued on the self-adjoint elements of A. Self-adjoint functionals are noncommutative analogues of signe
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https://en.wikipedia.org/wiki/Extreme%20point
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In mathematics, an extreme point of a convex set in a real or complex vector space is a point in that does not lie in any open line segment joining two points of In linear programming problems, an extreme point is also called vertex or corner point of
Definition
Throughout, it is assumed that is a real or complex vector space.
For any say that and if and there exists a such that
If is a subset of and then is called an of if it does not lie between any two points of That is, if there does exist and such that and The set of all extreme points of is denoted by
Generalizations
If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of A 0-dimensional support variety is called an extreme point of
Characterizations
The of two elements and in a vector space is the vector
For any elements and in a vector space, the set is called the or between and The or between and is when while it is when The points and are called the of these interval. An interval is said to be a or a if its endpoints are distinct. The is the midpoint of its endpoints.
The closed interval is equal to the convex hull of if (and only if) So if is convex and then
If is a nonempty subset of and is a nonempty subset of then is called a of if whenever a point lies betwe
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https://en.wikipedia.org/wiki/160%20%28number%29
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160 (one hundred [and] sixty) is the natural number following 159 and preceding 161.
In mathematics
160 is the sum of the first 11 primes, as well as the sum of the cubes of the first three primes.
Given 160, the Mertens function returns 0. 160 is the smallest number n with exactly 12 solutions to the equation φ(x) = n.
In telecommunications
The number of characters permitted in a standard short message service
The number for Dial-a-Disc (1966–1991), a telephone number operated by the General Post Office in the United Kingdom, which enabled callers to hear the latest chart hits
See also
160s
List of highways numbered 160
United Nations Security Council Resolution 160
United States Supreme Court cases, Volume 160
Article 160 of the Constitution of Malaysia
Norris School District 160, Lancaster County, Nebraska
References
External links
Number Facts and Trivia: 160
Integers
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https://en.wikipedia.org/wiki/170%20%28number%29
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170 (one hundred [and] seventy) is the natural number following 169 and preceding 171.
In mathematics
170 is the smallest n for which φ(n) and σ(n) are both square (64 and 324 respectively). But 170 is never a solution for φ(x), making it a nontotient. Nor is it ever a solution to x - φ(x), making it a noncototient.
170 is a repdigit in base 4 (2222) and base 16 (AA), as well as in bases 33, 84, and 169. It is also a sphenic number.
170 is the largest integer for which its factorial can be stored in IEEE 754 double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306.
There are 170 different cyclic Gilbreath permutations on 12 elements, and therefore there are 170 different real periodic points of order 12 on the Mandelbrot set.
See also
170s
E170 (disambiguation)
F170 (disambiguation)
List of highways numbered 170
United States Supreme Court cases, Volume 170
United Nations Security Council Resolution 170
Pennsylvania House of Representatives, District 170
References
External links
The Number 170
Number Facts and Trivia: 170
The Positive Integer 170
Prime curiosities: 170
Integers
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https://en.wikipedia.org/wiki/180%20%28number%29
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180 (one hundred [and] eighty) is the natural number following 179 and preceding 181.
In mathematics
180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number.
180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3.
180 is a 61-gonal number, while 61 is the 18th prime number.
Half a circle has 180 degrees, and thus a U-turn is also referred to as a 180.
Summing Euler's totient function φ(x) over the first + 24 integers gives 180.
In binary it is a digitally balanced number, since its binary representation has the same number of zeros as ones (10110100).
A triangle has three interior angles that collectively total 180 degrees. In general, the interior angles of an -sided polygon add to degrees.
In religion
The Book of Genesis says that Isaac died at the age of 180.
Other
180 is the highest s
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https://en.wikipedia.org/wiki/190%20%28number%29
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190 (one hundred [and] ninety) is the natural number following 189 and preceding 191.
In mathematics
190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties. It is also a truncated square pyramid number.
Integers from 191 to 199
191
191 is a prime number.
192
192 = 26 × 3 is a 3-smooth number, the smallest number with 14 divisors.
193
193 is a prime number.
194
194 = 2 × 97 is a Markov number, the smallest number written as the sum of three squares in five ways, and the number of irreducible representations of the Monster group.
195
195 = 3 × 5 × 13 is the smallest number expressed as a sum of distinct squares in 16 different ways.
196
196 = 22 × 72 is a square number.
197
197 is a prime number and a Schröder–Hipparchus number.
198
198 = 2 × 32 × 11 is the smallest number written as the sum of four squares in ten ways.
No integer factorial ever ends in exactly 198 zeroes in base 10 or in base 12.
There are 198 ridges on a U.S. dollar coin.
199
199 is a prime number and a centered triangular number.
In other fields
190 is the telephonic number of the 27 Brazilian Military Polices.
See also
190 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/M.%20King%20Hubbert
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Marion King Hubbert (October 5, 1903 – October 11, 1989) was an American geologist and geophysicist. He worked at the Shell research lab in Houston, Texas. He made several important contributions to geology, geophysics, and petroleum geology, most notably the Hubbert curve and Hubbert peak theory (a basic component of peak oil), with important political ramifications. He was often referred to as "M. King Hubbert" or "King Hubbert".
Biography
Hubbert was born in San Saba, Texas. He attended the University of Chicago, where he received a Bachelor of Science in 1926, a Master of Science in 1928, and a Doctor of Philosophy in 1937, studying geology, mathematics, and physics. He worked as an assistant geologist for the Amerada Petroleum Company for two years while pursuing the PhD, additionally teaching geophysics at Columbia University. He also served as a senior analyst at the Board of Economic Warfare. He joined the Shell Oil Company in 1943, retiring from that firm in 1964. After he retired from Shell, he became a senior research geophysicist for the United States Geological Survey until his retirement in 1976. He also held positions as a professor of geology and geophysics at Stanford University from 1963 to 1968, and as a professor at UC Berkeley from 1973 to 1976.
Hubbert was an avid technocrat. He co-founded Technocracy Incorporated with Howard Scott. Hubbert wrote a study course that was published without attribution called the Technocracy Study Course, which advoca
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https://en.wikipedia.org/wiki/Sensitivity%20%28electronics%29
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The sensitivity of an electronic device, such as a communications system receiver, or detection device, such as a PIN diode, is the minimum magnitude of input signal required to produce a specified output signal having a specified signal-to-noise ratio, or other specified criteria.
In signal processing, sensitivity also relates to bandwidth and noise floor.
Sensitivity is sometimes improperly used as a synonym for responsivity.
Electroacoustics
The sensitivity of a microphone is usually expressed as the sound field strength in decibels (dB) relative to 1 V/Pa (Pa = N/m2) or as the transfer factor in millivolts per pascal (mV/Pa) into an open circuit or into a 1 kiloohm load.
The sensitivity of a loudspeaker is usually expressed as dB / 2.83 VRMS at 1 metre. This is not the same as the electrical efficiency; see Efficiency vs sensitivity.
The sensitivity of a hydrophone is usually expressed as dB re 1 V/μPa.
Receivers
Sensitivity in a receiver, such a radio receiver, indicates its capability to extract information from a weak signal, quantified as the lowest signal level that can be useful. It is mathematically defined as the minimum input signal required to produce a specified signal-to-noise S/N ratio at the output port of the receiver and is defined as the mean noise power at the input port of the receiver times the minimum required signal-to-noise ratio at the output of the receiver:
where
= sensitivity [W]
= Boltzmann constant
= equivalent noise temperatur
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https://en.wikipedia.org/wiki/Data%20engineering
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Data engineering refers to the building of systems to enable the collection and usage of data. This data is usually used to enable subsequent analysis and data science; which often involves machine learning. Making the data usable usually involves substantial compute and storage, as well as data processing
History
Around the 1970s/1980s the term information engineering methodology (IEM) was created to describe database design and the use of software for data analysis and processing. These techniques were intended to be used by database administrators (DBAs) and by systems analysts based upon an understanding of the operational processing needs of organizations for the 1980s. In particular, these techniques were meant to help bridge the gap between strategic business planning and information systems. A key early contributor (often called the "father" of information engineering methodology) was the Australian Clive Finkelstein, who wrote several articles about it between 1976 and 1980, and also co-authored an influential Savant Institute report on it with James Martin. Over the next few years, Finkelstein continued work in a more business-driven direction, which was intended to address a rapidly changing business environment; Martin continued work in a more data processing-driven direction. From 1983 to 1987, Charles M. Richter, guided by Clive Finkelstein, played a significant role in revamping IEM as well as helping to design the IEM software product (user data), which help
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https://en.wikipedia.org/wiki/Guram%20Mchedlidze
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Guram I. Mchedlidze (Georgian: გურამ ი. მჭედლიძე) (September 27, 1931, Tbilisi – 2009, Tbilisi) was a Georgian Palaeobiologist, Corresponding Member of the Georgian National Academy of Sciences (GNAS), Doctor of Biological Sciences (Dr. Habil.), Professor.
Education and career
In 1954 he graduated from the Faculty of Biology of the Tbilisi State University (TSU). Since 1973 he was Professor of this Faculty.
In 1962 Mchedlidze received a PhD degree in Biology, in 1973 a degree of the Doctor of Biological Sciences. In 1983 he was elected as Corresponding Member of the Georgian National Academy of Sciences (GNAS).
In 1979-1989 he was a Deputy Director of the L. Davitashvili Institute of Palaeobiology, in 1989-2009 a Director of this Institute.
In 1971 he was elected as a Fellow of the American Society of Paleontology.
Main fields of scientific activity of Guram Mchedlidze were: a fossil dolphin, Tertiary cetaceans, phylogenesis of cetaceans, ancient mammals, etc. He was author of more than 100 scientific-research works (among them 5 monographs).
Mchedlidze was organizer and participant of many important scientific events in Georgia and abroad.
Some of main scientific works of Guram Mchedlidze
"Fossil Cetacea of the Caucasus" (a monograph), Publishing House "Metsniereba", Tbilisi, 1964, 145 pp. (in Russian, Georgian and English summaries)
"Some general features of the historic development of cetaceans" (a monograph), Publishing House "Metsniereba", 1970, 112 pp. (in Russ
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https://en.wikipedia.org/wiki/Henry%20Gilman
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Henry Gilman (May 9, 1893 – November 7, 1986) was an American organic chemist known as the father of organometallic chemistry, the field within which his most notable work was done. He discovered the Gilman reagent, which bears his name.
Early life and education (1893-1918)
Henry Gilman was born in Boston, Massachusetts, as the son of a tailor. He was the third of eight children. Gilman graduated from a Boston high school and later attended Harvard University where he graduated summa cum laude with a Bachelor of Science degree in 1915. During his final year as an undergraduate at Harvard, Gilman researched with Roger Adams. During this time, the two worked on the synthesis of substituted phenyl esters of oxalic acids. Gilman worked hard on his research describing it as 'a sheer delight' and often worked until midnight 'without any compulsion-just for the joy of it' and claimed this was an important step toward his interest in research. After undergraduate work Gilman was invited to stay for graduate work with the head of the Harvard department of chemistry, E.P. Kohler. Based on his work, he received a Master of Arts degree in 1917(a year late) and a PhD in 1918.
While attending graduate school, Gilman had an opportunity to travel in Europe as a recipient of the Sheldon Fellowship. He spent time at both the Polytechnicum in Zurich and at Oxford in England. During his time in Europe, Gilman met Madame Curie at the Sorbonne, the historic University of Paris. While staying in
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https://en.wikipedia.org/wiki/Rigid%20body
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In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.
In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).
Kinematics
Linear and angular position
The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by:
the linear posi
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https://en.wikipedia.org/wiki/Von%20Neumann%20bicommutant%20theorem
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In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory.
The formal statement of the theorem is as follows:
Von Neumann bicommutant theorem. Let be an algebra consisting of bounded operators on a Hilbert space , containing the identity operator, and closed under taking adjoints. Then the closures of in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant of .
This algebra is called the von Neumann algebra generated by .
There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies.
It is related to the Jacobson density theorem.
Proof
Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains t
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https://en.wikipedia.org/wiki/Operator%20algebra
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In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
Overview
Operator algebras can be used to study arbitrary sets of operators with little algebraic relation simultaneously. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings.
An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of continuous linear operators. In particular, it is a set of operators with both algebraic and topological closure properties. In some disciplines such properties are axiomized and algebras with certain topological structure become the subject of the research.
Though algebras of operators are studied in various contexts (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to alge
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https://en.wikipedia.org/wiki/Incomplete%20gamma%20function
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In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals.
Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.
Definition
The upper incomplete gamma function is defined as:
whereas the lower incomplete gamma function is defined as:
In both cases is a complex parameter, such that the real part of is positive.
Properties
By integration by parts we find the recurrence relations
and
Since the ordinary gamma function is defined as
we have
and
Continuation to complex values
The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive and , can be developed into holomorphic functions, with respect both to and , defined for almost all combinations of complex and . Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.
Lower incomplete gamma function
Holomorphic extension
Repeated application of the recurrence relation for the lower
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https://en.wikipedia.org/wiki/Gaussian%20period
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In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods.
History
As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which
is an example involving the seventeenth root of unity
General definition
Given an integer n > 1, let H be any subgroup of the multiplicative group
of invertible residues modulo n, and let
A Gaussian period P is a sum of the primitive n-th roots of unity , where runs through all of the elements in a fixed coset of H in G.
The definition of P can also be stated in terms of the field trace. We have
for some subfield L of Q(ζ) and some j coprime to n. This corresponds to the previous definition by identifying G and H with the Galois groups of Q(ζ)/Q and Q(ζ)/L, respectively. The choice of j determines the choice of coset of H in G in the previous definition.
Example
The situation is simplest when n is a prime number p > 2. In that case G is cyclic of order p − 1, and h
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https://en.wikipedia.org/wiki/496%20%28number%29
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496 (four hundred [and] ninety-six) is the natural number following 495 and preceding 497.
In mathematics
496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.
A triangular number and a hexagonal number, 496 is also a centered nonagonal number. Being the 31st triangular number, 496 is the smallest counterexample to the hypothesis that one more than an even triangular prime-indexed number is a prime number. It is the largest happy number less than 500.
There is no solution to the equation φ(x) = 496, making 496 a nontotient.
E8 has real dimension 496.
In physics
The number 496 is a very important number in superstring theory. In 1984, Michael Green and John H. Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
Telephone numbers
The UK's Ofcom rese
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https://en.wikipedia.org/wiki/Organic%20geochemistry
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Organic geochemistry is the study of the impacts and processes that organisms have had on the Earth. It is mainly concerned with the composition and mode of origin of organic matter in rocks and in bodies of water. The study of organic geochemistry is traced to the work of Alfred E. Treibs, "the father of organic geochemistry." Treibs first isolated metalloporphyrins from petroleum. This discovery established the biological origin of petroleum, which was previously poorly understood. Metalloporphyrins in general are highly stable organic compounds, and the detailed structures of the extracted derivatives made clear that they originated from chlorophyll.
Applications
Energy
Petroleum
The relationship between the occurrence of organic compounds in sedimentary deposits and petroleum deposits has long been of interest. Studies of ancient sediments and rock provide insights into the origins and sources of oil and petroleum, as well as the biochemical antecedents of life. Oil spills in particular have been of interest to geochemists in regards to the impact of petroleum and oil on the current geological environment. Following the Exxon Valdez Oil Spill, organic geochemistry knowledge on oil-spill chemistry bloomed with the analyses of samples from the spill.
Geochemists study petroleum-inclusions in geological samples to compare present-day fluid-inclusions to dated samples. This analysis provides insight into the age of the petroleum samples and the surrounding rock. Spectrog
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https://en.wikipedia.org/wiki/Colligative%20properties
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In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present.<ref>McQuarrie, Donald, et al. Colligative properties of Solutions" General Chemistry Mill Valley: Library of Congress, 2011. .</ref> The number ratio can be related to the various units for concentration of a solution such as molarity, molality, normality (chemistry), etc. The assumption that solution properties are independent of nature of solute particles is exact only for ideal solutions, which are solutions that exhibit thermodynamic properties analogous to those of an ideal gas, and is approximate for dilute real solutions. In other words, colligative properties are a set of solution properties that can be reasonably approximated by the assumption that the solution is ideal.
Only properties which result from the dissolution of a nonvolatile solute in a volatile liquid solvent are considered. They are essentially solvent properties which are changed by the presence of the solute. The solute particles displace some solvent molecules in the liquid phase and thereby reduce the concentration of solvent and increase its entropy, so that the colligative properties are independent of the nature of the solute. The word colligative is derived from the Latin colligatus meaning bound together. This indicates that all colligative properties have a com
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https://en.wikipedia.org/wiki/Stellar%20dynamics
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Stellar dynamics is the branch of astrophysics which describes in a statistical way the collective motions of stars subject to their mutual gravity. The essential difference from celestial mechanics is that the number of body
Typical galaxies have upwards of millions of macroscopic gravitating bodies and countless number of neutrinos and perhaps other dark microscopic bodies. Also each star contributes more or less equally to the total gravitational field, whereas in celestial mechanics the pull of a massive body dominates any satellite orbits.
Connection with fluid dynamics
Stellar dynamics also has connections to the field of plasma physics. The two fields underwent significant development during a similar time period in the early 20th century, and both borrow mathematical formalism originally developed in the field of fluid mechanics.
In accretion disks and stellar surfaces, the dense plasma or gas particles collide very frequently, and collisions result in equipartition and perhaps viscosity under magnetic field. We see various sizes for accretion disks and stellar atmosphere, both made of enormous number of microscopic particle mass,
at stellar surfaces,
around Sun-like stars or km-sized stellar black holes,
around million solar mass black holes (about AU-sized) in centres of galaxies.
The system crossing time scale is long in stellar dynamics, where it is handy to note that
The long timescale means that, unlike gas particles in accretion disks, stars in gal
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https://en.wikipedia.org/wiki/Second%20%28disambiguation%29
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A second is the base unit of time in the International System of Units (SI).
Second, Seconds or 2nd may also refer to:
Mathematics
2 (number), as an ordinal (also written as 2nd or 2d)
Second of arc, an angular measurement unit, of a degree
Seconds (angle), units of angular measurement
Music
Notes and intervals
Augmented second, an interval in classical music
Diminished second, unison
Major second, a whole tone
Minor second, semitone
Neutral second one-and-a-half semitones
Albums and EPs
2nd (The Rasmus EP), 1996
Second (Baroness EP), 2005
Second (Raye EP), 2014
The Second, second studio album by rock band Steppenwolf
Seconds (The Dogs D'Amour album), released in 2000
Seconds (Kate Rogers album), released in 2005
Seconds (Tim Berne album), released in 2007
The 2nd (album), a 2006 album by Hater
Songs
"Second" (song), a 2021 song by Hyoyeon
"Second", a 2020 song by Hope D
"Second", a 2019 song by Erika Costell
"Second", a song from Sleaford Mods' 2020 compilation album All That Glue
"Seconds", from The Human League's 1981 album Dare
"Seconds" (song), from U2's 1983 album War
"Seconds", from Le Tigre's 2004 album This Island
Film
The 2nd (film), an American 2020 film starring Ryan Phillippe
Seconds (1966 film), a US thriller directed by John Frankenheimer
Seconds (2014 film), an Indian Malayalam-language thriller film by Aneesh Upasana
"Seconds" (The Batman), an episode in the American animated TV series
"Seconds", an episode of the Ame
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https://en.wikipedia.org/wiki/Willis%20Lamb
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Willis Eugene Lamb Jr. (; July 12, 1913 – May 15, 2008) was an American physicist who won the Nobel Prize in Physics in 1955 "for his discoveries concerning the fine structure of the hydrogen spectrum." The Nobel Committee that year awarded half the prize to Lamb and the other half to Polykarp Kusch, who won "for his precision determination of the magnetic moment of the electron." Lamb was able to precisely determine a surprising shift in electron energies in a hydrogen atom (see Lamb shift). Lamb was a professor at the University of Arizona College of Optical Sciences.
Biography
Lamb was born in Los Angeles, California, United States and attended Los Angeles High School. First admitted in 1930, he received a Bachelor of Science in chemistry from the University of California, Berkeley in 1934. For theoretical work on scattering of neutrons by a crystal, guided by J. Robert Oppenheimer, he received the Ph.D. in physics in 1938. Because of limited computational methods available at the time, this research narrowly missed revealing the Mössbauer Effect, 19 years before its recognition by Mössbauer. He worked on nuclear theory, laser physics, and verifying quantum mechanics.
Lamb was a physics professor at Stanford from 1951 to 1956. Lamb was the Wykeham Professor of Physics at the University of Oxford from 1956 to 1962, and also taught at Yale, Columbia and the University of Arizona. He was elected a Fellow of the American Academy of Arts and Sciences in 1963.
Lamb is reme
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https://en.wikipedia.org/wiki/YDS
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YDS or yds may refer to:
YDS (Language Proficiency Test administered in Turkey)
Yards
YDS algorithm in computer science
Yosemite Decimal System
Young Democratic Socialists, US
Yiddish Sign Language's ISO 639 code.
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https://en.wikipedia.org/wiki/Galois%20extension
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In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
Characterization of Galois extensions
An important theorem of Emil Artin states that for a finite extension each of the following statements is equivalent to the statement that is Galois:
is a normal extension and a separable extension.
is a splitting field of a separable polynomial with coefficients in
that is, the number of automorphisms equals the degree of the extension.
Other equivalent statements are:
Every irreducible polynomial in with at least one root in splits over and is separable.
that is, the number of automorphisms is at least the degree of the extension.
is the fixed field of a subgroup of
is the fixed field of
There is a one-to-one correspondence between subfields of and subgroups of
Examples
There are two basic ways to construct examples of Galois extensions.
Take any field , any finite subgroup of , and let be the fixed field.
Take any field , any separable polynomial in , and let be
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https://en.wikipedia.org/wiki/Field%20trace
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In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.
Definition
Let K be a field and L a finite extension (and hence an algebraic extension) of K. L can be viewed as a vector space over K. Multiplication by α, an element of L,
,
is a K-linear transformation of this vector space into itself. The trace, TrL/K(α), is defined as the trace (in the linear algebra sense) of this linear transformation.
For α in L, let σ(α), ..., σ(α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
If L/K is separable then each root appears only once (however this does not mean the coefficient above is one; for example if α is the identity element 1 of K then the trace is [L:K ] times 1).
More particularly, if L/K is a Galois extension and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.,
where Gal(L/K) denotes the Galois group of L/K.
Example
Let be a quadratic extension of . Then a basis of is If then the matrix of is:
,
and so, . The minimal polynomial of α is .
Properties of the trace
Several properties of the trace function hold for any finite extension.
The trace is a K-linear map (a K-linear functional), that is
.
If then
Additionally, trace behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from
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https://en.wikipedia.org/wiki/Alexandre-Th%C3%A9ophile%20Vandermonde
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Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician, and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.
Biography
Vandermonde was a violinist, and became engaged with mathematics only around 1770. In Mémoire sur la résolution des équations (1771) he reported on symmetric functions and solution of cyclotomic polynomials; this paper anticipated later Galois theory (see also abstract algebra for the role of Vandermonde in the genesis of group theory). In Remarques sur des problèmes de situation (1771) he studied knight's tours, and presaged the development of knot theory by explicitly noting the importance of topological features when discussing the properties of knots:
"Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the theads are interlaced"
The same year he was elected to the French Academy of Sciences. Mémoire sur des irrationnelles de différents ordres avec une application au cercle (1772) was on combinatorics, and Mémoire sur l'élimination (1772) on the foundations of determinant the
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https://en.wikipedia.org/wiki/Pure%20mathematics
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Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law
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https://en.wikipedia.org/wiki/1739%20in%20science
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The year 1739 in science and technology involved some significant events.
Earth sciences
Plinian eruption of Mount Tarumae volcano in Japan.
Exploration
January 1 – Bouvet Island is discovered by French explorer Jean-Baptiste Charles Bouvet de Lozier in the South Atlantic Ocean.
Mathematics
Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.
Euler invents the tonnetz (German for "tone-network"), a conceptual lattice diagram that shows a two-dimensional tonal pitch space created by the network of relationships between musical pitches in just intonation.
Physics
Émilie du Châtelet publishes Dissertation sur la nature et la propagation du feu.
Awards
Copley Medal: Stephen Hales
Societies
June 2 – The Royal Swedish Academy of Sciences is founded in Stockholm by Linnaeus, Mårten Triewald and others.
Births
November 14 – William Hewson, English surgeon, anatomist and physiologist, "father of haematology" (died 1774)
December 14 – Pierre Samuel du Pont de Nemours, French industrialist (died 1817)
Israel Lyons, English mathematician and botanist (died 1775)
Deaths
April 19 – Nicholas Saunderson, English scientist and mathematician (born 1682)
April 27 – Nicolas Sarrabat, French scientist, astronomer and mathematician (born 1698)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/Merck%20Index
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The Merck Index is an encyclopedia of chemicals, drugs and biologicals with over 10,000 monographs on single substances or groups of related compounds published online by the Royal Society of Chemistry.
History
The first edition of the Merck's Index was published in 1889 by the German chemical company Emanuel Merck and was primarily used as a sales catalog for Merck's growing list of chemicals it sold. The American subsidiary was established two years later and continued to publish it. During World War I the US government seized Merck's US operations and made it a separate American "Merck" company that continued to publish the Merck Index.
In 2012 the Merck Index was licensed to the Royal Society of Chemistry. An online version of The Merck Index, including historic records and new updates not in the print edition, is commonly available through research libraries. It also includes an appendix with monographs on organic named reactions.
The 15th edition was published in April 2013.
Monographs in The Merck Index typically contain:
a CAS registry number
synonyms of the substance, such as trivial names and International Union of Pure and Applied Chemistry nomenclature
a chemical formula
molecular weight
percent composition
a structural formula
a description of the substance's appearance
melting point and boiling point
solubility in solvents commonly used in the laboratory
citations to other literature regarding the compound's chemical synthesis
a therapeutic catego
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https://en.wikipedia.org/wiki/Block%20matrix
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In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.
This notion can be made more precise for an by matrix by partitioning into a collection , and then partitioning into a collection . The original matrix is then considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some , where and .
Block matrix algebra arises in general from biproducts in categories of matrices.
Example
The matrix
can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
Block matrix multiplication
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices and such that all submatrix products that will be used are defined. Given an matrix with row partitions and column partitions
and a matrix with row partitions and column partition
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https://en.wikipedia.org/wiki/Joseph%20Black
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Joseph Black (16 April 1728 – 6 December 1799) was a Scottish physicist and chemist, known for his discoveries of magnesium, latent heat, specific heat, and carbon dioxide. He was Professor of Anatomy and Chemistry at the University of Glasgow for 10 years from 1756, and then Professor of Medicine and Chemistry at the University of Edinburgh from 1766, teaching and lecturing there for more than 30 years.
The chemistry buildings at both the University of Edinburgh and the University of Glasgow are named after Black.
Early life and education
Black was born "on the banks of the river Garonne" in Bordeaux, France, the sixth of the 12 children of Margaret Gordon (d. 1747) and John Black. His mother was from an Aberdeenshire family that had connections with the wine business and his father was from Belfast, Ireland, and worked as a factor in the wine trade. He was educated at home until the age of 12, after which he attended grammar school in Belfast. In 1746, at the age of 18, he entered the University of Glasgow, studying there for four years before spending another four at the University of Edinburgh, furthering his medical studies. During his studies he wrote a doctorate thesis on the treatment of kidney stones with the salt magnesium carbonate.
Scientific Studies
Chemical principles
Like most 18th-century experimentalists, Black's conceptualisation of chemistry was based on five principles of matter: Water, Salt, Earth, Fire and Metal. He added the principle of Air when hi
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https://en.wikipedia.org/wiki/Spline%20%28mathematics%29
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In mathematics, a spline is a special function defined piecewise by polynomials.
In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees.
In the computer science subfields of computer-aided design and computer graphics, the term spline more frequently refers to a piecewise polynomial (parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.
The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes.
Introduction
The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The data may be either one-dimensional or multi-dimensional. Spline functions for interpolation are normally determined as the minimizers of suitable measures of roughness (for example integral squared curvature) subject to the interpolation constraints. Smoothing splines may be viewed as generalizations of interpolation splines where the functions are determined to minimize a weighted combination of the average squared approximation error over observed data and the roughness measure. For a number of meaningful defin
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https://en.wikipedia.org/wiki/153%20%28number%29
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153 (one hundred [and] fifty-three) is the natural number following 152 and preceding 154.
In mathematics
The number 153 is associated with the geometric shape known as the Vesica Piscis or Mandorla. Archimedes, in his Measurement of a Circle, referred to this ratio (153/265), as constituting the "measure of the fish", this ratio being an imperfect representation of .
As a triangular number, 153 is the sum of the first 17 integers, and is also the sum of the first five positive factorials:.
The number 153 is also a hexagonal number, and a truncated triangle number, meaning that 1, 15, and 153 are all triangle numbers.
The distinct prime factors of 153 add up to 20, and so do the ones of 154, hence the two form a Ruth-Aaron pair.
Since , it is a 3-narcissistic number, and it is also the smallest three-digit number which can be expressed as the sum of cubes of its digits. Only five other numbers can be expressed as the sum of the cubes of their digits: 0, 1, 370, 371 and 407. It is also a Friedman number, since 153 = 3 × 51.
The Biggs–Smith graph is a symmetric graph with 153 edges, all equivalent.
Another feature of the number 153 is that it is the limit of the following algorithm:
Take a random positive integer, divisible by three
Split that number into its base 10 digits
Take the sum of their cubes
Go back to the second step
An example, starting with the number 84:
There are 153 uniform polypeta that are generated from four different fundamental Coxeter groups
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https://en.wikipedia.org/wiki/Paleoecology
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Paleoecology (also spelled palaeoecology) is the study of interactions between organisms and/or interactions between organisms and their environments across geologic timescales. As a discipline, paleoecology interacts with, depends on and informs a variety of fields including paleontology, ecology, climatology and biology.
Paleoecology emerged from the field of paleontology in the 1950s, though paleontologists have conducted paleoecological studies since the creation of paleontology in the 1700s and 1800s. Combining the investigative approach of searching for fossils with the theoretical approach of Charles Darwin and Alexander von Humboldt, paleoecology began as paleontologists began examining both the ancient organisms they discovered and the reconstructed environments in which they lived. Visual depictions of past marine and terrestrial communities have been considered an early form of paleoecology. The term "paleo-ecology" was coined by Frederic Clements in 1916.
Overview of paleoecological approaches
Classic paleoecology uses data from fossils and subfossils to reconstruct the ecosystems of the past. It involves the study of fossil organisms and their associated remains (such as shells, teeth, pollen, and seeds), which can help in the interpretation of their life cycle, living interactions, natural environment, communities, and manner of death and burial. Such interpretations aid the reconstruction of past environments (i.e., paleoenvironments). Paleoecologists have
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https://en.wikipedia.org/wiki/Stasis
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Stasis (from Greek στάσις "a standing still") may refer to:
A state in stability theory, in which all forces are equal and opposing, therefore they cancel out each other
Stasis (political history), a period of civil war within an ancient Greek city-state
Stasis (biology), a block of little or no evolutionary change in a species, in the punctuated equilibrium model of evolutionary biology
Stasis (fiction), a concept in science fiction in which time or motion is stopped
Stasis (film), a 2017 science-fiction film starring Anna Harr
Stasis (liturgy), a division of a Kathisma or other liturgical verses
Stasis (medicine), a state in which the normal flow of a body liquid stops, for example the flow of blood through vessels or of intestinal contents through the digestive tract
Stasis, moniker of Steve Pickton, British techno musician
Stasis (music), a technique or form used in minimalist music, and also any other style that may use slow musical development
Stasis (rhetoric), a rhetoric technique
Stasis (The UA Years 1971–1975), a compilation album by Hawkwind
"Stasis" (The Outer Limits), an episode of the television show
Stasis (video game), a science fiction point-and-click crowdfunded video game released in 2015
Venous stasis in medicine may refer to venous insufficiency
See also
Hemostasis, a process to prevent and stop bleeding
Homeostasis, a state of steady internal, physical, and chemical conditions maintained by living systems
Stasi, the official state secur
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https://en.wikipedia.org/wiki/Rotation%20%28disambiguation%29
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Rotation is a circular motion of a body about a center.
Rotation may also refer to:
Science, mathematics and computing
Rotation (anatomy)
Rotation (mathematics)
Rotation (medicine), medical student training
Rotation (physics), ratio between a given angle and a full turn of 2π radians
Bitwise rotation, a mathematical operator on bit patterns
Curl (mathematics), a vector operator
Differential rotation, objects rotating at different speeds
Display rotation, of a computer monitor or display
Earth's rotation
Improper rotation or rotoreflection, a rotation and reflection in one
Internal rotation, a term in anatomy
Optical rotation, rotation acting on polarized light
Rotation around a fixed axis
Rotational spectroscopy, a spectroscopy technique
Tree rotation, a well-known method used in order to make a tree balanced.
Arts, entertainment, and media
Music
Rotation (Cute Is What We Aim For album), 2008
Rotation (Joe McPhee album), 1977
Rotation (music), the repeated airing of a limited playlist of songs on a radio station
"Rotate" (song), a song on the album Channel 10 by Capone-N-Noreaga
"Rotation", a song on the 1979 album Rise by Herb Alpert
"Rotation (LOTUS-2)", a song on the 2000 album Philosopher's Propeller by Susumu Hirasawa
Other uses in arts, entertainment, and media
Rotation (film), a 1949 East German film
Rotation (pool), a type of pocket billiards game
Politics
Rotation government, the practice of a government switching Prime Ministers mid-
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https://en.wikipedia.org/wiki/Stanford%20University%20School%20of%20Engineering
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Stanford University School of Engineering is one of the schools of Stanford University. The current dean is Jennifer Widom, the former senior associate dean of faculty affairs and computer science chair. She is the school's 10th dean.
Organization and academics
The school of engineering was established in 1925, when Stanford organized the previously independent academic departments into a school. The original departments in the school were:
Civil engineering, one of the original university departments (1891), later to become civil and environmental engineering
Electrical engineering, taught as a subject prior to being established as a department in 1894
Mechanical engineering, one of the original university departments (1891)
Mining and metallurgy, established in 1918 and eventually disbanded in 1945
Departments added afterwards
Aeronautics and astronautics, started as aeronautical engineering in 1958
Chemical engineering in 1961 (split from chemistry)
Computer science, established in 1965 in the school of humanities and sciences, but moved to the school of engineering in 1985
Materials science and engineering in 1961 (originally known as materials science)
Management science and engineering in the 1950s (originally industrial engineering)
Bioengineering in 2002
Current departments at the school
Aeronautics and astronautics
Bioengineering and chemical engineering (also in the School of Medicine)
Chemical engineering
Civil and environmental engineering
Com
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https://en.wikipedia.org/wiki/Conjugate%20element%20%28field%20theory%29
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In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally itself is included in the set of conjugates of .
Equivalently, the conjugates of are the images of under the field automorphisms of that leave fixed the elements of . The equivalence of the two definitions is one of the starting points of Galois theory.
The concept generalizes the complex conjugation, since the algebraic conjugates over of a complex number are the number itself and its complex conjugate.
Example
The cube roots of the number one are:
The latter two roots are conjugate elements in with minimal polynomial
Properties
If K is given inside an algebraically closed field C, then the conjugates can be taken inside C. If no such C is specified, one can take the conjugates in some relatively small field L. The smallest possible choice for L is to take a splitting field over K of pK,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting field.
Given then a normal extension L of K, with automorphism group Aut(L/K) = G, and containing α, any element g(α) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other
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https://en.wikipedia.org/wiki/137%20%28number%29
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137 (one hundred [and] thirty-seven) is the natural number following 136 and preceding 138.
Mathematics
the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime.
an Eisenstein prime with no imaginary part and a real part of the form .
the fourth Stern prime.
a Pythagorean prime: a prime number of the form , where () or the sum of two squares .
a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.
a strictly non-palindromic number and a primeval number.
a factor of 10001 (the other being 73) and the repdigit 11111111 (= 10001 × 1111).
using two radii to divide a circle according to the golden ratio yields sectors of approximately 137.51° (the golden angle) and 222° in degree system so 137 is the largest integer before it.
In decimal notation, 1/137 = 0.007299270072992700..., so its period value happens to be palindromic and has a period length of only 8. However, this is only special to decimal, as in pentadecimal it (1/92) has a period length of twenty-four (24) and the period value is not at all palindromic.
Physics
Since the early 1900s, physicists have postulated that the number could lie at the heart of a grand unified theory, relating theories of electromagnetism, quantum mechanics and, especially, gravity.
1/137 was once believed to be the exact value of the fine-structure constant. The fine-structure constant, a dimensionless physical constant, is approxi
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https://en.wikipedia.org/wiki/Dimensionless%20physical%20constant
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In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used.
The concept should not be confused with dimensionless numbers, that are not universally constant, and remain constant only for a particular phenomena. In aerodynamics for example, if one considers one particular airfoil, the Reynolds number value of the laminar–turbulent transition is one relevant dimensionless number of the problem. However, it is strictly related to the particular problem: for example, it is related to the airfoil being considered and also to the type of fluid in which it moves.
The term fundamental physical constant is used to refer to some dimensionless constants. Perhaps the best-known example is the fine-structure constant, α, which has an approximate value of .
Terminology
It has been argued the term fundamental physical constant should be restricted to the dimensionless universal physical constants that currently cannot be derived from any other source; this stricter definition is followed here.
However, the term fundamental physical constant has also been used occasionally to refer to certain universal dimensioned physical constants, such as the speed of light c, vacuum permittivity ε0, Planck constant h, and the gravitational constant G, that appear in the most basic theories of physics. NIST and CODATA sometimes used the
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https://en.wikipedia.org/wiki/1735%20in%20science
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The year 1735 in science and technology involved some significant events.
Astronomy
July 11 - Pluto (not known at this time) enters a fourteen-year period inside the orbit of Neptune, which will not recur until 1979.
Biology
Carl Linnaeus publishes his Systema Naturae.
Chemistry
Cobalt is discovered and isolated by Georg Brandt. This is the first metal discovered since ancient times.
Earth sciences
May – French Geodesic Mission (including Charles Marie de La Condamine, Pierre Bouguer, Louis Godin, Jorge Juan, Antonio de Ulloa, Joseph de Jussieu and Jean Godin) sets out for Ecuador.
Mathematics
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem.
Meteorology
May 22 – George Hadley publishes the first explanation of the trade winds.
Physiology and medicine
December 6 – The second successful appendectomy is performed by naturalised British surgeon Claudius Aymand at St George's Hospital in London (the first was in 1731).
Births
April 21 – Ivan Petrovich Kulibin, Russian inventor (died 1818)
May 17 (bapt.) – John Brown, Scottish physician (died 1788)
August 7 – Claudine Picardet, French, chemist, mineralogist, meteorologist and scientific translator (died 1820)
September 6 – John Joseph Merlin, Liégeois-born inventor (died 1803)
October 6 – Jesse Ramsden, English scientific instrument maker (died 1800)
December 4 – Josephus Nicolaus Laurenti, Viennese herpetologist (died 1805)
Deaths
Fe
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https://en.wikipedia.org/wiki/1734%20in%20science
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The year 1734 in science and technology involved some significant events.
Mathematics
George Berkeley publishes The Analyst, an empiricist critique of the foundations of infinitesimal calculus, influential in the development of mathematics.
Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations.
Technology
James Short constructs a Gregorian reflecting telescope with an aperture of .
Zoology
René Antoine Ferchault de Réaumur begins publication of Mémoires pour servir à l'histoire des insectes in Amsterdam.
Awards
Copley Medal: John Theophilus Desaguliers
Births
January 23 – Wolfgang von Kempelen, Hungarian inventor (died 1804)
April 18 – Elsa Beata Bunge, Swedish botanist (died 1819)
May 23 – Franz Mesmer, German physician (died 1815)
September 3 – Joseph Wright, English painter of scientific subjects (died 1797)
Deaths
February 1 – John Floyer, English physician (born 1649)
April 25 – Johann Konrad Dippel, German theologian, alchemist and physician (born 1673)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/1732%20in%20science
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The year 1732 in science and technology involved some significant events.
Chemistry
Herman Boerhaave publishes the authorized edition of his Elementa chemiae in Leiden.
Exploration
March 3 – English Captain Charles Gough rediscovers Gough Island in the South Atlantic.
August – Mikhail Gvozdev with navigator Ivan Fyodorov in the Sviatoi Gavriil make the first known crossing of the Bering Strait, from Cape Dezhnev to Cape Prince of Wales, and explore the Alaskan coast.
Metrology
French astronomer in Russian service Joseph-Nicolas Delisle invents the Delisle scale for measuring temperature (recalibrated in 1738).
Technology
Henri Pitot develops the Pitot tube used for measuring flow velocity under the Seine bridges.
The world's first lightship is moored at the Nore in the Thames Estuary of England.
Awards
Copley Medal: Stephen Gray
Births
January 11 – Peter Forsskål, Finnish naturalist (died 1763)
October 6 – Nevil Maskelyne, English Astronomer Royal (died 1811)
October 24 – Cristina Roccati, Italian scholar in physics (died 1797)
Maria Christina Bruhn, Swedish inventor, (died 1802)
Deaths
January 17 – John Horsley, British archaeologist (born c. 1685)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/Brinell%20scale
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The Brinell scale characterizes the indentation hardness of materials through the scale of penetration of an indenter, loaded on a material test-piece. It is one of several definitions of hardness in materials science.
History
Proposed by Swedish engineer Johan August Brinell in 1900, it was the first widely used and standardised hardness test in engineering and metallurgy. The large size of indentation and possible damage to test-piece limits its usefulness. However, it also had the useful feature that the hardness value divided by two gave the approximate UTS in ksi for steels. This feature contributed to its early adoption over competing hardness tests.
Test details
The typical test uses a diameter steel ball as an indenter with a force. For softer materials, a smaller force is used; for harder materials, a tungsten carbide ball is substituted for the steel ball. The indentation is measured and hardness calculated as:
where:
BHN = Brinell Hardness Number (kgf/mm)
P = applied load in kilogram-force (kgf)
D = diameter of indenter (mm)
d = diameter of indentation (mm)
Brinell hardness is sometimes quoted in megapascals; the Brinell hardness number is multiplied by the acceleration due to gravity, 9.80665 m/s2, to convert it to megapascals.
The BHN can be converted into the ultimate tensile strength (UTS), although the relationship is dependent on the material, and therefore determined empirically. The relationship is based on Meyer's index (n) from Meyer's law. If Mey
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https://en.wikipedia.org/wiki/1730%20in%20science
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The year 1730 in science and technology involved some significant events.
Astronomy
The analemma is developed by the French astronomer Grandjean de Fouchy.
Mathematics
James Stirling publishes Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum.
Physics
The Reaumur scale is developed by French naturalist René Antoine Ferchault de Réaumur, with 0° = the freezing point of water and 80° = the boiling point.
Technology
Joseph Foljambe of Rotherham, England, produces the iron-clad Rotherham swing plough.
Births
April 15 – Moses Harris, English entomologist and engraver (died c. 1788)
July 12 – Anna Barbara Reinhart, Swiss mathematician (died 1796)
June 26 – Charles Messier, French astronomer (died 1817)
August 12 – Edmé-Louis Daubenton, French naturalist (died 1785)
December 8
Johann Hedwig, Transylvanian-born German botanist (died 1799)
Jan Ingenhousz, Dutch physiologist (died 1799)
Maria Angela Ardinghelli, Italian scientific translator (died 1825)
between 1730 and 1732 – William Hudson, English botanist (died 1793)
Deaths
January 18 – Antonio Vallisneri, Italian physician and natural scientist (born 1661)
April 21 - Jan Palfijn, Flemish surgeon and obstetrician (born 1650)
December 5 (bur.) – Alida Withoos, Dutch botanical artist (born c. 1661/1662)
References
18th century in science
1730s in science
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https://en.wikipedia.org/wiki/Strong%20operator%20topology
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In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space H induced by the seminorms of the form , as x varies in H.
Equivalently, it is the coarsest topology such that, for each fixed x in H, the evaluation map (taking values in H) is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets (where T0 is any bounded operator on H, x is any vector and ε is any positive real number).
In concrete terms, this means that in the strong operator topology if and only if for each x in H.
The SOT is stronger than the weak operator topology and weaker than the norm topology.
The SOT lacks some of the nicer properties that the weak operator topology has, but being stronger, things are sometimes easier to prove in this topology. It can be viewed as more natural, too, since it is simply the topology of pointwise convergence.
The SOT topology also provides the framework for the measurable functional calculus, just as the norm topology does for the continuous functional calculus.
The linear functionals on the set of bounded operators on a Hilbert space that are continuous in the SOT are precisely those continuous in the weak operator topology (WOT). Because of this, the closure of a convex set of operators in the WOT is the same as the closure of that set in the SOT.
This lan
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https://en.wikipedia.org/wiki/Predual
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In mathematics, the predual of an object D is an object P whose dual space is D.
For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions.
Abstract algebra
Functional analysis
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https://en.wikipedia.org/wiki/1729%20in%20science
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The year 1729 in science and technology involved some significant events.
Astronomy
January 9 & 16 – James Bradley, in a letter written to Edmond Halley and read before the Royal Society, describes his discovery of aberration of starlight.
August 1 – Fr. Nicolas Sarrabat, a professor of mathematics at Marseille, discovers the Comet of 1729, possibly the largest comet, with the highest apparent magnitude, on record.
English optician Chester Moore Hall (1703–1771) develops an achromatic lens (or achromat) commonly used as the objective of small refracting telescopes.
Biology
June 8 – The Botanic Gardens of Pamplemousses on Mauritius are started by Pierre Barmond.
Mark Catesby begins part publication in London of The Natural History of Carolina, Florida and the Bahama Islands, containing the figures of birds, beasts, fishes, serpents, insects, and plants ... together with their descriptions in English and French, the first published account of the flora and fauna of North America, and the first work of natural history to use folio-size coloured plates.
Mathematics
Andrew Motte publishes The Mathematical Principles of Natural Philosophy in London, the first English translation of Isaac Newton's Philosophiæ Naturalis Principia Mathematica (originally published 1687; Motte translated the 1726 edition).
Medicine
August 6 – Royal Infirmary of Edinburgh established as the "Hospital for the Sick Poor" or "Physicians' Hospital" in Edinburgh (Scotland).
Physics
Stephen Gray
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https://en.wikipedia.org/wiki/Pointer%20%28computer%20programming%29
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In computer science, a pointer is an object in many programming languages that stores a memory address. This can be that of another value located in computer memory, or in some cases, that of memory-mapped computer hardware. A pointer references a location in memory, and obtaining the value stored at that location is known as dereferencing the pointer. As an analogy, a page number in a book's index could be considered a pointer to the corresponding page; dereferencing such a pointer would be done by flipping to the page with the given page number and reading the text found on that page. The actual format and content of a pointer variable is dependent on the underlying computer architecture.
Using pointers significantly improves performance for repetitive operations, like traversing iterable data structures (e.g. strings, lookup tables, control tables and tree structures). In particular, it is often much cheaper in time and space to copy and dereference pointers than it is to copy and access the data to which the pointers point.
Pointers are also used to hold the addresses of entry points for called subroutines in procedural programming and for run-time linking to dynamic link libraries (DLLs). In object-oriented programming, pointers to functions are used for binding methods, often using virtual method tables.
A pointer is a simple, more concrete implementation of the more abstract reference data type. Several languages, especially low-level languages, support some type of
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https://en.wikipedia.org/wiki/Ectoplasm
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Ectoplasm may refer to:
Biology
Ectoplasm (cell biology), the outer part of the cytoplasm
Ectoplasm, outer layer of soft tissue in foraminiferans
Art and entertainment
Ectoplasm (radio show), BBC Radio 4 comedy series
Ectoplasm (My Hero Academia), a character in the manga series My Hero Academia
Other uses
Ectoplasm (paranormal), physically sensible phenomenon claimed to be due to "energy" described as paranormal
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https://en.wikipedia.org/wiki/List%20of%20Indian%20Americans
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Indian Americans are citizens or residents of the United States of America who trace their family descent to India. Notable Indian Americans include:
Academics
Nobel Prize recipients
Har Gobind Khorana (1922-2011), Nobel Prize in Medicine, 1968
Subramanyan Chandrasekhar (1910-1995), Nobel Prize for Physics, 1983
Venkatraman Ramakrishnan (b. 1952), Nobel Prize in Chemistry, 2009; Former President of the Royal Society, (2015-2020)
Abhijit Banerjee (b. 1961), Nobel Memorial Prize in Economic Sciences, 2019; Ford Foundation International Professor of Economics at Massachusetts Institute of Technology
Deans and presidents
Rakesh Khurana (born 1967), dean of Harvard College
Neeli Bendapudi (born 1962), president of University of Louisville
Jamshed Bharucha (born 1956), former president of Cooper Union, (2011-2015); former dean of arts & sciences at Dartmouth College and former provost at Tufts University
Vijay K. Dhir (born 1943), former dean of the UCLA Henry Samueli School of Engineering and Applied Science, (2003-2016)
Ravi V. Bellamkonda (born 1968), Vinik Dean of Engineering at Duke University Edmund T. Pratt Jr. School of Engineering
Dinesh D'Souza (born 1961), former president of The King's College, New York, (2010-2012)
Anjli Jain (born 1981), executive director of Campus Consortium
Dipak C. Jain (born 1957), former dean of INSEAD, (2011-2013); former dean of the Kellogg School of Management at Northwestern University, (2001-2009)
Vistasp Karbhari, for
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https://en.wikipedia.org/wiki/Ternary%20operation
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In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.
In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.
Examples
The function is an example of a ternary operation on the integers (or on any structure where and are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.
In the Euclidean plane with points a, b, c referred to an origin, the ternary operation has been used to define free vectors. Since (abc) = d implies a – b = c – d, these directed segments are equipollent and are associated with the same free vector. Any three points in the plane a, b, c thus determine a parallelogram with d at the fourth vertex.
In projective geometry, the process of finding a projective harmonic conjugate is a ternary operation on three points. In the diagram, points A, B and P determine point V, the harmonic conjugate of P with respect to A and B. Point R and the line through P can be selected arbitrarily, determining C and D. Drawing AC and BD produces the intersection Q, and RQ then yields V.
Suppose A and B are given sets and is the collection of binary relations between A and B. Composition of relations is always defined when A = B, but otherwise a ternary composition can be defined by where is the
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https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Soldner%20constant
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In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner.
Its value is approximately μ ≈ 1.45136923488338105028396848589202744949303228…
Since the logarithmic integral is defined by
then using we have
thus easing calculation for numbers greater than μ. Also, since the exponential integral function satisfies the equation
the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866…
External links
Mathematical constants
Srinivasa Ramanujan
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https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s%20Cat%20Trilogy
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The Schrödinger's Cat Trilogy is a trilogy of novels by American writer Robert Anton Wilson consisting of The Universe Next Door (1979), The Trick Top Hat (1980), and The Homing Pigeons (1981), each illustrating a different interpretation of quantum physics. They were collected into an omnibus edition in 1988.
Wilson is also co-author of The Illuminatus! Trilogy (1975), and Schrödinger's Cat is a sequel of sorts, re-using several of the same characters and carrying on many of the themes of the earlier work.
The name Schrödinger's Cat comes from a thought experiment in quantum mechanics. The first book, The Universe Next Door, takes place in different universes in accord with the many worlds interpretation of quantum physics; in the second, The Trick Top Hat, characters are unknowingly connected through non-locality, i.e., having once crossed paths they are joined in quantum entanglement; and the third book, The Homing Pigeons, places characters in an "observer-created universe" in which consciousness causes the collapse of the wavefunction.
Taking place in Unistat, which is the novel's parallel to the United States, the novels have intertwining plots involving a wide array of characters, including:
Epicene Wildeblood, a.k.a. Mary Margaret Wildeblood, a transsexual woman who throws great parties
Frank Dashwood, president of Orgasm Research
Markoff Chaney, a prankster
Hugh Crane, a.k.a. Cagliostro the Great, a mystic and magician
Furbish Lousewart V, author and Preside
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https://en.wikipedia.org/wiki/Dirichlet%27s%20unit%20theorem
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In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a positive real number that determines how "dense" the units are.
The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to
where is the number of real embeddings and the number of conjugate pairs of complex embeddings of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that
Note that if is Galois over then either or .
Other ways of determining and are
use the primitive element theorem to write , and then is the number of conjugates of that are real, the number that are complex; in other words, if is the minimal polynomial of over , then is the number of real roots and is the number of non-real complex roots of (which come in complex conjugate pairs);
write the tensor product of fields as a product of fields, there being copies of and copies of .
As an example, if is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of Pel
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https://en.wikipedia.org/wiki/Resultant%20force
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In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces. Calculating and visualizing the resultant force on a body is done through computational analysis, or (in the case of sufficiently simple systems) a free body diagram.
The point of application of the resultant force determines its associated torque. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term resultant force–torque.
Illustration
The diagram illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems.
Lines of application of the actual forces and in the leftmost illustration intersect. After vector addition is performed "at the location of ", the net force obtained is translated so that its line of application passes through the common intersection point. With respect to that point all torques are zero, so the torque of the resultant force is equal to the sum of the torques of the actual forces.
Illustration in the middle of the diagram shows two parallel actual forces. After vector addition "at the location of ", the net force is translated to the appropriate line of application, where
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https://en.wikipedia.org/wiki/Nuclear%20reaction
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In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. If a nucleus interacts with another nucleus or particle and they then separate without changing the nature of any nuclide, the process is simply referred to as a type of nuclear scattering, rather than a nuclear reaction.
In principle, a reaction can involve more than two particles colliding, but because the probability of three or more nuclei to meet at the same time at the same place is much less than for two nuclei, such an event is exceptionally rare (see triple alpha process for an example very close to a three-body nuclear reaction). The term "nuclear reaction" may refer either to a change in a nuclide induced by collision with another particle or to a spontaneous change of a nuclide without collision.
Natural nuclear reactions occur in the interaction between cosmic rays and matter, and nuclear reactions can be employed artificially to obtain nuclear energy, at an adjustable rate, on-demand. Nuclear chain reactions in fissionable materials produce induced nuclear fission. Various nuclear fusion reactions of light elements power the energy production of the Sun and stars.
History
In 1919, Ernest Rutherford was able to accomplish transmutation of nitrogen into oxygen at the University of Manches
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https://en.wikipedia.org/wiki/Nest%20algebra
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In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by and have many interesting properties. They are non-selfadjoint algebras, are closed in the weak operator topology and are reflexive.
Nest algebras are among the simplest examples of commutative subspace lattice algebras. Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion and is also a complete lattice. Since the orthogonal projections corresponding to the subspaces in a nest commute, nests are commutative subspace lattices.
By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the -dimensional complex vector space , and let be the standard basis. For , let be the -dimensional subspace of spanned by the first basis vectors . Let
then N is a subspace nest, and the corresponding nest algebra of n × n complex matrices M leaving each subspace in N invariant that is, satisfying for each S in N – is precisely the set of upper-triangular matrices.
If we omit one or more of the subspaces Sj from N then the corresponding nest algebra consists of block upper-triangular matrices.
Properties
Nest algebras are hyperreflexive with distance constant 1.
See also
fl
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https://en.wikipedia.org/wiki/1724%20in%20science
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The year 1724 in science and technology involved some significant events.
Astronomy
May 22 – Giacomo F. Maraldi concludes, from his observations during an eclipse, that the corona is part of the Sun.
Mathematics
Daniel Bernoulli expresses the numbers of the Fibonacci sequence in terms of the golden ratio.
Isaac Watts publishes Logic, or The Right Use of Reason in the Enquiry After Truth With a Variety of Rules to Guard Against Error in the Affairs of Religion and Human Life, as well as in the Sciences.
Medicine
Herman Boerhaave describes Boerhaave syndrome, a fatal tearing of the esophagus.
Institutions
January 28 – The Saint Petersburg Academy of Sciences is founded by Peter I of Russia.
Births
March 27 – Jane Colden, American botanist (died 1766)
June 8 – John Smeaton, English civil engineer (died 1792)
July 10 – Eva Ekeblad, Swedish agronomist, first woman in the Swedish Royal Academy of Science (died 1786)
September 27 – Anton Friedrich Busching, German geographer (died 1793)
December 25 – John Michell, English scientist (died 1793)
Date unknown – Marie Anne Victoire Pigeon, French mathematician (died 1767)
Deaths
October 18 - Jean de Hautefeuille, French inventor (born 1647)
References
18th century in science
1720s in science
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https://en.wikipedia.org/wiki/1723%20in%20science
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The year 1723 in science and technology involved some significant events.
Geophysics
George Graham discovers diurnal variation in Earth's magnetic field.
Antoine de Jussieu publishes De l'Origine et des usages de la Pierre de Foudre on the origins of fossils, prehistoric stone tools and meteorites.
Optics
Giacomo F. Maraldi makes the first observation of the Arago spot, unrecognized at this time.
Births
January 5 – Nicole-Reine Lepaute, French astronomer (died 1788)
January 31 – Petronella Johanna de Timmerman, Dutch scientist (died 1786)
February 17 – Tobias Mayer, German cartographer, astronomer and physicist (died 1762)
April 30 – Mathurin Jacques Brisson, French zoologist (died 1806)
November 12 – Saverio Manetti, Italian natural historian (died 1785)
Deaths
August 26 – Anton van Leeuwenhoek, Dutch pioneer of the microscope (born 1632)
References
18th century in science
1720s in science
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https://en.wikipedia.org/wiki/Ring%20of%20integers
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In mathematics, the ring of integers of an algebraic number field is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of , the ring is always a subring of .
The ring of integers is the simplest possible ring of integers. Namely, where is the field of rational numbers. And indeed, in algebraic number theory the elements of are often called the "rational integers" because of this.
The next simplest example is the ring of Gaussian integers , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, is a Euclidean domain.
The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.
Properties
The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as
with . The rank of as a free -module is equal to the degree of over .
Examples
Computational tool
A useful tool for computing the integral closure of the ring of integers in an algebraic field is the discriminant. If is of deg
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