BARNES WALL LATTICE
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In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940.
In connection with: Leech lattice
Title combos: Leech lattice
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Sound amplification by stimulated emission of radiation
Sound amplification by stimulated emission of radiation (SASER) refers to a device that emits acoustic radiation. It focuses sound waves in a way that they can serve as accurate and high-speed carriers of information in many kinds of applications—similar to uses of laser light. Acoustic radiation (sound waves) can be emitted by using the process of sound amplification based on stimulated emission of phonons. Sound (or lattice vibration) can be described by a phonon just as light can be considered as photons, and therefore one can state that SASER is the acoustic analogue of the laser. In a SASER device, a source (e.g., an electric field as a pump) produces sound waves (lattice vibrations, phonons) that travel through an active medium. In this active medium, a stimulated emission of phonons leads to amplification of the sound waves, resulting in a sound beam coming out of the device. The sound wave beams emitted from such devices are highly coherent. The first successful SASERs were developed in 2009.
In connection with: Sound amplification by stimulated emission of radiation
Title combos: emission stimulated amplification stimulated by emission stimulated by amplification
Description combos: therefore radiation to state that can in of travel waves laser applications phonons an phonons emitted the sound In similar and light high out The coming 2009 of sound light described in It pump laser an is waves way light high based emission in to waves similar stimulated produces or of be vibration Sound sound produces lattice therefore emitted travel emission sound high of by described radiation first laser that phonon based on of electric amplification as laser speed first is
In mathematics, the Coxeter–Todd lattice K12, discovered by Coxeter and Todd (1953), is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice).
In connection with: Coxeter–Todd lattice
Title combos: Coxeter Todd Coxeter lattice Todd
Description combos: the Wall nonzero to discriminant the group norm and order analogous nonzero mathematics Todd norm of Wall this 12 1953 K12 order lattice Leech this K12 36 sublattice the In Leech lattice is by the dimensional is dimensional 756 and is 1953 lattice In Todd lattice lattice certain order by analogous sublattice norm automorphism nonzero Coxeter lattice 756 K12 this lattice and lattice vectors analogous certain discovered by dimensional integral fixed discriminant the Todd is of dimensional fixed is discriminant Coxeter
In mathematics, the Barnes–Wall lattice B W 16 {\displaystyle BW_{16}} , discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter–Todd lattice. The automorphism group of the Barnes–Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2). There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice). The genus of the Barnes–Wall lattice was described by Scharlau & Venkov (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16. The Barnes–Wall lattice is described in detail in (Conway & Sloane 1999, section 4.10). While Λ16 is often referred to as the Barnes-Wall lattice, their original article in fact construct a family of lattices of increasing dimension n=2k for any integer k, and increasing normalized minimal distance, namely n1/4. This is to be compared to the normalized minimal distance of 1 for the trivial lattice Z n {\displaystyle \mathbb {Z} ^{n}} , and an upper bound of 2 ⋅ Γ ( n 2 + 1 ) 1 / n / π = 2 n π e + o ( n ) {\displaystyle 2\cdot \Gamma \left({\frac {n}{2}}+1\right)^{1/n}{\big /}{\sqrt {\pi }}={\sqrt {\frac {2n}{\pi e}}}+o({\sqrt {n}})} given by Minkowski's theorem applied to Euclidean balls. Interestingly, this family comes with a polynomial time decoding algorithm by Micciancio & Nicolesi (2008).
In connection with: Barnes–Wall lattice
Title combos: Barnes lattice Barnes Wall lattice
Description combos: is no be detail lattice BW in The Nicolesi the article theorem Nicolesi of distance referred Wall has lattice contains increasing of is Sloane 1959 In vectors lattice Venkov group automorphism Wall the BW lattice In of integer Venkov in system Coxeter and is section increasing original Barnes 89181388800 sqrt is the the trivial to Barnes lattices 16 and definite mathematics Leech in the for Barnes Coxeter lattice definite Wall 16 is Stephen Wall of Conway is Barnes Nicolesi as 1999
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions between such states they interact with a very specific frequency of electromagnetic radiation. This phenomenon serves as the basis for the International System of Units' (SI) definition of a second: The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, Δ ν Cs {\displaystyle \Delta \nu _{\text{Cs}}} , the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s−1. This definition is the basis for the system of International Atomic Time (TAI), which is maintained by an ensemble of atomic clocks around the world. The system of Coordinated Universal Time (UTC) that is the basis of civil time implements leap seconds to allow clock time to track changes in Earth's rotation to within one second while being based on clocks that are based on the definition of the second, though leap seconds will be phased out in 2035. The accurate timekeeping capabilities of atomic clocks are also used for navigation by satellite networks such as the European Union's Galileo Programme and the United States' GPS. The timekeeping accuracy of the involved atomic clocks is important because the smaller the error in time measurement, the smaller the error in distance obtained by multiplying the time by the speed of light is (a timing error of a nanosecond or 1 billionth of a second (10−9 or 1⁄1,000,000,000 second) translates into an almost 30-centimetre (11.8 in) distance and hence positional error). The main variety of atomic clock uses caesium atoms cooled to temperatures that approach absolute zero. The primary standard for the United States, the National Institute of Standards and Technology (NIST)'s caesium fountain clock named NIST-F2, measures time with an uncertainty of 1 second in 300 million years (relative uncertainty 10−16). NIST-F2 was brought online on 3 April 2014.
In connection with: Atomic clock
Title combos: clock Atomic
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Ossicles are small calcareous elements embedded in the dermis of the body wall of echinoderms. They form part of the endoskeleton and provide rigidity and protection. They are found in different forms and arrangements in sea urchins, starfish, brittle stars, sea cucumbers, and crinoids. The ossicles and spines (which are specialised sharp ossicles) are the only parts of the animal likely to be fossilized after an echinoderm dies.
In connection with: Ossicle (echinoderm)
Title combos: Ossicle echinoderm
Description combos: dermis protection of embedded calcareous animal Ossicles They elements sea are specialised fossilized specialised starfish parts cucumbers which fossilized brittle the are Ossicles spines urchins parts cucumbers sharp They elements are urchins urchins wall in They which and different elements echinoderms calcareous calcareous forms in different protection echinoderms sea Ossicles of different forms and The the only an ossicles the sea They forms and They protection are and starfish They be form crinoids after urchins provide protection embedded echinoderms in rigidity
Eric Stephen Barnes (1924–2000), was an Australian pure mathematician. He was awarded the Thomas Ranken Lyle Medal in 1959, and was (Sir Thomas) Elder Professor of Mathematics at the University of Adelaide. He was elected a Fellow of the Australian Academy of Science in 1954. He was born in Cardiff, Wales, 16 January 1924 and died 16 October 2000 in Adelaide, South Australia. He was educated at the Universities of Sydney and Cambridge. He held appointments as a Fellow of Trinity College, Cambridge 1950–1954; assistant lecturer, Cambridge 1951–1953; reader in pure mathematics, University of Sydney 1953–1958; Elder Professor of Mathematics, University of Adelaide 1959–1974; Secretary (Physical Sciences) Australian Academy of Science 1972–1976; Deputy Vice-chancellor University of Adelaide 1975–1980; Professor of Pure Mathematics University of Adelaide 1981–1983.
In connection with: Eric Stephen Barnes
Title combos: Barnes Stephen Eric Stephen Barnes
Description combos: of 1953 mathematics lecturer of of Cambridge Secretary reader 1981 1953 Adelaide 2000 in Deputy of 2000 Eric was of the an of 1954 the and died 1953 appointments Science in Cambridge Physical Eric and Mathematics University Barnes Cambridge of He the of of an 2000 Sir Adelaide Trinity Thomas Eric Thomas Stephen pure Professor and University in the of Stephen Academy He appointments Sydney of as Vice Eric an He Stephen elected Elder Elder of Australia and January Fellow Australian
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