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33,300 | The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121738 in the OEIS): 1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16... | F4 lattice | 0.803498 |
33,301 | Kato's Euler system consists of certain elements occurring in the algebraic K-theory of modular curves. These elements—named Beilinson elements after Alexander Beilinson who introduced them in Beilinson (1984)—were used by Kazuya Kato in Kato (2004) to prove one divisibility in Barry Mazur's main conjecture of Iwasawa ... | Euler systems | 0.803496 |
33,302 | In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems ... | Euler systems | 0.803496 |
33,303 | An important parameter for transmission lines is the mode of transmission employed. The mode describes the electromagnetic field patterns caused by the geometry of the transmission structure. It is possible for more than one mode to exist simultaneously on the same line. Usually, steps are taken to suppress all modes e... | Planar transmission line | 0.803495 |
33,304 | The characteristic impedance of a line is the impedance encountered by a wave travelling along the line; it depends only on the line geometry and materials and is not changed by the line termination. It is necessary to match the characteristic impedance of the planar line to the impedance of the systems to which it is ... | Planar transmission line | 0.803495 |
33,305 | If the wire is longer than a large fraction of a wavelength (one tenth is often used as a rule of thumb), these assumptions are no longer true and transmission line theory must be used instead. With transmission lines, the geometry of the line is precisely controlled (in most cases, the cross-section is kept constant a... | Planar transmission line | 0.803495 |
33,306 | The basic advantage of using an unbalanced circuit topology, as compared to an equivalent balanced circuit, is that far fewer components are required. The difficulties come when a port of the circuit is to be connected to a transmission line or to an external device that's designed for balanced operation. Many transmis... | Unbalanced circuit | 0.803493 |
33,307 | In electrical engineering, an unbalanced circuit is one in which the transmission properties between the ports of the circuit are different for the two poles of each port. It is usually taken to mean that one pole of each port is bonded to a common potential (single-ended signalling) but more complex topologies are pos... | Unbalanced circuit | 0.803493 |
33,308 | Pythia takes as input the damaged text, and is trained to return hypothesised restorations of ancient Greek inscriptions, working as an assistive aid for ancient historians. Its neural network architecture works at both the character- and word-level, thereby effectively handling long-term context information, and deali... | Pythia (machine learning) | 0.803492 |
33,309 | Pythia is an ancient text restoration model that recovers missing characters from a damaged text input using deep neural networks. It was created by Yannis Assael, Thea Sommerschield, and Jonathan Prag, researchers from Google DeepMind and the University of Oxford.To study the society and the history of ancient civilis... | Pythia (machine learning) | 0.803492 |
33,310 | The upper bound of the Furstenberg–Sárközy theorem can be generalized from sets that avoid square differences to sets that avoid differences in p ( N ) {\displaystyle p(\mathbb {N} )} , the values at integers of a polynomial p {\displaystyle p} with integer coefficients, as long as the values of p {\displaystyle p} inc... | Square-difference-free set | 0.803491 |
33,311 | As Golomb observed, the cold positions are infinite, and more strongly the number of cold positions up to n {\displaystyle n} is at least proportional to n {\displaystyle {\sqrt {n}}} . For, if there were fewer cold positions, there wouldn't be enough of them to supply a winning move to each hot position. The Furstenbe... | Square-difference-free set | 0.803491 |
33,312 | According to the Furstenberg–Sárközy theorem, if S {\displaystyle S} is a square-difference-free set, then the natural density of S {\displaystyle S} is zero. That is, for every ε > 0 {\displaystyle \varepsilon >0} , and for all sufficiently large n {\displaystyle n} , the fraction of the numbers up to n {\displaystyle... | Square-difference-free set | 0.803491 |
33,313 | Subsequently, Ruzsa's construction has been improved by using a different base, b = 205 {\displaystyle b=205} , to give square-difference-free sets with size When applied to the base b = 2 {\displaystyle b=2} , the same construction generates the Moser–de Bruijn sequence multiplied by two, a square-difference-free set ... | Square-difference-free set | 0.803491 |
33,314 | In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large. In the game of ... | Square-difference-free set | 0.803491 |
33,315 | The Journal of the Brazilian Computer Society (JBCS) is the flagship scientific journal of the Brazilian Computer Society (SBC). It is a quarterly journal. Its aim is to publish original research papers, serving as a forum for disseminating innovative research in all aspects of computer science. The priorities of the j... | Journal of the Brazilian Computer Society | 0.803486 |
33,316 | The covariance between two complex random variables Z , W {\displaystyle Z,W} is defined as: 119 Notice the complex conjugation of the second factor in the definition. In contrast to real random variables, we also define a pseudo-covariance (also called complementary variance): The second order statistics are fully cha... | Complex random variable | 0.803481 |
33,317 | In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and ima... | Complex random variable | 0.803481 |
33,318 | The Quadrics name was first used in 1993 for a commercialized version of the APE100 SIMD parallel computer produced by Alenia Spazio and originally developed by INFN, the Italian National Institute of Nuclear Physics. In 1996, a new Alenia subsidiary, Quadrics Supercomputers World (QSW) was formed, based in Bristol, UK... | Quadrics (company) | 0.80348 |
33,319 | Multiple, parallel QsNet networks can be employed in a system to maintain the compute to communications ratio where high CPU count SMP nodes are employed. The fat tree topology is resilient with large amounts of redundancy in the higher levels of the switch. Performance depends on platform used and configuration of the... | Quadrics (company) | 0.80348 |
33,320 | Elan4 is a communication processor that forms the interface between a high-performance multistage network and a processing node with one or more CPUs. Elite4 is a switching component that can switch eight bidirectional communications links, each of which carrying data in both directions simultaneously at 1.3 GB/s. Quad... | Quadrics (company) | 0.80348 |
33,321 | This correspondence is a defining feature of algebraic geometry. Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic ... | Projective algebraic set | 0.803476 |
33,322 | The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensat... | Projective algebraic set | 0.803476 |
33,323 | For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. | Projective algebraic set | 0.803476 |
33,324 | Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while a... | Projective algebraic set | 0.803476 |
33,325 | Let V1, V2 be algebraic varieties. We say V1 and V2 are isomorphic, and write V1 ≅ V2, if there are regular maps φ: V1 → V2 and ψ: V2 → V1 such that the compositions ψ ∘ φ and φ ∘ ψ are the identity maps on V1 and V2 respectively. | Projective algebraic set | 0.803476 |
33,326 | Then from the multivariable chain rule, Further, one may take p = p ( t ) {\displaystyle p=p(t)} and q = q ( t ) {\displaystyle q=q(t)} to be solutions to Hamilton's equations; that is, Then Thus, the time evolution of a function f {\displaystyle f} on a symplectic manifold can be given as a one-parameter family of sym... | Poisson brackets | 0.803475 |
33,327 | Given a smooth vector field X {\displaystyle X} on the configuration space, let P X {\displaystyle P_{X}} be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Lie bracket to the Poisson bracket: This important result is worth a short proof. Write a vector field X {\displ... | Poisson brackets | 0.803475 |
33,328 | Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, ħ → 0) yields the above Lie algebra. To state this more explicitl... | Poisson brackets | 0.803475 |
33,329 | JES is the flagship journal of The Electrochemical Society. Published continuously from 1902 to the present, JES is one of the most highly-cited journals in electrochemistry and solid state science and technology. | Electrochemical Technology | 0.803474 |
33,330 | The Little Box Challenge was an engineering competition run by Google and the IEEE's Power Electronics Society. The original challenge was posted on July 22, 2014 with modifications on December 16, 2014 and March 23, 2015. Testing was in October 2015 at the National Renewable Energy Laboratory. From the 18 finalists, C... | Little Box Challenge | 0.803474 |
33,331 | A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph K2,4). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph. Tutt... | Planarity (graph theory) | 0.803473 |
33,332 | In nuclear physics, the semi-empirical mass formula (SEMF) (sometimes also called the Weizsäcker formula, Bethe–Weizsäcker formula, or Bethe–Weizsäcker mass formula to distinguish it from the Bethe–Weizsäcker process) is used to approximate the mass of an atomic nucleus from its number of protons and neutrons. As the n... | Liquid drop model | 0.803472 |
33,333 | In linear algebra, reduction refers to applying simple rules to a series of equations or matrices to change them into a simpler form. In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. ... | Reduction (mathematics) | 0.803472 |
33,334 | In dynamic analysis, static reduction refers to reducing the number of degrees of freedom. Static reduction can also be used in finite element analysis to refer to simplification of a linear algebraic problem. Since a static reduction requires several inversion steps it is an expensive matrix operation and is prone to ... | Reduction (mathematics) | 0.803472 |
33,335 | In calculus, reduction refers to using the technique of integration by parts to evaluate integrals by reducing them to simpler forms. | Reduction (mathematics) | 0.803472 |
33,336 | In the 9th century, Persian mathematician Al-Khwarizmi's Al-Jabr introduced the fundamental concepts of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation and the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Kh... | Reduction (mathematics) | 0.803472 |
33,337 | This clarifies the fact that in this context the generalized principle reflects the linearity of quantum mechanics and the fact that the quantum mechanics equations are first order in time. Finally only in this case the superposition principle fully apply, i.e. the wave function in a point P can be expanded as a superp... | Huygens' Principle | 0.803472 |
33,338 | Many books and references e.g. and refer to the Generalized Huygens' Principle as the one referred by Feynman in this publication.Feynman defines the generalized principle in the following way: "Actually Huygens’ principle is not correct in optics. It is replaced by Kirchoff’s modification which requires that both the... | Huygens' Principle | 0.803472 |
33,339 | Generalised: This manner utilises a premise on a sample set to extract a conclusion about a population. Statistical: This is a method that utilises statistics based on a large and viable random sample set that is quantifiable to strengthen conclusions and observations. Bayesian: This form adapts statistical reasoning t... | Analytical skill | 0.803471 |
33,340 | To strengthen the conclusion, it is made more probable, for example, ‘All the swans I have seen are white. (Premise) Therefore most swans are probably white (Conclusion)’. Inductive reasoning is an analytical skill common in many professions such as the corporate environment, where statistics and data are constantly an... | Analytical skill | 0.803471 |
33,341 | People with motor impairments, such as Parkinson's disease, may not be able to navigate devices precisely using the traditional mouse pointer for input. To overcome these barriers, researchers continue to explore ways to make modern computer systems more accessible. Recent developments in artificial intelligence, speci... | WIMP (computing) | 0.803471 |
33,342 | Multiple studies have explored the possibilities of moving past the WIMP interface, such as using reality-based interaction, making the interface "three-dimensional" by adding visual depth through the use of monocular cues, and even combining depth with physics. The latter resulted in the development of BumpTop desktop... | WIMP (computing) | 0.803471 |
33,343 | Suppose we have a three-dimensional wave function in position space ψ(r), then we can write this functions as a weighted sum of orthogonal basis functions ψj(r): or, in the continuous case, as an integral It is clear that if we specify the set of functions ψ k ( r ) {\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )} , ... | Position and momentum space | 0.803471 |
33,344 | The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor g μ ν {\displaystyle g_{\mu \nu }} , since both the Ricci tensor and scalar curvature depend on the metric in a complic... | Einstein Equations | 0.80347 |
33,345 | The method of moments was introduced by Pafnuty Chebyshev for proving the central limit theorem; Chebyshev cited earlier contributions by Irénée-Jules Bienaymé. More recently, it has been applied by Eugene Wigner to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrice... | Method of moments (probability theory) | 0.803466 |
33,346 | In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences. Suppose X is a random variable and that all of the moments E ( X k ) {\displaystyle \operatorname {E} (X^{k})\,} exist. Further suppose the probability distribution of... | Method of moments (probability theory) | 0.803466 |
33,347 | The geometry of the copper binding site is described as a ‘distorted trigonal pyramidal’. The Cu-S (cys) contact is much shorter (207 picometers) than Cu-S (met) (282 pm) bond. The elongated Cu-thioether bond appears to destabilise the CuII state thereby enhancing its oxidizing power. | Plastocyanin | 0.803466 |
33,348 | None were, though it sometimes took a long while to establish this. In the 19th century, the general standard improved somewhat (e.g., works by Auguste Kerckhoffs, Friedrich Kasiski, and Étienne Bazeries). Colonel Parker Hitt and William Friedman in the early 20th century also wrote books on cryptography. These authors... | Applied Cryptography | 0.803466 |
33,349 | This bond also shortens the Cu(I)-SCys bond (2.1Å), increasing its electron donating strength. Overall, plastocyanin exhibits a lower reorganization energy due to the entatic state of the protein ligand enforcing the same distorted tetrahedral geometry in both the Cu(II) and Cu(I) oxidation states, enabling it to perfo... | Plastocyanin | 0.803466 |
33,350 | In ordinary copper complexes involved in Cu(I)/Cu(II) redox coupling without a constraining protein environment, their ligand geometry changes significantly, and typically corresponds to the presence of a Jahn-Teller distorting force. However, the Jahn-Teller distorting force is not present in plastocyanin due to a lar... | Plastocyanin | 0.803466 |
33,351 | This unusual geometry is induced by the rigid “pre-organized” conformation of the ligand donors by the protein, which is an entatic state. Plastocyanin performs electron transfer with the redox between Cu(I) and Cu(II), and it was first theorized that its entatic state was a result of the protein imposing an undistorte... | Plastocyanin | 0.803466 |
33,352 | This method was used to determine that plastocyanin has an entatic strain energy of about 10 kcal/mol (42 kJ/mol). Four-coordinate copper complexes often exhibit square planar geometry, however plastocyanin has a trigonally distorted tetrahedral geometry. This distorted geometry is less stable than ideal tetrahedral ge... | Plastocyanin | 0.803466 |
33,353 | A catalyst's function is to increase the speed of the electron transfer (redox) reaction. Plastocyanin is believed to work less like an enzyme where enzymes decrease the transition energy needed to transfer the electron. Plastocyanin works more on the principles of entatic states where it increases the energy of the re... | Plastocyanin | 0.803466 |
33,354 | Books on cryptography have been published sporadically and with highly variable quality for a long time. This is despite the tempting, though superficial, paradox that secrecy is of the essence in sending confidential messages — see Kerckhoffs' principle. In contrast, the revolutions in cryptography and secure communic... | Applied Cryptography | 0.803466 |
33,355 | Prados, John – Combined Fleet Decoded, An account of cryptography in the Pacific Theatre of World War II with special emphasis on the Japanese side. Reflects extensive research in Japanese sources and recently available US material. | Applied Cryptography | 0.803466 |
33,356 | Some surprising content, e.g., in the discussion of President Edgar Hoover's Secretary of State, Henry Stimson. Budiansky, Stephen, Battle of Wits: a one-volume history of cryptography in WW II. It is well written, well researched, and responsible. | Applied Cryptography | 0.803466 |
33,357 | The included cryptanalytic contest has been won and the prize awarded, but the cyphertexts are still worth attempting. Bauer, F L, Decrypted Secrets, This book is unusual. It is both a history of cryptography, and a discussion of mathematical topics related to cryptography. | Applied Cryptography | 0.803466 |
33,358 | Levy, Steven – Crypto: How the Code Rebels Beat the Government—Saving Privacy in the Digital Age (2001) (ISBN 0-14-024432-8): a journalistic overview of the development of public cryptographic techniques and the US regulatory context for cryptography. This is an account of a major policy conflict. Singh, Simon, The Cod... | Applied Cryptography | 0.803466 |
33,359 | Kahn has written other books and articles on cryptography, and on cryptographic history. They are very highly regarded. Kozaczuk, Władysław, Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War II, edited and translated by Christopher Kasparek, Frederick, MD, 1984: a history ... | Applied Cryptography | 0.803466 |
33,360 | James Gannon, Stealing Secrets, Telling Lies: How Spies and Codebreakers Helped Shape the Twentieth Century, Washington, D.C., Brassey's, 2001: an overview of major 20th-century episodes in cryptology and espionage, particularly strong regarding the misappropriation of credit for conspicuous achievements. Kahn, David –... | Applied Cryptography | 0.803466 |
33,361 | Gustave Bertrand, Enigma ou la plus grande énigme de la guerre 1939–1945 (Enigma: the Greatest Enigma of the War of 1939–1945), Paris, 1973. The first public disclosure in the West of the breaking of Enigma, by the chief of French military cryptography prior to WW II. The first public disclosure anywhere was made in th... | Applied Cryptography | 0.803466 |
33,362 | An early example of a book about cryptography was a Roman work, now lost and known only by references. Many early cryptographic works were esoteric, mystical, and/or reputation-promoting; cryptography being mysterious, there was much opportunity for such things. At least one work by Trithemius was banned by the Catholi... | Applied Cryptography | 0.803466 |
33,363 | Max Hernandez - Thieves Emporium (2013), a novel that examines how the world will change if cryptography makes fully bi-directional anonymous communications possible. A technically accurate document, it shows the effects of crypto from the citizen's standpoint rather than the NSA. Barry Eisler, Fault Line (2009) ISBN 9... | Applied Cryptography | 0.803466 |
33,364 | Ken Follett – The Key to Rebecca (1980), World War II spy novel whose plot revolves around the heroes' efforts to cryptanalyze a book cipher with time running out. Clifford B. Hicks – Alvin's Secret Code (1963), a children's novel which introduces some basics of cryptography and cryptanalysis. Robert Harris – Enigma (1... | Applied Cryptography | 0.803466 |
33,365 | UAVs may exploit distributed neural networks. | Autonomous aircraft | 0.803465 |
33,366 | Remote network system, such as satellite duplex data links for some military powers. Downstream digital video over mobile networks has also entered consumer markets, while direct UAV control uplink over the cellular mesh and LTE have been demonstrated and are in trials. Another aircraft, serving as a relay or mobile co... | Autonomous aircraft | 0.803465 |
33,367 | A necessary assumption to derive Bell's theorem is that the hidden variables are not correlated with the measurement settings. This assumption has been justified on the grounds that the experimenter has "free will" to choose the settings, and that such is necessary to do science in the first place. A (hypothetical) the... | Bell tests | 0.803465 |
33,368 | In philosophy, absolute theory (or absolutism) usually refers to a theory based on concepts (such as the concept of space) that exist independently of other concepts and objects. The absolute point of view was advocated in physics by Isaac Newton. It is one of the traditional views of space along with relational theory... | Absolute theory | 0.803465 |
33,369 | Functional programming languages often allow the subtyping of records. Consequently, simply typed lambda calculus extended with record types is perhaps the simplest theoretical setting in which a useful notion of subtyping may be defined and studied. Because the resulting calculus allows terms to have more than one typ... | Subtyping | 0.803465 |
33,370 | In the example shown below, the meso form of tartaric acid forms a diastereomeric pair with both levo- and dextro-tartaric acids, which form an enantiomeric pair. The D- and L- labeling of the isomers above is not the same as the d- and l- labeling more commonly seen, explaining why these may appear reversed to those f... | Configurational isomer | 0.803463 |
33,371 | In stereochemistry, stereoisomerism, or spatial isomerism, is a form of isomerism in which molecules have the same molecular formula and sequence of bonded atoms (constitution), but differ in the three-dimensional orientations of their atoms in space. This contrasts with structural isomers, which share the same molecul... | Configurational isomer | 0.803463 |
33,372 | The BioWall is composed of an array of small, locally connected elements and as a result is also an ideal platform to prototype many different kinds of two-dimensional cellular systems and bio-inspired systems, such as: Conway's Game of Life (>>>) Self-replicating Loops (>>>) Turing Artificial Neural Networks (>>>) Von... | BioWall | 0.803461 |
33,373 | More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of perturbation theory by introducing additional fields (the Faddeev–Popov ghosts) and counterterms motivated by anomaly cancellation, in an approach known as BRST quanti... | Gauge connection | 0.803457 |
33,374 | Other than these classical continuum field theories, the most widely known gauge theories are quantum field theories, including quantum electrodynamics and the Standard Model of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant action ... | Gauge connection | 0.803457 |
33,375 | In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are points in X, and Nx is the set of all neighborhoods that contain x, and Ny is the set of all neighborhoods that contain y, then x and y are "topologically indistingui... | Topologically indistinguishable | 0.803457 |
33,376 | So the "I" is merely syntactic sugar. Since I is optional, the system is also referred as SK calculus or SK combinator calculus. It is possible to define a complete system using only one (improper) combinator. An example is Chris Barker's iota combinator, which can be expressed in terms of S and K as follows: ιx = xSKI... | SKI combinator calculus | 0.803455 |
33,377 | As no further rule can be applied, the computation halts here. For all trees x and all trees y, SKxy will always evaluate to y in two steps, Ky(xy) = y, so the ultimate result of evaluating SKxy will always equal the result of evaluating y. We say that SKx and I are "functionally equivalent" because they always yield t... | SKI combinator calculus | 0.803455 |
33,378 | The reduced Kronecker coefficients are structure constants of Deligne categories of representations of S n {\displaystyle S_{n}} with n ∈ C − N {\displaystyle n\in \mathbb {C} -\mathbb {N} } .In contrast to Kronecker coefficients, reduced Kronecker coefficients are defined for any triple of Young diagrams, not necessar... | Representation theory of symmetric groups | 0.803455 |
33,379 | If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KSn is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). However... | Representation theory of symmetric groups | 0.803455 |
33,380 | In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules ... | Representation theory of symmetric groups | 0.803455 |
33,381 | As of 2014, HyungTae Kim held the role of chief executive officer (CEO) at Macrogen. By early 2015, however, the CEO role had been filled by Hyon-yong Chong.With respect to the company's Board, founder Jeong-sun Seo remained Chairman as of 2015.According to KOSDAQ statistics, over 90% of the company's outstanding stock... | Macrogen | 0.803455 |
33,382 | Macrogen, Inc. is a South Korean public biotechnology company. The company's headquarters are located in Seoul. The company was founded in 1997 by Jeong-sun Seo, a professor at Seoul National University. It was venture capital-backed until its initial public offering (IPO) and subsequent listing on the KOSDAQ market in... | Macrogen | 0.803455 |
33,383 | Macrogen's business model focuses on providing research services to other firms engaged in biotechnology research and development. As of 2005, 50% of the firm's revenue was derived from DNA sequencing services. Prior to its 2000 IPO, the company aimed to be the low-cost leader in providing research services; after the ... | Macrogen | 0.803455 |
33,384 | In addition to the automorphism α, there are two antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products of vectors: Since the ideal IQ is invariant under this reversal, this operation ... | Clifford Algebra | 0.803455 |
33,385 | For dim V = 1, if Q has diagonalization diag(a), that is there is a non-zero vector x such that Q(x) = a, then Cl(V, Q) is algebra-isomorphic to a K-algebra generated by an element x satisfying x2 = a, the quadratic algebra K / (X2 − a). In particular, if a = 0 (that is, Q is the zero quadratic form) then Cl(V, Q) is a... | Clifford Algebra | 0.803455 |
33,386 | The Lipschitz group Γ is defined to be the set of invertible elements x that stabilize the set of vectors under this action, meaning that for all v in V we have: This formula also defines an action of the Lipschitz group on the vector space V that preserves the quadratic form Q, and so gives a homomorphism from the Lip... | Clifford Algebra | 0.803455 |
33,387 | The universal characterization of the Clifford algebra shows that the construction of Cl(V, Q) is functorial in nature. Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebra... | Clifford Algebra | 0.803455 |
33,388 | The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Each of the algebras Clp,q(R) and Cln(C) is isomorphic to A or A ⊕ A, where A is a full matrix ring with entries from R, C, or H. For a complete classification of these algebras see Classifica... | Clifford Algebra | 0.803454 |
33,389 | For reference, the spin Lie algebra is given by This is in the (3, 1) convention, hence fits in Cl3,1(R)C.The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the alge... | Clifford Algebra | 0.803454 |
33,390 | The Clifford algebra of spacetime used in physics thus has more structure than Cl4(C). It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforward... | Clifford Algebra | 0.803454 |
33,391 | Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices γ0, ..., γ3 called Dirac matrices which have the property that where η is the matrix of a quadratic form of signature (1, 3) (or (3, 1) corresponding ... | Clifford Algebra | 0.803454 |
33,392 | On restriction to the spin group Spinp,q(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2n. More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: when... | Clifford Algebra | 0.803454 |
33,393 | Clifford algebras Clp,q(C), with p + q = 2n even, are matrix algebras which have a complex representation of dimension 2n. By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the spin representation. If we restrict this to the spin group Spinp,q(R) then i... | Clifford Algebra | 0.803454 |
33,394 | The class of Lipschitz groups (a.k.a. Clifford groups or Clifford–Lipschitz groups) was discovered by Rudolf Lipschitz.In this section we assume that V is finite-dimensional and the quadratic form Q is nondegenerate. An action on the elements of a Clifford algebra by its group of units may be defined in terms of a twis... | Clifford Algebra | 0.803454 |
33,395 | Since V comes equipped with a quadratic form Q, in characteristic not equal to 2 there exist bases for V that are orthogonal. An orthogonal basis is one such that for a symmetric bilinear form for i ≠ j {\displaystyle i\neq j} , and The fundamental Clifford identity implies that for an orthogonal basis for i ≠ j {\disp... | Clifford Algebra | 0.803454 |
33,396 | It is then straightforward to show that Cl(V, Q) contains V and satisfies the above universal property, so that Cl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Cl(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear sub... | Clifford Algebra | 0.803454 |
33,397 | Let V be a vector space over a field K, and let Q: V → K be a quadratic form on V. In most cases of interest the field K is either the field of real numbers R, or the field of complex numbers C, or a finite field. A Clifford algebra Cl(V, Q) is a pair (A, i), where A is a unital associative algebra over K and i is a li... | Clifford Algebra | 0.803454 |
33,398 | There are no Arctic-wide or Antarctic-wide measurements of the volume of sea ice, but the volume of the Arctic sea ice is calculated using the Pan-Arctic Ice Ocean Modeling and Assimilation System (PIOMAS) developed at the University of Washington Applied Physics Laboratory/Polar Science Center. PIOMAS blends satellite... | Measurement of sea ice | 0.803454 |
33,399 | Starting in 1958 U. S. Navy submarines collected upward-looking sonar profiles, for navigation and defense, and converted the information into estimates of ice thickness. Data from U. S. and Royal Navy submarines available from the NSIDC includes maps showing submarine tracks. Data are provided as ice draft profiles an... | Measurement of sea ice | 0.803454 |
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