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32,000 | In computer science, the Sharp Satisfiability Problem (sometimes called Sharp-SAT, #SAT or model counting) is the problem of counting the number of interpretations that satisfy a given Boolean formula, introduced by Valiant in 1979. In other words, it asks in how many ways the variables of a given Boolean formula can b... | Sharp-SAT | 0.804154 |
32,001 | Often the unital associative algebra is the algebra of endomorphisms of a vector space V {\displaystyle V} over a field k {\displaystyle k} , that is, A = End ( V ) {\displaystyle A={\text{End}}(V)} . With respect to a basis { e i } {\displaystyle \{e_{i}\}} of V {\displaystyle V} , the components of the matrices R ∈ E... | Star-triangle relation | 0.804152 |
32,002 | In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a mat... | Star-triangle relation | 0.804152 |
32,003 | To make routing fast enough for long paths (continental size) and avoid heuristical approaches GraphHopper uses contraction hierarchies by default. In the Java Magazine from Oracle, the author, Peter Karich, describes the techniques necessary to make the system memory efficient and fast. Furthermore, GraphHopper is bui... | GraphHopper | 0.804152 |
32,004 | The GraphHopper Directions API is an offering of the GraphHopper GmbH and includes a Geocoding API, a Distance Matrix API, a Map Matching API, an Isochrone API and a Route Optimization API besides the Routing API | GraphHopper | 0.804152 |
32,005 | The chemistry of thioesters and acid halides is similar, the reactivity being reminiscent of, but milder, than acid chlorides. | Nucleophilic acyl substitution | 0.804149 |
32,006 | The chemistry of acid halides and anhydrides is similar. While anhydrides cannot be converted to acid halides, they can be converted to the remaining acyl derivatives. Anhydrides also participate in Schotten–Baumann-type reactions to furnish esters and amides from alcohols and amines, and water can hydrolyze anhydrides... | Nucleophilic acyl substitution | 0.804149 |
32,007 | In the late 1990s the Bethesda-based Cystic Fibrosis Foundation, encouraged by then-President Robert Beall, began investing in Vertex— when it was a small start-up biotechnology company— to help fund the development of Kalydeco in the form of venture philanthropy. The total investment amounted to $150 million. In 2014,... | Vertex Pharmaceuticals | 0.804148 |
32,008 | Vertex Pharmaceuticals is an American biopharmaceutical company based in Boston, Massachusetts. It was one of the first biotech firms to use an explicit strategy of rational drug design rather than combinatorial chemistry. It maintains headquarters in South Boston, Massachusetts, and three research facilities, in San D... | Vertex Pharmaceuticals | 0.804148 |
32,009 | (See ) Let z i := p ( x ¯ i ) − f ( x ¯ i ) {\displaystyle z_{i}:=p({\bar {x}}_{i})-f({\bar {x}}_{i})} . Each amplitude | z i | {\displaystyle |z_{i}|} is greater than or equal to E. The Theorem of de La Vallée Poussin and its proof also apply to z 0 , . . | Remez algorithm | 0.804147 |
32,010 | {\displaystyle p(x_{i})-f(x_{i})\ =\ -(-1)^{i}E,\ \ i=0,...,n\!+\!1.} The theorem of de La Vallée Poussin states that under this condition no polynomial of degree n exists with error less than E. Indeed, if such a polynomial existed, call it p ~ ( x ) {\displaystyle {\tilde {p}}(x)} , then the difference p ( x ) − p ~ ... | Remez algorithm | 0.804147 |
32,011 | The Remez algorithm or Remez exchange algorithm, published by Evgeny Yakovlevich Remez in 1934, is an iterative algorithm used to find simple approximations to functions, specifically, approximations by functions in a Chebyshev space that are the best in the uniform norm L∞ sense. It is sometimes referred to as Remes a... | Remez algorithm | 0.804147 |
32,012 | The Chebyshev nodes are a common choice for the initial approximation because of their role in the theory of polynomial interpolation. For the initialization of the optimization problem for function f by the Lagrange interpolant Ln(f), it can be shown that this initial approximation is bounded by ‖ f − L n ( f ) ‖ ∞ ≤ ... | Remez algorithm | 0.804147 |
32,013 | The answer is yes. The first step is to observe that X 1 {\displaystyle X^{1}} and the attaching maps to construct X 2 {\displaystyle X^{2}} from X 1 {\displaystyle X^{1}} form a group presentation. The Tietze theorem for group presentations states that there is a sequence of moves we can perform to reduce this group p... | Cell complex | 0.804147 |
32,014 | On the other hand, the product of X and Y in the category of compactly generated spaces agrees with the weak topology and therefore defines a CW complex. Let X and Y be CW complexes. Then the function spaces Hom(X,Y) (with the compact-open topology) are not CW complexes in general. If X is finite then Hom(X,Y) is homot... | Cell complex | 0.804147 |
32,015 | In addition, the weak topology on this set often agrees with the more familiar product topology on X × Y, for example if either X or Y is finite. However, the weak topology can be finer than the product topology, for example if neither X nor Y is locally compact. In this unfavorable case, the product X × Y in the produ... | Cell complex | 0.804147 |
32,016 | The product of two CW complexes can be made into a CW complex. Specifically, if X and Y are CW complexes, then one can form a CW complex X × Y in which each cell is a product of a cell in X and a cell in Y, endowed with the weak topology. The underlying set of X × Y is then the Cartesian product of X and Y, as expected... | Cell complex | 0.804147 |
32,017 | A compact subspace of a CW complex is always contained in a finite subcomplex. CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. A covering space of a CW complex is also a CW complex. | Cell complex | 0.804147 |
32,018 | The CW complex construction is a straightforward generalization of the following process: A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology). A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of... | Cell complex | 0.804147 |
32,019 | It is a fast flow tube/ion swarm method to react positive or negative ions with atoms and molecules under truly thermalised conditions over a wide range of temperatures. It has been used extensively to study ion-molecule reaction kinetics. Its application to ionospheric and interstellar ion chemistry over a 20-year per... | SIFT-MS selected ion flow tube mass spectrometry | 0.804144 |
32,020 | The stages involved include: Attempt Fail Implicitly analyze the result Implicitly decide how to change the next attempt so that success is achievedThe stages are repeated over and over until the learner builds or remodels the neural network to guide an activity appropriately and accurately without conscious thought. T... | Procedural learning | 0.804143 |
32,021 | McDougall (1923) first made the distinction between explicit and implicit memory. In the 1970s procedural and declarative knowledge was distinguished in literature on artificial intelligence. Studies in the 1970s divided and moved towards two areas of work: one focusing on animal studies and the other to amnesic patien... | Procedural learning | 0.804143 |
32,022 | In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian pro... | Triadic relation | 0.804142 |
32,023 | Solutions to the problem use a variety of techniques, including both mathematically exact solutions and a variety of heuristic solutions using decomposition, parallel computing, stochastic optimization, genetic algorithms, colony optimization, simulated annealing, quantum annealing Tabu search, and coordinate descent.B... | Nurse scheduling problem | 0.80414 |
32,024 | In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by Steinitz (1901) but forgotten until it was rediscovered by Philip Hall (1959), both of whom published no more than brief summaries of their work. The Hall po... | Hall algebra | 0.804139 |
32,025 | {\displaystyle g_{\mu ,\nu }^{\lambda }(q)\in \mathbb {Z} .\,} Hall next constructs an associative ring H {\displaystyle H} over Z {\displaystyle \mathbb {Z} } , now called the Hall algebra. This ring has a basis consisting of the symbols u λ {\displaystyle u_{\lambda }} and the structure constants of the multiplicati... | Hall algebra | 0.804139 |
32,026 | The Molecular Conceptor Learning Series, produced by Synergix Ltd, is an interactive computer-based learning suite that teaches the principles and techniques used in everyday drug discovery. The Molecular Conceptor Learning Series comprises five modules each of which are designed to give students and professionals in t... | Molecular Conceptor | 0.804139 |
32,027 | The International Journal of Information Technology & Decision Making was founded in 2002 and is published by World Scientific. It is a peer-reviewed scientific journal covering the application of information technology to decision-making, in areas such as bio-informatics, fuzzy logic, neural networks, and online busin... | International Journal of Information Technology & Decision Making | 0.804136 |
32,028 | But although this scenario salvages conservation of energy, it violates gauge symmetry. Gauge symmetry requires that the laws of physics be invariant under the transformation V → V + C {\displaystyle V\rightarrow V+C} , which implies that no experiment should be able to measure the absolute potential, without reference... | Introduction to gauge theory | 0.804136 |
32,029 | If the potential V {\displaystyle V} is a solution to Maxwell's equations then, after this gauge transformation, the new potential V → V + C {\displaystyle V\rightarrow V+C} is also a solution to Maxwell's equations and no experiment can distinguish between these two solutions. In other words, the laws of physics gover... | Introduction to gauge theory | 0.804136 |
32,030 | Until the advent of quantum mechanics, the only well known example of gauge symmetry was in electromagnetism, and the general significance of the concept was not fully understood. For example, it was not clear whether it was the fields E and B or the potentials V and A that were the fundamental quantities; if the forme... | Introduction to gauge theory | 0.804136 |
32,031 | This idea, dubbed Yang–Mills theory, later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. The importance of gauge theories for physics stems from their tremendous success in providing a unified framework to describe the quantum-mecha... | Introduction to gauge theory | 0.804136 |
32,032 | : 5, 12 Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. After the development of quantum mechanics, Weyl, Fock and London modified their gauge choice by replacing the scale factor with a change of wave phase, and applying it successfully to electromagnetism. Gauge symmetry was... | Introduction to gauge theory | 0.804136 |
32,033 | Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B. With the advent of quantum mechanics in the 1920s, and with successive advances in quantum field theory, the importance of gauge transformations has steadily grown. Gauge theories constrain the laws of physics, becaus... | Introduction to gauge theory | 0.804136 |
32,034 | A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition). Modern theories describe physi... | Introduction to gauge theory | 0.804136 |
32,035 | The description of the electrons in the subsection above as little clocks is in effect a statement of the mathematical rules according to which the phases of electrons are to be added and subtracted: they are to be treated as ordinary numbers, except that in the case where the result of the calculation falls outside th... | Introduction to gauge theory | 0.804136 |
32,036 | This oversimplification is the reason that the gauge field θ comes out to be a scalar, whereas the electromagnetic field is actually represented by a vector consisting of V and A.) The result is that we have an explanation for the presence of electromagnetic interactions: if we try to construct a gauge-symmetric theory... | Introduction to gauge theory | 0.804136 |
32,037 | The details of how this is represented mathematically depend on technical issues relating to the spins of the particles, but for our present purposes we consider a spinless particle, for which it turns out that the mixing can be specified by some arbitrary choice of gauge θ(x), where an angle θ = 0° represents 100% A a... | Introduction to gauge theory | 0.804136 |
32,038 | Both observers discover the same laws of physics. Because light from hydrogen atoms in distant galaxies may reach the earth after having traveled across space for billions of years, in effect one can do such observations covering periods of time almost all the way back to the Big Bang, and they show that the laws of ph... | Introduction to gauge theory | 0.804136 |
32,039 | In summary, gauge symmetry attains its full importance in the context of quantum mechanics. In the application of quantum mechanics to electromagnetism, i.e., quantum electrodynamics, gauge symmetry applies to both electromagnetic waves and electron waves. These two gauge symmetries are in fact intimately related. If a... | Introduction to gauge theory | 0.804136 |
32,040 | The part of the electron wave passing through that side oscillates at a different rate, since its energy has had −eV added to it, where −e is the charge of the electron and V the electrical potential. The results of the experiment will be different, because phase relationships between the two parts of the electron wave... | Introduction to gauge theory | 0.804136 |
32,041 | In quantum mechanics, a particle such as an electron is also described as a wave. For example, if the double-slit experiment is performed with electrons, then a wave-like interference pattern is observed. The electron has the highest probability of being detected at locations where the parts of the wave passing through... | Introduction to gauge theory | 0.804136 |
32,042 | Constructivism is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the law of the excluded middle, the axiom of infinity, and the axiom of choice, and induces a different meaning for some terminology (f... | Non-constructive proof | 0.804135 |
32,043 | In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a part... | Non-constructive proof | 0.804135 |
32,044 | The following nerve theorem uses the homology groups of intersections of sets in the cover. For each finite J ⊂ I {\displaystyle J\subset I} , denote H J , j := H ~ j ( ⋂ i ∈ J U i ) = {\displaystyle H_{J,j}:={\tilde {H}}_{j}(\bigcap _{i\in J}U_{i})=} the j-th reduced homology group of ⋂ i ∈ J U i {\displaystyle \bigca... | Nerve complex | 0.804132 |
32,045 | In topology, the nerve complex of a set family is an abstract complex that records the pattern of intersections between the sets in the family. It was introduced by Pavel Alexandrov and now has many variants and generalisations, among them the Čech nerve of a cover, which in turn is generalised by hypercoverings. It ca... | Nerve complex | 0.804132 |
32,046 | A cohort study is a particular form of longitudinal study that samples a cohort (a group of people who share a defining characteristic, typically those who experienced a common event in a selected period, such as birth or graduation), performing a cross-section at intervals through time. It is a type of panel study whe... | Cohort studies | 0.804131 |
32,047 | From Büchi automatonMcNaughton's theorem provides a procedure to transform any non-deterministic Büchi automaton into a deterministic Muller automaton. | Muller automaton | 0.804128 |
32,048 | The Muller automata are equally expressive as parity automata, Rabin automata, Streett automata, and non-deterministic Büchi automata, to mention some, and strictly more expressive than the deterministic Büchi automata. The equivalence of the above automata and non-deterministic Muller automata can be shown very easily... | Muller automaton | 0.804128 |
32,049 | Various media sources and scientific groups have noted separate trends in differing areas which might together result in greater robotic functionalities and autonomy, and which pose some inherent concerns.A survey of AI experts estimated that the chance of human-level machine learning having an "extremely bad (e.g., hu... | Global catastrophe scenarios | 0.804124 |
32,050 | In Bostrom's book Superintelligence, he defines this as the control problem. Physicist Stephen Hawking, Microsoft founder Bill Gates, and SpaceX founder Elon Musk have echoed these concerns, with Hawking theorizing that such an AI could "spell the end of the human race".In 2009, the Association for the Advancement of A... | Global catastrophe scenarios | 0.804124 |
32,051 | David Pimentel, professor of ecology and agriculture at Cornell University, and Mario Giampietro, senior researcher at the National Research Institute on Food and Nutrition (INRAN), place in their 1994 study Food, Land, Population and the U.S. Economy the maximum U.S. population for a sustainable economy at 200 million... | Global catastrophe scenarios | 0.804124 |
32,052 | Experts at the Future of Humanity Institute at the University of Oxford and the Centre for the Study of Existential Risk at the University of Cambridge prioritize anthropogenic over natural risks due to their much greater estimated likelihood. They are especially concerned by, and consequently focus on, risks posed by ... | Global catastrophe scenarios | 0.804124 |
32,053 | However, given current development, more risk from novel, engineered pathogens is to be expected in the future. Exponential growth has been observed in the biotechnology sector, and Noun and Chyba predict that this will lead to major increases in biotechnological capabilities in the coming decades. They argue that risk... | Global catastrophe scenarios | 0.804124 |
32,054 | The technological means to genetically modify virus characteristics are likely to become more widely available in the future if not properly regulated.Biological weapons, whether used in war or terrorism, could result in human extinction. Terrorist applications of biotechnology have historically been infrequent. To wha... | Global catastrophe scenarios | 0.804124 |
32,055 | Biotechnology can pose a global catastrophic risk in the form of bioengineered organisms (viruses, bacteria, fungi, plants, or animals). In many cases the organism will be a pathogen of humans, livestock, crops, or other organisms we depend upon (e.g. pollinators or gut bacteria). However, any organism able to catastro... | Global catastrophe scenarios | 0.804124 |
32,056 | Population decline through a preference for fewer children. If developing world demographics are assumed to become developed world demographics, and if the latter are extrapolated, some projections suggest an extinction before the year 3000. John A. Leslie estimates that if the reproduction rate drops to the German or ... | Global catastrophe scenarios | 0.804124 |
32,057 | Nick Bostrom suggested that in the pursuit of knowledge, humanity might inadvertently create a device that could destroy Earth and the Solar System. Investigations in nuclear and high-energy physics could create unusual conditions with catastrophic consequences. All of these worries have so far proven unfounded. | Global catastrophe scenarios | 0.804124 |
32,058 | International institutions dedicated specifically to nanotechnology (perhaps analogously to the International Atomic Energy Agency IAEA) or general arms control may also be designed. One may also jointly make differential technological progress on defensive technologies, a policy that players should usually favour. The... | Global catastrophe scenarios | 0.804124 |
32,059 | Being equipped with compact computers and motors these could be increasingly autonomous and have a large range of capabilities.Chris Phoenix and Treder classify catastrophic risks posed by nanotechnology into three categories: From augmenting the development of other technologies such as AI and biotechnology. By enabli... | Global catastrophe scenarios | 0.804124 |
32,060 | Many nanoscale technologies are in development or currently in use. The only one that appears to pose a significant global catastrophic risk is molecular manufacturing, a technique that would make it possible to build complex structures at atomic precision. Molecular manufacturing requires significant advances in nanot... | Global catastrophe scenarios | 0.804124 |
32,061 | Kenneth Arrow's monograph Social Choice and Individual Values (1951, 2nd ed., 1963, 3rd ed., 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somewhat formally, the "social choice" in the title refers to Arrow's representat... | Social Choice and Individual Values | 0.804118 |
32,062 | Pareto and Unrestricted Domain for a constitution imply that such a set would at least include the entire set of voters. By Nondictatorship, the set must have at least 2 voters. Among all such sets, postulate a set such that no other set is smaller. Such a set can be constructed with Unrestricted Domain and an adaptati... | Social Choice and Individual Values | 0.804118 |
32,063 | Nondictatorship D: No voter in the society is a dictator. That is, there is no voter i in the society such that for every set of orderings in the domain of the constitution and every pair of distinct social states x and y, if voter i strictly prefers x over y, x is socially selected over y.Arrow's Theorem: The constitu... | Social Choice and Individual Values | 0.804118 |
32,064 | 3. The (weak) Pareto Principle P: For any x and y in the set of social states, if all prefer x over y, then x is socially selected over y.#The conditions, particularly the second and third, may seem minimal, but jointly they are harsh, as may be represented in either of two ways. Arrow's Theorem: The 3 conditions of th... | Social Choice and Individual Values | 0.804118 |
32,065 | The theorem might seem to have unravelled a skein of behavior-based social-ethical theory from Adam Smith and Bentham on. But Arrow himself expresses hope at the end of his Nobel prize lecture that, though the philosophical and distributive implications of the paradox of social choice were "still not clear," others wou... | Social Choice and Individual Values | 0.804118 |
32,066 | In so doing, it also ruled out any one consistent social ordering to which an agent or official might appeal in trying to implement social welfare through the votes of others under the constitution. The result generalizes and deepens the voting paradox to any voting rule satisfying the conditions, however complex or co... | Social Choice and Individual Values | 0.804118 |
32,067 | The outer fixed member is attached to the rack and the inner moving member is generally screwed to the side of the mounted equipment. Rack slides are typically compound or 3-part slides allowing full extension of the mounted equipment and generally include provision for sliding the inner member completely free to allow... | Linear-motion bearing | 0.804117 |
32,068 | Many HFAU units require isobutene-to-olefin ratios on the order of 13 - 15/1 to produce an acceptable octane product. Other HFAU and most of SAAU develop conditions of mixing and recycle optimization such that they produce similar octane products with isobutane to olefin ratios on the order of 7 - 9/1. Clearly the latt... | Alkylation unit | 0.804114 |
32,069 | Dr. J. Halcombe Laning, with the help of Phil Hankins and Charlie Werner, initiated work on MAC, an algebraic programming language for the IBM 650, which was completed by early spring of 1958. MAC became the work-horse of the MIT lab. MAC is an extremely readable language having a three-line format, vector-matrix notat... | Guidance systems | 0.804114 |
32,070 | Funded by Army Ordnance in 1942, JPL's early efforts would eventually involve technologies beyond those of aerodynamics and propellant chemistry. The result of the Army Ordnance effort was JPL's answer to the German V-2 missile, named MGM-5 Corporal, first launched in May 1947. On December 3, 1958, two months after the... | Guidance systems | 0.804114 |
32,071 | The term "correspondence principle" is used in a more general sense to mean the reduction of a new scientific theory to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "co... | Correspondence principle | 0.804112 |
32,072 | If an electron in an atom is moving on an orbit with period T, classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit does not decay very much in one cycle, the radiation will be emitted in a pattern which repeats every ... | Correspondence principle | 0.804112 |
32,073 | In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws. T... | Correspondence principle | 0.804112 |
32,074 | Arnold Sommerfeld referred to the principle as "Bohrs Zauberstab" (Bohr's magic wand) in 1921.The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum nu... | Correspondence principle | 0.804112 |
32,075 | The rules of quantum mechanics are highly successful in describing microscopic objects, atoms and elementary particles. But macroscopic systems, like springs and capacitors, are accurately described by classical theories like classical mechanics and classical electrodynamics. If quantum mechanics were to be applicable ... | Correspondence principle | 0.804112 |
32,076 | Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior. Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values, E = ( n + 1 / 2 ) ℏ ω , n = 0 , 1 ... | Correspondence principle | 0.804112 |
32,077 | One of these results was that the z-component of the angular momentum, the classical inclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotational invariance. This was called space quantization for a while, but this term fell out of favor with the new... | Correspondence principle | 0.804112 |
32,078 | Bohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define a quantity J(E) which is a function only of the energy, and has the property that d J d E = T {\displaystyle {dJ \over dE}=T} This is the analogue of the angular momentum in the case of the circular orbit... | Correspondence principle | 0.804112 |
32,079 | In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says that for large orbits and for large energies, quantum calculations must agr... | Correspondence principle | 0.804112 |
32,080 | One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as C = R / ( x 2 + 1 ) , {\displaystyle \mathbb {C} =\mathbb {R} /(x^{2}+1),} which corresponds t... | Representation of an associative algebra | 0.804108 |
32,081 | The statistical interpretation is related to quantum mechanics in describing the way that energy is distributed (quantized) amongst molecules on specific energy levels, with all the energy of the macrostate always in only one microstate at one instant. Entropy is described as measuring the energy dispersal for a system... | Energy dispersal | 0.804108 |
32,082 | Increase of entropy in a thermodynamic process can be described in terms of "energy dispersal" and the "spreading of energy," while avoiding mention of "disorder" except when explaining misconceptions. All explanations of where and how energy is dispersing or spreading have been recast in terms of energy dispersal, so ... | Energy dispersal | 0.804108 |
32,083 | While at the level of manifolds and Lie groups, the construction of the projective plane P2(K ⊗ K′) of two normed division algebras does not work, the corresponding construction at the level of Lie algebras does work. That is, if one decomposes the Lie algebra of infinitesimal isometries of the projective plane P2(K) a... | Freudenthal magic square | 0.804107 |
32,084 | The split forms of the composition algebras and Lie algebras can be defined over any field K. This yields the following magic square. There is some ambiguity here if K is not algebraically closed. In the case K = C, this is the complexification of the Freudenthal magic squares for R discussed so far. | Freudenthal magic square | 0.804107 |
32,085 | The Barton–Sudbery construction extends this to the other Lie algebras in the magic square. In particular, for the exceptional Lie algebras in the last row (or column), the symmetric decompositions are: f 4 ≅ s o 9 ⊕ Δ 16 {\displaystyle {\mathfrak {f}}_{4}\cong {\mathfrak {so}}_{9}\oplus \Delta ^{16}} e 6 ≅ ( s o 10 ⊕ ... | Freudenthal magic square | 0.804107 |
32,086 | It is denoted Tri(A). The following table compares its Lie algebra to the Lie algebra of derivations. Barton and Sudbery then identify the magic square Lie algebra corresponding to (A,B) with a Lie algebra structure on the vector space t r i ( A ) ⊕ t r i ( B ) ⊕ ( A 1 ⊗ B 1 ) ⊕ ( A 2 ⊗ B 2 ) ⊕ ( A 3 ⊗ B 3 ) . | Freudenthal magic square | 0.804107 |
32,087 | A more recent construction, due to Pierre Ramond (Ramond 1976) and Bruce Allison (Allison 1978) and developed by Chris Barton and Anthony Sudbery, uses triality in the form developed by John Frank Adams; this was presented in (Barton & Sudbery 2000), and in streamlined form in (Barton & Sudbery 2003). Whereas Vinberg's... | Freudenthal magic square | 0.804107 |
32,088 | The Cosmic Radiation Satellite (CORSA, also CORSA-A) was a Japanese space telescope. It was supposed to be Japan's first X-ray astronomy satellite but was lost due to failure of its Mu-3 launch vehicle on 4 Feb 1976. A replacement satellite Hakucho (CORSA-b) was later launched. | Cosmic Radiation Satellite | 0.804106 |
32,089 | In terms of the query, however, the complexity of this process is generally nonelementary. Thanks to Courcelle's theorem, we can also evaluate a Boolean MSO formula in linear time on an input graph if the treewidth of the graph is bounded by a constant. For MSO formulas that have free variables, when the input data is ... | Monadic second-order | 0.804106 |
32,090 | Existential monadic second-order logic (EMSO) is the fragment of MSO in which all quantifiers over sets must be existential quantifiers, outside of any other part of the formula. The first-order quantifiers are not restricted. By analogy to Fagin's theorem, according to which existential (non-monadic) second-order logi... | Monadic second-order | 0.804106 |
32,091 | Monadic second-order logic comes in two variants. In the variant considered over structures such as graphs and in Courcelle's theorem, the formula may involve non-monadic predicates (in this case the binary edge predicate E ( x , y ) {\displaystyle E(x,y)} ), but quantification is restricted to be over monadic predicat... | Monadic second-order | 0.804106 |
32,092 | In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order f... | Monadic second-order | 0.804106 |
32,093 | In recent years a number of research and education areas have been grouped under the umbrella term "Design Computing", namely: Artificial Intelligence in Design Artificial Architecture Expert and Knowledge-based Systems Computational creativity Computer-Aided Design Responsive computer-aided design Digital morphogenesi... | Design computing | 0.804105 |
32,094 | The terms design computing and other relevant terms including design and computation and computational design refer to the study and practice of design activities through the application and development of novel ideas and techniques in computing. One of the early groups to coin this term was the Key Centre of Design Co... | Design computing | 0.804105 |
32,095 | The main research groups working in this area span from Faculties of Architecture, Engineering and Computer Science. Australia has been a pioneer in this area. For the last five decades the Key Centre of Design Computing and Cognition (KCDC), currently known as the Design Lab, at the University of Sydney has been activ... | Design computing | 0.804105 |
32,096 | The Paczyński–Wiita potential is an approximation of the gravitational potential around a non-rotating black hole. It was introduced by Bohdan Paczyński and Paul Wiita in 1980. The article is one of the 40 most-cited from the first 40 years of the journal Astronomy & Astrophysics. | Paczyński–Wiita potential | 0.804103 |
32,097 | In nuclear physics, the symmetry energy reflects the variation of the binding energy of the nucleons in the nuclear matter depending on its neutron to proton ratio as a function of baryon density. Symmetry energy is an important parameter in the equation of state describing the nuclear structure of heavy nuclei and neu... | Symmetry energy | 0.804103 |
32,098 | In 1997, TB accounted for almost half of all Russian prison deaths, and as Bobrik et al. point out in their public health study, the 90% reduction in TB incidence contributed to a consequential fall in the prisoner death rate in the years following 1997. Baussano et al. articulate that concerning statistics like these ... | Multi-drug-resistant tuberculosis | 0.804103 |
32,099 | Magnetometric surveys can be useful in defining magnetic anomalies which represent ore (direct detection), or in some cases gangue minerals associated with ore deposits (indirect or inferential detection). This includes iron ore, magnetite, hematite, and often pyrrhotite. Developed countries such as Australia, Canada a... | Vector magnetometer | 0.804103 |
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