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81,200 | In this form, it is often called the Hahn–Kolmogorov theorem. Let Σ 0 {\displaystyle \Sigma _{0}} be an algebra of subsets of a set X . {\displaystyle X.} | Carathéodory extension theorem | 0.784676 |
81,201 | Several very similar statements of the theorem can be given. A slightly more involved one, based on semi-rings of sets, is given further down below. A shorter, simpler statement is as follows. | Carathéodory extension theorem | 0.784676 |
81,202 | In measure theory, we are not interested in semi-rings and rings themselves, but rather in σ-algebras generated by them. The idea is that it is possible to build a pre-measure on a semi-ring S {\displaystyle S} (for example Stieltjes measures), which can then be extended to a pre-measure on R ( S ) , {\displaystyle R(S... | Carathéodory extension theorem | 0.784676 |
81,203 | {\displaystyle \cup _{n=1}^{\infty }A_{n}\in \Sigma _{0}.} (Functions μ 0 {\displaystyle \mu _{0}} obeying these two properties are known as pre-measures.) Then, μ 0 {\displaystyle \mu _{0}} extends to a measure defined on the σ {\displaystyle \sigma } -algebra Σ {\displaystyle \Sigma } generated by Σ 0 {\displaystyle ... | Carathéodory extension theorem | 0.784676 |
81,204 | The superposition theorem of the Poisson point process says that the superposition of independent Poisson point processes N 1 , N 2 … {\displaystyle \textstyle N_{1},N_{2}\dots } with mean measures Λ 1 , Λ 2 , … {\displaystyle \textstyle \Lambda _{1},\Lambda _{2},\dots } will also be a Poisson point process with mean m... | Poisson processes | 0.784675 |
81,205 | A mathematical model may require randomly moving points of a point process to other locations on the underlying mathematical space, which gives rise to a point process operation known as displacement or translation. The Poisson point process has been used to model, for example, the movement of plants between generation... | Poisson processes | 0.784675 |
81,206 | More generally, the intensity measures is a realization of a non-negative locally finite random measure. Cox point processes exhibit a clustering of points, which can be shown mathematically to be larger than those of Poisson point processes. The generality and tractability of Cox processes has resulted in them being u... | Poisson processes | 0.784675 |
81,207 | As a result, it does not suffer as much loss of throughput as TCP does on networks with high latency or high packet loss and avoids the overhead of naive "UDP data blaster" protocols. The protocol innovates upon naive "data blaster" protocols through an optimal control-theoretic retransmission algorithm and implementat... | Fast and Secure Protocol | 0.784674 |
81,208 | Medical and biological applications of EPR also exist. Although radicals are very reactive, so they do not normally occur in high concentrations in biology, special reagents have been developed to attach "spin labels", also called "spin probes", to molecules of interest. Specially-designed nonreactive radical molecules... | Electron spin resonance spectroscopy | 0.784674 |
81,209 | In the field of quantum computing, pulsed EPR is used to control the state of electron spin qubits in materials such as diamond, silicon and gallium arsenide. | Electron spin resonance spectroscopy | 0.784674 |
81,210 | Spectral objects known as "Be stars" are massive yet non-supergiant entities that notably have, or had at some time, 1 or more Balmer lines in emission, with the hydrogen-related electromagnetic radiation series projected out by the stars being of particular scientific interest. Be stars are generally thought to featur... | B-type star | 0.78467 |
81,211 | The characteristic polynomial of a matrix A is a scalar-valued polynomial, defined by p A ( t ) = det ( t I − A ) {\displaystyle p_{A}(t)=\det \left(tI-A\right)} . The Cayley–Hamilton theorem states that if this polynomial is viewed as a matrix polynomial and evaluated at the matrix A itself, the result is the zero mat... | Matrix polynomial | 0.784669 |
81,212 | The reliance on theories related to environmental factors to place blame on the mother reflects many Ugandan's poor knowledge of how the disease is acquired as it is determined by genetics, not environment. Mothers of children with sickle cell disease are also often left with very limited resources to safeguard their f... | Sickle cell disease | 0.784668 |
81,213 | If an algebra A {\displaystyle A} is Koszul, there is an equivalence between certain subcategories of the derived categories of graded A {\displaystyle A} - and A ! {\displaystyle A^{!}} -modules. These subcategories are defined by certain boundedness conditions on the grading vs. the cohomological degree of a complex. | Koszul duality | 0.784667 |
81,214 | A different reformulation gives an equivalence between the derived category of A {\displaystyle A} and the 'coderived' category of the coalgebra ( A ! ) ∗ {\displaystyle (A^{! })^{*}} . An extension of Koszul duality to D-modules states a similar equivalence of derived categories between dg-modules over the dg-algebra ... | Koszul duality | 0.784667 |
81,215 | The special case of associative algebras gives back the functor A ↦ A ! {\displaystyle A\mapsto A^{!}} mentioned above. | Koszul duality | 0.784667 |
81,216 | Yet another example is the Lie operad whose algebras are Lie algebras. The quadratic duality mentioned above is such that the associative operad is self-dual, while the commutative and the Lie operad correspond to each other under this duality. Koszul duality for operads states an equivalence between algebras over dual... | Koszul duality | 0.784667 |
81,217 | An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad. Very roughly, an operad is an algebraic structure consisting of an object of n-ary operations for all n. An algebra over an operad... | Koszul duality | 0.784667 |
81,218 | In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototype example, due to Joseph Bernstein, Israel Gelfand,... | Koszul duality | 0.784667 |
81,219 | The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space V over a field k, with dual vector space V ∗ {\displaystyle V^{*}} , the exterior algebra of V has two non-trivial components, namely ⋀ 1 V = V , ⋀ 0 V = k . {\displaystyle \bigwedge ^{1}V=V,\quad \bigwe... | Koszul duality | 0.784667 |
81,220 | = Λ ( V ∗ ) {\displaystyle A^{! }=\Lambda (V^{*})} , the exterior algebra on the dual of V. In general, the dual of a Koszul algebra is again a Koszul algebra. Its opposite ring is given by the graded ring of self-extensions of the underlying field k, thought of as an A-module: ( A ! ) | Koszul duality | 0.784667 |
81,221 | Koszul duality, as treated by Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel can be formulated using the notion of Koszul algebra. An example of such a Koszul algebra A is the symmetric algebra S ( V ) {\displaystyle S(V)} on a finite-dimensional vector space. More generally, any Koszul algebra can be shown... | Koszul duality | 0.784667 |
81,222 | In the United Kingdom, aggregate supply data is published in the Office for National Statistics' Input–output supply and use tables. | Aggregate supply | 0.784666 |
81,223 | Descartes's ambitions were far more modest than Leibniz's, and also far more successful, as shown by his wedding of algebra and geometry to yield what we now know as analytic geometry. Decades of research on symbolic artificial intelligence have not brought Leibniz's dream of a characteristica any closer to fruition. O... | Universal language | 0.784665 |
81,224 | In the vast writings of Gottfried Leibniz can be found many elements relating to a possible universal language, specifically a constructed language, a concept that gradually came to replace that of a rationalized Latin as the natural basis for a projected universal language. Leibniz conceived of a characteristica unive... | Universal language | 0.784665 |
81,225 | Suppose you have some logical formalism that contains certain rules allowing you to derive new formulas from given ones. Consider the set F of all possible formulas, and let P be the power set of F, ordered by ⊆. For a set X of formulas, let cl(X) be the set of all formulas that can be derived from X. Then cl is a clos... | Closure operator | 0.784665 |
81,226 | But then C is not a sublattice of the lattice P(S). Given a finitary closure operator on a set, the closures of finite sets are exactly the compact elements of the set C of closed sets. It follows that C is an algebraic poset. Since C is also a lattice, it is often referred to as an algebraic lattice in this context. C... | Closure operator | 0.784665 |
81,227 | While Darboux's theorem identifies locally a symplectic manifold M {\displaystyle M} with T ∗ L {\displaystyle T^{*}L} , Weinstein's theorem identifies locally a Lagrangian L {\displaystyle L} with the zero section of T ∗ L {\displaystyle T^{*}L} . More preciselyLet ( M , ω ) {\displaystyle (M,\omega )} be a symplectic... | Weinstein's tubular neighbourhood theorem | 0.784665 |
81,228 | This statement relies on the Weinstein's Lagrangian neighbourhood theorem, as well as on the standard tubular neighbourhood theorem. == References == | Weinstein's tubular neighbourhood theorem | 0.784665 |
81,229 | The statement (and the proof) of Darboux-Moser-Weinstein theorem can be generalised in presence of a symplectic action of a Lie group.Let M {\displaystyle M} be a smooth manifold of dimension 2 n {\displaystyle 2n} , and ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} two symplectic forms on M {\dis... | Weinstein's tubular neighbourhood theorem | 0.784665 |
81,230 | In symplectic geometry, a branch of mathematics, Weinstein's neighbourhood theorem refers to a few distinct but related theorems, involving the neighbourhoods of submanifolds in symplectic manifolds and generalising the classical Darboux's theorem. They were proved by Alan Weinstein in 1971. | Weinstein's tubular neighbourhood theorem | 0.784665 |
81,231 | A further application of streptavidin is for purification and detection of proteins genetically modified with the Strep-tag peptide. Streptavidin is widely used in Western blotting and immunoassays conjugated to some reporter molecule, such as horseradish peroxidase. Streptavidin has also been used in the developing fi... | Streptavidin | 0.784663 |
81,232 | Streptavidin is a 52 kDa protein (tetramer) purified from the bacterium Streptomyces avidinii. Streptavidin homo-tetramers have an extraordinarily high affinity for biotin (also known as vitamin B7 or vitamin H). With a dissociation constant (Kd) on the order of ≈10−14 mol/L, the binding of biotin to streptavidin is on... | Streptavidin | 0.784663 |
81,233 | High-valency streptavidinsStreptavidins of higher valency has been obtained by utilizing the chemistry of isopeptide bond conjugation using the SpyTag/SpyCatcher technology. This involves having a streptavidin tetramer with three biotin binding sites and a dead streptavidin fused to either SpyTag or SpyCatcher. When th... | Streptavidin | 0.784663 |
81,234 | The Wöhler synthesis is the conversion of ammonium cyanate into urea. This chemical reaction was described in 1828 by Friedrich Wöhler. It is often cited as the starting point of modern organic chemistry. Although the Wöhler reaction concerns the conversion of ammonium cyanate, this salt appears only as an (unstable) i... | Wöhler synthesis | 0.78466 |
81,235 | Daniel Jackson (born 1963) is a professor of Computer Science at the Massachusetts Institute of Technology (MIT). He is the principal designer of the Alloy modelling language, and author of the book Software Abstractions: Logic, Language, and Analysis. | Daniel Jackson (computer scientist) | 0.78466 |
81,236 | Jackson was born in London, England, in 1963. He studied physics at the University of Oxford, receiving an MA in 1984. After completing his MA, Jackson worked for two years as a software engineer at Logica UK Ltd. He then returned to academia to study computer science at MIT, where he received an SM in 1988, and a PhD ... | Daniel Jackson (computer scientist) | 0.78466 |
81,237 | Following the completion of his doctorate Jackson took up a position as an Assistant Professor of Computer Science at Carnegie Mellon University, which he held until 1997. He has been on the faculty of the Department of Electrical Engineering and Computer Science at MIT since 1997. In 2017 he became a Fellow of the Ass... | Daniel Jackson (computer scientist) | 0.78466 |
81,238 | NESSUS is a general-purpose, probabilistic analysis program that simulates variations and uncertainties in loads, geometry, material behavior and other user-defined inputs to compute probability of failure and probabilistic sensitivity measures of engineered systems. Because NESSUS uses highly efficient and accurate pr... | NESSUS Probabilistic Analysis Software | 0.784658 |
81,239 | In organic chemistry, an acyl chloride (or acid chloride) is an organic compound with the functional group −C(=O)Cl. Their formula is usually written R−COCl, where R is a side chain. They are reactive derivatives of carboxylic acids (R−C(=O)OH). A specific example of an acyl chloride is acetyl chloride, CH3COCl. Acyl c... | Acid chloride | 0.784658 |
81,240 | The Dashboard is also used to provide public access to information from EPA Action Plans, e.g. around perfluorinated alkylated substances.Originally titled the Chemistry Dashboard, the first version was released in 2016. The latest release of the database (version 3.0.5) contains manually curated data for over 875,000 ... | CompTox Chemicals Dashboard | 0.784657 |
81,241 | They are the primary antagonists to the naturally evolved humans of the Twelve Colonies, who are descendants of a single race of humans from the planet Kobol. One of the series’ core themes is that of historic recurrence, made explicit in the line "All of this has happened before and will happen again". Characters with... | Cylons | 0.784656 |
81,242 | Cylons in Battlestar Galactica continuity post-Galactica 1980 are a fictional, artificially-intelligent race of machines envisioned in the Battlestar Galactica science fiction series and related franchises. Originally created as purely mechanical devices to serve human needs, they eventually evolve into sentient, self-... | Cylons | 0.784656 |
81,243 | Unaware of the events on Kobol or with the Thirteenth Tribe, the Twelve Colonies developed self-aware, artificial intelligence in the form of the Cylon race, which rebelled and initiated a decade-long war across the colonies. Colonial veterans of this conflict remarked that the Cylons were relentless, highly adaptable,... | Cylons | 0.784656 |
81,244 | The Final Five are unique, and are the last survivors of the Thirteenth Tribe. They developed the Seven (originally eight) in conjunction with the Centurions of the Twelve Colonies. Each copy shares biology and general personality throughout their model, but they develop into distinct individuals. | Cylons | 0.784656 |
81,245 | Although they made only a single appearance in the original continuity (the episode "The Night the Cylons Landed" from Galactica 1980) humanoid Cylons are the focus of the reimagining. They are usually known as "the Cylons", or derogatorily as "skinjobs" or "toasters". Humanoid Cylons are indistinguishable from humans ... | Cylons | 0.784656 |
81,246 | The "final five" Cylons are the last five survivors of the earliest identified race of Cylons. Approximately 4,000 years prior to the events of the show, humans on Kobol developed artificial intelligence, which became the original race of Cylons. This race rebelled and fought in a war against the twelve tribes of human... | Cylons | 0.784656 |
81,247 | Recent years has seen interests in resurrection plants other than their ability to withstand extreme dryness. The metabolites, sugar alcohols, and sugar acids present in these plants may be applied as natural products for medicinal purposes and in biotechnology. During desiccation, the levels of the sugars sucrose, raf... | Xerophyte | 0.784655 |
81,248 | If we now chose x = c − δ 2 {\displaystyle x=c-{\frac {\delta }{2}}} , then g ( x ) > 0 {\displaystyle g(x)>0} and a < x < c {\displaystyle a 0 {\displaystyle g(c)>0} is impossible. If we combine both results, we get that g ( c ) = 0 {\displaystyle g(c)=0} or f ( c ) = u {\displaystyle f(c)=u} is the only remaining pos... | Bolzano's theorem | 0.784655 |
81,249 | The intermediate value theorem states the following: Consider an interval I = {\displaystyle I=} of real numbers R {\displaystyle \mathbb {R} } and a continuous function f: I → R {\displaystyle f\colon I\to \mathbb {R} } . Then Version I. if u {\displaystyle u} is a number between f ( a ) {\displaystyle f(a)} and f ( ... | Bolzano's theorem | 0.784655 |
81,250 | A similar result is the Borsuk–Ulam theorem, which says that a continuous map from the n {\displaystyle n} -sphere to Euclidean n {\displaystyle n} -space will always map some pair of antipodal points to the same place. In general, for any continuous function whose domain is some closed convex n {\displaystyle n} -dime... | Bolzano's theorem | 0.784655 |
81,251 | Simon Stevin proved the intermediate value theorem for polynomials (using a cubic as an example) by providing an algorithm for constructing the decimal expansion of the solution. The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration. Before ... | Bolzano's theorem | 0.784655 |
81,252 | A form of the theorem was postulated as early as the 5th century BCE, in the work of Bryson of Heraclea on squaring the circle. Bryson argued that, as circles larger than and smaller than a given square both exist, there must exist a circle of equal area. The theorem was first proved by Bernard Bolzano in 1817. Bolzano... | Bolzano's theorem | 0.784655 |
81,253 | Another, more complicated example is given by the Conway base 13 function. In fact, Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property (even though they need not be continuous). Historically, this intermediate val... | Bolzano's theorem | 0.784655 |
81,254 | A Darboux function is a real-valued function f that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values a and b in the domain of f, and any y between f(a) and f(b), there is some c between a and b with f(c) = y. The intermediate value theorem ... | Bolzano's theorem | 0.784655 |
81,255 | Efficiency should be understood as if we were to find some other estimator β ~ {\displaystyle \scriptstyle {\tilde {\beta }}} which would be linear in y and unbiased, then Var − Var ≥ 0 {\displaystyle \operatorname {Var} -\operatorname {Var} \geq 0} in the sense that this is a nonnegative-definite matrix. This th... | Least-squares normal matrix | 0.784655 |
81,256 | {\displaystyle \operatorname {Cov} =0.} The Gauss–Markov theorem states that under the spherical errors assumption (that is, the errors should be uncorrelated and homoscedastic) the estimator β ^ {\displaystyle \scriptstyle {\hat {\beta }}} is efficient in the class of linear unbiased estimators. This is called the bes... | Least-squares normal matrix | 0.784655 |
81,257 | The OLS estimator is consistent for the level-one fixed effects when the regressors are exogenous and forms perfect colinearity (rank condition), consistent for the variance estimate of the residuals when regressors have finite fourth moments and—by the Gauss–Markov theorem—optimal in the class of linear unbiased estim... | Least-squares normal matrix | 0.784654 |
81,258 | In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differen... | Least-squares normal matrix | 0.784654 |
81,259 | Telescope objective lenses for wide-band digital imaging in astronomy must have apochromatic correction, as the optical sensitivity of typical CCD imaging arrays can extend from the ultraviolet through the visible spectrum and into the near infrared wavelength range. Apochromatic lenses for astrophotography in the 60–1... | Apochromat | 0.784654 |
81,260 | Independent tests can be used to demonstrate that the "APO" designation is used rather loosely by some photographic lens manufacturers to describe the color accuracy of their lenses, as comparable lenses have shown superior color accuracy even though they did not carry the "APO" designation.Also, when considering lens ... | Apochromat | 0.784654 |
81,261 | In the 1980s, the Graphical Kernel System (GKS) library, based on a 1970s specification with a similar basic geometry and command structure to NAPLPS, was widely implemented on microcomputers, and became the basis of Digital Research's GSX graphics system used in their GEM GUI. GKS was later extended into a 3D version,... | NAPLPS | 0.784654 |
81,262 | Model checking of finite state systems can often be translated into various operations on Büchi automata. In addition to the closure operations presented above, the following are some useful operations for the applications of Büchi automata. DeterminizationSince deterministic Büchi automata are strictly less expressive... | Büchi automaton | 0.784654 |
81,263 | Analysis of synthetic lethality Predict adaptive evolution outcomes Use in metabolic engineering for high value outputsReconstructions and their corresponding models allow the formulation of hypotheses about the presence of certain enzymatic activities and the production of metabolites that can be experimentally tested... | Metabolic network modelling | 0.784653 |
81,264 | A metabolic reconstruction provides a highly mathematical, structured platform on which to understand the systems biology of metabolic pathways within an organism. The integration of biochemical metabolic pathways with rapidly available, annotated genome sequences has developed what are called genome-scale metabolic mo... | Metabolic network modelling | 0.784653 |
81,265 | A different technique to simulate the metabolic network is to perform flux balance analysis. This method uses linear programming, but in contrast to elementary mode analysis and extreme pathways, only a single solution results in the end. Linear programming is usually used to obtain the maximum potential of the objecti... | Metabolic network modelling | 0.784653 |
81,266 | The EcoCyc database can serve as a paradigm and model for any reconstruction. Additionally, MetaCyc, an encyclopedia of experimentally defined metabolic pathways and enzymes, contains 2,100 metabolic pathways and 11,400 metabolic reactions (Oct 2013). ENZYME: An enzyme nomenclature database (part of the ExPASy proteono... | Metabolic network modelling | 0.784653 |
81,267 | Kyoto Encyclopedia of Genes and Genomes (KEGG): a bioinformatics database containing information on genes, proteins, reactions, and pathways. The ‘KEGG Organisms’ section, which is divided into eukaryotes and prokaryotes, encompasses many organisms for which gene and DNA information can be searched by typing in the enz... | Metabolic network modelling | 0.784653 |
81,268 | Dynamic causal modeling (DCM) is a framework for specifying models, fitting them to data and comparing their evidence using Bayesian model comparison. It uses nonlinear state-space models in continuous time, specified using stochastic or ordinary differential equations. DCM was initially developed for testing hypothese... | Dynamic causal modeling | 0.784653 |
81,269 | Model inversion or estimation is implemented in DCM using variational Bayes under the Laplace assumption. This provides two useful quantities: the log marginal likelihood or model evidence ln p ( y | m ) {\displaystyle \ln {p(y|m)}} is the probability of observing of the data under a given model. Generally, this cann... | Dynamic causal modeling | 0.784653 |
81,270 | In the case that dμ = f(x) dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the characteristic function, but the typical conventions in pr... | Fourier uncertainty principle | 0.784652 |
81,271 | Taking the completion with respect to the largest possibly C*-norm gives its enveloping C*-algebra, called the group C*-algebra C*(G) of G. (Any C*-norm on L1(G) is bounded by the L1 norm, therefore their supremum exists.) Given any abelian C*-algebra A, the Gelfand transform gives an isomorphism between A and C0(A^), ... | Fourier uncertainty principle | 0.784652 |
81,272 | With convolution as multiplication, L1(G) is an abelian Banach algebra. It also has an involution * given by f ∗ ( g ) = f ( g − 1 ) ¯ . {\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.} | Fourier uncertainty principle | 0.784652 |
81,273 | Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time–frequency analysis. In physics, this transform was introduced by Edward Condon. This ... | Fourier uncertainty principle | 0.784652 |
81,274 | The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still ... | Fourier uncertainty principle | 0.784652 |
81,275 | Let f(x) and g(x) be integrable, and let f̂(ξ) and ĝ(ξ) be their Fourier transforms. If f(x) and g(x) are also square-integrable, then the Parseval formula follows: ⟨ f , g ⟩ L 2 = ∫ − ∞ ∞ f ( x ) g ( x ) ¯ d x = ∫ − ∞ ∞ f ^ ( ξ ) g ^ ( ξ ) ¯ d ξ , {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(... | Fourier uncertainty principle | 0.784652 |
81,276 | Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L2(R). On L1(R) ∩ L2(R), this extension agrees with original Fourier transform defined on L1(R), thus enlarging the domain of the Fourier transform to L1(R) + L2(R) (and consequently to Lp(R) for 1... | Fourier uncertainty principle | 0.784652 |
81,277 | Nitrous oxide is one of the most prominent anthropogenic ozone-depleting gases in the atmosphere. It is released into the atmosphere primarily through natural sources such as soil and rock, as well as anthropogenic process like farming. Atmospheric nitrous oxide is also created in the atmosphere as a product of a react... | Greenhouse gas monitoring | 0.78465 |
81,278 | A primary reinforcer, sometimes called an unconditioned reinforcer, is a stimulus that does not require pairing with a different stimulus in order to function as a reinforcer and most likely has obtained this function through the evolution and its role in species' survival. Examples of primary reinforcers include food,... | Superimposed schedules of reinforcement | 0.784649 |
81,279 | The 1952 Miller–Urey experiment demonstrated the synthesis of organic compounds, such as amino acids, formaldehyde and sugars, from the original inorganic precursors the researchers presumed to have been present in the primordial soup (but is no longer considered likely). This experiment inspired many others. In 1961, ... | PAH world hypothesis | 0.784649 |
81,280 | In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal conta... | Minimal right ideal | 0.784648 |
81,281 | However the presence of a second conductor in the case of a transmission line reduces the extent of the external magnetic field (and of the total self-inductance) regardless of the wire's length, so that the inductance decrease due to skin effect can still be important. For instance, in the case of a telephone twisted ... | Skin effect | 0.784646 |
81,282 | Grushko's theorem is, in a sense, a starting point in Dunwoody's theory of accessibility for finitely generated and finitely presented groups. Since the ranks of the free factors are smaller than the rank of a free product, Grushko's theorem implies that the process of iterated splitting of a finitely generated group G... | Grushko theorem | 0.784645 |
81,283 | Algorithmically computing the Grushko decomposition for specific classes of groups is a difficult problem which primarily requires being able to determine if a given group is freely decomposable. Positive results are available for some classes of groups such as torsion-free word-hyperbolic groups, certain classes of re... | Grushko theorem | 0.784645 |
81,284 | A useful consequence of the original Grushko theorem is the so-called Grushko decomposition theorem. It asserts that any nontrivial finitely generated group G can be decomposed as a free product G = A1∗A2∗...∗Ar∗Fs, where s ≥ 0, r ≥ 0,where each of the groups Ai is nontrivial, freely indecomposable (that is, it cannot ... | Grushko theorem | 0.784645 |
81,285 | Stallings' proof of Grushko Theorem follows from the following lemma. | Grushko theorem | 0.784645 |
81,286 | So, the components of Z 1 ∩ Z 2 {\displaystyle Z_{1}\cap Z_{2}} are contractible. In the case where Z 1 ∩ Z 2 {\displaystyle Z_{1}\cap Z_{2}} has only one component, by Van Kampen's theorem, we are done, as in that case,: F = Π 1 ( Z 1 ) ∗ Π 1 ( Z 2 ) {\displaystyle F=\Pi _{1}(Z_{1})\ast \Pi _{1}(Z_{2})} . The general ... | Grushko theorem | 0.784645 |
81,287 | 2. There exists a two complex X = X 1 ∪ X 2 {\displaystyle X=X_{1}\cup X_{2}} where { p } = X 1 ∩ X 2 {\displaystyle \{p\}=X_{1}\cap X_{2}} is a point on a one cell of X such that X1 and X2 are two complexes with fundamental groups G1 and G2 respectively. Note that by the Van Kampen theorem, this implies that the funda... | Grushko theorem | 0.784645 |
81,288 | Scott (1974) gave another topological proof of Grushko's theorem, inspired by the methods of 3-manifold topology Imrich (1984) gave a version of Grushko's theorem for free products with infinitely many factors. A 1976 paper of Chiswell gave a relatively straightforward proof of Grushko's theorem, modelled on Stallings'... | Grushko theorem | 0.784645 |
81,289 | After the original proofs of Grushko (1940) and Neumann(1943), there were many subsequent alternative proofs, simplifications and generalizations of Grushko's theorem. A close version of Grushko's original proof is given in the 1955 book of Kurosh.Like the original proofs, Lyndon's proof (1965) relied on length-functio... | Grushko theorem | 0.784645 |
81,290 | 1. There is a one or two dimensional CW complex, Z with fundamental group F. By Van Kampen theorem, the wedge of n circles is one such space. | Grushko theorem | 0.784645 |
81,291 | In molecular biology, the HAMP domain (present in Histidine kinases, Adenylate cyclases, Methyl accepting proteins and Phosphatases) is an approximately 50-amino acid alpha-helical region that forms a dimeric, four-helical coiled coil. It is found in bacterial sensor and chemotaxis proteins and in eukaryotic histidine ... | HAMP domain | 0.784642 |
81,292 | We briefly present some facts from hyperbolic geometry which are helpful in understanding prime geodesics. | Prime geodesic | 0.784641 |
81,293 | The importance of prime geodesics comes from their relationship to other branches of mathematics, especially dynamical systems, ergodic theory, and number theory, as well as Riemann surfaces themselves. These applications often overlap among several different research fields. | Prime geodesic | 0.784641 |
81,294 | Each hyperbolic element h of Γ determines a closed geodesic of Γ\H: first, by connecting the geodesic semicircle joining the fixed points of h, we get a geodesic on H called the axis of h, and by projecting this geodesic to M, we get a geodesic on Γ\H. This geodesic is closed because 2 points which are in the same orbi... | Prime geodesic | 0.784641 |
81,295 | Consider the Poincaré half-plane model H of 2-dimensional hyperbolic geometry. Given a Fuchsian group, that is, a discrete subgroup Γ of PSL(2, R), Γ acts on H via linear fractional transformation. Each element of PSL(2, R) in fact defines an isometry of H, so Γ is a group of isometries of H. There are then 3 types of ... | Prime geodesic | 0.784641 |
81,296 | Also, there is a Selberg zeta function which is formally similar to the usual Riemann zeta function and shares many of its properties. Algebraically, prime geodesics can be lifted to higher surfaces in much the same way that prime ideals in the ring of integers of a number field can be split (factored) in a Galois exte... | Prime geodesic | 0.784641 |
81,297 | thesis, Grigory Margulis proved a similar result for surfaces of variable negative curvature, while in his 1980 Ph.D. thesis, Peter Sarnak proved an analogue of Chebotarev's density theorem. There are other similarities to number theory — error estimates are improved upon, in much the same way that error estimates of t... | Prime geodesic | 0.784641 |
81,298 | In number theory, various "prime geodesic theorems" have been proved which are very similar in spirit to the prime number theorem. To be specific, we let π(x) denote the number of closed geodesics whose norm (a function related to length) is less than or equal to x; then π(x) ∼ x/ln(x). This result is usually credited ... | Prime geodesic | 0.784641 |
81,299 | In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem. | Prime geodesic | 0.784641 |
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