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90,100 | After some years of running, any particle physics experiment typically begins to suffer from diminishing returns: as the key results reachable by the device begin to be completed, later years of operation discover proportionately less than earlier years. A common response is to upgrade the devices involved, typically i... | Large Hadron Collider | 0.781009 |
90,101 | The collider has four crossing points where the accelerated particles collide. Nine detectors, each designed to detect different phenomena, are positioned around the crossing points. The LHC primarily collides proton beams, but it can also accelerate beams of heavy ions: lead–lead collisions and proton–lead collisions ... | Large Hadron Collider | 0.781009 |
90,102 | In 2016, the machine operators focused on increasing the luminosity for proton–proton collisions. The design value was first reached 29 June, and further improvements increased the collision rate to 40% above the design value. The total number of collisions in 2016 exceeded the number from Run 1 – at a higher energy pe... | Large Hadron Collider | 0.781009 |
90,103 | The upgrades culminated in colliding protons together with a combined energy of 13 TeV. On 3 June 2015, the LHC started delivering physics data after almost two years offline. In the following months, it was used for proton–proton collisions, while in November, the machine switched to collisions of lead ions and in Dec... | Large Hadron Collider | 0.781009 |
90,104 | On 20 November 2009, low-energy beams circulated in the tunnel for the first time since the incident, and shortly after, on 30 November, the LHC achieved 1.18 TeV per beam to become the world's highest-energy particle accelerator, beating the Tevatron's previous record of 0.98 TeV per beam held for eight years.The earl... | Large Hadron Collider | 0.781009 |
90,105 | Many physicists hope that the Large Hadron Collider will help answer some of the fundamental open questions in physics, which concern the basic laws governing the interactions and forces among the elementary objects, the deep structure of space and time, and in particular the interrelation between quantum mechanics and... | Large Hadron Collider | 0.781009 |
90,106 | LHC became operational again on 22 April 2022 with a new maximum beam energy of 6.8 TeV (13.6 TeV collision energy), which was first achieved on 25 April. It officially commenced its run 3 physics season on 5 July 2022. This round is expected to continue until 2026. In addition to a higher energy the LHC is expected to... | Large Hadron Collider | 0.781009 |
90,107 | We consider the problem of finding a line parallel to two given lines, a and a'. There are three cases: a and a' intersect at a point O, a and a' are parallel to each other, and a and a' are ultraparallel to each other.Case 1: a and a' intersect at a point O, Bisect one of the angles made by these two lines and name th... | Constructions in hyperbolic geometry | 0.781008 |
90,108 | For the purposes of the following definitions, the following assumptions will be made, which usually cannot be made in hyperbolic geometry Three distinct points create a unique circle Given any two lines, they meet at a unique point (normally, this would contradict the parallel axiom of hyperbolic geometry, since there... | Constructions in hyperbolic geometry | 0.781008 |
90,109 | Suppose you have a line a and a point A on that line, and you want to construct a line perpendicular to a and through A. Then let a' be a line through A where a and a' are two distinct lines. Then you will have one of two cases.Case 1: a is perpendicular to a' In this case, we already have the line perpendicular to a t... | Constructions in hyperbolic geometry | 0.781008 |
90,110 | Construct the common perpendicular to II' and AB, which can be done using the ultraparallel theorem because II' and AB are ultraparallel to each other. Name this line CC'. C now ends up being the midpoint of AB. | Constructions in hyperbolic geometry | 0.781008 |
90,111 | Let dE(A,B) denote the pseudolength for a given hyperbolic line segment AB. Let a transformation move A to the center of a Poincaré disk with a radius equalling 1. The pseudolength dE(A,B) is the length of this segment in Euclidean geometry. | Constructions in hyperbolic geometry | 0.781008 |
90,112 | There are a couple of models for hyperbolic geometry, which can make performing and visualizing constructions easier. Portions of the hyperbolic plane can be placed onto a pseudosphere and maintain angles and hyperbolic distances, as well as be bent around the pseudosphere and still keep its properties. However, not th... | Constructions in hyperbolic geometry | 0.781008 |
90,113 | In hyperbolic geometry, one can use the standard ruler and compass that is often used in Euclidean plane geometry. However, there are a variety of compasses and rulers developed for hyperbolic constructions. A hypercompass can be used to construct a hypercycle given the central line and radius. A horocompass can be use... | Constructions in hyperbolic geometry | 0.781008 |
90,114 | Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are para... | Constructions in hyperbolic geometry | 0.781008 |
90,115 | The Three Dunce Caps Theorem then says that P1, P2, and P3 all lie on the same line.Proof: Construct a sphere on top of each circle and then construct a plane tangent to these three spheres. The plane intersects the plane that the circles lies on at a straight line containing P1, P2, and P3. These points are also the c... | Constructions in hyperbolic geometry | 0.781008 |
90,116 | Then, line OA is the angle bisector for ᗉ IAI'. Case 2c: IB' is ultraparallel to I'B. Using the ultraparallel theorem, construct the common perpendicular of IB' and I'B, CC'. Let the intersection of CB" and BC' be D. As a result, AD will be the angle bisector of ᗉ BDB'. We then find that the line through OD is also the... | Constructions in hyperbolic geometry | 0.781008 |
90,117 | Algebraically, hyperbolic and spherical geometry have the same structure. This allows us to apply concepts and theorems to one geometry to the other. Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to ... | Constructions in hyperbolic geometry | 0.781008 |
90,118 | A point in Teichmüller space T ( S ) {\displaystyle T(S)} can alternatively be seen as a faithful representation of the fundamental group π 1 ( S ) {\displaystyle \pi _{1}(S)} into the isometry group P S L 2 ( R ) {\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )} of the hyperbolic plane H 2 {\displaystyle \mathbb {H} ^{... | Thurston boundary | 0.781007 |
90,119 | A diffeomorphism S → S {\displaystyle S\to S} is called pseudo-Anosov if there exists two transverse measured foliations, such that under its action the underlying foliations are preserved, and the measures are multiplied by a factor λ , λ − 1 {\displaystyle \lambda ,\lambda ^{-1}} respectively for some λ > 1 {\display... | Thurston boundary | 0.781007 |
90,120 | The map x ↦ ℓ ( x , ⋅ ) {\displaystyle x\mapsto \ell (x,\cdot )} is an embedding of T ( S ) {\displaystyle T(S)} into R + S {\displaystyle \mathbb {R} _{+}^{\mathcal {S}}} , which can be used to give the Teichmüller space a topology (the right-hand side being given the product topology). In fact, the map to the project... | Thurston boundary | 0.781007 |
90,121 | Google announced the LaMDA conversational large language model during the Google I/O keynote on May 18, 2021, powered by artificial intelligence. The acronym stands for "Language Model for Dialogue Applications". Built on the seq2seq architecture, transformer-based neural networks developed by Google Research in 2017, ... | LaMDA | 0.781007 |
90,122 | University of California, Santa Cruz professor Max Kreminski noted that LaMDA's architecture did not "support some key capabilities of human-like consciousness" and that its neural network weights were "frozen", assuming it was a typical large language model. IBM Watson lead developer David Ferrucci compared how LaMDA ... | LaMDA | 0.781007 |
90,123 | Internal controversy instigated by the incident prompted Google executives to decide against releasing LaMDA to the public, which it had previously been considering.Lemoine's claims were widely pushed back by the scientific community. Many experts ridiculed the idea that a language model could be self-aware, including ... | LaMDA | 0.781007 |
90,124 | On June 11, 2022, The Washington Post reported that Google engineer Blake Lemoine had been placed on paid administrative leave after Lemoine told company executives Blaise Agüera y Arcas and Jen Gennai that LaMDA had become sentient. Lemoine came to this conclusion after the chatbot made questionable responses to quest... | LaMDA | 0.781007 |
90,125 | On January 28, 2020, Google unveiled Meena, a neural network-powered chatbot with 2.6 billion parameters, which Google claimed to be superior to all other existing chatbots. The company previously hired computer scientist Ray Kurzweil in 2012 to develop multiple chatbots for the company, including one named Danielle. T... | LaMDA | 0.781007 |
90,126 | LaMDA (Language Model for Dialogue Applications) is a family of conversational large language models developed by Google. Originally developed and introduced as Meena in 2020, the first-generation LaMDA was announced during the 2021 Google I/O keynote, while the second generation was announced the following year. In Ju... | LaMDA | 0.781007 |
90,127 | Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hyp... | Lie sphere transformation | 0.781006 |
90,128 | Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres). | Lie sphere transformation | 0.781006 |
90,129 | Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite... | Lie sphere transformation | 0.781006 |
90,130 | This identification is not invariant under Lie transformations: in Lie invariant terms, Z 2n – 1 is the space of (projective) lines on the Lie quadric. Any immersed oriented hypersurface in n-dimensional space has a contact lift to Z 2n – 1 determined by its oriented tangent spaces. There is no longer a preferred Lie c... | Lie sphere transformation | 0.781006 |
90,131 | The incidence of cycles in Lie sphere geometry provides a simple solution to the problem of Apollonius. This problem concerns a configuration of three distinct circles (which may be points or lines): the aim is to find every other circle (including points or lines) which is tangent to all three of the original circles.... | Lie sphere transformation | 0.781006 |
90,132 | The fact that Lie transformations do not preserve points in general can also be a hindrance to understanding Lie sphere geometry. In particular, the notion of a curve is not Lie invariant. This difficulty can be mitigated by the observation that there is a Lie invariant notion of contact element. An oriented contact el... | Lie sphere transformation | 0.781006 |
90,133 | Lie sphere geometry in n-dimensions is obtained by replacing R3,2 (corresponding to the Lie quadric in n = 2 dimensions) by Rn + 1, 2. This is Rn + 3 equipped with the symmetric bilinear form ( x 0 , x 1 , … x n , x n + 1 , x n + 2 ) ⋅ ( y 0 , y 1 , … y n , y n + 1 , y n + 2 ) {\displaystyle (x_{0},x_{1},\ldots x_{n},x... | Lie sphere transformation | 0.781006 |
90,134 | In the case n=3, the quadric Q3 in P(R4,2) describes the (Lie) geometry of spheres in Euclidean 3-space. Lie noticed a remarkable similarity with the Klein correspondence for lines in 3-dimensional space (more precisely in RP3).Suppose , ∈ RP3, with homogeneous coordinates (x0,x1,x2,x3) and (y0,y1,y2,y3). Put pij = xi... | Lie sphere transformation | 0.781006 |
90,135 | Nevertheless, Künneth theorems in just the same form have been proved in very many cases by various other methods. The first were Michael Atiyah's Künneth theorem for complex K-theory and Pierre Conner and Edwin E. Floyd's result in cobordism. A general method of proof emerged, based upon a homotopical theory of module... | Kunneth formula | 0.781004 |
90,136 | There are many generalized (or "extraordinary") homology and cohomology theories for topological spaces. K-theory and cobordism are the best-known. Unlike ordinary homology and cohomology, they typically cannot be defined using chain complexes. Thus Künneth theorems can not be obtained by the above methods of homologic... | Kunneth formula | 0.781004 |
90,137 | In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces X and Y... | Kunneth formula | 0.781004 |
90,138 | For cellular chains on CW complexes, it is a straightforward isomorphism. Then the homology of the tensor product on the right is given by the spectral Künneth formula of homological algebra.The freeness of the chain modules means that in this geometric case it is not necessary to use any hyperhomology or total derived... | Kunneth formula | 0.781004 |
90,139 | The chain complex of the space X × Y is related to the chain complexes of X and Y by a natural quasi-isomorphism C ∗ ( X × Y ) ≅ C ∗ ( X ) ⊗ C ∗ ( Y ) . {\displaystyle C_{*}(X\times Y)\cong C_{*}(X)\otimes C_{*}(Y).} For singular chains this is the theorem of Eilenberg and Zilber. | Kunneth formula | 0.781004 |
90,140 | Let X and Y be two topological spaces. In general one uses singular homology; but if X and Y happen to be CW complexes, then this can be replaced by cellular homology, because that is isomorphic to singular homology. The simplest case is when the coefficient ring for homology is a field F. In this situation, the Künnet... | Kunneth formula | 0.781004 |
90,141 | The theory is applicable to all organizationally closed or homeostatic processes that produce enduring and coherent products (where spins have a fixed average phase relationship and also in the sense of Nicholas Rescher's coherence theory of truth with the proviso that the sets and their members exert repulsive forces ... | Self-organization in cybernetics | 0.780997 |
90,142 | His strict definition of concept "a procedure to bring about a relation" permitted his theorem "Like concepts repel, unlike concepts attract" to state a general spin based principle of self-organization. His edict, an exclusion principle, "There are No Doppelgangers" means no two concepts can be the same (all interacti... | Self-organization in cybernetics | 0.780997 |
90,143 | Ashby used his state counting measure of variety to describe stable states and produced the "Good Regulator" theorem which requires internal models for self-organized endurance and stability (e.g. Nyquist stability criterion). Warren McCulloch proposed "Redundancy of Potential Command" as characteristic of the organiza... | Self-organization in cybernetics | 0.780997 |
90,144 | The biophysicist Henri Atlan developed such a concept by proposing the principle of "complexity from noise" (French: le principe de complexité par le bruit) first in the 1972 book L'organisation biologique et la théorie de l'information and then in the 1979 book Entre le cristal et la fumée. The thermodynamicist Ilya P... | Self-organization in cybernetics | 0.780997 |
90,145 | By the compactness theorem, there is a model ∗ R {\displaystyle {}^{*}\mathbb {R} } that satisfies Σ {\displaystyle \Sigma } and also contains an infinitesimal element ε . {\displaystyle \varepsilon .} A similar argument, this time adjoining the axioms ω > 0 , ω > 1 , … , {\displaystyle \omega >0,\;\omega >1,\ldots ,} ... | Compactness theorem | 0.780997 |
90,146 | A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let Σ {\displaystyle \Sigma } be a first-order axiomatization of the theory of the real number... | Compactness theorem | 0.780997 |
90,147 | A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'na... | Compactness theorem | 0.780997 |
90,148 | A first-order sentence φ {\displaystyle \varphi } in the language of rings is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes p {\displaystyle p} for which φ {\displaystyle \varphi } is t... | Compactness theorem | 0.780997 |
90,149 | The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation. Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant p {\displaystyle p} such that the sentence holds for every field of characteristic larger t... | Compactness theorem | 0.780997 |
90,150 | The compactness theorem has many applications in model theory; a few typical results are sketched here. | Compactness theorem | 0.780997 |
90,151 | {\displaystyle A_{i}.} Now for any formula φ {\displaystyle \varphi } in Σ: {\displaystyle \Sigma :} the set A { φ } {\displaystyle A_{\{\varphi \}}} is in U {\displaystyle U} whenever j ∈ A { φ } , {\displaystyle j\in A_{\{\varphi \}},} then φ ∈ j , {\displaystyle \varphi \in j,} hence φ {\displaystyle \varphi } holds... | Compactness theorem | 0.780997 |
90,152 | One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fa... | Compactness theorem | 0.780997 |
90,153 | Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936. | Compactness theorem | 0.780997 |
90,154 | Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along wit... | Compactness theorem | 0.780997 |
90,155 | In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that... | Compactness theorem | 0.780997 |
90,156 | Some space and ground-based telescopes that have studied high energy astronomy include the following: | High-energy astrophysics | 0.780996 |
90,157 | Global organizations like the United Nations have included sustainable development goals (SDGs) to set a standard of 17 goals for countries around the world to take action on climate change and other related issues. Businesses primarily focus on the return of investment of changes such as their source of energy, consum... | Micro-sustainability | 0.780991 |
90,158 | Renewable energy can include hydropower, biomass energy, geothermal energy, wind power, and solar energy. Additionally, communities can educate and promote individual sustainable practices mentioned in part 2. This can be in the form of providing information such as directions to resources and household energy performa... | Micro-sustainability | 0.780991 |
90,159 | Institute of Laser Physics of the Siberian Branch of the RAS (Russian: Институт лазерной физики СО РАН) is a research institute in Akademgorodok of Novosibirsk, Russia. It was founded in 1991. | Institute of Laser Physics | 0.780991 |
90,160 | The history of the institute begins in the second half of the 1960s, when the Department of Laser Physics was created, headed by V. P. Chebotayev. At first, the department became part of the Institute of Semiconductor Physics, and since 1978 it has been included in the Institute of Thermophysics. In 1991, the Institute... | Institute of Laser Physics | 0.780991 |
90,161 | Irkutsk Division of Institute of Laser Physics of the Siberian Branch of the RAS | Institute of Laser Physics | 0.780991 |
90,162 | E ‖ P n − P ‖ C → 0 , {\displaystyle \ \operatorname {\mathbb {E} } {\Bigl \|}\mathbb {P} _{n}-\mathbb {P} {\Bigr \|}_{\mathcal {C}}\to 0\ ,} as n → ∞ {\displaystyle \ n\to \infty \ } (convergence in mean). The Glivenko–Cantelli classes of functions are defined similarly.A class is called a universal Glivenko–Cantell... | Glivenko–Cantelli theorem | 0.780989 |
90,163 | The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case. An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. (p 268) See asymptotic properties of the empirical distribution... | Glivenko–Cantelli theorem | 0.780989 |
90,164 | Let S = R {\displaystyle S=\mathbb {R} } and C = { ( − ∞ , t ]: t ∈ R } {\displaystyle {\mathcal {C}}=\{(-\infty ,t]:t\in {\mathbb {R} }\}} . The classical Glivenko–Cantelli theorem implies that this class is a universal GC class. Furthermore, by Kolmogorov's theorem, sup P ∈ P ( S , A ) ‖ P n − P ‖ C ∼ n − 1 / 2 {\dis... | Glivenko–Cantelli theorem | 0.780989 |
90,165 | In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distri... | Glivenko–Cantelli theorem | 0.780989 |
90,166 | In evolutionary biology, robustness of a biological system (also called biological or genetic robustness) is the persistence of a certain characteristic or trait in a system under perturbations or conditions of uncertainty. Robustness in development is known as canalization. According to the kind of perturbation involv... | Robustness (evolution) | 0.780989 |
90,167 | Biological processes at the molecular scale are inherently stochastic. They emerge from a combination of stochastic events that happen given the physico-chemical properties of molecules. For instance, gene expression is intrinsically noisy. This means that two cells in exactly identical regulatory states will exhibit d... | Robustness (evolution) | 0.780989 |
90,168 | Patterning mechanisms such as those described by the French flag model can be perturbed at many levels (production and stochasticity of the diffusion of the morphogen, production of the receptor, stochastic of the signaling cascade, etc). Patterning is therefore inherently noisy. Robustness against this noise and genet... | Robustness (evolution) | 0.780989 |
90,169 | In varying environments, perfect adaptation to one condition may come at the expense of adaptation to another. Consequently, the total selection pressure on an organism is the average selection across all environments weighted by the percentage time spent in that environment. Variable environment can therefore select f... | Robustness (evolution) | 0.780989 |
90,170 | Fermat's Last Theorem established that A n + B n = C n {\displaystyle A^{n}+B^{n}=C^{n}} has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat... | Beal conjecture | 0.780987 |
90,171 | (In fact, the maximum common prime factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.) The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided ( −... | Beal conjecture | 0.780987 |
90,172 | The Beal conjecture is the following conjecture in number theory: If A x + B y = C z , {\displaystyle A^{x}+B^{y}=C^{z},} where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.Equivalently, The equation A x + B y = C z {\displaystyle A^{x}+B^{y}=C^{z}} has no sol... | Beal conjecture | 0.780987 |
90,173 | The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y, z), there are at most finitely many coprime solutions for (A, B, C). : p. | Beal conjecture | 0.780987 |
90,174 | In the cases below where n is an exponent, multiples of n are also proven, since a kn-th power is also an n-th power. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat–Catalan conjecture#Known solutions. All cases of the form (2, 3, n) or (2, n, 3) have the solution... | Beal conjecture | 0.780987 |
90,175 | EHRs may include a range of data, including demographics, medical history, medication and allergies, immunization status, laboratory test results, radiology images, vital signs, personal statistics like age and weight, and billing information.For several decades, electronic health records (EHRs) have been touted as key... | Electronic Medical Records | 0.780987 |
90,176 | As mobile systems become more prevalent, practices will need comprehensive policies that govern security measures and patient privacy regulations.Other advanced computational techniques have allowed EHRs to be evaluated at a much quicker rate. Natural language processing is increasingly used to search EMRs, especially ... | Electronic Medical Records | 0.780987 |
90,177 | Philosophical anthropology Philosophical anthropology is the philosophical study of human nature. In terms of learning, examples of descriptive theories of the learner are: a mind, soul, and spirit capable of emulating the Absolute Mind (Idealism); an orderly, sensing, and rational being capable of understanding the wo... | Educational theorist | 0.780986 |
90,178 | Hang Gliding; Karate; Robin Leach; Elephant (October 15, 1994) Sled Dogs; Arctic Travel; Life In Camp; Arctic Weather (October 22, 1994) Aircraft Carrier; Brain; Carrier Life (October 29, 1994) Brain Mapping; Garlic; Sunscreens; Tasmanian Devil (November 5, 1994) Movie Dinosaurs; Bread Chemistry; Scott Hamilton; Wallab... | Newton's Apple | 0.780983 |
90,179 | Disney Cartoons; Skateboards; Smothers & Yo-Yos; "Lemons"; Turtles (October 13, 1990) Juggling; Greenwich; Echoes; Flies (October 20, 1990) Everglades; Tomorrow's World; Recycling Chat; Everglades Students (October 27, 1990) Demolition Derby; Stress; Topology Chat; Eagles (November 3, 1990) Kidney; Steve Allen & Music;... | Newton's Apple | 0.780983 |
90,180 | Plastic Surgery; High-Speed Bicycles; "Lemons"; Ailing Whale (October 10, 1987) Treasure Diving; Blackboard Chat; "Lemons"; Mothballs & Wintergreen (October 17, 1987) Polar Expedition; Inventors' Fair; Sea Otter (October 24, 1987) Submarine; Budgies In A Truck; Uvula Chat; Lemons, Sea World Science Day (October 31, 198... | Newton's Apple | 0.780983 |
90,181 | Earnie; Rainbow; Popcorn Chat; Pygmy Hippo (October 12, 1985) Dinosaurs; Bullet Proof Vest; Heartburn Chat; Killer Whales (October 22, 1985) Space Shuttle; Rube Goldberg; Crazy Inventions (October 29, 1985) Chimp Communication; Bends; Sweating Chat; Cold Remedies (November 2, 1985) Boomerangs; Cooking Chemistry; Bears ... | Newton's Apple | 0.780983 |
90,182 | Aerobic Exercise; Swimming & Weightlifting; Luge & Skating; Softball & Racewalking (October 12, 1991) Solar Cars; Steroids; The Amazing Kreskin; How A Foot Falls Asleep; Miniature Horses (October 19, 1991) Murder Mystery; Ted Nugent; Bats (October 26, 1991) Cancer Causes; Frisbees; Computer Virus (November 2, 1991) Sew... | Newton's Apple | 0.780983 |
90,183 | Television; Studio Tour; Post-Production; Satellites (October 10, 1992) Movie Stunts; Household Chemistry; Coffee & Cream; Musk Ox (October 17, 1992) Election Polls & Surveys; Electric Car; Ceramics; Cougars (October 24, 1992) Monster Make-Up; Ozone; Rearview Mirrors; Artificial Sweeteners (October 31, 1992) Oil Spill;... | Newton's Apple | 0.780983 |
90,184 | The axioms I0, I1, I2, and I3 were at first suspected to be inconsistent (in ZFC) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent. ... | Rank-into-rank | 0.780982 |
90,185 | In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubne... | Primorial | 0.780981 |
90,186 | This helps to define value in the text. The importance of relations of paradigmatic opposition is to help generate an order of dynamic complexity of experience in the past. People have believed in binary opposition since at least classical times: e.g. in Aristotle's physics of four elements earth, air, fire and water, ... | Paradigmatic analysis | 0.78098 |
90,187 | Thus, the network stack should not be composed at development time, but at runtime and needs the possibility to be adapted if needed. Additionally the decoupling relaxes the chains of the ISO OSI network layer model, and thus enables alternative concepts of communication stacks. Heuschkel et al. proposes the concept of... | Network Protocol Virtualization | 0.78098 |
90,188 | Having stipulated conventionally that one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in R N {\displaystyle R^{\mathbb {N} }} is defined and a topology on R N {\displaystyle R^{\mathbb {N} }} is constructed. There are several equivalent ways to ... | Formal power series ring | 0.780979 |
90,189 | This is because for i ≥ 2 {\displaystyle i\geq 2} the coefficient ( n i ) / n i {\displaystyle {\tbinom {n}{i}}/n^{i}} of X i {\displaystyle X^{i}} does not stabilize as n → ∞ {\displaystyle n\to \infty } . It does however converge in the usual topology of R {\displaystyle \mathbb {R} } , and in fact to the coefficient... | Formal power series ring | 0.780979 |
90,190 | So ∑ i = 0 ∞ X Y i {\displaystyle \sum _{i=0}^{\infty }XY^{i}} converges (and its sum can be written as X 1 − Y {\displaystyle {\tfrac {X}{1-Y}}} ); however ∑ i = 0 ∞ X i Y {\displaystyle \sum _{i=0}^{\infty }X^{i}Y} would be considered to be divergent, since every term affects the coefficient of Y {\displaystyle Y} . ... | Formal power series ring | 0.780979 |
90,191 | The above topology is the finest topology for which ∑ i = 0 ∞ a i X i {\displaystyle \sum _{i=0}^{\infty }a_{i}X^{i}} always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wis... | Formal power series ring | 0.780979 |
90,192 | More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutativ... | Formal power series ring | 0.780979 |
90,193 | In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose ... | Formal power series ring | 0.780979 |
90,194 | The topology has the useful property that an infinite summation converges if and only if the sequence of its terms converges to 0, which just means that any fixed power of X {\displaystyle X} occurs in only finitely many terms. The topological structure allows much more flexible usage of infinite summations. For instan... | Formal power series ring | 0.780979 |
90,195 | We may give R N {\displaystyle R^{\mathbb {N} }} the I-adic topology, where I = ( X ) {\displaystyle I=(X)} is the ideal generated by X {\displaystyle X} , which consists of all sequences whose first term a 0 {\displaystyle a_{0}} is zero. The desired topology could also be derived from the following metric. The distan... | Formal power series ring | 0.780979 |
90,196 | The last one being valid in the ring Q ] . {\displaystyle \mathbb {Q} ].} For K a field, the ring K ] {\displaystyle K]} is often used as the "standard, most general" complete local ring over K in algebra. | Formal power series ring | 0.780979 |
90,197 | Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions. One can use formal power series to prove several relations familiar from analysis i... | Formal power series ring | 0.780979 |
90,198 | of exp ( X ) {\displaystyle \exp(X)} . Therefore, if one would give R ] {\displaystyle \mathbb {R} ]} the product topology of R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} where the topology of R {\displaystyle \mathbb {R} } is the usual topology rather than the discrete one, then the above limit would converge to... | Formal power series ring | 0.780979 |
90,199 | This topology, which is also the I {\displaystyle I} -adic topology, where I = ( X , Y ) {\displaystyle I=(X,Y)} is the ideal generated by X {\displaystyle X} and Y {\displaystyle Y} , still enjoys the property that a summation converges if and only if its terms tend to 0. The same principle could be used to make other... | Formal power series ring | 0.780979 |
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