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89,500 | {\displaystyle f(z)=(z-\lambda _{i})^{\nu _{i}}.} The spectral mapping theorem tells us f ( T ) e i ( T ) = ( T − λ i ) ν i e i ( T ) {\displaystyle f(T)e_{i}(T)=(T-\lambda _{i})^{\nu _{i}}e_{i}(T)} has spectrum {0}. By property 1, f(T) can be directly computed in the Jordan form, and by inspection, we see that the ope... | Jordan canonical form | 0.781279 |
89,501 | In the finite-dimensional case, σ(T) = {λi} is a finite discrete set in the complex plane. Let ei be the function that is 1 in some open neighborhood of λi and 0 elsewhere. By property 3 of the functional calculus, the operator e i ( T ) {\displaystyle e_{i}(T)} is a projection. Moreover, let νi be the index of λi and ... | Jordan canonical form | 0.781279 |
89,502 | In molecular biology, Small nucleolar RNA Z50 is a non-coding RNA (ncRNA) molecule which functions in the modification of other small nuclear RNAs (snRNAs). This type of modifying RNA is usually located in the nucleolus of the eukaryotic cell which is a major site of snRNA biogenesis. It is known as a small nucleolar R... | Small nucleolar RNA Z50 | 0.781278 |
89,503 | If two methods are consistent, and one sums more series than the other, the one summing more series is stronger. There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants, as well as ... | Abelian mean | 0.781277 |
89,504 | In 1987, the Swiss company Demaurex purchased a license for the delta robot and started the production of delta robots for the packaging industry. In 1991, Reymond Clavel presented his doctoral thesis 'Conception d'un robot parallèle rapide à 4 degrés de liberté', and received the golden robot award in 1999 for his wor... | Delta robot | 0.781276 |
89,505 | All quantum-mechanical observables may be simultaneously assigned definite values (this is the realism postulate, which is false in standard quantum mechanics, since there are observables which are indefinite in every given quantum state). These global value assignments may deterministically depend on some 'hidden' cla... | Quantum contextuality | 0.781276 |
89,506 | The need for contextuality was discussed informally in 1935 by Grete Hermann, but it was more than 30 years later when Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for system... | Quantum contextuality | 0.781276 |
89,507 | The Kochen–Specker theorem proves that quantum mechanics is incompatible with realistic noncontextual hidden variable models. On the other hand Bell's theorem proves that quantum mechanics is incompatible with factorisable hidden variable models in an experiment in which measurements are performed at distinct spacelike... | Quantum contextuality | 0.781276 |
89,508 | A representative example is Hardy's nonlocality proof of nonlocality. Strong contextuality is a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be reproduced by a mixture of global value assignments, strong contextuality arises when no global value assignme... | Quantum contextuality | 0.781276 |
89,509 | (Probabilistic) contextuality may be witnessed in measurement statistics, e.g. by the violation of an inequality. A representative example is the KCBS proof of contextuality. Logical contextuality may be witnessed in the 'possibilistic' information about which outcome events are possible and which are not possible. | Quantum contextuality | 0.781276 |
89,510 | The sheaf-theoretic, or Abramsky–Brandenburger, approach to contextuality initiated by Samson Abramsky and Adam Brandenburger is theory-independent and can be applied beyond quantum theory to any situation in which empirical data arises in contexts. As well as being used to study forms of contextuality arising in quant... | Quantum contextuality | 0.781276 |
89,511 | This stochasticity is however epistemic and not ontic as in the standard formulation of quantum mechanics. Value assignments pre-exist and are independent of the choice of any other observables which, in standard quantum mechanics, are described as commuting with the measured observable, and they are also measured. Som... | Quantum contextuality | 0.781276 |
89,512 | In 2013, Robert Raussendorf showed more generally that access to strongly contextual measurement statistics is necessary and sufficient for an l2-MBQC to compute a non-linear function. He also showed that to compute non-linear Boolean functions with sufficiently high probability requires contextuality. | Quantum contextuality | 0.781276 |
89,513 | This equivalence class is called an extravalence class, and the associated transfer of certainty between contexts is called extracontextuality. As a simple example, the usual singlet state for two spins 1/2 can be found in the (non commuting) CSCOs associated with the measurement of the total spin (with S = 0 , m = 0 {... | Quantum contextuality | 0.781276 |
89,514 | Spekkens' contextuality can be motivated using Leibniz's law of the identity of indiscernibles. The law applied to physical systems in this framework mirrors the entended definition of noncontextuality. This was further explored by Simmons et al, who demonstrated that other notions of contextuality could also be motiva... | Quantum contextuality | 0.781276 |
89,515 | Adán Cabello, Simone Severini, and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories. Within this framework experimental scenarios are described by graphs, and certain invariants of these graphs were shown have particular physical significance. One w... | Quantum contextuality | 0.781276 |
89,516 | Here normalisation means that violations are expressed as fractions of the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for which this violation is attained. In this sense the contextual fraction is a more neutral measure ... | Quantum contextuality | 0.781276 |
89,517 | A number of methods exist for quantifying contextuality. One approach is by measuring the degree to which some particular noncontextuality inequality is violated, e.g. the KCBS inequality, the Yu–Oh inequality, or some Bell inequality. A more general measure of contextuality is the contextual fraction.Given a set of me... | Quantum contextuality | 0.781276 |
89,518 | A further generalization and refinement of these results due to Samson Abramsky, Rui Soares Barbosa and Shane Mansfield appeared in 2017, proving a precise quantifiable relationship between the probability of successfully computing any given non-linear function and the degree of contextuality present in the l2-MBQC as ... | Quantum contextuality | 0.781276 |
89,519 | Recently, quantum contextuality has been investigated as a source of quantum advantage and computational speedups in quantum computing. | Quantum contextuality | 0.781276 |
89,520 | The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the per... | Quantum contextuality | 0.781276 |
89,521 | Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) o... | Quantum contextuality | 0.781276 |
89,522 | Measurement-based quantum computation (MBQC) is a model for quantum computing in which a classical control computer interacts with a quantum system by specifying measurements to be performed and receiving measurement outcomes in return. The measurement statistics for the quantum system may or may not exhibit contextual... | Quantum contextuality | 0.781276 |
89,523 | Magic state distillation is a scheme for quantum computing in which quantum circuits constructed only of Clifford operators, which by themselves are fault-tolerant but efficiently classically simulable, are injected with certain "magic" states that promote the computational power to universal fault-tolerant quantum com... | Quantum contextuality | 0.781276 |
89,524 | CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is strongly consistently connected, which means that the joint distributions of { R q 1 c , … , R q k c } {\displaystyle \left\{R_{q_{1}}^{c},\ldots ,R_{q_{k}}^{c}\right\}} and { R q 1 c ′ , … , R q k c ′ } {\disp... | Quantum contextuality | 0.781276 |
89,525 | Except for certain logical issues, in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to D ( C n ) ≤ n − 2 , {\displaystyle D\left({\mathcal {C}}_{n}\right)\leq n-2,} which incl... | Quantum contextuality | 0.781276 |
89,526 | In plasma physics, a wave can be collisionlessly damped by particles in the plasma whose velocity matches the wave's phase velocity. This is typically represented as a spatially dispersive loss in the plasma's permittivity. | Spatial dispersion | 0.781275 |
89,527 | Another machining process to increase the surface quality and dimensional accuracy of a bore hole is the use of single-bladed reamers. Reaming is the ountersinking of a pre-drilled hole, where the tool is supported by the guide pads themselves. Therefore, the tool geometry of these reamers is very similar to single-lip... | Deep hole drilling | 0.781275 |
89,528 | In the physics of continuous media, spatial dispersion is a phenomenon where material parameters such as permittivity or conductivity have dependence on wavevector. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials. Spatial dispersi... | Spatial dispersion | 0.781275 |
89,529 | The drill head is usually made of carbides of the ISO cutting application group K 10 to K 20 and is coated if required. In special applications, PCD, cermets, ceramics or high-speed steels are also used. The choice of the drill head geometry is made depending on the existing machining situation. | Deep hole drilling | 0.781275 |
89,530 | The ocean's chemistry is changing due to the uptake of anthropogenic carbon dioxide (CO2). : 395 Ocean pH, carbonate ion concentrations (), and calcium carbonate mineral saturation states (Ω) have been declining as a result of the uptake of approximately 30% of the anthropogenic carbon dioxide emissions over the past 2... | Oceanic acidification | 0.781274 |
89,531 | Therefore, greater seawater warming could limit CO2 absorption and lead to a smaller change in pH for a given increase in CO2. The difference in changes in temperature between basins is one of the main reasons for the differences in acidification rates in different localities. Current rates of ocean acidification have ... | Oceanic acidification | 0.781274 |
89,532 | These expected changes are considered unprecedented in the geological record. In combination with other ocean biogeochemical changes, this drop in pH value could undermine the functioning of marine ecosystems and disrupt the provision of many goods and services associated with the ocean beginning as early as 2100.The e... | Oceanic acidification | 0.781274 |
89,533 | However, the additional CO2 in the ocean results in a wholesale shift in seawater acid-base chemistry toward more acidic, lower pH conditions and lower saturation states for carbonate minerals used in many marine organism shells and skeletons.Cumulated since 1850, the ocean sink holds up to 175 ± 35 gigatons of carbon,... | Oceanic acidification | 0.781274 |
89,534 | As a result, when sediment is transported, rare-earth element concentrations are unaffected by the fluid and instead the rock retains the rare-earth element concentration from its source.Sea and river waters typically have low rare-earth element concentrations. However, aqueous geochemistry is still very important. In ... | Rare earth element | 0.781274 |
89,535 | The anomalies represent enrichment (positive anomalies) or depletion (negative anomalies) of specific elements along the series and are graphically recognizable as positive or negative “peaks” along the REE patterns. The anomalies can be numerically quantified as the ratio between the normalized concentration of the el... | Rare earth element | 0.781274 |
89,536 | The application of rare-earth elements to geology is important to understanding the petrological processes of igneous, sedimentary and metamorphic rock formation. In geochemistry, rare-earth elements can be used to infer the petrological mechanisms that have affected a rock due to the subtle atomic size differences bet... | Rare earth element | 0.781274 |
89,537 | At the other extreme from indecomposability is infinite divisibility. Cramér's theorem shows that while the normal distribution is infinitely divisible, it can only be decomposed into normal distributions. Cochran's theorem shows that the terms in a decomposition of a sum of squares of normal random variables into sums... | Indecomposable distribution | 0.781274 |
89,538 | In probability theory, an indecomposable distribution is a probability distribution that cannot be represented as the distribution of the sum of two or more non-constant independent random variables: Z ≠ X + Y. If it can be so expressed, it is decomposable: Z = X + Y. If, further, it can be expressed as the distributio... | Indecomposable distribution | 0.781274 |
89,539 | The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups. | Weyl representation | 0.78127 |
89,540 | It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly. : Section 14.4 The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from L2(Rn) to the Segal–Bargmann space that intertwines the usual an... | Weyl representation | 0.78127 |
89,541 | These properties follow from the orthogonality relations for characters of representations of finite groups. For any non-zero h in K define the representation Uh on the finite-dimensional inner product space ℓ2(Kn) by It follows that By the orthogonality relations for characters of representations of finite groups this... | Weyl representation | 0.78127 |
89,542 | The Heisenberg group Hn(K) is defined for any commutative ring K. In this section let us specialize to the field K = Z/pZ for p a prime. This field has the property that there is an embedding ω of K as an additive group into the circle group T. Note that Hn(K) is finite with cardinality |K|2n + 1. For finite Heisenberg... | Weyl representation | 0.78127 |
89,543 | The idea of the Stone–von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples. : Example 14.5 To obta... | Weyl representation | 0.78127 |
89,544 | Further analysis shows that any two self-adjoint operators satisfying the above commutation relation cannot be both bounded (in fact, a theorem of Wielandt shows the relation cannot be satisfied by elements of any normed algebra). For notational convenience, the nonvanishing square root of ℏ may be absorbed into the no... | Weyl representation | 0.78127 |
89,545 | The operators x {\displaystyle x} , p {\displaystyle p} satisfy the canonical commutation relation Lie algebra, Already in his classic book, Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ℏ vanishes. This is apparent from ta... | Weyl representation | 0.78127 |
89,546 | In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line R {\displaystyle \mathbb {R} } , there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a part... | Weyl representation | 0.78127 |
89,547 | He was born at Kilbehenny in Limerick, Ireland and educated locally at Mitchelstown, before becoming a teacher under the Board of National Education. He later became headmaster of the Central Model Schools in Kilkenny City. He subsequently entered Trinity College Dublin in 1858, where he was elected a Scholar in 1861 a... | John Casey (mathematician) | 0.781269 |
89,548 | 1880: On Cubic Transformations 1881: On Cyclides and Sphero-quartics, from Internet Archive 1882: The First Six Books of the Elements of Euclid, link from Project Gutenberg 1885: A Treatise on the Analytic Geometry of the Point, Line, Circle and Conic Sections, Second edition, 1893, links from Internet Archive 1886 A S... | John Casey (mathematician) | 0.781269 |
89,549 | John Casey (12 May 1820, Kilbehenny, County Limerick, Ireland – 3 January 1891, Dublin) was a respected Irish geometer. He is most famous for Casey's theorem on a circle that is tangent to four other circles, an extension of Ptolemy's theorem. However, he contributed several novel proofs and perspectives on Euclidean g... | John Casey (mathematician) | 0.781269 |
89,550 | GEMPACK (General Equilibrium Modelling PACKage) is a modeling system for CGE economic models, used at the Centre of Policy Studies (CoPS) in Melbourne, Australia, and sold to other CGE modellers. Some of the more well-known CGE models solved using GEMPACK are the GTAP model of world trade, and the MONASH, MMRF, ORANI-G... | RunGEM | 0.781269 |
89,551 | The original equation system is linearized (reformulated as a system of first-order partial differential equations). If most variables are expressed in terms of percentage changes (akin to log changes) the coefficients of the linearized system are usually very simple functions of database value flows. Computer algebra ... | RunGEM | 0.781269 |
89,552 | In the meantime, many researchers still call this model the "Prandtl–Tomlinson Model". In Russia this model was introduced by the Soviet physicists Yakov Frenkel and T. Kontorova. The Frenkel defect became firmly fixed in the physics of solids and liquids. In the 1930s, this research was supplemented with works on the ... | Tomlinson model | 0.781266 |
89,553 | In monodromy terms, the question is of identifying the cases of finite monodromy group. By reformulation and passing to a larger system, the essential case is for rational functions in A and rational number coefficients. Then a necessary condition is that for almost all prime numbers p, the system defined by reduction ... | Grothendieck conjecture | 0.781266 |
89,554 | In a simplest possible statement the conjecture can be stated in its essentials for a vector system written as d v / d z = A ( z ) v {\displaystyle dv/dz=A(z)v} for a vector v of size n, and an n×n matrix A of algebraic functions with algebraic number coefficients. The question is to give a criterion for when there is ... | Grothendieck conjecture | 0.781266 |
89,555 | In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case. It is a conjecture of Alexander Gro... | Grothendieck conjecture | 0.781266 |
89,556 | Gaussian measure on Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (with its Borel topology and σ {\displaystyle \sigma } -algebra) is strictly positive. Wiener measure on the space of continuous paths in R n {\displaystyle \mathbb {R} ^{n}} is a strictly positive measure — Wiener measure is an example of a Gauss... | Strictly positive measure | 0.781264 |
89,557 | Counting measure on any set X {\displaystyle X} (with any topology) is strictly positive. Dirac measure is usually not strictly positive unless the topology T {\displaystyle T} is particularly "coarse" (contains "few" sets). For example, δ 0 {\displaystyle \delta _{0}} on the real line R {\displaystyle \mathbb {R} } wi... | Strictly positive measure | 0.781264 |
89,558 | Let ( X , T ) {\displaystyle (X,T)} be a Hausdorff topological space and let Σ {\displaystyle \Sigma } be a σ {\displaystyle \sigma } -algebra on X {\displaystyle X} that contains the topology T {\displaystyle T} (so that every open set is a measurable set, and Σ {\displaystyle \Sigma } is at least as fine as the Borel... | Strictly positive measure | 0.781264 |
89,559 | Poisson clumping is named for 19th-century French mathematician Siméon Denis Poisson, known for his work on definite integrals, electromagnetic theory, and probability theory, and after whom the Poisson distribution is also named. | Poisson clumping | 0.781264 |
89,560 | Not long after Hoffbauer's and Reibisch's findings were published, aging was used in fishery assessments of the early 1900s. One of the first to focus on the applications of fish aging was the Norwegian fisheries scientist Johan Hjort. Focusing on fish scales, Hjort developed an extensive aging program collecting stati... | Age determination in fish | 0.781262 |
89,561 | In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the ... | Offset logarithmic integral | 0.781262 |
89,562 | The American Board of Science in Nuclear Medicine, founded in 1976, also known as the ABSNM is an organization in North America that offers certification to specialty areas of advanced nuclear medicine. The board offers certification in the following areas: Nuclear Medicine Physics and Instrumentation Radiopharmaceutic... | American Board of Science in Nuclear Medicine | 0.78126 |
89,563 | On macOS, BSD systems, Linux distributions, and Microsoft Windows: To display the statistics for only the TCP or UDP protocols, type one of the following commands: netstat -sp tcp netstat -sp udp | Netstat | 0.78126 |
89,564 | In computing, netstat (network statistics) is a command-line network utility that displays network connections for Transmission Control Protocol (both incoming and outgoing), routing tables, and a number of network interface (network interface controller or software-defined network interface) and network protocol stati... | Netstat | 0.78126 |
89,565 | Netstat provides statistics for the following: Proto – The name of the protocol (TCP or UDP). Local Address – The IP address of the local computer and the port number being used. The name of the local computer that corresponds to the IP address and the name of the port is shown unless the -n parameter is specified. An ... | Netstat | 0.78126 |
89,566 | Under Linux, raw data can often be obtained from the /proc/net/dev to work around the printf output corruption arising in netstat's network interface statistics summary, netstat -i, until such time as the problem is corrected.On the Windows platform, netstat information can be retrieved by calling the GetTcpTable and G... | Netstat | 0.78126 |
89,567 | The Low-Frequency Array, or LOFAR, is a large radio telescope, with an antenna network located mainly in the Netherlands, and spreading across 7 other European countries as of 2019. Originally designed and built by ASTRON, the Netherlands Institute for Radio Astronomy, it was first opened by Queen Beatrix of The Nether... | Low-Frequency Array | 0.78126 |
89,568 | A consequence of Cauchy's postulate is Cauchy's Fundamental Lemma, also called the Cauchy reciprocal theorem,: p.103–130 which states that the stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy's fundamental lemma is equivalent to Newton's third law of m... | Principal stress | 0.781259 |
89,569 | For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum S p e c ( R ) . {\displaystyle \mathrm {Spec} (R).} If R is an Artinian ring, then this map becomes a bijection. Matlis' Theorem: For a commutative Noetherian ring R, the map ... | Associated prime | 0.781258 |
89,570 | A nonzero R module N is called a prime module if the annihilator A n n R ( N ) = A n n R ( N ′ ) {\displaystyle \mathrm {Ann} _{R}(N)=\mathrm {Ann} _{R}(N')\,} for any nonzero submodule N' of N. For a prime module N, A n n R ( N ) {\displaystyle \mathrm {Ann} _{R}(N)\,} is a prime ideal in R.An associated prime of an R... | Associated prime | 0.781258 |
89,571 | Taylor received a degree in physics from the University of Liverpool. Taylor earned a PhD in space physics which focused on Magnetohydrodynamics modelling of astrophysical plasma in the magnetosphere from Imperial College London. | Matt Taylor (scientist) | 0.781258 |
89,572 | Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms by Diane Ravitch: There was the nonstandard analysis movement for teaching elementary calculus. Its stock rose a bit before the movement collapsed from inner complexity and scant necessity.Nonstandard calculus in the classroom has ... | Criticism of nonstandard analysis | 0.781256 |
89,573 | Ian Stewart (1986) compared Halmos' asking Bishop to review Keisler's book, to inviting Margaret Thatcher to review Das Kapital. Katz & Katz (2010) point out that Bishop is criticizing apples for not being oranges: the critic (Bishop) and the criticized (Robinson's non-standard analysis) do not share a common foundatio... | Criticism of nonstandard analysis | 0.781256 |
89,574 | 192). After evoking the "success" of nonstandard analysis at the most elementary level at which it could be introduced—namely, at which calculus is taught for the first time,Dauben stated: there is also a deeper level of meaning at which nonstandard analysis operates.Dauben mentioned "impressive" applications in physic... | Criticism of nonstandard analysis | 0.781256 |
89,575 | In his response in The Notices, Keisler (1977, p. 269) asked: why did Paul Halmos, the Bulletin book review editor, choose a constructivist as the reviewer?Comparing the use of the law of excluded middle (rejected by constructivists) to wine, Keisler likened Halmos' choice with "choosing a teetotaller to sample wine". ... | Criticism of nonstandard analysis | 0.781256 |
89,576 | In "Brisure de symétrie spontanée et géométrie du point de vue spectral", Journal of Geometry and Physics 23 (1997), 206–234, Alain Connes wrote: "The answer given by non-standard analysis, namely a nonstandard real, is equally disappointing: every non-standard real canonically determines a (Lebesgue) non-measurable su... | Criticism of nonstandard analysis | 0.781256 |
89,577 | (Even the notorious (ε, δ)-definition of limit is common sense, and moreover it is central to the important practical problems of approximation and estimation.) They do not believe me. In fact the idea makes them uncomfortable because it contradicts their previous experience. Now we have a calculus text that can be use... | Criticism of nonstandard analysis | 0.781256 |
89,578 | No invocation of Newton and Leibniz is going to justify developing calculus using axioms V* and VI*-on the grounds that the usual definition of a limit is too complicated! and Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. | Criticism of nonstandard analysis | 0.781256 |
89,579 | However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line.The review criticized Keisler's text for not providing evidence to support these statements, and for adopting an axiomatic approach when it was not clear to the students there was any system that satisfied th... | Criticism of nonstandard analysis | 0.781256 |
89,580 | Bishop reviewed the book Elementary Calculus: An Infinitesimal Approach by Howard Jerome Keisler, which presented elementary calculus using the methods of nonstandard analysis. Bishop was chosen by his advisor Paul Halmos to review the book. The review appeared in the Bulletin of the American Mathematical Society in 19... | Criticism of nonstandard analysis | 0.781256 |
89,581 | Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below. | Criticism of nonstandard analysis | 0.781256 |
89,582 | Leibniz historian Henk Bos (1974) acknowledged that Robinson's hyperreals provide preliminary explanation of why the calculus could develop on the insecure foundation of the acceptance of infinitely small and infinitely large quantities.F. Medvedev (1998) further points out that onstandard analysis makes it possible t... | Criticism of nonstandard analysis | 0.781256 |
89,583 | My interest in non-standard analysis is that attempts are being made to introduce it into calculus courses. It is difficult to believe that debasement of meaning could be carried so far.Katz & Katz (2010) note that a number of criticisms were voiced by the participating mathematicians and historians following Bishop's ... | Criticism of nonstandard analysis | 0.781256 |
89,584 | In molecular biology, gel extraction or gel isolation is a technique used to isolate a desired fragment of intact DNA from an agarose gel following agarose gel electrophoresis. After extraction, fragments of interest can be mixed, precipitated, and enzymatically ligated together in several simple steps. This process, u... | Gel extraction | 0.781254 |
89,585 | When the amount of one factor of production increases, the production of the good that uses that particular production factor intensively increases relative to the increase in the factor of production, as the H–O model assumes perfect competition where price is equal to the costs of factors of production. This theorem ... | Heckscher–Ohlin model | 0.781253 |
89,586 | The original Heckscher–Ohlin model and extended model such as the Vanek model performs poorly, as it is shown in the section "Econometric testing of H–O model theorems". Daniel Trefler and Susan Chun Zhu summarizes their paper that "It is hard to believe that factor endowments theory could offer an adequate explanatio... | Heckscher–Ohlin model | 0.781253 |
89,587 | The assumptions of H–O are unrealistic with respect to north–south trade. Income differences between North and South is the concern that third world cares most. The factor price equalization theorem has not shown a sign of realization, even for a long time lag of a half century. | Heckscher–Ohlin model | 0.781253 |
89,588 | The factor equalization theorem (FET) applies only to the most advanced countries. The average wage in Japan was once as big as 70 times the wage in Vietnam. These wage discrepancies are not normally in the scope of the H–O model analysis.Heckscher–Ohlin theory is badly adapted to the analyze south–north trade problems... | Heckscher–Ohlin model | 0.781253 |
89,589 | Under constant returns to scale, doubling both capital and labor leads to a doubling of the output. Since outputs are increasing in both factors of production, doubling capital while holding labor constant leads to less than doubling of an output. Diminishing returns to capital and diminishing returns to labor are cruc... | Heckscher–Ohlin model | 0.781253 |
89,590 | Heckscher and Ohlin considered the Factor-Price Equalization theorem an econometric success because the large volume of international trade in the late 19th and early 20th centuries coincided with the convergence of commodity and factor prices worldwide. Modern econometric estimates have shown the model to perform poor... | Heckscher–Ohlin model | 0.781253 |
89,591 | Free and competitive trade makes factor prices converge along with traded goods prices. The FPE theorem is the most significant conclusion of the H–O model, but also has found the least agreement with the economic evidence. Neither the rental return to capital, nor the wage rates seem to consistently converge between t... | Heckscher–Ohlin model | 0.781253 |
89,592 | The Magnification effect on production quantity-shifts induced by endowment changes (via the Rybczynski theorem) predicts a larger proportionate shift in output-quantity than in the corresponding endowment factor shift that induced it. This has implications to both labor and capital: Assuming fixed capital, population ... | Heckscher–Ohlin model | 0.781253 |
89,593 | The Stolper–Samuelson theorem concerns nominal rents and wages. The Magnification effect on prices considers the effect of output-goods price-changes on the real return to capital and labor. This is done by dividing the nominal rates with a price index, but took thirty years to develop completely because of the theoret... | Heckscher–Ohlin model | 0.781253 |
89,594 | The Leontief paradox, presented by Wassily Leontief in 1953, found that the U.S. (the most capital-abundant country in the world by any criterion) exported labor-intensive commodities and imported capital-intensive commodities, contrary to the Heckscher–Ohlin theory.However, if labor is separated into two distinct fact... | Heckscher–Ohlin model | 0.781253 |
89,595 | The heralded single-photon sources are also used to examine the fundamental physics laws in quantum mechanics. There are two commonly used types of heralded single-photon sources: spontaneous parametric down-conversion and spontaneous four-wave mixing. The first source has line-width around THz and the second one has l... | Single-photon source | 0.781252 |
89,596 | In organic chemistry, a sulfone is a organosulfur compound containing a sulfonyl (R−S(=O)2−R’) functional group attached to two carbon atoms. The central hexavalent sulfur atom is double-bonded to each of two oxygen atoms and has a single bond to each of two carbon atoms, usually in two separate hydrocarbon substituent... | Sulfone | 0.781252 |
89,597 | In reviews dated 2009 and 2011, Guralnik states that in the GHK model the boson is massless only in a lowest-order approximation, but it is not subject to any constraint and acquires mass at higher orders, and adds that the GHK paper was the only one to show that there are no massless Goldstone bosons in the model and ... | Introduction to the Higgs field | 0.78125 |
89,598 | The three papers written in 1964 were each recognised as milestone papers during Physical Review Letters's 50th anniversary celebration. Their six authors were also awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics for this work. (A controversy also arose the same year, because in the event of a Nob... | Introduction to the Higgs field | 0.78125 |
89,599 | Additionally, solutions based on spontaneous symmetry breaking appeared to fail, seemingly an inevitable result of Goldstone's theorem. Because there is no potential energy cost to moving around the complex plane's "circular valley" responsible for spontaneous symmetry breaking, the resulting quantum excitation is pure... | Introduction to the Higgs field | 0.78125 |
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