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27,400 | As above with radio astronomy, there are certain limitations, e.g. N2 is difficult to detect by either IR or radio astronomy. Such IR observations have determined that in dense clouds (where there are enough particles to attenuate the destructive UV radiation) thin ice layers coat the microscopic particles, permitting ... | Interstellar chemistry | 0.806664 |
27,401 | The PAHs are thought to form in hot circumstellar environments (around dying, carbon-rich red giant stars). Infrared astronomy has also been used to assess the composition of solid materials in the interstellar medium, including silicates, kerogen-like carbon-rich solids, and ices. This is because unlike visible light,... | Interstellar chemistry | 0.806664 |
27,402 | While radio astronomy was developed in the 1930s, it was not until 1937 that any substantial evidence arose for the conclusive identification of an interstellar molecule – up until this point, the only chemical species known to exist in interstellar space were atomic. These findings were confirmed in 1940, when McKella... | Interstellar chemistry | 0.806664 |
27,403 | A derivative is an instrument that allows for buying and selling of the same asset on two markets – the spot market and the derivatives market. Mathematical finance assumes that any imbalance between the two markets will be arbitraged away. Thus, in a correctly priced derivative contract, the derivative price, the stri... | Rational pricing | 0.806663 |
27,404 | Note that above, the risk neutral formula does not refer to the expected or forecast return of the underlying, nor its volatility – p as solved, relates to the risk-neutral measure as opposed to the actual probability distribution of prices. Nevertheless, both arbitrage free pricing and risk neutral valuation deliver i... | Rational pricing | 0.806663 |
27,405 | Before the multitudinous components of present biochemistry could come to lean together they had to lean on something else." However, neither Muller or Cairns-Smith claimed their ideas as evidence of something supernatural.An early concept of irreducibly complex systems comes from Ludwig von Bertalanffy (1901–1972), an... | Irreducible complexity | 0.806662 |
27,406 | In the late 19th century, in a dispute between supporters of the adequacy of natural selection and those who held for inheritance of acquired characteristics, one of the arguments made repeatedly by Herbert Spencer, and followed by others, depended on what Spencer referred to as co-adaptation of co-operative parts, as ... | Irreducible complexity | 0.806662 |
27,407 | The appearance of design in aspects of biology is overwhelming. Since nothing other than an intelligent cause has been demonstrated to be able to yield such a strong appearance of design, Darwinian claims notwithstanding, the conclusion that the design seen in life is real design is rationally justified. (18:90–91, 18:... | Irreducible complexity | 0.806662 |
27,408 | At the 2005 Kitzmiller v. Dover Area School District trial, expert witness testimony defending ID and IC was given by Behe and Scott Minnich, who had been one of the "Johnson-Behe cadre of scholars" at Pajaro Dunes in 1993, was prominent in ID, and was now a tenured associate professor in microbiology at the University... | Irreducible complexity | 0.806662 |
27,409 | The April 7, 2006 issue of Science published this research.Irreducible complexity may not actually exist in nature, and the examples given by Behe and others may not in fact represent irreducible complexity, but can be explained in terms of simpler precursors. The theory of facilitated variation challenges irreducible ... | Irreducible complexity | 0.806662 |
27,410 | Evolution can act to simplify as well as to complicate. This raises the possibility that seemingly irreducibly complex biological features may have been achieved with a period of increasing complexity, followed by a period of simplification. A team led by Joseph Thornton, assistant professor of biology at the Universit... | Irreducible complexity | 0.806662 |
27,411 | The model was first introduced by Pat Moran in 1954 where a discrete-time model was considered. Fluid queues allow arrivals to be continuous rather than discrete, as in models like the M/M/1 and M/G/1 queues. Fluid queues have been used to model the performance of a network switch, a router, the IEEE 802.11 protocol, A... | Fluid queue | 0.806662 |
27,412 | The eluted unbound proteins are highly concentrated with the N-terminal peptides and neo-N-terminal peptides. This eluted sample is then quantified and analysis completed by MS/MS. The final step in TAILS involves bioinformatics. Using a hierarchical substrate winnowing process that discriminates from background proteo... | Terminal amine isotopic labeling of substrates | 0.80666 |
27,413 | Pulse computation is a hybrid of digital & analog computation that uses aperiodic electrical spikes, as opposed to the periodic voltages in a digital computer or the continuously varying voltages in an analog computer. Pulse streams are unclocked, so they can arrive at arbitrary times and can be generated by analog pro... | Pulse computation | 0.806659 |
27,414 | The generalization of the Nevanlinna–Pick theorem became an area of active research in operator theory following the work of Donald Sarason on the Sarason interpolation theorem. Sarason gave a new proof of the Nevanlinna–Pick theorem using Hilbert space methods in terms of operator contractions. Other approaches were d... | Pick matrix | 0.806657 |
27,415 | The Nevanlinna–Pick theorem states the following. Given λ 1 , … , λ n , z 1 , … , z n ∈ D {\displaystyle \lambda _{1},\ldots ,\lambda _{n},z_{1},\ldots ,z_{n}\in \mathbb {D} } , there exists a holomorphic function φ: D → D ¯ {\displaystyle \varphi :\mathbb {D} \to {\overline {\mathbb {D} }}} such that φ ( λ i ) = z i {... | Pick matrix | 0.806657 |
27,416 | However, one can avoid using this lemma by proving the above inequality first for the dyadic maximal functions M Δ α f ( x ) = sup x ∈ Q ∈ Δ α 1 | Q | ∫ Q | f ( x ) | d x . {\displaystyle M_{\Delta ^{\alpha }}f(x)=\sup _{x\in Q\in \Delta ^{\alpha }}{\frac {1}{|Q|}}\int _{Q}|f(x)|dx.} The proof is similar to the proof o... | One-third trick | 0.806657 |
27,417 | The appeal of the one-third trick is that one can first prove dyadic versions of a theorem and then deduce "non-dyadic" theorems from those. For example, recall the Hardy-Littlewood Maximal function M f ( x ) = sup r > 0 1 | B ( x , r ) | ∫ B ( x , r ) | f ( y ) | d y {\displaystyle Mf(x)=\sup _{r>0}{\frac {1}{|B(x,r)|... | One-third trick | 0.806657 |
27,418 | In computer programming languages, a switch statement is a type of selection control mechanism used to allow the value of a variable or expression to change the control flow of program execution via search and map. Switch statements function somewhat similarly to the if statement used in programming languages like C/C+... | Switch statement | 0.806656 |
27,419 | Optimizing compilers such as GCC or Clang may compile a switch statement into either a branch table or a binary search through the values in the cases. A branch table allows the switch statement to determine with a small, constant number of instructions which branch to execute without having to go through a list of com... | Switch statement | 0.806656 |
27,420 | Many languages evaluate expressions inside switch blocks at runtime, allowing a number of less obvious uses for the construction. This prohibits certain compiler optimizations, so is more common in dynamic and scripting languages where the enhanced flexibility is more important than the performance overhead. | Switch statement | 0.806656 |
27,421 | Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the physical quantities (observables) of A have been measured (for instance, coordinates o... | Hidden variable theory | 0.806655 |
27,422 | The debates between Bohr and Einstein essentially concluded in 1935, when Einstein finally expressed what is widely considered his best argument for the incompleteness of quantum mechanics. Einstein, Podolsky, and Rosen had proposed their definition of a "complete" description as one that uniquely determines the values... | Hidden variable theory | 0.806655 |
27,423 | In solid-state physics, band bending refers to the process in which the electronic band structure in a material curves up or down near a junction or interface. It does not involve any physical (spatial) bending. When the electrochemical potential of the free charge carriers around an interface of a semiconductor is dis... | Band bending | 0.806653 |
27,424 | The electric field in the depletion region separates the electrons and holes generating a current when the two sides of the p-n diode are connected. Photovoltaic cells are an important supplier of renewable energy. They are a promising source of reliable clean energy. | Band bending | 0.806653 |
27,425 | The names for chemical elements in East Asian languages, along with those for some chemical compounds (mostly organic), are among the newest words to enter the local vocabularies. Except for those metals well-known since antiquity, the names of most elements were created after modern chemistry was introduced to East As... | Chemical elements in East Asian languages | 0.806649 |
27,426 | It is the root of the word 化学 (huàxué) 'chemistry'. For example, P4S10 is called 十硫化四磷 (shíliúhuàsìlín) (literally: 'ten sulfur of four phosphorus', 'decasulfide of tetraphosphorus'). As in English nomenclature, if m = 1, the numerical prefix of E is usually dropped in covalent compounds. | Chemical elements in East Asian languages | 0.806649 |
27,427 | Let F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} be the set of all functions from a set X to real numbers R {\displaystyle \mathbb {R} } . Because R {\displaystyle \mathbb {R} } is a field, F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} may be turned into a vector space and a commutative a... | Real-valued function | 0.806645 |
27,428 | The σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that the preimage f −1(B) of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a vector space and an algebra as explained above in § Algebraic s... | Real-valued function | 0.806645 |
27,429 | Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space (which yields a real multivariable function), a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as ... | Real-valued function | 0.806645 |
27,430 | Other contexts where real-valued functions and their special properties are used include monotonic functions (on ordered sets), convex functions (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one or more real variables), algebraic functions (o... | Real-valued function | 0.806645 |
27,431 | In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object ... | Real-valued function | 0.806645 |
27,432 | Liquid ammonia is the best-known and most widely studied nonaqueous ionising solvent. Its most conspicuous property is its ability to dissolve alkali metals to form highly coloured, electrically conductive solutions containing solvated electrons. Apart from these remarkable solutions, much of the chemistry in liquid am... | Ammonia | 0.806645 |
27,433 | Percent concentration does not refer to this quantity. This improper name persists, especially in elementary textbooks. In biology, the unit "%" is sometimes (incorrectly) used to denote mass concentration, also called mass/volume percentage. A solution with 1 g of solute dissolved in a final volume of 100 mL of soluti... | Mass fraction (chemistry) | 0.806644 |
27,434 | The maximum number of edges for an undirected graph is ( | V | 2 ) = | V | ( | V | − 1 ) 2 {\displaystyle {\binom {|V|}{2}}={\frac {|V|(|V|-1)}{2}}} , so the maximal density is 1 (for complete graphs) and the minimal density is 0 (Coleman & Moré 1983). For families of graphs of increasing size, one often calls them spa... | Dense graph | 0.806643 |
27,435 | In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing. Toughness is the strength with which the material opposes rupture. One definition of material toughness is the amount of energy per unit volume that a material can absorb before ruptur... | Toughness | 0.806643 |
27,436 | Upper density is an extension of the concept of graph density defined above from finite graphs to infinite graphs. Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper ... | Dense graph | 0.806643 |
27,437 | A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations). The language of a propositional calculus consists of a set of primitive symbols, variously referred to as atomic formulas, placehol... | Propositional Calculus | 0.806642 |
27,438 | In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.... | Propositional Calculus | 0.806642 |
27,439 | The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of: their language (i.e., the particular collection of primitive symbols and operator symbols), the set of axioms, or distinguished formulas, and the set of infere... | Propositional Calculus | 0.806642 |
27,440 | (This is usually the much harder direction of proof.) We adopt the same notational conventions as above. We want to show: If G implies A, then G proves A. We proceed by contraposition: We show instead that if G does not prove A then G does not imply A. If we show that there is a model where A does not hold despite G be... | Propositional Calculus | 0.806642 |
27,441 | Theorem. If the Abelian von Neumann algebra A is unitarily equivalent to both L∞μ(X) and L∞ν(Y) acting on the direct integral spaces ∫ X ⊕ H x d μ ( x ) , ∫ Y ⊕ K y d ν ( y ) {\displaystyle \int _{X}^{\oplus }H_{x}d\mu (x),\quad \int _{Y}^{\oplus }K_{y}d\nu (y)} and μ, ν are standard measures, then there is a Borel iso... | Decomposable operator | 0.806641 |
27,442 | Note that this asserts more than just the algebraic equivalence of A with the algebra of diagonal operators. This version however does not explicitly state how the underlying standard Borel space X is obtained. There is a uniqueness result for the above decomposition. | Decomposable operator | 0.806641 |
27,443 | The spectral theorem has many variants. A particularly powerful version is as follows: Theorem. For any Abelian von Neumann algebra A on a separable Hilbert space H, there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L∞μ(X) acting on a direct integral o... | Decomposable operator | 0.806641 |
27,444 | Let A be a separable C*-algebra and π a non-degenerate involutive representation of A on a separable Hilbert space H. Let W*(π) be the von Neumann algebra generated by the operators π(a) for a ∈ A. Then corresponding to any central decomposition of W*(π) over a standard measure space (X, μ) (which as stated is unique i... | Decomposable operator | 0.806641 |
27,445 | Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings. Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the resu... | Decomposable operator | 0.806641 |
27,446 | In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in ... | Decomposable operator | 0.806641 |
27,447 | If A is a separable C*-algebra, we can consider measurable families of non-degenerate *-representations of A; recall that in case A has a unit, non-degeneracy is equivalent to unit-preserving. By the general correspondence that exists between strongly continuous unitary representations of a locally compact group G and ... | Decomposable operator | 0.806641 |
27,448 | A workshop titled "Semantic Representation of Mathematical Knowledge" was held at the Fields Institute in Toronto during February 3–5, 2016. The goal of the workshop was to lay down the foundations of a prototype semantic representation language for the GDML. The workshop's organizers recognized that the extremely wide... | Global Digital Mathematics Library | 0.806637 |
27,449 | In the spring of 2014, the Committee on Planning a Global Library of the Mathematical Sciences released a comprehensive study entitled “Developing a 21st Century Global Library for Mathematics Research.” This report states in its Strategic Plan section, “There is a compelling argument that through a combination of mach... | Global Digital Mathematics Library | 0.806637 |
27,450 | The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|, instead of ord ( ⟨ a ⟩ ) , {\displaystyle \operatorname {ord} (\langle a\rangle ),} where the brackets denote the generated group. Lagrange's theorem states that for any subgroup H of a finite group G, the o... | Finite order | 0.806634 |
27,451 | A simple method for its determination from 13-C NMR line intensities has also been proposed.It has been proposed that some biological systems might lie near critical points. Examples include neural networks in the salamander retina, bird flocks gene expression networks in Drosophila, and protein folding. However, it is... | First order phase transition | 0.806633 |
27,452 | The following table shows the effective areas of differing valve quantities as proportion of cylinder bore. These percentages are based on simple geometry and do not take into account orifices for spark plugs or injectors, but these voids will usually be sited in the "dead space" unavailable for valves. Also, in practi... | Multi-valve | 0.806633 |
27,453 | A multi-valve engine design has three, four, or five valves per cylinder to achieve improved performance. Any four-stroke internal combustion engine needs at least two valves per cylinder: one for intake of air (and often fuel), and another for exhaust of combustion gases. Adding more valves increases valve area and im... | Multi-valve | 0.806633 |
27,454 | In physics, a front can be understood as an interface between two different possible states (either stable or unstable) in a physical system. For example, a weather front is the interface between two different density masses of air, in combustion where the flame is the interface between burned and unburned material or ... | Front (physics) | 0.806633 |
27,455 | This is usually done by calculating the acceptance rate, which is the fraction of proposed samples that is accepted in a window of the last N {\displaystyle N} samples. The desired acceptance rate depends on the target distribution, however it has been shown theoretically that the ideal acceptance rate for a one-dimens... | Metropolis-Hastings algorithm | 0.806632 |
27,456 | In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a histogram... | Metropolis-Hastings algorithm | 0.806632 |
27,457 | Gibbs paradox: In an ideal gas, is entropy an extensive variable? Loschmidt's paradox: Why is there an inevitable increase in entropy when the laws of physics are invariant under time reversal? The time reversal symmetry of physical laws appears to contradict the second law of thermodynamics. Maxwell's demon: The secon... | Self-referential paradoxes | 0.806632 |
27,458 | Faraday paradox (electrochemistry): Diluted nitric acid will corrode steel, while concentrated nitric acid will not. Levinthal paradox: The length of time that it takes for a protein chain to find its folded state is many orders of magnitude shorter than it would be if it freely searched all possible configurations. SA... | Self-referential paradoxes | 0.806632 |
27,459 | Mott problem: Spherically symmetric wave functions, when observed, produce linear particle tracks. Quantum Zeno effect: (Turing paradox) echoing the Zeno paradox, a quantum particle that is continuously observed cannot change its state Schrödinger's cat paradox: According to the Copenhagen interpretation of quantum mec... | Self-referential paradoxes | 0.806632 |
27,460 | Aharonov–Bohm effect: A charged particle is affected by an electromagnetic field even though it has no local contact with that field. Bell's theorem: Why do measured quantum particles not satisfy mathematical probability theory? Double-slit experiment: Matter and energy can act as a wave or as a particle depending on t... | Self-referential paradoxes | 0.806632 |
27,461 | Jevons paradox: Increases in efficiency lead to even larger increases in demand. Leontief paradox: Some countries export labor-intensive commodities and import capital-intensive commodities, in contradiction with the Heckscher–Ohlin theorem. Lerner paradox: The imposition of a tariff on imports may raise the relative w... | Self-referential paradoxes | 0.806632 |
27,462 | By conformal equivalence, the period ω can be further restricted to satisfy |Re ω| ≤ 1/2 and |ω| ≥ 1. As Dirichlet showed ("Dirichlet's hexagon theorem", 1850), for almost all ω the fundamental domain is a hexagon. | Fundamental polygon | 0.806631 |
27,463 | Fedorov's theorem, established by the Russian crystallographer Evgraf Fedorov in 1891, asserts that parallelograms and centrally symmetric hexagons are the only convex polygons that are fundamental domains. There are several proofs of this, some of the more recent ones related to results in convexity theory, the geomet... | Fundamental polygon | 0.806631 |
27,464 | The area of the standard fundamental polygon is 4 π ( g − 1 ) {\displaystyle 4\pi (g-1)} where g is the genus of the Riemann surface (equivalently, where 4g is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of th... | Fundamental polygon | 0.806631 |
27,465 | Moreover, it also follows from the theory of quasiconformal mappings that two compact Riemann surfaces are diffeomorphic if and only if they are homeomorphic. Consequently, two closed oriented 2-manifolds are homeomorphic if and only if they are diffeomorphic. Such a result can also be proved using the methods of diffe... | Fundamental polygon | 0.806631 |
27,466 | In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface ... | Fundamental polygon | 0.806631 |
27,467 | After a suitable normalisation and labelling of the generators of the fundamental group, the Fricke polygon is uniquely determined and the 6g – 6 real parameters describing it can be used as global real analytic parameters for Teichmüller space in genus g.These results are tied up with the interrelation between homeomo... | Fundamental polygon | 0.806631 |
27,468 | It is easy to see that two Riemann surfaces with the same genus are homeomorphic since as topological space since they are obtained by identifying sides of a 4g-sided polygon—a Euclidean polygon in the Klein model—by diffeomorphisms between paired sides. Applying this construction to the regular 4g-sided polygon allows... | Fundamental polygon | 0.806631 |
27,469 | For a compact Riemann surface a conformal metric can also be introduced which is conformal, so that in holomorphic coordinates the metric takes the form ρ(z) |dz|2. Once this metric has been chosen, locally biholomorphic mappings are precisely orientation-preserving diffeomorphisms that are conformal, i.e. scale the me... | Fundamental polygon | 0.806631 |
27,470 | These not only allow the group to be recovered but provide an explicit presentation of the group by generators and relations. Conversely Poincaré proved that any such polygon gives rise to a compact Riemann surface; in fact, Poincaré's polygon theorem applied to more general polygons, where the polygon was allowed to h... | Fundamental polygon | 0.806631 |
27,471 | Every compact Riemann surface X has a universal covering surface which is a simply connected Riemann surface X. The fundamental group of X acts as deck transformations of X and can be identified with a subgroup Γ of the group of biholomorphisms of X. The group Γ thus acts freely on X with compact quotient space X/Γ, wh... | Fundamental polygon | 0.806631 |
27,472 | This is what is meant when we say that the eigenvalues of the Casimir operator is used to classify the irreducible representations of a Lie algebra (and of an associated Lie Group Lie Group): two irreducible representations of a Lie Algebra are equivalent if and only if their Casimir element have the same eigenvalue. I... | Casimir operator | 0.806626 |
27,473 | This constant can be computed explicitly, giving the following result L 2 = L x 2 + L y 2 + L z 2 = ℓ ( ℓ + 1 ) I . {\displaystyle L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2}=\ell (\ell +1)I.} In quantum mechanics, the scalar value ℓ {\displaystyle \ell } is referred to as the total angular momentum. | Casimir operator | 0.806626 |
27,474 | The Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} is the Lie algebra of SO(3), the rotation group for three-dimensional Euclidean space. It is simple of rank 1, and so it has a single independent Casimir. The Killing form for the rotation group is just the Kronecker delta, and so the Casimir invariant is si... | Casimir operator | 0.806626 |
27,475 | In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space. | Cartan model | 0.806625 |
27,476 | Mathematical Biosciences is a monthly peer-reviewed scientific journal publishing work that provides new concepts or new understanding of biological systems using mathematical models, or methodological articles likely to find application to multiple biological systems. Papers are expected to present a major research fi... | Mathematical Biosciences | 0.806625 |
27,477 | In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. For example, in the book Topology and Groupoids, where the cond... | Pasting lemma | 0.806623 |
27,478 | For instance, the inclusion map ι: Z → R {\displaystyle \iota :Z\to R} from the integers to the real line (with the integers equipped with the cofinite topology) is continuous when restricted to an integer, but the inverse image of a bounded open set in the reals with this map is at most a finite number of points, so n... | Pasting lemma | 0.806623 |
27,479 | Methods for object detection generally fall into either neural network-based or non-neural approaches. For non-neural approaches, it becomes necessary to first define features using one of the methods below, then using a technique such as support vector machine (SVM) to do the classification. On the other hand, neural ... | Object detection | 0.806622 |
27,480 | It is widely used in computer vision tasks such as image annotation, vehicle counting, activity recognition, face detection, face recognition, video object co-segmentation. It is also used in tracking objects, for example tracking a ball during a football match, tracking movement of a cricket bat, or tracking a person ... | Object detection | 0.806622 |
27,481 | Object detection is a computer technology related to computer vision and image processing that deals with detecting instances of semantic objects of a certain class (such as humans, buildings, or cars) in digital images and videos. Well-researched domains of object detection include face detection and pedestrian detect... | Object detection | 0.806622 |
27,482 | A Boolean algebra A {\displaystyle A} has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set X {\displaystyle X} such that A {\displaystyle A} is isomorphic to the finite–cofinite algebra on X . {\displaystyle X.} I... | Cofinite subset | 0.806621 |
27,483 | The set of all subsets of X {\displaystyle X} that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the finite–cofinite algebra on X . {\displaystyle X.} | Cofinite subset | 0.806621 |
27,484 | The space is compact as the product of two compact spaces; alternatively, it is compact because each nonempty open set contains all but finitely many points. For an example of the countable double-pointed cofinite topology, the set Z {\displaystyle \mathbb {Z} } of integers can be given a topology such that every even ... | Cofinite subset | 0.806621 |
27,485 | The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of each doublet are topologically indistinguishable. It is, however, R0 since to... | Cofinite subset | 0.806621 |
27,486 | {\displaystyle X.} Symbolically, one writes the topology as This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field K {\displaystyle K} are zero on finite sets, or the whole of K , {\displaystyle K,} the Zariski topology on K {\displaystyle K} (considered as... | Cofinite subset | 0.806621 |
27,487 | The cofinite topology (sometimes called the finite complement topology) is a topology that can be defined on every set X . {\displaystyle X.} It has precisely the empty set and all cofinite subsets of X {\displaystyle X} as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, ... | Cofinite subset | 0.806621 |
27,488 | Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence without an outside argument. The values of the loadings L {\displaystyle L} , the averages μ {\displaystyle \mu } , and the variances of the "errors" ε {\displaystyle \varepsi... | Factor loading | 0.80662 |
27,489 | Factor analysis has also been widely used in physical sciences such as geochemistry, hydrochemistry, astrophysics and cosmology, as well as biological sciences, such as ecology, molecular biology, neuroscience and biochemistry. In groundwater quality management, it is important to relate the spatial distribution of dif... | Factor loading | 0.80662 |
27,490 | 1965 - Andrei Kolmogorov independently develops the theory of Kolmogorov complexity and uses it to analyze the concept of randomness. 1966 - Grothendieck proves the Ax-Grothendieck theorem: any injective polynomial self-map of algebraic varieties over algebraically closed fields is bijective. 1968 - James Ax independen... | Timeline of mathematical logic | 0.806617 |
27,491 | 1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory. 1963 - Saul Kripke extends his possible-world semantics to normal modal logics. 1965 - Michael D. Morley introduces the beginnings of stable theory ... | Timeline of mathematical logic | 0.806617 |
27,492 | 1947 - Andrey Markov Jr. and Emil Post independently prove the undecidability of the word problem for semigroups. 1948 - McKinsey and Alfred Tarski study closure algebras for S4 and intuitionistic logic. | Timeline of mathematical logic | 0.806617 |
27,493 | 1943 - Stephen Kleene introduces the assertion he calls "Church's Thesis" asserting the identity of general recursive functions with effective calculable ones. 1944 - McKinsey and Alfred Tarski study the relationship between topological closure and Boolean closure algebras. 1944 - Emil Leon Post introduces the partial ... | Timeline of mathematical logic | 0.806617 |
27,494 | Alan Turing introduces the Turing machine model proves the existence of universal Turing machines, and uses these results to settle the Entscheidungsproblem by proving it equivalent to (what is now called) the halting problem. 1936 - Anatoly Maltsev proves the full compactness theorem for first-order logic, and the "up... | Timeline of mathematical logic | 0.806617 |
27,495 | 1933 - Kurt Gödel develops two interpretations of intuitionistic logic in terms of a provability logic, which would become the standard axiomatization of S4. 1934 - Thoralf Skolem constructs a non-standard model of arithmetic. 1936 - Alonzo Church develops the lambda calculus. | Timeline of mathematical logic | 0.806617 |
27,496 | 1930 - Arend Heyting develops an intuitionistic propositional calculus. 1931 – Kurt Gödel proves his incompleteness theorem which shows that every axiomatic system for mathematics is either incomplete or inconsistent. 1932 - C. I. Lewis and C. H. Langford's Symbolic Logic contains descriptions of the modal logic system... | Timeline of mathematical logic | 0.806617 |
27,497 | 1920 - Thoralf Skolem proves the (downward) Löwenheim-Skolem theorem using the axiom of choice explicitly. 1922 - Thoralf Skolem proves a weaker version of the Löwenheim-Skolem theorem without the axiom of choice. 1929 - Mojzesj Presburger introduces Presburger arithmetic and proving its decidability and completeness. | Timeline of mathematical logic | 0.806617 |
27,498 | 1904 - Edward Vermilye Huntington develops the back-and-forth method to prove Cantor's result that countable dense linear orders (without endpoints) are isomorphic. 1908 – Ernst Zermelo axiomatizes set theory, thus avoiding Cantor's contradictions. 1915 - Leopold Löwenheim publishes a proof of the (downward) Löwenheim-... | Timeline of mathematical logic | 0.806617 |
27,499 | 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining what is now called Boolean algebra. 1854 – George Boole perfects his ideas, with the publication of An Investigation of the Laws of Thought. 1874 – Georg Cantor proves that the set of all real numbers is uncountably infinite but... | Timeline of mathematical logic | 0.806617 |
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