id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
90,600 | The Dortmund Data Bank was founded in the 1970s at the University of Dortmund in Germany. The original reason for starting a vapor–liquid phase equilibria data collection was the development of the group contribution method UNIFAC which allows to estimate vapor pressures of mixtures. The DDB has since been extended to ... | Dortmund Data Bank | 0.78083 |
90,601 | The Dortmund Data Bank (short DDB) is a factual data bank for thermodynamic and thermophysical data. Its main usage is the data supply for process simulation where experimental data are the basis for the design, analysis, synthesis, and optimization of chemical processes. The DDB is used for fitting parameters for ther... | Dortmund Data Bank | 0.78083 |
90,602 | Wall–Sun–Sun primes are named after Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's Last Theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's Last Theorem, t... | Wall–Sun–Sun prime | 0.78083 |
90,603 | In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known. | Wall–Sun–Sun prime | 0.78083 |
90,604 | We now explain in greater detail the structure of the proof of Mercer's theorem, particularly how it relates to spectral theory of compact operators. The map K ↦ TK is injective. TK is a non-negative symmetric compact operator on L2; moreover K(x, x) ≥ 0.To show compactness, show that the image of the unit ball of L2 u... | Mercer's theorem | 0.78083 |
90,605 | A positive constant function K ( x , y ) = c {\displaystyle K(x,y)=c\,} satisfies Mercer's condition, as then the integral becomes by Fubini's theorem ∬ g ( x ) c g ( y ) d x d y = c ∫ g ( x ) d x ∫ g ( y ) d y = c ( ∫ g ( x ) d x ) 2 {\displaystyle \iint g(x)\,c\,g(y)\,dxdy=c\int \!g(x)\,dx\int \!g(y)\,dy=c\left(\int ... | Mercer's theorem | 0.78083 |
90,606 | Now apply the spectral theorem for compact operators on Hilbert spaces to TK to show the existence of the orthonormal basis {ei}i of L2 λ i e i ( t ) = ( t ) = ∫ a b K ( t , s ) e i ( s ) d s . {\displaystyle \lambda _{i}e_{i}(t)=(t)=\int _{a}^{b}K(t,s)e_{i}(s)\,ds.} If λi ≠ 0, the eigenvector (eigenfunction) ei is se... | Mercer's theorem | 0.78083 |
90,607 | Since TK is a linear operator, we can talk about eigenvalues and eigenfunctions of TK. Theorem. Suppose K is a continuous symmetric positive-definite kernel. Then there is an orthonormal basis {ei}i of L2 consisting of eigenfunctions of TK such that the corresponding sequence of eigenvalues {λi}i is nonnegative. The ei... | Mercer's theorem | 0.78083 |
90,608 | To explain Mercer's theorem, we first consider an important special case; see below for a more general formulation. A kernel, in this context, is a symmetric continuous function K: × → R {\displaystyle K:\times \rightarrow \mathbb {R} } where symmetric means that K ( x , y ) = K ( y , x ) {\displaystyle K(x,y)=K(y,x)... | Mercer's theorem | 0.78083 |
90,609 | The following is immediate: Theorem. Suppose K is a continuous symmetric positive-definite kernel; TK has a sequence of nonnegative eigenvalues {λi}i. Then ∫ a b K ( t , t ) d t = ∑ i λ i . {\displaystyle \int _{a}^{b}K(t,t)\,dt=\sum _{i}\lambda _{i}.} This shows that the operator TK is a trace class operator and trace... | Mercer's theorem | 0.78083 |
90,610 | Those eigenvectors that correspond to non-zero eigenvalues can be arranged in a sequence {ei}i (regardless of separability). Theorem. If K is a symmetric positive-definite kernel on (X, M, μ), then K ( y , x ) = ∑ i ∈ N λ i e i ( y ) e i ( x ) {\displaystyle K(y,x)=\sum _{i\in \mathbb {N} }\lambda _{i}e_{i}(y)e_{i}(x)}... | Mercer's theorem | 0.78083 |
90,611 | {\displaystyle \langle T_{K}\varphi ,\psi \rangle =\int _{X\times X}K(y,x)\varphi (y)\psi (x)\,d(y,x).} TK is a compact operator (actually it is even a Hilbert–Schmidt operator). If the kernel K is symmetric, by the spectral theorem, TK has an orthonormal basis of eigenvectors. | Mercer's theorem | 0.78083 |
90,612 | Mercer's theorem itself is a generalization of the result that any symmetric positive-semidefinite matrix is the Gramian matrix of a set of vectors. The first generalization replaces the interval with any compact Hausdorff space and Lebesgue measure on is replaced by a finite countably additive measure μ on the Borel... | Mercer's theorem | 0.78083 |
90,613 | In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in (Mercer 1909), is one of the most notable results of the work of James Mercer (1883–1932). It is... | Mercer's theorem | 0.78083 |
90,614 | For example, in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative. Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a... | Derivative (generalizations) | 0.780829 |
90,615 | There is no completely satisfactory analog of the first-order derivative or gradient. The Carlitz derivative is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to local fields of positive characteristic in the form of formal Laurent series with coefficients in... | Derivative (generalizations) | 0.780829 |
90,616 | The H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus. It is used in the study of stochastic processes. Laplacians and differential equations using the Laplacian can be defined on fractals. | Derivative (generalizations) | 0.780829 |
90,617 | In commutative algebra, Kähler differentials are universal derivations of a commutative ring or module. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary algebraic varieties, instead of just smooth manifolds. | Derivative (generalizations) | 0.780829 |
90,618 | In functional analysis, the functional derivative defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative to an infinite dimensional vector space. An important case is the variational derivative in the calculus of variations. The subd... | Derivative (generalizations) | 0.780829 |
90,619 | In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves. This extends the directional derivative of scalar functions to sections of vector bundles or principal bundles. In Riemannian geometry, the existence of a metric chooses a unique preferred... | Derivative (generalizations) | 0.780829 |
90,620 | This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4x − 1, by analogy with ordinary integration, and formally write Combining derivatives of different variables results in a notion of a partial differential operator. The linear operator... | Derivative (generalizations) | 0.780829 |
90,621 | A differential operator combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary linear differential equations with constant coefficients. For example, if f(x) is a twice differentiable function of one variable, the differential equation ... | Derivative (generalizations) | 0.780829 |
90,622 | In type theory, many abstract data types can be described as the algebra generated by a transformation that maps structures based on the type back into the type. For example, the type T of binary trees containing values of type A can be represented as the algebra generated by the transformation 1+A×T2→T. The "1" repres... | Derivative (generalizations) | 0.780829 |
90,623 | For an advanced application of this analysis to topology of manifolds, see Morse theory. In addition to n th derivatives for any natural number n, there are various ways to define derivatives of fractional or negative orders, which are studied in fractional calculus. The −1 order derivative corresponds to the integral,... | Derivative (generalizations) | 0.780829 |
90,624 | In the real numbers one can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. Higher derivatives can also be defined for functions of several variables, studied in multivariable calculus. In this case, instead of repeatedly applying the der... | Derivative (generalizations) | 0.780829 |
90,625 | In algebra, generalizations of the derivative can be obtained by imposing the Leibniz rule of differentiation in an algebraic structure, such as a ring or a Lie algebra. | Derivative (generalizations) | 0.780829 |
90,626 | Then the α th {\textstyle \alpha ^{\text{th}}} weak derivative of u {\displaystyle u} exists if there is a function v: R n → R {\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} } such that for all test functions φ {\displaystyle \varphi } , we have ∫ R n u D α φ d x = ( − 1 ) | α | ∫ R n v φ d x {\displaystyle \int _{\m... | Derivative (generalizations) | 0.780829 |
90,627 | Some of these operators are so important that they have their own names: The Laplace operator or Laplacian on R3 is a second-order partial differential operator Δ given by the divergence of the gradient of a scalar function of three variables, or explicitly as Analogous operators can be defined for functions of any num... | Derivative (generalizations) | 0.780829 |
90,628 | A direct sum of algebras X {\displaystyle X} and Y {\displaystyle Y} is the direct sum as vector spaces, with product ( x 1 + y 1 ) ( x 2 + y 2 ) = ( x 1 x 2 + y 1 y 2 ) . {\displaystyle (x_{1}+y_{1})(x_{2}+y_{2})=(x_{1}x_{2}+y_{1}y_{2}).} Consider these classical examples: R ⊕ R {\displaystyle \mathbf {R} \oplus \math... | Direct sum of Hilbert spaces | 0.780828 |
90,629 | Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: λ ( x ⊕ y ) = λ x ⊕ λ y {\displaystyle \lambda (x\oplus y)=\lambda x\oplus \lambda y} while for the direct product a scalar factor may be collected alternatel... | Direct sum of Hilbert spaces | 0.780828 |
90,630 | C ⊕ C {\displaystyle \mathbf {C} \oplus \mathbf {C} } is the algebra of tessarines introduced by James Cockle in 1848. H ⊕ H , {\displaystyle \mathbf {H} \oplus \mathbf {H} ,} called the split-biquaternions, was introduced by William Kingdon Clifford in 1873.Joseph Wedderburn exploited the concept of a direct sum of al... | Direct sum of Hilbert spaces | 0.780828 |
90,631 | If we now replace the paraboloid by an arbitrary convex function, we obtain a different dual mapping that retains the incidence and above-below properties of the standard projective dual. This implies that natural dual concepts in computational geometry like Voronoi diagrams and Delaunay triangulations retain their mea... | Bregman distance | 0.780827 |
90,632 | A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. There are numerous analytical forms of the projective dual: one common form maps the point p = ( p 1 , … p d ) {\displaystyle p=(p_{1},\ldots... | Bregman distance | 0.780827 |
90,633 | However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares... | Bregman distance | 0.780827 |
90,634 | In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as... | Bregman distance | 0.780827 |
90,635 | The developing sciences of chemistry and Paleontology, combined with the discovery of coprolites in commercial quantities in East Anglia, led Fisons and Packard to develop sulfuric acid and fertilizer plants at Bramford, and Snape, Suffolk in the 1850s to create superphosphates, which were shipped around the world from... | History of fertilizer | 0.780827 |
90,636 | In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal. | Grothendieck connection | 0.780827 |
90,637 | The sheaf I {\displaystyle I} of ideals of Δ {\displaystyle \Delta } in M × M {\displaystyle M\times M} consists of functions on M × M {\displaystyle M\times M} which vanish along the diagonal. Much of the infinitesimal geometry of M {\displaystyle M} can be realized in terms of I . {\displaystyle I.} | Grothendieck connection | 0.780827 |
90,638 | The Grothendieck connection is a generalization of the Gauss–Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of geometric invariance, which may be regarded as the analog of covariance for a... | Grothendieck connection | 0.780827 |
90,639 | Conservation biology is the study of the conservation of nature and of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction and the erosion of biotic interactions. It is an interdisciplinary subject drawing on natural and social sciences, and the pra... | Biological conservation | 0.780827 |
90,640 | Population level biodiversity (mostly in coldspots) are disappearing at a rate that is ten times that at the species level. The level of importance in addressing biomass versus endemism as a concern for conservation biology is highlighted in literature measuring the level of threat to global ecosystem carbon stocks tha... | Biological conservation | 0.780827 |
90,641 | s l 2 {\displaystyle {\mathfrak {sl}}_{2}} Michio Jimbo considered the algebras with three generators related by the three commutators = 2 e , = − 2 f , = sinh ( η h ) / sinh η . {\displaystyle =2e,\ =-2f,\ =\sinh(\eta h)/\sinh \eta .} When η → 0 {\displaystyle \eta \to 0} , these reduce to the commutators that ... | Quantized enveloping algebra | 0.780826 |
90,642 | In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and indepen... | Quantized enveloping algebra | 0.780826 |
90,643 | The area K of a trapezoid is given by K = a + b 2 ⋅ h = m h {\displaystyle K={\frac {a+b}{2}}\cdot h=mh} where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a gr... | Right trapezoid | 0.780825 |
90,644 | In order to obtain universally applicable designs, mild assumptions can be made about waveform format, channel statistics and the coding/decoding scheme. Both heuristic and optimal design methodologies can be used, and excellent performance is possible with low complexity as long as the transceiver operates with a reas... | Filter bank | 0.780825 |
90,645 | , c N ( z ) {\displaystyle c_{1}(z),...,c_{N}(z)} are polynomials. The Module is analogous to the span of a set of vectors in linear algebra. The theory of Gröbner bases implies that the Module has a unique reduced Gröbner basis for a given order of power products in polynomials. | Filter bank | 0.780825 |
90,646 | FIR filter are more popular since it is easier to implement. For 1-D oversampled FIR filter banks, the Euclidean algorithm plays a key role in the matrix inverse problem. However, the Euclidean algorithm fails for multidimensional (MD) filters. For MD filter, we can convert the FIR representation into a polynomial repr... | Filter bank | 0.780825 |
90,647 | The method is flexible towards frequency support configurations. 2D filter banks designed by optimization in the frequency domain has been used in Wei and Lu. In Nguyen's paper, the proposed method is not limited to two-channel 2D filter banks design; the approach is generalized to M-channel filter banks with any criti... | Filter bank | 0.780825 |
90,648 | Many of the existing methods for designing 2-channel filter banks are based on transformation of variable technique. For example, McClellan transform can be used to design 1-D 2-channel filter banks. Though the 2-D filter banks have many similar properties with the 1-D prototype, but it is difficult to extend to more t... | Filter bank | 0.780825 |
90,649 | Laughlin's first book “Total Immersion: The Revolutionary Way to Swim Better, Faster and Easier”, was published in 1996 by Simon & Schuster, and has sold more than 275,000 copies as an e-book and trade paperback.Trainer Christopher Drozd described the book as "gold" in a review at SportFit.com, "Just as there is more t... | Total Immersion | 0.780822 |
90,650 | dollars. Westinghouse sold Unimation to Stäubli Faverges SCA of France in 1988, which is still making articulated robots for general industrial and cleanroom applications and even bought the robotic division of Bosch in late 2004. Only a few non-Japanese companies ultimately managed to survive in this market, the major... | Teach pendant | 0.78082 |
90,651 | The first two IRB 6 robots were sold to Magnusson in Sweden for grinding and polishing pipe bends and were installed in production in January 1974. Also in 1973 KUKA Robotics built its first robot, known as FAMULUS, also one of the first articulated robots to have six electromechanically driven axes. Interest in roboti... | Teach pendant | 0.78082 |
90,652 | Scheinman, after receiving a fellowship from Unimation to develop his designs, sold those designs to Unimation who further developed them with support from General Motors and later marketed it as the Programmable Universal Machine for Assembly (PUMA). Industrial robotics took off quite quickly in Europe, with both ABB ... | Teach pendant | 0.78082 |
90,653 | Chris Shute built a complete replica of the robot in 1997. George Devol applied for the first robotics patents in 1954 (granted in 1961). The first company to produce a robot was Unimation, founded by Devol and Joseph F. Engelberger in 1956. | Teach pendant | 0.78082 |
90,654 | On October 5, 2017, OSHA, NIOSH and RIA signed an alliance to work together to enhance technical expertise, identify and help address potential workplace hazards associated with traditional industrial robots and the emerging technology of human-robot collaboration installations and systems, and help identify needed res... | Teach pendant | 0.78082 |
90,655 | The International Federation of Robotics has predicted a worldwide increase in adoption of industrial robots and they estimated 1.7 million new robot installations in factories worldwide by 2020 Archived 2017-02-11 at the Wayback Machine. Rapid advances in automation technologies (e.g. fixed robots, collaborative and ... | Teach pendant | 0.78082 |
90,656 | When the gasket includes two lines on the x {\displaystyle x} -axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the y {\displaystyle y} -axis, then the circles that are tangent to the x {\displaystyle x} -axis are the Ford circles, important in number theory. The Apollonian ga... | Apollonian gasket | 0.78082 |
90,657 | In the limit, this set of circles is an Apollonian gasket. In it, each pair of tangent circles has an infinite Pappus chain of circles tangent to both circles in the pair. The size of each new circle is determined by Descartes' theorem, which states that, for any four mutually tangent circles, the radii r i {\displayst... | Apollonian gasket | 0.78082 |
90,658 | Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry. The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian grou... | Apollonian gasket | 0.78082 |
90,659 | Night Sky Network is an educational effort sponsored by NASA to help educate the public through astronomy clubs across the United States. | Night Sky Network | 0.780818 |
90,660 | Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix M ∈ C M × N {\displaystyle M\in \mathbb {C} ^{M\times N}} of size M × N {\displaystyle M\times N} in given subspaces by turning the singu... | Rayleigh–Ritz method | 0.780814 |
90,661 | The name Rayleigh–Ritz is being debated vs. the Ritz method after Walther Ritz, since the numerical procedure has been published by Walther Ritz in 1908-1909. According to A. W. Leissa, Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places... | Rayleigh–Ritz method | 0.780814 |
90,662 | The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. I... | Rayleigh–Ritz method | 0.780814 |
90,663 | In numerical linear algebra, the Rayleigh–Ritz method is commonly applied to approximate an eigenvalue problem for the matrix A ∈ C N × N {\displaystyle A\in \mathbb {C} ^{N\times N}} of size N {\displaystyle N} using a projected matrix of a smaller size m < N {\displaystyle m | Rayleigh–Ritz method | 0.780814 |
90,664 | From 2021, Luck served as founding Director of the King's Institute for Artificial Intelligence. Luck has undertaken research in the area of intelligent agents. He is the co-author of the books Understanding Agent Systems and Agent-Based Software Development. | Michael Luck (computer scientist) | 0.780812 |
90,665 | In 2013 he was appointed as Executive Dean of the Faculty of Natural and Mathematical Sciences, staying in that role until 2020. In that time he also led the UKRI Centre for Doctoral Training in Safe and Trusted Artificial Intelligence, serving as founding Director from 2019 until 2023. | Michael Luck (computer scientist) | 0.780812 |
90,666 | From 1993 to 2000, Michael Luck was a lecturer based in the Department of Computer Science at the University of Warwick. From 2000 to 2006, Luck was a professor in the School of Electronics and Computer Science at the University of Southampton. While there, he led the AgentLink European Co-ordination Action for Agent-B... | Michael Luck (computer scientist) | 0.780812 |
90,667 | Michael Luck is a professor of computer science based at the Department of Informatics, King's College London, in central London, England. His main research area is in intelligent agents and multi-agent systems. | Michael Luck (computer scientist) | 0.780812 |
90,668 | Luck is a Fellow of the European Association for Artificial Intelligence (EurAI), and a British Computer Society (BCS) Fellow. In 2021, he was awarded Fellow of King’s College London. == References == | Michael Luck (computer scientist) | 0.780812 |
90,669 | SPE provides scholarships and fellowships to undergraduate and graduate students. According to the United States Department of Labor's Bureau of Labor Statistics, petroleum engineers are required to have a bachelor's degree in engineering, generally a degree focused on petroleum engineering is preferred, but degrees in... | Petroleum engineering | 0.78081 |
90,670 | The profession got its start in 1914 within the American Institute of Mining, Metallurgical and Petroleum Engineers (AIME). The first Petroleum Engineering degree was conferred in 1915 by the University of Pittsburgh. Since then, the profession has evolved to solve increasingly difficult situations. Improvements in com... | Petroleum engineering | 0.78081 |
90,671 | Petroleum geology is an interdisciplinary field composed of geophysics, geochemistry, and paleontology. The main focus of petroleum geology is the exploration and appraisal of reservoirs containing hydrocarbons via technical forms of analysis. | Petroleum engineering | 0.78081 |
90,672 | The combined efforts of geologists and petroleum engineers throughout the life of a hydrocarbon accumulation determine the way in which a reservoir is developed and depleted, and usually they have the highest impact on field economics. Petroleum engineering requires a good knowledge of many other related disciplines, s... | Petroleum engineering | 0.78081 |
90,673 | Petroleum engineering is a field of engineering concerned with the activities related to the production of Hydrocarbons, which can be either crude oil or natural gas. Exploration and production are deemed to fall within the upstream sector of the oil and gas industry. Exploration, by earth scientists, and petroleum eng... | Petroleum engineering | 0.78081 |
90,674 | Petroleum engineering has historically been one of the highest-paid engineering disciplines, although there is a tendency for mass layoffs when oil prices decline and waves of hiring as prices rise. In 2020, the United States Department of Labor's Bureau of Labor Statistics reported the median pay for petroleum enginee... | Petroleum engineering | 0.78081 |
90,675 | Petroleum Engineering, like most forms of engineering, requires a strong foundation in physics, chemistry, and mathematics. Other fields pertinent to petroleum engineering include geology, formation evaluation, fluid flow in porous media, well drilling technology, economics, geostatistics, etc. | Petroleum engineering | 0.78081 |
90,676 | Geostatistics as applied to petroleum engineering uses statistical analysis to characterize reservoirs and create flow simulations that quantify uncertainties of the location of oil and gas. | Petroleum engineering | 0.78081 |
90,677 | In the spirit of Palms' budget, the board of directors moved to transform university land on Assembly Street into an "innovation district" called Innovista that will develop four areas: biomedicine, nanotechnology, environmental science and alternative fuels. Innovista is a partnered development with the City of Columb... | University of South Carolina | 0.78081 |
90,678 | Particles satisfying the case of +1 are called bosons, and particles satisfying the case of −1 are called fermions. The spin–statistics theorem dictates that all particles with integer spin are bosons whereas all particles with half-integer spin are fermions. The exchange operator commutes with the Hamiltonian and is t... | Exchange operator | 0.780809 |
90,679 | A modified exchange operator is defined in the Hartree–Fock method of quantum chemistry, in order to estimate the exchange energy arising from the exchange statistics described above. In this method, one often defines an energetic exchange operator as: K ^ j ( x 1 ) f i ( x 1 ) = ϕ j ( x 1 ) ∫ ϕ j ∗ ( x 2 ) f i ( x 2 )... | Exchange operator | 0.780809 |
90,680 | In quantum mechanics, the exchange operator P ^ {\displaystyle {\hat {P}}} , also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state | x 1 , x 2... | Exchange operator | 0.780809 |
90,681 | The alcohol reacts with the phosphine to create a good leaving group then undergoes an inversion of stereochemistry in classic SN2 fashion as the nucleophile displaces it. A common side-product is produced when the azodicarboxylate displaces the leaving group instead of the desired nucleophile. | Mitsunobu reaction | 0.780809 |
90,682 | Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II gave examples of derivatives and used what is now known as Rolle's theorem.In the 14th century, Madhava of Sangamagrama developed... | Non-classical analysis | 0.780807 |
90,683 | Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The hist... | Non-classical analysis | 0.780807 |
90,684 | Convex analysis, the study of convex sets and functions. Idempotent analysis – analysis in the context of an idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. Tropical analysis – analysis of the idempotent semiring called the tropical semiring (or max-p... | Non-classical analysis | 0.780807 |
90,685 | Stochastic calculus – analytical notions developed for stochastic processes. Set-valued analysis – applies ideas from analysis and topology to set-valued functions. | Non-classical analysis | 0.780807 |
90,686 | Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves. Geometric analysis involves the use of geometrical methods in the study of partial different... | Non-classical analysis | 0.780807 |
90,687 | Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calc... | Non-classical analysis | 0.780807 |
90,688 | Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain... | Non-classical analysis | 0.780807 |
90,689 | Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theo... | Non-classical analysis | 0.780807 |
90,690 | The n {\displaystyle n} -dimensional hyperbolic space or Hyperbolic n {\displaystyle n} -space, usually denoted H n {\displaystyle \mathbb {H} ^{n}} , is the unique simply connected, n {\displaystyle n} -dimensional complete Riemannian manifold with a constant negative sectional curvature equal to -1. The unicity means... | Hyperbolic space | 0.780807 |
90,691 | Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometrical space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimension... | Hyperbolic space | 0.780807 |
90,692 | It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperb... | Hyperbolic space | 0.780807 |
90,693 | The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere ... | Hairy ball theorem | 0.780805 |
90,694 | Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What is true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeli... | Hairy ball theorem | 0.780805 |
90,695 | At least one point on the surface at which vectors of electric and magnetic fields equal zero will necessarily appear. On certain 2-spheres of parameter space for electromagnetic waves in plasmas (or other complex media), these type of "cowlicks" or "bald points" also appear, which indicates that there exists topologic... | Hairy ball theorem | 0.780805 |
90,696 | The hairy ball theorem has numerous physical exemplifications. For example, rotation of a rigid ball around its fixed axis gives rise to a continuous tangential vector field of velocities of the points located on its surface. This field has two zero-velocity points, which disappear after drilling the ball completely th... | Hairy ball theorem | 0.780805 |
90,697 | Every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus... | Hairy ball theorem | 0.780805 |
90,698 | A common problem in computer graphics is to generate a non-zero vector in R3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sph... | Hairy ball theorem | 0.780805 |
90,699 | Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general Poincaré-Hopf index theorem. | Hairy ball theorem | 0.780805 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.