id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
81,000 | INSPIRE-HEP – The INSPIRE High-Energy Physics literature database provides BibTeX format citations for over one million high-energy physics papers. JabRef – Open-source cross-platform software application for creating, editing, managing, and searching BibTeX and BibLaTeX files. MathSciNet – Database by the American Mat... | BibTeX | 0.784747 |
81,001 | Google Research – Housed within the artificial intelligence division of Google is a compilation of publications by Google staff with BibTeX citation links. HubMed – A versatile PubMed interface including BibTeX output. | BibTeX | 0.784747 |
81,002 | Digital Bibliography & Library Project – A bibliography website that lists more than 910,000 articles in the computer science field. Google Books – The bibliographic information for each book is exportable in BibTeX format via the 'Export Citation' feature. Google Scholar – Google's system for searching scholarly liter... | BibTeX | 0.784747 |
81,003 | CiteULike (discontinued) – A community based bibliography database that had BibTeX input and output. The Collection of Computer Science Bibliographies – uses BibTeX as internal data format, search results and contributions primarily in BibTeX. Connotea – Open-source social bookmark style publication management system. | BibTeX | 0.784747 |
81,004 | Astrophysics Data System – The NASA ADS is an online database of over eight million astronomy and physics papers and provides BibTeX format citations. BibDesk – Open-source software application for macOS for creating, editing, managing, and searching BibTeX files. BibSonomy – A social bookmark and publication managemen... | BibTeX | 0.784747 |
81,005 | Note that this is not the "article number" assigned by some journals. organization The conference sponsor pages Page numbers, separated either by commas or double-hyphens. publisher The publisher's name school The school where the thesis was written series The series of books the book was published in (e.g. "The Hardy ... | BibTeX | 0.784747 |
81,006 | A commonly asked question in inferential statistics is whether the parameter is included within a confidence interval. The only way to answer this question is for a census to be conducted. Referring to the example given above, the probability that the population proportion is in the range of the confidence interval is ... | Population proportion | 0.784745 |
81,007 | One of the main focuses of study in inferential statistics is determining the "true" value of a parameter. Generally, the actual value for a parameter will never be found, unless a census is conducted on the population of study. However, there are statistical methods that can be used to get a reasonable estimation for ... | Population proportion | 0.784745 |
81,008 | In statistics, a population proportion, generally denoted by P {\displaystyle P} or the Greek letter π {\displaystyle \pi } , is a parameter that describes a percentage value associated with a population. For example, the 2010 United States Census showed that 83.7% of the American population was identified as not being... | Population proportion | 0.784745 |
81,009 | In order to derive the formula for the one-sample proportion in the Z-interval, a sampling distribution of sample proportions needs to be taken into consideration. The mean of the sampling distribution of sample proportions is usually denoted as μ p ^ = P {\displaystyle \mu _{\hat {p}}=P} and its standard deviation is ... | Population proportion | 0.784745 |
81,010 | Boyer's law was named by Hubert Kennedy in 1972. It says, "Mathematical formulas and theorems are usually not named after their original discoverers" and was named after Carl Boyer, whose book A History of Mathematics contains many examples of this law. Kennedy observed that "it is perhaps interesting to note that this... | Stigler's Law | 0.784745 |
81,011 | Historical acclaim for discoveries is often assigned to persons of note who bring attention to an idea that is not yet widely known, whether or not that person was its original inventor – theories may be named long after their discovery. In the case of eponymy, the idea becomes named after that person, even if that per... | Stigler's Law | 0.784745 |
81,012 | Stigler's law of eponymy, proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication Stigler’s law of eponymy, states that no scientific discovery is named after its original discoverer. Examples include Hubble's law, which was derived by Georges Lemaître two years before Edwin Hubbl... | Stigler's Law | 0.784745 |
81,013 | These elements generate the dihedral group of the square. The linear subspace with basis {1, i, hj, hk} thus is closed under multiplication, and forms the coquaternion algebra. In the context of quantum mechanics and spinor algebra, the biquaternions hi, hj, and hk (or their negatives), viewed in the M2(C) representati... | Biquaternion | 0.784744 |
81,014 | Furthermore, { x + y j: x , y ∈ C } {\displaystyle \{x+y\mathbf {j} :x,y\in \mathbb {C} \}} is a subalgebra isomorphic to the tessarines. A third subalgebra called coquaternions is generated by hj and hk. It is seen that (hj)(hk) = (−1)i, and that the square of this element is −1. | Biquaternion | 0.784744 |
81,015 | Considering the biquaternion algebra over the scalar field of real numbers R, the set { 1 , h , i , h i , j , h j , k , h k } {\displaystyle \{\mathbf {1} ,h,\mathbf {i} ,h\mathbf {i} ,\mathbf {j} ,h\mathbf {j} ,\mathbf {k} ,h\mathbf {k} \}} forms a basis so the algebra has eight real dimensions. The squares of the ele... | Biquaternion | 0.784744 |
81,016 | . Indeed, M cannot form an algebra if it is not even a magma. | Biquaternion | 0.784744 |
81,017 | Consider now the linear subspace M = { q: q ∗ = q ¯ } = { t + x ( h i ) + y ( h j ) + z ( h k ): t , x , y , z ∈ R } . {\displaystyle M=\lbrace q\colon q^{*}={\bar {q}}\rbrace =\lbrace t+x(h\mathbf {i} )+y(h\mathbf {j} )+z(h\mathbf {k} )\colon t,x,y,z\in \mathbb {R} \rbrace .} M is not a subalgebra since it is not clos... | Biquaternion | 0.784744 |
81,018 | The new methods were founded on basis vectors in the set { q: q q ∗ = 0 } = { w + x i + y j + z k: w 2 + x 2 + y 2 + z 2 = 0 } {\displaystyle \{q\ :\ qq^{*}=0\}=\left\{w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} \ :\ w^{2}+x^{2}+y^{2}+z^{2}=0\right\}} which is called the complex light cone. The above representation of t... | Biquaternion | 0.784744 |
81,019 | There has been considerable work associating this "velocity space" with the hyperboloid model of hyperbolic geometry. In special relativity, the hyperbolic angle parameter of a hyperbolic versor is called rapidity. Thus we see the biquaternion group G provides a group representation for the Lorentz group.After the intr... | Biquaternion | 0.784744 |
81,020 | {\displaystyle G\cap M.} When equipped with the commutator, A forms the Lie algebra of G. Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, G is called the special linear group SL(2,C) in M(2, C). | Biquaternion | 0.784744 |
81,021 | The space of biquaternions has a natural topology through the Euclidean metric on 8-space. With respect to this topology, G is a topological group. | Biquaternion | 0.784744 |
81,022 | {\displaystyle \exp(ahr)\exp(bhr)=\exp((a+b)hr).} Hence these algebraic operators on the hyperbola are called hyperbolic versors. | Biquaternion | 0.784744 |
81,023 | But this subgroup of G is not a normal subgroup, so no quotient group can be formed. To view G ∩ M {\displaystyle G\cap M} it is necessary to show some subalgebra structure in the biquaternions. Let r represent an element of the sphere of square roots of minus one in the real quaternion subalgebra H. Then (hr)2 = +1 an... | Biquaternion | 0.784744 |
81,024 | As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group G = { g: g g ∗ = 1 } {\displaystyle G=\lbrace g:gg^{*}=1\rbrace } has two parts, G ∩ H {\displaysty... | Biquaternion | 0.784744 |
81,025 | When (a,b)* is written as a 4-vector of ordinary complex numbers, ( u , v , w , z ) ∗ = ( u , − v , − w , − z ) . {\displaystyle (u,v,w,z)^{*}=(u,-v,-w,-z).} The biquaternions form an example of a quaternion algebra, and it has norm N ( u , v , w , z ) = u 2 + v 2 + w 2 + z 2 . {\displaystyle N(u,v,w,z)=u^{2}+v^{2}+w^{... | Biquaternion | 0.784744 |
81,026 | A biquaternion is either a unit or a zero divisor. The algebra of biquaternions forms a composition algebra and can be constructed from bicomplex numbers. See § As a composition algebra below. | Biquaternion | 0.784744 |
81,027 | Hamilton introduced the terms bivector, biconjugate, bitensor, and biversor to extend notions used with real quaternions H. Hamilton's primary exposition on biquaternions came in 1853 in his Lectures on Quaternions. The editions of Elements of Quaternions, in 1866 by William Edwin Hamilton (son of Rowan), and in 1899, ... | Biquaternion | 0.784744 |
81,028 | In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions are isomorphic to the algebra of 2 × 2 complex matrices M2(C). They are also isomorphic to several Clifford algebras including H(C) = Cℓ03(C) = Cℓ2(C) = Cℓ1,2(R),: 112, 113 the Pauli algebr... | Biquaternion | 0.784744 |
81,029 | Some of the more prominent proponents of these biquaternions include Alexander Macfarlane, Arthur W. Conway, Ludwik Silberstein, and Cornelius Lanczos. As developed below, the unit quasi-sphere of the biquaternions provides a representation of the Lorentz group, which is the foundation of special relativity. The algebr... | Biquaternion | 0.784744 |
81,030 | In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and... | Biquaternion | 0.784744 |
81,031 | Although W.R. Hamilton introduced biquaternions in the 19th century, its delineation of its mathematical structure as a special type of algebra over a field was accomplished in the 20th century: the biquaternions may be generated out of the bicomplex numbers in the same way that Adrian Albert generated the real quatern... | Biquaternion | 0.784744 |
81,032 | Watson's natural language, hypothesis generation, and evidence-based learning capabilities are being investigated to see how Watson may contribute to clinical decision support systems, and the increase in artificial intelligence in healthcare for use by medical professionals. To aid physicians in the treatment of their... | IBM Watson Health | 0.784743 |
81,033 | Google's DeepMind platform is being used by the UK National Health Service (NHS) to detect certain health risks through data collected via a mobile app. A second project with the NHS involves analysis of medical images collected from NHS patients to develop computer vision algorithms to detect cancerous tissues. | IBM Watson Health | 0.784743 |
81,034 | Recent advances have suggested the use of AI to describe and evaluate the outcome of maxillo-facial surgery or the assessment of cleft palate therapy in regard to facial attractiveness or age appearance.In 2018, a paper published in the journal of Annals of Oncology mentioned that skin cancer could be detected more acc... | IBM Watson Health | 0.784743 |
81,035 | Large technology companies such as IBM and Google, and startups such as Welltok and Ayasdi, have also developed AI algorithms for healthcare. Additionally, hospitals are looking to AI solutions to support operational initiatives that increase cost saving, improve patient satisfaction, and satisfy their staffing and wor... | IBM Watson Health | 0.784743 |
81,036 | AI does this through machine learning algorithms. These algorithms can recognize patterns in behavior and create its own logic. In order to reduce the margin of error, AI algorithms need to be tested repeatedly. | IBM Watson Health | 0.784743 |
81,037 | Intel's venture capital arm (Intel Capital) recently invested in startup Lumiata, which uses AI to identify at-risk patients and develop care options.Artificial intelligence in healthcare is the use of complex algorithms and software to emulate human cognition in the analysis of complicated medical data. Specifically, ... | IBM Watson Health | 0.784743 |
81,038 | Some of these techniques discussed by Jiang, et al. include: Support vector machines, neural networks, decision trees, and many more. Each of these techniques are described as having a “training goal” so “classifications agree with the outcomes as much as possible…”.To demonstrate some specifics for disease diagnosis/c... | IBM Watson Health | 0.784743 |
81,039 | Competitors were tested on their ability to accurately determine, from the text, which drugs were shown to interact and what the characteristics of their interactions were. Researchers continue to use this corpus to standardize the measure of the effectiveness of their algorithms.Other algorithms identify drug-drug int... | IBM Watson Health | 0.784743 |
81,040 | Improvements in Natural Language Processing led to the development of algorithms to identify drug-drug interactions in medical literature. Drug-drug interactions pose a threat to those taking multiple medications simultaneously, and the danger increases with the number of medications being taken. To address the difficu... | IBM Watson Health | 0.784743 |
81,041 | Electronic health records are crucial to the digitalization and information spread of the healthcare industry. However logging all of this data comes with its own problems like cognitive overload and burnout for users. EHR developers are now automating much of the process and even starting to use natural language proce... | IBM Watson Health | 0.784743 |
81,042 | Merative L.P., formerly IBM Watson Health, is an American medical technology company that provides products and services that help clients facilitate medical research, clinical research, real world evidence, and healthcare services, through the use of artificial intelligence, data analytics, cloud computing, and other ... | IBM Watson Health | 0.784743 |
81,043 | In the structural view, this is displayed with gates and flip-flops. In the geometric domain, the logical level is described by standard cells. The behaviour of the circuit level is described by mathematics using differential equations or logical equations. This corresponds to transistors and capacitors up to crystal l... | Gajski–Kuhn chart | 0.784741 |
81,044 | The Kahn–Kalai conjecture, also known as the expectation threshold conjecture, is a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2006. | Kahn–Kalai conjecture | 0.784736 |
81,045 | In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Thaine (1988). Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem (Washington 1997), to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theore... | Thaine's theorem | 0.784735 |
81,046 | In January, AnandTech assumed that Vega would finally catch up with Nvidia regarding energy efficiency optimizations due to the new "DSBR (Draw Stream Binning Rasterizer)" to be introduced with Vega.It also added support for a new shader stage – Primitive Shaders. Primitive shaders provide more flexible geometry proces... | AMD ZeroCore Power | 0.784735 |
81,047 | The geometry processor contains a Geometry Assembler, a Tesselator, and a Vertex Assembler. The Tesselator is capable of doing tessellation in hardware as defined by Direct3D 11 and OpenGL 4.5 (see AMD January 21, 2017), and succeeded ATI TruForm and hardware tessellation in TeraScale as AMD's then-latest semiconductor... | AMD ZeroCore Power | 0.784735 |
81,048 | The fourth generation GCN instruction set architecture is compatible with the third generation. It is an optimization for 14 nm FinFET process enabling higher GPU clock speeds than with the 3rd GCN generation. Architectural improvements include new hardware schedulers, a new primitive discard accelerator, a new display... | AMD ZeroCore Power | 0.784735 |
81,049 | A Shader Engine comprises one geometry processor, up to 44 CUs (Hawaii chip), rasterizers, ROPs, and L1 cache. Not part of a Shader Engine is the Graphics Command Processor, the 8 ACEs, the L2 cache and memory controllers as well as the audio and video accelerators, the display controllers, the 2 DMA controllers and th... | AMD ZeroCore Power | 0.784735 |
81,050 | Howe (1980) noticed that there was a natural unitary isomorphism between H2(S) and L2(Rn) carrying the Toeplitz operators T j = T ( z j ) {\displaystyle {T_{j}=T(z_{j})}} onto the operators ( P j + i Q j ) Δ − 1 / 2 . {\displaystyle {(P_{j}+iQ_{j})\Delta ^{-1/2}.}} These are examples of zeroth order operators construct... | Weyl calculus | 0.784734 |
81,051 | Software developers have also published libraries or modules in several languages (see list below). Many of these packages are free and open source, which means they can be expanded or adapted by those willing and able to write or modify code. Finally, there are web applications for delineating watersheds. Some of thes... | Watershed delineation | 0.784734 |
81,052 | {\displaystyle \Delta _{n}=\sum _{i=1}^{n}-{\partial ^{2} \over \partial x_{i}^{2}}+x_{i}^{2}.} The Weyl calculus similarly extends to the n-dimensional case. The complexification Sp(2n,C) of the symplectic group is defined by the same relation, but allowing the matrices A, B, C and D to be complex. | Weyl calculus | 0.784734 |
81,053 | By the Stone–von Neumann theorem there is a unitary operator U {\displaystyle {\mathcal {U}}} from L2(Rn) onto F n {\displaystyle {\mathcal {F}}_{n}} intertwining the two representations. It is given by the Bargmann transform U f ( z ) = 1 ( 2 π ) n / 2 ∫ B ( z , t ) f ( t ) d t , {\displaystyle {{\mathcal {U}}f(z)={1 ... | Weyl calculus | 0.784734 |
81,054 | The closed semigroup H ¯ {\displaystyle {\overline {H}}} is maximal in SL(2,C): any larger semigroup must be the whole of SL(2,C).Using computations motivated by theoretical physics, Ferrara et al. (1973) introduced the semigroup H {\displaystyle H} , defined through a set of inequalities. Without identification H {\di... | Weyl calculus | 0.784734 |
81,055 | Its Lie algebra consists of matrices of the form Z = ( z 0 w − z ) . {\displaystyle Z={\begin{pmatrix}z&0\\w&-z\end{pmatrix}}.} | Weyl calculus | 0.784734 |
81,056 | Similarly the same formula defines a period two automorphism σ of the Lie algebra g c {\displaystyle {\mathfrak {g}}_{c}} of Gc, the complex matrices with trace zero. A standard basis of g c {\displaystyle {\mathfrak {g}}_{c}} over C is given by L 0 = ( 1 2 0 0 − 1 2 ) , L − 1 = ( 0 1 0 0 ) , L 1 = ( 0 0 − 1 0 ) . {\di... | Weyl calculus | 0.784734 |
81,057 | {\begin{pmatrix}a&0\\b&a^{-1}\end{pmatrix}}\right|a,b\in \mathbf {R} ,a>0\right\}.} Indeed, the orbit of the vector v = ( 0 1 ) {\displaystyle v={\begin{pmatrix}0\\1\end{pmatrix}}} under the subgroup generated by these matrices is easily seen to be the whole of R2 and the stabilizer of v in G1 lies in inside this subgr... | Weyl calculus | 0.784734 |
81,058 | Helton & Howe (1975) gave an analytic way to establish this index theorem, simplied later by Howe. Their proof relies on the fact if f is smooth then the index is given by the formula of McKean and Singer: ind T ( f ) = Tr ( I − T ( f − 1 ) T ( f ) ) n − Tr ( I − T ( f ) T ( f − 1 ) ) n . {\displaystyle \operator... | Weyl calculus | 0.784734 |
81,059 | Weil (1964) noted that the formalism of the Stone–von Neumann theorem and the oscillator representation of the symplectic group extends from the real numbers R to any locally compact abelian group. A particularly simple example is provided by finite abelian groups, where the proofs are either elementary or simplificati... | Weyl calculus | 0.784734 |
81,060 | {\displaystyle B(z_{1},z_{2})=x_{1}\cdot y_{2}-y_{1}\cdot x_{2}=\Im \,z_{1}\cdot {\overline {z_{2}}}.} The symplectic group Sp (2n,R) is defined to be group of automorphisms g of R2n preserving the form B. It follows from the Stone–von Neumann theorem that for each such g there is a unitary π(g) on L2(R) satisfying the... | Weyl calculus | 0.784734 |
81,061 | {\displaystyle {{\widehat {f}}(t)={1 \over (2\pi )^{n/2}}\int _{{\mathbf {R} }^{n}}f(x)e^{-ix\cdot t}\,dx.}} The Fourier inversion formula f ( x ) = 1 ( 2 π ) n / 2 ∫ R n f ^ ( t ) e i x ⋅ t d t {\displaystyle {f(x)={1 \over (2\pi )^{n/2}}\int _{{\mathbf {R} }^{n}}{\widehat {f}}(t)e^{ix\cdot t}\,dt}} shows that the Fou... | Weyl calculus | 0.784734 |
81,062 | By the Stone–von Neumann theorem, there is a unique irreducible representation corresponding to this cocycle. It follows that if g is an automorphism of R2 preserving the form B, i.e. an element of SL(2,R), then there is a unitary π(g) on L2(R) satisfying the covariance relation π ( g ) W ( z ) π ( g ) ∗ = W ( g ( z ) ... | Weyl calculus | 0.784734 |
81,063 | On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and su... | Weyl calculus | 0.784734 |
81,064 | It is an industry-led international research program with eight sub-topics including artificial intelligence hardware; analog mixed-signal circuits; computer-aided design and test; environment safety and health; hardware security; logic and memory devices; nanomanufacturing materials and processes; and packaging. | Semiconductor Research Corporation | 0.784734 |
81,065 | Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of... | Möbius transformation | 0.784732 |
81,066 | Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: Firstly, the projective linear group PGL(2, K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one... | Möbius transformation | 0.784732 |
81,067 | A particularly important discrete subgroup of the Möbius group is the modular group; it is central to the theory of many fractals, modular forms, elliptic curves and Pellian equations. Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving map... | Möbius transformation | 0.784732 |
81,068 | This group is called the Möbius group, and is sometimes denoted Aut ( C ^ ) {\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})} . The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifold... | Möbius transformation | 0.784732 |
81,069 | Möbius transformations are defined on the extended complex plane C ^ = C ∪ { ∞ } {\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}} (i.e., the complex plane augmented by the point at infinity). Stereographic projection identifies C ^ {\displaystyle {\widehat {\mathbb {C} }}} with a sphere, which is ... | Möbius transformation | 0.784732 |
81,070 | In molecular biology, olfactory marker protein is a protein involved in signal transduction. It is a highly expressed, cytoplasmic protein found in mature olfactory sensory receptor neurons of all vertebrates. OMP is a modulator of the olfactory signal transduction cascade. The crystal structure of OMP reveals a beta s... | Olfactory marker protein | 0.784732 |
81,071 | For example, a program written primarily in the C programming language, might have portions written in assembly language for optimization purposes. It is also possible for some components of a piece of software to be written and compiled separately, in an arbitrary programming language, and later integrated into the so... | Source files | 0.784731 |
81,072 | Stallman's definition thus contemplates JavaScript and HTML's source-target ambivalence, as well as contemplating possible future forms of software production, like visual programming languages, or datasets in Machine Learning.Other broader interpretations, however, consider source code to include the machine code alon... | Source files | 0.784731 |
81,073 | The SND will cover a wide range of physics, such as detecting all three neutrino flavors in the pseudorapidity (angular) range that has never been explored before. Along with the FASERnu detector at LHC, it will be the first experiment to observe and study the collider neutrinos. It will also search for Beyond Standard... | SND experiment (CERN) | 0.784731 |
81,074 | SND will primarily observe neutrinos in the pseudorapidity range of 7.2 to 8.6. It will detect the scattering properties of the neutrinos in this yet unexplored range and complement the observation range of FASERnu. The neutrinos in this range come from the decay of heavy quarks such as charm decays (c → s + W ± {\text... | SND experiment (CERN) | 0.784731 |
81,075 | The International Prize in Statistics is awarded every two years to an individual or team "for major achievements using statistics to advance science, technology and human welfare". The International Prize in Statistics, along with the COPSS Presidents' Award, are the two highest honours in the field of Statistics. The... | International Prize in Statistics | 0.784729 |
81,076 | The prize is awarded by the International Prize in Statistics Foundation, which comprises representatives of the following major learned societies: American Statistical Association International Biometric Society Institute of Mathematical Statistics International Statistical Institute Royal Statistical SocietyIn additi... | International Prize in Statistics | 0.784729 |
81,077 | The image f ( X ) {\displaystyle f(X)} of a local homeomorphism f: X → Y {\displaystyle f:X\to Y} is necessarily an open subset of its codomain Y {\displaystyle Y} and f: X → f ( X ) {\displaystyle f:X\to f(X)} will also be a local homeomorphism (that is, f {\displaystyle f} will continue to be a local homeomorphism wh... | Locally homeomorphic | 0.784728 |
81,078 | One corollary is that every continuous open surjection f {\displaystyle f} between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that O f {\displaystyle O_{f}} is a dense open subset of its domain). For example, the map f: R... | Locally homeomorphic | 0.784728 |
81,079 | Using the inverse function theorem one can show that a continuously differentiable function f: U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} (where U {\displaystyle U} is an open subset of R n {\displaystyle \mathbb {R} ^{n}} ) is a local homeomorphism if the derivative D x f {\displaystyle D_{x}f} is an invertible l... | Locally homeomorphic | 0.784728 |
81,080 | If f: X → Y {\displaystyle f:X\to Y} is continuous while both g: Y → Z {\displaystyle g:Y\to Z} and g ∘ f: X → Z {\displaystyle g\circ f:X\to Z} are local homeomorphisms, then f {\displaystyle f} is also a local homeomorphism. Inclusion maps If U ⊆ X {\displaystyle U\subseteq X} is any subspace (where as usual, U {\dis... | Locally homeomorphic | 0.784728 |
81,081 | Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates. | Calculus on Manifolds (book) | 0.784728 |
81,082 | Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f: Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fub... | Calculus on Manifolds (book) | 0.784728 |
81,083 | Calculus on Manifolds aims to present the topics of multivariable and vector calculus in the manner in which they are seen by a modern working mathematician, yet simply and selectively enough to be understood by undergraduate students whose previous coursework in mathematics comprises only one-variable calculus and int... | Calculus on Manifolds (book) | 0.784728 |
81,084 | A more recent textbook which also covers these topics at an undergraduate level is the text Analysis on Manifolds by James Munkres (366 pp.). At more than twice the length of Calculus on Manifolds, Munkres's work presents a more careful and detailed treatment of the subject matter at a leisurely pace. Nevertheless, Mun... | Calculus on Manifolds (book) | 0.784728 |
81,085 | Magnesium bromide can be synthesized by treating with magnesium oxide (and related basic salts) with hydrobromic acid. It can also be made by reacting magnesium carbonate and hydrobromic acids, and collecting the solid left after evaporation.As suggested by its easy conversion to various hydrates, anhydrous MgBr2 is a ... | Magnesium bromide | 0.784727 |
81,086 | Assuming that the sigma algebra is not trivial, that is, if B ≠ { X , ∅ } {\displaystyle {\mathcal {B}}\neq \{X,\varnothing \}} , then K ≠ T K . {\displaystyle {\mathcal {K}}\neq T{\mathcal {K}}.} It follows that K-automorphisms are strong mixing. All Bernoulli automorphisms are K-automorphisms, but not vice versa. Kol... | Kolmogorov automorphism | 0.784727 |
81,087 | Let ( X , B , μ ) {\displaystyle (X,{\mathcal {B}},\mu )} be a standard probability space, and let T {\displaystyle T} be an invertible, measure-preserving transformation. Then T {\displaystyle T} is called a K-automorphism, K-transform or K-shift, if there exists a sub-sigma algebra K ⊂ B {\displaystyle {\mathcal {K}}... | Kolmogorov automorphism | 0.784727 |
81,088 | In organic chemistry, carbon–hydrogen bond functionalization (C−H functionalization) is a type of organic reaction in which a carbon–hydrogen bond is cleaved and replaced with a C−X bond (where X is usually carbon, oxygen, or nitrogen). The term usually implies that a transition metal is involved in the C−H cleavage pr... | C-H functionalization | 0.784727 |
81,089 | In some cases, differential equations can be represented in a format that looks like delay differential equations. Example 1 Consider an equation Introduce y ( t ) = ∫ − ∞ 0 x ( t + τ ) e λ τ d τ {\displaystyle y(t)=\int _{-\infty }^{0}x(t+\tau )e^{\lambda \tau }\,d\tau } to get a system of ODEs Example 2 An equation i... | Delay differential equation | 0.784727 |
81,090 | Ferromagnetic resonance is a convenient laboratory method for determining the effect of magnetocrystalline anisotropy on the dispersion of spin waves. One group at the Max Planck Institute of Microstructure Physics in Halle, Germany proved that by using spin polarized electron energy loss spectroscopy (SPEELS), very hi... | Spin waves | 0.784726 |
81,091 | Theorem 4 Let the differential operator L {\displaystyle L} be defined by where A i ∈ Q ( x , y ) {\displaystyle A_{i}\in \mathbb {Q} (x,y)} for all i {\displaystyle i} . L x m ( L ) {\displaystyle \mathbb {L} _{x^{m}}(L)} and L y n ( L ) {\displaystyle \mathbb {L} _{y^{n}}(L)} as well as l m {\displaystyle {\mathfrak ... | Loewy decomposition | 0.784724 |
81,092 | There follows Consequently, a differential fundamental system is F {\displaystyle F} and G {\displaystyle G} are undetermined functions. If the only second-order derivative of an operator is ∂ x y {\displaystyle \partial _{xy}} , its possible decompositions involving only principal divisors may be described as follows.... | Loewy decomposition | 0.784724 |
81,093 | The existence of factors in a universal field, i.e. absolute irreducibility, may always be decided. The above theorem may be applied for solving reducible equations in closed form. | Loewy decomposition | 0.784724 |
81,094 | These results show that factorization provides an algorithmic scheme for solving reducible linear ode's. Whenever an equation of order 2 factorizes according to one of the types defined above the elements of a fundamental system are explicitly known, i.e. factorization is equivalent to solving it. A similar scheme may ... | Loewy decomposition | 0.784724 |
81,095 | The complete answer for n = 2 {\displaystyle n=2} is detailed in the following corollary to the above theorem.Corollary 1 Let L {\displaystyle L} be a second-order operator. Its possible Loewy decompositions are denoted by L 0 2 , … , L 3 2 {\displaystyle {\mathcal {L}}_{0}^{2},\ldots ,{\mathcal {L}}_{3}^{2}} , they ma... | Loewy decomposition | 0.784724 |
81,096 | Due to the lowering of order in each step, this proceeding terminates after a finite number of iterations and the desired decomposition is obtained. Based on these considerations, Loewy obtained the following fundamental result. The decomposition determined in this theorem is called the Loewy decomposition of L {\displ... | Loewy decomposition | 0.784724 |
81,097 | An operator L {\displaystyle L} of order n {\displaystyle n} is called reducible if it may be represented as the product of two operators L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} , both of order lower than n {\displaystyle n} . Then one writes L = L 1 L 2 {\displaystyle L=L_{1}L_{2}} , i.e. juxtaposition... | Loewy decomposition | 0.784724 |
81,098 | Factorizations and Loewy decompositions turned out to be an extremely useful method for determining solutions of linear differential equations in closed form, both for ordinary and partial equations. It should be possible to generalize these methods to equations of higher order, equations in more variables and system o... | Loewy decomposition | 0.784724 |
81,099 | Their general solutions contain not only constants but undetermined functions of varying numbers of arguments; determining them is part of the solution procedure. For equations with leading derivative ∂ x x {\displaystyle \partial _{xx}} Loewy's results may be generalized as follows. Theorem 2 Let the differential oper... | Loewy decomposition | 0.784724 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.