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79,700 | A Morpholino, also known as a Morpholino oligomer and as a phosphorodiamidate Morpholino oligomer (PMO), is a type of oligomer molecule (colloquially, an oligo) used in molecular biology to modify gene expression. Its molecular structure contains DNA bases attached to a backbone of methylenemorpholine rings linked thro... | Morpholino | 0.785255 |
79,701 | Network tomography may be able to infer network topology using end-to-end probes. Topology discovery is a tradeoff between accuracy vs. overhead. With network tomography, the emphasis is to achieve as accurate a picture of the network with minimal overhead. In comparison, other network topology discovery techniques usi... | Network tomography | 0.785252 |
79,702 | Network tomography seeks to map the path data takes through the Internet by examining information from “edge nodes,” the computers in which the data are originated and from which they are requested. The field is useful for engineers attempting to develop more efficient computer networks. Data derived from network tomog... | Network tomography | 0.785252 |
79,703 | Network tomography is the study of a network's internal characteristics using information derived from end point data. The word tomography is used to link the field, in concept, to other processes that infer the internal characteristics of an object from external observation, as is done in MRI or PET scanning (even tho... | Network tomography | 0.785252 |
79,704 | 2004 Levon Khachigian (UNSW) for developing DNA drugs with potential in cancer treatment. 2003 Mark Rowe (UNSW) for determining how sensations are processed and transmitted in the brains of mammals. 2002 Mark Hindell (University of Tasmania) for research on the behaviour of southern elephant seals and other marine pred... | Australasian Science | 0.785251 |
79,705 | The small Veblen ordinal θ Ω ω ( 0 ) {\displaystyle \theta _{\Omega ^{\omega }}(0)} or ψ ( Ω Ω ω ) {\displaystyle \psi (\Omega ^{\Omega ^{\omega }})} is the limit of ordinals that can be described using a version of Veblen functions with finitely many arguments. It is the ordinal that measures the strength of Kruskal's... | Small Veblen ordinal | 0.785251 |
79,706 | In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson. In other words, it is the probability distribution of the number of... | Poisson binomial distribution | 0.785251 |
79,707 | In the field of molecular biology, myocyte enhancer factor-2 (Mef2) proteins are a family of transcription factors which through control of gene expression are important regulators of cellular differentiation and consequently play a critical role in embryonic development. In adult organisms, Mef2 proteins mediate the s... | Mef2 | 0.785248 |
79,708 | A related example is that of van Kampen theorems for categories of covering morphisms in lextensive categories. Other reports of generalisations of the van Kampen theorem include statements for 2-categories and a topos of topoi . Important results in higher-dimensional algebra are also the extensions of the Galois theo... | Nonabelian algebraic topology | 0.785247 |
79,709 | Cubical omega-groupoids, higher homotopy groupoids, crossed modules, crossed complexes and Galois groupoids are key concepts in developing applications related to homotopy of filtered spaces, higher-dimensional space structures, the construction of the fundamental groupoid of a topos E in the general theory of topoi, a... | Nonabelian algebraic topology | 0.785247 |
79,710 | An important part of nonabelian algebraic topology is concerned with the properties and applications of homotopy groupoids and filtered spaces. Noncommutative double groupoids and double algebroids are only the first examples of such higher-dimensional structures that are nonabelian. The new methods of Nonabelian Algeb... | Nonabelian algebraic topology | 0.785247 |
79,711 | In mathematics, nonabelian algebraic topology studies an aspect of algebraic topology that involves (inevitably noncommutative) higher-dimensional algebras. Many of the higher-dimensional algebraic structures are noncommutative and, therefore, their study is a very significant part of nonabelian category theory, and al... | Nonabelian algebraic topology | 0.785247 |
79,712 | The circuits, or minimal dependent sets, of this matroid are the bicircular graphs (or bicycles, but that term has other meanings in graph theory); these are connected graphs whose circuit rank is exactly two. There are three distinct types of bicircular graph: The theta graph consists of three paths joining the same t... | Bicircular matroid | 0.785246 |
79,713 | John von Neumann posed a question about the Bernoulli process regrading the possibility of a given process being isomorphic to another, in the sense of the isomorphism of dynamical systems. The question long defied analysis, but was finally and completely answered with the Ornstein isomorphism theorem. This breakthroug... | Bernoulli sequence | 0.785245 |
79,714 | In this case, one may make use of Stirling's approximation to the factorial, and write n ! = 2 π n n n e − n ( 1 + O ( 1 n ) ) {\displaystyle n!={\sqrt {2\pi n}}\;n^{n}e^{-n}\left(1+{\mathcal {O}}\left({\frac {1}{n}}\right)\right)} Inserting this into the expression for P(k,n), one obtains the Normal distribution; this... | Bernoulli sequence | 0.785245 |
79,715 | The macrosystem is broader taking into account social economic status, culture, beliefs, customs and morals (example: a child from a wealthier family sees a peer from a less wealthy family as inferior for that reason). Lastly, the chronosystem refers to the chronological nature of life events and how they interact and ... | Child psychology | 0.785244 |
79,716 | Genetic factors can influence the timing and pattern of neural development, as well as the susceptibility to certain developmental disorders, such as autism spectrum disorder and attention-deficit/hyperactivity disorder.Research finds that the adolescent brain undergoes significant changes in neural connectivity and pl... | Child psychology | 0.785244 |
79,717 | These changes include the formation of neurons, the development of neural networks, and the establishment of synaptic connections. The formation of neurons and the establishment of basic neural circuits in the developing brain are crucial for laying the foundation of the brain's structure and function, and disruptions ... | Child psychology | 0.785244 |
79,718 | According to the report, the production and incineration of plastics will contribute in the equivalent of 850 million tonnes of carbon dioxide (CO2) to the atmosphere in 2019. With the current trend, annual life cycle greenhouse gas emissions of plastics will grow to 1.34 billion tonnes by 2030. By 2050, the life cycle... | Anthropogenic greenhouse gases | 0.785243 |
79,719 | In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in term of its number of edges.The conjecture states that if G is a simple undirected graph and L(G) its Laplacian matrix, ... | Brouwer's conjecture | 0.785242 |
79,720 | Long undersea or underground high-voltage cables have a high electrical capacitance compared with overhead transmission lines, since the live conductors within the cable are surrounded by a relatively thin layer of insulation (the dielectric), and a metal sheath. The geometry is that of a long coaxial capacitor. The to... | Green power superhighway | 0.785242 |
79,721 | On 7 September 2016, Micro Focus announced its intent to merge with Hewlett Packard Enterprise’s software business segment. The merger was completed on 1 September 2017.On 4 October 2016, Micro Focus announced it had acquired Gwava Inc., an enterprise information archiving business.On 19 March 2018, Micro Focus shares ... | Micro Focus | 0.785242 |
79,722 | Micro Focus was founded in 1976. In 1981, it became the first company to win the Queen's Award for Industry purely for developing a software product. The product was CIS COBOL, a standard-compliant COBOL implementation for microcomputers.In 1998, the company acquired Intersolv, Inc., an applications enablement business... | Micro Focus | 0.785242 |
79,723 | This electromagnetic radiation is at the precise frequency of 1420.40575177 MHz, which is equivalent to the vacuum wavelength of 21.10611405413 cm in free space. This wavelength or frequency falls within the microwave radio region of the electromagnetic spectrum, and it is observed frequently in radio astronomy, since ... | Primeval Structure Telescope | 0.785241 |
79,724 | Cognitive science is better understood as predominantly concerned with a much broader scope, with links to philosophy, linguistics, anthropology, neuroscience, and particularly with artificial intelligence. It could be said that cognitive science provides the corpus of information feeding the theories used by cognitive... | Cognitive psychologist | 0.785241 |
79,725 | In the early years of cognitive psychology, behaviorist critics held that the empiricism it pursued was incompatible with the concept of internal mental states. However, cognitive neuroscience continues to gather evidence of direct correlations between physiological brain activity and mental states, endorsing the basis... | Cognitive psychologist | 0.785241 |
79,726 | Developments in computer science would lead to parallels being drawn between human thought and the computational functionality of computers, opening entirely new areas of psychological thought. Allen Newell and Herbert Simon spent years developing the concept of artificial intelligence (AI) and later worked with cognit... | Cognitive psychologist | 0.785241 |
79,727 | They were successful in most cases, and this facilitated developing approximate solutions, efficiently evaluating chaos, and constructing Lypapunov functions. Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and quasi-steady stat... | Differential ring | 0.785239 |
79,728 | The differential ideal generated by S {\displaystyle S} is the set of the finite linear combinations of elements of S {\displaystyle S} and of the derivatives of any order of these elements; it is commonly denoted as . {\displaystyle .} When S {\displaystyle S} is finite, {\displaystyle } is generally not finitely ge... | Differential ring | 0.785239 |
79,729 | : 61–62 It follows that, given a subset S {\displaystyle S} of a differential ring, there are three ideals generated by it, which are the intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it. : 61–62: 21 The algebraic ideal generated by S {\d... | Differential ring | 0.785239 |
79,730 | A prime differential ideal is always a radical differential ideal. A discovery of Ritt is that, although the classical theory of algebraic ideals does not work for differential ideals, a large part of it can be extended to radical differential ideals, and this makes them fundamental in differential algebra. The interse... | Differential ring | 0.785239 |
79,731 | A differential ideal is said proper if it is not the whole ring. For avoiding confusion, an ideal that is not a differential ideal is sometimes called an algebraic ideal. The radical of a differential ideal is the same as its radical as an algebraic ideal, that is, the set of the ring elements that have a power in the ... | Differential ring | 0.785239 |
79,732 | Using differential Gröbner bases, researchers have investigated non-classical symmetry properties of non-linear differential equations. Other applications include control theory, model theory, and algebraic geometry. Differential algebra also applies to differential-difference equations. | Differential ring | 0.785239 |
79,733 | Another approach is to create a class of differential equations with a known solution form; matching a differential equation to its class identifies the equation's solution. Methods are available to facilitate the numerical integration of a differential-algebraic system of equations. : 41–47 In a study of non-linear dy... | Differential ring | 0.785239 |
79,734 | Differential algebra can determine if a set of differential polynomial equations has a solution. A total order ranking may identify algebraic constraints. An elimination ranking may determine if one or a selected group of independent variables can express the differential equations. Using triangular decomposition and e... | Differential ring | 0.785239 |
79,735 | The Gmelin database is a large database of organometallic and inorganic compounds updated quarterly. It is based on the German publication Gmelins Handbuch der anorganischen Chemie ("Gmelin's Handbook of Inorganic Chemistry") which was originally published by Leopold Gmelin in 1817; the last print edition, the 8th, app... | Gmelin database | 0.785237 |
79,736 | This method consists of relaxing the rank constraint r a n k ( L ) {\displaystyle rank(L)} in the optimization problem to the nuclear norm ‖ L ‖ ∗ {\displaystyle \|L\|_{*}} and the sparsity constraint ‖ S ‖ 0 {\displaystyle \|S\|_{0}} to ℓ 1 {\displaystyle \ell _{1}} -norm ‖ S ‖ 1 {\displaystyle \|S\|_{1}} . The result... | Robust principal component analysis | 0.785235 |
79,737 | Some recent works propose RPCA algorithms with learnable/training parameters. Such a learnable/trainable algorithm can be unfolded as a deep neural network whose parameters can be learned via machine learning techniques from a given dataset or problem distribution. The learned algorithm will have superior performance o... | Robust principal component analysis | 0.785235 |
79,738 | Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observations. A number of different approaches exist for Robust PCA, including an idealized version of Robust PCA, which aims to ... | Robust principal component analysis | 0.785235 |
79,739 | RSL-CV 2015: Workshop on Robust Subspace Learning and Computer Vision in conjunction with ICCV 2015 (For more information: http://rsl-cv2015.univ-lr.fr/workshop/) RSL-CV 2017: Workshop on Robust Subspace Learning and Computer Vision in conjunction with ICCV 2017 (For more information: http://rsl-cv.univ-lr.fr/2017/) RS... | Robust principal component analysis | 0.785235 |
79,740 | The LRS Library (developed by Andrews Sobral) provides a collection of low-rank and sparse decomposition algorithms in MATLAB. The library was designed for moving object detection in videos, but it can be also used for other computer vision / machine learning tasks. Currently the LRSLibrary offers more than 100 algorit... | Robust principal component analysis | 0.785235 |
79,741 | RPCA has many real life important applications particularly when the data under study can naturally be modeled as a low-rank plus a sparse contribution. Following examples are inspired by contemporary challenges in computer science, and depending on the applications, either the low-rank component or the sparse componen... | Robust principal component analysis | 0.785235 |
79,742 | Black polyethylene microspheres can have magnetic or conductive functionality, and have uses in electronic devices, EMI shielding and microscopy techniques.Fluorescent polyethylene microspheres are commonly used to run blind tests on laboratory and industrial processes, in order to develop proper methods and minimize c... | Microbead (research) | 0.785234 |
79,743 | Gene expression analysis also benefits from microbeads, such as isolating mRNA for transcriptional analysis. There are many uses for microbeads, mostly for biotechnology and biomedical research. Microbeads and magnetic separation technology have enabled a range of innovative methods to benefit research on disease preve... | Microbead (research) | 0.785234 |
79,744 | A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space. There are many ways to form Lusin spaces. In particular: Every Polish space is Lusin A subspace of a Lusin space is Lusin if and only if it is a Borel set. Any countable union or intersection of Lusin subspaces of a H... | Suslin space | 0.785234 |
79,745 | The Magellanic Premium, also known as the Magellanic Gold Medal and Magellanic Prize is awarded for major contributions in the field of navigation (whether by sea, air, or in space), astronomy, or natural philosophy. The Premium was established in 1786 through a grant by Jean-Hyacinthe Magellan (Portuguese: João Jacint... | Magellanic Premium | 0.785233 |
79,746 | A basic example of maps between manifolds are scalar-valued functions on a manifold, f: M → R {\displaystyle \scriptstyle f\colon M\to \mathbb {R} } or f: M → C , {\displaystyle \scriptstyle f\colon M\to \mathbb {C} ,} sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algeb... | Maps of manifolds | 0.785231 |
79,747 | In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In complex geometry, ramified covering spac... | Maps of manifolds | 0.785231 |
79,748 | Just as there are various types of manifolds, there are various types of maps of manifolds. In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and ... | Maps of manifolds | 0.785231 |
79,749 | In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed | Maps of manifolds | 0.785231 |
79,750 | Dual to scalar-valued functions – maps M → R {\displaystyle \scriptstyle M\to \mathbb {R} } – are maps R → M , {\displaystyle \scriptstyle \mathbb {R} \to M,} which correspond to curves or paths in a manifold. One can also define these where the domain is an interval , {\displaystyle ,} especially the unit interval ,... | Maps of manifolds | 0.785231 |
79,751 | In particle physics, a symmetry that remains after spontaneous symmetry breaking that can prevent higher-order radiative corrections from spoiling some property of a theory is called a custodial symmetry. | Custodial symmetry | 0.785229 |
79,752 | In the Standard Model of particle physics, the custodial symmetry is a residual global SU(2) symmetry of the Higgs potential V S M = − μ ( H † H ) − λ ( H † H ) 2 {\displaystyle V_{SM}=-\mu (H^{\dagger }H)-\lambda (H^{\dagger }H)^{2}} beyond the basic SU(2)×U(1) gauge symmetry of the Weak Interaction that prevents high... | Custodial symmetry | 0.785229 |
79,753 | The Higher Education Statistics Agency (HESA) classifies higher and further education courses using a more detailed framework using letter codes based on the original FHEQ. This separates postgraduate courses into research and taught, based on the largest component in terms of student effort. Doctorate-level courses ar... | Regulated Qualifications Framework | 0.785229 |
79,754 | When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry. When a surface has a constant positive Gaussian curvature, then the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there e... | Liebmann's theorem | 0.785228 |
79,755 | If a sphere is deformed, it does not remain a sphere, proving that a sphere is rigid. A standard proof uses Hilbert's lemma that non-umbilical points of extreme principal curvature have non-positive Gaussian curvature. Hilbert's theorem (1901) states that there exists no complete analytic (class Cω) regular surface in ... | Liebmann's theorem | 0.785228 |
79,756 | Liebmann's theorem (1900) answered Minding's question. The only regular (of class C2) closed surfaces in R3 with constant positive Gaussian curvature are spheres. | Liebmann's theorem | 0.785228 |
79,757 | Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the q... | Liebmann's theorem | 0.785228 |
79,758 | In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, the types of decomposition of module... | Azumaya's theorem | 0.785227 |
79,759 | Then, since L 1 {\displaystyle L_{1}} is a direct summand of L {\displaystyle L} , we can write L = L 1 ⊕ L 1 ′ {\displaystyle L=L_{1}\oplus L_{1}'} and then ⊕ j ∈ F M j ≃ H ⊕ L 1 ′ {\displaystyle \oplus _{j\in F}M_{j}\simeq H\oplus L_{1}'} , which implies, since F is finite, that H ≃ ⊕ j ∈ J M j {\displaystyle H\simeq... | Azumaya's theorem | 0.785227 |
79,760 | In the simplest form, Azumaya's theorem states: given a decomposition M = ⨁ i ∈ I M i {\displaystyle M=\bigoplus _{i\in I}M_{i}} such that the endomorphism ring of each M i {\displaystyle M_{i}} is local (so the decomposition is indecomposable), each indecomposable decomposition of M is equivalent to this given decompo... | Azumaya's theorem | 0.785227 |
79,761 | R: The sf package implements Simple Features and contains functions that bind to GDAL for reading and writing data, to GEOS for geometrical operations, and to PROJ for projection conversions and datum transformations. The GDAL library implements the Simple Features data model in its OGR component. The Java-based deegre... | Simple Features | 0.785225 |
79,762 | Part 2 is a SQL binding to Part 1, providing a translation of the interface to non-object-oriented environments. For example, instead of a someGeometryObject.isEmpty() as in Part 1, SQL/MM uses a ST_IsEmpty(...) function in SQL. | Simple Features | 0.785225 |
79,763 | The geometries are associated with spatial reference systems. The standard also specifies attributes, methods and assertions with the geometries, in the object-oriented style. In general, a 2D geometry is simple if it contains no self-intersection. The specification defines DE-9IM spatial predicates and several spatial... | Simple Features | 0.785225 |
79,764 | Simple Features (officially Simple Feature Access) is a set of standards that specify a common storage and access model of geographic features made of mostly two-dimensional geometries (point, line, polygon, multi-point, multi-line, etc.) used by geographic information systems. It is formalized by both the Open Geospat... | Simple Features | 0.785225 |
79,765 | ⟩ {\textstyle \sigma _{N,{\rm {classical}}}={\sqrt {\langle N\rangle }}} , and with the most-probable N value being near ⟨ N ⟩ {\displaystyle \langle N\rangle } .) | Bose statistics | 0.785224 |
79,766 | Fermi–Dirac statistics arises when considering the effect of the Pauli exclusion principle: whilst the number of fermions occupying the same single-particle state can only be either 1 or 0, the number of bosons occupying a single particle state may be any integer. Thus, the grand partition function for bosons can be co... | Bose statistics | 0.785224 |
79,767 | Modern flat Earth beliefs are promoted by organizations and individuals advocating that the Earth is flat while denying the Earth's sphericity, contrary to over two millennia of scientific consensus. Flat Earth beliefs are pseudoscience; the hypotheses and assertions are not based on scientific knowledge. Flat Earth ad... | Zetetic Astronomy | 0.785223 |
79,768 | Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and acco... | Bayesian hierarchical modeling | 0.785221 |
79,769 | The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options.Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having sur... | Bayesian hierarchical modeling | 0.785221 |
79,770 | The Dartmouth Summer Research Project on Artificial Intelligence was a 1956 summer workshop widely considered to be the founding event of artificial intelligence as a field. The project lasted approximately six to eight weeks and was essentially an extended brainstorming session. Eleven mathematicians and scientists or... | Dartmouth workshop | 0.78522 |
79,771 | We think that a significant advance can be made in one or more of these problems if a carefully selected group of scientists work on it together for a summer. The proposal goes on to discuss computers, natural language processing, neural networks, theory of computation, abstraction and creativity (these areas within th... | Dartmouth workshop | 0.78522 |
79,772 | The Proposal states: We propose that a 2-month, 10-man study of artificial intelligence be carried out during the summer of 1956 at Dartmouth College in Hanover, New Hampshire. The study is to proceed on the basis of the conjecture that every aspect of learning or any other feature of intelligence can in principle be s... | Dartmouth workshop | 0.78522 |
79,773 | He chose the name partly for its neutrality; avoiding a focus on narrow automata theory, and avoiding cybernetics which was heavily focused on analog feedback, as well as him potentially having to accept the assertive Norbert Wiener as guru or having to argue with him.In early 1955, McCarthy approached the Rockefeller ... | Dartmouth workshop | 0.78522 |
79,774 | In the early 1950s, there were various names for the field of "thinking machines": cybernetics, automata theory, and complex information processing. The variety of names suggests the variety of conceptual orientations. In 1955, John McCarthy, then a young Assistant Professor of Mathematics at Dartmouth College, decided... | Dartmouth workshop | 0.78522 |
79,775 | He also eventually died in 1994. In the same way some of the persons who worked in the basement of the wrecked Chernobyl have built up doses of 10 Gy, these workers received these doses in small fractions so the acute effects were avoided. It has been found in radiation biology experiments that if a group of cells are ... | Radioresistance | 0.785219 |
79,776 | In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including ... | Sum-of-divisors function | 0.785219 |
79,777 | This is often accomplished by the use of a preservation theorem such as: Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve ω 1 {\displaystyle \omega _{1}} . Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and t... | Iterated forcing | 0.785218 |
79,778 | The 4 k B T R {\displaystyle 4k_{\text{B}}TR} voltage noise described above is a special case for a purely resistive component for low frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the fluctuation-dissip... | Johnson–Nyquist noise | 0.785214 |
79,779 | In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental equation in the study of minimal submanifolds. It was discovered by James Simons in 1968. It can be viewed as a formula for the Laplacian of the seco... | Simons' formula | 0.785214 |
79,780 | The OnlineHPC was a free public web service that supplied tools to deal with high performance computers and online workflow editor. OnlineHPC allowed users to design and execute workflows using the online workflow designer and to work with high performance computers – clusters and clouds. Access to high performance res... | OnlineHPC | 0.785213 |
79,781 | After a while, it became obvious that engineering and scientific tasks require a more elaborate tool suit that enables researchers to execute the flow of tasks – workflows. Unless there is a number of scientific workflow implementations, they are almost all desktop applications and thus the aim was creating an online e... | OnlineHPC | 0.785213 |
79,782 | In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: F ( x ) = 2 π arcsin ( x ) = arcsin ( 2 x − 1 ) π + 1 2 {\displaystyle F(x)={\frac {2}{\pi }}\arcsin \left({\sqrt {x}}\right)={\frac {\arcsin(2x-1)}{\pi }}... | Arcsine transformation | 0.785213 |
79,783 | In the mathematical theory of probability, the Heyde theorem is the characterization theorem concerning the normal distribution (the Gaussian distribution) by the symmetry of one linear form given another. This theorem was proved by C. C. Heyde. | Heyde theorem | 0.785213 |
79,784 | To prove existence, it can be assumed after an affine transformation that f ( 0 ) = 0 {\displaystyle f(0)=0} and f ′ ( 0 ) = I {\displaystyle f^{\prime }(0)=I} , so that a = b = 0 {\displaystyle a=b=0} . By the mean value theorem for vector-valued functions, for a function u: → R m {\displaystyle u:\to \mathbb {R} ^{m... | Inverse function theorem | 0.785208 |
79,785 | Thus, ∂ f j − 1 ∂ w ¯ k ( w ) = 0 {\displaystyle {\frac {\partial f_{j}^{-1}}{\partial {\overline {w}}_{k}}}(w)=0} for each k {\displaystyle k} . ◻ {\displaystyle \square } Similarly, there is the implicit function theorem for holomorphic functions.As already noted earlier, it can happen that an injective smooth functi... | Inverse function theorem | 0.785208 |
79,786 | Then we have det J R ( f ) = | det J ( f ) | 2 {\displaystyle \det J_{\mathbb {R} }(f)=|\det J(f)|^{2}} , which is nonzero by assumption. Hence, by the usual inverse function theorem, f {\displaystyle f} is injective near 0 {\displaystyle 0} with continuously differentiable inverse. By chain rule, with w = f ( z ) {\di... | Inverse function theorem | 0.785208 |
79,787 | There is a version of the inverse function theorem for holomorphic maps. The theorem follows from the usual inverse function theorem. Indeed, let J R ( f ) {\displaystyle J_{\mathbb {R} }(f)} denote the Jacobian matrix of f {\displaystyle f} in variables x i , y i {\displaystyle x_{i},y_{i}} and J ( f ) {\displaystyle ... | Inverse function theorem | 0.785208 |
79,788 | Thus the constant rank theorem applies to a generic point of the domain. When the derivative of F is injective (resp. surjective) at a point p, it is also injective (resp. surjective) in a neighborhood of p, and hence the rank of F is constant on that neighborhood, and the constant rank theorem applies. | Inverse function theorem | 0.785208 |
79,789 | The inverse function theorem (and the implicit function theorem) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point. Specifically, if F: M → N {\displaystyle F:M\to N} has constant rank near a ... | Inverse function theorem | 0.785208 |
79,790 | Hence, f − 1 ( b ) {\displaystyle f^{-1}(b)} is a manifold. ◻ {\displaystyle \square } (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.) More generally, the theorem shows that if a smooth map f: P → E {\displaystyle f:P\... | Inverse function theorem | 0.785208 |
79,791 | That is, after the change of coordinates by G {\displaystyle G} , f {\displaystyle f} is a coordinate projection (this fact is known as the submersion theorem). Moreover, since G: V × W → U ′ = G ( V × W ) {\displaystyle G:V\times W\to U'=G(V\times W)} is bijective, the map g = G ( b , ⋅ ): W → f − 1 ( b ) ∩ U ′ , ( x ... | Inverse function theorem | 0.785208 |
79,792 | Hence, by the inverse function theorem, we find the smooth inverse G {\displaystyle G} of F {\displaystyle F} defined in a neighborhood V × W {\displaystyle V\times W} of ( b , a r + 1 , … , a n ) {\displaystyle (b,a_{r+1},\dots ,a_{n})} . We then have x = ( F ∘ G ) ( x ) = ( f ( G ( x ) ) , G r + 1 ( x ) , … , G n ( x... | Inverse function theorem | 0.785208 |
79,793 | In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold. Indeed, let f: U → R r {\displaystyle f:U\to \mathbb {R} ^{r}} be such a smooth map from an open subset of R n {\displaystyle \mathbb {R} ^{n}} (since the result is local, there... | Inverse function theorem | 0.785208 |
79,794 | If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unkn... | Inverse function theorem | 0.785208 |
79,795 | Define h ( x ) = a + T x {\displaystyle h(x)=a+Tx} so that we have: ( f ∘ h ) ′ ( 0 ) = f ′ ( a ) ∘ T = I . {\displaystyle (f\circ h)'(0)=f'(a)\circ T=I.} Thus, by the inverse function theorem, f ∘ h {\displaystyle f\circ h} has inverse near 0 {\displaystyle 0} ; i.e., f ∘ h ∘ ( f ∘ h ) − 1 = I {\displaystyle f\circ h\... | Inverse function theorem | 0.785208 |
79,796 | The statement in the first case is a special case of the submersion theorem. These variants are restatements of the inverse functions theorem. Indeed, in the first case when f ′ ( a ) {\displaystyle f'(a)} is surjective, we can find an (injective) linear map T {\displaystyle T} such that f ′ ( a ) ∘ T = I {\displaystyl... | Inverse function theorem | 0.785208 |
79,797 | There are two variants of the inverse function theorem. Given a continuously differentiable map f: U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , the first is The derivative f ′ ( a ) {\displaystyle f'(a)} is surjective (i.e., the Jacobian matrix representing it has rank m {\displaystyle m} ) if and only if there ex... | Inverse function theorem | 0.785208 |
79,798 | {\displaystyle Jf^{-1}(b)=Jf(a)^{-1}.} The hard part of the theorem is the existence and differentiability of f − 1 {\displaystyle f^{-1}} . | Inverse function theorem | 0.785208 |
79,799 | Moreover, the theorem says that the inverse function f − 1: V → U {\displaystyle f^{-1}:V\to U} is continuously differentiable, and its derivative at b = f ( a ) {\displaystyle b=f(a)} is the inverse map of f ′ ( a ) {\displaystyle f'(a)} ; i.e., ( f − 1 ) ′ ( b ) = f ′ ( a ) − 1 . {\displaystyle (f^{-1})'(b)=f'(a)^{-1... | Inverse function theorem | 0.785208 |
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