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88,400 | {\displaystyle S.} The family of seminorms { q U: U ∈ U } {\displaystyle \left\{q_{U}:U\in {\mathfrak {U}}\right\}} generates a topology making S {\displaystyle S} into a locally convex space. The vector space S {\displaystyle S} endowed with this topology will be denoted by l 1 ( A , X ) . | Injective tensor product | 0.781685 |
88,401 | This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define (from scratch) a topology on such a tensor product is frequently equivalent to the injective or projective tensor product topology. Let U {\displaystyl... | Injective tensor product | 0.781685 |
88,402 | {\displaystyle S.} We now define a topology on S {\displaystyle S} in a very natural way. This topology turns out to be the injective topology taken from l 1 ( A ) ⊗ ^ ε X {\displaystyle l^{1}(A){\widehat {\otimes }}_{\varepsilon }X} and transferred to S {\displaystyle S} via a canonical vector space isomorphism (the o... | Injective tensor product | 0.781685 |
88,403 | The set of separately continuous bilinear maps X × Y → Z {\displaystyle X\times Y\to Z} (that is, continuous in each variable when the other variable is fixed) will be denoted by B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z)} where if Z {\displaystyle Z} is the scalar field then we may instead write B ( X , Y ) ... | Injective tensor product | 0.781685 |
88,404 | Throughout let X , Y , {\displaystyle X,Y,} and Z {\displaystyle Z} be topological vector spaces and L: X → Y {\displaystyle L:X\to Y} be a linear map. L: X → Y {\displaystyle L:X\to Y} is a topological homomorphism or homomorphism, if it is linear, continuous, and L: X → Im L {\displaystyle L:X\to \operatorname {Im}... | Injective tensor product | 0.781685 |
88,405 | The strongest locally convex topology on B ( X σ ′ , Y σ ′ ) = X ⊗ Y {\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)=X\otimes Y} making the canonical map ⋅ ⊗ ⋅: X × Y → B ( X σ ′ , Y σ ′ ) {\displaystyle \cdot \otimes \cdot :X\times Y\to B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)... | Injective tensor product | 0.781685 |
88,406 | Henceforth, all topological vector spaces considered will be assumed to be locally convex. If Z {\displaystyle Z} is any locally convex topological vector space, then for any equicontinuous subsets G ⊆ X ′ {\displaystyle G\subseteq X^{\prime }} and H ⊆ Y ′ , {\displaystyle H\subseteq Y^{\prime },} and any neighborhood ... | Injective tensor product | 0.781685 |
88,407 | {\displaystyle J(X,Y).} The elements of J ( X , Y ) {\displaystyle J(X,Y)} are called integral (bilinear) forms on X × Y . {\displaystyle X\times Y.} The following theorem justifies the word integral. | Injective tensor product | 0.781685 |
88,408 | The canonical map ⋅ ⊗ ⋅: X × Y → B ( X σ ′ , Y σ ′ ) {\displaystyle \cdot \otimes \cdot :X\times Y\to {\mathcal {B}}\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)} is always continuous and the ε-topology is always finer than the π-topology and coarser than the inductive topology (which is the finest locally c... | Injective tensor product | 0.781685 |
88,409 | Suppose that denotes the TVS-embedding of X ⊗ ε Y {\displaystyle X\otimes _{\varepsilon }Y} into its completion and let be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of X ⊗ ε Y {\displaystyle X\otimes _{\varepsilon }Y} as being identical to the continuous dual space of... | Injective tensor product | 0.781685 |
88,410 | If X {\displaystyle X} is a normable space then a subset H ⊆ X ′ {\displaystyle H\subseteq X^{\prime }} is equicontinuous if and only if it is strongly bounded (that is, bounded in X b ′ {\displaystyle X_{b}^{\prime }} ). If X {\displaystyle X} is a barrelled space then for any subset H ⊆ X ′ , {\displaystyle H\subsete... | Injective tensor product | 0.781685 |
88,411 | If X {\displaystyle X} is separable, Y {\displaystyle Y} is metrizable, and D {\displaystyle D} is a dense subset of X , {\displaystyle X,} then the topology of pointwise convergence on D {\displaystyle D} makes L ( X ; Y ) {\displaystyle L(X;Y)} metrizable so that in particular, the subspace topology that any equicont... | Injective tensor product | 0.781685 |
88,412 | So in particular, H {\displaystyle H} will also bounded in every TVS topology that is coarser than the topology of bounded convergence. If X {\displaystyle X} is a barrelled space and Y {\displaystyle Y} is locally convex then for any subset H ⊆ L ( X ; Y ) , {\displaystyle H\subseteq L(X;Y),} the following are equival... | Injective tensor product | 0.781685 |
88,413 | If H ⊆ L ( X ; Y ) {\displaystyle H\subseteq L(X;Y)} is equicontinuous then the subspace topologies that H {\displaystyle H} inherits from the following topologies on L ( X ; Y ) {\displaystyle L(X;Y)} are identical:the topology of precompact convergence; the topology of compact convergence; the topology of pointwise c... | Injective tensor product | 0.781685 |
88,414 | Furthermore, the topology of a locally convex Hausdorff space X {\displaystyle X} is identical to the topology of uniform convergence on the equicontinuous subsets of X ′ . {\displaystyle X^{\prime }.} | Injective tensor product | 0.781685 |
88,415 | Specifically, distinct LCTVS topologies on X {\displaystyle X} produce distinct collections of equicontinuous subsets and conversely, given any such collection of equicontinuous sets, the TVS's original topology can be recovered by taking the polar of every (equicontinuous) set in the collection. Thus through this iden... | Injective tensor product | 0.781685 |
88,416 | One reason for converging on equicontinuous subsets (of all possibilities) is the following important fact: A set of continuous linear functionals H {\displaystyle H} on a TVS X {\displaystyle X} is equicontinuous if and only if it is contained in the polar of some neighborhood U {\displaystyle U} of 0 {\displaystyle 0... | Injective tensor product | 0.781685 |
88,417 | As usual, if X ′ {\displaystyle X^{\prime }} is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be b ( X ′ , X ) . {\displaystyle b\left(X^{\prime },X\right).} τ ( X , X ′ ) {\displaystyle \tau \left(X,X^{\prime }\right)} den... | Injective tensor product | 0.781685 |
88,418 | σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} is the coarsest topology on X′ making all such maps continuous. b ( X , X ′ ) {\displaystyle b\left(X,X^{\prime }\right)} denotes the topology of bounded convergence on X {\displaystyle X} and X b ( X , X ′ ) {\displaystyle X_{b\left(X,X^{\prime }\right)}}... | Injective tensor product | 0.781685 |
88,419 | {\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right).} The following theorem may be used to verify that B ( X σ ′ , Y σ ′ ) {\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)} together with the above map ⊗ {\displaystyle \,\otimes \,} is a tensor product of X {\displaystyle X} ... | Injective tensor product | 0.781685 |
88,420 | Although the question of whether or not one vector space is a tensor product of two other vector spaces is a purely algebraic one (that is, the answer does not depend on the topologies of X {\displaystyle X} or Y {\displaystyle Y} ), nevertheless the vector space B ( X σ ′ , Y σ ′ ) {\displaystyle B\left(X_{\sigma }^{\... | Injective tensor product | 0.781685 |
88,421 | In older landfills and those with no membrane between the waste and the underlying geology, leachate is free to leave the waste and flow directly into the groundwater. In such cases, high concentrations of leachate are often found in nearby springs and flushes. As leachate first emerges it can be black in colour, anoxi... | Leachate | 0.781685 |
88,422 | John Gow was a mechanical engineering graduate of the University of Leeds who had gone into computer programming and then became a software support manager at a Lancashire office of the British subsidiary of Digital Equipment Corporation (DEC). He also did some hardware sales work and realised that few of the customers... | Systime Computers | 0.781683 |
88,423 | Gow emphasized that Systime would provide not just hardware but also software applications, systems engineering, and support. By 1975, Systime had £2.75 million in turnover and profits of £300,000 and was already opening offices and subsidiaries overseas.Musichire's stake in the company impeded the company's ability to... | Systime Computers | 0.781683 |
88,424 | IPFN is organized in seven research groups: Engineering and Systems Integration, Experimental Physics, Materials Processing and Characterisation, Theory and Modelling, Lasers and Plasmas, Gas Discharges and Gaseous Electronics, and High Pressure Plasmas. The activities in the frame of the Associate Laboratory are evalu... | Instituto de Plasmas e Fusão Nuclear | 0.781682 |
88,425 | IPFN activities are centered on the following competences: Magnetic Confinement Fusion Devices Fusion Engineering Systems Fusion Theory and Modelling Inertial fusion Laser-Plasma Accelerators High-Performance Computing Relativistic Astrophysics Novel Radiation Sources Ultra Intense Photonics Space Physics Environmental... | Instituto de Plasmas e Fusão Nuclear | 0.781682 |
88,426 | Instituto de Plasmas e Fusão Nuclear (IPFN) (Institute for Plasmas and Nuclear Fusion) is a research unit of Instituto Superior Técnico (IST), Lisbon, and a leading Portuguese institution in physics research. IPFN has the status of Associate Laboratory in the thematic areas of controlled nuclear fusion, plasma technolo... | Instituto de Plasmas e Fusão Nuclear | 0.781682 |
88,427 | The Brumer Stark conjecture is known to be true for extensions K/k where K/Q is cyclotomic: this follows from Stickelberger's theorem K is abelian over Q K/k is a quadratic extension K/k is a biquadratic extensionSamit Dasgupta and Mahesh Kakde posted an article on Annals of Mathematics about the conjecture. | Brumer–Stark conjecture | 0.781679 |
88,428 | The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark. It arises as a special ... | Brumer–Stark conjecture | 0.781679 |
88,429 | Let μK be the group of roots of unity in K. The group G acts on μK; let A be the annihilator of μK as a Z-module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that θ(0) is actually in Q. A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Dani... | Brumer–Stark conjecture | 0.781679 |
88,430 | At a given trophic level there may be one species or a group of species with the same predators and prey.In 1927, Charles Elton published an influential synthesis on the use of food webs, which resulted in them becoming a central concept in ecology. In 1966, interest in food webs increased after Robert Paine's experime... | Ecological theory | 0.781679 |
88,431 | One class of computational models that is becoming increasingly popular are the agent-based models. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in the ... | Ecological theory | 0.781679 |
88,432 | A population ecology concept is r/K selection theory, one of the first predictive models in ecology used to explain life-history evolution. The premise behind the r/K selection model is that natural selection pressures change according to population density. For example, when an island is first colonized, density of in... | Ecological theory | 0.781679 |
88,433 | Fisher published his classic The Genetical Theory of Natural Selection, which introduced the idea that frequency-dependent fitness brings a strategic aspect to evolution, where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable of... | Ecological theory | 0.781679 |
88,434 | These model ecological processes that can be described as occurring over discrete time steps. Matrix algebra is often used to investigate the evolution of age-structured or stage-structured populations. The Leslie matrix, for example, mathematically represents the discrete time change of an age structured population.Mo... | Ecological theory | 0.781679 |
88,435 | Markov chain models are stochastic.Species can be modelled in continuous or discrete time. Continuous time is modelled using differential equations. Discrete time is modelled using difference equations. | Ecological theory | 0.781679 |
88,436 | The arctangent function of trigonometry can be expanded in a power series: arctan ( z ) = z − z 3 3 + z 5 5 − z 7 7 + ⋯ . {\displaystyle \arctan(z)=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots .} It is easy to apply the root test in this case to find that the radius of convergence is 1. | Radius of convergence | 0.781678 |
88,437 | Instead, the affine forms of them contain a Hadamard-like gate as a sub-matrix, which means, in the Heisenberg picture, they are able to generate superpositions of Pauli operators. This fact also generalises to the qudit case, with Hadamard replaced by Fourier transform operators. This serves as an intriguing fact rega... | Choi–Jamiołkowski isomorphism | 0.781678 |
88,438 | It is clear to see that a gen-extreme but not extreme channel is a convex sum of extreme channels of lower ranks. It has been conjectured and numerically supported that an arbitrary channel can be written as a convex sum of at most d {\displaystyle d} gen-extreme channels E = ∑ i = 1 d p i E g i {\displaystyle E=\sum _... | Choi–Jamiołkowski isomorphism | 0.781678 |
88,439 | Lafont and Streicher (1991) point out that Chu spaces over 2 realize both topological spaces and coherent spaces (introduced by J.-Y. Girard (1987) to model linear logic), while Chu spaces over K realize any category of vector spaces over a field whose cardinality is at most that of K. This was extended by Vaughan Prat... | Chu space | 0.781677 |
88,440 | The achieved acceleration is remarkable, but the reaction inevitably turns from neutral (pH) to acidic. Comparable acceleration is achieved with acetic anhydride, which is used in sugar and nucleoside chemistry. Reaction acceleration depends not only on the additives but also on their form, so all reagents are preferre... | Pyridinium dichromate | 0.781676 |
88,441 | The Nordic Institute for Theoretical Physics, or NORDITA, or Nordita (Danish: Nordisk Institut for Teoretisk Fysik), is an international organisation for research in theoretical physics. It was established as Nordisk Institut for Teoretisk Atomfysik in 1957 by Niels Bohr and the Swedish physicist Torsten Gustafson. Nor... | Nordic Institute for Theoretical Physics | 0.781675 |
88,442 | Since Nordita's establishment in 1957 the original focus on research in atomic and nuclear physics has been broadened. Research carried out by Nordita's academic staff presently includes astrophysics, biological physics, hard condensed matter physics and materials physics, soft condensed matter physics, cosmology, stat... | Nordic Institute for Theoretical Physics | 0.781675 |
88,443 | Aryabhatta Research Institute of Observational Sciences (ARIES) is a research institute in Nainital, Kumaon, India which specializes in astronomy, solar physics, astrophysics and atmospheric science. It is an autonomous body under the Department of Science and Technology, Government of India. The institute is situated ... | Aryabhatta Research Institute of Observational Sciences | 0.781675 |
88,444 | The Institute has in-house workshops to meet the requirements of electronic, mechanical, and optical maintenance of the instruments. ARIES has a modern computer center with Internet and a library with more than 10,000 volumes of research journals and a collection of books on astronomy, astrophysics and atmospheric scie... | Aryabhatta Research Institute of Observational Sciences | 0.781675 |
88,445 | Because of its geographical location, ARIES has made contributions to many areas of astronomical research, particularly those involving time-critical phenomena (e.g., the first successful attempt in the country to observe optical afterglow of GRBs was carried out from ARIES). Many eclipsing binaries, variable stars, st... | Aryabhatta Research Institute of Observational Sciences | 0.781675 |
88,446 | In organosulfur chemistry, organosulfates are a class of organic compounds sharing a common functional group with the structure R−O−SO−3. The SO4 core is a sulfate group and the R group is any organic residue. All organosulfates are formally esters derived from alcohols and sulfuric acid (H2SO4) although many are not p... | Alkyl sulfate | 0.781674 |
88,447 | These committees include the Audit Committee, which handles accounting issues with the company including auditing and reporting; the Compensation Committee, which approves compensation for the CEO and other employees of the company; the Governance and Nominating Committee, which handles various corporate matters includ... | Microsoft | 0.781674 |
88,448 | In August 2018, Toyota Tsusho began a partnership with Microsoft to create fish farming tools using the Microsoft Azure application suite for Internet of things (IoT) technologies related to water management. Developed in part by researchers from Kindai University, the water pump mechanisms use artificial intelligence ... | Microsoft | 0.781674 |
88,449 | In organic chemistry, xylene or xylol (from Greek ξύλον (xylon) 'wood'; IUPAC name: dimethylbenzene) are any of three organic compounds with the formula (CH3)2C6H4. They are derived from the substitution of two hydrogen atoms with methyl groups in a benzene ring; which hydrogens are substituted determines which of thre... | Xylene | 0.781672 |
88,450 | The MIPS product was one which DDC-I emphasised, with engineering efforts that included automatic recognition of certain tasking optimizations, and work in the U.S. Air Force-sponsored Common Ada Runtime System (CARTS) project towards providing standard interfaces into Ada runtime environments.At the end of 1993, the N... | DDC-I | 0.781671 |
88,451 | The early OEM customers included the University of Lund, Defence Materiel Administration, and Ericsson Radio Systems in Sweden; Softplan and Nokia Information Systems in Finland; Selenia and Olivetti in Italy; ICL Defence Systems and STL Ltd in the United Kingdom; Aitech Software Engineering in Israel; and Advanced Com... | DDC-I | 0.781671 |
88,452 | government mandate for Ada use was in effect, albeit with some waivers granted. Many of the advantages of the language for general-purpose programming were not seen as such by the general software engineering community or by educators. The sales situation was challenging, with periodic small layoffs. | DDC-I | 0.781671 |
88,453 | The Ministry is responsible for a number of parastatals, or government-owned agencies: National Board For Technology Incubation (NBTI) Energy Commission of Nigeria (ECN) The Nigerian Institute of Science Laboratory Technology (NISLT) - www.nislt.gov.ng Nigerian Institute For Trypanosomiasis And Onchocerciasis (NITR) Na... | Federal Ministry of Innovation, Science and Technology | 0.781671 |
88,454 | In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. | Fatou's theorem | 0.781671 |
88,455 | A given geometry of cavity will resonate at a characteristic frequency. The resonance effect can be used to selectively pass certain frequencies. Their use in a filter structure requires that some of the wave is allowed to pass out of one cavity into another through a coupling structure. | Waveguide filter | 0.781671 |
88,456 | Then it would be expected that the values of the extension of f {\displaystyle f} onto the circle should be the limit of these functions, and so the question reduces to determining when f r {\displaystyle f_{r}} converges, and in what sense, as r → 1 {\displaystyle r\to 1} , and how well defined is this limit. In parti... | Fatou's theorem | 0.781671 |
88,457 | The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle. The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way. | Fatou's theorem | 0.78167 |
88,458 | A USB system consists of a host with one or more downstream facing ports (DFP), and multiple peripherals, forming a tiered-star topology. Additional USB hubs may be included, allowing up to five tiers. A USB host may have multiple controllers, each with one or more ports. | USB device | 0.78167 |
88,459 | Data access: working with relational database management systems on the Java platform using Java Database Connectivity (JDBC) and object-relational mapping tools and with NoSQL databases. The spring-jdbc is an artifact found in the JDBC module while supports JDBC access by including datasource setup classes. | Spring Framework | 0.78167 |
88,460 | Spring's Remote Access framework is an abstraction for working with various RPC (remote procedure call)-based technologies available on the Java platform both for client connectivity and marshalling objects on servers. The most important feature offered by this framework is to ease configuration and usage of these tech... | Spring Framework | 0.78167 |
88,461 | Spring Batch is a framework for batch processing that provides reusable functions that are essential in processing large volumes of records, including: logging/tracing transaction management job processing statistics job restartIt provides more advanced technical services and features that enables extremely high-volume... | Spring Framework | 0.78167 |
88,462 | Molecular cohesion and gravity are shown to be contributing factors in the operation of a siphon; the presence of a positive atmospheric pressure is not required".Writing in Physics Today in 2011, J. Dooley from Millersville University stated that both a pressure differential within the siphon tube and the tensile stre... | Inverted siphon | 0.781669 |
88,463 | The action depends upon the influence of gravity (not, as sometimes thought, on the difference in atmospheric pressure; a siphon will work in a vacuum) and upon the cohesive forces that prevent the columns of liquid in the legs of the siphon from breaking under their own weight. At sea level, water can be lifted a litt... | Inverted siphon | 0.781669 |
88,464 | The above results point therefore to a failure of naive perturbation theory in describing correctly the physical behavior of the O(n)-symmetric model above two dimensions, and to the need for more sophisticated non-perturbative methods such as the lattice formulation. This means they can only arise as effective field t... | Nonlinear σ-model | 0.781667 |
88,465 | This model can also be considered in 1+2 dimensions, where the topology now comes only from the spatial slices. These are modelled as R^2 with a point at infinity, and hence have the same topology as the O(3) instantons in 1+1 dimensions. They are called sigma model lumps. | Nonlinear σ-model | 0.781667 |
88,466 | With the accommodation facilities at its hotel and its teaching facilities and equipment, the SBR provides conditions for teaching a range of subjects including zoology, phycology and coastal oceanography. Teaching at the SBR includes courses that form part of the UPMC Master's program and the European Socrates. The SB... | Station biologique de Roscoff | 0.781665 |
88,467 | The Station biologique de Roscoff (SBR) is a French marine biology and oceanography research and teaching center. Founded by Henri de Lacaze-Duthiers (1821–1901) in 1872, it is at the present time affiliated to the Sorbonne University (SU) and the Centre National de la Recherche Scientifique (CNRS). | Station biologique de Roscoff | 0.781665 |
88,468 | Performance optimization allows the FMS to determine the best or most economical speed to fly in level flight. This is often called the ECON speed. This is based on the cost index, which is entered to give a weighting between speed and fuel efficiency. | Flight Management System | 0.781664 |
88,469 | In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for X = | Y | {\displaystyle X=|Y|\!} where Y is a logistic random variable, X is a half-logistic random... | Half-logistic distribution | 0.781664 |
88,470 | So they are relatively easy targets in which to establish attacker tools. Inside of these medical devices, the cyber attacker now finds safe harbor in which to establish a backdoor (command and control). Since medical devices are FDA certified, hospital and cybersecurity team personnel cannot access the internal softwa... | Medical device hijack | 0.781663 |
88,471 | The Russell Berrie Nanotechnology Institute (RBNI) was established in January 2005 as a joint endeavour of the Russell Berrie Foundation, the government of Israel and the Technion, Israel Institute of Technology. It is one of the largest academic programs in Israel and is among the largest nanotechnology centers in the... | Russell Berrie Nanotechnology Institute | 0.781662 |
88,472 | The key research areas are the various fields of nanoscience and nanotechnology: Nanoelectronics, Nanooptics, Nanomaterials, Nanomechanics, Nanomedicine. | Russell Berrie Nanotechnology Institute | 0.781661 |
88,473 | The "nano bible" created at the Russell Berrie Nanotechnology Institute was originally produced to demonstrate the current miniaturization capabilities as part of the effort to attract high school students to science and technology. It was later given by Israel President Shimon Peres to Pope Benedict XVI, on his visit ... | Russell Berrie Nanotechnology Institute | 0.781661 |
88,474 | The 'Norman Seiden Multidisciplinary Graduate Program in Nanoscience and Nanotechnology was launched in October 2005 and expanded in October 2006 to include a PhD program.The science involving nanometric systems and the technology derived from them involves the convergence of know-how from numerous fields. The Program ... | Russell Berrie Nanotechnology Institute | 0.781661 |
88,475 | RBNI winter schools in Nanoscience and Nanotechnology attract participants from around the world. Following the success of the 2008 winter school, the second winter school was held in February 2010 in Ein Gedi near the Dead Sea. Both were organized by a dedicated committee of 15 lecturers, and approximately 140 graduat... | Russell Berrie Nanotechnology Institute | 0.781661 |
88,476 | The laws of the Society provided for a ‘Special Observer’. The Observer was to have control of the Society’s Observatory to 1 Corroborate the observations of Society Members. 2 Assist Members requiring practical help in observational Astronomy. 3 Undertake systematic observations on behalf of the Society.T H E C Espin ... | Liverpool Astronomical Society | 0.781661 |
88,477 | The Liverpool Astronomical Society was founded in 1881 in Liverpool, England, as a society to promote and coordinate amateur astronomy. In 1893 the Society was gifted a 5" (125mm) aperture Cooke equatorial telescope and a 2” (50mm) transit telescope by Thomas Rylands. An observatory was built for it on the roof of the ... | Liverpool Astronomical Society | 0.781661 |
88,478 | This ends in a difficult optimization problem in order to identify a large number of material parameters. On the other hand, over the years, the concept of fractional derivatives has been introduced to the theory of viscoelasticity. Among these models, the fractional Zener model was found to be very effective to predic... | Mittag-Leffler function | 0.781661 |
88,479 | In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for compact self-adjoint operators is virtually the same as in the finite-dimensional case. Theorem. Suppose A is a compact self-adjoint operator on a (real or complex) Hilbert space V. Then there ... | Spectral expansion | 0.78166 |
88,480 | Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a projection-valued measure. One formulation of the spectral theorem expresses the operator A as an integral of the coordinate function over the operator's spectrum σ ( A ) {\displays... | Spectral expansion | 0.78166 |
88,481 | In the absence of (true) eigenvectors, one can look for subspaces consisting of almost eigenvectors. In the above example, for example, where ( t ) = t φ ( t ) , {\displaystyle (t)=t\varphi (t),\;} we might consider the subspace of functions supported on a small interval {\displaystyle } inside {\displaystyle } . Th... | Spectral expansion | 0.78166 |
88,482 | An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator. The significance of this result is that multiplication operators are in many ways easy to understand. The spectral theorem is the beginning of the vast research area of ... | Spectral expansion | 0.78166 |
88,483 | One important application of the spectral theorem (in whatever form) is the idea of defining a functional calculus. That is, given a function f {\displaystyle f} defined on the spectrum of A {\displaystyle A} , we wish to define an operator f ( A ) {\displaystyle f(A)} . If f {\displaystyle f} is simply a positive powe... | Spectral expansion | 0.78166 |
88,484 | Microcosm was a hypermedia system, originally developed in 1988 by the Department of Electronics and Computer Science at the University of Southampton, with a small team of researchers in the Computer Science group: Wendy Hall, Andrew Fountain, Hugh Davis and Ian Heath. The system pre-dates the web and builds on early ... | Microcosm (hypermedia system) | 0.78166 |
88,485 | The Joseph Weber Award for Astronomical Instrumentation is awarded by the American Astronomical Society to an individual for the design, invention or significant improvement of instrumentation leading to advances in astronomy. It is named after physicist Joseph Weber. The awards tend to be for a career of instrument de... | Joseph Weber Award for Astronomical Instrumentation | 0.78166 |
88,486 | We prove the following theorem on concentration of measure, following notes by Barvinok and notes by Lap Chi Lau. See also Concentration of measure#Concentration on the sphere. Theorem: Let S {\textstyle S} be the unit sphere in R n {\textstyle \mathbb {R} ^{n}} . Let X ⊆ S {\textstyle X\subseteq S} . | Brunn–Minkowski theorem | 0.781659 |
88,487 | Grunbaum's theorem implies that if player 1 chooses the center of mass, then the worst that an adversarial player 2 can do is give him a piece of cake with volume at least a 1 / e {\textstyle 1/e} fraction of the total. In dimensions 2 and 3, the most common dimensions for cakes, the bounds given by the theorem are app... | Brunn–Minkowski theorem | 0.781659 |
88,488 | Grunbaum's theorem can be proven using Brunn–Minkowski inequality, specifically the convexity of the Brunn–Minkowski symmetrization. See these lecture notes for a proof sketch. Grunbaum's inequality has the following fair cake cutting interpretation. | Brunn–Minkowski theorem | 0.781659 |
88,489 | Theorem (Grunbaum's theorem): Consider a convex body K ⊆ R n {\textstyle K\subseteq \mathbb {R} ^{n}} . Let H {\textstyle H} be any half-space containing the center of mass of K {\textstyle K} ; that is, the expected location of a uniform point sampled from K . {\textstyle K.} Then μ ( H ∩ K ) ≥ ( n n + 1 ) n μ ( K ) ≥... | Brunn–Minkowski theorem | 0.781659 |
88,490 | The proof of the Brunn–Minkowski theorem establishes that the function A ↦ 1 / n {\displaystyle A\mapsto ^{1/n}} is concave in the sense that, for every pair of nonempty compact subsets A and B of Rn and every 0 ≤ t ≤ 1, 1 / n ≥ t 1 / n + ( 1 − t ) 1 / n . {\displaystyle \left^{1/n}\geq t^{1/n}+(1-t)^{1/n}.} For co... | Brunn–Minkowski theorem | 0.781659 |
88,491 | By convexity of K, we have that K ( λ x + ( 1 − λ ) y ) ⊇ λ K ( x ) + ( 1 − λ ) K ( y ) {\textstyle K(\lambda x+(1-\lambda )y)\supseteq \lambda K(x)+(1-\lambda )K(y)} . Applying the Brunn–Minkowski inequality gives r ( K ( λ x + ( 1 − λ ) y ) ) ≥ λ r ( K ( x ) ) + ( 1 − λ ) r ( K ( y ) ) {\textstyle r(K(\lambda x+(1-\l... | Brunn–Minkowski theorem | 0.781659 |
88,492 | Consider a convex body K ⊆ R n {\textstyle K\subseteq \mathbb {R} ^{n}} . Let K ( x ) = K ∩ { x 1 = x } {\textstyle K(x)=K\cap \{x_{1}=x\}} be vertical slices of K. Define r ( x ) = μ ( K ( x ) ) 1 n − 1 {\textstyle r(x)=\mu (K(x))^{\frac {1}{n-1}}} to be the radius function; if the slices of K are discs, then r(x) giv... | Brunn–Minkowski theorem | 0.781659 |
88,493 | The Brunn–Minkowski inequality gives much insight into the geometry of high dimensional convex bodies. In this section we sketch a few of those insights. | Brunn–Minkowski theorem | 0.781659 |
88,494 | The Brunn–Minkowski inequality continues to be relevant to modern geometry and algebra. For instance, there are connections to algebraic geometry, and combinatorial versions about counting sets of points inside the integer lattice. | Brunn–Minkowski theorem | 0.781659 |
88,495 | The Brunn–Minkowski inequality can be used to deduce the following inequality V ( K , … , K , L ) n ≥ V ( K ) n − 1 V ( L ) {\textstyle V(K,\ldots ,K,L)^{n}\geq V(K)^{n-1}V(L)} , where the V ( K , … , K , L ) {\textstyle V(K,\ldots ,K,L)} term is a mixed-volume. Equality holds iff K,L are homothetic. (See theorem 3.4.3... | Brunn–Minkowski theorem | 0.781659 |
88,496 | Let n ≥ 1 and let μ denote the Lebesgue measure on Rn. Let A and B be two nonempty compact subsets of Rn. Then the following inequality holds: 1 / n ≥ 1 / n + 1 / n , {\displaystyle ^{1/n}\geq ^{1/n}+^{1/n},} where A + B denotes the Minkowski sum: A + B := { a + b ∈ R n ∣ a ∈ A , b ∈ B } . {\displaystyle A+B:=\{\,a+... | Brunn–Minkowski theorem | 0.781659 |
88,497 | In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the gene... | Brunn–Minkowski theorem | 0.781659 |
88,498 | 2020: The establishment of the Non-Human Party, for the rights of robots, animals, and the wider non-human environment. 2021: We Robot 2021 (23 Sep 2021, University of Miami, FL, USA) 2021: IROS 2021 Workshop - Building and Evaluating Ethical Robotic SystemsComputer scientist Virginia Dignum noted in a March 2018 issue... | Robot ethics | 0.781657 |
88,499 | Additionally, the juxtaposition of the rights afforded to Saudi women against the Saudi consideration of a robot to be a citizen will soon attract criticism from several news providers (25 October 2017, Riyadh, Saudi Arabia). 2017: The European Parliament passed a resolution addressed to the European Commission concern... | Robot ethics | 0.781657 |
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