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96,100 | It is conjectured that the projectors H ∗(X) ↠ Hi(X) ↣ H ∗(X)are algebraic, i.e. induced by a cycle π i ⊂ X × X with rational coefficients. This implies that the motive of any smooth projective variety (and more generally, every pure motive) decomposes as h ( X ) = ⨁ i = 0 2 d i m ( X ) h i ( X ) . {\displaystyle h(X)=... | Standard conjectures on algebraic cycles | 0.778425 |
96,101 | For two algebraic varieties X and Y, Arapura (2006) has introduced a condition that Y is motivated by X. The precise condition is that the motive of Y is (in André's category of motives) expressible starting from the motive of X by means of sums, summands, and products. For example, Y is motivated if there is a surject... | Standard conjectures on algebraic cycles | 0.778425 |
96,102 | The Hodge standard conjecture is not to be confused with the Hodge conjecture which states that for smooth projective varieties over C, every rational (p, p)-class is algebraic. The Hodge conjecture implies the Lefschetz and Künneth conjectures and conjecture D for varieties over fields of characteristic zero. The Tate... | Standard conjectures on algebraic cycles | 0.778425 |
96,103 | The Hodge standard conjecture is modelled on the Hodge index theorem. It states the definiteness (positive or negative, according to the dimension) of the cup product pairing on primitive algebraic cohomology classes. If it holds, then the Lefschetz conjecture implies Conjecture D. In characteristic zero the Hodge stan... | Standard conjectures on algebraic cycles | 0.778425 |
96,104 | Katz & Messing (1974) have used the Weil conjectures to show the conjecture for algebraic varieties defined over finite fields, in arbitrary dimension. Šermenev (1974) proved the Künneth decomposition for abelian varieties A. Deninger & Murre (1991) refined this result by exhibiting a functorial Künneth decomposition o... | Standard conjectures on algebraic cycles | 0.778425 |
96,105 | In organic chemistry, thiocarboxylic acids or carbothioic acids are organosulfur compounds related to carboxylic acids by replacement of one of the oxygen atoms with a sulfur atom. Two tautomers are possible: a thione form (RC(S)OH) and a thiol form (RC(O)SH). These are sometimes also referred to as "carbothioic O-acid... | Dithiocarboxylic acid | 0.778424 |
96,106 | The impacts of research on Planctomycetota and their uses might be of global significance with regards to nutrient cycling processes and assist in furthering understanding for global marine biogeochemistry. However, with Planctomycetota's growing influences on metabolic processes involving water and air, it may also ha... | Planctomycetota | 0.778423 |
96,107 | Recently, interest has arisen in examining the Planctomycetota regarding their potential roles in biotechnology, mainly as a source of bioactive molecules, of interest mainly to the pharmaceutical industry. Bioactive compounds are mainly present as secondary metabolites, although little is known about Planctomycetota s... | Planctomycetota | 0.778423 |
96,108 | Members of the Planctomycetota also display distinct reproductive habits, with many species dividing by budding, in contrast to all other free-living bacteria, which divide by binary fission.Interest is growing in the Planctomycetota regarding biotechnology and human applications, mainly as a source of bioactive molecu... | Planctomycetota | 0.778423 |
96,109 | Planctomycetota have a significant impact on global biogeochemistry and climate, with their ability to mineralize and break down detritus particles in the water column.Planctomycetota play a considerable role in the global carbon cycle. As both obligate and facultative aerobic chemoheterotrophs, the primary source of c... | Planctomycetota | 0.778423 |
96,110 | Jet aerators do not require any external air source (i.e. compressor), except for the surrounding atmosphere. Jet aerators can be installed either as submersible units or piped through the tank wall using an external dry-installed chopper pump to feed the aspirating ejector(s). Jet aerators are easily configured into a... | Jet aerators | 0.778422 |
96,111 | In organic chemistry, a hydrocarbon is an organic compound consisting entirely of hydrogen and carbon. : 620 Hydrocarbons are examples of group 14 hydrides. Hydrocarbons are generally colourless and hydrophobic; their odor is usually faint, and may be similar to that of gasoline or lighter fluid. | Hydrocarbon | 0.77842 |
96,112 | The Atkinson–Stiglitz theorem is a theorem of public economics which states that "where the utility function is separable between labor and all commodities, no indirect taxes need be employed." Non-linear income taxation can be used by the government and was developed in a seminal article by Joseph Stiglitz and Anthony... | Atkinson–Stiglitz theorem | 0.778419 |
96,113 | Joseph Stiglitz explains why indirect taxation is unnecessary, viewing the Atkinson-Stiglitz theorem from a different perspective. | Atkinson–Stiglitz theorem | 0.778419 |
96,114 | "The Climate Investment Funds (CIF) comprises two Trust Funds, each with a specific scope and objective and its own governance structure: the Clean Technology Fund (CTF) and the Strategic Climate Fund (SCF). The CTF promotes investments to initiate a shift towards clean technologies. The CTF seeks to fill a gap in the ... | Energy burden | 0.778418 |
96,115 | Invar, also known generically as FeNi36 (64FeNi in the US), is a nickel–iron alloy notable for its uniquely low coefficient of thermal expansion (CTE or α). The name Invar comes from the word invariable, referring to its relative lack of expansion or contraction with temperature changes, and is a registered trademark o... | Invar | 0.778418 |
96,116 | After some controversy and patient deaths, new laparoscopic training centres were set up as most surgeons lacked the necessary specialised training to perform laparoscopic surgery. The first opened in Dundee in 1991 and became the Cuschieri Skills Centre at Ninewells Hospital in 2004. As of 2008, 40 specialist centres ... | Laparoscopic surgery | 0.778417 |
96,117 | Subsequently, Semm submitted a paper on laparoscopic appendectomy to the American Journal of Obstetrics and Gynecology, at first rejected as unacceptable for publication on the grounds that the technique reported on was "unethical," but finally published in the journal Endoscopy. The abstract of his paper on endoscopic... | Laparoscopic surgery | 0.778417 |
96,118 | The first publication on modern diagnostic laparoscopy by Raoul Palmer appeared in 1947, followed by the publication of Hans Frangenheim and Kurt Semm, who both practised CO2 hysteroscopy from the mid-1970s.Patrick Steptoe, one of the pioneers of IVF, was important in popularizing laparoscopy in the UK. He published a ... | Laparoscopic surgery | 0.778417 |
96,119 | In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ: U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that ... | Łojasiewicz inequality | 0.778414 |
96,120 | Some of the theory developed by Alain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular in ergodic theory. The proposal of George Mackey to create a virtual subgroup theory, with respect to which ergodic group actions would become homogeneous spaces of an extended ... | Connes connection | 0.778414 |
96,121 | It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L2(M, E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L2(M, E) with compact resolvent (e.g. the signature operator), ... | Connes connection | 0.778413 |
96,122 | A smooth Riemannian manifold M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. | Connes connection | 0.778413 |
96,123 | Matched filters play a central role in gravitational-wave astronomy. The first observation of gravitational waves was based on large-scale filtering of each detector's output for signals resembling the expected shape, followed by subsequent screening for coincident and coherent triggers between both instruments. False-... | Matched filter | 0.778411 |
96,124 | The optimization problem assumes its maximum when equality holds in this expression. According to the properties of the Cauchy–Schwarz inequality, this is only possible when h j − k = ν ⋅ s k = κ ⋅ f k − j 0 . {\displaystyle \ h_{j-k}=\nu \cdot s_{k}=\kappa \cdot f_{k-j_{0}}.} | Matched filter | 0.778411 |
96,125 | After reversing the sign, we obtain the equivalent optimization problem j ∗ , μ ∗ = arg max j , μ . {\displaystyle \ j^{*},\mu ^{*}=\arg \max _{j,\mu }\left.} Setting the derivative w.r.t. | Matched filter | 0.778411 |
96,126 | The last term has constant expected value because the noise is uncorrelated and has zero mean. We can therefore drop both terms from the optimization. | Matched filter | 0.778411 |
96,127 | A UHF C*-algebra is the direct limit of an inductive system {An, φn} where each An is a finite-dimensional full matrix algebra and each φn: An → An+1 is a unital embedding. Suppressing the connecting maps, one can write A = ∪ n A n ¯ . {\displaystyle A={\overline {\cup _{n}A_{n}}}.} | Uniformly hyperfinite algebra | 0.778407 |
96,128 | Therefore, the CAR algebra has supernatural number 2∞. This identification also yields that its K0 group is the dyadic rationals. == References == | Uniformly hyperfinite algebra | 0.778407 |
96,129 | One example of a UHF C*-algebra is the CAR algebra. It is defined as follows: let H be a separable complex Hilbert space H with orthonormal basis fn and L(H) the bounded operators on H, consider a linear map α: H → L ( H ) {\displaystyle \alpha :H\rightarrow L(H)} with the property that { α ( f n ) , α ( f m ) } = 0 an... | Uniformly hyperfinite algebra | 0.778407 |
96,130 | In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras. | Uniformly hyperfinite algebra | 0.778407 |
96,131 | If δ(A) is finite, then A is the full matrix algebra Mδ(A). A UHF algebra is said to be of infinite type if each tp in δ(A) is 0 or ∞. In the language of K-theory, each supernatural number δ ( A ) = ∏ p p t p {\displaystyle \delta (A)=\prod _{p}p^{t_{p}}} specifies an additive subgroup of Q that is the rational numbers... | Uniformly hyperfinite algebra | 0.778407 |
96,132 | If A n ≃ M k n ( C ) , {\displaystyle A_{n}\simeq M_{k_{n}}(\mathbb {C} ),} then rkn = kn + 1 for some integer r and ϕ n ( a ) = a ⊗ I r , {\displaystyle \phi _{n}(a)=a\otimes I_{r},} where Ir is the identity in the r × r matrices. The sequence ...kn|kn + 1|kn + 2... determines a formal product δ ( A ) = ∏ p p t p {\di... | Uniformly hyperfinite algebra | 0.778407 |
96,133 | Chem. 372 1, 2002 F. Gesellchen, B. Zimmermann, F. W. Herberg. Methods in Molecular Biology, 2005 T. Nagel, E. Ehrentreich-Forster, M. Singh, et al. Sensors and Actuators B-Chemical 129 2, 2008 P. Fechner, F. Pröll, M. Carlquist and G. Proll. | Reflectometric interference spectroscopy | 0.778407 |
96,134 | RIfS is used especially as a detection method in chemo- and biosensors. Chemosensors are particularly suitable for measurements under difficult conditions and in the gaseous phase. As sensitive layers, mostly non-selective measuring polymers are used which sort the analytes according to size (the so-called molecular si... | Reflectometric interference spectroscopy | 0.778407 |
96,135 | Possibly the most exciting potential use for PECO is producing hydrogen to be used as a source of renewable energy. Photoelectrochemical oxidation reactions that take place within PEC cells are the key to water splitting for hydrogen production. While the main concern with this technology is stability, systems that use... | Photoelectrochemical oxidation | 0.778406 |
96,136 | PECO represents a type of photocatalysis whereby semiconductor-based electrochemistry catalyzes an oxidation reaction—for example, the oxidative degradation of an airborne contaminant in air purification systems. The principal objective of photoelectrocatalysis is to provide low-energy activation pathways for the passa... | Photoelectrochemical oxidation | 0.778406 |
96,137 | In order to make antihydrogen cold enough to be trapped the ALPHA collaboration has implemented a novel technique, well known from atomic physics, called evaporative cooling. The motivation for this is that one of the main challenges of trapping antihydrogen is to make it cold enough. State-of-the art minimum-B traps l... | ALPHA Collaboration | 0.778405 |
96,138 | If variable X follows a normal distribution N ( μ , σ 2 ) {\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} , then the inverse or reciprocal Y=1/X follows a reciprocal normal distribution: f ( y ) = 1 2 π σ y 2 e − 1 2 ( 1 / y − μ σ ) 2 . {\displaystyle f(y)={\frac {1}{{\sqrt {2\pi }}\sigma y^{2}}}e^{-{\frac {1}{2}}\lef... | Reciprocal normal distribution | 0.778404 |
96,139 | In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions a... | Reciprocal normal distribution | 0.778404 |
96,140 | The Turing Year was coordinated by the Turing Centenary Advisory Committee (TCAC), representing a range of expertise and organisational involvement in the 2012 celebrations. Members of TCAC include Honorary President, Sir John Dermot Turing; The Chair and founder of the committee, mathematician and author of Alan Turin... | Alan Turing Year | 0.778401 |
96,141 | The Alan Turing Year, 2012, marked the celebration of the life and scientific influence of Alan Turing during the centenary of his birth on 23 June 1912. Turing had an important influence on computing, computer science, artificial intelligence, developmental biology, and the mathematical theory of computability and mad... | Alan Turing Year | 0.778401 |
96,142 | Artists, musicians and poets took part in the celebrations internationally. Events included the 2012 Computability in Europe conference, as well as Turing Centenary activities organized or sponsored by the British Computer Society, the Association for Symbolic Logic, British Colloquium for Theoretical Computer Science,... | Alan Turing Year | 0.778401 |
96,143 | Szemerédi's regularity lemma is one of the most powerful tools in extremal graph theory, particularly in the study of large dense graphs. It states that the vertices of every large enough graph can be partitioned into a bounded number of parts so that the edges between different parts behave almost randomly. According ... | Szemerédi's regularity lemma | 0.7784 |
96,144 | For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ1 as a filter which retains only those particles such that σ1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ. However, ... | Quantum indeterminacy | 0.778399 |
96,145 | Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed, according to the Kochen–Specker theorem, in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable proper... | Quantum indeterminacy | 0.778399 |
96,146 | In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes. It was first obtained by Hiroshi Kunita and Shinzo Watanabe and plays a fundamental role in their extension of Ito's stochastic integral to square-integrable martingales. | Kunita–Watanabe inequality | 0.778399 |
96,147 | The story is highlighted on 13 October 2022 at Exceptional Cosmic Blast. Wind celebrated the 28th anniversary of its launch on 1 November 2022. On 21 February 2023 the Wind review paper published in Reviews of Geophysics was awarded as a Top Cited Article 2021-2022 by the journal. | Wind (spacecraft) | 0.778398 |
96,148 | For a complete list of refereed publications directly or indirectly using data from the Wind spacecraft, see https://wind.nasa.gov/bibliographies.php. Wind continues to produce relevant research, with its data having contributed to over 4300 publications since 1 January 2010 and over 2480 publications prior. As of 26 A... | Wind (spacecraft) | 0.778398 |
96,149 | The arrays of detectors are mounted on two opposing booms, each 0.5 m (1 ft 8 in) in length. The top-hat ES analyzers are composed of four separate detectors, each with different geometry factors to cover different ranges of energies. The electron detectors, EESA, and ion detectors, PESA, are each separated into low (L... | Wind (spacecraft) | 0.778398 |
96,150 | The aim of the International Solar-Terrestrial Physics Science Initiative is to understand the behaviour of the solar-terrestrial plasma environment, in order to predict how the Earth's atmosphere will respond to changes in solar wind conditions. Wind's objective is to measure the properties of the solar wind before it... | Wind (spacecraft) | 0.778398 |
96,151 | First observation of a giant flare — emission of greater apparent intensity than gamma ray bursts with an average occurrence rate of once per decade — within the nearby Sculptor Galaxy. The press release can be found at Giant Flare in Nearby Galaxy. This work led to at least six papers published in Nature.A comprehensi... | Wind (spacecraft) | 0.778398 |
96,152 | In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure. In statistics and machine learning, it is called a log-linear model. In deep learning, the Boltzmann distribution is used in the sampling distribution of stochastic neural networks such as the Boltzmann machine, restri... | Boltzmann distribution | 0.778398 |
96,153 | : 332 From uniqueness of factorization of ideals in Q(ξ) it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and (x, y), (y, z) and (z, x) are elements of Hp. : 333 Granville and Monagan showed that (1, 1) ∈ Hp if and only if p is a Wieferich prime. : 333 | Wieferich prime | 0.778397 |
96,154 | Suzuki extended the proof to all primes m ≤ 113.Let Hp be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, (x + y)p−1 ≡ 1 (mod p2), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the field extension obtained by adjoi... | Wieferich prime | 0.778397 |
96,155 | The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909: Let p be prime, and let x, y, z be integers such that xp + yp + zp = 0. Furthermore, assume that p does not divide the product xyz. Then p is a Wieferich prime.The above case (where p does not divide any of x, y... | Wieferich prime | 0.778397 |
96,156 | It is known that the nth Mersenne number Mn = 2n − 1 is prime only if n is prime. Fermat's little theorem implies that if p > 2 is prime, then Mp−1 (= 2p − 1 − 1) is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime, A prime divisor p of Mq, where q is prime, is a Wieferich prime if ... | Wieferich prime | 0.778397 |
96,157 | If a Borel function happens to be a section of a map Y → π X , {\displaystyle Y\xrightarrow {~\pi ~} X,} it is called a Borel section. A Lebesgue measurable function is a measurable function f: ( R , L ) → ( C , B C ) , {\displaystyle f:(\mathbb {R} ,{\mathcal {L}})\to (\mathbb {C} ,{\mathcal {B}}_{\mathbb {C} }),} whe... | Measurable function | 0.778395 |
96,158 | Random variables are by definition measurable functions defined on probability spaces. If ( X , Σ ) {\displaystyle (X,\Sigma )} and ( Y , T ) {\displaystyle (Y,T)} are Borel spaces, a measurable function f: ( X , Σ ) → ( Y , T ) {\displaystyle f:(X,\Sigma )\to (Y,T)} is also called a Borel function. Continuous function... | Measurable function | 0.778395 |
96,159 | Let ( X , Σ ) {\displaystyle (X,\Sigma )} and ( Y , T ) {\displaystyle (Y,\mathrm {T} )} be measurable spaces, meaning that X {\displaystyle X} and Y {\displaystyle Y} are sets equipped with respective σ {\displaystyle \sigma } -algebras Σ {\displaystyle \Sigma } and T . {\displaystyle \mathrm {T} .} A function f: X → ... | Measurable function | 0.778395 |
96,160 | Although the materials' Seebeck coefficents S {\displaystyle S} are nonlinearly temperature dependent and different for the two materials, the open-circuit condition means that ∇ V = − S ∇ T {\displaystyle \nabla V=-S\nabla T} everywhere. Therefore (see the thermocouple article for more details) the voltage measured at... | Peltier cell | 0.778395 |
96,161 | The choice of σ {\displaystyle \sigma } -algebras in the definition above is sometimes implicit and left up to the context. For example, for R , {\displaystyle \mathbb {R} ,} C , {\displaystyle \mathbb {C} ,} or other topological spaces, the Borel algebra (generated by all the open sets) is a common choice. Some author... | Measurable function | 0.778395 |
96,162 | In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topologic... | Measurable function | 0.778395 |
96,163 | Make (stylized as Make: or MAKE:) is an American magazine published since June 2019 by Make: Community LLC which focuses on Do It Yourself (DIY) and/or Do It With Others (DIWO) projects involving computers, electronics, metalworking, robotics, woodworking and other disciplines. The magazine is marketed to people who en... | Make Controller Kit | 0.778395 |
96,164 | The G − H {\displaystyle {\mathcal {G}}-{\mathcal {H}}} -topology on B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z)} will be compatible with the vector space structure of B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z)} if both G {\displaystyle {\mathcal {G}}} and H {\displaystyle {\mathcal {H}}} consists of... | Space of linear maps | 0.778393 |
96,165 | However, as before, this topology is not necessarily compatible with the vector space structure of B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z)} or of B ( X , Y ; Z ) {\displaystyle B(X,Y;Z)} without the additional requirement that for all bilinear maps, b {\displaystyle b} in this space (that is, in B ( X , Y ... | Space of linear maps | 0.778393 |
96,166 | {\displaystyle H\in {\mathcal {H}}.} We can place on Z X × Y {\displaystyle Z^{X\times Y}} the G × H {\displaystyle {\mathcal {G}}\times {\mathcal {H}}} -topology, and consequently on any of its subsets, in particular on B ( X , Y ; Z ) {\displaystyle B(X,Y;Z)} and on B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z... | Space of linear maps | 0.778393 |
96,167 | We will let B ( X , Y ; Z ) {\displaystyle {\mathcal {B}}(X,Y;Z)} denote the space of separately continuous bilinear maps and B ( X , Y ; Z ) {\displaystyle B(X,Y;Z)} denote the space of continuous bilinear maps, where X , Y , {\displaystyle X,Y,} and Z {\displaystyle Z} are topological vector space over the same field... | Space of linear maps | 0.778393 |
96,168 | {\displaystyle F.} One may also replace G {\displaystyle {\mathcal {G}}} with the collection of all subsets of all finite unions of elements of G {\displaystyle {\mathcal {G}}} without changing the resulting G {\displaystyle {\mathcal {G}}} -topology on F . {\displaystyle F.} | Space of linear maps | 0.778393 |
96,169 | A subset G 1 {\displaystyle {\mathcal {G}}_{1}} of G {\displaystyle {\mathcal {G}}} is said to be fundamental with respect to G {\displaystyle {\mathcal {G}}} if each G ∈ G {\displaystyle G\in {\mathcal {G}}} is a subset of some element in G 1 . {\displaystyle {\mathcal {G}}_{1}.} In this case, the collection G {\displ... | Space of linear maps | 0.778393 |
96,170 | The following sets will constitute the basic open subsets of topologies on spaces of linear maps. For any subsets G ⊆ T {\displaystyle G\subseteq T} and N ⊆ Y , {\displaystyle N\subseteq Y,} let The family forms a neighborhood basis at the origin for a unique translation-invariant topology on F , {\displaystyle F,} whe... | Space of linear maps | 0.778393 |
96,171 | One important difference is that polar topologies are always locally convex while G {\displaystyle {\mathcal {G}}} -topologies need not be. Polar topologies have stronger results than the more general topologies of uniform convergence described in this article and we refer the read to the main article: polar topology. ... | Space of linear maps | 0.778393 |
96,172 | If X {\displaystyle X} is a TVS whose bounded subsets are exactly the same as its weakly bounded subsets (e.g. if X {\displaystyle X} is a Hausdorff locally convex space), then a G {\displaystyle {\mathcal {G}}} -topology on X ′ {\displaystyle X^{\prime }} (as defined in this article) is a polar topology and conversely... | Space of linear maps | 0.778393 |
96,173 | Equicontinuous subsets The weak-closure of an equicontinuous subset of L ( X ; Y ) {\displaystyle L(X;Y)} is equicontinuous. If Y {\displaystyle Y} is locally convex, then the convex balanced hull of an equicontinuous subset of L ( X ; Y ) {\displaystyle L(X;Y)} is equicontinuous. Let X {\displaystyle X} and Y {\displa... | Space of linear maps | 0.778393 |
96,174 | {\displaystyle Y.} Suppose X {\displaystyle X} and Y {\displaystyle Y} are locally convex. Any simply bounded subset of L ( X ; Y ) {\displaystyle L(X;Y)} is bounded when L ( X ; Y ) {\displaystyle L(X;Y)} has the topology of uniform convergence on convex, balanced, bounded, complete subsets of X . | Space of linear maps | 0.778393 |
96,175 | {\displaystyle Y.} If L ( X ; Y ) {\displaystyle L(X;Y)} is given the topology of pointwise convergence then space of all linear maps (continuous or not) X {\displaystyle X} into Y {\displaystyle Y} is closed in Y X {\displaystyle Y^{X}} . In addition, L ( X ; Y ) {\displaystyle L(X;Y)} is dense in the space of all lin... | Space of linear maps | 0.778393 |
96,176 | The weak-topology on L ( X ; Y ) {\displaystyle L(X;Y)} has the following properties: If X {\displaystyle X} is separable (that is, it has a countable dense subset) and if Y {\displaystyle Y} is a metrizable topological vector space then every equicontinuous subset H {\displaystyle H} of L σ ( X ; Y ) {\displaystyle L_... | Space of linear maps | 0.778393 |
96,177 | By letting G {\displaystyle {\mathcal {G}}} be the set of all finite subsets of X , {\displaystyle X,} L ( X ; Y ) {\displaystyle L(X;Y)} will have the weak topology on L ( X ; Y ) {\displaystyle L(X;Y)} or the topology of pointwise convergence or the topology of simple convergence and L ( X ; Y ) {\displaystyle L(X;Y)... | Space of linear maps | 0.778393 |
96,178 | {\displaystyle X.} The next theorem gives ways in which G {\displaystyle {\mathcal {G}}} can be modified without changing the resulting G {\displaystyle {\mathcal {G}}} -topology on Y . {\displaystyle Y.} | Space of linear maps | 0.778393 |
96,179 | Assumptions that guarantee a vector topology ( G {\displaystyle {\mathcal {G}}} is directed): G {\displaystyle {\mathcal {G}}} will be a non-empty collection of subsets of X {\displaystyle X} directed by (subset) inclusion. That is, for any G , H ∈ G , {\displaystyle G,H\in {\mathcal {G}},} there exists K ∈ G {\display... | Space of linear maps | 0.778393 |
96,180 | Suppose that X {\displaystyle X} is a TVS whose bounded subsets are the same as its weakly bounded subsets. Notation: If Δ ( Y , X ) {\displaystyle \Delta (Y,X)} denotes a polar topology on Y {\displaystyle Y} then Y {\displaystyle Y} endowed with this topology will be denoted by Y Δ ( Y , X ) {\displaystyle Y_{\Delta ... | Space of linear maps | 0.778393 |
96,181 | The Computer museum Aachen (officially called Computer-Museum der RWTH Aachen) existed from 1987 to 2009. It was created in cooperation with the Rogowski Institute for Electrical Engineering of the RWTH Aachen University. It was housed in the RWTH's "Campus Melaten" until the demolition of the building at Sommerfeldstr... | Computer museum Aachen | 0.778393 |
96,182 | The museum was created in cooperation with the Rogowski Institute for Electrical Engineering at the RWTH Aachen and collected obsolete computer hardware from them as pieces for the museum. The main advocate and supporter was Walter Ameling (1926–2010). The extensive collection, which was created in 1965 with the suppor... | Computer museum Aachen | 0.778393 |
96,183 | Let M t {\displaystyle M_{t}} be a local martingale. In order to prove that it is a martingale it is sufficient to prove that M t τ k → M t {\displaystyle M_{t}^{\tau _{k}}\to M_{t}} in L1 (as k → ∞ {\displaystyle k\to \infty } ) for every t, that is, E | M t τ k − M t | → 0 ; {\displaystyle \operatorname {E} |M_{t}^... | Local martingale | 0.778392 |
96,184 | In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local ... | Local martingale | 0.778392 |
96,185 | In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral. Heron'... | Brahmagupta's formula | 0.778391 |
96,186 | Heron's formula for the area of a triangle is the special case obtained by taking d = 0. The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem. Increasingly complicated closed-form formulas exist for the area of general polyg... | Brahmagupta's formula | 0.778391 |
96,187 | Consequently, in the case of an inscribed quadrilateral, θ is 90°, whence the term a b c d cos 2 θ = a b c d cos 2 ( 90 ∘ ) = a b c d ⋅ 0 = 0 , {\displaystyle abcd\cos ^{2}\theta =abcd\cos ^{2}\left(90^{\circ }\right)=abcd\cdot 0=0,} giving the basic form of Brahmagupta's formula. It follows from the latter equatio... | Brahmagupta's formula | 0.778391 |
96,188 | The Moreau envelope can also be seen as the infimal convolution between f {\displaystyle f} and ( 1 / 2 ) ‖ ⋅ ‖ 2 2 {\displaystyle (1/2)\|\cdot \|_{2}^{2}} . The proximal operator of a function is related to the gradient of the Moreau envelope by the following identity: ∇ M λ f ( x ) = 1 λ ( x − p r o x λ f ( x ) ) {\d... | Moreau envelope | 0.778391 |
96,189 | The Moreau envelope (or the Moreau-Yosida regularization) M f {\displaystyle M_{f}} of a proper lower semi-continuous convex function f {\displaystyle f} is a smoothed version of f {\displaystyle f} . It was proposed by Jean-Jacques Moreau in 1965.The Moreau envelope has important applications in mathematical optimizat... | Moreau envelope | 0.778391 |
96,190 | Some writers make a distinction between "strong" and "weak" sustainability. Strong sustainable consumption refers to participating in viable environmental activities, such as consuming renewable and efficient goods and services (such as electric locomotive, cycling, renewable energy). Strong sustainable consumption als... | Sustainable consumption | 0.77839 |
96,191 | In materials science and solid mechanics, Poisson's ratio ν {\displaystyle \nu } (nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse str... | Poisson's Ratio | 0.778388 |
96,192 | The compound has been characterized by X-ray crystallography. The salt crystallizes in a cubic motif like that of ammonium hexachloroplatinate. The 2− centers adopt octahedral molecular geometry.The compound is prepared in the laboratory by the addition of ammonium chloride to an aqueous solution of sodium hexachloroir... | Ammonium hexachloroiridate | 0.778387 |
96,193 | In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an n {\displaystyle n} -dimensional compact torus which is locally standard with the orbit space a simple convex polytope.The aim is to do combinatoric... | Toric manifold | 0.778387 |
96,194 | This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polytope. == References == | Toric manifold | 0.778387 |
96,195 | The second group of these services is known as acute care in the digital health domain. These services include telemedicine which is defined as handling patients over some sort of streaming device and is targeted towards areas where the population is more widely scattered, medical devices that incorporate different asp... | Digital healthcare | 0.778387 |
96,196 | In 2019, the FDA published a Digital Health Innovation Action Plan that would reduce inefficiencies for physicians in an effort to cut overhead costs, improve access, increase quality of service, and make medicine more easily adapted for each person. Topics within the innovation plan are wireless devices, telemedicine,... | Digital healthcare | 0.778387 |
96,197 | At an intergovernmental level, the World Health Organization is the United Nations Specialized Agency for health, and the International Telecommunication Union is the UN Specialized Agency for ICTs, the Agencies collaborate in their work on digital health, such as the H.870 standard on safe listening, as well as the IT... | Digital healthcare | 0.778387 |
96,198 | The innovation process for digital health is an iterative cycle for technological solutions that can be classified into five main activity processes from the identification of the healthcare problem, research, digital solution, and evaluating the solution, to implementation in working clinical practices. Digital health... | Digital healthcare | 0.778387 |
96,199 | The World Health Organization also made guidelines in regulating certifying digital health medical devices similar to how medicine and vaccines are tested. The strategy was also made to list health data as a public health good, and made an outline for how research and data are shared, and how Artificial Intelligence is... | Digital healthcare | 0.778387 |
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