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29,500 | The number-theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory. In 1974, Donald Knuth said "...virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem ... | Number theorist | 0.80545 |
29,501 | Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals,... | Number theorist | 0.80545 |
29,502 | In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal m... | Number theorist | 0.80545 |
29,503 | Pell's equation, first misnamed by Euler. He wrote on the link between continued fractions and Pell's equation. First steps towards analytic number theory. | Number theorist | 0.80545 |
29,504 | The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach, pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' ... | Number theorist | 0.80545 |
29,505 | The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize. | Number theorist | 0.80545 |
29,506 | Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek... | Number theorist | 0.80545 |
29,507 | In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other componen... | Lack-of-fit sum of squares | 0.805447 |
29,508 | The Number Theory Foundation (NTF) is a non-profit organization based in the United States which supports research and conferences in the field of number theory, with a particular focus on computational aspects and explicit methods.The NTF funds the Selfridge prize awarded at each Algorithmic Number Theory Symposium (A... | Number Theory Foundation | 0.805447 |
29,509 | Practical engineers receive formal education that meets with criteria published by the National Institute for Technological Training. In sum around 2200 frontal academic hours are taught over a period of two to three years. From an academic point of view, the main things differentiating engineers from practical enginee... | Practical engineer | 0.805445 |
29,510 | The Java virtual machine is an abstract (virtual) computer defined by a specification. It is a part of java runtime environment. The garbage collection algorithm used and any internal optimization of the Java virtual machine instructions (their translation into machine code) are not specified. | Java Virtual Machine | 0.805445 |
29,511 | Taylor's theorem is a statement that includes the evaluation of the error term when the Taylor series is truncated. | Partial sums | 0.805444 |
29,512 | Otherwise, the series is said to be divergent.The notation ∑ i = 1 ∞ a i {\textstyle \sum _{i=1}^{\infty }a_{i}} denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generali... | Partial sums | 0.805444 |
29,513 | In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. | Partial sums | 0.805444 |
29,514 | In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite stru... | Partial sums | 0.805444 |
29,515 | The Hilbert–Poincaré series is a formal power series used to study graded algebras. Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition, multiplication, derivative, antiderivative for power series "formally", treat... | Partial sums | 0.805444 |
29,516 | Using Bayes' theorem, we can partition over y {\displaystyle y}: We use that: Pr = Pr = 1 2 {\displaystyle \Pr=\Pr={\frac {1}{2}}} Pr = Pr ≤ Pr = 1 2 {\displaystyle \Pr=\Pr\leq \Pr={\frac {1}{2}}} Plugging these in the equation (1), we get: Pr ≤ 1 2 ⋅ Pr + 1 2 ⋅ Pr = 1 2 ⋅ Pr + 1 2 ⋅ ( 1 − Pr ) = 1 2 {\displa... | Freivalds' algorithm | 0.805443 |
29,517 | The earliest exploitation of groundwater resources dates back to 3000 BC, and hydrogeology as a science was developed by hydrologists beginning in the 17th century. Ecohydrology is the study of ecological systems in the hydrosphere. It can be divided into the physical study of aquatic ecosystems and the biological stud... | Earth sciences | 0.805442 |
29,518 | Hydrogeology is the study of groundwater. It includes the mapping of groundwater supplies and the analysis of groundwater contaminants. Applied hydrogeology seeks to prevent contamination of groundwater and mineral springs and make it available as drinking water. | Earth sciences | 0.805442 |
29,519 | Applied hydrology involves engineering to maintain aquatic environments and distribute water supplies. Subdisciplines of hydrology include oceanography, hydrogeology, ecohydrology, and glaciology. Oceanography is the study of oceans. | Earth sciences | 0.805442 |
29,520 | Physical geography is the study of Earth's systems and how they interact with one another as part of a single self-contained system. It incorporates astronomy, mathematical geography, meteorology, climatology, geology, geomorphology, biology, biogeography, pedology, and soils geography. Physical geography is distinct f... | Earth sciences | 0.805442 |
29,521 | In statistics, verification bias is a type of measurement bias in which the results of a diagnostic test affect whether the gold standard procedure is used to verify the test result. This type of bias is also known as "work-up bias" or "referral bias".In clinical practice, verification bias is more likely to occur when... | Verification bias | 0.805442 |
29,522 | Some use their knowledge about Earth processes such as volcanoes, earthquakes, and hurricanes to help protect people from these dangerous events. Earth sciences can include the study of geology, the lithosphere, and the large-scale structure of Earth's interior, as well as the atmosphere, hydrosphere, and biosphere. Ty... | Earth sciences | 0.805442 |
29,523 | The equation is named after Harden M. McConnell of Stanford University, who first presented it in 1956 in an article in the Journal of Chemical Physics. | McConnell equation | 0.805441 |
29,524 | In physical chemistry, the McConnell equation gives the probability of an unpaired electron in an in aromatic radical compound (such as benzene radical anion C 6 H 6 − {\displaystyle C_{6}H_{6}^{-}} ) being on a particular atom. It relates this probability, known as the "spin density", to its proportional dependence on... | McConnell equation | 0.805441 |
29,525 | In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: | Additive arithmetic function | 0.805439 |
29,526 | The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if A is the adjacency matrix of an n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ... | All-ones vector | 0.805437 |
29,527 | In theoretical physics, a dual resonance model arose during the early investigation (1968–1973) of string theory as an S-matrix theory of the strong interaction. | Dual resonance model | 0.805437 |
29,528 | Revise all the probabilities continuously during the search. For example, if the hypotheses for location X imply the likely disintegration of the object and the search at location X has yielded no fragments, then the probability that the object is somewhere around there is greatly reduced (though not usually to zero) w... | Bayesian search theory | 0.805434 |
29,529 | Bayesian search theory is the application of Bayesian statistics to the search for lost objects. It has been used several times to find lost sea vessels, for example USS Scorpion, and has played a key role in the recovery of the flight recorders in the Air France Flight 447 disaster of 2009. It has also been used in th... | Bayesian search theory | 0.805434 |
29,530 | Suppose a grid square has a probability p of containing the wreck and that the probability of successfully detecting the wreck if it is there is q. If the square is searched and no wreck is found, then, by Bayes' theorem, the revised probability of the wreck being in the square is given by p ′ = p ( 1 − q ) ( 1 − p ) +... | Bayesian search theory | 0.805434 |
29,531 | Photolysis occurs in the atmosphere as part of a series of reactions by which primary pollutants such as hydrocarbons and nitrogen oxides react to form secondary pollutants such as peroxyacyl nitrates. See Photochemical smog. The two most important photodissociation reactions in the troposphere are firstly: O 3 + h ν ⟶... | Photodissociation | 0.805429 |
29,532 | Game designer and developer Robert Zubek defines game design by breaking it down into its elements, which he says are the following: Gameplay, which is the interaction between the player and the mechanics and systems Mechanics and systems, which are the rules and objects in the game Player experience, which is how user... | Game design | 0.805429 |
29,533 | Most games require multiple players. Single-player games are unique in respect to the type of challenges a player faces. Unlike a game with multiple players competing with or against each other to reach the game's goal, a single-player game is against an element of the environment, against one's own skills, against tim... | Game design | 0.805429 |
29,534 | They fracture—split into two or more pieces.Strength is a measure of the extent of a material's elastic range, or elastic and plastic ranges together. This is quantified as compressive strength, shear strength, tensile strength depending on the direction of the forces involved. Ultimate strength is an engineering measu... | Hardness tester | 0.805427 |
29,535 | In materials science, hardness (antonym: softness) is a measure of the resistance to localized plastic deformation induced by either mechanical indentation or abrasion. In general, different materials differ in their hardness; for example hard metals such as titanium and beryllium are harder than soft metals such as so... | Hardness tester | 0.805427 |
29,536 | Deformation in the plastic range is non-linear, and is described by the stress-strain curve. This response produces the observed properties of scratch and indentation hardness, as described and measured in materials science. Some materials exhibit both elasticity and viscosity when undergoing plastic deformation; this ... | Hardness tester | 0.805427 |
29,537 | Fisher's equation belongs to the class of reaction-diffusion equations: in fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term f ( u , x , t ) = r u ( 1 − u ) , {\displaystyle f(u,x,t)=ru(1-u),\,} which can exhibit traveling wave solutions that switch betwee... | KPP equation | 0.805427 |
29,538 | In the same year (1937) as Fisher, Kolmogorov, Petrovsky and Piskunov introduced the more general reaction-diffusion equation ∂ u ∂ t − ∂ 2 u ∂ x 2 = F ( u ) {\displaystyle {\frac {\partial u}{\partial t}}-{\frac {\partial ^{2}u}{\partial x^{2}}}=F(u)} where F {\displaystyle F} is a sufficiently smooth function with th... | KPP equation | 0.805427 |
29,539 | This investigation has shown how conventional chemistry, i.e. internal recombination of hydrogen and oxygen, accounted for the entire amount of apparent excess heat. The investigators concluded their publication with the following word of advice: "All reports claiming the observation of excess heat should be accompanie... | Faraday-efficiency effect | 0.805425 |
29,540 | The Faraday-efficiency effect refers to the potential for misinterpretation of data from experiments in electrochemistry through failure to take into account a Faraday efficiency of less than 100 percent. | Faraday-efficiency effect | 0.805425 |
29,541 | Lacking any other plausible explanation, the anomalous excess heat produced during such electrolysis was attributed by Pons and Fleischmann to cold fusion. Later, it was discovered that such excess heat can easily be the product of conventional chemistry, i.e. internal recombination of hydrogen and oxygen. Such recombi... | Faraday-efficiency effect | 0.805425 |
29,542 | In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa. | Elimination matrix | 0.805424 |
29,543 | In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function. Associated with any semiautomaton is a monoid called the char... | Semiautomaton | 0.805421 |
29,544 | Dyson and Lenard did not consider the extreme magnetic or gravitational forces that occur in some astronomical objects. In 1995 Elliott Lieb and coworkers showed that the Pauli principle still leads to stability in intense magnetic fields such as in neutron stars, although at a much higher density than in ordinary matt... | Pauli exclusive principle | 0.805419 |
29,545 | In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extende... | Pauli exclusive principle | 0.805419 |
29,546 | : 123–125 The Pauli exclusion principle underpins many properties of everyday matter, from its large-scale stability to the chemical behavior of atoms. Half-integer spin means that the intrinsic angular momentum value of fermions is ℏ = h / 2 π {\displaystyle \hbar =h/2\pi } (reduced Planck's constant) times a half-int... | Pauli exclusive principle | 0.805419 |
29,547 | 3. In computer science, denotes the (unknown) greatest lower bound for the exponent of the computational complexity of matrix multiplication. | Glossary of mathematical symbols | 0.805412 |
29,548 | Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of mathematical objects. As the number of these sorts has remarkably increased in modern mathematics, the Greek alphabet and so... | Glossary of mathematical symbols | 0.805412 |
29,549 | 5. In number theory, may denote the prime omega function. That is, ω ( n ) {\displaystyle \omega (n)} is the number of distinct prime factors of the integer n. | Glossary of mathematical symbols | 0.805412 |
29,550 | In number theory and field theory, F / E {\displaystyle F/E} denotes a field extension, where F is an extension field of the field E. 4. In probability theory, denotes a conditional probability. | Glossary of mathematical symbols | 0.805412 |
29,551 | Denotes multiplication and is read as times; for example, 3 ⋅ 2. 2. In geometry and linear algebra, denotes the dot product. | Glossary of mathematical symbols | 0.805412 |
29,552 | 2. In geometry and linear algebra, denotes the cross product. 3. | Glossary of mathematical symbols | 0.805412 |
29,553 | List of mathematical symbols by subject List of logic symbols List of mathematical constants Table of mathematical symbols by introduction date Blackboard bold Greek letters used in mathematics, science, and engineering List of mathematical uses of Latin letters List of common physics notations List of letters used in ... | Glossary of mathematical symbols | 0.805412 |
29,554 | Einstein believed these plane waves would gravitationally collapse into points; he had long hoped something like this would explain quantum mechanical wave-particle duality.Einstein and Rosen accordingly submitted a paper entitled Do gravitational waves exist? to a leading physics journal, Physical Review, in which the... | Sticky bead argument | 0.805411 |
29,555 | In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.The von Neumann group algebra of a group is a factor if and only if the group has the infinite conjugacy class property. It will then be, provi... | Infinite conjugacy class property | 0.80541 |
29,556 | Mutation testing was originally proposed by Richard Lipton as a student in 1971, and first developed and published by DeMillo, Lipton and Sayward. The first implementation of a mutation testing tool was by Timothy Budd as part of his PhD work (titled Mutation Analysis) in 1980 from Yale University.Recently, with the av... | Program mutation | 0.805409 |
29,557 | In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers. An unambiguous Turing machine is a special kind of non-deterministic Turing machine, which, in some sense, is similar to a deterministic Turing machine. | Unambiguous Turing machine | 0.805408 |
29,558 | Also see pixel connectivity. A digital space and its (digital-)connectivity determine a digital topology. | Digital surface | 0.805405 |
29,559 | The precise statements of the theorems are as follows. In general, the homotopy fiber of B f: B C → B D {\displaystyle Bf:BC\to BD} is not naturally the classifying space of a category: there is no natural category F f {\displaystyle Ff} such that F B f = B F f {\displaystyle FBf=BFf} . Theorem B constructs F f {\displ... | Quillen's theorems A and B | 0.805402 |
29,560 | In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play cent... | Quillen's theorems A and B | 0.805402 |
29,561 | A common problem presented in introductory physics classes is a block subject to friction as it slides up or down an inclined plane. This is shown in the free body diagram to the right. The component of the force of gravity in the direction of the incline is given by: F g = m g sin θ {\displaystyle F_{g}=mg\sin {\the... | Sliding (motion) | 0.805401 |
29,562 | In semiconductors with low levels of doping, the electron concentration in the conduction band is in general given by n e = N C exp ( − E C − E F k T ) {\displaystyle n_{\text{e}}=N_{\text{C}}\exp \left(-{\frac {E_{\text{C}}-E_{\text{F}}}{kT}}\right)} where EF is the Fermi level, EC is the minimum energy of the condu... | Effective electron mass | 0.805399 |
29,563 | This algebra "contains" all distributions T of D' via the injection j(T) = (φn ∗ T)n + N,where ∗ is the convolution operation, and φn(x) = n φ(nx).This injection is non-canonical in the sense that it depends on the choice of the mollifier φ, which should be C∞, of integral one and have all its derivatives at 0 vanishin... | Generalized functions | 0.805398 |
29,564 | These include: the convolution quotient theory of Jan Mikusinski, based on the field of fractions of convolution algebras that are integral domains; and the theories of hyperfunctions, based (in their initial conception) on boundary values of analytic functions, and now making use of sheaf theory. | Generalized functions | 0.805398 |
29,565 | V. Egorov. Another approach to construct associative differential algebras is based on J.-F. Colombeau's construction: see Colombeau algebra. These are factor spaces G = M / N {\displaystyle G=M/N} of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respec... | Generalized functions | 0.805398 |
29,566 | If (E,P) is a (pre-)sheaf of semi normed algebras on some topological space X, then Gs(E, P) will also have this property. This means that the notion of restriction will be defined, which allows to define the support of a generalized function w.r.t. a subsheaf, in particular: For the subsheaf {0}, one gets the usual su... | Generalized functions | 0.805398 |
29,567 | A simple example is obtained by using the polynomial scale on N, s = { a m: N → R , n ↦ n m ; m ∈ Z } {\displaystyle s=\{a_{m}:\mathbb {N} \to \mathbb {R} ,n\mapsto n^{m};~m\in \mathbb {Z} \}} . Then for any semi normed algebra (E,P), the factor space will be G s ( E , P ) = { f ∈ E N ∣ ∀ p ∈ P , ∃ m ∈ Z: p ( f n ) = o... | Generalized functions | 0.805398 |
29,568 | Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute. Few applications of the algebra were suggested. | Generalized functions | 0.805398 |
29,569 | The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function F = F ( x ) {\displaystyle F=F(x)} to its smooth F s m o o t h {\displaystyle F_{\rm {smooth}}} and its singular F s i n g u l a r {\displaystyle F_{\rm {singular}}} parts. The product of generalized functions... | Generalized functions | 0.805398 |
29,570 | They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are cl... | Generalized functions | 0.805398 |
29,571 | The Nested Grid Model (usually known as NGM for short) was a numerical weather prediction model run by the National Centers for Environmental Prediction, a division of the National Weather Service, in the United States. The NGM was, as its name suggested, derived from two levels of grids: a hemispheric-scale grid and a... | Nested Grid Model | 0.805397 |
29,572 | This is the basis of fluorescence resonance energy transfer, a technique that is used in molecular biology to study the interaction of suitable proteins.Several different kinds of hardware random number generators involve the detection of single photons. In one example, for each bit in the random sequence that is to be... | Light quanta | 0.805395 |
29,573 | In the same papers, Einstein extended Bose's formalism to material particles (bosons) and predicted that they would condense into their lowest quantum state at low enough temperatures; this Bose–Einstein condensation was observed experimentally in 1995. It was later used by Lene Hau to slow, and then completely stop, l... | Light quanta | 0.805395 |
29,574 | A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which of the two pa... | Light quanta | 0.805395 |
29,575 | Also, by arguments analogous to the derivation of Boltzmann statistics, the ratio of N i {\displaystyle N_{i}} and N j {\displaystyle N_{j}} is g i / g j exp ( E j − E i ) / ( k T ) , {\displaystyle g_{i}/g_{j}\exp {(E_{j}-E_{i})/(kT)},} where g i {\displaystyle g_{i}} and g j {\displaystyle g_{j}} are the degeneracy... | Light quanta | 0.805395 |
29,576 | However, there are also non-synchronous versions of the problem, where the robots may act at different times or may have different internal clocks . The deterministic rendezvous problem has applications in a wide range of fields, including robotics, distributed systems, and computer networks . It is often used as a ben... | Deterministic rendezvous problem | 0.805392 |
29,577 | The deterministic rendezvous problem is a problem in computer science and robotics that involves two or more robots or players that must find each other by following a predetermined set of instructions. The goal is for the robots to meet at a specific location, or rendezvous point, without knowing the location of the o... | Deterministic rendezvous problem | 0.805392 |
29,578 | In metaphysics, the A series and the B series are two different descriptions of the temporal ordering relation among events. The two series differ principally in their use of tense to describe the temporal relation between events and the resulting ontological implications regarding time. John McTaggart introduced these... | A series and B series | 0.805392 |
29,579 | However in Galois theory, a branch of algebra, and in this article, the term generic polynomial has a different, although related, meaning: a generic polynomial for a finite group G and a field F is a monic polynomial P with coefficients in the field of rational functions L = F(t1, ..., tn) in n indeterminates over F, ... | Generic polynomial | 0.805392 |
29,580 | Then it is a matter of measuring the different components, usually designated by subscripts. According to the ΛCDM model, there are important components of Ω due to baryons, cold dark matter and dark energy. The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat. | Density parameter | 0.805391 |
29,581 | {\displaystyle \Omega \equiv {\frac {\rho }{\rho _{c}}}={\frac {8\pi G\rho }{3H^{2}}}.} This term originally was used as a means to determine the spatial geometry of the universe, where ρc is the critical density for which the spatial geometry is flat (or Euclidean). Assuming a zero vacuum energy density, if Ω is large... | Density parameter | 0.805391 |
29,582 | The density parameter Ω is defined as the ratio of the actual (or observed) density ρ to the critical density ρc of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclide... | Density parameter | 0.805391 |
29,583 | Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density. Some cosmologists call ... | Density parameter | 0.805391 |
29,584 | We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices. There are two commonly used choices for a and k which describe the same physics: k = +1, 0 or −1 depending on whether the shape of the universe is a closed 3-sphere, flat (Euclidean space) or an open 3-h... | Density parameter | 0.805391 |
29,585 | The Banach fixed-point theorem allows one to obtain fixed-point iterations with linear convergence. The requirement that f is continuous is important, as the following example shows. The iteration converges to 0 for all values of x 0 {\displaystyle x_{0}} . However, 0 is not a fixed point of the function as this functi... | Fixed point algorithm | 0.80539 |
29,586 | This example does satisfy (at the latest after the first iteration step) the assumptions of the Banach fixed-point theorem. Hence, the error after n steps satisfies | x n − x | ≤ q n 1 − q | x 1 − x 0 | = C q n {\displaystyle |x_{n}-x|\leq {q^{n} \over 1-q}|x_{1}-x_{0}|=Cq^{n}} (where we can take q = 0.85 {\displaystyl... | Fixed point algorithm | 0.80539 |
29,587 | The Atherton-Todd reaction is a name reaction in organic chemistry, which goes back to the British chemists F. R. Atherton, H. T. Openshaw and A. R. Todd. These described the reaction for the first time in 1945 as a method of converting dialkyl phosphites into dialkyl chlorophosphates. The dialkyl chlorophosphates form... | Atherton–Todd reaction | 0.80539 |
29,588 | The issue with learning rate schedules is that they all depend on hyperparameters that must be manually chosen for each given learning session and may vary greatly depending on the problem at hand or the model used. To combat this, there are many different types of adaptive gradient descent algorithms such as Adagrad, ... | Adaptive learning rate | 0.805389 |
29,589 | The formula for factoring in the momentum is more complex than for decay but is most often built in with deep learning libraries such as Keras. Time-based learning schedules alter the learning rate depending on the learning rate of the previous time iteration. Factoring in the decay the mathematical formula for the lea... | Adaptive learning rate | 0.805389 |
29,590 | In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics: they take on lar... | Similarity measure | 0.805387 |
29,591 | Nano Research is a peer-reviewed scientific journal published by Springer Science+Business Media. It covers research in all areas of nanotechnology. | Nano Research | 0.805386 |
29,592 | The objective of the project is to conduct fusion plasma confinement research in a steady state in order to elucidate possible solutions to physics and engineering problems in helical plasma reactors. The LHD uses neutral beam injection, ion cyclotron radio frequency (ICRF), and electron cyclotron resonance heating (EC... | Large Helical Device | 0.805385 |
29,593 | Proteinase K is commonly used in molecular biology to digest protein and remove contamination from preparations of nucleic acid. Addition of Proteinase K to nucleic acid preparations rapidly inactivates nucleases that might otherwise degrade the DNA or RNA during purification. It is highly suited to this application si... | Proteinase K | 0.805384 |
29,594 | In molecular biology, Proteinase K (EC 3.4.21.64, protease K, endopeptidase K, Tritirachium alkaline proteinase, Tritirachium album serine proteinase, Tritirachium album proteinase K) is a broad-spectrum serine protease. The enzyme was discovered in 1974 in extracts of the fungus Parengyodontium album (formerly Engyodo... | Proteinase K | 0.805384 |
29,595 | When drawing the conjugate beam it is important that the shear and moment developed at the supports of the conjugate beam account for the corresponding slope and displacement of the real beam at its supports, a consequence of Theorems 1 and 2. For example, as shown below, a pin or roller support at the end of the real ... | Conjugate beam method | 0.805382 |
29,596 | A M/EI diagram is a moment diagram divided by the beam's Young's modulus and moment of inertia. To make use of this comparison we will now consider a beam having the same length as the real beam, but referred here as the "conjugate beam." The conjugate beam is "loaded" with the M/EI diagram derived from the load on the... | Conjugate beam method | 0.805382 |
29,597 | The conjugate-beam methods is an engineering method to derive the slope and displacement of a beam. A conjugate beam is defined as an imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjugate beam is equal to the bending moment at that point divided by EI.The c... | Conjugate beam method | 0.805382 |
29,598 | Standard monomial theory proves the Kempf vanishing theorem and other vanishing theorems for the higher cohomology of effective line bundles over Schubert varieties. Standard monomial theory gives explicit bases for some rings of invariants in invariant theory. Standard monomial theory gives generalizations of the Litt... | Standard monomial basis | 0.805381 |
29,599 | Since the sections of line bundles over generalized flag varieties tend to form irreducible representations of the corresponding algebraic groups, having an explicit basis of standard monomials allows one to give character formulas for these representations. Similarly one gets character formulas for Demazure modules. T... | Standard monomial basis | 0.805381 |
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