id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
28,900 | These machines define a probability distribution over all strings, and are related to other probabilistic models such as Markov decision processes and Markov chains. Weighted automata have applications in natural language processing where they are used to assign weights to words and sentences,: 571–596 as well as in im... | Weighted automaton | 0.805811 |
28,901 | In theoretical computer science and formal language theory, a weighted automaton or weighted finite-state machine is a generalization of a finite-state machine in which the edges have weights, for example real numbers or integers. Finite-state machines are only capable of answering decision problems; they take as input... | Weighted automaton | 0.805811 |
28,902 | The Tensorial Time Derivative ∇ ˙ {\displaystyle {\dot {\nabla }}} for a scalar field F defined on Σ t {\displaystyle \Sigma _{t}} is the rate of change in F {\displaystyle F} in the instantaneously normal direction: δ F δ t = lim h → 0 F ( P ∗ ) − F ( P ) h {\displaystyle {\frac {\delta F}{\delta t}}=\lim _{h\to 0}{\f... | Calculus of moving surfaces | 0.805809 |
28,903 | The velocity C {\displaystyle C} is a signed quantity: it is positive when P P ∗ ¯ {\displaystyle {\overline {PP^{*}}}} points in the direction of the chosen normal, and negative otherwise. The relationship between Σ t {\displaystyle \Sigma _{t}} and C {\displaystyle C} is analogous to the relationship between location... | Calculus of moving surfaces | 0.805809 |
28,904 | The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative ∇ ˙ {\displaystyle {\dot {\nabla }}} whose original definition was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant deriva... | Calculus of moving surfaces | 0.805809 |
28,905 | In natural language processing, multinomial LR classifiers are commonly used as an alternative to naive Bayes classifiers because they do not assume statistical independence of the random variables (commonly known as features) that serve as predictors. However, learning in such a model is slower than for a naive Bayes ... | Multinomial logistic regression | 0.805808 |
28,906 | Momentum has been used successfully by computer scientists in the training of artificial neural networks for several decades. The momentum method is closely related to underdamped Langevin dynamics, and may be combined with Simulated Annealing. In mid-1980s the method was modified by Yurii Nesterov to use the gradient ... | Stochastic gradient descent | 0.805806 |
28,907 | Further proposals include the momentum method or the heavy ball method, which in ML context appeared in Rumelhart, Hinton and Williams' paper on backpropagation learning and borrowed the idea from Soviet mathematician Boris Polyak's 1964 article on solving functional equations. Stochastic gradient descent with momentum... | Stochastic gradient descent | 0.805806 |
28,908 | Since this approximation does not capture the random fluctuations around the mean behavior of stochastic gradient descent solutions to stochastic differential equations (SDEs) have been proposed as limiting objects. More precisely, the solution to the SDE d W t = − ∇ ( Q ( W t ) + 1 4 η | ∇ Q ( W t ) | 2 ) d t + η Σ ( ... | Stochastic gradient descent | 0.805806 |
28,909 | Even though sign-based optimization goes back to the aforementioned Rprop, only in 2018 researchers tried to simplify Adam by removing the magnitude of the stochastic gradient from being taked into account and only considering its sign. | Stochastic gradient descent | 0.805806 |
28,910 | Averaged stochastic gradient descent, invented independently by Ruppert and Polyak in the late 1980s, is ordinary stochastic gradient descent that records an average of its parameter vector over time. That is, the update is the same as for ordinary stochastic gradient descent, but the algorithm also keeps track of w ¯ ... | Stochastic gradient descent | 0.805806 |
28,911 | The formula for an update is now w := w − η d i a g ( G ) − 1 2 ⊙ g {\displaystyle w:=w-\eta \,\mathrm {diag} (G)^{-{\frac {1}{2}}}\odot g} or, written as per-parameter updates, w j := w j − η G j , j g j . {\displaystyle w_{j}:=w_{j}-{\frac {\eta }{\sqrt {G_{j,j}}}}g_{j}.} Each {G(i,i)} gives rise to a scaling factor ... | Stochastic gradient descent | 0.805806 |
28,912 | AdaGrad (for adaptive gradient algorithm) is a modified stochastic gradient descent algorithm with per-parameter learning rate, first published in 2011. Informally, this increases the learning rate for sparser parameters and decreases the learning rate for ones that are less sparse. This strategy often improves converg... | Stochastic gradient descent | 0.805806 |
28,913 | Stochastic gradient descent is a popular algorithm for training a wide range of models in machine learning, including (linear) support vector machines, logistic regression (see, e.g., Vowpal Wabbit) and graphical models. When combined with the backpropagation algorithm, it is the de facto standard algorithm for trainin... | Stochastic gradient descent | 0.805806 |
28,914 | Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calcul... | Stochastic gradient descent | 0.805806 |
28,915 | Many improvements on the basic stochastic gradient descent algorithm have been proposed and used. In particular, in machine learning, the need to set a learning rate (step size) has been recognized as problematic. Setting this parameter too high can cause the algorithm to diverge; setting it too low makes it slow to co... | Stochastic gradient descent | 0.805806 |
28,916 | It may also result in smoother convergence, as the gradient computed at each step is averaged over more training samples. The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates η {\displaystyle \eta } d... | Stochastic gradient descent | 0.805806 |
28,917 | For example, in statistics, one-parameter exponential families allow economical function-evaluations and gradient-evaluations. However, in other cases, evaluating the sum-gradient may require expensive evaluations of the gradients from all summand functions. When the training set is enormous and no simple formulas exis... | Stochastic gradient descent | 0.805806 |
28,918 | In this Q i ( w ) {\displaystyle Q_{i}(w)} is the value of the loss function at i {\displaystyle i} -th example, and Q ( w ) {\displaystyle Q(w)} is the empirical risk. When used to minimize the above function, a standard (or "batch") gradient descent method would perform the following iterations: w := w − η ∇ Q ( w ) ... | Stochastic gradient descent | 0.805806 |
28,919 | However, in statistics, it has been long recognized that requiring even local minimization is too restrictive for some problems of maximum-likelihood estimation. Therefore, contemporary statistical theorists often consider stationary points of the likelihood function (or zeros of its derivative, the score function, and... | Stochastic gradient descent | 0.805806 |
28,920 | Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q ( w ) = 1 n ∑ i = 1 n Q i ( w ) , {\displaystyle Q(w)={\frac {1}{n}}\sum _{i=1}^{n}Q_{i}(w),} where the parameter w {\displaystyle w} that minimizes Q ( w ) {\displaystyle Q(w)} is to ... | Stochastic gradient descent | 0.805806 |
28,921 | A stochastic analogue of the standard (deterministic) Newton–Raphson algorithm (a "second-order" method) provides an asymptotically optimal or near-optimal form of iterative optimization in the setting of stochastic approximation. A method that uses direct measurements of the Hessian matrices of the summands in the emp... | Stochastic gradient descent | 0.805806 |
28,922 | The SETI project has for the past several decades been conducting a search for signals being transmitted by extraterrestrial life located outside the Solar System, primarily in the radio frequencies of the electromagnetic spectrum. Special attention has been given to the Water Hole, the frequency of one of neutral hydr... | Interstellar communication | 0.805805 |
28,923 | In December 2015, Sam Altman, Greg Brockman, Reid Hoffman, Jessica Livingston, Peter Thiel, Elon Musk, Amazon Web Services (AWS), Infosys, and YC Research announced the formation of OpenAI and pledged over $1 billion to the venture. According to an investigation led by TechCrunch, the non-profit's funding remains murky... | Open AI | 0.805803 |
28,924 | Some experts expressed skepticism that GPT-2 posed a significant threat. The Allen Institute for Artificial Intelligence responded to GPT-2 with a tool to detect "neural fake news". Other researchers, such as Jeremy Howard, warned of "the technology to totally fill Twitter, email, and the web up with reasonable-soundin... | Open AI | 0.805803 |
28,925 | Released in 2020, Microscope is a collection of visualizations of every significant layer and neuron of eight different neural network models which are often studied in interpretability. Microscope was created to analyze the features that form inside these neural networks easily. The models included are AlexNet, VGG 19... | Open AI | 0.805803 |
28,926 | Launched in November 2022, ChatGPT is an artificial intelligence tool built on top of GPT-3 that provides a conversational interface that allows users to ask questions in natural language. The system then responds with an answer within seconds. ChatGPT reached 1 million users 5 days after its launch.ChatGPT Plus is a $... | Open AI | 0.805803 |
28,927 | Key employees: CEO and co-founder: Sam Altman, former president of the startup accelerator Y Combinator President and co-founder: Greg Brockman, former CTO, 3rd employee of Stripe Chief Scientist and co-founder: Ilya Sutskever, a former Google expert on machine learning Chief Technology Officer: Mira Murati, previously... | Open AI | 0.805803 |
28,928 | OpenAI is an American artificial intelligence (AI) research laboratory consisting of the non-profit OpenAI, Inc. and its for-profit subsidiary corporation OpenAI, L.P.. OpenAI conducts research on artificial intelligence with the declared intention of developing "safe and beneficial" artificial general intelligence, wh... | Open AI | 0.805803 |
28,929 | "Musk and Altman's counter-intuitive strategy of trying to reduce the risk that AI will cause overall harm, by giving AI to everyone, is controversial among those who are concerned with existential risk from artificial intelligence. Philosopher Nick Bostrom is skeptical of Musk's approach: "If you have a button that co... | Open AI | 0.805803 |
28,930 | Artificial intelligence opponents often need hidden information or bonuses to provide a challenge to players.More recently, massively multiplayer online strategy games have appeared such as Shattered Galaxy from 2001. However, these games are relatively difficult to design and implement compared to other massively mult... | Computer strategy game | 0.805801 |
28,931 | Strategy games include single-player gameplay, multiplayer gameplay, or both. Single player games will sometimes feature a campaign mode, which involves a series of matches against several artificial intelligence opponents. Finishing each match or mission will advance the game's plot, often with cut scenes, and some ga... | Computer strategy game | 0.805801 |
28,932 | Type 1 computability is the naive form of computable analysis in which one restricts the inputs to a machine to be computable numbers instead of arbitrary real numbers. The difference between the two models lies in the fact that a program that is well-behaved over computable numbers (in the sense of being total) is not... | Computable analysis | 0.805799 |
28,933 | In mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and n... | Computable analysis | 0.805799 |
28,934 | In particular, the unit interval {\displaystyle } is computably compact. Discrete spaces in topology are analogous to sets in computability where equality between elements is semi-decidable. Hausdorff spaces in topology are analogous to sets in computability where inequality between elements is semi-decidable.The anal... | Computable analysis | 0.805799 |
28,935 | There is a computable analogue of topological compactness. Namely, a subset S {\displaystyle S} of R {\displaystyle \mathbb {R} } is computably compact if it there is a semi-decision procedure " ∀ S {\displaystyle \forall _{S}} " that, given a semidecidable predicate P {\displaystyle P} as input, semi-decides whether e... | Computable analysis | 0.805799 |
28,936 | One of the basic results of computable analysis is that every computable function from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } is continuous. Taking this further, this suggests that there is an analogy between basic notions in topology and basic notions in computability: Computable functions ar... | Computable analysis | 0.805799 |
28,937 | In the event that one is unhappy with using Turing machines (on the grounds that they are low level and somewhat arbitrary), there is a realisability topos called the Kleene–Vesley topos in which one can reduce computable analysis to constructive analysis. This constructive analysis includes everything that is valid in... | Computable analysis | 0.805799 |
28,938 | Program I investigates non-constructibility in geometry, classification of patterns in two dimensions, error-correcting codes, cryptography, and the analysis of the Rubik's Cube. The mathematics that is central to solving these problems comes from the areas of abstract algebra and number theory. Program II contains an ... | Stanford University Mathematics Camp | 0.805799 |
28,939 | SUMaC was founded in 1995 by Professors Rafe Mazzeo and Ralph Cohen of the Stanford Mathematics Department and has been directed from the beginning by Prof. Mazzeo, and Dr. Rick Sommer. Dr. Sommer was an assistant professor in the Stanford Mathematics Department and is currently a deputy director of the Education Progr... | Stanford University Mathematics Camp | 0.805799 |
28,940 | John Dumbleton of the 14th-century Oxford Calculators, was one of the first to express functional relationships in graphical form. He gave a proof of the mean speed theorem stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitativ... | Inverse square law | 0.805798 |
28,941 | The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics. In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). Such a high transition temperat... | Superconducting engineering | 0.805794 |
28,942 | An electric current through a loop of superconducting wire can persist indefinitely with no power source.The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and atomic spectral lines, superconductivity is a phenomenon which can only be explained by quan... | Superconducting engineering | 0.805794 |
28,943 | Lithium 12-hydroxystearate (C18H35LiO3) is a chemical compound classified as a lithium soap. In chemistry, "soap" refers to salts of fatty acids. Lithium 12-hydroxystearate is a white solid. Lithium soaps are key component of many lubricating greases. | Lithium 12-hydroxystearate | 0.805792 |
28,944 | Optimization methods such as genetic algorithm and differential evolution have several parameters that govern their behaviour and efficiency in optimizing a given problem and these parameters must be chosen by the practitioner to achieve satisfactory results. Selecting the behavioural parameters by hand is a laborious ... | Meta-optimization | 0.805788 |
28,945 | Birattari et al. meta-optimized ant colony optimization. Statistical models have also been used to reveal more about the relationship between choices of behavioural parameters and optimization performance, see for example Francois and Lavergne, and Nannen and Eiben. A comparison of various meta-optimization techniques ... | Meta-optimization | 0.805788 |
28,946 | A simple way of finding good behavioural parameters for an optimizer is to employ another overlaying optimizer, called the meta-optimizer. There are different ways of doing this depending on whether the behavioural parameters to be tuned are real-valued or discrete-valued, and depending on what performance measure is b... | Meta-optimization | 0.805788 |
28,947 | In numerical optimization, meta-optimization is the use of one optimization method to tune another optimization method. Meta-optimization is reported to have been used as early as in the late 1970s by Mercer and Sampson for finding optimal parameter settings of a genetic algorithm. Meta-optimization and related concept... | Meta-optimization | 0.805788 |
28,948 | Like the matrix multiplication exponent, the dual matrix multiplication exponent sometimes appears in the complexity of algorithms in numerical linear algebra and optimization.The first bound on α is by Coppersmith in 1982, who showed that α > 0.17227 {\displaystyle \alpha >0.17227} . The current best peer-reviewed bou... | Fast matrix multiplication | 0.805787 |
28,949 | Similar techniques also apply to rectangular matrix multiplication. The central object of study is ω ( k ) {\displaystyle \omega (k)} , which is the smallest c {\displaystyle c} such that one can multiply a matrix of size n × ⌈ n k ⌉ {\displaystyle n\times \lceil n^{k}\rceil } with a matrix of size ⌈ n k ⌉ × n {\displa... | Fast matrix multiplication | 0.805787 |
28,950 | Indeed, such second-order and higher-order perturbation calculations can give apparently infinite contributions to the sum. Such unphysical results are corrected for using the technique of renormalization.Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly t... | Photon | 0.805785 |
28,951 | He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A i j {\displaystyle A_{ij}} and B i j {\display... | Photon | 0.805785 |
28,952 | The intrinsic properties of particles, such as charge, mass, and spin, are determined by gauge symmetry. The photon concept has led to momentous advances in experimental and theoretical physics, including lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics... | Photon | 0.805785 |
28,953 | In 1926, Gilbert N. Lewis popularized the term photon for these energy units. Subsequently, many other experiments validated Einstein's approach.In the Standard Model of particle physics, photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every ... | Photon | 0.805785 |
28,954 | A photon (from Ancient Greek φῶς, φωτός (phôs, phōtós) 'light') is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always move at the speed of ligh... | Photon | 0.805785 |
28,955 | The uncertainty principle applies to situations where an experimenter has a choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle. According to the uncertainty principle, no matter how the particle is prepared, it is not possible to make a precise pre... | Photon | 0.805785 |
28,956 | Rather, the received photon acts like a point-like particle since it is absorbed or emitted as a whole by arbitrarily small systems, including systems much smaller than its wavelength, such as an atomic nucleus (≈10−15 m across) or even the point-like electron. While many introductory texts treat photons using the math... | Photon | 0.805785 |
28,957 | Photons obey the laws of quantum mechanics, and so their behavior has both wave-like and particle-like aspects. When a photon is detected by a measuring instrument, it is registered as a single, particulate unit. However, the probability of detecting a photon is calculated by equations that describe waves. This combina... | Photon | 0.805785 |
28,958 | Nevertheless, the failures of the BKS model inspired Werner Heisenberg in his development of matrix mechanics.A few physicists persisted in developing semiclassical models in which electromagnetic radiation is not quantized, but matter appears to obey the laws of quantum mechanics. Although the evidence from chemical a... | Photon | 0.805785 |
28,959 | Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the photoelectric effect); the energy of the ejected electron is related only to the light's frequency, not to its intensity.At the same time, investigations of black-body radiation carried out over four decade... | Photon | 0.805785 |
28,960 | For two sequences of measures ( P n ) and ( Q n ) {\displaystyle (P_{n}){\text{ and }}(Q_{n})} on measurable spaces ( Ω n , F n ) {\displaystyle (\Omega _{n},{\mathcal {F}}_{n})} the following statements are equivalent: P n ◃ Q n {\displaystyle P_{n}\triangleleft Q_{n}} d Q n d P n ⟶ P n U along a subsequence ⇒ P ( U >... | Contiguity (probability theory) | 0.805784 |
28,961 | By the aforementioned logic, this statement is also false. It is possible however that each of the measures Qn be absolutely continuous with respect to Pn, while the sequence Qn not being contiguous with respect to Pn. The fundamental Radon–Nikodym theorem for absolutely continuous measures states that if Q is absolute... | Contiguity (probability theory) | 0.805784 |
28,962 | In probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. Thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures. The concept was originally introduced by Le Cam (1960) as part of his foundational ... | Contiguity (probability theory) | 0.805784 |
28,963 | Roussas, George G. (1972), Contiguity of Probability Measures: Some Applications in Statistics, CUP, ISBN 978-0-521-09095-7. Scott, D.J. (1982) Contiguity of Probability Measures, Australian & New Zealand Journal of Statistics, 24 (1), 80–88. | Contiguity (probability theory) | 0.805784 |
28,964 | Quaternary is a term used in organic chemistry to classify various types of compounds (e. g. amines and ammonium salts). | Quaternary (chemistry) | 0.805782 |
28,965 | The infinitesimal form of this metric may be quickly obtained by taking φ = ψ + δ ψ {\displaystyle \varphi =\psi +\delta \psi } , or equivalently, W α = Z α + d Z α {\displaystyle W_{\alpha }=Z_{\alpha }+dZ_{\alpha }} to obtain d s 2 = ⟨ δ ψ | δ ψ ⟩ ⟨ ψ | ψ ⟩ − ⟨ δ ψ | ψ ⟩ ⟨ ψ | δ ψ ⟩ ⟨ ψ | ψ ⟩ 2 . {\displaystyle ds^{2... | Quantum angle | 0.805781 |
28,966 | In quantum mechanics, the Fubini–Study metric is also known as the Bures metric. However, the Bures metric is typically defined in the notation of mixed states, whereas the exposition below is written in terms of a pure state. The real part of the metric is (a quarter of) the Fisher information metric.The Fubini–Study ... | Quantum angle | 0.805781 |
28,967 | Here φ, θ are "mathematician's spherical coordinates" on S2 coming from the stereographic projection r tan(φ/2) = 1, tan θ = y/x. (Many physics references interchange the roles of φ and θ.) The Kähler form is K = i 2 d z ∧ d z ¯ ( 1 + z z ¯ ) 2 = d x ∧ d y ( 1 + x 2 + y 2 ) 2 {\displaystyle K={\frac {i}{2}}{\frac {dz\w... | Quantum angle | 0.805781 |
28,968 | Electronic nose technology allows fast and continuous measurement of microbial food spoilage by sensing odours produced by these volatile compounds. Electronic nose technology can thus be applied to detect traces of Pseudomonas milk spoilage and isolate the responsible Pseudomonas species. The gas sensor consists of a ... | Pseudomonas | 0.805781 |
28,969 | Although gallium has no natural function in biology, gallium ions interact with cellular processes in a manner similar to iron(III). When gallium ions are mistakenly taken up in place of iron(III) by bacteria such as Pseudomonas, the ions interfere with respiration, and the bacteria die. This happens because iron is re... | Pseudomonas | 0.805781 |
28,970 | Pseudomonas is a genus of Gram-negative bacteria belonging to the family Pseudomonadaceae in the class Gammaproteobacteria. The 313 members of the genus demonstrate a great deal of metabolic diversity and consequently are able to colonize a wide range of niches. Their ease of culture in vitro and availability of an inc... | Pseudomonas | 0.805781 |
28,971 | Like most bacterial genera, the pseudomonad last common ancestor lived hundreds of millions of years ago. They were initially classified at the end of the 19th century when first identified by Walter Migula. The etymology of the name was not specified at the time and first appeared in the seventh edition of Bergey's Ma... | Pseudomonas | 0.805781 |
28,972 | Molecular cloning is a set of experimental methods in molecular biology that are used to assemble recombinant DNA molecules and to direct their replication within host organisms. The use of the word cloning refers to the fact that the method involves the replication of one molecule to produce a population of cells with... | Molecular cloning | 0.80578 |
28,973 | Proton decay would be an example of such a process taking place, but has never been observed. The conservation of baryon number is not consistent with the physics of black hole evaporation via Hawking radiation. It is expected in general that quantum gravitational effects violate the conservation of all charges associa... | Quark number | 0.805776 |
28,974 | In software engineering, a project fork happens when developers take a copy of source code from one software package and start independent development on it, creating a distinct and separate piece of software. The term often implies not merely a development branch, but also a split in the developer community; as such, ... | Software fork | 0.805775 |
28,975 | CALYPSO - The Crystal structure AnaLYsis by Particle Swarm Optimization, implementing the particle swarm optimization (PSO) algorithm to identify/determine the crystal structure. As with other codes, knowledge of the structure can be used to design multi-functional materials (e.g.,superconductive, thermoelectric, super... | Crystal structure prediction | 0.805774 |
28,976 | The problem has two facets: combinatorics (the "search phase space", in practice most acute for inorganic crystals), and energetics (or "stability ranking", most acute for molecular organic crystals). For complex non-molecular crystals (where the "search problem" is most acute), major recent advances have been the deve... | Crystal structure prediction | 0.805774 |
28,977 | Ether cleavage refers to chemical substitution reactions that lead to the cleavage of ethers. Due to the high chemical stability of ethers, the cleavage of the C-O bond is uncommon in the absence of specialized reagents or under extreme conditions.In organic chemistry, ether cleavage is an acid catalyzed nucleophilic s... | Ether cleavage | 0.805771 |
28,978 | Paula Y. Bruice: Organic Chemistry, Prentice Hall. ISBN 978-0321697684. == References == | Ether cleavage | 0.805771 |
28,979 | Early radio astronomy observations, performed by Seth Shostak, later SETI Institute Senior Astronomer, showed a half-dozen galaxies spun too fast in their outer regions, pointing to the existence of dark matter as a means of creating the gravitational pull needed to keep the stars in their orbits. | Dark matter problem | 0.805771 |
28,980 | Dark matter regularly appears as a topic in hybrid periodicals that cover both factual scientific topics and science fiction, and dark matter itself has been referred to as "the stuff of science fiction". Mention of dark matter is made in works of fiction. In such cases, it is usually attributed extraordinary physical ... | Dark matter problem | 0.805771 |
28,981 | Dark matter does not bend light itself; mass (in this case the mass of the dark matter) bends spacetime. Light follows the curvature of spacetime, resulting in the lensing effect.In May 2021, a new detailed dark matter map was revealed by the Dark Energy Survey Collaboration. In addition, the map revealed previously un... | Dark matter problem | 0.805771 |
28,982 | By measuring the distortion geometry, the mass of the intervening cluster can be obtained. In the dozens of cases where this has been done, the mass-to-light ratios obtained correspond to the dynamical dark matter measurements of clusters. Lensing can lead to multiple copies of an image. | Dark matter problem | 0.805771 |
28,983 | (In a turbulent flow, the object might oscillate indefinitely. )Levitation techniques are useful tools in physics research. For example, levitation methods are useful for high-temperature melt property studies because they eliminate the problem of reaction with containers and allow deep undercooling of melts. The conta... | Levitation (physics) | 0.805769 |
28,984 | The Institute of Electrical and Electronics Engineers (IEEE) style is a widely accepted format for writing research papers, commonly used in technical fields, particularly in computer science. IEEE style is based on the Chicago Style. In IEEE style, citations are numbered, but citation numbers are included in the text ... | IEEE format | 0.805769 |
28,985 | The Serine-Histidine-Aspartate motif is one of the most thoroughly characterised catalytic motifs in biochemistry. The triad is exemplified by chymotrypsin, a model serine protease from the PA superfamily which uses its triad to hydrolyse protein backbones. The aspartate is hydrogen bonded to the histidine, increasing ... | Catalytic dyad | 0.805767 |
28,986 | The MEROPS classification system in the 1990s and 2000s began classing proteases into structurally related enzyme superfamilies and so acts as a database of the convergent evolution of triads in over 20 superfamilies. Understanding how chemical constraints on evolution led to the convergence of so many enzyme families ... | Catalytic dyad | 0.805767 |
28,987 | If A is associative then so is any homotope of A, and any mutation of A is Lie-admissible. If A is alternative then so is any homotope of A, and any mutation of A is Malcev-admissible. Any isotope of a Hurwitz algebra is isomorphic to the original. A homotope of a Bernstein algebra by an element of non-zero weight is a... | Mutation (algebra) | 0.805763 |
28,988 | A Jordan algebra is a commutative algebra satisfying the Jordan identity ( x y ) ( x x ) = x ( y ( x x ) ) {\displaystyle (xy)(xx)=x(y(xx))} . The Jordan triple product is defined by { a , b , c } = ( a b ) c + ( c b ) a − ( a c ) b . {\displaystyle \{a,b,c\}=(ab)c+(cb)a-(ac)b.\,} For y in A the mutation or homotope Ay... | Mutation (algebra) | 0.805763 |
28,989 | Let A be an algebra over a field F with multiplication (not assumed to be associative) denoted by juxtaposition. For an element a of A, define the left a-homotope A ( a ) {\displaystyle A(a)} to be the algebra with multiplication x ∗ y = ( x a ) y . {\displaystyle x*y=(xa)y.\,} Similarly define the left (a,b) mutation ... | Mutation (algebra) | 0.805763 |
28,990 | In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original. | Mutation (algebra) | 0.805763 |
28,991 | Prior to the proof of the four color theorem, he proved that every nearly planar graph can be colored with at most four colors, except for the graphs formed from a wheel graph with an odd outer cycle by replacing the hub vertex with two adjacent vertices, which require five colors. Additionally, he proved that, with a ... | Apex graph | 0.805759 |
28,992 | Every apex graph has chromatic number at most five: the underlying planar graph requires at most four colors by the four color theorem, and the remaining vertex needs at most one additional color. Robertson, Seymour & Thomas (1993a) used this fact in their proof of the case k = 6 of the Hadwiger conjecture, the stateme... | Apex graph | 0.805759 |
28,993 | Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph. For, if G is an apex graph with apex v, then any contraction or removal that does not involve v preserves the planarity of the remaining graph, as does any edge removal of an e... | Apex graph | 0.805759 |
28,994 | In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K5 or K3,3, every v... | Apex graph | 0.805759 |
28,995 | With this terminology, the connection between apex graphs and local treewidth can be restated as the fact that apex-minor-free graph families are the same as minor-closed graph families with bounded local treewidth. The concept of bounded local treewidth forms the basis of the theory of bidimensionality, and allows for... | Apex graph | 0.805759 |
28,996 | Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters. 12 (3): 260–262. | Theory of Special Relativity | 0.805759 |
28,997 | Special relativity uses a "flat" 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time. In 3D space, the differential of distance (line element) ds is defined by wher... | Theory of Special Relativity | 0.805759 |
28,998 | If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space we see that the null geodesics lie along a dual-cone (see Fig. 10-2) defined by the equation; or simply which is the equation of a circle of radius c dt. | Theory of Special Relativity | 0.805759 |
28,999 | Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote: The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are... | Theory of Special Relativity | 0.805759 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.