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23,400 | However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theori... | Mathematical discipline | 0.808826 |
23,401 | Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and scientific ... | Mathematical discipline | 0.808826 |
23,402 | These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. Al... | Mathematical discipline | 0.808826 |
23,403 | Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysi... | Mathematical discipline | 0.808826 |
23,404 | In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining m... | Mathematical discipline | 0.808825 |
23,405 | This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted... | Mathematical discipline | 0.808825 |
23,406 | It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these ... | Mathematical discipline | 0.808825 |
23,407 | The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject ... | Mathematical discipline | 0.808825 |
23,408 | The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that element... | Mathematical discipline | 0.808825 |
23,409 | The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in comm... | Mathematical discipline | 0.808825 |
23,410 | Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.Statistical theory studies decision problems such as minimizing the risk (expected loss) of a st... | Mathematical discipline | 0.808825 |
23,411 | The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments. The design of a statistical sample or experi... | Mathematical discipline | 0.808825 |
23,412 | The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.Discrete mathematics includes: Combinatorics, the art of enumerating mathematical objects that satisfy some given constrai... | Mathematical discipline | 0.808825 |
23,413 | Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms—especially their implementation and computational... | Mathematical discipline | 0.808825 |
23,414 | This is particularly acute when the results of modeling influence political decisions; the existence of contradictory models could allow nations to choose the most favorable model.Genotype evolution can be modeled with the Hardy-Weinberg principle.Phylogeography uses probabilistic models.Medicine uses statistical hypot... | Mathematical discipline | 0.808825 |
23,415 | Biology uses probability extensively – for example, in ecology or neurobiology. Most of the discussion of probability in biology, however, centers on the concept of evolutionary fitness.Ecology heavily uses modeling to simulate population dynamics, study ecosystems such as the predator-prey model, measure pollution dif... | Mathematical discipline | 0.808825 |
23,416 | (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.) Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, ... | Mathematical discipline | 0.808825 |
23,417 | The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algeb... | Mathematical discipline | 0.808825 |
23,418 | Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas. Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic e... | Mathematical discipline | 0.808825 |
23,419 | Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in the title of his main treatise. Al... | Mathematical discipline | 0.808825 |
23,420 | Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. | Mathematical discipline | 0.808825 |
23,421 | Structural geology and climatology use probabilistic models to predict the risk of natural catastrophes. Similarly, meteorology, oceanography, and planetology also use mathematics due to their heavy use of models. | Mathematical discipline | 0.808825 |
23,422 | In the controversial book Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of scientific terminology, particularly from mathematics or physics, in the social sciences. The study of complex systems (evolution of unemployment, business capital, demographic evolution of a population, ... | Mathematical discipline | 0.808825 |
23,423 | Areas of mathematics used in the social sciences include probability/statistics and differential equations (stochastic or deterministic). These areas are used in fields such as sociology, psychology, economics, finance, and linguistics. The fundamental postulate of mathematical economics is that of the rational individ... | Mathematical discipline | 0.808825 |
23,424 | If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. | Mathematical discipline | 0.808825 |
23,425 | Given a topological space X, a topological distance is given by d(x, A) = 0 if x ∈ A, and d(x, A) = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances. | Approach space | 0.808823 |
23,426 | Every ∞pq-metric space (X, d) can be distanced to (X, d), as described at the beginning of the definition. Given a set X, the discrete distance is given by d(x, A) = 0 if x ∈ A and d(x, A) = ∞ if x ∉ A. The induced topology is the discrete topology. Given a set X, the indiscrete distance is given by d(x, A) = 0 if A is... | Approach space | 0.808823 |
23,427 | Given a metric space (X, d), or more generally, an extended pseudoquasimetric (which will be abbreviated ∞pq-metric here), one can define an induced map d: X × P(X) → by d(x, A) = inf{d(x, a): a ∈ A}. With this example in mind, a distance on X is defined to be a map X × P(X) → satisfying for all x in X and A, B ⊆ X, ... | Approach space | 0.808823 |
23,428 | In topology, a branch of mathematics, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989, in a series of papers on approach theory between 1988 and 1995. | Approach space | 0.808823 |
23,429 | Prior to Megiddo's result showing that the smallest-circle problem may be solved in linear time, several algorithms of higher complexity appeared in the literature. A naive algorithm solves the problem in time O(n4) by testing the circles determined by all pairs and triples of points. An algorithm of Chrystal and Peirc... | Bounding circle | 0.808821 |
23,430 | The eigenvalues of the structure tensor play a significant role in many image processing algorithms, for problems like corner detection, interest point detection, and feature tracking. The structure tensor also plays a central role in the Lucas-Kanade optical flow algorithm, and in its extensions to estimate affine sha... | Structure Tensor | 0.80882 |
23,431 | The elegance of the complex representation stems from that the two components of the structure tensor can be obtained as averages and independently. In turn, this means that κ 20 {\displaystyle \kappa _{20}} and κ 11 {\displaystyle \kappa _{11}} can be used in a scale space representation to describe the evidence for p... | Structure Tensor | 0.80882 |
23,432 | In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is... | Structure Tensor | 0.80882 |
23,433 | This branch of systems medicine, going back to the traditions of Ludwig von Bertalanffy's systems theory and biological cybernetics is a top-down strategy that starts with the description of large, complex processing structures (i.e. neural networks, feedback loops and other motifs) and tries to find sufficient and nec... | Systems medicine | 0.808813 |
23,434 | Systems medicine is an interdisciplinary field of study that looks at the systems of the human body as part of an integrated whole, incorporating biochemical, physiological, and environment interactions. Systems medicine draws on systems science and systems biology, and considers complex interactions within the human b... | Systems medicine | 0.808813 |
23,435 | In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} and x + 2 x 2 − 3 {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} . Algebraic fractions are subject to the same law... | Algebraic fraction | 0.808809 |
23,436 | If the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction or simply rational fraction. Rational fractions are also known as rational expressions. A rational fraction f ( x ) g ( x ) {\displaystyle {\tfrac {f(x)}{g(x)}}} is called proper if deg f ( x ) < deg g ( x ) {... | Algebraic fraction | 0.808809 |
23,437 | A fraction is in lowest terms if the only factor common to the numerator and the denominator is 1. An expression which is not in fractional form is an integral expression. An integral expression can always be written in fractional form by giving it the denominator 1. A mixed expression is the algebraic sum of one or mo... | Algebraic fraction | 0.808809 |
23,438 | In the algebraic fraction a b {\displaystyle {\tfrac {a}{b}}} , the dividend a is called the numerator and the divisor b is called the denominator. The numerator and denominator are called the terms of the algebraic fraction. A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction.... | Algebraic fraction | 0.808809 |
23,439 | In 2015, Polina Anikeeva's research group demonstrated that similar concept can enhance the neuronal signals in mammalian brain. In 2021, Jinwoo Cheon's research group has successfully developed the magneto-mechanical genetics which uses magnetic stimulation derived mechanical force in mammalian. In this study, magneti... | Magnetogenetics | 0.808807 |
23,440 | On the other hand, nanoparticles have been suggested as possible candidates that can function as the transducer of magnetic field to the stimulus cue. Based on this concept, next generation of magnetogenetics technique is being developed. In 2010, Arnd Pralle and colleges showed that the first in vivo magneto-thermal s... | Magnetogenetics | 0.808807 |
23,441 | To date, several technologies are adapted with genetics (e.g. optogenetics, chemogenetics, etc.), and each technology has strengths and limits. For example, optogenetics has advantages in that it can provide temporally and spatially precise manipulation of neurons. | Magnetogenetics | 0.808807 |
23,442 | The development of genetic technologies that can modulate cellular processes has greatly contributed to biological research. A representative example is the development of optogenetics, which is a neuromodulation tool kit that involves light-sensitive proteins such as opsins. This progress provided the grounds for a br... | Magnetogenetics | 0.808807 |
23,443 | The superparamagnetic nanocomposites generate thermal energy under alternating magnetic fields. This process is closely related to the Neel relaxation theory of magnetic energy. Generated heat from magnetic nanocomposites is generally used for the magnetic hyperthermia. In magnetogenetics, the temperature increase caus... | Magnetogenetics | 0.808807 |
23,444 | In 1980, Young and colleagues have shown that magnetic fields with magnitudes in millitesla range are able to penetrate into the brain without attenuation of the signal or side effects because of the negligible magnetic susceptibility and low conductivity of biological tissue. Early attempts to manipulate electrical si... | Magnetogenetics | 0.808807 |
23,445 | The working mechanism of magnetogenetics can be classified by how magnetic stimulation is mediated to activate bio systems, such as ion channels. Heat mediated (magneto-thermal genetics) activation and force mediated activation (magneto-mechanical genetics) are two main categories of magnetogenetics. In magneto-mechani... | Magnetogenetics | 0.808807 |
23,446 | There are many ways to utilize magnetogenetics in research. One of the main challenges of magnetogenetics is in introducing the magnetic component into a specific system (i.e. protein or cells) or regions (e.g. mouse brain region) of the organism in question. One approach involves synthesizing the fusion protein of ion... | Magnetogenetics | 0.808807 |
23,447 | The field of magnetogenetics is a relatively new area of research that has the possibility to further the scientific understanding of how specific cells and genes contribute to the function of biological tissues such as neural circuits in vivo. It also has great potential for clinical applications as many medical tools... | Magnetogenetics | 0.808807 |
23,448 | Experiments can be devised where the magnetic stimulation triggers some sort of cellular (in vitro) or behavioral (in vivo) response. A U.S. patent awarded to Henricus Loos specifies that such experiments are envisioned to be utilized in animal control, noting that "The nervous system of mammals is similar to that of h... | Magnetogenetics | 0.808807 |
23,449 | The currently available magnetogenetics techniques combine targeted magnetic particle delivery with magnetic stimulation. This allows for the accurate spatial control of the targeted intervention (i.e. inhibition or excitation of the target neurons in target brain regions). The temporal precision seems to vary dependin... | Magnetogenetics | 0.808807 |
23,450 | One of the main issues in magnetogenetics is related the physical properties of the ferritin. The ferritin is composed of 24 subunits of protein complex and a small iron oxide core. The core of the ferritin is in the form of ferric hydroxide which has antiferromagnetic properties. | Magnetogenetics | 0.808807 |
23,451 | Magnetogenetics, freely moving mammalian behavior, and immunohistochemistry have been integrated to investigate the efficacy and validity of magnetogenetics as a neuroscientific tool. Magnetogenetic activation of the mice motor cortex has been shown to be effective in inducing locomotor changes in freely moving mice. T... | Magnetogenetics | 0.808807 |
23,452 | As such, magnetogenetics is a cellular modulation method that uses a combination of techniques from magnetism and genetics to control activities of individual cells in living tissue – even within freely moving animals and people. This technique is comparable to optogenetics, which is the manipulation of cell behavior u... | Magnetogenetics | 0.808807 |
23,453 | Magnetogenetics refers to a biological technique that involves the use of magnetic fields to remotely control cell activity. In most cases, magnetic stimulation is transformed into either force (magneto-mechanical genetics) or heat (magneto-thermal genetics), which depends on the applied magnetic field. Therefore, cell... | Magnetogenetics | 0.808807 |
23,454 | Visible-light astronomy has existed as long as people have been looking up at the night sky, although it has since improved in its observational capabilities since the invention of the telescope, which is commonly credited to Hans Lippershey, a German-Dutch spectacle-maker, although Galileo played a large role in the d... | Visible-light astronomy | 0.8088 |
23,455 | Visible-light astronomy encompasses a wide variety of observations via telescopes that are sensitive in the range of visible light (optical telescopes). Visible-light astronomy is part of optical astronomy, and differs from astronomies based on invisible types of light in the electromagnetic radiation spectrum, such as... | Visible-light astronomy | 0.8088 |
23,456 | The Hubble Space Telescope is a space telescope created by NASA, and was launched into low Earth orbit in 1990. It is still in operation today. The Hubble Space Telescope's four main instruments observe in the near ultraviolet, visible, and near infrared spectra. Hubble's images are some of the most detailed images eve... | Visible-light astronomy | 0.8088 |
23,457 | Peptide synthesis can be easily conducted by the established method of solid-phase chemistry in grams or kilogram quantities. The d-isomer conformation can be used for peptide synthesis. Nanostructures can be made by dissolving dipeptides in 1,1,1,3,3, 3-hexafluoro-2-propanol at 100 mg/ml and then diluting it with wate... | Self-assembling peptides | 0.8088 |
23,458 | With these integers uniquely determined for each such α we can define the height of α to be H ( α ) = max { | x | , | y | , | z | } . {\displaystyle H(\alpha )=\max\{|x|,|y|,|z|\}.} The theorem then says that for any real number ξ which is neither rational nor a quadratic irrational, we can find infinitely many real nu... | Davenport–Schmidt theorem | 0.8088 |
23,459 | In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irra... | Davenport–Schmidt theorem | 0.8088 |
23,460 | In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents th... | Explained sum of squares | 0.808796 |
23,461 | In the power systems analysis field of electrical engineering, a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit do not change when they are referred from one side of a transformer to the other. Thi... | Per-unit system | 0.808795 |
23,462 | In geometry, specifically projective geometry, a blocking set is a set of points in a projective plane that every line intersects and that does not contain an entire line. The concept can be generalized in several ways. Instead of talking about points and lines, one could deal with n-dimensional subspaces and m-dimensi... | Blocking set | 0.808792 |
23,463 | Points on the line X = 0 have coordinates of the form (0,1,a), and those on the line Y = 0 have coordinates of the form (1,0,b). Points of the line X = Y have coordinates which may be written as (1,1,c). Three points, one from each of these lines, are collinear if and only if a = b + c. By selecting all the points on t... | Blocking set | 0.808792 |
23,464 | By choosing all of the points where a, b and c are nonzero squares of GF(q), the condition in the definition of a projective triangle is satisfied.Theorem: In PG(2,q) with q even, there exists a projective triad of side (q + 2)/2 which is a blocking set of size (3q + 2)/2. The construction is similar to the above, but ... | Blocking set | 0.808792 |
23,465 | In any projective plane of order n (each line contains n + 1 points), the points on the lines forming a triangle without the vertices of the triangle (3(n - 1) points) form a minimal blocking set (if n = 2 this blocking set is trivial) which in general is not a committee. Another general construction in an arbitrary pr... | Blocking set | 0.808792 |
23,466 | In a projective plane a complete k-arc is a set of k points, no three collinear, which can not be extended to a larger arc (thus, every point not on the arc is on a secant line of the arc–a line meeting the arc in two points.) Theorem: Let K be a complete k-arc in Π = PG(2,q) with k < q + 2. The dual in Π of the set of... | Blocking set | 0.808792 |
23,467 | Moreover, if this upper bound is reached, then n is necessarily a square and the blocking set consists of the points of some unital embedded in π. When n is not a square less can be said about the smallest sized nontrivial blocking sets. One well known result due to Aart Blokhuis is:Theorem: A nontrivial blocking set i... | Blocking set | 0.808792 |
23,468 | Thus the Chow group of an elliptic curve X {\displaystyle X} is closely related to the group X ( k ) {\displaystyle X(k)} of k {\displaystyle k} -rational points of X {\displaystyle X} . When k {\displaystyle k} is a number field, X ( k ) {\displaystyle X(k)} is called the Mordell–Weil group of X {\displaystyle X} , an... | Cycle class map | 0.808791 |
23,469 | For other algebraic varieties, Chow groups can have richer behavior. For example, let X {\displaystyle X} be an elliptic curve over a field k {\displaystyle k} . Then the Chow group of zero-cycles on X {\displaystyle X} fits into an exact sequence 0 → X ( k ) → C H 0 ( X ) → Z → 0. {\displaystyle 0\rightarrow X(k)\righ... | Cycle class map | 0.808791 |
23,470 | The Chow ring of projective space P n {\displaystyle \mathbb {P} ^{n}} over any field k {\displaystyle k} is the ring C H ∗ ( P n ) ≅ Z / ( H n + 1 ) , {\displaystyle CH^{*}(\mathbb {P} ^{n})\cong \mathbf {Z} /(H^{n+1}),} where H {\displaystyle H} is the class of a hyperplane (the zero locus of a single linear functio... | Cycle class map | 0.808791 |
23,471 | β-Carbon elimination (beta-carbon elimination) is a type of reaction in organometallic chemistry wherein an allyl ligand bonded to a metal center is broken into the corresponding metal-bonded alkyl (aryl) ligand and an alkene. It is a subgroup of elimination reactions. Though less common and less understood than β-hydr... | Β-Carbon elimination | 0.80879 |
23,472 | In terms of kinetics, steric effects of ligands could play a role in increasing the energy barrier of β-H elimination relative to β-alkyl elimination, specifically when the ligand is Cp*. A model was proposed to illustrate this effect: In both β-methyl elimination (A) and β-hydride elimination (B), the transferring gro... | Β-Carbon elimination | 0.80879 |
23,473 | Let us take a second look at Euclid's theorem – Book IX, Proposition 20: Prime numbers are more than any assigned multitude of prime numbers.We may read the statement as saying that for every finite list of primes, there is another prime not on that list, which is arguably closer to and in the same spirit as Euclid's o... | Proof by Contradiction | 0.808784 |
23,474 | NASA is planning the largest ever balloon observatory in 2020 with a 400 foot balloon and 2.5 metre telescope. ASTHROS (Astrophysics Stratospheric Telescope for High Spectral Resolution Observations at Submillimeter-wavelengths) will launch from the Antarctic and is envisioned to last three weeks. | Flying telescope | 0.808784 |
23,475 | Like the theory of algebraically closed fields, the theory DCF0 of differentially closed fields of characteristic 0 eliminates quantifiers. The geometric content of this statement is that the projection of a d-constructible set is d-constructible. It also eliminates imaginaries, is complete, and model complete. In char... | Differentially closed field | 0.808782 |
23,476 | In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues for differential equations o... | Differentially closed field | 0.808782 |
23,477 | Any differentially perfect field K has a differential closure, a prime model extension, which is differentially closed. Shelah showed that the differential closure is unique up to isomorphism over K. Shelah also showed that the prime differentially closed field of characteristic 0 (the differential closure of the ratio... | Differentially closed field | 0.808782 |
23,478 | In characteristic 0 this implies that it is algebraically closed, but in characteristic p>0 differentially closed fields are never algebraically closed. Unlike the complex numbers in the theory of algebraically closed fields, there is no natural example of a differentially closed field. | Differentially closed field | 0.808782 |
23,479 | The differential Nullstellensatz is the analogue in differential algebra of Hilbert's nullstellensatz. A differential ideal or ∂-ideal is an ideal closed under ∂. An ideal is called radical if it contains all roots of its elements.Suppose that K is a differentially closed field of characteristic 0. . Then Seidenberg's ... | Differentially closed field | 0.808782 |
23,480 | The Kolchin topology on K m is defined by taking sets of solutions of systems of differential equations over K in m variables as basic closed sets. Like the Zariski topology, the Kolchin topology is Noetherian. A d-constructible set is a finite union of closed and open sets in the Kolchin topology. Equivalently, a d-co... | Differentially closed field | 0.808782 |
23,481 | The hydroamination reaction is approximately thermochemically neutral. The reaction however suffers from a high activation barrier, perhaps owing to the repulsion of the electron-rich substrate and the amine nucleophile. The intermolecular reaction also is accompanied by highly negative changing entropy, making it unfa... | Hydroamination | 0.808779 |
23,482 | In 1992, Marks developed the first chiral hydroamination catalysts by using a chiral auxiliary, which were the first hydroamination catalysts to favor only one specific stereoisomer. Chiral auxiliaries on the metallocene ligands were used to dictate the stereochemistry of the product. The first non-metallocene chiral c... | Hydroamination | 0.808779 |
23,483 | In organic chemistry, hydroamination is the addition of an N−H bond of an amine across a carbon-carbon multiple bond of an alkene, alkyne, diene, or allene. In the ideal case, hydroamination is atom economical and green. Amines are common in fine-chemical, pharmaceutical, and agricultural industries. | Hydroamination | 0.808779 |
23,484 | Open XML Paper Specification (also referred to as OpenXPS) is an open specification for a page description language and a fixed-document format. Microsoft developed it as the XML Paper Specification (XPS). In June 2009, Ecma International adopted it as international standard ECMA-388.The format is intentionally restric... | Comparison of e-book formats | 0.808779 |
23,485 | The concept smart for a smart physical object simply means that it is active, digital, networked, can operate to some extent autonomously, is reconfigurable and has local control of the resources it needs such as energy, data storage, etc. Note, a smart object does not necessarily need to be intelligent as in exhibitin... | Smart object | 0.808777 |
23,486 | The opposite approach to smart objects is 'plain' objects that do not provide this information. The additional information provided by this concept enables far more general interaction schemes, and can greatly simplify the planner of an artificial intelligence agent.In contrast to smart virtual objects used in virtual ... | Smart object | 0.808777 |
23,487 | That is, if the three vertices are L = ( x L , y L ) , {\displaystyle L=(x_{L},y_{L}),} M = ( x M , y M ) , {\displaystyle M=(x_{M},y_{M}),} and N = ( x N , y N ) , {\displaystyle N=(x_{N},y_{N}),} then the centroid (denoted C here but most commonly denoted G in triangle geometry) is The centroid is therefore at 1 3: 1... | Geometric center | 0.808777 |
23,488 | Nonetheless, the center of gravity of figures was studied extensively in Antiquity; Bossut credits Archimedes (287–212 BCE) with being the first to find the centroid of plane figures, although he never defines it. A treatment of centroids of solids by Archimedes has been lost.It is unlikely that Archimedes learned the ... | Geometric center | 0.808777 |
23,489 | In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in n-dimensional Euclidean space.In geometry, one often assumes unifor... | Geometric center | 0.808777 |
23,490 | Some dialects of BASIC supported matrices and matrix operations, which can be used to solve sets of simultaneous linear algebraic equations. These dialects would directly support matrix operations such as assignment, addition, multiplication (of compatible matrix types), and evaluation of a determinant. Many microcompu... | ANSI BASIC | 0.808775 |
23,491 | If c = ħ = 1, where c is the speed of light and ħ is the reduced Planck constant, and a suitable fixed unit of energy is chosen, then all quantities of time T, length L and mass M can be expressed (dimensionally) as a power of energy E, because length, mass and time can be expressed using speed v, action S, and energy ... | Orientational analysis | 0.808774 |
23,492 | Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force. | Orientational analysis | 0.808774 |
23,493 | While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change). Cons... | Orientational analysis | 0.808774 |
23,494 | More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches, which optimize a function along one... | Numerical optimization | 0.808772 |
23,495 | Mathematical optimization is used in much modern controller design. High-level controllers such as model predictive control (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plan... | Numerical optimization | 0.808772 |
23,496 | S-matrix theory was a proposal for replacing local quantum field theory as the basic principle of elementary particle physics. It avoided the notion of space and time by replacing it with abstract mathematical properties of the S-matrix. In S-matrix theory, the S-matrix relates the infinite past to the infinite future ... | Landau principle | 0.808772 |
23,497 | Suppose we want to solve the following stochastic optimization problem where g ( θ ) = E {\textstyle g(\theta )=\operatorname {E} } is differentiable and convex, then this problem is equivalent to find the root θ ∗ {\displaystyle \theta ^{*}} of ∇ g ( θ ) = 0 {\displaystyle \nabla g(\theta )=0} . Here Q ( θ , X ) {\... | Robbins–Monro algorithm | 0.808769 |
23,498 | In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f n = f n − 1 ± f n − 2 {\displaystyle f_{n}=f_{n-1}\pm f_{n-2}} , where the signs + or − are chosen at random with equal probability 1 2 {\displaystyle {\tfrac {1}{2}}} , independently fo... | Embree–Trefethen constant | 0.808768 |
23,499 | It gives a simple orbital picture equivalent to Lewis structures. Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. | Sp hybridization | 0.808767 |
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