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22,200 | It has been shown to give true atomic resolution in ultra-high vacuum (UHV) and, more recently, in liquid environments. High resolution AFM is comparable in resolution to scanning tunneling microscopy and transmission electron microscopy. AFM can also be combined with a variety of optical microscopy and spectroscopy te... | Atomic Force Microscope | 0.809621 |
22,201 | The AFM was invented by IBM scientists in 1985. The precursor to the AFM, the scanning tunneling microscope (STM), was developed by Gerd Binnig and Heinrich Rohrer in the early 1980s at IBM Research – Zurich, a development that earned them the 1986 Nobel Prize for Physics. Binnig invented the atomic force microscope an... | Atomic Force Microscope | 0.809621 |
22,202 | These decision tree models, defined by Rabin and Reingold, are often used for proving lower bounds in computational geometry. For example, Ben-Or showed that element uniqueness (the task of computing f: R n → { 0 , 1 } {\displaystyle f:\mathbb {R} ^{n}\to \{0,1\}} , where f ( x ) {\displaystyle f(x)} is 0 if and only i... | Sensitivity conjecture | 0.809619 |
22,203 | From its definition, linear decision trees can only specify functions f {\displaystyle f} whose fibers can be constructed by taking unions and intersections of half-spaces. Algebraic decision trees are a generalization of linear decision trees that allow the test functions to be polynomials of degree d {\displaystyle d... | Sensitivity conjecture | 0.809619 |
22,204 | In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actu... | Configuration space (physics) | 0.809618 |
22,205 | For a robotic arm consisting of numerous rigid linkages, the configuration space consists of the location of each linkage (taken to be a rigid body, as in the section above), subject to the constraints of how the linkages are attached to each other, and their allowed range of motion. Thus, for n {\displaystyle n} linka... | Configuration space (physics) | 0.809618 |
22,206 | In quantum mechanics, the analogous concept is called the state space. A rather different set of formalisms and notation are used in this case. The analog of a "point particle" becomes a single point in C P 1 {\displaystyle \mathbb {CP} ^{1}} , the complex projective line, also known as the Bloch sphere. It is complex,... | Configuration space (physics) | 0.809618 |
22,207 | In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountered in models of turbulent flows, particularly the Reynolds-averaged Navie... | Reynolds operator | 0.809617 |
22,208 | In functional analysis a Reynolds operator is a linear operator R acting on some algebra of functions φ, satisfying the Reynolds identity R ( ϕ ψ ) = R ( ϕ ) R ( ψ ) + R ( ( ϕ − R ( ϕ ) ) ( ψ − R ( ψ ) ) ) for all ϕ , ψ {\textstyle R(\phi \psi )=R(\phi )R(\psi )+R\left(\left(\phi -R(\phi )\right)\left(\psi -R(\psi )\ri... | Reynolds operator | 0.809617 |
22,209 | Reynolds operators are used in fluid dynamics, functional analysis, and invariant theory, and the notation and definitions in these areas differ slightly. A Reynolds operator acting on φ is sometimes denoted by R ( ϕ ) , P ( ϕ ) , ρ ( ϕ ) , ⟨ ϕ ⟩ {\displaystyle R(\phi ),P(\phi ),\rho (\phi ),\langle \phi \rangle } or ϕ... | Reynolds operator | 0.809617 |
22,210 | Reynolds operators are often given by projecting onto an invariant subspace of a group action. The "Reynolds operator" considered by Reynolds (1895) was essentially the projection of a fluid flow to the "average" fluid flow, which can be thought of as projection to time-invariant flows. Here the group action is given b... | Reynolds operator | 0.809617 |
22,211 | Kernel functions can be grouped under another criterion: low-level and high-level ones. The low-level operations include constructing curves and surfaces (canonical objects, NURBS, offset curves and surfaces, and so on), projecting points and curves on surfaces, intersecting and extending curves and surfaces, modifying... | Russian Geometric Kernel | 0.809614 |
22,212 | (Note that the Simple Examples below are not quotes from the Focal Points, but are based on the descriptions of activities found in the Focal Points.) The Focal Points define not only the recommended curriculum emphases, but also the ways in which students should learn them, as in the PSSM. An example of a complete des... | NCTM standards | 0.809609 |
22,213 | Geometry: The overall goals for learning geometry are to "analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; specify locations and describe spatial relationships using coordinate geometry and other representational syst... | NCTM standards | 0.809609 |
22,214 | for easy calculation. Formally, this is described in algebraic notation like this: (19 + 1) + (15 − 1) = x, but even a young student might use this technique without calling it algebra. The PSSM recommends that all students complete pre-algebra coursework by the end of eighth grade and take an algebra class during high... | NCTM standards | 0.809609 |
22,215 | The resulting document sets forth a set of six principles (Equity, Curriculum, Teaching, Learning, Assessment, and Technology) that describe NCTM's recommended framework for mathematics programs, and ten general strands or standards that cut across the school mathematics curriculum. These strands are divided into mathe... | NCTM standards | 0.809609 |
22,216 | These skills allow students to make sense of critical information, such as medical statistics and the results of political surveys. These skills are increasingly important as statistical data are used selectively by manufacturers to promote products. While young students learn simple skills such as ways to represent th... | NCTM standards | 0.809609 |
22,217 | (See van Hiele model.) The PSSM promotes the appropriate use of physical objects, drawings, and computer software for teaching geometry. Measurement: Measurement skills have many practical applications, as well as providing opportunities for advancing mathematical understand and for practicing other mathematical skills... | NCTM standards | 0.809609 |
22,218 | Very simple algebra skills are often taught to young children. For example, a student might convert an addition equation such as 19 + 15 = ? into a simpler equation, 20 + 14 = ? | NCTM standards | 0.809609 |
22,219 | Number and Operations: These are the fundamental basis of all mathematics, and teaching this critical area is the first content standard. All students must be taught to "understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they rela... | NCTM standards | 0.809609 |
22,220 | On 14 October 2017, astrophysics blogger Steinn Sigurðsson publicly revealed that ScienceBlogs was due to be shut down, and David Gorski, author of the "Respectful Insolence" blog under his pseudonym Orac, stated that ScienceBlogs had "barely existed as an entity for a few years". Astrophysics blogger Ethan Siegel repo... | ScienceBlogs | 0.809607 |
22,221 | Knapsack cryptosystems are cryptosystems whose security is based on the hardness of solving the knapsack problem. They remain quite unpopular because simple versions of these algorithms have been broken for several decades. However, that type of cryptosystem is a good candidate for post-quantum cryptography.The most fa... | Knapsack cryptosystems | 0.809606 |
22,222 | Probability amplitudes have special significance because they act in quantum mechanics as the equivalent of conventional probabilities, with many analogous laws, as described above. For example, in the classic double-slit experiment, electrons are fired randomly at two slits, and the probability distribution of detecti... | Probability amplitude | 0.809605 |
22,223 | Rock paper scissors incorporated into an evolutionary game has been used for modelling natural processes in the study of ecology. Using experimental economics methods, scientists have used RPS games to test human social evolutionary dynamical behaviours in laboratories. The social cyclic behaviours, predicted by evolut... | Evolutionary Game Theory | 0.809604 |
22,224 | Geographic factors in evolution include gene flow and horizontal gene transfer. Spatial game models represent geometry by putting contestants in a lattice of cells: contests take place only with immediate neighbours. Winning strategies take over these immediate neighbourhoods and then interact with adjacent neighbourho... | Evolutionary Game Theory | 0.809604 |
22,225 | In the context of animal behavior, this usually means such strategies are programmed and heavily influenced by genetics, thus making any player or organism's strategy determined by these biological factors.Evolutionary games are mathematical objects with different rules, payoffs, and mathematical behaviours. Each "game... | Evolutionary Game Theory | 0.809604 |
22,226 | This implements a distribution of bids for a resource of specific value V, where the bid for any specific contest is chosen at random from that distribution. The distribution (an ESS) can be computed using the Bishop-Cannings theorem, which holds true for any mixed-strategy ESS. The distribution function in these conte... | Evolutionary Game Theory | 0.809604 |
22,227 | Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as ... | Evolutionary Game Theory | 0.809604 |
22,228 | Maynard Smith realised that an evolutionary version of game theory does not require players to act rationally—only that they have a strategy. The results of a game show how good that strategy was, just as evolution tests alternative strategies for the ability to survive and reproduce. In biology, strategies are genetic... | Evolutionary Game Theory | 0.809604 |
22,229 | Fibre Channel – Arbitrated Loop (FC-AL) initiators use the LIP (Loop Initialization Protocol) to interrogate each device port for its WWN (World Wide Name). For iSCSI, because of the unlimited scope of the (IP) network, the process is quite complicated. These discovery processes occur at power-on/initialization time an... | Small Computer Systems Interface | 0.809602 |
22,230 | A disjunctive population, in ecology, is a colony of plants or animals, whose geographical locus is severed from the continuous range of the bulk of the species distribution. Although a disjunctive population may sometimes occur on an island, which creates physical separation via water, a large percentage of disjunctiv... | Disjunctive population | 0.809602 |
22,231 | The complexity (in terms of calculating the forward and inverse kinematics) of the chain is determined by the following factors: Its topology: a serial chain, a parallel manipulator, a tree structure, or a graph. Its geometrical form: how are neighbouring joints spatially connected to each other?Explanation Two or more... | Kinematic chain | 0.8096 |
22,232 | The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form algebraic equations that are solved to determine the configuration of the chain associated with specific values of input parameters, called degrees of freedom. The const... | Kinematic chain | 0.8096 |
22,233 | In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system. As the word chain suggests, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open... | Kinematic chain | 0.8096 |
22,234 | Several studies propose either non-intelligent estimators which depend on the model, such as the extended Kalman filter (EKF) and Luenberger's observer, or intelligent estimators such as cascade-forward neural network (CFNN) and quasi-Newton BFGS backpropagation . | Brushed motor | 0.809599 |
22,235 | The Process Window Index is calculated as the worst case (i.e. highest number) in the set of thermal profile data. For each profile statistic the percentage used of the respective process window is calculated, and the worst case (i.e. highest percentage) is the PWI. For example, a thermal profile with three thermocoupl... | Control limits | 0.809598 |
22,236 | Keeping track of non-decimal notation also explains the need to distinguish numbers (values) from the numerals that represent them.Topics introduced in the New Math include set theory, modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, and abstract algebra.All of... | New Math | 0.809595 |
22,237 | After the Sputnik launch in 1957, the U.S. National Science Foundation funded the development of several new curricula in the sciences, such as the Physical Science Study Committee high school physics curriculum, Biological Sciences Curriculum Study in biology, and CHEM Study in chemistry. Several mathematics curriculu... | New Math | 0.809595 |
22,238 | In an effort to learn the material, many parents attended their children's classes. In the end, it was concluded that the experiment was not working, and New Math fell out of favor before the end of the 1960s, though it continued to be taught for years thereafter in some school districts.In the Algebra preface of his b... | New Math | 0.809595 |
22,239 | Some compromises have since been required, given that discrete mathematics is the basic language of computing. Teaching in the USSR did not experience such extreme upheavals, while being kept in tune, both with the applications and academic trends: "Under A. N. Kolmogorov, the mathematics committee declared a reform of... | New Math | 0.809595 |
22,240 | In the broader context, reform of school mathematics curricula was also pursued in European countries, such as the United Kingdom (particularly by the School Mathematics Project), and France due to concerns that mathematics as taught in schools was becoming too disconnected from mathematics research, in particular that... | New Math | 0.809595 |
22,241 | The committee found the type of reform in progress in Western countries to be unacceptable; for example, no special topic for sets was accepted for inclusion in school textbooks. Transformation approaches were accepted in teaching geometry, but not to such sophisticated level presented in the textbook produced by Vlad... | New Math | 0.809595 |
22,242 | : 237–238 An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component de... | Stray capacitance | 0.809592 |
22,243 | In the past, alternate subunits were used in old historical texts; "mf" and "mfd" for microfarad (µF); "mmf", "mmfd", "pfd", "µµF" for picofarad (pF); but are now considered obsolete.Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are... | Stray capacitance | 0.809592 |
22,244 | Life Sciences Greenhouse of Central Pennsylvania (LSGPA) is a biotechnology initiative and non-profit organization based in Harrisburg, Pennsylvania. It was founded in 2001. It focuses on in the advancement of life sciences through technology to improve the healthcare and economic opportunities of Pennsylvanians. | Life Sciences Greenhouse of Central Pennsylvania | 0.809591 |
22,245 | If a finite partially ordered set is totally ordered, then it has only one linear extension, but otherwise it will have more than one. The 1/3–2/3 conjecture states that one can choose two elements x and y such that, among this set of possible linear extensions, between 1/3 and 2/3 of them place x earlier than y, and s... | 1/3–2/3 conjecture | 0.80959 |
22,246 | Organic Letters is a biweekly peer-reviewed scientific journal covering research in organic chemistry. It was established in 1999 and is published by the American Chemical Society. In 2014, the journal moved to a hybrid open access publishing model. The founding editor-in-chief was Amos Smith. The current editor-in-chi... | Organic Letters | 0.809585 |
22,247 | Phosphoamino acid analysis, or PAA, is an experimental technique used in molecular biology to determine which amino acid or acids are phosphorylated in a protein. | Phosphoamino acid analysis | 0.809582 |
22,248 | Observations of atmospheric chemistry are made in observatories such as that on Mauna Loa and on mobile platforms such as aircraft (e.g. the UK's Facility for Airborne Atmospheric Measurements), ships and balloons. Observations of atmospheric composition are increasingly made by satellites with important instruments su... | Atmospheric Chemistry | 0.809582 |
22,249 | Observations of atmospheric chemistry are essential to our understanding. Routine observations of chemical composition tell us about changes in atmospheric composition over time. One important example of this is the Keeling Curve - a series of measurements from 1958 to today which show a steady rise in of the concentra... | Atmospheric Chemistry | 0.809582 |
22,250 | Zero shot learning has been applied to the following fields: image classification semantic segmentation image generation object detection natural language processing computational biology | Zero-shot learning | 0.809581 |
22,251 | This approach was also extended to multilingual domains, fine entity typing and other problems. Moreover, beyond relying solely on representations, the computational approach has been extended to depend on transfer from other tasks, such as textual entailment and question answering.The original paper also points out th... | Zero-shot learning | 0.809581 |
22,252 | In natural language processing, the key technical direction developed builds on the ability to "understand the labels"—represent the labels in the same semantic space as that of the documents to be classified. This supports the classification of a single example without observing any annotated data, the purest form of ... | Zero-shot learning | 0.809581 |
22,253 | The first paper on zero-shot learning in natural language processing appeared in 2008 at the AAAI’08, but the name given to the learning paradigm there was dataless classification. The first paper on zero-shot learning in computer vision appeared at the same conference, under the name zero-data learning. The term zero-... | Zero-shot learning | 0.809581 |
22,254 | While this approach was used mostly in computer vision, there are some examples for it also in natural language processing. Learning from textual description. As pointed out above, this has been the key direction pursued in natural language processing. | Zero-shot learning | 0.809581 |
22,255 | Zero-shot learning (ZSL) is a problem setup in deep learning where, at test time, a learner observes samples from classes which were not observed during training, and needs to predict the class that they belong to. Zero-shot methods generally work by associating observed and non-observed classes through some form of au... | Zero-shot learning | 0.809581 |
22,256 | In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single ... | Line-plane intersection | 0.80958 |
22,257 | In the ray tracing method of computer graphics a surface can be represented as a set of pieces of planes. The intersection of a ray of light with each plane is used to produce an image of the surface. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called trian... | Line-plane intersection | 0.80958 |
22,258 | The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz.The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. | K-k-m theorem | 0.809578 |
22,259 | According to Zang, numerical simulation can be aided by considering the skew-symmetric form for the advection operator. where and u {\displaystyle \mathbf {u} } is the same as above. Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in nume... | Advection equation | 0.809578 |
22,260 | In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is also a fluid. The properties that are carried with the advected substance a... | Advection equation | 0.809578 |
22,261 | Algae biofuels such as butanol and methanol are produced both at laboratory and commercial scales. This method has benefited from the development of synthetic biology, which is also being explored by the J. Craig Venter Institute to produce a synthetic organism capable of biofuel production. In 2017, an efficient proce... | Artificial Photosynthesis | 0.809576 |
22,262 | However, these catalysts are still inefficient.Photoelectrochemical cells that reduce carbon dioxide into carbon monoxide (CO), formic acid (HCOOH) and methanol (CH3OH) are under development. Similar to natural photosynthesis, such artificial leaves can use a tandem of light absorbers for overall water splitting or CO2... | Artificial Photosynthesis | 0.809576 |
22,263 | Artificial photosynthesis was first anticipated by the Italian chemist Giacomo Ciamician during 1912. In a lecture that was later published in Science he proposed a switch from the use of fossil fuels to radiant energy provided by the sun and captured by technical photochemistry devices. In this switch he saw a possibi... | Artificial Photosynthesis | 0.809576 |
22,264 | Artificial photosynthesis is a chemical process that biomimics the natural process of photosynthesis to convert sunlight, water, and carbon dioxide into carbohydrates and oxygen. The term artificial photosynthesis is commonly used to refer to any scheme for capturing and storing the energy from sunlight in the chemical... | Artificial Photosynthesis | 0.809576 |
22,265 | However, other energy-demanding metabolic pathways can compete with the necessary electrons for proton reduction, decreasing the efficiency of the overall process; also, these hydrogenases are very sensitive to oxygen.Several carbon-based biofuels have also been produced using cyanobacteria, such as 1-butanol.Synthetic... | Artificial Photosynthesis | 0.809576 |
22,266 | Research in artificial photosynthesis is necessarily a multidisciplinary topic, requiring a multitude of different expertise. Some techniques employed in making and investigating catalysts and solar cells include: Organic and inorganic chemical synthesis. Electrochemistry methods, such as photoelectrochemistry, cyclic ... | Artificial Photosynthesis | 0.809576 |
22,267 | The American Society for Precision Engineering (ASPE) focuses on many areas that are important in the research, design, development, manufacture and measurement of high accuracy components and systems. This collective discipline is known as Precision engineering, and includes precision controls, metrology, interferomet... | American Society for Precision Engineering | 0.809574 |
22,268 | Nano-Structures & Nano-Objects is an interdisciplinary peer-reviewed scientific journal devoted to all aspects of the synthesis and properties of the nanotechnology. The journal focuses on novel architecture at the nanolevel with an emphasis on new synthesis and characterization methods. The journal focused on objects ... | Nano-Structures & Nano-Objects | 0.809573 |
22,269 | In mathematics, specifically abstract algebra, a square class of a field F {\displaystyle F} is an element of the square class group, the quotient group F × / F × 2 {\displaystyle F^{\times }/F^{\times 2}} of the multiplicative group of nonzero elements in the field modulo the square elements of the field. Each square ... | Square class | 0.809572 |
22,270 | In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest desc... | Cycloidal pendulum | 0.80957 |
22,271 | Wallis published Wren's proof (crediting Wren) in Wallis's Tractatus Duo, giving Wren priority for the first published proof.Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in th... | Cycloidal pendulum | 0.80957 |
22,272 | According to his student Evangelista Torricelli, in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discove... | Cycloidal pendulum | 0.80957 |
22,273 | Aminode combines multiple alignments with phylogenetic analysis to analyze changes in homologous proteins and produce a plot that indicates the local rates of evolutionary changes. This approach identifies the Evolutionarily Constrained Regions in a protein, which are segments that are subject to purifying selection an... | Conserved sequences | 0.80957 |
22,274 | "Equation of State Calculations by Fast Computing Machines" is a scholarly article published by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller in the Journal of Chemical Physics in 1953. This paper proposed what became known as the Metropolis Monte Carlo algorit... | Equations of State Calculations by Fast Computing Machines | 0.809569 |
22,275 | Additional clarification of attribution is given in connection with the Metropolis–Hastings algorithm. The Rosenbluths would subsequently publish two additional, lesser-known papers using the Monte Carlo method, while the other authors would not continue to work on the topic. Already in 1953, however, Marshall was recr... | Equations of State Calculations by Fast Computing Machines | 0.809569 |
22,276 | Lattice models in biophysics represent a class of statistical-mechanical models which consider a biological macromacromolecule (such as DNA, protein, actin, etc.) as a lattice of units, each unit being in different states or conformations. For example, DNA in chromatin can be represented as a one-dimensional lattice, w... | Lattice model (biophysics) | 0.809569 |
22,277 | For SOLiD data, the format is modified to a color space FASTQ sequence (CSFASTQ), where bases in the sequence are combined with the numbers 0, 1, 2, and 3, indicating how bases are modified relative to the previous base in the sequence (0: no change; 1: transition; 2: non-complementary transversion; 3: complementary tr... | FASTQ format | 0.809568 |
22,278 | The error rate of 'B' reads was roughly 3 phred scores lower the mean observed score of a given run. Starting in Illumina 1.8, the quality scores have basically returned to the use of the Sanger format (Phred+33).For raw reads, the range of scores will depend on the technology and the base caller used, but will typical... | FASTQ format | 0.809568 |
22,279 | For example, SCALCE reduces the alphabet size based on the observation that “neighboring” quality values are similar in general. For a benchmark, see.As of the HiSeq 2500 Illumina gives the option to output qualities that have been coarse grained into quality bins. The binned scores are computed directly from the empir... | FASTQ format | 0.809568 |
22,280 | Then by the extreme value theorem, Φ {\displaystyle \Phi } attains its global minimum. The final step is to show that a minimum of Φ {\displaystyle \Phi } is indeed a Nash equilibrium. | Congestion game | 0.809567 |
22,281 | More general Iwasawa algebras are of the form Λ ( G ) := lim ← H Z p {\displaystyle \Lambda (G):=\varprojlim _{H}\mathbf {Z} _{p}} where G is a compact p-adic Lie group. The case above corresponds to G = Z p {\displaystyle G=\mathbf {Z} _{p}} . A classification of modules over Λ ( G ) {\displaystyle \Lambda (G)} up ... | Iwasawa algebra | 0.809565 |
22,282 | In the special case when the profinite group G is isomorphic to the additive group of the ring of p-adic integers Zp, the Iwasawa algebra Λ(G) is isomorphic to the ring of the formal power series Zp] in one variable over Zp. The isomorphism is given by identifying 1 + T with a topological generator of G. This ring is a... | Iwasawa algebra | 0.809565 |
22,283 | Write νn for the element 1+γ+γ2+...+γpn–1 where γ is a topological generator of Γ. Iwasawa (1959) showed that if X is a finitely generated torsion module over the Iwasawa algebra and X/νnX has order pen then e n = μ p n + λ n + c {\displaystyle e_{n}=\mu p^{n}+\lambda n+c} for n sufficiently large, where μ, λ, and c de... | Iwasawa algebra | 0.809565 |
22,284 | The rank of a finitely generated module is the number of times the module Zp] occurs in it. This is well-defined and is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens if and only if the suppor... | Iwasawa algebra | 0.809565 |
22,285 | In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H runs through the open normal subgroups of G. Commutative Iwasawa algebras wer... | Iwasawa algebra | 0.809565 |
22,286 | the real algebraic numbers the computable numbers the definable numbers the real numbers superreal numbers hyperreal numbers the Puiseux series with real coefficients the surreal numbers | Real closed fields | 0.809564 |
22,287 | For example, the real closure of the ordered field of rational numbers is the field R a l g {\displaystyle \mathbb {R} _{\mathrm {alg} }} of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926. If (F, P) is an ordered field, and E is a Galois extension of F, then by Zorn... | Real closed fields | 0.809564 |
22,288 | (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.) There is an ordering on F making it an ordered field such that, in this ordering, the intermediate value theorem holds for all polynomials over F with degree ≥ 0. F is a weakly o-minimal ordered field.If ... | Real closed fields | 0.809564 |
22,289 | F is not algebraically closed, but its algebraic closure is a finite extension. F is not algebraically closed but the field extension F ( − 1 ) {\displaystyle F({\sqrt {-1}}\,)} is algebraically closed. There is an ordering on F that does not extend to an ordering on any proper algebraic extension of F. F is a formally... | Real closed fields | 0.809564 |
22,290 | Tarski's original algorithm for quantifier elimination has nonelementary computational complexity, meaning that no tower 2 2 ⋅ ⋅ ⋅ n {\displaystyle 2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}} can bound the execution time of the algorithm if n is the size of the input formula. The cylindrical algebraic decomposition, introduc... | Real closed fields | 0.809564 |
22,291 | In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. | Real closed fields | 0.809564 |
22,292 | Insights from this work lead him and Roald Hoffmann to formulate the Woodward–Hoffmann rules for elucidating the stereochemistry of the products of organic reactions. Frederick Sanger demonstrates the dideoxy- or chain termination method for determining DNA sequences (1975). Kary Mullis demonstrates the polymerase chai... | List of experiments | 0.809562 |
22,293 | Edward Jenner tests his hypothesis for the protective action of mild cowpox infection for smallpox, the first vaccine (1796). Gregor Mendel's experiments with the garden pea led him to surmise many of the fundamental laws of genetics (dominant vs recessive genes, the 1–2–1 ratio, see Mendelian inheritance) (1856–1863).... | List of experiments | 0.809562 |
22,294 | Finite-dimensional Riesz spaces are entirely classified by the Archimedean property: Theorem: Suppose that X {\displaystyle X} is a vector lattice of finite-dimension n . {\displaystyle n.} If X {\displaystyle X} is Archimedean ordered then it is (a vector lattice) isomorphic to R n {\displaystyle \mathbb {R} ^{n}} und... | Vector lattice | 0.80956 |
22,295 | {\displaystyle P_{B}(f)=u.} The collection of all projection bands in a Riesz space forms a Boolean algebra. Some spaces do not have non-trivial projection bands (for example, C ( ) {\displaystyle C()} ), so this Boolean algebra may be trivial. | Vector lattice | 0.80956 |
22,296 | In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wid... | Vector lattice | 0.80956 |
22,297 | The concept of hydrostatic equilibrium has also become important in determining whether an astronomical object is a planet, dwarf planet, or small Solar System body. According to the definition of planet adopted by the International Astronomical Union in 2006, one defining characteristic of planets and dwarf planets is... | Hydrostatic Balance | 0.809559 |
22,298 | Some have kept Hindi as a compulsory subject while some have kept it optional. Some schools do not offer it. Subjects like Business Administration, Economics, Music, Fine Arts, Physical Education, National Cadets Corps, Geography, Computer Science, Information Technology, Artificial Intelligence, Foreign languages are ... | Board examination | 0.809553 |
22,299 | The main subjects of this stream are English, Accountancy, Economics and Business Studies. Vocational subjects like Marketing, Retail, Taxation, Banking, Entrepreneurship, Mathematics , Computer Science etc...are offers by schools. In Humanities, students study social sciences and liberal arts. | Board examination | 0.809553 |
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