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33,600 | A magnetic gear resembles the traditional mechanical gear in geometry and function, using magnets instead of teeth. As two opposing magnets approach each other, they repel; when placed on two rings the magnets will act like teeth. As opposed to conventional hard contact backlash in a spur gear, where a gear may rotate ... | Magnetic gear | 0.80336 |
33,601 | Frances H. Arnold, (2001) Giorgio Carta, (2002) Rashid Bashir, (2010) Ravi V. Bellamkonda, (2006) President 2014-2016 Emery N. Brown, (2006) Shu Chien, Founding Fellow(1992), President 2000, recipient of the 2004 Pierre Galletti Award Dorin Comaniciu, (2013) Tejal A. Desai, (2012) Joseph DeSimone, (2007) Anthony Guisep... | American Institute for Medical and Biological Engineering | 0.803359 |
33,602 | AIMBE is one of many organizations that has a Network of Experts Agreement with the U.S. Food and Drug Administration (FDA).It also partners with the FDA for its Scholars Program, which places post-doctorates in biomedical engineering fields in one-year positions at the FDA's Center for Devices and Radiological Health ... | American Institute for Medical and Biological Engineering | 0.803359 |
33,603 | The Industry Council consists of corporate leaders in the biomedical engineering field. | American Institute for Medical and Biological Engineering | 0.803359 |
33,604 | In practice the point 0 is often transformed into the point ∞ of the Riemann sphere. This gives a version of the principle that applies to strips, for example bounded by two lines of constant real part in the complex plane. This special case is sometimes known as Lindelöf's theorem. Carlson's theorem is an application ... | Phragmén–Lindelöf principle | 0.803358 |
33,605 | Many universities teach functional programming. Some treat it as an introductory programming concept while others first teach imperative programming methods.Outside of computer science, functional programming is used to teach problem-solving, algebraic and geometric concepts. It has also been used to teach classical me... | Functional programming language | 0.803356 |
33,606 | Some compilers, such as gcc, add extra keywords for a programmer to explicitly mark external functions as pure, to enable such optimizations. Fortran 95 also lets functions be designated pure. C++11 added constexpr keyword with similar semantics. | Functional programming language | 0.803356 |
33,607 | If there is no data dependency between two pure expressions, their order can be reversed, or they can be performed in parallel and they cannot interfere with one another (in other terms, the evaluation of any pure expression is thread-safe). If the entire language does not allow side-effects, then any evaluation strate... | Functional programming language | 0.803356 |
33,608 | Pure functions (or expressions) have no side effects (memory or I/O). This means that pure functions have several useful properties, many of which can be used to optimize the code: If the result of a pure expression is not used, it can be removed without affecting other expressions. If a pure function is called with ar... | Functional programming language | 0.803356 |
33,609 | Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, ... | Associative superalgebra | 0.803355 |
33,610 | In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix super- comes from the theory of supersymmetry in theoretical phy... | Associative superalgebra | 0.803355 |
33,611 | Model-free static controllers: these approaches learn directly, via machine learning techniques (e.g. regression methods and neural networks), the inverse kinematic or the direct kinematic representation of the continuum robot from collected data, and they are also known as data-driven methods. Even though these contro... | Continuum robot | 0.803348 |
33,612 | The control strategies can be distinguished in static and dynamic; the first one is based on the steady-state assumption, while the latter also considers the dynamic behaviour of the continuum robot. We can also differentiate between model-based controllers, that depend on a model of the robot, and model-free, that lea... | Continuum robot | 0.803348 |
33,613 | As of 2021, they are in the early stage, as they require high computational power and high-dimensional sensory feedback. With improvements in computational power and sensing capabilities they could be crucial in industrial applications of continuum robots, where time and cost are also relevant along with accuracy. Mode... | Continuum robot | 0.803348 |
33,614 | In materials science the flow stress, typically denoted as Yf (or σ f {\displaystyle \sigma _{\text{f}}} ), is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-st... | Flow stress | 0.803348 |
33,615 | In the context of free and open-source software, proprietary software only available as a binary executable is referred to as a blob or binary blob. The term usually refers to a device driver module loaded into the kernel of an open-source operating system, and is sometimes also applied to code running outside the kern... | Binary blobs | 0.803347 |
33,616 | In theoretical physics, a gauge anomaly is an example of an anomaly: it is a feature of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e. of a gauge theory.All gauge anomalies must cancel out. Anomalies in gauge symmetries lead to an inconsistency, since a ... | Gauge anomalies | 0.803346 |
33,617 | A spot assay or spot test can also refer to a specific test in microbiology. This test is often used to check the growth rate of bacterial or yeast cells on different media or to perform serial dilution tests of micro-organisms. Usually a 96-pinner (often called frogger) is used to perform these spot assay. | Spot analysis | 0.803345 |
33,618 | Spot analysis, spot test analysis, or spot test is a chemical test, a simple and efficient technique where analytic assays are executed in only one, or a few drops, of a chemical solution, preferably in a great piece of filter paper, without using any sophisticated instrumentation. The development and popularization of... | Spot analysis | 0.803345 |
33,619 | Pick's theorem may also be proved based on complex integration of a doubly periodic function related to Weierstrass elliptic functions. Applying the Poisson summation formula to the characteristic function of the polygon leads to another proof.Pick's theorem was included in a 1999 web listing of the "top 100 mathematic... | Pick's formula | 0.803345 |
33,620 | Generalizations to Pick's theorem to non-simple polygons are more complicated and require more information than just the number of interior and boundary vertices. For instance, a polygon with h {\displaystyle h} holes bounded by simple integer polygons, disjoint from each other and from the boundary, has area It is als... | Pick's formula | 0.803345 |
33,621 | Alternative proofs of Pick's theorem that do not use Euler's formula include the following. One can recursively decompose the given polygon into triangles, allowing some triangles of the subdivision to have area larger than 1/2. Both the area and the counts of points used in Pick's formula add together in the same way ... | Pick's formula | 0.803345 |
33,622 | Each edge interior to the polygon is the side of two triangles. However, there are b {\displaystyle b} edges of triangles that lie along the polygon's boundary and form part of only one triangle. Therefore, the number of sides of triangles obeys the equation 6 A = 2 E − b {\displaystyle 6A=2E-b} , from which one can so... | Pick's formula | 0.803345 |
33,623 | A different proof that these triangles have area 1 2 {\displaystyle {\tfrac {1}{2}}} is based on the use of Minkowski's theorem on lattice points in symmetric convex sets. This already proves Pick's formula for a polygon that is one of these special triangles. Any other polygon can be subdivided into special triangles:... | Pick's formula | 0.803345 |
33,624 | One proof of this theorem involves subdividing the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly 1 2 {\displaystyle {\tfrac {1}{2}}} . Therefore, the area of the whole polygon equals half the number of triangles in the s... | Pick's formula | 0.803345 |
33,625 | A 3D Content Retrieval system is a computer system for browsing, searching and retrieving three dimensional digital contents (e.g.: Computer-aided design, molecular biology models, and cultural heritage 3D scenes, etc.) from a large database of digital images. The most original way of doing 3D content retrieval uses me... | 3D Content Retrieval | 0.803344 |
33,626 | In Purdue University, researchers led by Professor Karthik Ramani at the Research and Education Center for Information created a 3D search engine called the 3D Engineering Search System (3DESS). It is designed to find computer-generated engineering parts. The mechanism behind this search engine is that it starts from a... | 3D Content Retrieval | 0.803344 |
33,627 | The skeleton 3D retrieval method needs more time and effort before it can be used widely. Compute a feature vector based on statistics Unlike Skeleton modeling, which requires a high quality standard for the input source, statistical methods do not put restriction on the validity of an input source. Shape histograms, f... | 3D Content Retrieval | 0.803344 |
33,628 | In mathematical analysis, the initial value theorem is a theorem used to relate frequency domain expressions to the time domain behavior as time approaches zero.Let F ( s ) = ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaysty... | Initial value theorem | 0.803343 |
33,629 | The theorem assuming just that f ( t ) = O ( e c t ) {\displaystyle f(t)=O(e^{ct})} follows from the theorem for bounded f {\displaystyle f}: Define g ( t ) = e − c t f ( t ) {\displaystyle g(t)=e^{-ct}f(t)} . Then g {\displaystyle g} is bounded, so we've shown that g ( 0 + ) = lim s → ∞ s G ( s ) {\displaystyle g(0^{+... | Initial value theorem | 0.803343 |
33,630 | Suppose first that f {\displaystyle f} is bounded, i.e. lim t → 0 + f ( t ) = α {\displaystyle \lim _{t\to 0^{+}}f(t)=\alpha } . A change of variable in the integral ∫ 0 ∞ f ( t ) e − s t d t {\displaystyle \int _{0}^{\infty }f(t)e^{-st}\,dt} shows that s F ( s ) = ∫ 0 ∞ f ( t s ) e − t d t {\displaystyle sF(s)=\int _{... | Initial value theorem | 0.803343 |
33,631 | Of course we don't really need DCT here, one can give a very simple proof using only elementary calculus: Start by choosing A {\displaystyle A} so that ∫ A ∞ e − t d t < ϵ {\displaystyle \int _{A}^{\infty }e^{-t}\,dt<\epsilon } , and then note that lim s → ∞ f ( t s ) = α {\displaystyle \lim _{s\to \infty }f\left({\fra... | Initial value theorem | 0.803343 |
33,632 | In the field of cell biology, the method of partial cloning (PCL) converts a fully differentiated old somatic cell into a partially reprogrammed young cell that retains all the specialised functions of the differentiated old cell but is simply younger. The method of PCL reverses characteristics associated with old cell... | Partial cloning | 0.803341 |
33,633 | In the field of mathematics called combinatorial optimization, the method of symmetry-breaking constraints can be used to take advantage of symmetries in many constraint satisfaction and optimization problems, by adding constraints that eliminate symmetries and reduce the search space size. Symmetries in a combinatoria... | Symmetry-breaking constraints | 0.803338 |
33,634 | Alternative statements of the paradox include "If given the axioms of Euclidean geometry, can an omnipotent being create a triangle whose angles do not add up to 180 degrees?" and "Can God create a prison so secure that he cannot escape from it? ". | Omnipotence paradox | 0.803336 |
33,635 | Modern physics indicates that the choice of phrasing about lifting stones should relate to acceleration; however, this does not in itself of course invalidate the fundamental concept of the generalized omnipotence paradox. However, one could easily modify the classic statement as follows: "An omnipotent being creates a... | Omnipotence paradox | 0.803336 |
33,636 | This can be done on a sphere, and not on a flat surface. The later invention of non-Euclidean geometry does not resolve this question; for one might as well ask, "If given the axioms of Riemannian geometry, can an omnipotent being create a triangle whose angles do not add up to more than 180 degrees?" In either case, t... | Omnipotence paradox | 0.803336 |
33,637 | In the 6th century, Pseudo-Dionysius claims that a version of the omnipotence paradox constituted the dispute between Paul the Apostle and Elymas the Magician mentioned in Acts 13:8, but it is phrased in terms of a debate as to whether God can "deny himself" a'la 2 Tim 2:13. In the 11th century, Anselm of Canterbury ar... | Omnipotence paradox | 0.803336 |
33,638 | LUVOIR would be a Large Strategic Science Mission and was considered for a development start sometime in the 2020s. The LUVOIR Study Team, under Study Scientist Aki Roberge, has produced designs for two variants of LUVOIR: one with a 15.1 m diameter telescope mirror (LUVOIR-A) and one with an 8 m diameter mirror (LUVOI... | Large UV Optical Infrared Surveyor | 0.803336 |
33,639 | The Large Ultraviolet Optical Infrared Surveyor, commonly known as LUVOIR (), is a multi-wavelength space telescope concept being developed by NASA under the leadership of a Science and Technology Definition Team. It is one of four large astrophysics space mission concepts studied in preparation for the National Academ... | Large UV Optical Infrared Surveyor | 0.803336 |
33,640 | LUVOIR's multi-wavelength capability would also provide key information to help understand how a host star's UV radiation regulates the atmospheric photochemistry on habitable planets. LUVOIR will also observe large numbers of exoplanets spanning a wide range of characteristics (mass, host star type, age, etc.), with t... | Large UV Optical Infrared Surveyor | 0.803336 |
33,641 | Second, the telescope and coronagraph both incorporate several layers of wavefront control through active optics. Third, the telescope is actively heated to a precise 270 K (−3 °C; 26 °F) to control thermal disturbances. The LUVOIR technology development plan is supported with funding from NASA's Astrophysics Strategic... | Large UV Optical Infrared Surveyor | 0.803336 |
33,642 | Besides modules that contain a single power electronic switch (as MOSFET, IGBT, BJT, Thyristor, GTO or JFET) or diode, classical power modules contain multiple semiconductor dies that are connected to form an electrical circuit of a certain structure, called topology. Modules also contain other components such as ceram... | Power module | 0.803334 |
33,643 | Another problem connected to parasitics is their negative impact on the switching behavior and the switching loss of the power semiconductors. Therefore, manufacturers work on minimizing the parasitic elements of their modules while keeping cost low and maintain a high degree of interchangeability of their modules with... | Power module | 0.803334 |
33,644 | More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave... | Invariant set | 0.803333 |
33,645 | In particular, when verifying an imperative program using the Hoare calculus, a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs. In the context of the above MU puzzle example, there is currently no general... | Invariant set | 0.803333 |
33,646 | (Some authors use the terminology setwise invariant, vs. pointwise invariant, to distinguish between these cases.) For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space. An invariant set of an ope... | Invariant set | 0.803333 |
33,647 | Although prototype elements consist of AT-rich regions several hundred base pairs in length, the overall base composition is definitely not the primary determinant of their activity. Instead, their function requires a pattern of "AT-patches" that confer the propensity for local strand unpairing under torsional strain. ... | Scaffold attachment region | 0.803333 |
33,648 | Open synthetic biology is the idea that scientific knowledge and data should be openly accessible through common rights licensing to enable the rapid development of safe, effective and commercially viable synthetic biology applications. | Open synthetic biology | 0.803332 |
33,649 | One example of open synthetic biology is when DNA2.0 donated several artificial gene sequences into an open-access repository run by the BioBricks Foundation. | Open synthetic biology | 0.803332 |
33,650 | Its foundational concepts are open science and the Bermuda Principles.Open science is the idea that scientific research should be openly shared to enable massive collaboration (e.g., the Polymath Project). The Bermuda Principles is a private accord declaring that all DNA sequence data should be released in publicly acc... | Open synthetic biology | 0.803332 |
33,651 | In the theory of algebraic plane curves, a general quartic plane curve has 28 bitangent lines, lines that are tangent to the curve in two places. These lines exist in the complex projective plane, but it is possible to define quartic curves for which all 28 of these lines have real numbers as their coordinates and ther... | Bitangents of a quartic | 0.803332 |
33,652 | In molecular biology, the EcoEI R protein C-terminal domain is a protein domain found at the C-terminus of both the R subunit of type I restriction enzymes and the Res subunit of type III restriction enzymes. The type I enzymes include EcoEI, which recognises 5'-GAGN(7)ATGC-3; the R protein (HsdR) is required for both ... | EcoEI R protein C-terminal domain | 0.803332 |
33,653 | The Erdős–Ko–Rado theorem states that the independence number of the Kneser graph K(n, k) for n ≥ 2 k {\displaystyle n\geq 2k} is | Kneser graphs | 0.803331 |
33,654 | It used nanostructured components as the basis for creating printable semiconductors, printable transparent electrodes, novel forms of advanced nanocomposite solar-cell design and powerful new forms of barrier films.According to the company at the time, "leveraging recent science advances in nanostructured materials, N... | Nanosolar | 0.803329 |
33,655 | Although RISKS is a forum of a computer science association, most contributions are readable and informative to anyone with an interest in the subject. It is heavily read by system administrators, and computer security managers, as well as computer scientists and engineers. The RISKS Digest is published on a frequent b... | RISKS Digest | 0.803325 |
33,656 | Other recurring subjects include cryptography and the effects of technically ill-considered public policies. RISKS also publishes announcements and Calls for Papers from various technical conferences, and technical book reviews (usually by Rob Slade, though occasionally by others). | RISKS Digest | 0.803325 |
33,657 | In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, fair, good, excellent). Similarly, the widely used logit method also has a... | Ordered probit | 0.803324 |
33,658 | Spaun ("Semantic Pointer Architecture Unified Network") is a cognitive architecture pioneered by Chris Eliasmith of the University of Waterloo Centre for Theoretical Neuroscience. It consists of 2.5 million simulated neurons organized into subsystems that resemble specific brain regions, such as the prefrontal cortex, ... | Spaun (Semantic Pointer Architecture Unified Network) | 0.803324 |
33,659 | It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomial could be expressed in terms of a finite number of radicals and elementary operations). However, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), ... | Fourth root | 0.803324 |
33,660 | Suppose that R = R {\displaystyle R=\mathbb {R} } . If the semialgebraic set W = { x ∈ R n ∣ ∀ f ∈ F , f ( x ) ≥ 0 } {\displaystyle W=\{x\in \mathbb {R} ^{n}\mid \forall f\in F,\,f(x)\geq 0\}} is compact, then each polynomial p ∈ R {\displaystyle p\in \mathbb {R} } that is strictly positive on W {\displaystyle W} can ... | Krivine–Stengle Positivstellensatz | 0.803322 |
33,661 | In real algebraic geometry, Krivine–Stengle Positivstellensatz (German for "positive-locus-theorem") characterizes polynomials that are positive on a semialgebraic set, which is defined by systems of inequalities of polynomials with real coefficients, or more generally, coefficients from any real closed field. It can b... | Krivine–Stengle Positivstellensatz | 0.803322 |
33,662 | The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average translational kinetic energy, 3/2kBT. However, this is true only for ideal gas, and the same result can be derived from the Maxwell–Boltzmann distribution. First,... | Equipartition of energy | 0.803321 |
33,663 | Since the divergence of the position vector q is ∇ ⋅ q = ∂ q x ∂ q x + ∂ q y ∂ q y + ∂ q z ∂ q z = 3 , {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {q} ={\frac {\partial q_{x}}{\partial q_{x}}}+{\frac {\partial q_{y}}{\partial q_{y}}}+{\frac {\partial q_{z}}{\partial q_{z}}}=3,} the divergence theorem implies th... | Equipartition of energy | 0.803321 |
33,664 | An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of 3N independent simple harmonic oscillators, where N denotes the number of atoms in the lattice... | Equipartition of energy | 0.803321 |
33,665 | The (Newtonian) kinetic energy of a particle of mass m, velocity v is given by H kin = 1 2 m | v | 2 = 1 2 m ( v x 2 + v y 2 + v z 2 ) , {\displaystyle H_{\text{kin}}={\tfrac {1}{2}}m|\mathbf {v} |^{2}={\tfrac {1}{2}}m\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right),} where vx, vy and vz are the Cartesian components of the v... | Equipartition of energy | 0.803321 |
33,666 | More generally, the equipartition theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax2, where A is a constant, has an average energy of 1⁄2kBT in thermal equilibrium. In this case the equipartition theorem may be derived from the partition function Z(β), wh... | Equipartition of energy | 0.803321 |
33,667 | Equipartition was used above to derive the classical ideal gas law from Newtonian mechanics. However, relativistic effects become dominant in some systems, such as white dwarfs and neutron stars, and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the correspondin... | Equipartition of energy | 0.803321 |
33,668 | However, once coupled with the fundamental theorem of algebra it says that the disk contains in fact at least one solution. More generally, a bound can be given directly in terms of any p-norm of the n-vector of coefficients a := ( a 0 , a 1 , … , a n − 1 ) , {\displaystyle a:=(a_{0},a_{1},\ldots ,a_{n-1}),} that is |ζ... | Fundamental Theorem of Algebra | 0.803318 |
33,669 | While the fundamental theorem of algebra states a general existence result, it is of some interest, both from the theoretical and from the practical point of view, to have information on the location of the zeros of a given polynomial. The simpler result in this direction is a bound on the modulus: all zeros ζ of a mon... | Fundamental Theorem of Algebra | 0.803318 |
33,670 | Then ∫ c ( r ) R ( z ) d z = ∑ k = 0 ∞ ∫ c ( r ) d z z k + 1 A k = 2 π i I n {\displaystyle \int _{c(r)}R(z)dz=\sum _{k=0}^{\infty }\int _{c(r)}{\frac {dz}{z^{k+1}}}A^{k}=2\pi iI_{n}} (in which only the summand k = 0 has a nonzero integral). This is a contradiction, and so A has an eigenvalue. Finally, Rouché's theorem... | Fundamental Theorem of Algebra | 0.803318 |
33,671 | Since, by assumption, A has no eigenvalues, the function R(z) is an entire function and Cauchy theorem implies that ∫ c ( r ) R ( z ) d z = 0. {\displaystyle \int _{c(r)}R(z)\,dz=0.} On the other hand, R(z) expanded as a geometric series gives: R ( z ) = z − 1 ( I n − z − 1 A ) − 1 = z − 1 ∑ k = 0 ∞ 1 z k A k ⋅ {\displ... | Fundamental Theorem of Algebra | 0.803318 |
33,672 | The numerator of the rational expression being integrated has degree at most n − 1 and the degree of the denominator is n + 1. Therefore, the number above tends to 0 as r → +∞. But the number is also equal to N − n and so N = n. Another complex-analytic proof can be given by combining linear algebra with the Cauchy the... | Fundamental Theorem of Algebra | 0.803318 |
33,673 | Applying Liouville's theorem, which states that a bounded entire function must be constant, this would imply that 1/p is constant and therefore that p is constant. This gives a contradiction, and hence p(z0) = 0.Yet another analytic proof uses the argument principle. Let R be a positive real number large enough so that... | Fundamental Theorem of Algebra | 0.803318 |
33,674 | If k = 0, then n is odd, and therefore p(z) has a real root. Now, suppose that n = 2km (with m odd and k > 0) and that the theorem is already proved when the degree of the polynomial has the form 2k − 1m′ with m′ odd. For a real number t, define: q t ( z ) = ∏ 1 ≤ i < j ≤ n ( z − z i − z j − t z i z j ) . {\displaystyl... | Fundamental Theorem of Algebra | 0.803318 |
33,675 | Suppose the minimum of |p(z)| on the whole complex plane is achieved at z0; it was seen at the proof which uses Liouville's theorem that such a number must exist. We can write p(z) as a polynomial in z − z0: there is some natural number k and there are some complex numbers ck, ck + 1, ..., cn such that ck ≠ 0 and: p ( ... | Fundamental Theorem of Algebra | 0.803318 |
33,676 | Another proof of this kind was obtained by Hellmuth Kneser in 1940 and simplified by his son Martin Kneser in 1981. Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers (which are not constructively equivalen... | Fundamental Theorem of Algebra | 0.803318 |
33,677 | None of the proofs mentioned so far is constructive. It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra. He presented his solution, which amounts in modern terms to a combination of the Durand–Kerner meth... | Fundamental Theorem of Algebra | 0.803318 |
33,678 | Gauss produced two other proofs in 1816 and another incomplete version of his original proof in 1849. The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821). It contained Argand's proof, although Argand is not credited for it. | Fundamental Theorem of Algebra | 0.803318 |
33,679 | One of them, due to James Wood and mainly algebraic, was published in 1798 and it was totally ignored. Wood's proof had an algebraic gap. The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, only filled by Alexander Ostrowski in 1920, as discussed in Smale (1981).The f... | Fundamental Theorem of Algebra | 0.803318 |
33,680 | {\displaystyle x^{4}+a^{4}=\left(x^{2}+a{\sqrt {2}}\cdot x+a^{2}\right)\left(x^{2}-a{\sqrt {2}}\cdot x+a^{2}\right).} A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux's theorem), which would n... | Fundamental Theorem of Algebra | 0.803318 |
33,681 | {\displaystyle -1-i{\sqrt {2}}.} As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 Leibniz erroneously said tha... | Fundamental Theorem of Algebra | 0.803318 |
33,682 | It was first formulated by H. A. Schwarz and served as a theoretical tool: its convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin. | Schwarz alternating method | 0.803316 |
33,683 | From 1870 onwards Carl Neumann also contributed to this theory. In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of an elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves so... | Schwarz alternating method | 0.803316 |
33,684 | The Society has now more than 1900 Individual Members, 15 Corporate Members (Universities and non-profit research organizations from Belgium, Croatia, Finland, Germany, India, Italy, New Zealand, Poland, Spain, Switzerland and Serbia) and 16 Corporate Sustaining Members. ISE has also 8 Divisions and Regional Representa... | International Society of Electrochemistry | 0.803314 |
33,685 | The International Society of Electrochemistry (ISE) is a global scientific society founded in 1949. The Head Office of ISE is located now in Lausanne, Switzerland. ISE is a Member Organization of IUPAC. | International Society of Electrochemistry | 0.803314 |
33,686 | January 21, 2008TCC Number Theory Event Day was held at Bristol and students were given a chance to talk about their research to their peers and share ideas.May 21, 2013Graduate TCC Conference in Number Theory was a successful continuation of the aforementioned event. | Taught Course Centre | 0.803314 |
33,687 | The five collaborating universities are: University of Bath University of Bristol Imperial College London University of Oxford University of WarwickLectures are given at all five universities and, using Access Grid technology, students at each of the other four institutes may participate with each lecture. The TCC was ... | Taught Course Centre | 0.803314 |
33,688 | Every profinite group occurs as a Galois group of some Galois extension, however not every profinite group occurs as an absolute Galois group. For example, the Artin–Schreier theorem asserts that the only finite absolute Galois groups are either trivial or of order 2, that is only two isomorphism classes. Every project... | Absolute Galois group | 0.803314 |
33,689 | The absolute Galois group of an algebraically closed field is trivial. The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and = 2. The absolute Galois group of a finite field K is isomorphic to the group Z ^ ... | Absolute Galois group | 0.803314 |
33,690 | No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the dessins d'enfants of Grothendieck (maps on surfaces), enabling us to "see" the Galois theory of algebraic number fields. Let K b... | Absolute Galois group | 0.803314 |
33,691 | Then the absolute Galois group of K = C(x) is free of rank equal to the cardinality of C. This result is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden using algebraic methods. Let K be a finite extension of the p-adic numbers Qp. For p ≠ 2, its absolute Galois group is g... | Absolute Galois group | 0.803313 |
33,692 | )The absolute Galois group of the field of rational functions with complex coefficients is free (as a profinite group). This result is due to Adrien Douady and has its origins in Riemann's existence theorem. More generally, let C be an algebraically closed field and x a variable. | Absolute Galois group | 0.803313 |
33,693 | In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. ... | Absolute Galois group | 0.803313 |
33,694 | This is a result of Uwe Jannsen and Kay Wingberg. Some results are known in the case p = 2, but the structure for Q2 is not known. Another case in which the absolute Galois group has been determined is for the largest totally real subfield of the field of algebraic numbers. | Absolute Galois group | 0.803313 |
33,695 | The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J by a similar equation: The g-factor gJ is known as the Landé g-factor, which can be related to gL and gS by quantum mechanics. See Landé g-factor for details. | Electron magnetic dipole moment | 0.803313 |
33,696 | In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magnetic moment (symbol μe) is −9.2847647043(28)×10−24 J⋅T−1. In units of the Bo... | Electron magnetic dipole moment | 0.803313 |
33,697 | The entire Dirac spinor represents an irreducible whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime - antimatter and the idea of creation and annihilation of particles. In a general case (if a certain linear function of electromagnetic ... | Electron magnetic dipole moment | 0.803313 |
33,698 | A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, a... | Electron magnetic dipole moment | 0.803313 |
33,699 | Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology — Dirac now had a theoretical argument that implied that spin was somehow the consequence of incorporating relativity into quantum mechanics. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as min... | Electron magnetic dipole moment | 0.803313 |
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