id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
26,800 | For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a sha... | Facet of a polyhedron | 0.806992 |
26,801 | As Branko Grünbaum observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... at each stage ... the writers failed to define what are the polyhedra". Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be desc... | Facet of a polyhedron | 0.806992 |
26,802 | An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. (Jessen's icosahedron provides an example of a polyhedron meeting one but not both of these two conditions.) Aside from the rectangular cuboids, orthogonal polyhedra are... | Facet of a polyhedron | 0.806992 |
26,803 | Regarding the sticking case, the friction force is set-valued and determined according to an additional algebraic constraint. To conclude, the non-smooth approach changes the underlying mathematical structure if required and leads to a proper description of mechanical systems with unilateral contacts and friction. As a... | Contact dynamics | 0.806992 |
26,804 | The integration of regularized models can be done by standard stiff solvers for ordinary differential equations. However, oscillations induced by the regularization can occur. Considering non-smooth models of mechanical systems with unilateral contacts and friction, two main classes of integrators exist, the event-driv... | Contact dynamics | 0.806992 |
26,805 | Event-driven integrators distinguish between smooth parts of the motion in which the underlying structure of the differential equations does not change, and in events or so-called switching points at which this structure changes, i.e. time instants at which a unilateral contact closes or a stick slip transition occurs.... | Contact dynamics | 0.806992 |
26,806 | The first application of singular traces was the noncommutative residue, a trace on classical pseudo-differential operators on a compact manifold that vanishes on trace class pseudo-differential operators of order less than the negative of the dimension of the manifold, introduced Mariusz Wodzicki and Victor Guillemin ... | Singular trace | 0.806991 |
26,807 | The qualification is relevant; as a von Neumann algebra B(H) admits semifinite (strong-densely defined) traces.Modern examination of the commutator subspace involves checking its spectral characterisation. The following ideals have no traces since the Cesàro means of positive sequences from the Calkin corresponding seq... | Singular trace | 0.806991 |
26,808 | By 1950 French mathematician Jacques Dixmier, a founder of the semifinite theory of von Neumann algebras, thought that a trace on the bounded operators of a separable Hilbert space would automatically be normal up to some trivial counterexamples. : 217 Over the course of 15 years Dixmier, aided by a suggestion of Nachm... | Singular trace | 0.806991 |
26,809 | Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes. In heuristic terms, a singular trace corresponds to a way of summing numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to... | Singular trace | 0.806991 |
26,810 | In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operato... | Singular trace | 0.806991 |
26,811 | Artificial neural networks extend regression and clustering methods to non-linear multivariate models. Statistical graphics such as tours, parallel coordinate plots, scatterplot matrices can be used to explore multivariate data. Simultaneous equations models involve more than one regression equation, with different dep... | Multivariate analyses | 0.806986 |
26,812 | Often, studies that wish to use multivariate analysis are stalled by the dimensionality of the problem. These concerns are often eased through the use of surrogate models, highly accurate approximations of the physics-based code. Since surrogate models take the form of an equation, they can be evaluated very quickly. T... | Multivariate analyses | 0.806986 |
26,813 | Multivariate analysis (MVA) is based on the principles of multivariate statistics. Typically, MVA is used to address the situations where multiple measurements are made on each experimental unit and the relations among these measurements and their structures are important. A modern, overlapping categorization of MVA in... | Multivariate analyses | 0.806986 |
26,814 | A database server is a server which uses a database application that provides database services to other computer programs or to computers. Database management systems (DBMSs) frequently provide database-server functionality, and some database management systems (such as MySQL) rely exclusively on the client–server mod... | Computer types | 0.806985 |
26,815 | Although we now could use the algebraic update equations given previously, it may be preferable to rewrite them. Let ρ = 1 − c s , {\displaystyle \rho ={\frac {1-c}{s}},} so that ρ = tan(θ/2). Then the revised update equations are a h k ′ = a k h ′ = a h k − s ( a h ℓ + ρ a h k ) {\displaystyle a'_{hk}=a'_{kh}=a_{hk}-s... | Jacobi rotation | 0.806984 |
26,816 | {\displaystyle q_{ij}=\delta _{ij}+(\delta _{ik}\delta _{jk}+\delta _{i\ell }\delta _{j\ell })(c-1)+(\delta _{ik}\delta _{j\ell }-\delta _{i\ell }\delta _{jk})s.\,\!} Suppose h is an index other than k or ℓ (which must themselves be distinct). Then the similarity update produces, algebraically, a h k ′ = a k h ′ = c a ... | Jacobi rotation | 0.806984 |
26,817 | In numerical linear algebra, a Jacobi rotation is a rotation, Qkℓ, of a 2-dimensional linear subspace of an n-dimensional inner product space, chosen to zero a symmetric pair of off-diagonal entries of an n×n real symmetric matrix, A, when applied as a similarity transformation: A ↦ Q k ℓ T A Q k ℓ = A ′ . {\displaysty... | Jacobi rotation | 0.806984 |
26,818 | Keepers are substances (typically solvents, but sometimes adsorbent solids) added in relatively small quantities during an evaporative procedure in analytical chemistry, such as concentration of an analyte-solvent mixture by rotary evaporation. The purpose of a keeper is to reduce losses of a target analyte during the ... | Keeper (chemistry) | 0.806983 |
26,819 | Now consider the decomposition of another positive number, b = p 1 b 1 p 2 b 2 ⋯ {\displaystyle b=p_{1}^{b_{1}}p_{2}^{b_{2}}\cdots } The multiplication a b {\displaystyle ab} corresponds point-wise addition of the exponents: a b = p 1 a 1 + b 1 p 2 a 2 + b 2 ⋯ {\displaystyle ab=p_{1}^{a_{1}+b_{1}}p_{2}^{a_{2}+b_{2}}\cd... | Skolem arithmetic | 0.806982 |
26,820 | In more detail, according to the fundamental theorem of arithmetic, a positive integer a > 1 {\displaystyle a>1} can be represented as a product of prime powers: a = p 1 a 1 p 2 a 2 ⋯ {\displaystyle a=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots } If a prime number p k {\displaystyle p_{k}} does not appear as a factor, we define i... | Skolem arithmetic | 0.806982 |
26,821 | The truth value of formulas of Skolem arithmetic can be reduced to the truth value of sequences of non-negative integers constituting their prime factor decomposition, with multiplication becoming point-wise addition of sequences. The decidability then follows from the Feferman–Vaught theorem that can be shown using Qu... | Skolem arithmetic | 0.806982 |
26,822 | Thanks to the above reduction using Feferman–Vaught theorem, we can obtain first-order theories whose open formulas define a larger set of relations if we strengthen the theory of multisets of prime factors. For example, consider the relation a ∼ b {\displaystyle a\sim b} that is true if and only if a {\displaystyle a}... | Skolem arithmetic | 0.806982 |
26,823 | A digital environment in business includes every resource that is either a computer-, mobile device-, or electronically-based resource in the organization in an integrated system. If an organization conducts business activities through the Internet or any other electronic-based communications system, including websites... | Digital environments | 0.806982 |
26,824 | Nanoindentation experiments are generally run with the 2D material as a linear strip clamped on both ends experiencing indentation by a wedge, or with the 2D material as a circular membrane clamped around the circumference experiencing indentation by a curbed tip in the center. The strip geometry is difficult to prepar... | Single-layer materials | 0.806981 |
26,825 | Single and double atom layers of platinum in a two-dimensional film geometry has been demonstrated. These atomically thin platinum films are epitaxially grown on graphene which imposes a compressive strain that modifies the surface chemistry of the platinum, while also allowing charge transfer through the graphene. Sin... | Single-layer materials | 0.806981 |
26,826 | Insect physiology includes the physiology and biochemistry of insect organ systems.Although diverse, insects are quite similar in overall design, internally and externally. The insect is made up of three main body regions (tagmata), the head, thorax and abdomen. The head comprises six fused segments with compound eyes,... | Insect physiology | 0.806981 |
26,827 | Robert Boyle emphasized the importance of not boring the reader with a dull, flat style.Because most scientific journals accept manuscripts only in English, an entire industry has developed to help non-native English speaking authors improve their text before submission. It is just now becoming an accepted practice to ... | Scientific writing | 0.80698 |
26,828 | Additionally, the use of passive voice allows the writer to focus on the subject being studied (the focus of the communication in science) rather than the author. Similarly, some use of first-person pronouns is acceptable (such as "we" or "I," which depends on the number of authors). According to some journal editors, ... | Scientific writing | 0.80698 |
26,829 | Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract a... | Subtraction | 0.806979 |
26,830 | In systems engineering, the tricotyledon theory of system design (T3SD) is a mathematical theory of system design developed by A. Wayne Wymore. T3SD consists of a language for describing systems and requirements, which is based on set theory, a mathematical systems model based on port automata, and a precise definition... | Tricotyledon theory of system design | 0.806977 |
26,831 | Plan one offers 36% renewable at a cost 1% cheaper than Southern California Edison (SCE). The second plan has 50% renewable energy and is the same cost as SCE; the third provides 100% renewable power at a cost 9% higher than SCE. All the rates are set by the CPA board, whose 31 directors represent each city and county ... | Community Choice Aggregation | 0.806977 |
26,832 | California CCAs typically offer customers several energy programs to choose from - a default program (Usually an energy mix similar to that provided by the incumbent utility), a program based on solar energy and a more expensive, 100% Renewable program. This renewable program is backed by unbundled Renewable Energy Cre... | Community Choice Aggregation | 0.806977 |
26,833 | Between 2011 and 2018, CCAs in California directly procured 24 terawatt-hours (TWh) of RPS-eligible electricity, of which 11 TWh is voluntary or in excess of RPS compliance, according to the UCLA Luskin Center for Innovation. Further, of the 72 cities and counties that are powered by 100% clean energy in the U.S., 64 a... | Community Choice Aggregation | 0.806977 |
26,834 | Additionally, it plans to own and manage six solar plants in the Petaluma area.Nationwide, only 13% of CCAs offer “green power,” as their primary focus is to reduce electrical power costs to their customers. California CCAs lead in offering renewable electricity because they are so required by state law. However, CCAs ... | Community Choice Aggregation | 0.806977 |
26,835 | CCAs are therefore already conspicuous leaders in green power innovation, receiving U.S. Environmental Protection Agency's “green power leadership awards” for achievements in renewable energy (MCE Clean Energy; Oak Park, IL, Cincinnati, OH). By 2019 Sonoma Clean Power was providing 91% carbon-free power and had accepte... | Community Choice Aggregation | 0.806977 |
26,836 | They are the primary procurers of California renewable energy portfolio-eligible renewable energy in the State. All CCAs in California are entering into contracts for new renewables. In 2018, six of 19 California CCAs had Power Purchase Agreements for new renewables, accounting for approximately 10% of CCAs total load ... | Community Choice Aggregation | 0.806977 |
26,837 | They engage with numerous state and federal agencies, including their regulatory overseer, the California Public Utilities Commission, and also the California Air Resources Board, California Energy Commission, and the Federal Energy Regulatory Commission.CCAs are well established in California with the first one having... | Community Choice Aggregation | 0.806977 |
26,838 | The California Renewables Portfolio Standard (RPS) is a standard established by the State of California that requires electricity service providers in the State to include certain minimum levels of defined renewable energy in their power mixes. In California all CCAs meet or exceed these minimum standards.CCAs are regu... | Community Choice Aggregation | 0.806977 |
26,839 | As of the end of its first full year of operations in 2016 Lancaster Choice Energy had 55,000 accounts in the City of Lancaster. LCE customers receive a minimum of 36% renewable energy through the standard Clear Choice product, with many opting up to 100% renewable Smart Choice. In addition, LCE's first solar energy pl... | Community Choice Aggregation | 0.806977 |
26,840 | Whether a CCA offers lower or higher rates than its incumbent utility is based on the size of the PCIA (exit fee), whether the incumbent utility changes its rates, and the specific “Renewable Energy” package the customer wants. CCA websites usually compare the CCA's rates to that the incumbent utility. For example, Cle... | Community Choice Aggregation | 0.806977 |
26,841 | This also increases the use of affordable renewable energy, provided through wind, solar, and geothermal steam. This provides environmental benefits for communities because it reduces natural gas consumption and greenhouse gas emissions.There are also disadvantages associated with the implementation of CCAs. Potential ... | Community Choice Aggregation | 0.806977 |
26,842 | There are advantages and disadvantages associated with the implementation of Community Choice Aggregation across different localities. CCA provides benefits like providing a customer choice, reduced energy costs, renewable energy, and environmental benefits.By providing customer choice, customers have the ability to be... | Community Choice Aggregation | 0.806977 |
26,843 | The state of Illinois adopted a CCA law in 2009, which has led to an increase of communities providing electricity services to over 2/3 of the state's population as of 2014, including the city of Chicago, whose former mayor Rahm Emanuel focused the program on reducing coal power production and increasing renewable ener... | Community Choice Aggregation | 0.806977 |
26,844 | In Ohio, the nation's largest CCA was formed in 2000 when the state legislature adopted a CCA law that formed the Northeast Ohio Public Energy Council (NOPEC), made up of approximately 500,000 customers in 112 cities and towns across 8 counties, procured a power supply contract that switched electric generation fuel su... | Community Choice Aggregation | 0.806977 |
26,845 | New jobs have been created constructing and operating clean energy generation facilities for energy suppliers of CCAs, and providing call center and other services for CCAs. CCAs in California support union jobs and wages by signing power purchase agreements with new energy facilities that are built with union labor. C... | Community Choice Aggregation | 0.806977 |
26,846 | This is critical as San Diego's climate plan calls for 100% renewable energy by 2035; more ambitious than the state of California's by ten years. A 150 MW solar farm with 4 hours of battery is scheduled for 2023.Critics of the CCA say cities have little or no experience in purchasing or generating electricity and the u... | Community Choice Aggregation | 0.806977 |
26,847 | CCAs have set a number of national green power and climate protection records while reducing power bills, a rare combination that has won National Renewable Energy Laboratory (NREL) and Environmental Protection Agency (EPA) recognition for achieving significantly higher renewable energy portfolios while maintaining rat... | Community Choice Aggregation | 0.806977 |
26,848 | Marin County launched California's first CCA program, MCE, on May 7, 2010, offering 60%-100% renewable energy at competitive and stable rate significantly reducing energy-related greenhouse gas emissions and enabling millions of dollars of reinvestment in local energy programs. MCE is a load-serving entity supporting a... | Community Choice Aggregation | 0.806977 |
26,849 | Maharam's theorem states that every complete measure space is decomposable into measures on continua, and a finite or countable counting measure. | Complete measure | 0.806975 |
26,850 | The Alloy Analyzer was specifically developed to support so-called "lightweight formal methods". As such, it is intended to provide fully automated analysis, in contrast to the interactive theorem proving techniques commonly used with specification languages similar to Alloy. Development of the Analyzer was originally ... | Alloy Analyzer | 0.806974 |
26,851 | The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point of its domain, then its derivative must be zero at that point. For problems where a smooth function must be minimized subject to constraints which can be expressed in the form of oth... | Variational analysis | 0.806972 |
26,852 | In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized der... | Variational analysis | 0.806972 |
26,853 | In the introduction to The Cost of Natural Selection Haldane writes that it is difficult for breeders to simultaneously select all the desired qualities, partly because the required genes may not be found together in the stock; but, he writes, especially in slowly breeding animals such as cattle, one cannot cull even h... | Haldane's dilemma | 0.806969 |
26,854 | In probability theory, a tree diagram may be used to represent a probability space. A tree diagram may represent a series of independent events (such as a set of coin flips) or conditional probabilities (such as drawing cards from a deck, without replacing the cards). Each node on the diagram represents an event and is... | Tree diagram (probability theory) | 0.806969 |
26,855 | The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: doubling the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited t... | Euclidean number | 0.806968 |
26,856 | Trigonometric numbers are the cosines or sines of angles that are rational multiples of π {\displaystyle \pi } . These numbers are always algebraic, but they may not be constructible. The cosine or sine of the angle 2 π / n {\displaystyle 2\pi /n} is constructible only for certain special numbers n {\displaystyle n}: T... | Euclidean number | 0.806968 |
26,857 | Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number. Alhazen's problem was not... | Euclidean number | 0.806968 |
26,858 | Alhazen's problem is also not one of the classic three problems, but despite being named after Ibn al-Haytham (Alhazen), a medieval Islamic mathematician, it already appear's in Ptolemy's work on optics from the second century.Pierre Wantzel (1837) proved algebraically that the problems of doubling the cube and trisect... | Euclidean number | 0.806968 |
26,859 | Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a ... | Euclidean number | 0.806968 |
26,860 | Analogously, the algebraically constructible complex numbers are the subset of complex numbers that have formulas of the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the principal square... | Euclidean number | 0.806968 |
26,861 | The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas l... | Euclidean number | 0.806968 |
26,862 | The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the complex plane given by algebraically constructible complex numbers. By the equivalence between the two defin... | Euclidean number | 0.806968 |
26,863 | In 1905 Study wrote "Kürzeste Wege im komplexen Gebiet" (Shortest paths in the complex domain) for Mathematische Annalen (60:321–378). Some of its contents were anticipated by Guido Fubini a year before. The distance Study refers to is a Hermitian form on complex projective space. Since then this metric has been called... | Eduard Study | 0.806965 |
26,864 | This article, "Foundations and goals of analytical kinematics" develops the field of kinematics, in particular exhibiting an element of E(3) as a homography of dual quaternions. Study's use of abstract algebra was noted in A History of Algebra (1985) by B. L. van der Waerden. On the other hand, Joe Rooney recounts thes... | Eduard Study | 0.806965 |
26,865 | Eduard Study cites William Kingdon Clifford as an earlier source on these biquaternions. In 1901 Study published Geometrie der Dynamen also using dual quaternions. In 1913 he wrote a review article treating both E(3) and elliptic geometry. | Eduard Study | 0.806965 |
26,866 | In 1891 Eduard Study published "Of Motions and Translations, in two parts". It treats the Euclidean group E(3). The second part of his article introduces the associative algebra of dual quaternions, that is numbers q = a + b i + c j + d k {\displaystyle q=a+bi+cj+dk\!} where a, b, c, and d are dual numbers and {1, i, j... | Eduard Study | 0.806965 |
26,867 | Eduard Study began his university career in Jena, Strasbourg, Leipzig, and Munich. He loved to study biology, especially entomology. He was awarded the doctorate in mathematics at the University of Munich in 1884. Paul Gordan, an expert in invariant theory was at Leipzig, and Study returned there as Privatdozent. | Eduard Study | 0.806965 |
26,868 | Eduard Study ( SHTOO-dee), more properly Christian Hugo Eduard Study (23 March 1862 – 6 January 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and critic... | Eduard Study | 0.806965 |
26,869 | 610). In 2006 Wormer and Paldus summarized Study's role as follows: The analogy, lacking a physical basis at the time, was criticised heavily by the mathematician E. Study and ignored completely by the chemistry community of the 1890s. After the advent of quantum mechanics it became clear, however, that chemical valenc... | Eduard Study | 0.806965 |
26,870 | Somewhat surprisingly Eduard Study is known by practitioners of quantum chemistry. Like James Joseph Sylvester, Paul Gordan believed that invariant theory could contribute to the understanding of chemical valence. In 1900 Gordan and his student G. Alexejeff contributed an article on an analogy between the coupling prob... | Eduard Study | 0.806965 |
26,871 | Study's work with dual numbers and line coordinates was noted by Heinrich Guggenheimer in 1963 in his book Differential Geometry (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in R3 are in one-to-one correspondence with the points of the dual unit sphere in D3. Later he says "... | Eduard Study | 0.806965 |
26,872 | Semiquaternions are often called 'Study's quaternions'.In 1985 Helmut Karzel and Günter Kist developed "Study's quaternions" as the kinematic algebra corresponding to the group of motions of the Euclidean plane. These quaternions arise in "Kinematic algebras and their geometries" alongside ordinary quaternions and the ... | Eduard Study | 0.806965 |
26,873 | Cartan acknowledged Eduard Study's guidance, in his title, with the words "after Eduard Study". In the 1993 biography of Cartan by Akivis and Rosenfeld, one reads: defined the algebra °H of 'semiquaternions' with the units 1, i, ε, η having the properties i 2 = − 1 , ε 2 = 0 , i ε = − ε i = η . {\displaystyle i^{2}=-1... | Eduard Study | 0.806965 |
26,874 | In compiler construction, strength reduction is a compiler optimization where expensive operations are replaced with equivalent but less expensive operations. The classic example of strength reduction converts strong multiplications inside a loop into weaker additions – something that frequently occurs in array address... | Strength reduction | 0.806965 |
26,875 | There are more optimizations to do. Register r3 is the main variable in the innermost loop (0140-0260); it gets incremented by 1 each time through the loop. Register r8 (which is invariant in the innermost loop) is added to r3. | Strength reduction | 0.806965 |
26,876 | The compiler will start doing many optimizations – not just strength reduction. Expressions that are constant (invariant) within a loop will be hoisted out of the loop. Constants can be loaded outside of both loops—such as floating point registers fr3 and fr4. Recognition that some variables don't change allows registe... | Strength reduction | 0.806965 |
26,877 | In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined. The sentence m... | Propositional function | 0.806964 |
26,878 | In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a nilpotent ideal in Ap. In non-commutative algebra and group theory, an algebra or group is locally nilpotent if and only if every finitely generated subalg... | Locally nilpotent | 0.80696 |
26,879 | Robophysics is an emerging scientific field to understand the physical principles of how robots move in the complex real world, analogous to biophysics to understand the motions of biological systems. This emerging area has demonstrated the need for a physics of robotics and reveal interesting problems at the interface... | Robophysics | 0.806959 |
26,880 | According to Tomita–Takesaki theory, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter modular automorphism group. The corresponding crossed product is a Type I I ∞ {\displaystyle II_{\infty }} von Neumann algebra and the corresponding dual action restricts to an ergodic action o... | Crossed product | 0.806958 |
26,881 | Suppose that A is a von Neumann algebra of operators acting on a Hilbert space H and G is a discrete group acting on A. We let K be the Hilbert space of all square summable H-valued functions on G. There is an action of A on K given by a(k)(g) = g−1(a)k(g)for k in K, g, h in G, and a in A, and there is an action of G o... | Crossed product | 0.806958 |
26,882 | Similar statements hold when G {\displaystyle G} is replaced by a non-Abelian locally compact group or more generally a locally compact quantum group, a class of Hopf algebra related to von Neumann algebras. An analogous theory has also been developed for actions on C* algebras and their crossed products. Duality first... | Crossed product | 0.806958 |
26,883 | If A {\displaystyle A} is a von Neumann algebra on which a locally compact Abelian G {\displaystyle G} acts, then Γ {\displaystyle \Gamma } , the dual group of characters χ {\displaystyle \chi } of G {\displaystyle G} , acts by unitaries on K {\displaystyle K}: ( χ ⋅ k ) ( h ) = χ ( h ) k ( h ) {\displaystyle (\chi \cd... | Crossed product | 0.806958 |
26,884 | We let G be an infinite countable discrete group acting on the abelian von Neumann algebra A. The action is called free if A has no non-zero projections p such that some nontrivial g fixes all elements of pAp. The action is called ergodic if the only invariant projections are 0 and 1. Usually A can be identified as the... | Crossed product | 0.806958 |
26,885 | If we take C {\displaystyle C} to be the complex numbers, then the crossed product C ⋊ G {\displaystyle C\rtimes G} is called the von Neumann group algebra of G. If G {\displaystyle G} is an infinite discrete group such that every conjugacy class has infinite order then the von Neumann group algebra is a factor of type... | Crossed product | 0.806958 |
26,886 | In physics, this structure appears in presence of the so called gauge group of the first kind. G is the gauge group, and N the "field" algebra. The observables are then defined as the fixed points of N under the action of G. A result by Doplicher, Haag and Roberts says that under some assumptions the crossed product ca... | Crossed product | 0.806958 |
26,887 | The crossed product of a von Neumann algebra by a group G acting on it is similar except that we have to be more careful about topologies, and need to construct a Hilbert space acted on by the crossed product. (Note that the von Neumann algebra crossed product is usually larger than the algebraic crossed product discus... | Crossed product | 0.806958 |
26,888 | Recall that if we have two finite groups G {\displaystyle G} and N with an action of G on N we can form the semidirect product N ⋊ G {\displaystyle N\rtimes G} . This contains N as a normal subgroup, and the action of G on N is given by conjugation in the semidirect product. We can replace N by its complex group algebr... | Crossed product | 0.806958 |
26,889 | In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, crossed product is the expected s... | Crossed product | 0.806958 |
26,890 | In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete logarithm in ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} where q {\displaystyle q} is a prime, index calculus leads to a family of algorithms adapted ... | Index calculus | 0.806957 |
26,891 | The third stage searches for a power s of the generator g which, when multiplied by the argument h, may be factored in terms of the factor base gsh = (−1)f0 2f1 3f2···prfr. Finally, in an operation too simple to really be called a fourth stage, the results of the second and third stages can be rearranged by simple alge... | Index calculus | 0.806957 |
26,892 | A system of hundreds of thousands or millions of equations is a significant computation requiring large amounts of memory, and it is not embarrassingly parallel, so a supercomputer is typically used. This was considered a minor step compared to the others for smaller discrete log computations. However, larger discrete ... | Index calculus | 0.806957 |
26,893 | Index Calculus inspired a large family of algorithms. In finite fields F q {\displaystyle \mathbb {F} _{q}} with q = p n {\displaystyle q=p^{n}} for some prime p {\displaystyle p} , the state-of-art algorithms are the Number Field Sieve for Discrete Logarithms, L q {\textstyle L_{q}\left{64/9}}\,\right]} , when p {\di... | Index calculus | 0.806957 |
26,894 | The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2θ, is the area of a spherical cap on a unit sphere For small θ such that cos θ ≈ 1 − θ2/2 this reduces to πθ2, the area of a circle. The above is found by computing the following double integral using the unit surface element i... | Square second | 0.806956 |
26,895 | In geometry, a solid angle (symbol: Ω) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, ... | Square second | 0.806956 |
26,896 | Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where a → , b → , c → {\displaystyle {\vec {a}}\ ,\,{\vec {b}}\ ,\,{\vec {c}}} are the vector positions of the vertices A, B and C. Define the vertex angle θa to be the angle BOC and define θb, θc correspondingly. Let ϕ a... | Square second | 0.806956 |
26,897 | Positive feedback (exacerbating feedback, self-reinforcing feedback) is a process that occurs in a feedback loop which exacerbates the effects of a small disturbance. That is, the effects of a perturbation on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn ... | Reinforcing loop | 0.806951 |
26,898 | The analogy of Evolutionary arms races provide further examples of positive feedback in biological systems. It has been shown that changes in biodiversity through the Phanerozoic correlate much better with hyperbolic model (widely used in demography and macrosociology) than with exponential and logistic models (traditi... | Reinforcing loop | 0.806951 |
26,899 | In projective geometry, a desmic system (from Greek δεσμός 'band, chain') is a set of three tetrahedra in 3-dimensional projective space, such that any two are desmic (related such that each edge of one cuts a pair of opposite edges of the other). It was introduced by Stephanos (1879). The three tetrahedra of a desmic ... | Desmic system | 0.80695 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.