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Gauge block wringing: What is the mechanism that allows gauge blocks to be wrung together?Fractional Hall effect: What mechanism explains the existence of the u = 5/2 state in the fractional quantum Hall effect? Does it describe quasiparticles with non-Abelian fractional statistics? Liquid crystals: Can the nematic to smectic (A) phase transition in liquid crystal states be characterized as a universal phase transition?
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Are there other theories of quantum gravity other than string theory that admit a holographic description? Problem of time: In quantum mechanics, time is a classical background parameter and the flow of time is universal and absolute. In general relativity time is one component of four-dimensional spacetime, and the flow of time changes depending on the curvature of spacetime and the spacetime trajectory of the observer. How can these two concepts of time be reconciled?
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The cosmic censorship hypothesis and the chronology protection conjecture: Can singularities not hidden behind an event horizon, known as "naked singularities", arise from realistic initial conditions, or is it possible to prove some version of the "cosmic censorship hypothesis" of Roger Penrose which proposes that this is impossible? Similarly, will the closed timelike curves which arise in some solutions to the equations of general relativity (and which imply the possibility of backwards time travel) be ruled out by a theory of quantum gravity which unites general relativity with quantum mechanics, as suggested by the "chronology protection conjecture" of Stephen Hawking? Holographic principle: Is it true that quantum gravity admits a lower-dimensional description that does not contain gravity?
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Black holes, black hole information paradox, and black hole radiation: Do black holes produce thermal radiation, as expected on theoretical grounds? Does this radiation contain information about their inner structure, as suggested by gauge–gravity duality, or not, as implied by Hawking's original calculation? If not, and black holes can evaporate away, what happens to the information stored in them (since quantum mechanics does not provide for the destruction of information)?
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Quantum gravity: Can quantum mechanics and general relativity be realized as a fully consistent theory (perhaps as a quantum field theory)? Is spacetime fundamentally continuous or discrete? Would a consistent theory involve a force mediated by a hypothetical graviton, or be a product of a discrete structure of spacetime itself (as in loop quantum gravity)? Are there deviations from the predictions of general relativity at very small or very large scales or in other extreme circumstances that flow from a quantum gravity mechanism?
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Do gluons form a dense system called colour glass condensate? What are the signatures and evidences for the Balitsky–Fadin–Kuarev–Lipatov, Balitsky–Kovchegov, Catani–Ciafaloni–Fiorani–Marchesini evolution equations? Nuclei and nuclear astrophysics: Why is there a lack of convergence in estimates of the mean lifetime of a free neutron based on two separate—and increasingly precise—experimental methods?
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What does the existence or absence of non-local phenomena imply about the fundamental structure of spacetime? How does this elucidate the proper interpretation of the fundamental nature of quantum physics? Quantum mind: Does quantum mechanical phenomena, such as entanglement and superposition, play an important part in the brain's function and can it explain critical aspects of consciousness?
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Locality: Are there non-local phenomena in quantum physics? If they exist, are non-local phenomena limited to the entanglement revealed in the violations of the Bell inequalities, or can information and conserved quantities also move in a non-local way? Under what circumstances are non-local phenomena observed?
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Interpretation of quantum mechanics: How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse or quantum decoherence, give rise to the reality we perceive? Another way of stating this question regards the measurement problem: What constitutes a "measurement" which apparently causes the wave function to collapse into a definite state? Unlike classical physical processes, some quantum mechanical processes (such as quantum teleportation arising from quantum entanglement) cannot be simultaneously "local", "causal", and "real", but it is not obvious which of these properties must be sacrificed, or if an attempt to describe quantum mechanical processes in these senses is a category error such that a proper understanding of quantum mechanics would render the question meaningless.
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Theory of everything: Is there a singular, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe? Dimensionless physical constants: At the present time, the values of various dimensionless physical constants cannot be calculated; they can be determined only by physical measurement. What is the minimum number of dimensionless physical constants from which all other dimensionless physical constants can be derived? Are dimensional physical constants necessary at all?
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The following is a list of notable unsolved problems grouped into broad areas of physics.Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon or experimental result. The others are experimental, meaning that there is a difficulty in creating an experiment to test a proposed theory or investigate a phenomenon in greater detail. There are still some questions beyond the Standard Model of physics, such as the strong CP problem, neutrino mass, matter–antimatter asymmetry, and the nature of dark matter and dark energy. Another problem lies within the mathematical framework of the Standard Model itself—the Standard Model is inconsistent with that of general relativity, to the point that one or both theories break down under certain conditions (for example within known spacetime singularities like the Big Bang and the centres of black holes beyond the event horizon).
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Shape of the universe: What is the 3-manifold of comoving space, i.e., of a comoving spatial section of the universe, informally called the "shape" of the universe? Neither the curvature nor the topology is presently known, though the curvature is known to be "close" to zero on observable scales. The cosmic inflation hypothesis suggests that the shape of the universe may be unmeasurable, but, since 2003, Jean-Pierre Luminet, et al., and other groups have suggested that the shape of the universe may be the Poincaré dodecahedral space.
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Or are most native conformations thermodynamically unstable, but kinetically trapped in metastable states? What keeps the high density of proteins present inside cells from precipitating? Quantum biology: Can coherence be maintained in biological systems at timeframes long enough to be functionally important? Are there non-trivial aspects of biology or biochemistry that can only be explained by the persistance of coherence as a mechanism?
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The injection problem: Fermi acceleration is thought to be the primary mechanism that accelerates astrophysical particles to high energy. However, it is unclear what mechanism causes those particles to initially have energies high enough for Fermi acceleration to work on them. Alfvénic turbulence: In the solar wind and the turbulence in solar flares, coronal mass ejections, and magnetospheric substorms are major unsolved problems in space plasma physics.
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Plasma physics and fusion power: Fusion energy may potentially provide power from an abundant resource (e.g. hydrogen) without the type of radioactive waste that fission energy currently produces. However, can ionized gases (plasma) be confined long enough and at a high enough temperature to create fusion power? What is the physical origin of H-mode?
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However, there are also long-duration GRBs that show evidence against an associated supernova, such as the Swift event GRB 060614. Solar neutrino problem (1968–2001): Solved by a new understanding of neutrino physics, requiring a modification of the Standard Model of particle physics—specifically, neutrino oscillation. Nature of quasars (1950s–1980s): The nature of quasars was not understood for decades.
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The Board of Genomics England includes a number of notable scientists: Sir John Chisholm (Executive Chair), former chair of the UK Medical Research Council Professor Sir Mark Caulfield (Chief Scientist), Queen Mary University of London Professor Sir John Bell, University of Oxford Dame Sally Davies (Non-executive Director), Chief Medical Officer (United Kingdom) Professor Ewan Birney (Non-executive Director), Director of the European Bioinformatics Institute (EMBL-EBI) Professor Sir Malcolm Grant CBE (Non-executive Director), Chair of the NHS England Board Professor Michael Parker (Non-executive Director; Chair of the Ethics Advisory Committee), University of Oxford Jon Symonds CBE (Non-executive Director), Chair of Innocoll AG, former Chief Financial Officer of Novartis AG Professor Dame Kay Davies (Non-executive Director), University of OxfordBaroness Nicola Blackwood of North Oxford became Chair of Genomics England in May 2020.
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Genomics England was formally established as a company on 17 April 2013 and was formally launched on 5 July 2013 as part of the celebrations for the 65th Birthday of the UK's National Health Service In August 2014, the Wellcome Trust announced that it was investing £27 million in a genome-sequencing hub for Genomics England, allowing the company to become part of the Wellcome Trust Genome Campus, home to the Sanger Institute. On the same date, Prime Minister David Cameron unveiled a new partnership between Genomics England and the sequencing firm Illumina. Illumina’s services for whole genome sequencing were secured in a deal worth around £78 million.The UK Government also committed £250 million to genomics in the 2015 Spending Review, which ensures the continued role of Genomics England to deliver the project, beyond the life of the project and up to 2021.On March 26, 2015, AstraZeneca announced that it has joined a public-private consortium with Genomics England to accelerate the development of new diagnostics and treatments arising from the 100,000 Genomes Project.In October 2018, the U.K. 's Secretary of State for Health and Social Care, Matt Hancock, announced that the program had been expanded with a new goal of sequencing five million genomes within five years. He also announced that starting in 2019, the NHS will offer whole genome sequencing (WGS) to all children suspected of having a rare genetic disease or with cancer.In July 2019, Genomics England announced Data Release 7, which included the 100,000th whole genome available to researchers.In June 2020, Genomics England announced a partnership with UK-based biotechnology company Lifebit to deploy a genomic research platform aimed at utilizing the genomic data generated through the 100,000 Genomes Project.
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Genomics England is a British company set up and owned by the United Kingdom Department of Health and Social Care to run the 100,000 Genomes Project. The project aimed in 2014 to sequence 100,000 genomes from NHS patients with a rare disease and their families, and patients with cancer. An infectious disease strand is being led by Public Health England.In the summer of 2019, Chris Wigley was appointed CEO of Genomics England, starting in October 2019. Wigley is a former McKinsey executive known for applying machine learning and artificial intelligence technology.
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Inner product space: an F vector space V with a definite bilinear form V × V → F. Bialgebra: an associative algebra with a compatible coalgebra structure. Lie bialgebra: a Lie algebra with a compatible bialgebra structure. Hopf algebra: a bialgebra with a connection axiom (antipode). Clifford algebra: an associative Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra additionally equipped with an exterior product from which several possible inner products may be derived. Exterior algebras and geometric algebras are special cases of this construction.
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Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. Graded algebra: a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of two elements a and b are known, then the grade of ab is known, and so the location of the product ab is determined in the decomposition.
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Lie algebra: a special type of nonassociative algebra whose product satisfies the Jacobi identity. Jordan algebra: a special type of nonassociative algebra whose product satisfies the Jordan identity. Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras.
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Associative algebra: an algebra over a ring such that the multiplication is associative. Nonassociative algebra: a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as alternation, the Jacobi identity, or the Jordan identity.
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This includes distributivity over addition and linearity with respect to multiplication by elements of R. Algebra over a field: This is a ring which is also a vector space over a field. Multiplication is usually assumed to be associative. The theory is especially well developed.
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These structures are defined over two sets, a ring R and an R-module M equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on R, two on M and one involving both R and M. Many of these structures are hybrid structures of the previously mentioned ones. Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure.
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Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. This can be shown to be equivalent with the ring-like structure of the same name above. Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by the infix operator →, and governed by the axioms: x → x = 1 x (x → y) = x y y (x → y) = y x → (y z) = (x → y) (x → z)
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The following module-like structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Modules, counting the ring operations, have at least three binary operations. Group with operators: a group G with a set Ω and a binary operation Ω × G → G satisfying certain axioms. Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra.
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Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
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Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors.
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From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition. Concrete examples of each structure will be found in the articles listed.
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In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms. Another branch of mathematics known as universal algebra studies algebraic structures in general.
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There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure. Topological vector spaces are vector spaces with a compatible topology. Lie groups: These are topological manifolds that also carry a compatible group structure. Ordered groups, ordered rings and ordered fields have algebraic structure compatible with an order on the set. Von Neumann algebras: these are *-algebras on a Hilbert space which are equipped with the weak operator topology.
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Robinson arithmetic with an axiom schema of induction. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.Lattice-like structures have two binary operations called meet and join, connected by the absorption law. Latticoid: meet and join commute but need not associate.
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Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties. *-algebra or *-ring: a ring with an additional unary operation (*) known as an involution, satisfying additional properties. Arithmetic: addition and multiplication on an infinite set, with an additional pointed unary structure.
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In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.
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Universal algebra studies algebraic structures abstractly, rather than specific types of structures. Varieties Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure. Example: The fundamental group of a topological space gives information about the topological space.
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Algebraic structures appear in most branches of mathematics, and one can encounter them in many different ways. Beginning study: In American universities, groups, vector spaces and fields are generally the first structures encountered in subjects such as linear algebra. They are usually introduced as sets with certain axioms. Advanced study: Abstract algebra studies properties of specific algebraic structures.
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Lie algebras Inner product spaces Kac–Moody algebra The quaternions and more generally geometric algebrasIn Mathematical logic: Boolean algebras are both rings and lattices, under their two operations. Heyting algebras are a special example of boolean algebras. Peano arithmetic Boundary algebra MV-algebraIn Computer science: Max-plus algebra Syntactic monoid Transition monoid
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Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields. In Physics: Lie groups are used extensively in physics. A few well-known ones include the orthogonal groups and the unitary groups.
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Set: a degenerate algebraic structure S having no operations. Pointed set: S has one or more distinguished elements, often 0, 1, or both. Unary system: S and a single unary operation over S. Pointed unary system: a unary system with S a pointed set.
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We can identify APS as a subalgebra of the spacetime algebra (STA) C ℓ 1 , 3 ( R ) {\displaystyle C\ell _{1,3}(\mathbb {R} )} , defining σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}} and I = γ 0 γ 1 γ 2 γ 3 {\displaystyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} . The γ μ {\displaystyle \gamma _{\mu }} s have the same algebraic properties of the gamma matrices but their matrix representation is not needed. The derivative is now The Riemann–Silberstein becomes a bivector and the charge and current density become a vector Owing to the identity Maxwell's equations reduce to the single equation
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It is of algebraic convenience that the geometric product is invertible, while the inner and outer products are not. As such, powerful techniques such as Green's functions can be used.
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This formulation uses the algebra that spacetime generates through the introduction of a distributive, associative (but not commutative) product called the geometric product. Elements and operations of the algebra can generally be associated with geometric meaning. The members of the algebra may be decomposed by grade (as in the formalism of differential forms) and the (geometric) product of a vector with a k-vector decomposes into a (k − 1)-vector and a (k + 1)-vector. The (k − 1)-vector component can be identified with the inner product and the (k + 1)-vector component with the outer product.
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In free space, where ε = ε0 and μ = μ0 are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgs-Gaussian units, not SI units are used. (To convert to SI, see here.) The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold.
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In advanced classical mechanics it is often useful, and in quantum mechanics frequently essential, to express Maxwell's equations in a potential formulation involving the electric potential (also called scalar potential) φ, and the magnetic potential (a vector potential) A. For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. The potentials can be introduced by using the Poincaré lemma on the homogeneous equations to solve them in a universal way (this assumes that we consider a topologically simple, e.g. contractible space). The potentials are defined as in the table above.
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Analogous to the tensor formulation, two objects, one for the electromagnetic field and one for the current density, are introduced. In geometric algebra (GA) these are multivectors.
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In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov–Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g., an iron wire magnetized longitudinally).
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These basis vectors share the algebra of the Pauli matrices, but are usually not equated with them, as they are different objects with different interpretations. After defining the derivative Maxwell's equations are reduced to the single equation In three dimensions, the derivative has a special structure allowing the introduction of a cross product: from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as
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In the Algebra of physical space (APS), also known as the Clifford algebra C ℓ 3 , 0 ( R ) {\displaystyle C\ell _{3,0}(\mathbb {R} )} , the field and current are represented by multivectors. The field multivector, known as the Riemann–Silberstein vector, is and the current multivector is using an orthonormal basis { σ k } {\displaystyle \{\sigma _{k}\}} . Similarly, the unit pseudoscalar is I = σ 1 σ 2 σ 3 {\displaystyle I=\sigma _{1}\sigma _{2}\sigma _{3}} , due to the fact that the basis used is orthonormal.
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While being a certified CIT may help in obtaining a chemistry related job, it is not typically required to do chemistry work. All work performed by an CIT must still be checked and certified by a P.Chem. == References ==
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A licensed CIT is certified to work in a chemistry profession. Upon completing a certain amount of work experience under the supervision of a Licensed Professional Chemist (P.Chem.) as determined by a licensing board, a CIT is then eligible to gain experience by supervision from a P.Chem.
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If we have a field extension F/K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product V F := V ⊗ K F {\displaystyle V_{F}:=V\otimes _{K}F} . So if A is an algebra over K, then A F {\displaystyle A_{F}} is an algebra over F.
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Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x, y in A. The space of all K-algebra homomorphisms between A and B is frequently written as H o m K -alg ( A , B ) . {\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).} A K-algebra isomorphism is a bijective K-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.
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For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K over a field. The construction of the unital zero algebra over a free R-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.
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If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K by the ideal generated by the EiEj for every pair (i, j). An example of unital zero algebra is the algebra of dual numbers, the unital zero R-algebra built from a one dimensional real vector space. These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or modules.
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An algebra is called zero algebra if uv = 0 for all u, v in the algebra, not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative. One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero. That is, if λ, μ ∈ K and u, v ∈ V, then (λ + u) (μ + v) = λμ + (λv + μu).
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{\displaystyle (k,a)\mapsto \eta (k)a.} Given two such associative unital K-algebras A and B, a unital K-algebra homomorphism f: A → B is a ring homomorphism that commutes with the scalar multiplication defined by η, which one may write as f ( k a ) = k f ( a ) {\displaystyle f(ka)=kf(a)} for all k ∈ K {\displaystyle k\in K} and a ∈ A {\displaystyle a\in A} . In other words, the following diagram commutes: K η A ↙ η B ↘ A f ⟶ B {\displaystyle {\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}}
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The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism η: K → Z ( A ) , {\displaystyle \eta \colon K\to Z(A),} where Z(A) is the center of A. Since η is a ring homomorphism, then one must have either that A is the zero ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication K × A → A {\displaystyle K\times A\to A} given by ( k , a ) ↦ η ( k ) a .
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Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (that is, if x and y are any two elements of A, then x · y is an element of A that is called the product of x and y). Then A is an algebra over K if the following identities hold for all elements x, y, z in A , and all elements (often called scalars) a and b in K: Right distributivity: (x + y) · z = x · z + y · z Left distributivity: z · (x + y) = z · x + z · y Compatibility with scalars: (ax) · (by) = (ab) (x · y).These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily associative, although some authors use the term algebra to refer to an associative algebra. When a binary operation on a vector space is commutative, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.
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The test had 50 multiple choice questions that were to be answered in one hour. All questions had five answer choices. Students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank. The questions covered a broad range of topics. Approximately 10-14% of questions focused on Numbers and Operations, 38-42% focused on Algebra and functions, 38-42% focused on Geometry (including Euclidean, coordinate, three-dimensional, and trigonometry), and 6-10% focused on Data analysis, Statistics, and probability.
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SAT Subject Test in Mathematics Level 1
| 0.839734
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1,360
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The SAT Subject Test in Mathematics Level 2 covered more advanced content. Generally you need to have completed a semester of a pre-calculus class with a solid “B” or better to feel comfortable on the Math 1, whereas the content of the Math 2 test extends through Algebra II and basic trigonometry, precalculus, and basic calculus.On January 19, 2021, the College Board discontinued all SAT Subject tests, including the SAT Subject Test in Mathematics Level 1. This was effective immediately in the United States, and the tests were to be phased out by the following summer for international students. This was done as a response to changes in college admissions due to the impact of the COVID-19 pandemic on education.
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SAT Subject Test in Mathematics Level 1
| 0.839734
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1,361
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The SAT Subject Test in Mathematics Level 1 (formerly known as Math I or MathIC (the "C" representing the use of a calculator)) was the name of a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and basic foundations of calculus by The College Board. A student chose whether to take the test depending upon college entrance requirements for the schools in which the student is planning to apply. Until 1994, the SAT Subject Tests were known as Achievement Tests; and from 1995 until January 2005, they were known as SAT IIs. Mathematics Level 1 was taken 109,048 times in 2006.
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SAT Subject Test in Mathematics Level 1
| 0.839734
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1,362
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The College Board suggested as preparation for the test three years of mathematics, including two years of algebra, and one year of geometry.
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SAT Subject Test in Mathematics Level 1
| 0.839734
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1,363
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For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under § Addition), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, a − b = a + (−b). The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved. For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others.
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Arithmetical operations
| 0.839727
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1,364
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Elementary arithmetic is typically taught at the primary or secondary school levels and is governed by local educational standards. In the United States and Canada, there has been debate about the content and methods used to teach elementary arithmetic. One issue has been the use of calculators versus manual computation, with some arguing that calculator use should be limited to promote mental arithmetic skills. Another debate has centered on the distinction between traditional and reform mathematics, with traditional methods often focusing more on basic computation skills and reform methods placing a greater emphasis on higher-level mathematical concepts such as algebra, statistics, and problem-solving. In the United States, the 1989 National Council of Teachers of Mathematics (NCTM) standards led to a shift in elementary school curricula that de-emphasized or omitted certain topics traditionally considered to be part of elementary arithmetic, in favor of a greater focus on college-level concepts such as algebra and statistics. This shift has been controversial, with some arguing that it has resulted in a lack of emphasis on basic computation skills that are important for success in later math classes.
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Elementary arithmetic
| 0.839705
|
1,365
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The asterisk notation is most commonly used in computer programming languages. In algebra, the multiplication symbol may be omitted; for example, xy represents x multiplied by y. The order in which two numbers are multiplied does not affect the result. This is known as the commutative property of multiplication.
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Elementary arithmetic
| 0.839705
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1,366
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A few things should be considered when improving the accuracy of the decision tree classifier. The following are some possible optimizations to consider when looking to make sure the decision tree model produced makes the correct decision or classification. Note that these things are not the only things to consider but only some.
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Decision trees
| 0.839698
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1,367
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Some graphing calculators have a computer algebra system (CAS), which means that they are capable of producing symbolic results. These calculators can manipulate algebraic expressions, performing operations such as factor, expand, and simplify. In addition, they can give answers in exact form without numerical approximations. Calculators that have a computer algebra system are called symbolic or CAS calculators.
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Graphing Calculators
| 0.839654
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1,368
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The book is aimed at students, written for a general audience, and does not require any background in mathematics beyond high school algebra. However, many of its chapters include exercises, making it suitable for teaching high school or undergraduate-level courses using it. It is also suitable for readers interested in recreational mathematics.
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The Mathematics of Games and Gambling
| 0.83962
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1,369
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The book has seven chapters. Its first gives a survey of the history of gambling games in western culture, including brief biographies of two famous gamblers, Gerolamo Cardano and Fyodor Dostoevsky, and a review of the games of chance found in Dostoevsky's novel The Gambler. The next four chapters introduce the basic concepts of probability theory, including expectation, binomial distributions and compound distributions, and conditional probability, through games including roulette, keno, craps, chuck-a-luck, backgammon, and blackjack.The sixth chapter of the book moves from probability theory to game theory, including material on tic-tac-toe, matrix representations of zero-sum games, nonzero-sum games such as the prisoner's dilemma, the concept of a Nash equilibrium, game trees, and the minimax method used by computers to play two-player strategy games. A final chapter, "Odds and ends", includes analyses of bluffing in poker, horse racing, and lotteries.The second edition adds material on online gambling systems, casino poker machines, and Texas hold 'em poker. It also adds links to online versions of the games, and expands the material on game theory.
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The Mathematics of Games and Gambling
| 0.83962
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1,370
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The Mathematics of Games and Gambling is a book on probability theory and its application to games of chance. It was written by Edward Packel, and published in 1981 by the Mathematical Association of America as volume 28 of their New Mathematical Library series, with a second edition in 2006.
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The Mathematics of Games and Gambling
| 0.83962
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1,371
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Chemistry sets may have been the first American toys marketed toward parents with the goal of "improving" children for success in later life. The target market for chemistry sets was almost exclusively boys, deemed "young men of science." However, during the 1950s, Gilbert introduced a set targeting girls.
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Chemistry set
| 0.839619
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1,372
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Robert Treat Johnson, noting the number of chemistry students at Yale whose interest in the science began with a chemistry set, argued the production of chemistry sets was a "patriotic duty. "Toy companies promoted chemistry sets through advertising campaigns, the "Chemcraft Chemist Club" and its accompanying "Chemcraft Science Magazine", comic books, and essay contests such as Porter's "Why I want to be a scientist". The goal of attracting students to a potential career in chemistry was often explicit in the sets' naming and promotion.
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Chemistry set
| 0.839619
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1,373
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Alfred Carlton Gilbert earned money by performing magic tricks while a medical student at Yale. He and John Petrie formed the Mysto Manufacturing Company (later the A. C. Gilbert Company) in 1909, and began selling boxed magic sets. By 1917, they sold chemistry sets, which they produced through World War II, in spite of restrictions on materials.
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Chemistry set
| 0.839619
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1,374
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Beginning in the early 1900s, modern chemistry sets targeted younger people with the intention to popularize chemistry. In the United States, Porter Chemical Company and the A. C. Gilbert Company produced the best known sets. Although Porter and Gilbert were the largest American producers of chemistry sets, other manufacturers such as the Skilcraft corporation were also active.John J. Porter and his brother Harold Mitchell Porter began The Porter Chemical Company in 1914. Their initial purpose was to sell packaged chemicals, but they soon introduced kits.
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Chemistry set
| 0.839619
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1,375
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A chemistry set is an educational toy allowing the user (typically a teenager) to perform simple chemistry experiments.
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Chemistry set
| 0.839619
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1,376
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Typical contents found in chemistry sets, including equipment and chemicals, might include:
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Chemistry set
| 0.839618
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1,377
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Primarily used for training druggists and medical students, they could also be carried and used in the field.Scientific kits also attracted well-educated members of the upper class who enjoyed experimenting and demonstrating their results. James Woodhouse of Philadelphia presented a Young Chemist's Pocket Companion (1797) with an accompanying portable laboratory, specifically targeted ladies and gentlemen. Jane Marcet's books on chemistry helped to popularize chemistry as a well-to-do pastime for both men and women.Beginning in the late 1850s John J. Griffin & Sons sold a line of "chemical cabinets", eventually offering 11 categories. These were marketed primarily to adults including elementary school teachers as well as students at the Royal Naval College, the Royal Agricultural Society, and the universities of Oxford and Cambridge.In the mid to late 1800s England, magic and illusion toys enabled children to make their own fireworks, create disappearing inks and cause changes in color, tricks which were mostly chemically based. The Columbian Cyclopedia of 1897 defines "CHEMISTRY TOYS" as "mostly pyrotechnic; recommended as illustrating to the young the rudiments of chemistry, but probably more dangerous than efficient for such use", listing a variety of hazardous examples.
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Chemistry set
| 0.839618
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1,378
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The forerunners of the chemistry set were 17th century books on "natural magick", "which all excellent wise men do admit and embrace, and worship with great applause; neither is there any thing more highly esteemed, or better thought of, by men of learning." Authors such as Giambattista della Porta included chemical magic tricks and scientific puzzles along with more serious topics.The earliest chemistry sets were developed in the 18th century in England and Germany to teach chemistry to adults. In 1791, Description of a portable chest of chemistry: or, Complete collection of chemical tests for the use of chemists, physicians, mineralogists, metallurgists, scientific artists, manufacturers, farmers, and the cultivators of natural philosophy by Johann Friedrich August Göttling, translated from German, was published in English. Friedrich Accum of London, England also sold portable chemistry sets and materials to refill them.
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Chemistry set
| 0.839618
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1,379
|
The experiments described in the instruction manual typically require a number of chemicals not shipped with the chemistry set, because they are common household chemicals: Acetic acid (in vinegar) Ammonium carbonate ("baker's ammonia" or "salts of hartshorn") Citric acid (in lemons) Ethanol (in denatured alcohol) Sodium bicarbonate (baking soda) Sodium chloride ("table salt")Other chemicals, including strong acids, bases and oxidizers cannot be safely shipped with the set and others having a limited shelf life have to be purchased separately from a drug store: Hydrochloric acid Hydrogen peroxide Silver nitrate Sodium hydroxide
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Chemistry set
| 0.839618
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1,380
|
They sold the set in an attractive pink box, but the set identified girls as "laboratory assistants" or "lab technicians," not scientists.In 1971, a Johnny Horizon Environmental Test Kit was licensed by the U. S. Department of the Interior and produced by Parker Brothers. It included four air pollution tests and six water pollution tests for young environmental scientists. The Johnny Horizon Environmental Test Kit was marketed to both boys and girls.Well-known chemistry sets from the United Kingdom include the 1960s and 1970s sets by Thomas Salter Science (produced in Scotland) and later Salter Science, then the "MERIT" sets through the 1970s and 1980s. Dekkertoys created a range of sets which were similar, complete with glass test tubes of dry chemicals.
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Chemistry set
| 0.839618
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1,381
|
Around the 1960s, safety concerns began to limit the range of materials and experiments available in chemistry sets. In the United States, the Federal Hazardous Substances Labeling Act of 1960, the Toy Safety Act of 1969, the Consumer Product Safety Commission, established in 1972, and the Toxic Substances Control Act of 1976 all introduced new levels of regulation, which led to the decline of chemistry sets' popularity during the 1970s and 1980s. The A. C. Gilbert Company went out of business in 1967, and the Porter Chemical Company went out of business in 1984.Modern chemistry sets, with a few exceptions, tend to include a more restricted range of chemicals and simplified instructions. Many chemistry kits are single use, containing only the types and amounts of chemicals for a specific application. Several authors note from the 1980s on, concerns about illegal drug production, terrorism, and legal liability have led to chemistry sets becoming increasingly bland and unexciting.Nonetheless, a GCSE equipment set was produced, offering students better equipment, and there is a more up-market range of sets available from Thames & Kosmos such as the C3000 Kit.
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Chemistry set
| 0.839618
|
1,382
|
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra.
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Mathematical diagram
| 0.839593
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1,383
|
For very short interatomic separations, important in radiation material science, the interactions can be described quite accurately with screened Coulomb potentials which have the general form V ( r i j ) = 1 4 π ε 0 Z 1 Z 2 e 2 r i j φ ( r / a ) {\displaystyle V(r_{ij})={1 \over 4\pi \varepsilon _{0}}{Z_{1}Z_{2}e^{2} \over r_{ij}}\varphi (r/a)} Here, φ ( r ) → 1 {\displaystyle \varphi (r)\to 1} when r → 0 {\displaystyle r\to 0} . Z 1 {\displaystyle Z_{1}} and Z 2 {\displaystyle Z_{2}} are the charges of the interacting nuclei, and a {\displaystyle a} is the so-called screening parameter. A widely used popular screening function is the "Universal ZBL" one. and more accurate ones can be obtained from all-electron quantum chemistry calculations In binary collision approximation simulations this kind of potential can be used to describe the nuclear stopping power.
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Interatomic potential
| 0.83957
|
1,384
|
The arguably simplest widely used interatomic interaction model is the Lennard-Jones potential V L J ( r ) = 4 ε {\displaystyle V_{\mathrm {LJ} }(r)=4\varepsilon \left} where ε {\displaystyle \textstyle \varepsilon } is the depth of the potential well and σ {\displaystyle \textstyle \sigma } is the distance at which the potential crosses zero. The attractive term proportional to 1 / r 6 {\displaystyle \textstyle 1/r^{6}} in the potential comes from the scaling of van der Waals forces, while the 1 / r 12 {\displaystyle \textstyle 1/r^{12}} repulsive term is much more approximate (conveniently the square of the attractive term). On its own, this potential is quantitatively accurate only for noble gases and has been extensively studied in the past decades, but is also widely used for qualitative studies and in systems where dipole interactions are significant, particularly in chemistry force fields to describe intermolecular interactions - especially in fluids.Another simple and widely used pair potential is the Morse potential, which consists simply of a sum of two exponentials. V M ( r ) = D e ( e − 2 a ( r − r e ) − 2 e − a ( r − r e ) ) {\displaystyle V_{\mathrm {M} }(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})} Here D e {\displaystyle \textstyle D_{e}} is the equilibrium bond energy and r e {\displaystyle \textstyle r_{e}} the bond distance.
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Interatomic potential
| 0.83957
|
1,385
|
These feature vectors are then used to directly predict the final potentials. In 2017, the first-ever MPNN model, a deep tensor neural network, was used to calculate the properties of small organic molecules. Advancements in this technology led to the development of Matlantis in 2022, which commercially applies machine learning potentials for new materials discovery. Matlantis, which can simulate 72 elements, handle up to 20,000 atoms at a time, and execute calculations up to 20 million times faster than density functional theory with almost indistinguishable accuracy, showcases the power of machine learning potentials in the age of artificial intelligence.
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Interatomic potential
| 0.83957
|
1,386
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Atomic coordinates are sometimes transformed with atom-centered symmetry functions or pair symmetry functions before being fed into neural networks. Encoding symmetry has been pivotal in enhancing machine learning potentials by drastically constraining the neural networks' search space.Conversely, Message-Passing Neural Networks (MPNNs), a form of graph neural networks, learn their own descriptors and symmetry encodings. They treat molecules as three-dimensional graphs and iteratively update each atom's feature vectors as information about neighboring atoms is processed through message functions and convolutions.
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Interatomic potential
| 0.83957
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1,387
|
However, with continuous advancements in artificial intelligence technology, machine learning methods have become significantly more accurate, positioning machine learning as a significant player in potential fitting.Modern neural networks have revolutionized the construction of highly accurate and computationally light potentials by integrating theoretical understanding of materials science into their architectures and preprocessing. Almost all are local, accounting for all interactions between an atom and its neighbor up to some cutoff radius. These neural networks usually intake atomic coordinates and output potential energies.
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Interatomic potential
| 0.83957
|
1,388
|
Since the 1990s, machine learning programs have been employed to construct potentials, mapping atomic structures to their potential energies. Such machine learning potentials help fill the gap between highly accurate but computationally intensive simulations like density functional theory and computationally lighter, but much less precise, empirical potentials. Early neural networks showed promise, but their inability to systematically account for interatomic energy interactions limited their applications to smaller, low-dimensional systems, keeping them largely within the confines of academia.
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Interatomic potential
| 0.83957
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1,389
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However, the accuracy of a machine-learning potential can be converged to be comparable with the underlying quantum calculations, unlike analytical models. Hence, they are in general more accurate than traditional analytical potentials, but they are correspondingly less able to extrapolate. Further, owing to the complexity of the machine-learning model and the descriptors, they are computationally far more expensive than their analytical counterparts. Non-parametric, machine learned potentials may also be combined with parametric, analytical potentials, for example to include known physics such as the screened Coulomb repulsion, or to impose physical constraints on the predictions.
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Interatomic potential
| 0.83957
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1,390
|
However, more complex many-body descriptors are needed to produce highly accurate potentials. It is also possible to use a linear combination of multiple descriptors with associated machine-learning models. Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including neural networks, Gaussian process regression, and linear regression.A non-parametric potential is most often trained to total energies, forces, and/or stresses obtained from quantum-level calculations, such as density functional theory, as with most modern potentials.
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Interatomic potential
| 0.83957
|
1,391
|
E {\displaystyle E} is a machine-learning model that provides a prediction for the energy of atom i {\displaystyle i} based on the descriptor output. An accurate machine-learning potential requires both a robust descriptor and a suitable machine learning framework. The simplest descriptor is the set of interatomic distances from atom i {\displaystyle i} to its neighbours, yielding a machine-learned pair potential.
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Interatomic potential
| 0.83957
|
1,392
|
It should first be noted that non-parametric potentials are often referred to as "machine learning" potentials. While the descriptor/mapping forms of non-parametric models are closely related to machine learning in general and their complex nature make machine learning fitting optimizations almost necessary, differentiation is important in that parametric models can also be optimized using machine learning. Current research in interatomic potentials involves using systematically improvable, non-parametric mathematical forms and increasingly complex machine learning methods. The total energy is then writtenwhere q i {\displaystyle \mathbf {q} _{i}} is a mathematical representation of the atomic environment surrounding the atom i {\displaystyle i} , known as the descriptor.
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Interatomic potential
| 0.83957
|
1,393
|
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.
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Interatomic potential
| 0.83957
|
1,394
|
In most practical cases deterministic algorithms or randomized algorithms are discussed, although theoretical computer science also considers nondeterministic algorithms and other advanced models of computation.
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Asymptotic time complexity
| 0.839565
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1,395
|
Statistical estimations of the likely election results from opinion polls also involve algorithmic calculations, but produces ranges of possibilities rather than exact answers. To calculate means to determine mathematically in the case of a number or amount, or in the case of an abstract problem to deduce the answer using logic, reason or common sense. The English word derives from the Latin calculus, which originally meant a small stone in the gall-bladder (from Latin calx).
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Calculation
| 0.839549
|
1,396
|
Aryabhata, in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits. It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they cannot be written exactly as m n {\displaystyle {\frac {m}{n}}} , where m and n are integers). This is the theorem Euclid X, 9, almost certainly due to Theaetetus dating back to circa 380 BC.
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Square root
| 0.839539
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1,397
|
An accurate model with relatively poor precision could be useful to study the evolutionary relationships between the structures of a set of proteins, whereas the rational drug design requires both precise and accurate models. A model that is not accurate, regardless of the degree of precision with which it was obtained will not be very useful.Since protein structures are experimental models that can contain errors, it is very important to be able to detect these errors. The process aimed at the detection of errors is known as validation. There are several methods to validate structures, some are statistical like PROCHECK and WHAT IF while others are based on physical principles as CheShift, or a mixture of statistical and physics principles PSVS.
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Protein nuclear magnetic resonance
| 0.839531
|
1,398
|
A typical study might involve how two proteins interact with each other, possibly with a view to developing small molecules that can be used to probe the normal biology of the interaction ("chemical biology") or to provide possible leads for pharmaceutical use (drug development). Frequently, the interacting pair of proteins may have been identified by studies of human genetics, indicating the interaction can be disrupted by unfavorable mutations, or they may play a key role in the normal biology of a "model" organism like the fruit fly, yeast, the worm C. elegans, or mice. To prepare a sample, methods of molecular biology are typically used to make quantities by bacterial fermentation. This also permits changing the isotopic composition of the molecule, which is desirable because the isotopes behave differently and provide methods for identifying overlapping NMR signals.
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Protein nuclear magnetic resonance
| 0.839531
|
1,399
|
This change of scale requires much higher sensitivity of detection and stability for long term measurement. In contrast to MRI, structural biology studies do not directly generate an image, but rely on complex computer calculations to generate three-dimensional molecular models.
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Protein nuclear magnetic resonance
| 0.839531
|
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