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26,400 | While traditional development statistics is mainly concerned with the representativeness of random survey samples, digital trace data is never a random sample. Generalizability. While observational data always represents this source very well, it only represents what it represents, and nothing more. | Big data analysis | 0.80716 |
26,401 | The Theorem Proving System (TPS) is an automated theorem proving system for first-order and higher-order logic. TPS has been developed at Carnegie Mellon University. An educational version of it is known as ETPS (Educational Theorem Proving System). | Theorem Proving System | 0.80716 |
26,402 | Roche is the only drug company authorized to manufacture the drug, which was discovered by Gilead Sciences. Roche purchased the rights to the drug in 1996, and in 2005, settled a royalty dispute, agreeing to pay Gilead tiered royalties of 14–22% of annual net sales without adjusting the payments for manufacturing costs... | Roche | 0.807159 |
26,403 | Roche is the fifth-largest pharmaceutical company in the world by revenue and the leading provider of cancer treatments globally.The company controls the American biotechnology company Genentech, which is a wholly owned affiliate, and the Japanese biotechnology company Chugai Pharmaceuticals, as well as the United Stat... | Roche | 0.807159 |
26,404 | Roche has two major divisions: Pharmaceuticals and Diagnostics. Roche Diagnostics manufactures diagnostic equipment and reagents for research and medical diagnostic applications. Internally, it is organised into five major business areas: Roche Applied Science, Roche Professional Diagnostics, Roche Diabetes Care, Roche... | Roche | 0.807159 |
26,405 | The MGU is largely derived from the earlier Radiobiology Unit (RBU). In 2010, work at the unit discovered that overexpression of the FTO gene led to obesity. | Mammalian Genetics Unit | 0.807155 |
26,406 | The unit carries out work into genetics and genome engineering. | Mammalian Genetics Unit | 0.807155 |
26,407 | The Mammalian Genetics Unit was a genetics and genomics research institute in Oxfordshire. | Mammalian Genetics Unit | 0.807155 |
26,408 | Earlier research on the same site at the Radiobiology Research Unit, which opened in 1954, in the 1950s was into cytogenetics, where Charles Edmund Ford and John Hamerton confirmed on 12 January 1956 the size of the human genome. In the early 1970s this unit led research into mutagenic effects of radiation on the human... | Mammalian Genetics Unit | 0.807155 |
26,409 | The Centro de Astrofísica da Universidade do Porto (Centre for Astrophysics of the University of Porto - CAUP) is the largest astronomy research institute in Portugal, with more than 60 people. Since 2000 it has been evaluated as Excellent by international panels, organized under the auspices of the national science fo... | Centre for Astrophysics of the University of Porto | 0.807155 |
26,410 | Origin and Evolution of Stars and Planets - Star Formation and Early Evolution; Planetary Systems (Exoplanets); Stellar Populations and Stellar Evolution Galaxies and Observational Cosmology - Physical properties of massive galaxies; Galaxy cluster astrophysics; Structure formation paradigms; Dynamical dark energy; Var... | Centre for Astrophysics of the University of Porto | 0.807155 |
26,411 | The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to se... | Q-difference equation | 0.807155 |
26,412 | In this sense, the function x n {\displaystyle x^{n}} is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of x n {\displaystyle x^{n}} is instead the falling factorial, ( x ) n := x ( x − 1 ) ⋯ ( x − n + 1 ) . {\displaystyle (x)_{n}:=x(x-1)\cdots (x-n+1).} One may proceed further ... | Q-difference equation | 0.807155 |
26,413 | The derivative of the function x n {\displaystyle x^{n}} (for some positive integer n {\displaystyle n} ) in the classical calculus is n x n − 1 {\displaystyle nx^{n-1}} . The corresponding expressions in q-calculus and h-calculus are D q ( x n ) = q n − 1 q − 1 x n − 1 = q x n − 1 {\displaystyle D_{q}(x^{n})={\frac {... | Q-difference equation | 0.807155 |
26,414 | Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula q = e i h = e ... | Q-difference equation | 0.807155 |
26,415 | In biology, polymorphism is the occurrence of two or more clearly different morphs or forms, also referred to as alternative phenotypes, in the population of a species. To be classified as such, morphs must occupy the same habitat at the same time and belong to a panmictic population (one with random mating).Put simply... | Monomorphism (biology) | 0.807154 |
26,416 | Polymorphism was crucial to research in ecological genetics by E. B. Ford and his co-workers from the mid-1920s to the 1970s (similar work continues today, especially on mimicry). The results had a considerable effect on the mid-century evolutionary synthesis, and on present evolutionary theory. The work started at a t... | Monomorphism (biology) | 0.807154 |
26,417 | Investigation of polymorphism requires use of both field and laboratory techniques. In the field: detailed survey of occurrence, habits and predation selection of an ecological area or areas, with well-defined boundaries capture, mark, release, recapture data relative numbers and distribution of morphs estimation of po... | Monomorphism (biology) | 0.807154 |
26,418 | First, coordinates of cluster peaks (e.g. the voxels where the differences between patients and healthy controls were highest), and statistical maps if available, are selected according to SDM inclusion criteria. Second, coordinates are used to recreate statistical maps, and effect-sizes maps and their variances are de... | Seed-based d mapping | 0.807152 |
26,419 | Such measurements are used in research on neuroscience and intelligence. Quite separately, certain artists from the 15th century onward made measurements of heads and skulls with a view to attaining greater accuracy in their representation of those parts of the human frame. Bernard Palissy and Albrecht Dürer were pione... | Craniometry | 0.807149 |
26,420 | More direct measurements involve examinations of brains from corpses, or more recently, imaging techniques such as MRI, which can be used on living persons. Such measurements are used in research on neuroscience and intelligence. Brain volume data and other craniometric data are used in mainstream science to compare mo... | Craniometry | 0.807149 |
26,421 | K acts unitarily on Hσ and A by a uniformly bounded representation on H 'σ. The action of the standard basis of the complexification Lie algebra on this basis can be computed: π s ( L 0 ) f m = m f m , π s ( L − 1 ) f m = − ( m + 1 / 2 + s ) f m + 1 , π s ( L 1 ) f m = − ( m − 1 / 2 − s ) f m − 1 . {\displaystyle \disp... | Dixmier Problem | 0.807149 |
26,422 | The representations are irreducible because the action of the Lie algebra on the basis vectors fm is irreducible. This family of irreducible unitary representations is called the complementary series. Ehrenpreis & Mautner (1955) constructed an analytic continuation of this family of representations as follows. | Dixmier Problem | 0.807149 |
26,423 | The chromosomes presented above are well suited for processing tasks of continuous, mixed-integer, pure-integer or combinatorial optimization. For a combination of these optimization areas, on the other hand, it becomes increasingly difficult to map them to simple strings of values, depending on the task. The following... | Chromosome (genetic algorithm) | 0.807147 |
26,424 | Cellular neuroscience is a branch of neuroscience concerned with the study of neurons at a cellular level. This includes morphology and physiological properties of single neurons. Several techniques such as intracellular recording, patch-clamp, and voltage-clamp technique, pharmacology, confocal imaging, molecular biol... | Cellular neuroscience | 0.807146 |
26,425 | The basic properties, activity and regulation of membrane currents, synaptic transmission and synaptic plasticity, neurotransmission, neuroregensis, synaptogenesis and ion channels of cells are a few other fields studied by cellular neuroscientists. Tissue, cellular and subcellular anatomy are studied to provide insigh... | Cellular neuroscience | 0.807146 |
26,426 | Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to examine the structure of groups. There are also applications in harmonic analysis and number theory. For example, representation theory is used in the modern approach to g... | Complex representations of finite groups | 0.807146 |
26,427 | Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of... | Complex representations of finite groups | 0.807146 |
26,428 | {\displaystyle \dim({\text{Hom}}_{G}(V_{\eta },V_{I}))=\langle V_{\eta },V_{I}\rangle _{G}.} Let ( ρ , V ρ ) {\displaystyle (\rho ,V_{\rho })} be a representation of G , {\displaystyle G,} then there exists a canonical isomorphism T: Hom G ( V ρ , I H G ( η ) ) → Hom H ( V ρ | H , V η ) . {\displaystyle T:{\text{Hom}}_... | Complex representations of finite groups | 0.807146 |
26,429 | Let χ {\displaystyle \chi } be the character of the representation V , {\displaystyle V,} then ( χ | χ ) {\displaystyle (\chi |\chi )} is a positive integer. Moreover ( χ | χ ) = 1 {\displaystyle (\chi |\chi )=1} if and only if V {\displaystyle V} is irreducible.Therefore, using the first theorem, the characters of irr... | Complex representations of finite groups | 0.807146 |
26,430 | {\displaystyle 1.} Theorem. Let V {\displaystyle V} be a representation of G {\displaystyle G} with character χ V . | Complex representations of finite groups | 0.807146 |
26,431 | We define the matrix coefficients τ k , l ( s ) = ⟨ τ ( s ) e k , e l ⟩ {\displaystyle \tau _{k,l}(s)=\langle \tau (s)e_{k},e_{l}\rangle } for k , l ∈ { 1 , … , dim ( τ ) } , s ∈ G . {\displaystyle k,l\in \{1,\ldots ,\dim(\tau )\},s\in G.} Then we have the following orthonormal basis of L 2 ( G ) {\displaystyle L^{2}... | Complex representations of finite groups | 0.807146 |
26,432 | Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. It is usually presented and proven in harmonic analysis, as it represents one of its central and fundamental statements. The Peter-Weyl Theorem. | Complex representations of finite groups | 0.807146 |
26,433 | Theorem. (i) V ρ ( τ ) {\displaystyle V_{\rho }(\tau )} is a closed invariant subspace of V ρ . {\displaystyle V_{\rho }.} | Complex representations of finite groups | 0.807146 |
26,434 | The map π ′ {\displaystyle \pi '} is again a continuous group homomorphism and thus a representation. On Hilbert spaces: π {\displaystyle \pi } is irreducible if and only if π ′ {\displaystyle \pi '} is irreducible. By transferring the results of the section decompositions to compact groups, we obtain the following the... | Complex representations of finite groups | 0.807146 |
26,435 | A representation ( π , V π ) {\displaystyle (\pi ,V_{\pi })} of a group G {\displaystyle G} extends to a ∗ {\displaystyle ^{*}} –algebra homomorphism π: L 1 ( G ) → End ( V π ) {\displaystyle \pi :L^{1}(G)\to {\text{End}}(V_{\pi })} by π ( δ s ) = π ( s ) . {\displaystyle \pi (\delta _{s})=\pi (s).} Since multiplicity ... | Complex representations of finite groups | 0.807146 |
26,436 | The involution f ∗ ( s ) = f ( s − 1 ) ¯ {\displaystyle f^{*}(s)={\overline {f(s^{-1})}}} turns L 1 ( G ) {\displaystyle L^{1}(G)} into a ∗ {\displaystyle ^{*}} –algebra. We have δ s ∗ = δ s − 1 . {\displaystyle \delta _{s}^{*}=\delta _{s^{-1}}.} | Complex representations of finite groups | 0.807146 |
26,437 | {\displaystyle \mathbb {C} .} Thus, the convolution algebra and the group algebra are isomorphic as algebras. | Complex representations of finite groups | 0.807146 |
26,438 | The convolution of two elements f , h ∈ L 1 ( G ) {\displaystyle f,h\in L^{1}(G)} defined by f ∗ h ( s ) := ∑ t ∈ G f ( t ) h ( t − 1 s ) {\displaystyle f*h(s):=\sum _{t\in G}f(t)h(t^{-1}s)} makes L 1 ( G ) {\displaystyle L^{1}(G)} an algebra. The algebra L 1 ( G ) {\displaystyle L^{1}(G)} is called the convolution alg... | Complex representations of finite groups | 0.807146 |
26,439 | {\displaystyle G.} The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid: Theorem. Let χ {\displaystyle \chi } and χ ′ {\displaystyle \chi '} be the characters of two non-isomorphic irreducible representatio... | Complex representations of finite groups | 0.807146 |
26,440 | First, we note that the direct product G 1 × G 2 {\displaystyle G_{1}\times G_{2}} of two compact groups is again a compact group when provided with the product topology. The Haar measure on the direct product is then given by the product of the Haar measures on the factor groups. For the dual representation on compact... | Complex representations of finite groups | 0.807146 |
26,441 | The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. The direct sum and the tensor product with a finite number of summands/factors are defined in exactly the same way as for finite groups. This i... | Complex representations of finite groups | 0.807146 |
26,442 | {\displaystyle s\in G,v\in V.} Theorem. The irreducible representations of G 1 × G 2 {\displaystyle G_{1}\times G_{2}} up to isomorphism are exactly the representations ρ 1 ⊗ ρ 2 {\displaystyle \rho _{1}\otimes \rho _{2}} in which ρ 1 {\displaystyle \rho _{1}} and ρ 2 {\displaystyle \rho _{2}} are irreducible represent... | Complex representations of finite groups | 0.807146 |
26,443 | For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary. Using the convolution algebra we can implement a Fourier transformation on a group G . | Complex representations of finite groups | 0.807146 |
26,444 | In the same way C {\displaystyle \mathbb {C} } as a right C {\displaystyle \mathbb {C} } –module corresponds to the right-regular representation. In the following we will define the convolution algebra: Let G {\displaystyle G} be a group, the set L 1 ( G ) := { f: G → C } {\displaystyle L^{1}(G):=\{f:G\to \mathbb {C}... | Complex representations of finite groups | 0.807146 |
26,445 | Vice versa we obtain a linear representation of G {\displaystyle G} starting from a K {\displaystyle K} –module V {\displaystyle V} . Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. | Complex representations of finite groups | 0.807146 |
26,446 | Let G {\displaystyle G} be a finite group, let K {\displaystyle K} be a commutative ring and let K {\displaystyle K} be the group algebra of G {\displaystyle G} over K . {\displaystyle K.} This algebra is free and a basis can be indexed by the elements of G . {\displaystyle G.} | Complex representations of finite groups | 0.807146 |
26,447 | Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations: The Fourier inversion formula: f ( s ) = 1 | G | ∑ ρ irr. rep. of G dim ( V ρ ) ⋅ Tr ( ρ ( s − 1 ) ⋅ f ^ ( ρ ) ) . | Complex representations of finite groups | 0.807146 |
26,448 | A given irreducible representation V {\displaystyle V} of G {\displaystyle G} is contained n {\displaystyle n} –times in the regular representation. In other words, if R {\displaystyle R} denotes the regular representation of G {\displaystyle G} then we have: R ≅ ⊕ ( W j ) ⊕ dim ( W j ) , {\displaystyle R\cong \oplus... | Complex representations of finite groups | 0.807146 |
26,449 | {\displaystyle (\chi |\chi )\in \mathbb {N} _{0}.} The case ( χ | χ ) = 1 {\displaystyle (\chi |\chi )=1} holds if and only if V {\displaystyle V} is irreducible. Therefore, using the first theorem, the characters of irreducible representations of G {\displaystyle G} form an orthonormal set on C class ( G ) {\displayst... | Complex representations of finite groups | 0.807146 |
26,450 | It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations. Theorem. | Complex representations of finite groups | 0.807146 |
26,451 | The set of all class functions is a C {\displaystyle \mathbb {C} } –algebra and is denoted by C class ( G ) {\displaystyle \mathbb {C} _{\text{class}}(G)} . Its dimension is equal to the number of conjugacy classes of G . {\displaystyle G.} | Complex representations of finite groups | 0.807146 |
26,452 | The current third-generation IGBTs are even better, with speed rivaling power MOSFETs, and excellent ruggedness and tolerance of overloads. Extremely high pulse ratings of second and third-generation devices also make them useful for generating large power pulses in areas including particle and plasma physics, where th... | IGBT transistor | 0.807145 |
26,453 | Computable topology is a discipline in mathematics that studies the topological and algebraic structure of computation. Computable topology is not to be confused with algorithmic or computational topology, which studies the application of computation to topology. | Computable topology | 0.807142 |
26,454 | y M 0 ⋯ M m − 1 {\displaystyle \cdots x_{n}.yM_{0}\cdots M_{m-1}}: BT(M)(< >) = λ x_{1} ⋯ x n . y {\displaystyle \cdots x_{n}.y} BT(M)( ⟨ k ⟩ ∗ α {\displaystyle \langle k\rangle *\alpha } ) = BT(M_k)( α {\displaystyle \alpha } ) ∀ α {\displaystyle \forall \alpha } and k < m= undefined ∀ α {\displaystyle \forall \alpha ... | Computable topology | 0.807142 |
26,455 | The result of a Scott continuous λ-calculus topology is a function space built upon a programming semantic allowing fixed point combinatorics, such as the Y combinator, and data types. By 1971, λ-calculus was equipped to define any sequential computation and could be easily adapted to parallel computations. The reducib... | Computable topology | 0.807142 |
26,456 | Type free λ-calculus treats functions as rules and does not differentiate functions and the objects which they are applied to, meaning λ-calculus is type free. A by-product of type free λ-calculus is an effective computability equivalent to general recursion and Turing machines. The set of λ-terms can be considered a f... | Computable topology | 0.807142 |
26,457 | The topology of λ-calculus is the Scott topology, and when restricted to continuous functions the type free λ-calculus amounts to a topological space reliant on the tree topology. Both the Scott and Tree topologies exhibit continuity with respect to the binary operators of application ( f applied to a = fa ) and abstra... | Computable topology | 0.807142 |
26,458 | As shown by Alan Turing and Alonzo Church, the λ-calculus is strong enough to describe all mechanically computable functions (see Church–Turing thesis). Lambda-calculus is thus effectively a programming language, from which other languages can be built. For this reason when considering the topology of computation it is... | Computable topology | 0.807142 |
26,459 | f ˇ: → {\displaystyle f. {\check {f}}:\rightarrow } is continuous.Proof (i): Let X ⊆ D be directed, then f ˇ {\displaystyle {\check {f}}} (sup(X)) = λ y.f( sup(X),y )= λ y. sup x ∈ X {\displaystyle \sup _{x\in X}} ( f(x,y) )= sup x ∈ X {\displaystyle \sup _{x\in X}} ( λy.f(x,y) )= sup( f ˇ {\displaystyle {\check {f}}... | Computable topology | 0.807142 |
26,460 | It has now been shown application is continuous under the Scott topology.In order to demonstrate the Scott topology is a suitable fit for λ-calculus it is necessary to prove abstraction remains continuous over the Scott topology. Once completed it will have been shown that the mathematical foundation of λ-calculus is a... | Computable topology | 0.807142 |
26,461 | Using the definition of application as follows: Ap: × D → D ′ {\displaystyle \times D\rightarrow D^{'}} where Ap(f,x) = f(x).Ap is continuous with respect to the Scott topology on the product ( × D → D ′ {\displaystyle \times D\rightarrow D^{'}} ): Proof: λx.f(x) = f is continuous. Let h = λ f.f(x). | Computable topology | 0.807142 |
26,462 | So there is a ∈ U ∩ D {\displaystyle U\cap D} and b ∈ V ∩ D {\displaystyle V\cap D} . Since D is directed there is c ∈ D with a ≤ c , b ≤ c {\displaystyle a\leq c,b\leq c} ; and since U and V are upper sets, c ∈ U ∩ V {\displaystyle U\cap V} as well.Though not shown here, it is the case that the map f: D → D ′ {\displa... | Computable topology | 0.807142 |
26,463 | We will re-iterate to familiarize ourselves with the notation to be used during the discussion of Scott topology. Complete partial orders are defined as follows: First, given the partially ordered set D=(D,≤), a nonempty subset X ⊆ D is directed if ∀ x,y ∈ X ∃ z ∈ X where x≤ z & y ≤ z. D is a complete partial order (cp... | Computable topology | 0.807142 |
26,464 | It is found that Böhm tree's allow for a continuous mapping from the tree topology to the Scott topology. More specifically: We begin with the cpo B = (B,⊆) on the Scott topology, with ordering of Böhm tree's denoted M⊆ N, meaning M, N are trees and M results from N. The tree topology on the set Ɣ is the smallest set a... | Computable topology | 0.807142 |
26,465 | Based on the operators within lambda calculus, application and abstraction, it is possible to develop an algebra whose group structure uses application and abstraction as binary operators. Application is defined as an operation between lambda terms producing a λ-term, e.g. the application of λ onto the lambda term a pr... | Computable topology | 0.807142 |
26,466 | Scott approaches his derivation using a complete lattice, resulting in a topology dependent on the lattice structure. It is possible to generalise Scott's theory with the use of complete partial orders. For this reason a more general understanding of the computational topology is provided through complete partial order... | Computable topology | 0.807142 |
26,467 | The Scott topology is essential in understanding the topological structure of computation as expressed through the λ-calculus. Scott found that after constructing a function space using λ-calculus one obtains a Kolmogorov space, a T o {\displaystyle T_{o}} topological space which exhibits pointwise convergence, in shor... | Computable topology | 0.807142 |
26,468 | {\displaystyle v_{0},v_{1},...} ) M,N ∈ Γ ( W ) ⇒ {\displaystyle \Gamma (W)\Rightarrow } (MN) ∈ Γ ( W ) {\displaystyle \Gamma (W)} M ∈ Γ ( W ) ⇒ {\displaystyle \Gamma (W)\Rightarrow } (λx.M) ∈ Γ ( W ) {\displaystyle \Gamma (W)} A mapping from the terms within Γ ( W ) {\displaystyle \Gamma (W)} to all λ terms within W, ... | Computable topology | 0.807142 |
26,469 | W must also demonstrate: ( β ) {\displaystyle (\beta )} (λ*x.A)x = A α 1 {\displaystyle \alpha _{1}} λ*x.A≡ λ*x.A provided y is not a free variable of A α 2 {\displaystyle \alpha _{2}} (λ*x.A)≡λ*x.A provided y,z ≠ x and z is not a free variable of ABefore defining the full λ-algebra we must introduce the following defi... | Computable topology | 0.807142 |
26,470 | bx = A(x, y → {\displaystyle {\vec {y}}} ).We then require b have a dependence on y → {\displaystyle {\vec {y}}} resulting in: ∀ x {\displaystyle \forall x} B( y → {\displaystyle {\vec {y}}} )x = A(x, y → {\displaystyle {\vec {y}}} ).B( y → {\displaystyle {\vec {y}}} ) becomes equivalent to a λ term, and is therefore s... | Computable topology | 0.807142 |
26,471 | However the existence of an applicative term A ( x , y → {\displaystyle A(x,{\vec {y}}} ) provides a good starting point to build a λ-calculus algebra. What is needed is the introduction of a lambda term, i.e. include λx.A(x, y → {\displaystyle {\vec {y}}} ). We begin by exploiting the fact that within a combinatory al... | Computable topology | 0.807142 |
26,472 | , x n {\displaystyle {x_{1},...,x_{n}}} then: ∃ f ∀ x 1 ⋅ ⋅ ⋅ x n {\displaystyle \exists f\forall x_{1}\cdot \cdot \cdot x_{n}} where f x 1 ⋅ ⋅ ⋅ x n = A {\displaystyle fx_{1}\cdot \cdot \cdot x_{n}=A} The combinatory algebra is: Never commutative Not associative. Never finite. Never recursive.Combinatory algebras rema... | Computable topology | 0.807142 |
26,473 | Relevant attributes of the combinatory algebra are: Within combinatory algebra there exists applicative structures. An applicative structure W is a combinatory algebra provided: ·W is non-trival, meaning W has cardinality > 1 ·W exhibits combinatory completeness (see completeness of the S-K basis). More specifically: f... | Computable topology | 0.807142 |
26,474 | The λ algebra becomes a combinatory algebra M combined with a syntactic operator λ* that transforms a term B(x,y), with constants in M, into C( y ^ {\displaystyle {\hat {y}}} )≡ λ* x.B(x, y ^ {\displaystyle {\hat {y}}} ). It is also possible to define an extension model to circumvent the need of the λ* operator by allo... | Computable topology | 0.807142 |
26,475 | New methods of interpretation of the λ-calculus are not only interesting in themselves but allow new modes of thought concerning the behaviors of computer science. The binary operator within the λ-algebra A is application. Application is denoted · and is said to give structure A = ( X , ⋅ ) {\displaystyle A=(X,\cdot )}... | Computable topology | 0.807142 |
26,476 | The theorem describing this effect is called the frozen-in-field theorem. Even in a fluid with a finite conductivity, new field is generated by stretching field lines as the fluid moves in ways that deform it. This process could go on generating new field indefinitely, were it not that as the magnetic field increases i... | Geomagnetic field | 0.807142 |
26,477 | Biological systems engineering or Biosystems engineering is a broad-based engineering discipline with particular emphasis on non-medical biology. It can be thought of as a subset of the broader notion of biological engineering or bio-technology though not in the respects that pertain to biomedical engineering as biosys... | Biosystems engineering | 0.807141 |
26,478 | Land and water resources engineering Food engineering and bioprocess engineering Machinery systems engineering Natural resources and environmental engineering Biomedical engineering | Biosystems engineering | 0.807141 |
26,479 | The model treated the electrons as a gas. Later in 1927, Arnold Sommerfeld applied the Pauli principle via Fermi-Dirac statistics to this electron gas model, computing the specific heat of metals; the result became Fermi gas model for metals. Sommerfeld called the low temperature region with quantum effects a "wholly d... | Degeneracy pressure | 0.807141 |
26,480 | Quantum mechanics uses the word 'degenerate' in two ways: degenerate energy levels and as the low temperature ground state limit for states of matter. : 437 The electron degeneracy pressure occurs in the ground state systems which are non-degenerate in energy levels. The term "degeneracy" derives from work on the speci... | Degeneracy pressure | 0.807141 |
26,481 | In some branches of physics, the universal constants c, the speed of light, and ħ, the reduced Planck constant, are treated as being unity (i.e. that c = ħ = 1), which leads to mass, energy, momentum, frequency and reciprocal length all having the same unit. As a result, reciprocal length is used as a measure of energy... | Reciprocal centimeter | 0.80714 |
26,482 | In the European Union, a biological medicinal product is one of the active substance(s) produced from or extracted from a biological (living) system, and requires, in addition to physicochemical testing, biological testing for full characterisation. The characterisation of a biological medicinal product is a combinatio... | Biologic medical product | 0.807138 |
26,483 | Physical pharmacy is the branch of pharmacy that concentrates on the applications of physics and chemistry to the study of pharmacy. In other words, it is the study of the effects that dosage forms have on their environment by addressing issues at the molecular level. It emphasis on the physical characteristics and act... | Physical pharmacy | 0.807138 |
26,484 | This analytic process provides several practical advantages: The database of cases provides a concrete representation of the concept of a multiplicity of plausible futures. Running a simulation multiple times in the forward direction can simplify the analytic challenge of representing adaptive strategies in many practi... | Robust decision-making | 0.807137 |
26,485 | A left-handed system would fix ε123 = −1 or equivalently ε321 = +1. The scalar triple product can now be written: with the geometric interpretation of volume (of the parallelepiped spanned by a, b, c) and algebraically is a determinant:: 23 This in turn can be used to rewrite the cross product of two vectors as follows... | Cartesian tensor | 0.807137 |
26,486 | D. C. Kay (1988). Tensor Calculus. Schaum's Outlines. McGraw Hill. | Cartesian tensor | 0.807137 |
26,487 | Cartesian tensors are as in tensor algebra, but Euclidean structure of and restriction of the basis brings some simplifications compared to the general theory. The general tensor algebra consists of general mixed tensors of type (p, q): with basis elements: the components transform according to: as for the bases: For C... | Cartesian tensor | 0.807137 |
26,488 | The notation used for functions in mathematical analysis leads us to write v − T(u), while the same idea can be expressed in matrix and index notations (including the summation convention), respectively: By "linear", if u = ρr + σs for two scalars ρ and σ and vectors r and s, then in function and index notations: and s... | Cartesian tensor | 0.807137 |
26,489 | A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b. Analogous to vectors, it can be written as a linear combination of the tensor basis ex ⊗ ex ≡ exx, ex ⊗ ey ≡ exy, ..., ez ⊗ ez ≡ ezz (the right-hand side of each identity is only an abbreviation, n... | Cartesian tensor | 0.807137 |
26,490 | The dot product and cross product occur very frequently, in applications of vector analysis to physics and engineering, examples include: power transferred P by an object exerting a force F with velocity v along a straight-line path: tangential velocity v at a point x of a rotating rigid body with angular velocity ω: p... | Cartesian tensor | 0.807137 |
26,491 | Other identities can be formed from the δ tensor and ε pseudotensor, a notable and very useful identity is one that converts two Levi-Civita symbols adjacently contracted over two indices into an antisymmetrized combination of Kronecker deltas: The index forms of the dot and cross products, together with this identity,... | Cartesian tensor | 0.807137 |
26,492 | ARC's mission is to ensure that powerful machine learning systems of the future are designed and developed safely and for the benefit of humanity. It was founded in April 2021 by Paul Christiano and other researchers focused on the theoretical challenges of AI alignment. They attempt to develop scalable methods for tra... | Alignment Research Center | 0.807136 |
26,493 | The Alignment Research Center (ARC) is a nonprofit research institute based Berkeley, California, dedicated to the alignment of advanced artificial intelligence with human values and priorities. Established by former OpenAI researcher Paul Christiano, ARC focuses on recognizing and comprehending the potentially harmful... | Alignment Research Center | 0.807136 |
26,494 | Hamilton showed mathematically that, because other members of a population may share one's genes, a gene can also increase its evolutionary success by indirectly promoting the reproduction and survival of other individuals who also carry that gene. This is variously called "kin theory", "kin selection theory" or "inclu... | Inclusive fitness | 0.807136 |
26,495 | In the context of sociobiology, Hamilton proposed that inclusive fitness offers a mechanism for the evolution of altruism. He claimed that this leads natural selection to favor organisms that behave in ways that correlate with maximizing their inclusive fitness. If a gene (or gene complex) promoting altruistic behavior... | Inclusive fitness | 0.807136 |
26,496 | In evolutionary biology, inclusive fitness is one of two metrics of evolutionary success as defined by W. D. Hamilton in 1964: Personal fitness is the number of offspring that an individual begets (regardless of who rescues/rears/supports them) Inclusive fitness is the number of offspring equivalents that an individual... | Inclusive fitness | 0.807136 |
26,497 | Using the big O notation an n {\displaystyle n} th-order accurate numerical method is notated as | | u − u h | | = O ( h n ) {\displaystyle ||u-u_{h}||=O(h^{n})} This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in genera... | Order of accuracy | 0.807134 |
26,498 | Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring). An automorphism of a group G is inner if and only if it extends to every group containing G.By associating the element a ∈ G with the inner autom... | Inner automorphisms | 0.80713 |
26,499 | In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or obje... | Minimal resolution (algebra) | 0.807128 |
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