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20,700 | {\displaystyle \mathbb {C} _{n}.} The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute ( i n i m + i m i n = 0 {\displaystyle i_{n}i_{m}+i_{m}i_{n}=0} when m ≠ n for Clifford). Because the multicomplex numbers hav... | Multicomplex number | 0.810731 |
20,701 | In the multicomplex number systems one also requires that i n i m = i m i n {\displaystyle i_{n}i_{m}=i_{m}i_{n}} (commutativity). Then C 1 {\displaystyle \mathbb {C} _{1}} is the complex number system, C 2 {\displaystyle \mathbb {C} _{2}} is the bicomplex number system, C 3 {\displaystyle \mathbb {C} _{3}} is the tric... | Multicomplex number | 0.810731 |
20,702 | One use of the concept of biocontainment is related to laboratory biosafety and pertains to microbiology laboratories in which the physical containment of pathogenic organisms or agents (bacteria, viruses, and toxins) is required, usually by isolation in environmentally and biologically secure cabinets or rooms, to pre... | Biological Containment | 0.810726 |
20,703 | Radio Objects with Continuous Optical Spectra, or ROCOSes, were discovered in the 1970s. Among the discoverers was a group of Soviet astrophysicists, who studied them at the Crimean Astrophysical Observatory and the Special Astrophysical Observatory of the Russian Academy of Science, using the former's 2.6-meter optica... | Radio object with continuous optical spectrum | 0.810725 |
20,704 | Hence, by the reflexive version of the Lumer–Phillips theorem A generates a contraction semigroup.There are many more examples where a direct application of the Lumer–Phillips theorem gives the desired result. In conjunction with translation, scaling and perturbation theory the Lumer–Phillips theorem is the main tool f... | Lumer–Phillips theorem | 0.810722 |
20,705 | In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup. | Lumer–Phillips theorem | 0.810722 |
20,706 | In 2013, The European Journal of Contraception & Reproductive Health Care published a review of 36 studies including 8,422 female subjects in total taking COCPs that found that 5,358 subjects (or 63.6 percent) reported no change in libido, 1,826 subjects (or 21.7 percent) reported an increase, and 1,238 subjects (or 14... | Combined oral contraceptive | 0.810721 |
20,707 | (But see Sexagesimal system below.) In general, the number of possible values that can be represented by a d {\displaystyle d} digit number in base r {\displaystyle r} is r d {\displaystyle r^{d}} . The common numeral systems in computer science are binary (radix 2), octal (radix 8), and hexadecimal (radix 16). | Place-value system | 0.810719 |
20,708 | Segmentation in biology is the division of some animal and plant body plans into a linear series of repetitive segments that may or maynot be interconnected to each other. This article focuses on the segmentation of animal body plans, specifically using the examples of the taxa Arthropoda, Chordata, and Annelida. These... | Segmentation (biology) | 0.810718 |
20,709 | For cyclic group, H0 is the same as H2, so H2(E/F) = |E/F| for all cyclic layers. Another theorem of group cohomology shows that since H1(E/F) = 1 for all normal layers and H2(E/F) ≤ |E/F| for all cyclic layers, we have H2(E/F)≤ |E/F|for all normal layers. (In fact, equality holds for all normal layers, but this takes ... | Class formation | 0.810717 |
20,710 | For cyclic layers, the two inequalities together prove that H1(E/F)|E/F| = H0(E/F) ≤ |E/F|so H0(E/F) = |E/F|and H1(E/F) = 1.Now a basic theorem about cohomology groups shows that since H1(E/F) = 1 for all cyclic layers, we have H1(E/F) = 1for all normal layers (so in particular the formation is a field formation). This... | Class formation | 0.810717 |
20,711 | In 1940 Chevalley found a purely algebraic proof of the second inequality, but it is longer and harder than Weber's original proof. Before about 1950, the second inequality was known as the first inequality; the name was changed because Chevalley's algebraic proof of it uses the first inequality. Takagi defined a class... | Class formation | 0.810717 |
20,712 | The second inequality of class field theory states that |H0(E/F)| ≤ |E/F|for all normal layers E/F. For local fields, this inequality follows easily from Hilbert's theorem 90 together with the first inequality and some basic properties of group cohomology. The second inequality was first proved for global fields by Web... | Class formation | 0.810717 |
20,713 | Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group. Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algeb... | Class formation | 0.810717 |
20,714 | The most important examples of class formations (arranged roughly in order of difficulty) are as follows: Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation. Finite fields: The module A i... | Class formation | 0.810717 |
20,715 | This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem. For general layers E,F there is an exact sequence 0 → H 2 ( E / F ) ∩ H 2 ( E / E ∩ G ∞ ) → H 2 ( E / E ∩ G ∞ ) → H 2 ( F / F ∩ G ∞ ) {\disp... | Class formation | 0.810717 |
20,716 | These homomorphisms are constructed using cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by the maximal unramified extensions, and for global fields they are slightly more complicated. As these extensions are given expli... | Class formation | 0.810717 |
20,717 | The main remaining theorem of class field theory is the Takagi existence theorem, which states that every finite index closed subgroup of the idele class group is the group of norms corresponding to some abelian extension. The classical way to prove this is to construct some extensions with small groups of norms, by fi... | Class formation | 0.810717 |
20,718 | In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. A class formation is a formation such that for every normal layer E/F H1(E/F) is trivial, and H2(... | Class formation | 0.810717 |
20,719 | Taking the inverse of this isomorphism gives a homomorphism AE → abelianization of E/F,and taking the limit over all open subgroups F gives a homomorphism AE → abelianization of E,called the Artin map. The Artin map is not necessarily surjective, but has dense image. By the existence theorem below its kernel is the con... | Class formation | 0.810717 |
20,720 | Tate's theorem in group cohomology is as follows. Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G H1(E,A) is trivial, and H2(E,A) is generated by Res(a) which has order E.Then cup product with a is an isomorphism Hn(G,Z) → Hn+2(G,A).If we apply the ca... | Class formation | 0.810717 |
20,721 | Inverse topology micellar cubic phases (such as the Fd3m phase) are observed for some type II amphiphiles at very high amphiphile concentrations. These aggregates, in which water is the minority phase, have a polar-apolar interface with a negative mean curvature. The structures of the normal topology micellar cubic pha... | Micellar cubic | 0.810716 |
20,722 | A micellar cubic phase is a lyotropic liquid crystal phase formed when the concentration of micelles dispersed in a solvent (usually water) is sufficiently high that they are forced to pack into a structure having long-ranged positional (translational) order. For example, spherical micelles a subic packing of a body-ce... | Micellar cubic | 0.810716 |
20,723 | Infectious bursal disease virus (IBDV) is the best-characterized member of the family Birnaviridae. These viruses have bipartite dsRNA genomes enclosed in single layered icosahedral capsids with T = 13l geometry. IBDV shares functional strategies and structural features with many other icosahedral dsRNA viruses, except... | Double-stranded RNA viruses | 0.810708 |
20,724 | σ = 1 − 1.98 z ( 1 − ϕ 2 cot β 2 ) {\displaystyle \sigma =1-{\frac {1.98}{z(1-\phi _{2}\cot \beta _{2})}}} where, z = number of blades, β2 varies from 450 to 900. For radial tip: β2 = 900 ∴ σ = 1 − 1.98 z = 1 − 0.63 ∗ p i z {\displaystyle \sigma =1-{\frac {1.98}{z}}=1-{\frac {0.63*pi}{z}}} Balje's formula: An approxi... | Slip factor | 0.810703 |
20,725 | Stodola's Equation: According to Stodola, it is the relative eddy that fills the entire exit session of the impeller passage. For a given flow geometry, the slip factor increases with the increase in the number of impeller blades, thus, accounts for one of the important parameter for losses. σ = 1 − π z sin β 2 ( 1 −... | Slip factor | 0.810703 |
20,726 | Relative eddy. Back eddy. Impeller design or geometryMean blade loading. | Slip factor | 0.810703 |
20,727 | When considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. This approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about mod... | Representation space | 0.810702 |
20,728 | If ( V , ψ ) {\displaystyle (V,\psi )} is a representation of (say) a group G {\displaystyle G} , and W {\displaystyle W} is a linear subspace of V {\displaystyle V} that is preserved by the action of G {\displaystyle G} in the sense that for all w ∈ W {\displaystyle w\in W} and g ∈ G {\displaystyle g\in G} , g ⋅ w ∈ W... | Representation space | 0.810702 |
20,729 | Important examples are linear algebraic groups over finite fields. The representation theory of linear algebraic groups and Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representations. The importance of character theory for finite groups has an an... | Representation space | 0.810702 |
20,730 | Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for compact topological groups (including compact Lie groups), using Haar me... | Representation space | 0.810702 |
20,731 | The finite-dimensional G-representations can be understood using character theory: the character of a representation φ: G → GL(V) is the class function χφ: G → F defined by χ φ ( g ) = T r ( φ ( g ) ) {\displaystyle \chi _{\varphi }(g)=\mathrm {Tr} (\varphi (g))} where T r {\displaystyle \mathrm {Tr} } is the trace. An... | Representation space | 0.810702 |
20,732 | First, the representations of G are semisimple (completely reducible). This is a consequence of Maschke's theorem, which states that any subrepresentation V of a G-representation W has a G-invariant complement. | Representation space | 0.810702 |
20,733 | Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to geometry and crystallography. Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation ... | Representation space | 0.810702 |
20,734 | A unitary representation of a group G is a linear representation φ of G on a real or (usually) complex Hilbert space V such that φ(g) is a unitary operator for every g ∈ G. Such representations have been widely applied in quantum mechanics since the 1920s, thanks in particular to the influence of Hermann Weyl, and this... | Representation space | 0.810702 |
20,735 | Modular representations of a finite group G are representations over a field whose characteristic is not coprime to |G|, so that Maschke's theorem no longer holds (because |G| is not invertible in F and so one cannot divide by it). Nevertheless, Richard Brauer extended much of character theory to modular representation... | Representation space | 0.810702 |
20,736 | Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a Z2-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras. | Representation space | 0.810702 |
20,737 | Applications arise in physics and chemistry, but also in fields such as sociology. For example, it is natural to describe a system of two political parties by an Ising model. Thereby, at a transition from one majority to the other, the above-mentioned critical phenomena may appear. | Critical phenomena | 0.810699 |
20,738 | In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such... | Critical phenomena | 0.810699 |
20,739 | The maximal ideal of this ring consists precisely of those germs f with f(0) = 0. Exactly the same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ... | Local algebra | 0.810696 |
20,740 | Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorph... | Local algebra | 0.810696 |
20,741 | Given a field K, which may or may not be a function field, we may look for local rings in it. If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty th... | Local algebra | 0.810696 |
20,742 | In differential geometry, the four-gradient (or 4-gradient) ∂ {\displaystyle {\boldsymbol {\partial }}} is the four-vector analogue of the gradient ∇ → {\displaystyle {\vec {\boldsymbol {\nabla }}}} from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties ... | Four-gradient | 0.810694 |
20,743 | The Greek tensor index ranges in {0, 1, 2, 3}, and represents a 4-vector, e.g. A μ = ( a 0 , a 1 , a 2 , a 3 ) = A {\displaystyle A^{\mu }=\left(a^{0},a^{1},a^{2},a^{3}\right)=\mathbf {A} } . In SR physics, one typically uses a concise blend, e.g. A = ( a 0 , a → ) {\displaystyle \mathbf {A} =\left(a^{0},{\vec {\mathbf... | Four-gradient | 0.810694 |
20,744 | There are alternate ways of writing four-vector expressions in physics: The four-vector style can be used: A ⋅ B {\displaystyle \mathbf {A} \cdot \mathbf {B} } , which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and ... | Four-gradient | 0.810694 |
20,745 | When the conservation of the stress–energy tensor ( ∂ ν T μ ν = 0 μ {\displaystyle \partial _{\nu }T^{\mu \nu }=0^{\mu }} ) for a perfect fluid is combined with the conservation of particle number density ( ∂ ⋅ N = 0 {\displaystyle {\boldsymbol {\partial }}\cdot \mathbf {N} =0} ), both utilizing the 4-gradient, one can... | Four-gradient | 0.810694 |
20,746 | The Hamilton–Jacobi equation (HJE) is a formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possi... | Four-gradient | 0.810694 |
20,747 | The 4-gradient is connected with quantum mechanics. The relation between the 4-momentum P {\displaystyle \mathbf {P} } and the 4-gradient ∂ {\displaystyle {\boldsymbol {\partial }}} gives the Schrödinger QM relations. : 3–5 The temporal component gives: E = i ℏ ∂ t {\displaystyle E=i\hbar \partial _{t}} The spatial com... | Four-gradient | 0.810694 |
20,748 | The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule". So, for example, if T μ ν , μ = 0 {\displaystyle T^{\mu \nu }{}_{,\mu }=0} in SR, then T... | Four-gradient | 0.810694 |
20,749 | The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:: 16 The comma in the last part above , μ {\displaystyle {}_{,\mu }} implies the partial differentiation with respect to 4-position X μ {\displaystyle X^{\mu }} . The contravariant components are:: 16 Alternative symbol... | Four-gradient | 0.810694 |
20,750 | Some use the metric (+ − − −), others use the metric (− + + +). Some don't use 4-vectors, but do everything as the old style E and 3-space vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style thr... | Four-gradient | 0.810694 |
20,751 | Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use m {\displaystyle m} for invariant rest mass, others use m 0 {\displaystyle m_{0}} for invariant rest mass and use m {\displaystyle m} for relativistic mass. Man... | Four-gradient | 0.810694 |
20,752 | In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vect... | Four-gradient | 0.810694 |
20,753 | In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). According to:: 4 Taking the spatial components, Since η μ ν = diag {\displaystyle \eta ... | Four-gradient | 0.810694 |
20,754 | In modern elementary particle physics, one can define a gauge covariant derivative which utilizes the extra RQM fields (internal particle spaces) now known to exist. The version known from classical EM (in natural units) is:: 39 The full covariant derivative for the fundamental interactions of the Standard Model that w... | Four-gradient | 0.810694 |
20,755 | In general characteristic, which we assume in this section, J-structures are related to quadratic Jordan algebras. We take a quadratic Jordan algebra to be a finite dimensional vector space V with a quadratic map Q from V to End(V) and a distinguished element e. We let Q also denote the bilinear map Q(x,y) = Q(x+y) − Q... | J-structure | 0.810693 |
20,756 | An alternative characterisation is that for all invertible x we have x − 1 ( x y ) = x ( x − 1 y ) . {\displaystyle x^{-1}(xy)=x(x^{-1}y).} If A is a Jordan algebra, then (A,i,e) is a J-structure. If (V,j,e) is a J-structure, then there exists a unique Jordan algebra structure on V with identity e with inversion j. | J-structure | 0.810693 |
20,757 | In characteristic not equal to 2, which we assume in this section, the theory of J-structures is essentially the same as that of Jordan algebras. Let A be a finite-dimensional commutative non-associative algebra over K with identity e. Let L(x) denote multiplication on the left by x. There is a unique birational map i ... | J-structure | 0.810693 |
20,758 | If we drop the condition on the distinguished element e, we obtain "J-structures without identity". These are related to isotopes of Jordan algebras. | J-structure | 0.810693 |
20,759 | A J-structure has a Peirce decomposition into subspaces determined by idempotent elements. Let a be an idempotent of the J-structure (V,j,e), that is, a2 = a. Let Q be the quadratic map. Define ϕ a ( t , u ) = Q ( t a + u ( e − a ) ) . {\displaystyle \phi _{a}(t,u)=Q(ta+u(e-a)).} This is invertible for non-zero t,u in ... | J-structure | 0.810693 |
20,760 | McCrimmon proposed a notion of H-structure by dropping the density axiom and strengthening the third (a form of Hua's identity) to hold in all isotopes. The resulting structure is categorically equivalent to a quadratic Jordan algebra. | J-structure | 0.810693 |
20,761 | In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by Springer (1973) to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is ... | J-structure | 0.810693 |
20,762 | The development of high-throughput methods for genome sequencing has brought about a wealth of data that requires sophisticated bioinformatics tools for their analysis and interpretation. Nowadays, numerous tools exist to study genomics sequence data and extract information from different perspectives. However, the lac... | MicrobesOnline | 0.810691 |
20,763 | MicrobesOnline has integrated numerous tools for analysing sequences, gene expression profiles and protein-protein interactions into an interface called Bioinformatics Workbench, which is accessed via gene carts. Analyses currently supported include multiple sequence alignments, construction of phylogenetic trees, moti... | MicrobesOnline | 0.810691 |
20,764 | Information of microbial genes stored in MicrobesOnline includes sequences (genes, transcripts and proteins), genomic loci, gene annotations and some statistics of sequences. This information can be accessed through three features displayed on the homepage of MicrobesOnline: sequence search and advanced search in the t... | MicrobesOnline | 0.810691 |
20,765 | The Centre for Genomic Research (CGR) (Liverpool) is located on the University of Liverpool Campus, and was established by the Medical Research Council in partnership with the Natural Environment Research Council (NERC). The goal of this research centre is to provide DNA sequencing and bioinformatics services to the sc... | Centre for Genomic Research (Liverpool) | 0.810688 |
20,766 | A generalized 2-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For n ≥ 3 {\displaystyle n\geq 3} a generalized n-gon is an incidence structure ( P , L , I {\displaystyle P,L,I} ), where P {\displaystyle P} is the set of points, L {\displaystyle L... | Generalized n-gons | 0.810687 |
20,767 | Each level after is generated by one more line than the last. If a level was generated with L {\displaystyle L} lines, then the next level has L {\displaystyle L} more vertices and 2 L − 1 {\displaystyle 2L-1} more edges. The best known algorithms from computational geometry for constructing the graphs of line arrangem... | Planarity | 0.810686 |
20,768 | Planarity is a puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University. The name comes from the concept of planar graphs in graph theory; these are graphs that can be embedded in the Euclidean plane so that no edges intersect. By Fáry's theorem, if a graph is planar, it... | Planarity | 0.810686 |
20,769 | In the field of computational geometry, the process of moving a subset of the vertices in a graph embedding to eliminate edge crossings has been studied by Pach and Tardos (2002), and others, inspired by the planarity puzzle. The results of these researchers shows that (in theory, assuming that the field of play is the... | Planarity | 0.810686 |
20,770 | The problem of determining whether a graph is planar can be solved in linear time, and any such graph is guaranteed to have a straight-line embedding by Fáry's theorem, that can also be found from the planar embedding in linear time. Therefore, any puzzle could be solved in linear time by a computer. However, these puz... | Planarity | 0.810686 |
20,771 | Microbial phylogenetics is the study of the manner in which various groups of microorganisms are genetically related. This helps to trace their evolution. To study these relationships biologists rely on comparative genomics, as physiology and comparative anatomy are not possible methods. | Microbial phylogenetics | 0.81068 |
20,772 | Microbial phylogenetics emerged as a field of study in the 1960s, scientists started to create genealogical trees based on differences in the order of amino acids of proteins and nucleotides of genes instead of using comparative anatomy and physiology.One of the most important figures in the early stage of this field i... | Microbial phylogenetics | 0.81068 |
20,773 | In the 1980s microbial phylogenetics went into its golden age, as the techniques for sequencing RNA and DNA improved greatly. For example, comparison of the nucleotide sequences of whole genes was facilitated by the development of the means to clone DNA, making possible to create many copies of sequences from minute sa... | Microbial phylogenetics | 0.81068 |
20,774 | SOLID is a mnemonic invented by Michael Feathers which spells out five software engineering design principles: Single responsibility principle Open/closed principle Liskov substitution principle Interface segregation principle Dependency inversion principleGRASP (General Responsibility Assignment Software Patterns) is ... | Object-oriented languages | 0.810679 |
20,775 | Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual processes by studying the eff... | Psychophysics | 0.810676 |
20,776 | Nevertheless, that sort of response has remained popular in applied psychophysics. Such multiple-category layouts are often misnamed Likert scaling after the question items used by Likert to create multi-item psychometric scales, e.g., seven phrases from "strongly agree" through "strongly disagree". Omar Khaleefa has a... | Psychophysics | 0.810676 |
20,777 | Many of the classical techniques and theories of psychophysics were formulated in 1860 when Gustav Theodor Fechner in Leipzig published Elemente der Psychophysik (Elements of Psychophysics). He coined the term "psychophysics", describing research intended to relate physical stimuli to the contents of consciousness such... | Psychophysics | 0.810676 |
20,778 | In psychophysics, experiments seek to determine whether the subject can detect a stimulus, identify it, differentiate between it and another stimulus, or describe the magnitude or nature of this difference. Software for psychophysical experimentation is overviewed by Strasburger. | Psychophysics | 0.810676 |
20,779 | and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon... | Uniform convergence (combinatorics) | 0.810673 |
20,780 | By Hoeffding's inequality, this is at most 2 e − m ε 2 / 8 {\displaystyle 2e^{-m\varepsilon ^{2}/8}} . Finally, combining all the three parts of the proof we get the Uniform Convergence Theorem. == References == | Uniform convergence (combinatorics) | 0.810673 |
20,781 | In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an important role in the second quantization formulation of quantum mechanics. T... | Photon number | 0.810671 |
20,782 | LMDB makes novel use of well-known computer science techniques such as copy-on-write semantics and B+ trees to provide atomicity and reliability guarantees as well as performance that can be hard to accept, given the library's relative simplicity and that no other similar key-value store database offers the same guaran... | Lightning Memory-Mapped Database | 0.810671 |
20,783 | In some branches of psychology, depending on school of thought, a physical object has physical properties, as compared to mental objects. In (reductionistic) behaviorism, objects and their properties are the (only) meaningful objects of study. While in the modern day behavioral psychotherapy it is still only the means ... | Physical object | 0.810671 |
20,784 | A physical body is an enduring object that exists throughout a particular trajectory of space and orientation over a particular duration of time, and which is located in the world of physical space (i.e., as studied by physics). This contrasts with abstract objects such as mathematical objects which do not exist at any... | Physical object | 0.810671 |
20,785 | The advantages of proper randomization in RCTs include: "It eliminates bias in treatment assignment," specifically selection bias and confounding. "It facilitates blinding (masking) of the identity of treatments from investigators, participants, and assessors." "It permits the use of probability theory to express the l... | Randomized Controlled Trial | 0.81067 |
20,786 | Even if the cost function has globally continuous gradient, good estimate of the Lipschitz constant for the cost functions in deep learning may not be feasible or desirable, given the very high dimensions of deep neural networks. Hence, there is a technique of fine-tuning of learning rates in applying standard GD or SG... | Backtracking line search | 0.810669 |
20,787 | For example, this is similar to an exercise in Bertsekas (2016), for the cost function f ( t ) = | t | 1.5 {\displaystyle f(t)=|t|^{1.5}\,} and for whatever constant learning rate one chooses, with a random initial point the sequence constructed by this special scheme does not converge to the global minimum 0. If one d... | Backtracking line search | 0.810669 |
20,788 | In (unconstrained) mathematical optimization, a backtracking line search is a line search method to determine the amount to move along a given search direction. Its use requires that the objective function is differentiable and that its gradient is known. The method involves starting with a relatively large estimate of... | Backtracking line search | 0.810669 |
20,789 | In the same reference, similarly convergence is guaranteed for other modifications of Backtracking line search (such as Unbounded backtracking gradient descent mentioned in the section "Upper bound for learning rates"), and even if the function has uncountably many critical points still one can deduce some non-trivial ... | Backtracking line search | 0.810669 |
20,790 | In the setting of deep learning, saddle points are also prevalent, see Dauphin et al. (2014). Thus, to apply in deep learning, one needs results for non-convex functions. For convergence to critical points: For example, if the cost function is a real analytic function, then it is shown in Absil, Mahony & Andrews (2005)... | Backtracking line search | 0.810669 |
20,791 | Compared with Wolfe's conditions, which is more complicated, Armijo's condition has a better theoretical guarantee. Indeed, so far backtracking line search and its modifications are the most theoretically guaranteed methods among all numerical optimization algorithms concerning convergence to critical points and avoida... | Backtracking line search | 0.810669 |
20,792 | {\displaystyle \gamma _{j}=\tau \,\gamma _{j-1}\,.} Return γ j {\displaystyle \gamma _{j}} for the learning rate α n {\displaystyle \alpha _{n}} . (In Nocedal & Wright (2000) one can find a description of an algorithm with 1), 3) and 4) above, which was not tested in deep neural networks before the cited paper.) | Backtracking line search | 0.810669 |
20,793 | It also depends on f {\displaystyle f} , α 0 {\displaystyle \alpha _{0}} , τ {\displaystyle \tau } and c {\displaystyle c} of course, although these dependencies can be left implicit if they are assumed to be fixed with respect to the optimization problem. The detailed steps are thus, see Armijo (1966), Bertsekas (2016... | Backtracking line search | 0.810669 |
20,794 | An argument against the use of Backtracking line search, in particular in Large scale optimisation, is that satisfying Armijo's condition is expensive. There is a way (so-called Two-way Backtracking) to go around, with good theoretical guarantees and has been tested with good results on deep neural networks, see Truong... | Backtracking line search | 0.810669 |
20,795 | An important property of K ( G , n ) {\displaystyle K(G,n)} 's is that for any abelian group G, and any based CW-complex X, the set {\displaystyle } of based homotopy classes of based maps from X to K ( G , n ) {\displaystyle K(G,n)} is in natural bijection with the n-th singular cohomology group H n ( X , G ) {\displ... | Eilenberg–MacLane spectra | 0.810667 |
20,796 | It follows from the universal coefficient theorem for cohomology that the Eilenberg MacLane space is a quasi-functor of the group; that is, for each positive integer n {\displaystyle n} if a: G → G ′ {\displaystyle a\colon G\to G'} is any homomorphism of abelian groups, then there is a non-empty set K ( a , n ) = { : f... | Eilenberg–MacLane spectra | 0.810667 |
20,797 | The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are im... | Eilenberg–MacLane spectra | 0.810667 |
20,798 | In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type K ( G , n ) {\displaystyle K(G,n)} , if it has n-th homotopy... | Eilenberg–MacLane spectra | 0.810667 |
20,799 | In software engineering, containerization is operating system-level virtualization or application-level virtualization over multiple network resources so that software applications can run in isolated user spaces called containers in any cloud or non-cloud environment, regardless of type or vendor. | Containerization (computing) | 0.810667 |
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