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In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and do not work for all inhomogeneous linear differential equations. Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named after Jean-Marie Duhamel (1797–1872) who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice versa. | https://en.wikipedia.org/wiki/Variation_of_parameters |
In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say s = ∑ n = 0 ∞ a n g n {\displaystyle s=\sum _{n=0}^{\infty }a_{n}g^{n}} ,into a convergent series in powers s = ∑ n = 0 ∞ b n / ( g ω ) n {\displaystyle s=\sum _{n=0}^{\infty }b_{n}/(g^{\omega })^{n}} ,where ω {\displaystyle \omega } is a critical exponent (the so-called index of "approach to scaling" introduced by Franz Wegner). This is possible with the help of variational parameters, which are determined by optimization order by order in g {\displaystyle g} . The partial sums are converted to convergent partial sums by a method developed in 1992.Most perturbation expansions in quantum mechanics are divergent for any small coupling strength g {\displaystyle g} . They can be made convergent by VPT (for details see the first textbook cited below). | https://en.wikipedia.org/wiki/Variational_perturbation_theory |
The convergence is exponentially fast.After its success in quantum mechanics, VPT has been developed further to become an important mathematical tool in quantum field theory with its anomalous dimensions. Applications focus on the theory of critical phenomena. It has led to the most accurate predictions of critical exponents. More details can be read here. | https://en.wikipedia.org/wiki/Variational_perturbation_theory |
In mathematics, vector algebra may mean: Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} of dot product and especially cross product. In this sense, vector algebra is contrasted with geometric algebra, which provides an alternative generalization to higher dimensions. An algebra over a field, a vector space equipped with a bilinear product Original vector algebras of the nineteenth century like quaternions, tessarines, or coquaternions, each of which has its own product. The vector algebras biquaternions and hyperbolic quaternions enabled the revolution in physics called special relativity by providing mathematical models. | https://en.wikipedia.org/wiki/Vector_algebra |
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points). Some foundational results on classification were known in the 1950s. The result of Grothendieck (1957), that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of Birkhoff (1909) on the Riemann–Hilbert problem. Atiyah (1957) gave the classification of vector bundles on elliptic curves. | https://en.wikipedia.org/wiki/Vector_bundles_on_algebraic_curves |
The Riemann–Roch theorem for vector bundles was proved by Weil (1938), before the 'vector bundle' concept had really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. | https://en.wikipedia.org/wiki/Vector_bundles_on_algebraic_curves |
He was seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank. This idea would prove fruitful, in terms of moduli spaces of vector bundles. following on the work in the 1960s on geometric invariant theory. | https://en.wikipedia.org/wiki/Vector_bundles_on_algebraic_curves |
In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector. | https://en.wikipedia.org/wiki/Vector_multiplication |
Thus, A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} ⋅ B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} = | A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} | | B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} | cos θMore generally, a bilinear product in an algebra over a field. Cross product – also known as the "vector product", a binary operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. | https://en.wikipedia.org/wiki/Vector_multiplication |
So, if n̂ is the unit vector perpendicular to the plane determined by vectors A and B, A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} × B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} = | A ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A}}} | | B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{B}}} | sin θ n̂More generally, a Lie bracket in a Lie algebra. Hadamard product – entrywise or elementwise product of vectors, where ( A ⊙ B ) i = A i B i {\displaystyle (A\odot B)_{i}=A_{i}B_{i}} . Outer product - where ( a ⊗ b ) {\displaystyle (\mathbf {a} \otimes \mathbf {b} )} with a ∈ R d , b ∈ R d {\displaystyle \mathbf {a} \in \mathbb {R} ^{d},\mathbf {b} \in \mathbb {R} ^{d}} results in a ( d × d ) {\displaystyle (d\times d)} matrix. Triple products – products involving three vectors. Quadruple products – products involving four vectors. | https://en.wikipedia.org/wiki/Vector_multiplication |
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors. | https://en.wikipedia.org/wiki/Vector_spherical_harmonics |
In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces. | https://en.wikipedia.org/wiki/Von_Neumann's_theorem |
In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras. These objects were introduced by Böhm, Nill and Szlachányi. | https://en.wikipedia.org/wiki/Weak_Hopf_algebra |
The first motivations for studying them came from quantum field theory and operator algebras. Weak Hopf algebras have quite interesting representation theory; in particular modules over a semisimple finite weak Hopf algebra is a fusion category (which is a monoidal category with extra properties). It was also shown by Etingof, Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra. | https://en.wikipedia.org/wiki/Weak_Hopf_algebra |
In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology. | https://en.wikipedia.org/wiki/Banach-Saks_theorem |
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. | https://en.wikipedia.org/wiki/Weak_limit |
In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in (Sullivan 2005). The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y. | https://en.wikipedia.org/wiki/Localization_of_a_topological_space |
In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation σ {\displaystyle \sigma } of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x, y of X such that x < y and σ(x) > σ(y). The sign, signature, or signum of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group Sn. | https://en.wikipedia.org/wiki/Parity_of_a_permutation |
Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (εσ), which is defined for all maps from X to X, and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as sgn(σ) = (−1)N(σ)where N(σ) is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as sgn(σ) = (−1)mwhere m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well-defined. | https://en.wikipedia.org/wiki/Parity_of_a_permutation |
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. | https://en.wikipedia.org/wiki/Pathological_(mathematics) |
In mathematics, when the elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle S} into equivalence classes. These equivalence classes are constructed so that elements a {\displaystyle a} and b {\displaystyle b} belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S {\displaystyle S} and an equivalence relation ∼ {\displaystyle \,\sim \,} on S , {\displaystyle S,} the equivalence class of an element a {\displaystyle a} in S , {\displaystyle S,} denoted by , {\displaystyle ,} is the set of elements which are equivalent to a . {\displaystyle a.} | https://en.wikipedia.org/wiki/Canonical_projection_map |
It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S . {\displaystyle S.} | https://en.wikipedia.org/wiki/Canonical_projection_map |
This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S {\displaystyle S} by ∼ , {\displaystyle \,\sim \,,} and is denoted by S / ∼ {\displaystyle S/{\sim }} . When the set S {\displaystyle S} has some structure (such as a group operation or a topology) and the equivalence relation ∼ {\displaystyle \,\sim \,} is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories. | https://en.wikipedia.org/wiki/Canonical_projection_map |
In mathematics, with special application to complex analysis, a normal family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are not widely spread out, but rather stick together in a somewhat "clustered" manner. Note that a compact family of continuous functions is automatically a normal family. | https://en.wikipedia.org/wiki/Normal_family |
Sometimes, if each function in a normal family F satisfies a particular property (e.g. is holomorphic), then the property also holds for each limit point of the set F. More formally, let X and Y be topological spaces. The set of continuous functions f: X → Y {\displaystyle f:X\to Y} has a natural topology called the compact-open topology. A normal family is a pre-compact subset with respect to this topology. If Y is a metric space, then the compact-open topology is equivalent to the topology of compact convergence, and we obtain a definition which is closer to the classical one: A collection F of continuous functions is called a normal family if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of X to a continuous function from X to Y. That is, for every sequence of functions in F, there is a subsequence f n ( x ) {\displaystyle f_{n}(x)} and a continuous function f ( x ) {\displaystyle f(x)} from X to Y such that the following holds for every compact subset K contained in X: lim n → ∞ sup x ∈ K d Y ( f n ( x ) , f ( x ) ) = 0 {\displaystyle \lim _{n\rightarrow \infty }\sup _{x\in K}d_{Y}(f_{n}(x),f(x))=0} where d Y {\displaystyle d_{Y}} is the metric of Y. | https://en.wikipedia.org/wiki/Normal_family |
In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems. | https://en.wikipedia.org/wiki/Zero_dynamics |
In mathematics, zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. This can be easily verified based on the definition of "even": it is an integer multiple of 2, specifically 0 × 2. As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if y is even then y + x has the same parity as x—indeed, 0 + x and x always have the same parity. | https://en.wikipedia.org/wiki/Zero_is_even |
Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. | https://en.wikipedia.org/wiki/Zero_is_even |
Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all.Among the general public, the parity of zero can be a source of confusion. | https://en.wikipedia.org/wiki/Zero_is_even |
In reaction time experiments, most people are slower to identify 0 as even than 2, 4, 6, or 8. Some teachers—and some children in mathematics classes—think that zero is odd, or both even and odd, or neither. | https://en.wikipedia.org/wiki/Zero_is_even |
Researchers in mathematics education propose that these misconceptions can become learning opportunities. Studying equalities like 0 × 2 = 0 can address students' doubts about calling 0 a number and using it in arithmetic. Class discussions can lead students to appreciate the basic principles of mathematical reasoning, such as the importance of definitions. Evaluating the parity of this exceptional number is an early example of a pervasive theme in mathematics: the abstraction of a familiar concept to an unfamiliar setting. | https://en.wikipedia.org/wiki/Zero_is_even |
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group A {\displaystyle A} ), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in A {\displaystyle A} ). It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv: for any 2 m − 1 {\displaystyle 2m-1} elements of Z m {\displaystyle \mathbb {Z} _{m}} , there is a subset of size m {\displaystyle m} that sums to zero. | https://en.wikipedia.org/wiki/Zero-sum_Ramsey_theory |
(This bound is tight, as a sequence of m − 1 {\displaystyle m-1} zeroes and m − 1 {\displaystyle m-1} ones cannot have any subset of size m {\displaystyle m} summing to zero.) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem, or the Chevalley–Warning theorem.Generalizing this result, one can define for any abelian group G the minimum quantity E G Z ( G ) {\displaystyle EGZ(G)} of elements of G such that there must be a subsequence of o ( G ) {\displaystyle o(G)} elements (where o ( G ) {\displaystyle o(G)} is the order of the group) which adds to zero. It is known that E G Z ( G ) ≤ 2 o ( G ) − 1 {\displaystyle EGZ(G)\leq 2o(G)-1} , and that this bound is strict if and only if G = Z m {\displaystyle G=\mathbb {Z} _{m}} . | https://en.wikipedia.org/wiki/Zero-sum_Ramsey_theory |
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.A σ-approximated summation for a series of period T can be written as follows: in terms of the normalized sinc function The term is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. It does not do so entirely, however, but one can square or even cube the expression to serially attenuate Gibbs phenomenon in the most extreme cases. | https://en.wikipedia.org/wiki/Sigma_approximation |
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem Π 1 1 {\displaystyle \Pi _{1}^{1}} -CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of I D < ω {\displaystyle {\mathsf {ID_{<\omega }}}} , the theory of finitely iterated inductive definitions, and of K P ℓ 0 {\displaystyle KP\ell _{0}} , a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by D 0 D ω 0 {\displaystyle D_{0}D_{\omega }0} in Buchholz's ordinal notation ( O T , < ) {\displaystyle {\mathsf {(OT,<)}}} . Lastly, it can be expressed as the limit of the sequence: ε 0 = ψ 0 ( Ω ) {\displaystyle \varepsilon _{0}=\psi _{0}(\Omega )} , B H O = ψ 0 ( Ω 2 ) {\displaystyle {\mathsf {BHO}}=\psi _{0}(\Omega _{2})} , ψ 0 ( Ω 3 ) {\displaystyle \psi _{0}(\Omega _{3})} , ... | https://en.wikipedia.org/wiki/Buchholz's_ordinal |
In mathematics, ℓ ∞ {\displaystyle \ell ^{\infty }} , the (real or complex) vector space of bounded sequences with the supremum norm, and L ∞ = L ∞ ( X , Σ , μ ) {\displaystyle L^{\infty }=L^{\infty }(X,\Sigma ,\mu )} , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces ℓ 1 {\displaystyle \ell _{1}} of absolutely summable sequences, and L 1 = L 1 ( X , Σ , μ ) {\displaystyle L^{1}=L^{1}(X,\Sigma ,\mu )} of absolutely integrable measurable functions (if the measure space fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras. | https://en.wikipedia.org/wiki/L-infinity |
In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0. | https://en.wikipedia.org/wiki/−1 |
In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols ♭ {\displaystyle \flat } (flat) and ♯ {\displaystyle \sharp } (sharp).In the notation of Ricci calculus, it is also known as raising and lowering indices. | https://en.wikipedia.org/wiki/Musical_isomorphism |
In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged. For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points. If every point of an oriented manifold (M, g) belongs to a Wiedersehen pair, then (M, g) is said to be a Wiedersehen manifold. | https://en.wikipedia.org/wiki/Blaschke_conjecture |
The concept was introduced by the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n). | https://en.wikipedia.org/wiki/Blaschke_conjecture |
In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ: M → N is said to be a geodesic map if φ is a diffeomorphism of M onto N; and the image under φ of any geodesic arc in M is a geodesic arc in N; and the image under the inverse function φ−1 of any geodesic arc in N is a geodesic arc in M. | https://en.wikipedia.org/wiki/Geodesic_map |
In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree. | https://en.wikipedia.org/wiki/Pettis_theorem |
In mating, the blennies form distinct pairs, and the females lay up to 7000 eggs in a sitting, which are then guarded in burrows by the males. Males are known to cannibalize dead eggs to prevent infection spread amongst the healthy eggs, although in smaller broods they have also been reported consuming living eggs. It is believed that this is due to limited feeding opportunities for the males during breeding, as a result of their restriction to the nests. FishBase considers the blennies to be of Low Vulnerability, with a reproductive doubling time of less than 15 months. | https://en.wikipedia.org/wiki/Sphinx_blenny |
In mating, the male deposits spermatophores on the underside of the female's thorax, between the walking legs. The female then extrudes eggs, which pass through the spermatophores. The female carries the fertilised eggs with her until they hatch; the time may vary, but is generally less than 3 weeks. Females lay 10,000–50,000 eggs up to five times per year.From these eggs hatch zoeae, the first larval stage of crustaceans. | https://en.wikipedia.org/wiki/Macrobrachium_rosenbergii |
They go through several larval stages in brackish water before metamorphosing into postlarvae, at which stage they are 0.28–0.39 in (7.1–9.9 mm) long and resemble adults. This metamorphosis usually takes place about 32 to 35 days after hatching. These postlarvae then migrate back into fresh water. | https://en.wikipedia.org/wiki/Macrobrachium_rosenbergii |
In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab+, in which b+ represents the pseudoinverse of b. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then b+ = 0. | https://en.wikipedia.org/wiki/Division_by_zero |
In matrix analysis Stahl's theorem is a theorem proved in 2011 by Herbert Stahl concerning Laplace transforms for special matrix functions. It originated in 1975 as the Bessis-Moussa-Villani (BMV) conjecture by Daniel Bessis, Pierre Moussa, and Marcel Villani. In 2004 Elliott H. Lieb and Robert Seiringer gave two important reformulations of the BMV conjecture. In 2015 Alexandre Eremenko gave a simplified proof of Stahl's theorem. | https://en.wikipedia.org/wiki/Stahl's_theorem |
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.If A is a differentiable map from the real numbers to n × n matrices, then d d t det A ( t ) = tr ( adj ( A ( t ) ) d A ( t ) d t ) = ( det A ( t ) ) ⋅ tr ( A ( t ) − 1 ⋅ d A ( t ) d t ) {\displaystyle {\frac {d}{dt}}\det A(t)=\operatorname {tr} \left(\operatorname {adj} (A(t))\,{\frac {dA(t)}{dt}}\right)=\left(\det A(t)\right)\cdot \operatorname {tr} \left(A(t)^{-1}\cdot \,{\frac {dA(t)}{dt}}\right)} where tr(X) is the trace of the matrix X. (The latter equality only holds if A(t) is invertible.) As a special case, ∂ det ( A ) ∂ A i j = adj ( A ) j i . | https://en.wikipedia.org/wiki/Jacobi's_formula |
{\displaystyle {\partial \det(A) \over \partial A_{ij}}=\operatorname {adj} (A)_{ji}.} Equivalently, if dA stands for the differential of A, the general formula is d det ( A ) = tr ( adj ( A ) d A ) . {\displaystyle d\det(A)=\operatorname {tr} (\operatorname {adj} (A)\,dA).} The formula is named after the mathematician Carl Gustav Jacob Jacobi. | https://en.wikipedia.org/wiki/Jacobi's_formula |
In matrix form, Oja's rule can be written d w ( t ) d t = w ( t ) Q − d i a g w ( t ) {\displaystyle \,{\frac {{\text{d}}w(t)}{{\text{d}}t}}~=~w(t)Q-\mathrm {diag} w(t)} ,and the Gram-Schmidt algorithm is Δ w ( t ) = − l o w e r w ( t ) {\displaystyle \,\Delta w(t)~=~-\mathrm {lower} w(t)} ,where w(t) is any matrix, in this case representing synaptic weights, Q = η x xT is the autocorrelation matrix, simply the outer product of inputs, diag is the function that diagonalizes a matrix, and lower is the function that sets all matrix elements on or above the diagonal equal to 0. We can combine these equations to get our original rule in matrix form, Δ w ( t ) = η ( t ) ( y ( t ) x ( t ) T − L T w ( t ) ) {\displaystyle \,\Delta w(t)~=~\eta (t)\left(\mathbf {y} (t)\mathbf {x} (t)^{\mathrm {T} }-\mathrm {LT} w(t)\right)} ,where the function LT sets all matrix elements above the diagonal equal to 0, and note that our output y(t) = w(t) x(t) is a linear neuron. | https://en.wikipedia.org/wiki/Generalized_Hebbian_Algorithm |
In matrix notation a linear mixed model can be represented as y = X β + Z u + ϵ {\displaystyle {\boldsymbol {y}}=X{\boldsymbol {\beta }}+Z{\boldsymbol {u}}+{\boldsymbol {\epsilon }}} where y {\displaystyle {\boldsymbol {y}}} is a known vector of observations, with mean E ( y ) = X β {\displaystyle E({\boldsymbol {y}})=X{\boldsymbol {\beta }}} ; β {\displaystyle {\boldsymbol {\beta }}} is an unknown vector of fixed effects; u {\displaystyle {\boldsymbol {u}}} is an unknown vector of random effects, with mean E ( u ) = 0 {\displaystyle E({\boldsymbol {u}})={\boldsymbol {0}}} and variance–covariance matrix var ( u ) = G {\displaystyle \operatorname {var} ({\boldsymbol {u}})=G} ; ϵ {\displaystyle {\boldsymbol {\epsilon }}} is an unknown vector of random errors, with mean E ( ϵ ) = 0 {\displaystyle E({\boldsymbol {\epsilon }})={\boldsymbol {0}}} and variance var ( ϵ ) = R {\displaystyle \operatorname {var} ({\boldsymbol {\epsilon }})=R} ; X {\displaystyle X} and Z {\displaystyle Z} are known design matrices relating the observations y {\displaystyle {\boldsymbol {y}}} to β {\displaystyle {\boldsymbol {\beta }}} and u {\displaystyle {\boldsymbol {u}}} , respectively. | https://en.wikipedia.org/wiki/Mixed_models |
In matrix notation, E ( X ) = n p , {\displaystyle \operatorname {E} (\mathbf {X} )=n\mathbf {p} ,\,} and Var ( X ) = n { diag ( p ) − p p T } , {\displaystyle \operatorname {Var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T}}\rbrace ,\,} with pT = the row vector transpose of the column vector p. | https://en.wikipedia.org/wiki/Multinomial_distribution |
In matrix notation, E ( X ) = n p , {\displaystyle \operatorname {E} (\mathbf {X} )=n\mathbf {p} ,\,} and var ( X ) = n { diag ( p ) − p p T } ( n + α 0 1 + α 0 ) , {\displaystyle \operatorname {var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T}}\rbrace \left({\frac {n+\alpha _{0}}{1+\alpha _{0}}}\right),\,} with pT = the row vector transpose of the column vector p. Letting α 0 = 1 − ρ 2 ρ 2 {\displaystyle \alpha _{0}={\frac {1-\rho ^{2}}{\rho ^{2}}}\,} , we can write alternatively var ( X ) = n { diag ( p ) − p p T } ( 1 + ρ 2 ( n − 1 ) ) , {\displaystyle \operatorname {var} (\mathbf {X} )=n\lbrace \operatorname {diag} (\mathbf {p} )-\mathbf {p} \mathbf {p} ^{\rm {T}}\rbrace (1+\rho ^{2}(n-1)),\,} The parameter ρ {\displaystyle \rho \!} is known as the "intra class" or "intra cluster" correlation. It is this positive correlation which gives rise to overdispersion relative to the multinomial distribution. | https://en.wikipedia.org/wiki/Multivariate_Pólya_distribution |
In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.Given an n-by-n matrix A {\displaystyle A} , let det ( A ) {\displaystyle \det(A)} denote its determinant. Choose a pair u = ( u 1 , … , u m ) , v = ( v 1 , … , v m ) ⊂ ( 1 , … , n ) {\displaystyle u=(u_{1},\dots ,u_{m}),v=(v_{1},\dots ,v_{m})\subset (1,\dots ,n)} of m-element ordered subsets of ( 1 , … , n ) {\displaystyle (1,\dots ,n)} , where m ≤ n. Let A v u {\displaystyle A_{v}^{u}} denote the (n−m)-by-(n−m) submatrix of A {\displaystyle A} obtained by deleting the rows in u {\displaystyle u} and the columns in v {\displaystyle v} . Define the auxiliary m-by-m matrix A ~ v u {\displaystyle {\tilde {A}}_{v}^{u}} whose elements are equal to the following determinants ( A ~ v u ) i j := det ( A v u ) , {\displaystyle ({\tilde {A}}_{v}^{u})_{ij}:=\det(A_{v}^{u}),} where u {\displaystyle u} , v {\displaystyle v} denote the m−1 element subsets of u {\displaystyle u} and v {\displaystyle v} obtained by deleting the elements u i {\displaystyle u_{i}} and v j {\displaystyle v_{j}} , respectively. | https://en.wikipedia.org/wiki/Sylvester's_determinant_identity |
Then the following is Sylvester's determinantal identity (Sylvester, 1851): det ( A ) ( det ( A v u ) ) m − 1 = det ( A ~ v u ) . {\displaystyle \det(A)(\det(A_{v}^{u}))^{m-1}=\det({\tilde {A}}_{v}^{u}).} When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851). | https://en.wikipedia.org/wiki/Sylvester's_determinant_identity |
In matrix theory, the Frobenius covariants of a square matrix A are special polynomials of it, namely projection matrices Ai associated with the eigenvalues and eigenvectors of A.: pp.403, 437–8 They are named after the mathematician Ferdinand Frobenius. Each covariant is a projection on the eigenspace associated with the eigenvalue λi. Frobenius covariants are the coefficients of Sylvester's formula, which expresses a function of a matrix f(A) as a matrix polynomial, namely a linear combination of that function's values on the eigenvalues of A. | https://en.wikipedia.org/wiki/Frobenius_covariant |
In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition); to demography (Leslie population age distribution model); to social networks (DeGroot learning process); to Internet search engines (PageRank); and even to ranking of football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. | https://en.wikipedia.org/wiki/Perron–Frobenius_theorem |
In matrix theory, the Rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus.Consider a 3 × 3 {\displaystyle 3\times 3} matrix M = , {\displaystyle M={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}},} then its determinant can be computed by the following scheme. Write out the first two columns of the matrix to the right of the third column, giving five columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields det ( M ) = det = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 31 a 22 a 13 − a 32 a 23 a 11 − a 33 a 21 a 12 . | https://en.wikipedia.org/wiki/Rule_of_Sarrus |
{\displaystyle {\begin{aligned}\det(M)&=\det {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}\\&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.\end{aligned}}} A similar scheme based on diagonals works for 2 × 2 {\displaystyle 2\times 2} matrices: det ( M ) = det = a 11 a 22 − a 21 a 12 . {\displaystyle \det(M)=\det {\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}=a_{11}a_{22}-a_{21}a_{12}.} Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived using the Laplace expansion of a 3 × 3 {\displaystyle 3\times 3} matrix.Another way of thinking of Sarrus' rule is to imagine that the matrix is wrapped around a cylinder, such that the right and left edges are joined. | https://en.wikipedia.org/wiki/Rule_of_Sarrus |
In matrix-assisted laser desorption and ionization (MALDI), the sample is incorporated in a chemical matrix that is capable of absorbing energy from a laser. Similar to SIMS, ionization happens in vacuum. Laser irradiation ablates the matrix material from the surface and results in charged gas phase matrix particles, the analyte molecules are ionized from this charged chemical matrix. Liu et al. used MALDI-MS to detect eight phospholipids from single A549 cells. | https://en.wikipedia.org/wiki/Single_cell_analysis |
In matroid theory, a Sylvester matroid is a matroid in which every pair of elements belongs to a three-element circuit (a triangle) of the matroid. | https://en.wikipedia.org/wiki/Sylvester_matroid |
In matroid theory, a binary matroid is a matroid that can be represented over the finite field GF(2). That is, up to isomorphism, they are the matroids whose elements are the columns of a (0,1)-matrix and whose sets of elements are independent if and only if the corresponding columns are linearly independent in GF(2). | https://en.wikipedia.org/wiki/Binary_matroid |
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by Hazel Perfect (1968), based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. Gammoids were given their name by Pym (1969) and studied in more detail by Mason (1972). | https://en.wikipedia.org/wiki/Gammoid |
In matroid theory, a mathematical discipline, the girth of a matroid is the size of its smallest circuit or dependent set. The cogirth of a matroid is the girth of its dual matroid. Matroid girth generalizes the notion of the shortest cycle in a graph, the edge connectivity of a graph, Hall sets in bipartite graphs, even sets in families of sets, and general position of point sets. It is hard to compute, but fixed-parameter tractable for linear matroids when parameterized both by the matroid rank and the field size of a linear representation. | https://en.wikipedia.org/wiki/Matroid_girth |
In matroid theory, an Eulerian matroid is a matroid whose elements can be partitioned into a collection of disjoint circuits. | https://en.wikipedia.org/wiki/Eulerian_matroid |
In matroid theory, the closure of X is the largest superset of X that has the same rank as X. The transitive closure of a set. The algebraic closure of a field. The integral closure of an integral domain in a field that contains it. | https://en.wikipedia.org/wiki/Closure_(mathematics) |
The radical of an ideal in a commutative ring. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language. In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection. | https://en.wikipedia.org/wiki/Closure_(mathematics) |
In matroid theory, the dual of a matroid M {\displaystyle M} is another matroid M ∗ {\displaystyle M^{\ast }} that has the same elements as M {\displaystyle M} , and in which a set is independent if and only if M {\displaystyle M} has a basis set disjoint from it.Matroid duals go back to the original paper by Hassler Whitney defining matroids. They generalize to matroids the notions of plane graph duality. | https://en.wikipedia.org/wiki/Dual_matroid |
In matters concerning the local, national and transboundary environment, the Aarhus convention grants the public rights regarding access to information, public participation and access to justice in governmental decision-making processes. It focuses on interactions between the public and public authorities. | https://en.wikipedia.org/wiki/Freedom_of_Information_legislation |
In matters of Islamic jurisprudence, Zaydīs follow the teachings of Zayd ibn ʿAlī, which are documented in his book Majmu'l Fiqh (in Arabic: مجموع الفِقه). Al-Ḥādī ila'l-Ḥaqq Yaḥyā, the first Zaydī Imam and founder of the Zaydī State in Yemen, is regarded as the codifier of Zaydī jurisprudence, and as such most Zaydī Shīʿas today are known as Hadawis. | https://en.wikipedia.org/wiki/Shia_Islam |
In matters of script, Nepali uses Devanagari. On this grammar page Nepali is written in "standard orientalist" transcription as outlined in Masica (1991:xv). Being "primarily a system of transliteration from the Indian scripts, based in turn upon Sanskrit" (cf. IAST), these are its salient features: subscript dots for retroflex consonants; macrons for etymologically, contrastively long vowels; h denoting aspirated plosives. Tildes denote nasalized vowels. | https://en.wikipedia.org/wiki/Nepali_grammar |
Vowels and consonants are outlined in the tables below. Hovering the mouse cursor over them will reveal the appropriate IPA symbol, while in the rest of the article hovering the mouse cursor over underlined forms will reveal the appropriate English translation. Like Bengali, Assamese, and Marathi and unlike Hindustani, Nepali may retain the final schwa in a word. | https://en.wikipedia.org/wiki/Nepali_grammar |
The following rules can be followed to figure out whether or not Nepali words retain the final schwa. 1) Schwa is retained if the final syllable is a conjunct consonant. अन्त (anta, 'end'), सम्बन्ध (sambandha, 'relation'), श्रेष्ठ (śreṣṭha, 'greatest'/a last name).Exceptions: conjuncts such as ञ्च ञ्ज in मञ्च (mañc, 'stage') गञ्ज (gañj, 'city') and occasionally the last name पन्त (panta/pant). | https://en.wikipedia.org/wiki/Nepali_grammar |
2) For any verb form the final schwa is always retained unless the schwa-cancelling halanta is present. हुन्छ (huncha, 'it happens'), भएर (bhaera, 'in happening so; therefore'), गएछ(gaecha, 'he apparently went'), but छन् (chan, 'they are'), गईन् (gain, 'she went'). Meanings may change with the wrong orthography: गईन (gaina, 'she didn't go') vs गईन् (gain, 'she went'). | https://en.wikipedia.org/wiki/Nepali_grammar |
3) Adverbs, onomatopoeia and postpositions usually maintain the schwa and if they don't, halanta is acquired: अब (aba 'now'), तिर (tira, 'towards'), आज (āja, 'today') सिम्सिम (simsim 'drizzle') vs झन् (jhan, 'more'). 4) Few exceptional nouns retain the schwa such as: दुख(dukha, 'suffering'), सुख (sukha, 'pleasure'). Note: Schwas are often retained in music and poetry to facilitate singing and recitation. | https://en.wikipedia.org/wiki/Nepali_grammar |
In matters of sexuality, several Evangelical churches promote the virginity pledge among young Evangelical Christians, who are invited to commit themselves during a public ceremony to sexual abstinence until Christian marriage. This pledge is often symbolized by a purity ring.In evangelical churches, young adults and unmarried couples are encouraged to marry early in order to live a sexuality according to the will of God.Although some churches are discreet on the subject, other evangelical churches in United States and Switzerland speak of a satisfying sexuality as a gift from God and a component of a harmonious Christian marriage, in messages during worship services or conferences. Many evangelical books and websites are specialized on the subject.The perceptions of homosexuality in the Evangelical Churches are varied. | https://en.wikipedia.org/wiki/Alternative_sexuality |
They range from liberal through moderate to conservative.The christian marriage is presented by some churches as a protection against sexual misconduct and a compulsory step to obtain a position of responsibility in the church. This concept, however, has been challenged by numerous sex scandals involving married evangelical leaders. Finally, evangelical theologians recalled that celibacy should be more valued in the Church today, since the gift of celibacy was taught and lived by Jesus Christ and Paul of Tarsus. | https://en.wikipedia.org/wiki/Alternative_sexuality |
In matters relating to quantum information theory, it is convenient to work with the simplest possible unit of information: the two-state system of the qubit. The qubit functions as the quantum analog of the classic computational part, the bit, as it can have a measurement value of both a 0 and a 1, whereas the classical bit can only be measured as a 0 or a 1. The quantum two-state system seeks to transfer quantum information from one location to another location without losing the information and preserving the quality of this information. This process involves moving the information between carriers and not movement of the actual carriers, similar to the traditional process of communications, as two parties remain stationary while the information (digital media, voice, text, etc.) is being transferred, contrary to the implications of the word "teleport." | https://en.wikipedia.org/wiki/Gate_teleportation |
The main components needed for teleportation include a sender, the information (a qubit), a traditional channel, a quantum channel, and a receiver. An interesting fact is that the sender does not need to know the exact contents of the information that is being sent. The measurement postulate of quantum mechanics—when a measurement is made upon a quantum state, any subsequent measurements will "collapse" or that the observed state will be lost—creates an imposition within teleportation: if a sender makes a measurement on their information, the state could collapse when the receiver obtains the information since the state has changed from when the sender made the initial measurement. | https://en.wikipedia.org/wiki/Gate_teleportation |
For actual teleportation, it is required that an entangled quantum state or Bell state be created for the qubit to be transferred. Entanglement imposes statistical correlations between otherwise distinct physical systems by creating or placing two or more separate particles into a single, shared quantum state. This intermediate state contains two particles whose quantum states are dependent on each other as they form a connection: if one particle is moved, the other particle will move along with it. | https://en.wikipedia.org/wiki/Gate_teleportation |
Any changes that one particle of the entanglement undergoes, the other particle will also undergo that change, causing the entangled particles to act as one quantum state. These correlations hold even when measurements are chosen and performed independently, out of causal contact from one another, as verified in Bell test experiments. | https://en.wikipedia.org/wiki/Gate_teleportation |
Thus, an observation resulting from a measurement choice made at one point in spacetime seems to instantaneously affect outcomes in another region, even though light hasn't yet had time to travel the distance; a conclusion seemingly at odds with special relativity. This is known as the EPR paradox. | https://en.wikipedia.org/wiki/Gate_teleportation |
However such correlations can never be used to transmit any information faster than the speed of light, a statement encapsulated in the no-communication theorem. Thus, teleportation as a whole can never be superluminal, as a qubit cannot be reconstructed until the accompanying classical information arrives. The sender will then combine the particle, of which the information is teleported, with one of the entangled particles, causing a change of the overall entangled quantum state. | https://en.wikipedia.org/wiki/Gate_teleportation |
Of this changed state, the particles in Alice's possession are then sent to an analyzer that will measure the change of the entangled state. The "change" measurement will allow the receiver to recreate the original information that the sender had resulting in the information being teleported or carried between two people that have different locations. Since the initial quantum information is "destroyed" as it becomes part of the entanglement state, the no-cloning theorem is maintained as the information is recreated from the entangled state and not copied during teleportation. | https://en.wikipedia.org/wiki/Gate_teleportation |
The quantum channel is the communication mechanism that is used for all quantum information transmission and is the channel used for teleportation (relationship of quantum channel to traditional communication channel is akin to the qubit being the quantum analog of the classical bit). However, in addition to the quantum channel, a traditional channel must also be used to accompany a qubit to "preserve" the quantum information. When the change measurement between the original qubit and the entangled particle is made, the measurement result must be carried by a traditional channel so that the quantum information can be reconstructed and the receiver can get the original information. | https://en.wikipedia.org/wiki/Gate_teleportation |
Because of this need for the traditional channel, the speed of teleportation can be no faster than the speed of light (hence the no-communication theorem is not violated). The main advantage with this is that Bell states can be shared using photons from lasers making teleportation achievable through open space having no need to send information through physical cables or optical fibers. Quantum states can be encoded in various degrees of freedom of atoms. | https://en.wikipedia.org/wiki/Gate_teleportation |
For example, qubits can be encoded in the degrees of freedom of electrons surrounding the atomic nucleus or in the degrees of freedom of the nucleus itself. Thus, performing this kind of teleportation requires a stock of atoms at the receiving site, available for having qubits imprinted on them.As of 2015, the quantum states of single photons, photon modes, single atoms, atomic ensembles, defect centers in solids, single electrons, and superconducting circuits have been employed as information bearers.Understanding quantum teleportation requires a good grounding in finite-dimensional linear algebra, Hilbert spaces and projection matrixes. A qubit is described using a two-dimensional complex number-valued vector space (a Hilbert space), which are the primary basis for the formal manipulations given below. A working knowledge of quantum mechanics is not absolutely required to understand the mathematics of quantum teleportation, although without such acquaintance, the deeper meaning of the equations may remain quite mysterious. | https://en.wikipedia.org/wiki/Gate_teleportation |
In mature adults, glycine is a inhibitory neurotransmitter found in the spinal cord and regions of the brain. As it binds to a glycine receptor, a conformational change is induced, and the channel created by the receptor opens. As the channel opens, chloride ions are able to flow into the cell which results in hyperpolarization. In addition to this hyperpolarization, which decreases the likelihood of action potential propagation, glycine is also responsible for decreasing the release of both inhibitory and excitatory neurotransmitters as it binds to its receptor. | https://en.wikipedia.org/wiki/Glycine_receptor |
This is called the "shunting" effect and can be explained by Ohm's Law. As the receptor is activated, the membrane conductance is increased and the membrane resistance is decreased. According to Ohm's Law, as resistance decreases, so does voltage. A decreased postsynaptic voltage results in a decreased release of neurotransmitters. | https://en.wikipedia.org/wiki/Glycine_receptor |
In mature mice brain tissue PGBD5 is found primarily in regions of the olfactory bulb, hippocampus, and cerebellum. In embryonic mice brain tissue PGBD5 is found not only in the medial pallium and prepontine isthmus, which are embryonic brain areas that give rise to the development of the hippocampus and cerebellum but also in areas in the embryonic brain that give rise to the hypothalamus and medulla. | https://en.wikipedia.org/wiki/PiggyBac_Transposable_Element_Derived_5 |
In maximum likelihood beamformer (DML), the noise is modeled as a stationary Gaussian white random processes while the signal waveform as deterministic (but arbitrary) and unknown. | https://en.wikipedia.org/wiki/Digital_antenna_array |
In maximum parsimony, Dollo parsimony refers to a model whereby a character is gained only one time and can never be regained if it is lost. For example, the evolution and repeated loss of teeth in vertebrates could be well-modeled under Dollo parsimony, whereby teeth made from hydroxyapatite evolved only once at the origin of vertebrates, and were then lost multiple times, in birds, turtles, and seahorses, among others.This also applies to molecular characters, such as losses or inactivation of individual genes themselves. The loss of gulonolactone oxidase, the final enzyme in the biosynthetic pathway of vitamin C, is responsible for the dietary requirement of vitamin C in humans, as well as many other animals. | https://en.wikipedia.org/wiki/Dollo's_law_of_irreversibility |
In may 2005 Swedish Radio first published podcast for the listerners to download. The first podcasts offered were Ekots lördagsintervju i P1, Spanarna i P1 and Salva i P3 but shortly after the offer was expanded to include several other titles. In February 2006, following London radio station LBC's successful launch of the first premium-podcasting platform, LBC Plus, there was widespread acceptance that podcasting had considerable commercial potential. UK comedian Ricky Gervais, whose first season of The Ricky Gervais Show became a big hit, launched a new series of the popular podcast. | https://en.wikipedia.org/wiki/Odiogo |
The second series of the podcast was distributed through audible.co.uk and was the first major podcast to charge consumers to download the show (at a rate of 95 pence per half-hour episode). The first series of The Ricky Gervais Show podcast had been freely distributed by the Positive Internet Company and marketed through The Guardian newspaper's website, and it was the world's most successful podcast for several years, eventually gaining more than 300 million unique downloads by March 2011. Even in its new subscription format, The Ricky Gervais Show was regularly the most-downloaded podcast on iTunes. | https://en.wikipedia.org/wiki/Odiogo |
The Adam Carolla Show claimed a new Guinness world record, with total downloads approaching 60 million, but Guinness failed to acknowledge that Gervais's podcast had more than 5 times as many downloads as Carolla's show at the time that this new record was supposedly set. In February 2006, LA podcaster Lance Anderson became nearly the first to take a podcast and create a live venue tour. The Lance Anderson Podcast Experment (sic) included a sold-out extravaganza in The Pilgrim, a central Liverpool (UK) venue (February 23, 2006), followed by a theatrical event at The Rose Theatre, Edge Hill University (February 24, 2006), which included appearances by Mark Hunter from The Tartan Podcast, Jon and Rob from Top of the Pods, Dan Klass from The Bitterest Pill via video link from Los Angeles, and live music from The Hotrod Cadets. | https://en.wikipedia.org/wiki/Odiogo |
In addition, Anderson was also invited to take part in the first-ever Podcast Forum at CARET, the Centre for Applied Research in Educational Technologies at the University of Cambridge (February 21, 2006). Organised and supported by Josh Newman, the university's Apple Campus Rep, Anderson was joined at this event by Dr. Chris Smith from the Naked Scientists podcast; Debbie McGowan, an Open University lecturer and advocate for podcasting in education; and Nigel Paice, a professional music producer and podcasting tutor. In March 2006, Canadian Prime Minister Stephen Harper became the second head of government to issue a podcast, the Prime Minister of Canada's Podcast (George W. Bush technically being the first one back in July 2005). | https://en.wikipedia.org/wiki/Odiogo |
In July 2009, the company VoloMedia is awarded the "Podcast patent" by the USPTO in patent number 7,568,213. Dave Winer, the co-inventor of podcasting (with Adam Curry), points out that his invention predated this patent by two years.On February 2, 2006, Virginia Tech (Virginia Polytechnic Institute and State University) launched the first regular schedule of podcast programming at the university. Having four regularly scheduled podcasts was a first for a major American university, which was launched as part of Virginia Tech's "Invent the Future" campaign.In April 2006, comedy podcast Never Not Funny began when Matt Belknap of ASpecialThing Records interviewed comedian Jimmy Pardo on the podcast for his popular alternative comedy forum A Special Thing. | https://en.wikipedia.org/wiki/Odiogo |
The two had previously discussed producing a podcast version of Jimmy's Los Angeles show "Running Your Trap", which he hosted at the Upright Citizens Brigade Theatre, but they hit it off so well on AST Radio that Pardo said "This is the show." Shortly after, Never Not Funny started simulcasting both a podcast stream and a paid video version. The podcast still uses this format, releasing two shows a week—one free and one paid—along with paid video feed. | https://en.wikipedia.org/wiki/Odiogo |
In October 2006, the This American Life radio program began to offer a podcast version to listeners. Since debuting, This American Life has consistently been one of the most-listened-to podcasts, averaging around 2.5 million downloads per episode. In March 2007, after being on-air talent and being fired from KYSR (STAR) in Los Angeles, California, Jack and Stench started their own subscription-based podcast. | https://en.wikipedia.org/wiki/Odiogo |
At $5.00 per subscription, subscribers had access to a one-hour podcast, free of any commercials. They had free local events at bars, ice cream parlors and restaurants all around Southern California. | https://en.wikipedia.org/wiki/Odiogo |
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