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Dobson unit
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Derivation
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A Dobson unit is the total amount of a trace gas per unit area. In atmospheric sciences, this is referred to as a column density. How, though, do we go from units of molecules per cubic meter, a volume, to molecules per square centimeter, an area? This must be done by integration. To get a column density, we must integrate the total column over a height. Per the definition of Dobson units, we see that 1 DU = 0.01 mm of trace gas when compressed down to sea level at standard temperature and pressure. So if we integrate our number density of air from 0 to 0.01 mm, we find the number density which is equal to 1 DU: mm 0.01 mm 2.69 10 25 molecules 2.69 10 25 molecules 0.01 mm 2.69 10 25 molecules mm 2.69 10 25 molecules 10 2.69 10 20 molecules ⋅m−2.
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Dobson unit
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Derivation
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And thus we come up with the value of 1 DU, which is 2.69×1020 molecules per meter squared.
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Indigenous bundle
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Indigenous bundle
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In mathematics, an indigenous bundle on a Riemann surface is a fiber bundle with a flat connection associated to some complex projective structure. Indigenous bundles were introduced by Robert C. Gunning (1967). Indigenous bundles for curves over p-adic fields were introduced by Shinichi Mochizuki (1996) in his study of p-adic Teichmüller theory.
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Jordan matrix
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Jordan matrix
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In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form:
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Jordan matrix
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Definition
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Every Jordan block is specified by its dimension n and its eigenvalue λ∈R , and is denoted as Jλ,n. It is an n×n matrix of zeroes everywhere except for the diagonal, which is filled with λ and for the superdiagonal, which is composed of ones.
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This (n1 + ⋯ + nr) × (n1 + ⋯ + nr) square matrix, consisting of r diagonal blocks, can be compactly indicated as Jλ1,n1⊕⋯⊕Jλr,nr or diag(Jλ1,n1,…,Jλr,nr) , where the i-th Jordan block is Jλi,ni.
For example, the matrix is a 10 × 10 Jordan matrix with a 3 × 3 block with eigenvalue 0, two 2 × 2 blocks with eigenvalue the imaginary unit i, and a 3 × 3 block with eigenvalue 7. Its Jordan-block structure is written as either J0,3⊕Ji,2⊕Ji,2⊕J7,3 or diag(J0,3, Ji,2, Ji,2, J7,3).
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Jordan matrix
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Linear algebra
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Any n × n square matrix A whose elements are in an algebraically closed field K is similar to a Jordan matrix J, also in Mn(K) , which is unique up to a permutation of its diagonal blocks themselves. J is called the Jordan normal form of A and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all 1 × 1.More generally, given a Jordan matrix J=Jλ1,m1⊕Jλ2,m2⊕⋯⊕JλN,mN , that is, whose kth diagonal block, 1≤k≤N , is the Jordan block Jλk,mk and whose diagonal elements λk may not all be distinct, the geometric multiplicity of λ∈K for the matrix J, indicated as gmul Jλ , corresponds to the number of Jordan blocks whose eigenvalue is λ. Whereas the index of an eigenvalue λ for J, indicated as idx Jλ , is defined as the dimension of the largest Jordan block associated to that eigenvalue.
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Jordan matrix
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Linear algebra
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The same goes for all the matrices A similar to J, so idx Aλ can be defined accordingly with respect to the Jordan normal form of A for any of its eigenvalues spec A . In this case one can check that the index of λ for A is equal to its multiplicity as a root of the minimal polynomial of A (whereas, by definition, its algebraic multiplicity for A, mul Aλ , is its multiplicity as a root of the characteristic polynomial of A; that is, det (A−xI)∈K[x] ). An equivalent necessary and sufficient condition for A to be diagonalizable in K is that all of its eigenvalues have index equal to 1; that is, its minimal polynomial has only simple roots.
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Jordan matrix
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Linear algebra
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Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices): the Jordan decomposition is, in general, a computationally challenging task. From the vector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (that is, via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for.
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Jordan matrix
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Functions of matrices
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Let A∈Mn(C) (that is, a n × n complex matrix) and C∈GLn(C) be the change of basis matrix to the Jordan normal form of A; that is, A = C−1JC. Now let f (z) be a holomorphic function on an open set Ω such that specA⊂Ω⊆C ; that is, the spectrum of the matrix is contained inside the domain of holomorphy of f. Let be the power series expansion of f around spec A , which will be hereinafter supposed to be 0 for simplicity's sake. The matrix f (A) is then defined via the following formal power series and is absolutely convergent with respect to the Euclidean norm of Mn(C) . To put it another way, f (A) converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of f around 0 and is uniformly convergent on any compact subsets of Mn(C) satisfying this property in the matrix Lie group topology.
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Jordan matrix
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Functions of matrices
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The Jordan normal form allows the computation of functions of matrices without explicitly computing an infinite series, which is one of the main achievements of Jordan matrices. Using the facts that the kth power ( k∈N0 ) of a diagonal block matrix is the diagonal block matrix whose blocks are the kth powers of the respective blocks; that is, (A1⊕A2⊕A3⊕⋯)k=A1k⊕A2k⊕A3k⊕⋯ , and that Ak = C−1JkC, the above matrix power series becomes where the last series need not be computed explicitly via power series of every Jordan block. In fact, if λ∈Ω , any holomorphic function of a Jordan block f(Jλ,n)=f(λI+Z) has a finite power series around λI because Zn=0 . Here, Z is the nilpotent part of J and Zk has all 0's except 1's along the th superdiagonal. Thus it is the following upper triangular matrix: As a consequence of this, the computation of any function of a matrix is straightforward whenever its Jordan normal form and its change-of-basis matrix are known. For example, using f(z)=1/z , the inverse of Jλ,n is: Also, spec f(A) = f (spec A); that is, every eigenvalue λ∈specA corresponds to the eigenvalue spec f(A) , but it has, in general, different algebraic multiplicity, geometric multiplicity and index. However, the algebraic multiplicity may be computed as follows: The function f (T) of a linear transformation T between vector spaces can be defined in a similar way according to the holomorphic functional calculus, where Banach space and Riemann surface theories play a fundamental role. In the case of finite-dimensional spaces, both theories perfectly match.
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Jordan matrix
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Dynamical systems
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Now suppose a (complex) dynamical system is simply defined by the equation where z:R+→R is the (n-dimensional) curve parametrization of an orbit on the Riemann surface R of the dynamical system, whereas A(c) is an n × n complex matrix whose elements are complex functions of a d-dimensional parameter c∈Cd Even if A∈Mn(C0(Cd)) (that is, A continuously depends on the parameter c) the Jordan normal form of the matrix is continuously deformed almost everywhere on Cd but, in general, not everywhere: there is some critical submanifold of Cd on which the Jordan form abruptly changes its structure whenever the parameter crosses or simply "travels" around it (monodromy). Such changes mean that several Jordan blocks (either belonging to different eigenvalues or not) join to a unique Jordan block, or vice versa (that is, one Jordan block splits into two or more different ones). Many aspects of bifurcation theory for both continuous and discrete dynamical systems can be interpreted with the analysis of functional Jordan matrices.
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Jordan matrix
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Dynamical systems
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From the tangent space dynamics, this means that the orthogonal decomposition of the dynamical system's phase space changes and, for example, different orbits gain periodicity, or lose it, or shift from a certain kind of periodicity to another (such as period-doubling, cfr. logistic map).
In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of A(c).
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Jordan matrix
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Linear ordinary differential equations
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The simplest example of a dynamical system is a system of linear, constant-coefficient, ordinary differential equations; that is, let A∈Mn(C) and z0∈Cn whose direct closed-form solution involves computation of the matrix exponential: Another way, provided the solution is restricted to the local Lebesgue space of n-dimensional vector fields z∈Lloc1(R+)n , is to use its Laplace transform Z(s)=L[z](s) . In this case The matrix function (A − sI)−1 is called the resolvent matrix of the differential operator {\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}-A} . It is meromorphic with respect to the complex parameter s∈C since its matrix elements are rational functions whose denominator is equal for all to det(A − sI). Its polar singularities are the eigenvalues of A, whose order equals their index for it; that is, ord(A−sI)−1λ=idxAλ
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Undertow (water waves)
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Undertow (water waves)
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In physical oceanography, undertow is the undercurrent that moves offshore while waves approach the shore. Undertow is a natural and universal feature for almost any large body of water; it is a return flow compensating for the onshore-directed average transport of water by the waves in the zone above the wave troughs. The undertow's flow velocities are generally strongest in the surf zone, where the water is shallow and the waves are high due to shoaling.In popular usage, the word undertow is often misapplied to rip currents. An undertow occurs everywhere underneath shore-approaching waves, whereas rip currents are localized narrow offshore currents occurring at certain locations along the coast.
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Undertow (water waves)
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Oceanography
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An "undertow" is a steady, offshore-directed compensation flow, which occurs below waves near the shore. Physically, nearshore, the wave-induced mass flux between wave crest and trough is onshore directed. This mass transport is localized in the upper part of the water column, i.e. above the wave troughs. To compensate for the amount of water being transported towards the shore, a second-order (i.e. proportional to the wave height squared), offshore-directed mean current takes place in the lower section of the water column. This flow – the undertow – affects the nearshore waves everywhere, unlike rip currents localized at certain positions along the shore.The term undertow is used in scientific coastal oceanography papers. The distribution of flow velocities in the undertow over the water column is important as it strongly influences the on- or offshore transport of sediment. Outside the surf zone there is a near-bed onshore-directed sediment transport induced by Stokes drift and skewed-asymmetric wave transport. In the surf zone, strong undertow generates a near-bed offshore sediment transport. These antagonistic flows may lead to sand bar formation where the flows converge near the wave breaking point, or in the wave breaking zone.
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Undertow (water waves)
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Oceanography
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Seaward mass flux An exact relation for the mass flux of a nonlinear periodic wave on an inviscid fluid layer was established by Levi-Civita in 1924. In a frame of reference according to Stokes' first definition of wave celerity, the mass flux Mw of the wave is related to the wave's kinetic energy density Ek (integrated over depth and thereafter averaged over wavelength) and phase speed c through: Mw=2Ekc.
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Undertow (water waves)
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Oceanography
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Similarly, Longuet Higgins showed in 1975 that – for the common situation of zero mass flux towards the shore (i.e. Stokes' second definition of wave celerity) – normal-incident periodic waves produce a depth- and time-averaged undertow velocity: u¯=−2Ekρch, with h the mean water depth and ρ the fluid density. The positive flow direction of u¯ is in the wave propagation direction.
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Undertow (water waves)
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Oceanography
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For small-amplitude waves, there is equipartition of kinetic ( Ek ) and potential energy ( Ep ): Ew=Ek+Ep≈2Ek≈2Ep, with Ew the total energy density of the wave, integrated over depth and averaged over horizontal space. Since in general the potential energy Ep is much easier to measure than the kinetic energy, the wave energy is approximately Ew≈18ρgH2 (with H the wave height). So u¯≈−18gH2ch.
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Undertow (water waves)
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Oceanography
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For irregular waves the required wave height is the root-mean-square wave height rms ≈8σ, with σ the standard deviation of the free-surface elevation.
The potential energy is Ep=12ρgσ2 and Ew≈ρgσ2.
The distribution of the undertow velocity over the water depth is a topic of ongoing research.
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Undertow (water waves)
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Confusion with rip currents
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In contrast to undertow, rip currents are responsible for the great majority of drownings close to beaches. When a swimmer enters a rip current, it starts to carry them offshore. The swimmer can exit the rip current by swimming at right angles to the flow, parallel to the shore, or by simply treading water or floating until the rip releases them. However, drowning can occur when swimmers exhaust themselves by trying unsuccessfully to swim directly against the flow of a rip.
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Undertow (water waves)
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Confusion with rip currents
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On the United States Lifesaving Association website, it is noted that some uses of the word "undertow" are incorrect: A rip current is a horizontal current. Rip currents do not pull people under the water—they pull people away from shore. Drowning deaths occur when people pulled offshore are unable to keep themselves afloat and swim to shore. This may be due to any combination of fear, panic, exhaustion, or lack of swimming skills.
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Undertow (water waves)
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Confusion with rip currents
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In some regions, rip currents are referred to by other, incorrect terms such as "rip tides" and "undertow". We encourage exclusive use of the correct term—rip currents. Use of other terms may confuse people and negatively impact public education efforts.
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Voice (phonetics)
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Voice (phonetics)
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Voice or voicing is a term used in phonetics and phonology to characterize speech sounds (usually consonants). Speech sounds can be described as either voiceless (otherwise known as unvoiced) or voiced.
The term, however, is used to refer to two separate concepts: Voicing can refer to the articulatory process in which the vocal folds vibrate, its primary use in phonetics to describe phones, which are particular speech sounds.
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Voice (phonetics)
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Voice (phonetics)
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It can also refer to a classification of speech sounds that tend to be associated with vocal cord vibration but may not actually be voiced at the articulatory level. That is the term's primary use in phonology: to describe phonemes; while in phonetics its primary use is to describe phones.For example, voicing accounts for the difference between the pair of sounds associated with the English letters ⟨s⟩ and ⟨z⟩. The two sounds are transcribed as [s] and [z] to distinguish them from the English letters, which have several possible pronunciations, depending on the context. If one places the fingers on the voice box (i.e., the location of the Adam's apple in the upper throat), one can feel a vibration while [z] is pronounced but not with [s]. (For a more detailed, technical explanation, see modal voice and phonation.) In most European languages, with a notable exception being Icelandic, vowels and other sonorants (consonants such as m, n, l, and r) are modally voiced.
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Voice (phonetics)
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Voice (phonetics)
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Yidiny has no underlyingly voiceless consonants, only voiced ones.When used to classify speech sounds, voiced and unvoiced are merely labels used to group phones and phonemes together for the purposes of classification.
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Voice (phonetics)
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Notation
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The International Phonetic Alphabet has distinct letters for many voiceless and voiced pairs of consonants (the obstruents), such as [p b], [t d], [k ɡ], [q ɢ]. In addition, there is a diacritic for voicedness: ⟨◌̬⟩. Diacritics are typically used with letters for prototypically voiceless sounds.
In Unicode, the symbols are encoded U+032C ◌̬ COMBINING CARON BELOW and U+0325 ◌̥ COMBINING RING BELOW.
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Voice (phonetics)
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Notation
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The extensions to the International Phonetic Alphabet have a notation for partial voicing and devoicing as well as for prevoicing: Partial voicing can mean light but continuous voicing, discontinuous voicing, or discontinuities in the degree of voicing. For example, ₍s̬₎ could be an [s] with (some) voicing in the middle and ₍z̥₎ could be [z] with (some) devoicing in the middle.
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Voice (phonetics)
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Notation
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Partial voicing can also be indicated in the normal IPA with transcriptions like [ᵇb̥iˑ] and [ædᵈ̥].
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Voice (phonetics)
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In English
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The distinction between the articulatory use of voice and the phonological use rests on the distinction between phone (represented between square brackets) and phoneme (represented between slashes). The difference is best illustrated by a rough example.
The English word nods is made up of a sequence of phonemes, represented symbolically as /nɒdz/, or the sequence of /n/, /ɒ/, /d/, and /z/. Each symbol is an abstract representation of a phoneme. That awareness is an inherent part of speakers' mental grammar that allows them to recognise words.
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Voice (phonetics)
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In English
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However, phonemes are not sounds in themselves. Rather, phonemes are, in a sense, converted to phones before being spoken. The /z/ phoneme, for instance, can actually be pronounced as either the [s] phone or the [z] phone since /z/ is frequently devoiced, even in fluent speech, especially at the end of an utterance. The sequence of phones for nods might be transcribed as [nɒts] or [nɒdz], depending on the presence or strength of this devoicing. While the [z] phone has articulatory voicing, the [s] phone does not have it.
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Voice (phonetics)
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In English
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What complicates the matter is that for English, consonant phonemes are classified as either voiced or voiceless even though it is not the primary distinctive feature between them. Still, the classification is used as a stand-in for phonological processes, such as vowel lengthening that occurs before voiced consonants but not before unvoiced consonants or vowel quality changes (the sound of the vowel) in some dialects of English that occur before unvoiced but not voiced consonants. Such processes allow English speakers to continue to perceive difference between voiced and voiceless consonants when the devoicing of the former would otherwise make them sound identical to the latter.
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Voice (phonetics)
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In English
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English has four pairs of fricative phonemes that can be divided into a table by place of articulation and voicing. The voiced fricatives can readily be felt to have voicing throughout the duration of the phone especially when they occur between vowels.
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Voice (phonetics)
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In English
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However, in the class of consonants called stops, such as /p, t, k, b, d, ɡ/, the contrast is more complicated for English. The "voiced" sounds do not typically feature articulatory voicing throughout the sound. The difference between the unvoiced stop phonemes and the voiced stop phonemes is not just a matter of whether articulatory voicing is present or not. Rather, it includes when voicing starts (if at all), the presence of aspiration (airflow burst following the release of the closure) and the duration of the closure and aspiration.
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Voice (phonetics)
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In English
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English voiceless stops are generally aspirated at the beginning of a stressed syllable, and in the same context, their voiced counterparts are voiced only partway through. In more narrow phonetic transcription, the voiced symbols are maybe used only to represent the presence of articulatory voicing, and aspiration is represented with a superscript h.
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Voice (phonetics)
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In English
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When the consonants come at the end of a syllable, however, what distinguishes them is quite different. Voiceless phonemes are typically unaspirated, glottalized and the closure itself may not even be released, making it sometimes difficult to hear the difference between, for example, light and like. However, auditory cues remain to distinguish between voiced and voiceless sounds, such as what has been described above, like the length of the preceding vowel.
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Voice (phonetics)
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In English
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Other English sounds, the vowels and sonorants, are normally fully voiced. However, they may be devoiced in certain positions, especially after aspirated consonants, as in coffee, tree, and play in which the voicing is delayed to the extent of missing the sonorant or vowel altogether.
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Voice (phonetics)
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Degrees of voicing
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There are two variables to degrees of voicing: intensity (discussed under phonation), and duration (discussed under voice onset time). When a sound is described as "half voiced" or "partially voiced", it is not always clear whether that means that the voicing is weak (low intensity) or if the voicing occurs during only part of the sound (short duration). In the case of English, it is the latter.
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Voice (phonetics)
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Degrees of voicing
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Juǀʼhoansi and some of its neighboring languages are typologically unusual in having contrastive partially-voiced consonants. They have aspirate and ejective consonants, which are normally incompatible with voicing, in voiceless and voiced pairs. The consonants start out voiced but become voiceless partway through and allow normal aspiration or ejection. They are [b͡pʰ, d͡tʰ, d͡tsʰ, d͡tʃʰ, ɡ͡kʰ] and [d͡tsʼ, d͡tʃʼ] and a similar series of clicks.
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Voice (phonetics)
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Voice and tenseness
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There are languages with two sets of contrasting obstruents that are labelled /p t k f s x …/ vs. /b d ɡ v z ɣ …/ even though there is no involvement of voice (or voice onset time) in that contrast. That happens, for instance, in several Alemannic German dialects. Because voice is not involved, this is explained as a contrast in tenseness, called a fortis and lenis contrast.
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Voice (phonetics)
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Voice and tenseness
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There is a hypothesis that the contrast between fortis and lenis consonants is related to the contrast between voiceless and voiced consonants. That relation is based on sound perception as well as on sound production, where consonant voice, tenseness and length are only different manifestations of a common sound feature.
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Coining (metalworking)
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Coining (metalworking)
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Coining is a form of precision stamping in which a workpiece is subjected to a sufficiently high stress to induce plastic flow on the surface of the material. A beneficial feature is that in some metals, the plastic flow reduces surface grain size, and work hardens the surface, while the material deeper in the part retains its toughness and ductility. The term comes from the initial use of the process: manufacturing of coins.
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Coining (metalworking)
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Coining (metalworking)
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Coining is used to manufacture parts for all industries and is commonly used when high relief or very fine features are required. For example, it is used to produce coins, badges, buttons, precision-energy springs and precision parts with small or polished surface features.
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Coining (metalworking)
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Coining (metalworking)
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Coining is a cold working process similar in other respects to forging, which takes place at elevated temperature; it uses a great deal of force to elastically deform a workpiece, so that it conforms to a die. Coining can be done using a gear driven press, a mechanical press, or more commonly, a hydraulically actuated press. Coining typically requires higher tonnage presses than stamping, because the workpiece is elastically deformed and not actually cut, as in some other forms of stamping. The coining process is preferred when there is a high tonnage.
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Coining (metalworking)
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Coining in electronic industry
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In soldering of electronic components, bumps are formed on bonding pads to enhance adhesion, which are further flattened by the coining process. Unlike typical coining applications, in this case the goal of coining is to create a flat, rather than patterned, surface.
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Tetrahemihexahedron
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Tetrahemihexahedron
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In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).
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Tetrahemihexahedron
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Tetrahemihexahedron
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It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.) It is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.
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Tetrahemihexahedron
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Tetrahemihexahedron
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The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.
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Tetrahemihexahedron
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Related surfaces
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It is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a projective polyhedron, yielding a representation of the real projective plane very similar to the Roman surface.
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Tetrahemihexahedron
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Related polyhedra
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It has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.
The dual figure is the tetrahemihexacron.
It is 2-covered by the cuboctahedron, which accordingly has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the abstract polyhedron hemi-cuboctahedron.
It may also be constructed as a crossed triangular cuploid. All cuploids and their duals are topologically projective planes.
Tetrahemihexacron The tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra.
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Tetrahemihexahedron
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Related polyhedra
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Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
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Tetrahemihexahedron
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Related polyhedra
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Topologically it is considered to contain seven vertices. The three vertices considered at infinity (the real projective plane at infinity) correspond directionally to the three vertices of the hemi-octahedron, an abstract polyhedron. The other four vertices exist at alternate corners of a central cube (a demicube, in this case a tetrahedron).
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Freudenthal suspension theorem
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Freudenthal suspension theorem
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In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal.
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Freudenthal suspension theorem
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Freudenthal suspension theorem
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The theorem is a corollary of the homotopy excision theorem.
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Freudenthal suspension theorem
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Statement of the theorem
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Let X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map X→Ω(ΣX) induces a map πk(X)→πk(Ω(ΣX)) on homotopy groups, where Ω denotes the loop functor and Σ denotes the reduced suspension functor. The suspension theorem then states that the induced map on homotopy groups is an isomorphism if k ≤ 2n and an epimorphism if k = 2n + 1.
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Freudenthal suspension theorem
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Statement of the theorem
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A basic result on loop spaces gives the relation πk(Ω(ΣX))≅πk+1(ΣX) so the theorem could otherwise be stated in terms of the map πk(X)→πk+1(ΣX), with the small caveat that in this case one must be careful with the indexing.
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Freudenthal suspension theorem
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Statement of the theorem
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Proof As mentioned above, the Freudenthal suspension theorem follows quickly from homotopy excision; this proof is in terms of the natural map πk(X)→πk+1(ΣX) . If a space X is n -connected, then the pair of spaces (CX,X) is (n+1) -connected, where CX is the reduced cone over X ; this follows from the relative homotopy long exact sequence. We can decompose ΣX as two copies of CX , say (CX)+,(CX)− , whose intersection is X . Then, homotopy excision says the inclusion map: ((CX)+,X)⊂(ΣX,(CX)−) induces isomorphisms on πi,i<2n+2 and a surjection on π2n+2 . From the same relative long exact sequence, πi(X)=πi+1(CX,X), and since in addition cones are contractible, πi(ΣX,(CX)−)=πi(ΣX).
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Freudenthal suspension theorem
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Statement of the theorem
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Putting this all together, we get πi(X)=πi+1((CX)+,X)=πi+1((ΣX,(CX)−)=πi+1(ΣX) for i+1<2n+2 , i.e. i⩽2n , as claimed above; for i=2n+1 the left and right maps are isomorphisms, regardless of how connected X is, and the middle one is a surjection by excision, so the composition is a surjection as claimed.
Corollary 1 Let Sn denote the n-sphere and note that it is (n − 1)-connected so that the groups πn+k(Sn) stabilize for n⩾k+2 by the Freudenthal theorem. These groups represent the kth stable homotopy group of spheres.
Corollary 2 More generally, for fixed k ≥ 1, k ≤ 2n for sufficiently large n, so that any n-connected space X will have corresponding stabilized homotopy groups. These groups are actually the homotopy groups of an object corresponding to X in the stable homotopy category.
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Cichoń's diagram
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Cichoń's diagram
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In set theory, Cichoń's diagram or Cichon's diagram is a table of 10 infinite cardinal numbers related to the set theory of the reals displaying the provable relations between these cardinal characteristics of the continuum. All these cardinals are greater than or equal to ℵ1 , the smallest uncountable cardinal, and they are bounded above by 2ℵ0 , the cardinality of the continuum. Four cardinals describe properties of the ideal of sets of measure zero; four more describe the corresponding properties of the ideal of meager sets (first category sets).
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Cichoń's diagram
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Definitions
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Let I be an ideal of a fixed infinite set X, containing all finite subsets of X. We define the following "cardinal coefficients" of I: add min {|A|:A⊆I∧⋃A∉I}.
The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least ℵ0 ; if I is a σ-ideal, then add(I) ≥ ℵ1 cov min {|A|:A⊆I∧⋃A=X}.
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I) ≤ cov(I).
non min {|A|:A⊆X∧A∉I}, The "uniformity number" of I (sometimes also written unif (I) ) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).
cof min {|A|:A⊆I∧(∀B∈I)(∃A∈A)(B⊆A)}.
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Cichoń's diagram
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Definitions
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The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I) ≤ cof(I) and cov(I) ≤ cof(I).Furthermore, the "bounding number" or "unboundedness number" b and the "dominating number" d are defined as follows: min {|F|:F⊆NN∧(∀g∈NN)(∃f∈F)(∃∞n∈N)(g(n)<f(n))}, min {|F|:F⊆NN∧(∀g∈NN)(∃f∈F)(∀∞n∈N)(g(n)<f(n))}, where " ∃∞n∈N " means: "there are infinitely many natural numbers n such that …", and " ∀∞n∈N " means "for all except finitely many natural numbers n we have …".
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Cichoń's diagram
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Diagram
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Let B be the σ-ideal of those subsets of the real line that are meager (or "of the first category") in the euclidean topology, and let L be the σ-ideal of those subsets of the real line that are of Lebesgue measure zero. Then the following inequalities hold: Where an arrow from x to y is to mean that x≤y . In addition, the following relations hold: It turns out that the inequalities described by the diagram, together with the relations mentioned above, are all the relations between these cardinals that are provable in ZFC, in the following limited sense. Let A be any assignment of the cardinals ℵ1 and ℵ2 to the 10 cardinals in Cichoń's diagram. Then if A is consistent with the diagram's relations, and if A also satisfies the two additional relations, then A can be realized in some model of ZFC.
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Cichoń's diagram
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Diagram
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For larger continuum sizes, the situation is less clear. It is consistent with ZFC that all of the Cichoń's diagram cardinals are simultaneously different apart from add (B) and cof (B) (which are equal to other entries), but (as of 2019) it remains open whether all combinations of the cardinal orderings consistent with the diagram are consistent.
Some inequalities in the diagram (such as "add ≤ cov") follow immediately from the definitions. The inequalities cov non (L) and cov non (B) are classical theorems and follow from the fact that the real line can be partitioned into a meager set and a set of measure zero.
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Cichoń's diagram
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Remarks
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The British mathematician David Fremlin named the diagram after the Polish mathematician from Wrocław, Jacek Cichoń.The continuum hypothesis, of 2ℵ0 being equal to ℵ1 , would make all of these relations equalities.
Martin's axiom, a weakening of the continuum hypothesis, implies that all cardinals in the diagram (except perhaps ℵ1 ) are equal to 2ℵ0 Similar diagrams can be drawn for cardinal characteristics of higher cardinals κ for κ strongly inaccessible, which assort various cardinals between κ+ and 2κ
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Mixed receptive-expressive language disorder
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Mixed receptive-expressive language disorder
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Mixed receptive-expressive language disorder (DSM-IV 315.32) is a communication disorder in which both the receptive and expressive areas of communication may be affected in any degree, from mild to severe. Children with this disorder have difficulty understanding words and sentences. This impairment is classified by deficiencies in expressive and receptive language development that is not attributed to sensory deficits, nonverbal intellectual deficits, a neurological condition, environmental deprivation or psychiatric impairments. Research illustrates that 2% to 4% of five year olds have mixed receptive-expressive language disorder. This distinction is made when children have issues in expressive language skills, the production of language, and when children also have issues in receptive language skills, the understanding of language. Those with mixed receptive-language disorder have a normal left-right anatomical asymmetry of the planum temporale and parietale. This is attributed to a reduced left hemisphere functional specialization for language. Taken from a measure of cerebral blood flow (SPECT) in phonemic discrimination tasks, children with mixed receptive-expressive language disorder do not exhibit the expected predominant left hemisphere activation. Mixed receptive-expressive language disorder is also known as receptive-expressive language impairment (RELI) or receptive language disorder.
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Mixed receptive-expressive language disorder
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Classification
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If assessed on the Wechsler Adult Intelligence Scale, for instance, symptoms of mixed receptive-expressive language disorder may show as relatively low scores for Information, Vocabulary and Comprehension (perhaps below the 25th percentile). If a person has difficulty with specific types of concepts, for example spatial terms, such as 'over', 'under', 'here' and 'there', they may also have difficulties with arithmetic, understanding word problems and instructions, or difficulties using words at all.
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Mixed receptive-expressive language disorder
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Classification
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They may also have a more general problem with words or sentences, both comprehension and orally. Some children will have issues with pragmatics – the use of language in social contexts as well; and therefore, will have difficulty with inferring meaning. Furthermore, they have severe impairment of spontaneous language production and for this reason, they have difficulty in formulating questions. Generally, children will have trouble with morphosyntax, which is word inflections. These children have difficulty understanding and applying grammatical rules, such as endings that mark verb tenses (e.g. -ed), third-person singular verbs (e.g. I think, he thinks), plurals (e.g. -s), auxiliary verbs that denote tenses (e.g. was running, is running), and with determiners (the, a). Moreover, children with mixed receptive-expressive language disorders have deficits in completing two cognitive operations at the same time and learning new words or morphemes under time pressure or when processing demands are high. These children also have auditory processing deficits in which they process auditory information at a slower rate and as a result, require more time for processing.
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Mixed receptive-expressive language disorder
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Presentation
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Related disorders Studies show that low receptive and expressive language at young ages was correlated to increased autism symptom severity in children in their early school years. Below is a chart depicting language deficits of children on the autistic spectrum. This table indicates the lower levels of language processing, receptive/expressive disorders, which is more severe in children with autism. When autistic children speak, they are often difficult to understand, their language is sparse and dysfluent, they speak in single, uninflected words or short phrases, and their supply of words is severely depleted. This leads to limited vocabulary while also having deficits in verbal short-term memory.
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Mixed receptive-expressive language disorder
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Management
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Children who demonstrate deficiencies early in their speech and language development are at risk for continued speech and language issues throughout later childhood. Similarly, even if these speech and language problems have been resolved, children with early language delay are more at risk for difficulties in phonological awareness, reading, and writing throughout their lives. Children with mixed receptive-expressive language disorder are often likely to have long-term implications for language development, literacy, behavior, social development, and even mental health problems. If suspected of having a mixed receptive-expressive language disorder, treatment is available from a speech therapist or pathologist. Most treatments are short term, and rely upon accommodations made within the environment, in order to minimize interfering with work or school. Programs that involve intervention planning that link verbal short-term memory with visual/nonverbal information may be helpful for these children. In addition, approaches such as parent training for language stimulation and monitoring language through the "watch and see" method are recommended. The watch-and-see technique advises children with mixed receptive-expressive language disorder who come from stable, middle-class homes without any other behavioral, medical, or hearing problems should be vigilantly monitored rather than receive intervention. It is often the case that children do not meet the eligibility criteria established through a comprehensive oral language evaluation; and as a result, are not best suited for early intervention programs and require a different approach besides the "one size fits all" model.
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Film recorder
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Film recorder
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A film recorder is a graphical output device for transferring images to photographic film from a digital source. In a typical film recorder, an image is passed from a host computer to a mechanism to expose film through a variety of methods, historically by direct photography of a high-resolution cathode ray tube (CRT) display. The exposed film can then be developed using conventional developing techniques, and displayed with a slide or motion picture projector. The use of film recorders predates the current use of digital projectors, which eliminate the time and cost involved in the intermediate step of transferring computer images to film stock, instead directly displaying the image signal from a computer. Motion picture film scanners are the opposite of film recorders, copying content from film stock to a computer system. Film recorders can be thought of as modern versions of Kinescopes.
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Film recorder
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Design
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Operation All film recorders typically work in the same manner. The image is fed from a host computer as a raster stream over a digital interface. A film recorder exposes film through various mechanisms; flying spot (early recorders); photographing a high resolution video monitor; electron beam recorder (Sony HDVS); a CRT scanning dot (Celco); focused beam of light from a light valve technology (LVT) recorder; a scanning laser beam (Arrilaser); or recently, full-frame LCD array chips.
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Film recorder
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Design
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For color image recording on a CRT film recorder, the red, green, and blue channels are sequentially displayed on a single gray scale CRT, and exposed to the same piece of film as a multiple exposure through a filter of the appropriate color. This approach yields better resolution and color quality than possible with a tri-phosphor color CRT. The three filters are usually mounted on a motor-driven wheel. The filter wheel, as well as the camera's shutter, aperture, and film motion mechanism are usually controlled by the recorder's electronics and/or the driving software. CRT film recorders are further divided into analog and digital types. The analog film recorder uses the native video signal from the computer, while the digital type uses a separate display board in the computer to produce a digital signal for a display in the recorder. Digital CRT recorders provide a higher resolution at a higher cost compared to analog recorders due to the additional specialized hardware. Typical resolutions for digital recorders were quoted as 2K and 4K, referring to 2048×1366 and 4096×2732 pixels, respectively, while analog recorders provided a resolution of 640×428 pixels in comparison.Higher-quality LVT film recorders use a focused beam of light to write the image directly onto a film loaded spinning drum, one pixel at a time. In one example, the light valve was a liquid-crystal shutter, the light beam was steered with a lens, and text was printed using a pre-cut optical mask. The LVT will record pixel beyond grain. Some machines can burn 120-res or 120 lines per millimeter. The LVT is basically a reverse drum scanner. The exposed film is developed and printed by regular photographic chemical processing.
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Film recorder
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Design
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Formats Film recorders are available for a variety of film types and formats. The 35mm negative film and transparencies are popular because they can be processed by any photo shop. Single-image 4×5 film and 8×10 are often used for high-quality, large format printing.Some models have detachable film holders to handle multiple formats with the same camera or with Polaroid backs to provide on-site review of output before exposing film.
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Film recorder
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Uses
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Film recorders are used in digital printing to generate master negatives for offset and other bulk printing processes. For preview, archiving, and small-volume reproduction, film recorders have been rendered obsolete by modern printers that produce photographic-quality hardcopies directly on plain paper.
They are also used to produce the master copies of movies that use computer animation or other special effects based on digital image processing. However, most cinemas nowadays use Digital Cinema Packages on hard drives instead of film stock.
Computer graphics Film recorders were among the earliest computer graphics output devices; for example, the IBM 740 CRT Recorder was announced in 1954.
Film recorders were also commonly used to produce slides for slide projectors; but this need is now largely met by video projectors that project images directly from a computer to a screen. The terms "slide" and "slide deck" are still commonly used in presentation programs.
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Film recorder
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Uses
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Current uses Currently, film recorders are primarily used in the motion picture film-out process for the ever increasing amount of digital intermediate work being done. Although significant advances in large venue video projection alleviates the need to output to film, there remains a deadlock between the motion picture studios and theater owners over who should pay for the cost of these very costly projection systems. This, combined with the increase in international and independent film production, will keep the demand for film recording steady for at least a decade.
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Film recorder
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Key manufacturers
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Traditional film recorder manufacturers have all but vanished from the scene or have evolved their product lines to cater to the motion picture industry. Dicomed was one such early provider of digital color film recorders. Polaroid, Management Graphics, Inc, MacDonald-Detwiler, Information International, Inc., and Agfa were other producers of film recorders. Arri is the only current major manufacturer of film recorders.
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Film recorder
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Key manufacturers
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Kodak Lightning I film recorder. One of the first laser recorders. Needed an engineering staff to set up.
Kodak Lightning II film recorder used both gas and diode laser to record on to film.
The last LVT machines produced by Kodak / Durst-Dice stopped production in 2002. There are no LVT film recorders currently being produced. LVT Saturn 1010 uses a LED exposure (RGB) to 8"x10" film at 1000-3000ppi.
LUX Laser Cinema Recorder from Autologic/Information International in Thousand Oaks, California. Sales end in March 2000. Used on the 1997 film “Titanic”.
Arri produces the Arrilaser line of laser-based motion picture film recorders.
MGI produced the Solitaire line of CRT-based motion picture film recorders.
Matrix, originally ImaPRO, a branch of Agfa Division, produced the QCR line of CRT-based motion picture film recorders.
CCG, formerly Agfa film recorders, has been a steady manufacturer of film recorders based in Germany.
In 2004 CCG introduced Definity, a motion picture film recorder utilizing LCD technology. In 2010 CCG introduced the first full LED LCD film recorder as a new step in film recording.
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Film recorder
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Key manufacturers
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Cinevator was made by Cinevation AS, in Drammen, Norway. The Cinevator was a real-time digital film recorder. It could record IN, IP and prints with and without sound Oxberry produced the Model 3100 film recorder camera system, with interchangeable pin-registered movements (shuttles) for 35mm (full frame/Silent, 1.33:1) and 16mm (regular 16, "2R"), and others have adapted the Oxberry movements for CinemaScope, 1.85:1, 1.75:1, 1.66:1, as well as Academy/Sound (1.37:1) in 35mm and Super-16 in 16mm ("1R"). For instance, the "Solitaire" and numerous others employed the Oxberry 3100 camera system.
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Film recorder
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History
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Before video tape recorders or VTRs were invented, TV shows were either broadcast live or recorded to film for later showing, using the Kinescope process. In 1967, CBS Laboratories introduced the Electronic Video Recording format, which used video and telecined-to-video film sources, which were then recorded with an electron-beam recorder at CBS' EVR mastering plant at the time to 35mm film stock in a rank of 4 strips on the film, which was then slit down to 4 8.75 mm (0.344 in) film copies, for playback in an EVR player.
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Film recorder
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History
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All types of CRT recorders were (and still are) used for film recording. Some early examples used for computer-output recording were the 1954 IBM 740 CRT Recorder, and the 1962 Stromberg-Carlson SC-4020, the latter using a Charactron CRT for text and vector graphic output to either 16mm motion picture film, 16mm microfilm, or hard-copy paper output. Later 1970 and 80s-era recording to B&W (and color, with 3 separate exposures for red, green, and blue)) 16mm film was done with an EBR (Electron Beam Recorder), the most prominent examples made by 3M), for both video and COM (Computer Output Microfilm) applications. Image Transform in Universal City, California used specially modified 3M EBR film recorders that could perform color film-out recording on 16mm by exposing three 16mm frames in a row (one red, one green and one blue). The film was then printed to color 16mm or 35mm film. The video fed to the recorder could either be NTSC, PAL or SECAM. Later, Image Transform used specially modified VTRs to record 24 frame for their "Image Vision" system. The modified 1 inch type B videotape VTRs would record and play back 24frame video at 10 MHz bandwidth, at about twice the normal NTSC resolution. Modified 24fps 10 MHz Bosch Fernseh KCK-40 cameras were used on the set. This was a custom pre-HDTV video system. Image Transform had modified other gear for this process. At its peak, this system was used in the production of the film "Monty Python Live at the Hollywood Bowl" in 1982. This was the first major pre-digital intermediate post production using a film recorder for film-out production.
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Film recorder
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History
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In 1988, companies in the United States collectively produced 715 million slides at a cost of $8.3 billion.
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Film recorder
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History
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Awards The Academy of Motion Picture Arts and Sciences awarded an Oscar to the makers of the Arrilaser film recorder. The Award of Merit Oscar from the Academy Scientific and Technical Award ceremony was given on 11 February 2012 to Franz Kraus, Johannes Steurer and Wolfgang Riedel. Steurer was awarded the Oskar Messter Memorial Medal two years later in 2014 for his role in the development of the Arrilaser.
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First principle
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First principle
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In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from first cause attitudes and taught by Aristotelians, and nuanced versions of first principles are referred to as postulates by Kantians.In mathematics and formal logic, first principles are referred to as axioms or postulates. In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting. "First principles thinking" consists of decomposing things down to the fundamental axioms in the given arena, before reasoning up by asking which ones are relevant to the question at hand, then cross referencing conclusions based on chosen axioms and making sure conclusions do not violate any fundamental laws. Physicists include counterintuitive concepts with reiteration.
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First principle
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In formal logic
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In a formal logical system, that is, a set of propositions that are consistent with one another, it is possible that some of the statements can be deduced from other statements. For example, in the syllogism, "All men are mortal; Socrates is a man; Socrates is mortal" the last claim can be deduced from the first two.
A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of geometric propositions can be deduced from a set of definitions, postulates, and common notions: all three types constitute first principles.
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First principle
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Philosophy
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In philosophy "first principles" are from first cause attitudes commonly referred to as a priori terms and arguments, which are contrasted to a posteriori terms, reasoning or arguments, in that the former is simply assumed and exist prior to the reasoning process and the latter are deduced or inferred after the initial reasoning process. First principles are generally treated in the realm of philosophy known as epistemology, but are an important factor in any metaphysical speculation.
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First principle
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Philosophy
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In philosophy "first principles" are often somewhat synonymous with a priori, datum and axiomatic reasoning.
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First principle
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Philosophy
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Ancient Greek philosophy In Ancient Greek philosophy, a first principle from which other principles are derived is called an arche and later "first principle" or "element". By extension, it may mean "first place", "method of government", "empire, realm", "authorities" The concept of an arche was adapted from the earliest cosmogonies of Hesiod and Orphism, through the physical theories of Pre-Socratic philosophy and Plato before being formalized as a part of metaphysics by Aristotle. Arche sometimes also transcribed as arkhé) is an Ancient Greek word with primary senses "beginning", "origin" or "source of action": from the beginning, οr the original argument,"command". The first principle or element corresponds to the "ultimate underlying substance" and "ultimate indemonstrable principle".
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First principle
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Philosophy
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Mythical cosmogonies The heritage of Greek mythology already embodied the desire to articulate reality as a whole and this universalizing impulse was fundamental for the first projects of speculative theorizing. It appears that the order of "being" was first imaginatively visualized before it was abstractly thought.In the mythological cosmogonies of the Near East, the universe is formless and empty and the only existing thing prior to creation was the water abyss. In the Babylonian creation story, Enuma Elish, the primordial world is described as a "watery chaos" from which everything else appeared. This watery chaos has similarities in the cosmogony of the Greek mythographer Pherecydes of Syros. In the mythical Greek cosmogony of Hesiod (8th to 7th century BC), the origin of the world is Chaos, considered as a divine primordial condition, from which everything else appeared. In the creation "chaos" is a gaping-void, but later the word is used to describe the space between the earth and the sky, after their separation. "Chaos" may mean infinite space, or a formless matter which can be differentiated. The notion of temporal infinity was familiar to the Greek mind from remote antiquity in the religious conception of immortality. The conception of the "divine" as an origin influenced the first Greek philosophers. In the Orphic cosmogony, the unaging Chronos produced Aether and Chaos and made in divine Aether a silvery egg, from which everything else appeared.
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First principle
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Philosophy
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Ionian school The earliest Pre-Socratic philosophers, the Ionian material monists, sought to explain all of nature (physis) in terms of one unifying arche. Among the material monists were the three Milesian philosophers: Thales, who believed that everything was composed of water; Anaximander, who believed it was apeiron; and Anaximenes, who believed it was air. This is considered as a permanent substance or either one or more which is conserved in the generation of rest of it. From this all things first come to be and into this they are resolved in a final state. This source of entity is always preserved. Although their theories were primitive, these philosophers were the first to give an explanation of the physical world without referencing the supernatural; this opened the way for much of modern science (and philosophy), which has the same goal of explaining the world without dependence on the supernatural.Thales of Miletus (7th to 6th century BC), the father of philosophy, claimed that the first principle of all things is water, and considered it as a substance that contains in it motion and change. His theory was supported by the observation of moisture throughout the world and coincided with his theory that the earth floated on water. His ideas were influenced by the Near-Eastern mythological cosmogony and probably by the Homeric statement that the surrounding Oceanus (ocean) is the source of all springs and rivers.Anaximander argued that water could not be the arche, because it could not give rise to its opposite, fire. Anaximander claimed that none of the elements (earth, fire, air, water) could be arche for the same reason. Instead, he proposed the existence of the apeiron, an indefinite substance from which all things are born and to which all things will return. Apeiron (endless or boundless) is something completely indefinite; and, Anaximander was probably influenced by the original chaos of Hesiod (yawning abyss). Anaximander was the first philosopher that used arche for that which writers from Aristotle onwards called "the substratum" (Simplicius Phys. 150, 22). He probably intended it to mean primarily "indefinite in kind" but assumed it also to be "of unlimited extent and duration". The notion of temporal infinity was familiar to the Greek mind from remote antiquity in the religious conception of immortality and Anaximander's description was in terms appropriate to this conception. This arche is called "eternal and ageless". (Hippolitus I,6,I;DK B2)Anaximenes, Anaximander's pupil, advanced yet another theory. He returns to the elemental theory, but this time posits air, rather than water, as the arche and ascribes to it divine attributes. He was the first recorded philosopher who provided a theory of change and supported it with observation. Using two contrary processes of rarefaction and condensation (thinning or thickening), he explains how air is part of a series of changes. Rarefied air becomes fire, condensed it becomes first wind, then cloud, water, earth, and stone in order. The arche is technically what underlies all of reality/appearances.
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First principle
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Philosophy
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Aristotle Terence Irwin writes: When Aristotle explains in general terms what he tries to do in his philosophical works, he says he is looking for "first principles" (or "origins"; archai): In every systematic inquiry (methodos) where there are first principles, or causes, or elements, knowledge and science result from acquiring knowledge of these; for we think we know something just in case we acquire knowledge of the primary causes, the primary first principles, all the way to the elements. It is clear, then, that in the science of nature as elsewhere, we should try first to determine questions about the first principles. The naturally proper direction of our road is from things better known and clearer to us, to things that are clearer and better known by nature; for the things that are known to us are not the same as the things known unconditionally (haplôs). Hence it is necessary for us to progress, following this procedure, from the things that are less clear by nature, but clearer to us, towards things that are clearer and better known by nature. (Phys. 184a10–21) The connection between knowledge and first principles is not axiomatic as expressed in Aristotle's account of a first principle (in one sense) as "the first basis from which a thing is known" (Met. 1013a14–15). For Aristotle, the arche is the condition necessary for the existence of something, the basis for what he calls "first philosophy" or metaphysics. The search for first principles is not peculiar to philosophy; philosophy shares this aim with biological, meteorological, and historical inquiries, among others. But Aristotle's references to first principles in this opening passage of the Physics and at the start of other philosophical inquiries imply that it is a primary task of philosophy.
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First principle
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Philosophy
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Modern philosophy Descartes Profoundly influenced by Euclid, Descartes was a rationalist who invented the foundationalist system of philosophy. He used the method of doubt, now called Cartesian doubt, to systematically doubt everything he could possibly doubt until he was left with what he saw as purely indubitable truths. Using these self-evident propositions as his axioms, or foundations, he went on to deduce his entire body of knowledge from them. The foundations are also called a priori truths. His most famous proposition is "Je pense, donc je suis" (I think, therefore I am, or Cogito ergo sum), which he indicated in his Discourse on the Method was "the first principle of the philosophy of which I was in search." Descartes describes the concept of a first principle in the following excerpt from the preface to the Principles of Philosophy (1644): I should have desired, in the first place, to explain in it what philosophy is, by commencing with the most common matters, as, for example, that the word philosophy signifies the study of wisdom, and that by wisdom is to be understood not merely prudence in the management of affairs, but a perfect knowledge of all that man can know, as well for the conduct of his life as for the preservation of his health and the discovery of all the arts, and that knowledge to subserve these ends must necessarily be deduced from first causes; so that in order to study the acquisition of it (which is properly called [284] philosophizing), we must commence with the investigation of those first causes which are called Principles. Now, these principles must possess two conditions: in the first place, they must be so clear and evident that the human mind, when it attentively considers them, cannot doubt their truth; in the second place, the knowledge of other things must be so dependent on them as that though the principles themselves may indeed be known apart from what depends on them, the latter cannot nevertheless be known apart from the former. It will accordingly be necessary thereafter to endeavor so to deduce from those principles the knowledge of the things that depend on them, as that there may be nothing in the whole series of deductions which is not perfectly manifest.
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First principle
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In physics
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In physics, a calculation is said to be from first principles, or ab initio, if it starts directly at the level of established laws of physics and does not make assumptions such as empirical model and fitting parameters.
For example, calculation of electronic structure using Schrödinger's equation within a set of approximations that do not include fitting the model to experimental data is an ab initio approach.
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Barnes G-function
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Barnes G-function
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In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function.
Formally, the Barnes G-function is defined in the following Weierstrass product form: exp exp (z22k−z)} where γ is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π denotes multiplication (capital pi notation).
As an entire function, G is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.
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Barnes G-function
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Functional equation and integer arguments
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The Barnes G-function satisfies the functional equation G(z+1)=Γ(z)G(z) with normalisation G(1) = 1. Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function: Γ(z+1)=zΓ(z).
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Barnes G-function
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Functional equation and integer arguments
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The functional equation implies that G takes the following values at integer arguments: if if n=1,2,… (in particular, G(0)=0,G(1)=1 and thus G(n)=(Γ(n))n−1K(n) where Γ(x) denotes the gamma function and K denotes the K-function. The functional equation uniquely defines the G function if the convexity condition, log (G(x))≥0 is added. Additionally, the Barnes G function satisfies the duplication formula, 11 12 πx−12G(2x)
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Barnes G-function
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Characterisation
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Similar to the Bohr-Mollerup theorem for the gamma function, for a constant c>0 , we have for f(x)=cG(x) f(x+1)=Γ(x)f(x) and for x>0 f(x+n)∼Γ(x)nn(x2)f(n) as n→∞
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Barnes G-function
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Value at 1/2
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24 24 e18π−14A−32, where A is the Glaisher–Kinkelin constant.
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Barnes G-function
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Reflection formula 1.0
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The difference equation for the G-function, in conjunction with the functional equation for the gamma function, can be used to obtain the following reflection formula for the Barnes G-function (originally proved by Hermann Kinkelin): log log log cot πxdx.
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Barnes G-function
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Reflection formula 1.0
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The logtangent integral on the right-hand side can be evaluated in terms of the Clausen function (of order 2), as is shown below: log log sin Cl 2(2πz) The proof of this result hinges on the following evaluation of the cotangent integral: introducing the notation Lc (z) for the logcotangent integral, and using the fact that log sin cot πx , an integration by parts gives Lc cot log sin log sin log sin log sin log log sin log sin πx)dx.
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Barnes G-function
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Reflection formula 1.0
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Performing the integral substitution y=2πx⇒dx=dy/(2π) gives log sin log sin y2)dy.
The Clausen function – of second order – has the integral representation Cl log sin x2|dx.
However, within the interval 0<θ<2π , the absolute value sign within the integrand can be omitted, since within the range the 'half-sine' function in the integral is strictly positive, and strictly non-zero. Comparing this definition with the result above for the logtangent integral, the following relation clearly holds: Lc log sin Cl 2(2πz).
Thus, after a slight rearrangement of terms, the proof is complete: log log sin Cl 2(2πz).◻ Using the relation G(1+z)=Γ(z)G(z) and dividing the reflection formula by a factor of 2π gives the equivalent form: log log sin log Cl 2(2πz) Ref: see Adamchik below for an equivalent form of the reflection formula, but with a different proof.
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Barnes G-function
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Reflection formula 2.0
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Replacing z with (1/2) − z'' in the previous reflection formula gives, after some simplification, the equivalent formula shown below (involving Bernoulli polynomials): log log log log tan πxdx
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