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Phi (faɪ; uppercase Φ, lowercase φ or ϕ; ϕεῖ pheî pʰéî̯; Modern Greek: φι fi fi) is the twenty-first letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([pʰ]), which was the origin of its usual romanization as . During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]). The romanization of the Modern Greek phoneme is therefore usually . It may be that phi originated as the letter qoppa (Ϙ, ϙ), and initially represented the sound /kʷʰ/ before shifting to Classical Greek pʰ. In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi. As with other Gre

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How Lagrangian States Evolve Into Random Waves

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In this paper, we consider a compact connected manifold (X, g) of negative curvature, and a family of semi-classical Lagrangian states f(h)(x) = a(x)e(i phi(x)/h) on X. For a wide family of phases phi, we show that f(h), when evolved by the semi-classical Schrodinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry's random waves conjecture for Lagrangian states.

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Formulation reproducing the ignition delays simulated by a detailed mechanism: Application to n-heptane combustion

This article is part of the project to model the kinetics of high-temperature combustions, occurring behind shock waves and in detonation waves. The "conventional" semi-empirical correlations of ignition delays have been reformulated, by keeping the Arrhenius equation form. It is shown how it polynomial with 3(N) Coefficients (where N is an element of [1, 4] is the number of adjustable kinetic parameters, likely to be simultaneously chosen among the temperature T, the pressure P, the inert fraction X-Ar, and the equivalence ratio Phi) can reproduce the delays predicted by the Curran et al. [H.J. Curran, P. Gaffuri, W.J. Pitz. C.K. Westbrook, Combust. Flame 129 (2002) 253-280] detailed mechanism (565 species and 22538 reactions), over it wide range of conditions (comparable with the validity domain). The deviations between the simulated times and their fits (typically 1%) are definitely lower than the Uncertainties related to the mechanism (at least 25%). In addition. using, this new formalism to evaluate these durations is about 10(6) times faster than simulating them With SENKIN (CHEMKIN III package) and only 10 times slower than using the classical correlations. The adaptation of the traditional method for predicting delays is interesting, for modeling. because those performances are difficult to obtain simultaneously with Other reduction methods (either purely mathematical, chemical, or even mixed). After a physical and mathematical justification of the proposed formalism, some of its potentialities for n-heptane combustion are presented. In particular, the trends of simulated delays and activation energies are shown for T is an element of [1500 K, 1900 K], P is an element of [10 kPa, 1 MPa] X-Ar is an element of [0, 0, 7], and Phi is an element of [0.25, 4.0]. (C) 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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The group of unimodular automorphisms of a principal bundle and the Euler-Yang-Mills equations

Given a principal bundle G hooked right arrow P -> B (each being compact, connected and oriented) and a G-invariant metric h(P) on P which induces a volume form mu(P), we consider the group of all unimodular automorphisms SAut(P, mu(P)) := {phi is an element of Diff(P) vertical bar phi*mu(P) = mu(P) and phi is G-equivariant) of P, and determines its Euler equation a la Arnold. The resulting equations turn out to be (a particular case of) the Euler-Yang-Mills equations of an incompressible classical charged ideal fluid moving on B. It is also shown that the group SAut(P, mu(P)) is an extension of a certain volume preserving diffeomorphisms group of B by the gauge group Gau(P) of P. (C) 2010 Elsevier B.V. All rights reserved.

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