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A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases is proposed, based on the Rykov model for diatomic gases. We adopt two velocity distribution functions (VDFs) to describe the system state; inelastic collisions are the same as in the Rykov model, but elastic collisions are modelled by the Boltzmann collision operator (BCO) for monatomic gases, so that the overall kinetic model equation reduces to the Boltzmann equation for monatomic gases in the limit of no translational–rotational energy exchange. The free parameters in the model are determined by comparing the transport coefficients, obtained by a Chapman–Enskog expansion, to values from experiment and kinetic theory. The kinetic model equations are solved numerically using the fast spectral method for elastic collision operators and the discrete velocity method for inelastic ones. The numerical results for normal shock waves and planar Fourier/Couette flows are in good agreement with both conventional direct simulation Monte Carlo (DSMC) results and experimental data. Poiseuille and thermal creep flows of polyatomic gases between two parallel plates are also investigated. Finally, we find that the spectra of both spontaneous and coherent Rayleigh–Brillouin scattering (RBS) compare well with DSMC results, and the computational speed of our model is approximately 300 times faster. Compared to the Rykov model, our model greatly improves prediction accuracy, and reveals the significant influence of molecular models. For coherent RBS, we find that the Rykov model could overpredict the bulk viscosity by a factor of two.

A two-temperature CFD (computational fluid dynamics) solver is a prerequisite to any spacecraft re-entry numerical study that aims at producing results with a satisfactory level of accuracy within realistic timescales. In this respect, a new two-temperature CFD solver, hy2Foam, has been developed within the framework of the open-source CFD platform OpenFOAM for the prediction of hypersonic reacting flows. This solver makes the distinct juncture between the trans-rotational and multiple vibrational-electronic temperatures. hy2Foam has the capability to model vibrational-translational and vibrational-vibrational energy exchanges in an eleven-species air mixture. It makes use of either the Park TTv model or the coupled vibration-dissociation-vibration (CVDV) model to handle chemistry-vibration coupling and it can simulate flows with or without electronic energy. Verification of the code for various zero-dimensional adiabatic heat baths of progressive complexity has been carried out. hy2Foam has been shown to produce results in good agreement with those given by the CFD code LeMANS (The Michigan Aerothermodynamic Navier-Stokes solver) and previously published data. A comparison is also performed with the open-source DSMC (direct simulation Monte Carlo) code dsmcFoam. It has been demonstrated that the use of the CVDV model and rates derived from Quantum-Kinetic theory promote a satisfactory consistency between the CFD and DSMC chemistry modules.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\\pi :D \\to \\Gamma \\backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\\Gamma < G$ , an arithmetic subgroup. Let $S' \\subseteq S$ be a special subvariety of S realized as $\\pi (D')$ for $D' \\subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \\subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\\pi (gD')$ is persistently likely as g ranges through G with $\\pi (gD')$ a special subvariety of S, meaning that whenever $\\zeta :S_1 \\to S$ and $\\xi :S_1 \\to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\\zeta $ finite, $\\dim \\xi \\zeta ^{-1} X + \\dim \\xi \\zeta ^{-1} \\pi (gD') \\geq \\dim \\xi S_1$ . Then $X \\cap \\bigcup _{g \\in G, \\pi (g D') \\text { is special }} \\pi (g D')$ is dense in X for the Euclidean topology.

hy2Foam is a newly-coded open-source two-temperature computational fluid dynamics (CFD) solver that has previously been validated for zero-dimensional test cases. It aims at (1) giving open-source access to a state-of-the-art hypersonic CFD solver to students and researchers; and (2) providing a foundation for a future hybrid CFD-DSMC (direct simulation Monte Carlo) code within the OpenFOAM framework. This paper focuses on the multi-dimensional verification of hy2Foam and firstly describes the different models implemented. In conjunction with employing the coupled vibration-dissociation-vibration (CVDV) chemistry–vibration model, novel use is made of the quantum-kinetic (QK) rates in a CFD solver. hy2Foam has been shown to produce results in good agreement with previously published data for a Mach 11 nitrogen flow over a blunted cone and with the dsmcFoam code for a Mach 20 cylinder flow for a binary reacting mixture. This latter case scenario provides a useful basis for other codes to compare against.

: Three contest scenes in Homer reveal a thematic concern with class tension: the two contests with Epeius in Iliad 23, Odysseus's encounter with Euryalus in Odyssey 8, and Odysseus's boxing match with Iros in Odyssey 18. Epeius is a comic scapegoat who succeeds in challenging the elite Euryalus, boasts ineptly, and is later ridiculed. Odysseus in Odyssey 8 is also challenged by a (different) nobleman named Euryalus, whom Odysseus rebukes, saying that a man cannot be skilled in all things and that one ought not judge by appearances. The ‘skilled man’ phrase found both in the Epeius episode and in that with Odysseus (Il. 23.670–71; Od. 8. 59–60), highlights the intertextuality and focuses on the theme of merit over appearances. Finally the Iros–Odysseus boxing match parodies and parallels the above epic‐challenge scenes. Each episode fosters consideration of the essential ambiguity of class relations in the period of transition to the polis c. 700 bce.

These essays in political philosophy by T. M. Scanlon, written between 1969 and 1999, examine the standards by which social and political institutions should be justified and appraised. Scanlon explains how the powers of just institutions are limited by rights such as freedom of expression, and considers why these limits should be respected even when it seems that better results could be achieved by violating them. Other topics which are explored include voluntariness and consent, freedom of expression, tolerance, punishment, and human rights. The collection includes the classic essays 'Preference and Urgency', 'A Theory of Freedom of Expression', and 'Contractualism and Utilitarianism', as well as a number of other essays that have hitherto not been easily accessible. It will be essential reading for all those studying these topics from the perspective of political philosophy, politics, and law.

We study algebraic dynamical systems (and, more generally, σ-varieties) $\\Phi :\\mathbb{A}^n_{\\mathbb{C}}\\rightarrow \\mathbb{A}^n_{\\mathbb{C}}$ given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of \"clusters\" from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism σ : ℂ → ℂ those algebraic varieties $X\\subseteq \\mathbb{A}^n_\\mathbb{C}$ for which Π(X) ⊆ Xσ. As a special case, we show that if f(x) ∈ ℂ[x] is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X\\subseteq \\mathbb{A}^2_\\mathbb{C} is an irreducible curve that is invariant under the action of (x,y) ↦ (f(x), f(y)) and projects dominantly in both directions, then X must be the graph of a polynomial that commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA0, a disintegrated set defined by σ(x) = f(x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of f is defined over a fixed field of a power σ, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of f is defined over a fixed field of a power of σ.

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable: