We classify multiply transitive homogeneous real (2, 3, 5) distributions up to local diffeomorphism equivalence.Proposition 2.1 ([18, Section 76]). Let (M, D) be a real (complex) (2, 3, 5) distribution and fix a point u ∈ M . There is a neighborhood U ⊆ M of u, a diffeomorphism (biholomorphism) Ψ : U → V ⊂ J 2,0 (F, F 2 ), and a smooth (complex-analytic) Homogeneous distributions Infinitesimal symmetriesAn infinitesimal symmetry of a (2, 3, 5) distribution (M, D) is a vector field ξ ∈ Γ(D) whose (local) flow preserves D, or equivalently for which L ξ η ∈ Γ(D) for all η ∈ Γ(D). We denote the Lie algebra of infinitesimal symmetries, called the (infinitesimal) symmetry algebra, by aut(D), and we say that (M, D) is infinitesimally homogeneous if aut(D) acts infinitesimally transitively, that is, if {ξ u : ξ ∈ aut(D)} = T u M for all u ∈ M . This article concerns (infinitesimally) multiply transitive homogeneous distributions, that is, infinitesimally homogeneous distributions for which the isotropy subalgebra k u < aut(D) of infinitesimal symmetries vanishing at any u ∈ M is nontrivial, or equivalently, for which dim aut(D) ≥ 6. Algebraic models for homogeneous distributionsFix a homogeneous (2, 3, 5) (real or complex) distribution (M, D) with transitive symmetry algebra h := aut(D), fix a point u ∈ M , and denote by k < h the subalgebra of vector fields in h vanishing at u and by d ⊂ h the subspace d := {ξ ∈ h : ξ u ∈ D u }. Then, k ⊆ d, [k, d] ⊆ d (we call this property k-invariance), and dim(d/k) = 2. The fact that D is a (2, 3, 5) distribution implies the genericity condition d + [d, d] + [d, [d, d]] = h. We call the triple (h, k; d) a (real or complex) algebraic model (of (M, D)).Given an algebraic model, we can reconstruct D up to local equivalence: For any groups H, K with K < H and respectively realizing h, k (for the groups that occur in the classification, we can choose K to be a closed subgroup of H), invoke the canonical identification T id ·K (H/K) ∼ = h/k to take D ⊂ T (H/K) to be the distribution with fibers D h·K = T id ·K L h · (d/k), where L h : H/K → H/K is the map L h : g · K → (hg) · K; by k-invariance this definition is independent of the coset representative h, and by genericity D is an H-invariant (2, 3, 5) distribution. Via the above identification, [D, D] id ·K = (d + [d, d])/k. 1 We declare two algebraic models (h, k; d), (h , k ; d ) to be equivalent iff there is a Lie algebra isomorphism α : h → h satisfying α(k) = k and α(d) = d . Unwinding definitions shows that equivalent algebraic models determine locally equivalent distributions. Multiply transitive homogeneous complex distributionsCartan showed that for all (2, 3, 5) distributions D, dim aut(D) ≤ 14, and that equality holds iff it is locally equivalent to the so-called flat distribution ∆ [9]; his argument applies to both the real and complex settings. We call the corresponding (complex) algebraic model O (see Section 4.1). In this case, aut(D) is isomorphic to the simple complex Lie algebra of type G 2 -we denote it by g 2 (C) -a...

A new model for bubbly, cavitating flow is validated and used to study the shock-induced oscillations of bubble clouds arising in shockwave lithotripsy and other applications. Compared to previous models based on volume and phase averaging, the new model extends the range of void fractions that can be reliably simulated and, for appropriately low void fractions, reproduces the results of the polydisperse phase-averaged model with much smaller computational expense.

We prove that certain quiver varieties are irreducible and therefore are isomorphic to Hilbert schemes of points of the total spaces of the bundles O P 1 (−n) for n ≥ 1.

Let S be a Clifford module for the complexified Clifford algebra C (R n ), S its dual, ρ and ρ be the corresponding representations of the spin group Spin(n). The group G = Spin(1, n + 1) is a (twofold) covering of the conformal group of R n . For λ, µ ∈ C, let π ρ,λ (resp. π ρ ,µ ) be the spinorial representation of G realized on a (subspace of)) which intertwines the representations π ρ,λ ⊗π ρ ,µ and π τ * k ,λ+µ+2m , where τ * k is the representation of Spin(n) on the space Λ * k (R n ) ⊗ C of complex-valued alternating k-forms on R n .

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