The Muller F element (4.2 Mb, ~80 protein-coding genes) is an unusual autosome of Drosophila melanogaster; it is mostly heterochromatic with a low recombination rate. To investigate how these properties impact the evolution of repeats and genes, we manually improved the sequence and annotated the genes on the D. erecta, D. mojavensis, and D. grimshawi F elements and euchromatic domains from the Muller D element. We find that F elements have greater transposon density (25–50%) than euchromatic reference regions (3–11%). Among the F elements, D. grimshawi has the lowest transposon density (particularly DINE-1: 2% vs. 11–27%). F element genes have larger coding spans, more coding exons, larger introns, and lower codon bias. Comparison of the Effective Number of Codons with the Codon Adaptation Index shows that, in contrast to the other species, codon bias in D. grimshawi F element genes can be attributed primarily to selection instead of mutational biases, suggesting that density and types of transposons affect the degree of local heterochromatin formation. F element genes have lower estimated DNA melting temperatures than D element genes, potentially facilitating transcription through heterochromatin. Most F element genes (~90%) have remained on that element, but the F element has smaller syntenic blocks than genome averages (3.4–3.6 vs. 8.4–8.8 genes per block), indicating greater rates of inversion despite lower rates of recombination. Overall, the F element has maintained characteristics that are distinct from other autosomes in the Drosophila lineage, illuminating the constraints imposed by a heterochromatic milieu.

We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if H ⊂ M is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions ϕ j | H of an orthonormal basis {ϕ j } of ∆-eigenfunctions of (M, g) to H are quantum ergodic on H. The condition on H is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface. A key step in the proof is that matrix elements F ϕ j , ϕ j of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.

Abstract. On any compact Riemannian manifold (M, g) of dimension n, the L 2 -normalized eigenfunctions ϕ λ satisfy ||ϕ λ ||∞ ≤ Cλ n−1 2where −∆ϕ λ = λ 2 ϕ λ . The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S n . But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori R n /Γ. We say that S n , but not R n /Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M, g) with maximal eigenfunction growth. In an earlier work, two of us showed that such an (M, g) must have a point x where the set Lx of geodesic loops at x has positive measure in S * x M . We strengthen this result here by showing that such a manifold must have a point where the set Rx of recurrent directions for the geodesic flow through x satisfies |Rx| > 0. We also show that if there are no such points, L 2 -normalized quasimodes have sup-norms that are o(λ n−1)/2 ), and, in the other extreme, we show that if there is a point blow-down x at which the first return map for the flow is the identity, then there is a sequence of quasi-modes with L ∞ -norms that are Ω(λ (n−1)/2 ).

In the last 2 decades many authors have described the use of an isolated lateral retinacular release for the treatment of patellar instability. This review analyzes the published long-term results of this procedure for the treatment of patellar instability. The isolated use of a lateral retinacular release of the patella has not proven to be of long-term benefit for the treatment of patellar instability. It may be used as an adjunct procedure to a proximal or distal realignment of the extensor mechanism. Various pitfalls of a lateral release for patellar instability are discussed.

We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction ϕ λ of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λ) where −∆ϕ λ = λ 2 ϕ λ . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(λ). It follows that the number of nodal lines of ϕ λ (components of the nodal set) which touch the boundary is of order λ. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains.