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By
Tanny, Shira; Yakovenko, Sergei
1 Citations
We study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the “type of differential equation” introduced by Frobenius. The main source of difficulties is the fact that there is no natural group action generating this classification. We establish a number of results on higher order equations which are similar but not completely parallel to the known results on local (holomorphic and meromorphic) gauge equivalence of systems of first order equations.
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By
ParisRomaskevich, Olga
Consider a swiveling arm on an oriented complete riemannian surface composed of three geodesic intervals, attached one to another in a chain. Each interval of the arm rotates with constant angular velocity around its extremity contributing to a common motion of the arm. Does the extremity of such a chain have an asymptotic velocity? This question for the motion in the euclidian plane, formulated by J.L. Lagrange, was solved by P. Hartman, E. R. Van Kampen, A. Wintner. We generalize their result to motions on any complete orientable surface of nonzero (and even nonconstant) curvature. In particular, we give the answer to Lagrange’s question for the movement of a swiveling arm on the hyperbolic plane. The question we study here can be seen as a dream about celestial mechanics on any riemannian surface: how many turns around the Sun a satellite of a planet in the geliocentric epicycle model would make in 1 billion years?
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By
Ebin, David G.; Preston, Stephen C.
4 Citations
Given an odddimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the
$$L^2$$
inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain rightinvariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an “Euler–Arnold” equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a “quasiLipschitz” estimate on the stream function, which leads to a Beale–Kato–Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler–Arnold equations are the Camassa–Holm equation (when the manifold is onedimensional) and the quasigeostrophic equation in geophysics.
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By
Misiołek, Gerard
3 Citations
We prove that the weakRiemannian exponential map of the
$$L^2$$
metric on the group of volumepreserving diffeomorphisms of a compact twodimensional manifold is not injective in any neighbourhood of its conjugate vectors. This can be viewed as a hydrodynamical analogue of the classical result of Morse and Littauer.
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By
Mani, Nitya; RubinsteinSalzedo, Simon
We show that the ndivision points of all rational hypocycloids are constructible with an unmarked straightedge and compass for all integers n, given a predrawn hypocycloid. We also consider the question of constructibility of ndivision points of hypocycloids without a predrawn hypocycloid in the case of a tricuspoid, concluding that only the 1, 2, 3, and 6division points of a tricuspoid are constructible in this manner.
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By
Ozbagci, Burak; PopescuPampu, Patrick
We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai’s credo “the Murasugi sum is a natural geometric operation” holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory and contact topology.
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By
Can, Mahir Bilen ; Uğurlu, Özlem
1 Citations
This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric spaces of classical types. Here, we determine the generating series the numbers of Borel orbits in
$${\mathbf {SO}}_{2n+1}/{\mathbf {S(O}}_{2p}\times {\mathbf {O}}_{2q+1} \mathbf {)}$$
(type BI) and in
$${\mathbf {Sp}}_n/{\mathbf {Sp}}_p \times {\mathbf {Sp}}_q$$
(type CII). In addition, we explore relations to lattice path enumeration.
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By
Kiritchenko, Valentina
1 Citations
For classical groups $$SL_n(\mathbb {C})$$, $$SO_n(\mathbb {C})$$ and $$Sp_{2n}(\mathbb {C})$$, we define uniformly geometric valuations on the corresponding complete flag varieties. The valuation in every type comes from a natural coordinate system on the open Schubert cell, and is combinatorially related to the Gelfand–Zetlin pattern in the same type. In types A and C, we identify the corresponding Newton–Okounkov polytopes with the Feigin–Fourier–Littelmann–Vinberg polytopes. In types B and D, we compute lowdimensional examples and formulate open questions.
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By
Butler, Clark; Chmutov, Sergei
We establish a relation between the Bollobás–Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.
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