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By
Tanny, Shira; Yakovenko, Sergei
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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
Breiding, Paul; Kozhasov, Khazhgali; Lerario, Antonio
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We investigate some geometric properties of the real algebraic variety
$$\Delta $$
of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart–Young–Mirskytype theorem for the distance function from a generic matrix to points in
$$\Delta $$
. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of
$$\Delta $$
) and random matrix theory.
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By
Ebin, David G.; Preston, Stephen C.
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2 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
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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
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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
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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
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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
Butler, Clark; Chmutov, Sergei
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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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By
Katthän, Lukas
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1 Citations
Let S be a polynomial ring and let
$$I \subseteq S$$
be a monomial ideal. In this short note, we propose the conjecture that the Betti poset of I determines the Stanley projective dimension of S / I or I. Our main result is that this conjecture implies the Stanley conjecture for I, and it also implies that
$${{\mathrm{sdepth}}}S/I \ge {{\mathrm{depth}}}S/I  1.$$
Recently, Duval et al. (A nonpartitionable Cohen–Macaulay simplicial complex,
arXiv:1504.04279
, 2015) found a counterexample to the Stanley conjecture, and their counterexample satisfies
$${{\mathrm{sdepth}}}S/I = {{\mathrm{depth}}}S/I  1$$
. So if our conjecture is true, then the conclusion is best possible.
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