Real projective line

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not intersect but seem to intersect "at infinity". For solving this problem, points at infinity have been introduced, in such a way that in a real projective plane, two distinct projective lines meet in exactly one point. The set of these points at infinity, the "horizon" of the visual perspective in the plane, is a real projective line. It is the set of directions emanating from an observer situated at any point, with opposite directions identified. An example of a real projective line is the projectively extended real line, which is often called the projective line. Formally, a real projective line P(R) is defined as the set of all one-dimensional linear subspaces of a two-dimensional vector space over the reals. The automorphisms of a real pr

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Fourier interpolation on the real line

Maryna Viazovska

In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0,+/- 1,+/- 2,+/- 3The functions in the interpolating basis are constructed in a closed form as an integral transform of weakly holomorphic modular forms for the theta subgroup of the modular group.

2019

Obstacles For Splitting Multidimensional Necklaces

Michal Lason

The well-known "necklace splitting theorem" of Alon (1987) asserts that every k-colored necklace can be fairly split into q parts using at most t cuts, provided k(q - 1) t + 2 is a sufficient condition (while k > t is necessary). We generalize this result to Euclidean spaces of arbitrary dimension d, and to arbitrary number of parts q. We prove that if k(q - 1) > t + d + q - 1, then there is a measurable k-coloring of R-d such that no axis-aligned cube has a fair q-splitting using at most t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k(q - 1) > t implied by Alon's 1987 work. Moreover, for d = 1, q = 2 we get exactly the result of the 2009 work. Additionally, we prove that if a stronger inequality k(q - 1) > dt + d + q - 1 is satisfied, then there is a measurable k-coloring of R-d with no axis-aligned cube having a fair q-splitting using at most t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.

Amer Mathematical Soc2015

Homogeneous Sobolev Metric of Order One on Diffeomorphism Groups on Real Line

Martin Bauer, Martins Bruveris

In this article we study Sobolev metrics of order one on diffeomorphism groups on the real line. We prove that the space equipped with the homogeneous Sobolev metric of order one is a flat space in the sense of Riemannian geometry, as it is isometric to an open subset of a mapping space equipped with the flat -metric. Here denotes the extension of the group of all compactly supported, rapidly decreasing, or -diffeomorphisms, which allows for a shift toward infinity. Surprisingly, on the non-extended group the Levi-Civita connection does not exist. In particular, this result provides an analytic solution formula for the corresponding geodesic equation, the non-periodic Hunter-Saxton (HS) equation. In addition, we show that one can obtain a similar result for the two-component HS equation and discuss the case of the non-homogeneous Sobolev one metric, which is related to the Camassa-Holm equation.

Springer Verlag2014

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