On triple intersections of three families of unit circles

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

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Deep Generative Model and Reinforcement Learning Solutions to Traveling Salesman Problems with Unit Circles

10.2514/6.2022-2209 ◽

2022 ◽

Author(s):

Jane Shin ◽

Frank Kim ◽

Silvia Ferrari

Keyword(s):

Reinforcement Learning ◽

Generative Model ◽

Traveling Salesman ◽

Traveling Salesman Problems ◽

Unit Circles

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Covering Discs in Minkowski Planes

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-046-2 ◽

2009 ◽

Vol 52 (3) ◽

pp. 424-434 ◽

Cited By ~ 2

Author(s):

Horst Martini ◽

Margarita Spirova

Keyword(s):

Essential Role ◽

Covering Problem ◽

Minkowski Geometry ◽

Minkowski Planes ◽

Strictly Convex ◽

Unit Circles ◽

Circle Covering ◽

Monotonicity Lemma

AbstractWe investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by k unit circles. In particular, we study the cases k = 3, k = 4, and k = 7. For k = 3 and k = 4, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, d-segments, and the monotonicity lemma.

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Unit Circles and Inverse Trigonometric Functions

Mathematics Teacher ◽

10.5951/mathteacher.108.2.0114 ◽

2014 ◽

Vol 108 (2) ◽

pp. 114-119

Author(s):

Azael Barrera

Keyword(s):

Unit Circle ◽

Trigonometric Functions ◽

Unit Circles ◽

Inverse Trigonometric Functions

A method to determine all the inverse trigonometric functions directly from the unit circle.

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On the Intersection Points of Unit Circles

American Mathematical Monthly ◽

10.2307/2324249 ◽

1992 ◽

Vol 99 (8) ◽

pp. 779

Author(s):

Andras Bezdek

Keyword(s):

Intersection Points ◽

Unit Circles

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On the Intersection Points of Unit Circles

American Mathematical Monthly ◽

10.1080/00029890.1992.11995930 ◽

1992 ◽

Vol 99 (8) ◽

pp. 779-780

Author(s):

András Bezdek

Keyword(s):

Intersection Points ◽

Unit Circles

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Certain Invariant Subspaces of H2 and L2 on a Bidisc

Canadian Journal of Mathematics ◽

10.4153/cjm-1988-055-6 ◽

1988 ◽

Vol 40 (5) ◽

pp. 1272-1280 ◽

Cited By ~ 9

Author(s):

Takahiko Nakazi

Keyword(s):

Haar Measure ◽

Fourier Coefficients ◽

Closed Subspace ◽

Cartesian Product ◽

Invariant Subspaces ◽

Explicit Description ◽

Lebesgue Spaces ◽

Unit Circles ◽

Image Position

We let T2 be the torus that is the cartesian product of 2 unit circles in C. The usual Lebesgue spaces, with respect to the Haar measure m of T2, are denoted by Lp = Lp(T2), and Hp = Hp(T2) is the space of all f in LP whose Fourier coefficientsare 0 as soon as at least one component of (j, ℓ) is negative.A closed subspace M of L2 is said to be invariant ifWhenever this is the case, it follows that fM ⊂ M for every f in H∞. One can ask for a classification or an explicit description (in some sense) of all invariant subspaces of L2, but this seems out of reach.

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