Invisibility in billiards

Plakhov, Alexander; Roshchina, Vera

doi:10.1088/0951-7715/24/3/007

Cited by 29 publications

(35 citation statements)

References 22 publications

(33 reference statements)

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“…✷ Remark 5. 18 As it was shown in loc.cit., an integral plane E 2 ⊂ T x Λ may exist only for x from an algebraic subset in Λ. For example, in the case, when η ′ = 0, it exists only if t 1 (x) = t 3 (x).…”

Section: Proposition 513mentioning

confidence: 82%

On 4-Reflective Complex Analytic Planar Billiards

Glutsyuk

2016

J Geom Anal

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The famous conjecture of V.Ya.Ivrii [14] says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result [7] classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C 4 -smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise C 4 -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

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Section: Proposition 513mentioning

confidence: 82%

On 4-Reflective Complex Analytic Planar Billiards

Glutsyuk

2016

J Geom Anal

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“…Substitute this segment with a finite sequence of hollows (circular arcs outside the half-plane). Take a coordinate system xOy such that the hollows are described by formula (8) and the half-plane takes the form y ≥ − √ r 2 − ε 2 , and assume that the length of the segment is a multiple of 2ε. Take several sequences of horizontal vectors V j and sequences of disjoint intervals U j , and consider the (ε, r, α j , V j , U j )-systems (j = 1, .…”

Section: Constructing a System Of Multilevel Reflecting Setsmentioning

confidence: 99%

Plane sets invisible in finitely many directions

Plakhov

2018

Nonlinearity

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We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with π, for any ε > 0 there exists a set invisible in the chosen directions that contains the circle and is contained in its ε-neighborhood. This set is the disjoint union of infinitely many domains with piecewise smooth boundary. Mathematics subject classifications: 49Q10, 49K30

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“…Even though research on invisibility is flourishing, and impressive progress has been made in the design of metamaterials (artificial materials engineered to bend electromagnetic waves around a concealed object), until very recently invisibility in mirror optics was largely overlooked both in engineering and mathematics. We refer the reader to our paper [3] for a brief overview of recent developments in the field and historical background. The focus of this note is solely on the mathematical aspects of billiard invisibility.…”

Section: Introductionmentioning

confidence: 99%