Rational Values of Trigonometric Functions

Olmsted, John M. H.

doi:10.2307/2304540

Cited by 22 publications

(19 citation statements)

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“…It is also a well-known fact that the only rational values of cos(x), where x is a rational multiple of π, are 0,±1/2,±1 (see [14,Theorem 6.16] or [20]). This fact implies that b ∈ {−3, −2, 0, 1}.…”

Section: The Non-elliptic Curves Casementioning

confidence: 99%

Rational periodic sequences for the Lyness recurrence

Gasull

Mañosa²,

Xarles

2012

Discrete &Amp; Continuous Dynamical Systems - A

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Section: The Non-elliptic Curves Casementioning

confidence: 99%

Rational periodic sequences for the Lyness recurrence

Gasull

Mañosa²,

Xarles

2012

Discrete &Amp; Continuous Dynamical Systems - A

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“…By doing a random walk, we are able to make the gate Λ(e iθ ), see for instance [8]. We conclude by using a statement of [20] which we recall below. Olmsted's theorem is also proven independently by Jack S. Calcut in a more recent American Mathematical Monthly [10] using Gaussian integers.…”

Section: Final Ancilla a Fmentioning

confidence: 98%

Topological quantum computation within the anyonic system the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4

Levaillant¹

2016

Quantum Inf Process

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The third and last chapter deals with realizing all the 2-qubit permutation gates, some of which are entangling gates (like the CNOT gate). The chapter also relies on other published protocols which are not part of this paper. Chapter 1From ancilla to quantum gate Note This chapter grew out of many inspiring discussions with Michael Freedman and is the first stone in a series of two chapters leading to the production of irrational phase gates in the Kauffman-Jones version of SU (2) Chern-Simons theory at level 4. AbstractWe provide a way to turn an ancilla into a gate for specific ancillas in the Kauffman-Jones version of SU (2) Chern-Simons theory at level 4. We deal with both the qubit 1221 and the qutrit 2222. Together with ancilla preparations, our protocols are later used to make irrational phase gates, leading to universal single qubit and qutrit gates. MOTIVATION OF THE WORK5 interferometry, see [4], [5], [6]. For basic facts about recoupling theory, we refer the reader to [13], except the theory which we use is unitary. In particular we deal with unitary theta symbols and unitary 6j-symbols (see [21] and Appendix of [14]). The value of the Kauffman constant is, using the same notations as in [13],The main four moves which we use throughout the paper are summarized below.• The "F -move"The brackets are called unitary 6j-symbols.• The "R-move"• The "theta move" NoteThe author thanks Michael Freedman for providing these thrilling problems, for guiding her with endless generosity in time and encouragements and for communicating his own excitement during the pursuit of this research. AbstractIn Chapter 1, it is shown with respect to two different anyonic systems, the same as those considered in this paper, how to fuse an ancilla into the input in order to form a gate. In each case, the ancilla must satisfy to some adequate amplitude ratios properties. In the present chapter, we show how to prepare such ancillas, leading to infinite order phase gates. We obtain universal 1-qubit and 1-qutrit gates. Together with [16] and [17] in which are respectively shown how to make a 2-qubit and a 2-qutrit entangling gate, we get qubit and qutrit gate sets that are universal for quantum computation.

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“…A folklore result, see e.g. [27], states that sin(πq), where q ∈ Q, is a rational number if and only if sin(πq) ∈ −1, − 1 2 , 0, 1 2 , 1 . Since p is odd, this cannot occur in our context.…”

Section: Figure 1: Concentric Annuli With Infinite Area But Small Lenmentioning

confidence: 99%

Open Sets Avoiding Integral Distances

Kurz

Mishkin

2013

Discrete Comput Geom

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We study open point sets in Euclidean spaces R d without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how large such sets can be in d-dimensional volume. We determine the exact values for the maximum volumes of the sets in terms of the number of their connected components and dimension. Here techniques from diophantine approximation, algebra and the theory of convex bodies come into play. Our problem can be viewed as a counterpart to known problems on sets with pairwise rational or integral distances. This reveals interesting links between discrete geometry, topology, and measure theory.

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Rational Values of Trigonometric Functions

Cited by 22 publications

References 0 publications

Rational periodic sequences for the Lyness recurrence

Rational periodic sequences for the Lyness recurrence

Topological quantum computation within the anyonic system the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4

Open Sets Avoiding Integral Distances

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