Sines and Cosines of Angles in Arithmetic Progression

Knapp, Michael P.

doi:10.4169/002557009x478436

Cited by 37 publications

(19 citation statements)

References 2 publications

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“…Hence 2n+1 k=1 cos 2kπ 2n + 1 = 0, a fact that has been explained in this and other ways in this MAGAZINE [1,2]. Separating the last term gives 2n k=1 cos 2kπ 2n + 1 = −1.…”

mentioning

confidence: 79%

A Generalization of the Identity

Packard

Reitenbach

2012

Mathematics Magazine

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“…Hence 2n+1 k=1 cos 2kπ 2n + 1 = 0, a fact that has been explained in this and other ways in this MAGAZINE [1,2]. Separating the last term gives 2n k=1 cos 2kπ 2n + 1 = −1.…”

mentioning

confidence: 79%

A Generalization of the Identity

Packard

Reitenbach

2012

Mathematics Magazine

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“…From Figure 4 it is not totally clear whether the eigenvalue will remain away from zero as M increases. Indeed, for r = 0.2 roughly speaking we can see that: • for M ∈ [10,20] we have dϑM dM ≈ ϑ20−ϑ10 20−10 ≈ −10 −3 , • for M ∈ [50, 60] we have dϑM dM ≈ ϑ60−ϑ50 60−50 ≈ −6 · 10 −5 , • for M ∈ [110, 120] we have dϑM dM ≈ ϑ120−ϑ110 120−110 ≈ −1.5 · 10 −5 , from which we see that dϑM dM is increasing, but it also seems that it increases too slowly. In any case it is clear that, in the Neumann case, the eigenvalue ϑ M presents a remarkably different behaviour, for the locations D = mxe and D = uni.…”

Section: Comparison Between the Different Locationsmentioning

confidence: 94%

“…Let us fix M ≥ 1. The result follows from the fact that, for constants a ∈ R and b ∈ R, we have the identity whose proof can be found in[10]. Indeed, it is enough to observe that M k=1 cos(mc k ) =…”

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confidence: 88%

On the explicit feedback stabilization of one-dimensional linear nonautonomous parabolic equations via oblique projections

Rodrigues

Sturm

2018

IMA Journal of Mathematical Control and Information

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In recently proposed stabilisation techniques for parabolic equations, a crucial role is played by a suitable sequence of oblique projections in Hilbert spaces, onto the linear span of a suitable set of M actuators, and along the subspace orthogonal to the space spanned by "the" first M eigenfunctions of the Laplacian operator. This new approach uses an explicit feedback law, which is stabilising provided that the sequence of operator norms of such oblique projections remains bounded.The main result of the paper is the proof that, for suitable explicitly given sequences of sets of actuators, the operator norm of the corresponding oblique projections remains bounded.In the final part of the paper we provide numerical results, showing the performance of the explicit feedback control for both Dirichlet and Neumann homogeneous boundary conditions. 2010 Mathematics Subject Classification. 93D15,47A75,47N70.. supplemented with either homogeneous Dirichlet or Neumann boundary conditions, has been proposed. The reaction coefficient a = a(x, t) ∈ R is given and allowed to be time and space dependent. The scalar functions u 1 (t), u 2 (t), . . . , u M (t) are time-dependent controls at our disposal. The indicator functions 1 ωi = 1 ωi (x) associated with domains ω i ⊂ (0, L), are our actuators. Note that in this way, our control on the right hand side of (1.1), is finite dimensional, that is, for all given time t ≥ 0 our control is a linear combination of our (finite number of) actuators. Further, it is supported (localised) in the union M j=1 ω j . In [13, Sect. 3 and 5] it has been proven that for all positive λ > 0, the explicit feedback control M j=1 u j (t)1 ωj defined by y → Ky := P E ⊥ M UM (−∆y + ay − λy) (1.2)

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“…The following lemmas are necessary for proving the main results. We will use the well-known trigonometric identities (see [6][7][8][9])…”

Section: Auxiliary Factsmentioning

confidence: 99%