text: (cid:112) ∈ (cid:107) − (cid:107) If θ Θ is such that θ θ > ε, then on the event A , we have 0 (cid:112) n(cid:16) (cid:17) (cid:107) (cid:107) ≥ (cid:107) (cid:107) (cid:98) D (θ) λ (θ) D (θ) n,k Ω(θ) (cid:112) 1 (cid:16) n,k (cid:17) ≥ (cid:107) − − − (cid:107) (cid:98) λ L(θ ) L(θ) L(θ ) L 1 0 0 n,k (cid:18) (cid:19) ≥ (cid:112) (cid:107) − (cid:107) − (cid:107) − (cid:107) λ L(θ ) L(θ) L(θ ) L 1 0 √ √0 n,k − δ √λ δ √λ 1 1 > λ δ = . 1 1 + λ 1 + λ 1 1 (cid:113) It follows that on A , (cid:98) n √ (cid:107) (cid:107) ≥ δ √λ (cid:107) − (cid:107) inf D (θ) 1 >(cid:98) λ (θ ) L(θ ) L (cid:98) (cid:107) − (cid:107) n,k Ω(θ) q 0 0 n,k θ: θ θ0 >ε 1 + λ1 ≥ (cid:107) − (cid:107) ≥ (cid:107) − (cid:107) L(θ ) L inf L(θ) L . (cid:98) 0 n,k Ω(θ0) θ:(cid:107)(cid:98)θ−θ0(cid:107)≤ε n,k Ω(θ) The inﬁmum on the right-hand side is actually a minimum since L is continuous and B (θ ) ⊂ ε 0 is compact. Hence on A the set Θ is non-empty and Θ(cid:98) B (θ ). To show (A.1), it P n → n,k→ ∞ n,k ε 0 remains to prove that [A ] 1 as n , but this follows from (2.10). n Next we will prove that, with probability tending to one, Θ has exactly one element, i.e., n,k ∈ the function f has a unique minimizer. To do so, we will show that there exists ε (0, ε ] n,k 1 0 such that, with probability tending to one, the Hessian of f is positive deﬁnite on B (θ ) n,k ∈ ε1 0 and thus f is strictly convex on B (θ ). In combination with (A.1) for ε (0, ε ], this will n,k ε1 0 1 yield the desired conclusion. ∈ (cid:18) (cid:19) (cid:18) × (cid:19) (cid:18)H (cid:19) (cid:0) (cid:1) For θ Θ, deﬁne the symmetric p p matrix (θ; θ ) by 0 H ∂(cid:18)L(θ) T(cid:19) ∂L(θ) − ∂2L(θ) (cid:18)T (cid:19) − ( (θ; θ )) := 2 Ω(θ) (cid:0) 2 (cid:1) Ω(θ) L(θ ) L(cid:0)(θ) (cid:1) 0 i,j ∂θ ∂θ ∂θ ∂θ 0 j i j i − ∂L(θ) T ∂Ω(θ) − − ∂L(θ) T ∂Ω(θ) − (cid:0) (cid:1) (cid:0) (cid:1) 2 L(θ ) L(θ) 2 L(θ ) L(θ) 0 0 ∂θ ∂θ ∂θ ∂θ i j j i − ∂2Ω(θ) − T + L(θ ) L(θ) L(θ ) L(θ) , 0 0 ∂θ ∂θ j i ∈ { } (cid:55)→ H for i, j 1, . . . , p . The map θ (θ; θ ) is continuous and 0 H (θ ; θ ) = 2 L˙ (θ )T Ω(θ ) L˙ (θ ), (A.2) 0 0 0 0 0 × × is a positive deﬁnite matrix. This p p matrix is non-singular, since the q q matrix × ≥ (cid:107) · (cid:107) Ω(θ ) is non-singular and the q p matrix L˙ (θ ) has rank p (recall q p). Let denote 0 0 the spectral norm of a matrix. From Weyl’s perturbation theorem (Jiang, 2010, page 145), ∈ R × (cid:107) − H (cid:107) ≤ there exists an η > 0 such that every symmetric matrix A p p with A (θ ; θ ) η ∈ 0 0 has positive eigenvalues and is therefore positive deﬁnite. Let ε (0, ε ] be suﬃciently small 1 0 such that the second-order partial derivatives of L and Ω are continuous on B (θ ) and such (cid:107)H − H (cid:107) ≤ ∈ ε1 0 that (θ; θ ) (θ ; θ ) η/2 for all θ B (θ ). H(cid:0) 0 ∈(cid:1)R0 ×0 (cid:2) ε1 0 (cid:3) Let (θ) p p denote the Hessian matrix of f . Its (i, j)-th element is n,k,Ω n,k H ∂2 (θ) = D (θ)T Ω(θ) D (θ) n,k,Ω ij ∂θ ∂θ n,k n,k j i 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: If

text: θ

text: Θ

text: is

text: such

text: that

text: θ

text: θ

text: ε,

text: then

text: on

text: the

text: event

text: A

text: we

text: have

text: D

text: (θ)

text: λ

text: (θ)

text: D

text: (θ)

text: n,k

text: Ω(θ)

text: n,k

text: λ

text: L(θ

text: L(θ)

text: L(θ

text: L

text: n,k

text: λ

text: L(θ

text: L(θ)

text: L(θ

text: L

text: √0

text: n,k

text: δ

text: √λ

text: δ

text: √λ

text: λ

text: δ

text: λ

text: λ

text: It

text: follows

text: that

text: on

text: A

text: n

text: δ

text: √λ

text: inf

text: D

text: (θ)

text: λ

text: (θ

text: L(θ

text: L

text: n,k

text: Ω(θ)

text: q

text: n,k

text: θ:

text: θ

text: θ0

text: >ε

text: λ1

text: L(θ

text: L

text: inf

text: L(θ)

text: L

text: n,k

text: n,k

text: Ω(θ)

text: The

text: inﬁmum

text: on

text: the

text: right-hand

text: side

text: is

text: actually

text: a

text: minimum

text: since

text: L

text: is

text: continuous

text: and

text: B

text: (θ

text: ε

text: is

text: compact.

text: Hence

text: on

text: A

text: the

text: set

text: Θ

text: is

text: non-empty

text: and

text: B

text: (θ

text: To

text: show

text: it

text: P

text: n

text: n,k→

text: n,k

text: ε

text: remains

text: to

text: prove

text: that

text: [A

text: as

text: n

text: but

text: this

text: follows

text: from

text: n

text: Next

text: we

text: will

text: prove

text: that,

text: with

text: probability

text: tending

text: to

text: one,

text: Θ

text: has

text: exactly

text: one

text: element,

text: i.e.,

text: n,k

text: the

text: function

text: f

text: has

text: a

text: unique

text: minimizer.

text: To

text: do

text: so,

text: we

text: will

text: show

text: that

text: there

text: exists

text: ε

text: ε

text: n,k

text: such

text: that,

text: with

text: probability

text: tending

text: to

text: one,

text: the

text: Hessian

text: of

text: f

text: is

text: positive

text: deﬁnite

text: on

text: B

text: (θ

text: n,k

text: ε1

text: and

text: thus

text: f

text: is

text: strictly

text: convex

text: on

text: B

text: (θ

text: In

text: combination

text: with

text: for

text: ε

text: ε

text: this

text: will

text: n,k

text: ε1

text: yield

text: the

text: desired

text: conclusion.

text: For

text: θ

text: Θ,

text: deﬁne

text: the

text: symmetric

text: p

text: p

text: matrix

text: (θ;

text: θ

text: by

text: H

text: ∂L(θ)

text: (θ;

text: θ

text: Ω(θ)

text: Ω(θ)

text: L(θ

text: i,j

text: ∂θ

text: ∂θ

text: ∂θ

text: ∂θ

text: j

text: i

text: j

text: i

text: ∂L(θ)

text: T

text: ∂Ω(θ)

text: ∂L(θ)

text: T

text: ∂Ω(θ)

text: L(θ

text: L(θ)

text: L(θ

text: L(θ)

text: ∂θ

text: ∂θ

text: ∂θ

text: ∂θ

text: i

text: j

text: j

text: i

text: ∂2Ω(θ)

text: T

text: L(θ

text: L(θ)

text: L(θ

text: L(θ)

text: ∂θ

text: ∂θ

text: j

text: i

text: H

text: for

text: i,

text: j

text: p

text: The

text: map

text: θ

text: (θ;

text: θ

text: is

text: continuous

text: and

text: H

text: (θ

text: θ

text: L˙

text: (θ

text: )T

text: Ω(θ

text: L˙

text: (θ

text: is

text: a

text: positive

text: deﬁnite

text: matrix.

text: This

text: p

text: p

text: matrix

text: is

text: non-singular,

text: since

text: the

text: q

text: q

text: matrix

text: Ω(θ

text: is

text: non-singular

text: and

text: the

text: q

text: p

text: matrix

text: L˙

text: (θ

text: has

text: rank

text: p

text: (recall

text: q

text: p).

text: Let

text: denote

text: the

text: spectral

text: norm

text: of

text: a

text: matrix.

text: From

text: Weyl’s

text: perturbation

text: theorem

text: (Jiang,

text: page

text: R

text: H

text: there

text: exists

text: an

text: η

text: such

text: that

text: every

text: symmetric

text: matrix

text: A

text: p

text: p

text: with

text: A

text: (θ

text: θ

text: η

text: has

text: positive

text: eigenvalues

text: and

text: is

text: therefore

text: positive

text: deﬁnite.

text: Let

text: ε

text: ε

text: be

text: suﬃciently

text: small

text: such

text: that

text: the

text: second-order

text: partial

text: derivatives

text: of

text: L

text: and

text: Ω

text: are

text: continuous

text: on

text: B

text: (θ

text: and

text: such

text: H

text: ε1

text: that

text: (θ;

text: θ

text: (θ

text: θ

text: for

text: all

text: θ

text: B

text: (θ

text: ×0

text: ε1

text: Let

text: (θ)

text: p

text: p

text: denote

text: the

text: Hessian

text: matrix

text: of

text: f

text: Its

text: (i,

text: j)-th

text: element

text: is

text: n,k,Ω

text: n,k

text: H

text: ∂2

text: (θ)

text: D

text: (θ)T

text: Ω(θ)

text: D

text: (θ)

text: n,k,Ω

text: ij

text: ∂θ

text: ∂θ

text: n,k

text: n,k

text: j

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Convergence in distribution to the stated normal distribution follows from (2.11) and Slutsky’s (cid:98) lemma. − (cid:0) (cid:1) (cid:98) (cid:98) Proof of Corollary 2.4. Since D (θ) = L L(θ), we have n,k n,k √ √ √ − − k D (θ ) = k D (θ ) k L(θ ) L(θ ) . n,k n,k n,k 0 n,k 0 (cid:0) (cid:1) (cid:98) (cid:98) By (2.12) and the delta method, we have √ √ − − k L(θ ) L(θ ) = L˙ k(θ θ ) +√o (1) n,k 0 n,k 0 p − = L˙ (L˙ T Ω√L˙ ) 1L˙ T Ω k D (θ ) + o (1) n,k 0 p → ∞ = P (θ ) k D (θ ) + o (1), as n , 0 n,k 0 p where L˙ and Ω are eva(cid:98)luated at θ . Combination of the two previous displays yields √ 0 √ − → ∞ k D (θ ) = (I P (θ )) k D (θ ) + o (1), as n . n,k n,k q 0 n,k 0 p By (2.11) and Slutsky’s lemma, we arrive at (2.15), as required. × × × The q q matrix P has rank p since the q p matrix L˙ has rank p and the q q matrix Ω − − − is non-singular. Since P 2 = P , it follows that rank(I P ) = rank(I ) rank(P ) = q p. q q (cid:1) Proof of Corollary 2.5. Equation (2.11) can be written as √ −→ ∼ N → ∞ d Z := k D (θ ) Z (0, Σ(θ ) , as n . n,k n,k 0 q 0 − In view of (2.15) and Ω(θ) = Σ(θ) 1, we ﬁnd, by Slutsky’s lemma and the continuous mapping (cid:98) (cid:98) (cid:98) (cid:98) theorem, −(cid:98) k f (θ ) = k D (θ )T Σ(θ ) 1 D (θ ) n,k n,k n,k n,k n,k n,k n,k − − − = ZT (I P (θ ))T Σ(θ ) 1 (I P (θ )) Z + o (1) n,k q 0 n,k q 0 n,k p −→ − − − → ∞ d ZT (I P (θ ))T Σ(θ ) 1 (I P (θ )) Z, as n ; q 0 0 q 0 − − − here P = L˙ (L˙ T Σ 1L˙ ) 1 L˙ T Σ 1, with L˙ and Σ evaluated at θ . 0 It remains to identify the distribution of the limit random variable. The random vector Z ∼ N is equal in distribution to Σ1/2Y , where Y (0, I ) and where Σ1/2 is a symmetric square q q root of Σ. Straightforward calculation yields − − − − ZT (I P )T Σ 1 (I P ) Z =d Y T (I B)Y q q q − − − − where B = Σ 1/2L˙ (L˙ T Σ 1L˙ ) 1 L˙ T Σ 1/2. It is easily checked that B is a projection matrix − (B = BT = B2). Moreover, B has rank p. It follows that I B is a projection matrix too − q and that it has rank q p. The distribution of the limit random variable now follows by standard properties of quadratic forms of normal random vectors. ∼ N √ Proof of Corollary 2.6. Let Z (0, Σ(θ )), which by (2.11) is the limit in distribution of q 0 → ∞ k D (θ ). By (2.15) and the continuous mapping theorem, we have, as n , n,k 0 (cid:98) (cid:98) (cid:98) −→ − − k D (θ )T A(θ ) D (θ ) d ZT (I P (θ ))T A(θ ) (I P (θ )) Z. (A.8) n,k n,k n,k n,k n,k q 0 0 q 0 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Convergence

label: paragraph

text: in

text: distribution

text: to

text: the

text: stated

text: normal

text: distribution

text: follows

text: from

text: and

text: Slutsky’s

text: lemma.

text: Proof

text: of

text: Corollary

text: Since

text: D

text: (θ)

text: L

text: L(θ),

text: we

text: have

text: n,k

text: n,k

text: k

text: D

text: (θ

text: k

text: D

text: (θ

text: k

text: L(θ

text: L(θ

text: n,k

text: n,k

text: n,k

text: n,k

text: By

text: and

text: the

text: delta

text: method,

text: we

text: have

text: k

text: L(θ

text: L(θ

text: L˙

text: k(θ

text: θ

text: +√o

text: n,k

text: n,k

text: p

text: L˙

text: (L˙

text: T

text: Ω√L˙

text: T

text: Ω

text: k

text: D

text: (θ

text: o

text: n,k

text: p

text: P

text: (θ

text: k

text: D

text: (θ

text: o

text: as

text: n

text: n,k

text: p

text: where

text: L˙

text: and

text: Ω

text: are

text: at

text: θ

text: Combination

text: of

text: the

text: two

text: previous

text: displays

text: yields

text: k

text: D

text: (θ

text: (I

text: P

text: (θ

text: k

text: D

text: (θ

text: o

text: as

text: n

text: n,k

text: n,k

text: q

text: n,k

text: p

text: By

text: and

text: Slutsky’s

text: lemma,

text: we

text: arrive

text: at

text: as

text: required.

text: The

text: q

text: q

text: matrix

text: P

text: has

text: rank

text: p

text: since

text: the

text: q

text: p

text: matrix

text: L˙

text: has

text: rank

text: p

text: and

text: the

text: q

text: q

text: matrix

text: Ω

text: is

text: non-singular.

text: Since

text: P

text: P

text: it

text: follows

text: that

text: rank(I

text: P

text: rank(I

text: rank(P

text: q

text: p.

text: q

text: q

text: Proof

text: of

text: Corollary

text: Equation

text: can

text: be

text: written

text: as

text: N

text: d

text: Z

text: k

text: D

text: (θ

text: Z

text: Σ(θ

text: as

text: n

text: n,k

text: n,k

text: q

text: In

text: view

text: of

text: and

text: Ω(θ)

text: Σ(θ)

text: we

text: ﬁnd,

text: by

text: Slutsky’s

text: lemma

text: and

text: the

text: continuous

text: mapping

text: theorem,

text: k

text: f

text: (θ

text: k

text: D

text: (θ

text: )T

text: Σ(θ

text: D

text: (θ

text: n,k

text: n,k

text: n,k

text: n,k

text: n,k

text: n,k

text: n,k

text: ZT

text: (I

text: P

text: (θ

text: ))T

text: Σ(θ

text: (I

text: P

text: (θ

text: Z

text: o

text: n,k

text: q

text: n,k

text: q

text: n,k

text: p

text: d

text: ZT

text: (I

text: P

text: (θ

text: ))T

text: Σ(θ

text: (I

text: P

text: (θ

text: Z,

text: as

text: n

text: q

text: q

text: here

text: P

text: L˙

text: (L˙

text: T

text: Σ

text: L˙

text: T

text: Σ

text: with

text: L˙

text: and

text: Σ

text: evaluated

text: at

text: θ

text: It

text: remains

text: to

text: identify

text: the

text: distribution

text: of

text: the

text: limit

text: random

text: variable.

text: The

text: random

text: vector

text: Z

text: N

text: is

text: equal

text: in

text: distribution

text: to

text: where

text: Y

text: I

text: and

text: where

text: is

text: a

text: symmetric

text: square

text: q

text: q

text: root

text: of

text: Σ.

text: Straightforward

text: calculation

text: yields

text: ZT

text: (I

text: P

text: )T

text: Σ

text: (I

text: P

text: Z

text: =d

text: Y

text: T

text: (I

text: B)Y

text: q

text: q

text: q

text: where

text: B

text: Σ

text: (L˙

text: T

text: Σ

text: L˙

text: T

text: Σ

text: It

text: is

text: easily

text: checked

text: that

text: B

text: is

text: a

text: projection

text: matrix

text: (B

text: BT

text: Moreover,

text: B

text: has

text: rank

text: p.

text: It

text: follows

text: that

text: I

text: B

text: is

text: a

text: projection

text: matrix

text: too

text: q

text: and

text: that

text: it

text: has

text: rank

text: q

text: p.

text: The

text: distribution

text: of

text: the

text: limit

text: random

text: variable

text: now

text: follows

text: by

text: standard

text: properties

text: of

text: quadratic

text: forms

text: of

text: normal

text: random

text: vectors.

text: N

text: Proof

text: of

text: Corollary

text: Let

text: Z

text: Σ(θ

text: which

text: by

text: is

text: the

text: limit

text: in

text: distribution

text: of

text: q

text: k

text: D

text: (θ

text: By

text: and

text: the

text: continuous

text: mapping

text: theorem,

text: we

text: have,

text: as

text: n

text: n,k

text: k

text: D

text: (θ

text: )T

text: A(θ

text: D

text: (θ

text: d

text: ZT

text: (I

text: P

text: (θ

text: ))T

text: A(θ

text: (I

text: P

text: (θ

text: Z.

text: n,k

text: n,k

text: n,k

text: n,k

text: n,k

text: q

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: − ∼ N We can represent (I P )Z as V D1/2Y , with Y (0, I ). The limiting random variable q q q in (A.8) is then given by (cid:80) − Y T D1/2V T V D 1V T V D1/2Y. s s s Since V is an orthogonal matrix, this expression simpliﬁes to s Y 2, which has the stated j=1 j χ2 distribution. s Proof of Remark 2.1. Inspection of the proofs of Corollaries 2.5 and 2.6 shows that the dif- ference between the two test statistics converges in distribution to the random variable (cid:0) (cid:1) (cid:1) (cid:0) (cid:1) ZT R(θ ) Z, where Z is a certain q-variate normal random vector and where 0 − − − − R(θ ) = I P (θ ) T (Σ(θ ) 1 A(θ ) I P (θ ) . 0 q 0 0 0 q 0 The matrix R(θ ) can be shown to be equal to zero, proving the claim of the remark. To see 0 − why R(θ ) is zero, note ﬁrst that, suppressing θ and writing Q = I P , we have Q2 = Q 0 0 q and ΣQT = QΣ = QΣQT . Recall the eigenvalue equation QΣQT v = ν v for j = 1, . . . , q. ≤ ≥ j j j Note that ν > 0 if j s and ν = 0 if j s + 1. The eigenvalue equation implies that j ≤ j ≥ Qv = v for j s while QΣv = 0 for j s + 1. Since the vectors v , . . . , v are orthogonal, j j j 1 q we ﬁnd that the vectors v , . . . , v , Σv , . . . , Σv are linearly independent. It then suﬃces 1 ≤ s s+1 q ≥ to show that Rv = 0 for all j s and RΣv = 0 for all j s + 1. The ﬁrst property follows j − − j − ≤ from the fact that Σ 1v = ν 1QT v and Av = ν 1v for j s (use the eigenvalue equation j j j j j j ≥ again), while the second property follows from QΣv = 0 for j s + 1. j References Abadir, K. M. and J. R. Magnus (2005). Matrix Algebra, Volume 1. Cambridge University Press. Beirlant, J., M. Escobar-Bach, Y. Goegebeur, and A. Guillou (2015). Bias-corrected estimation of the stable tail dependence function. Available at https://hal. archives-ouvertes.fr/hal-01115538/. Beirlant, J., Y. Goegebeur, J. Segers, and J. Teugels (2004). Statistics of Extremes: Theory and Applications. Wiley. Bu¨cher, A. and J. Segers (2014). Extreme value copula estimation based on block maxima of a multivariate stationary time series. Extremes 17(3), 495–528. Can, S. U., J. H. J. Einmahl, E. V. Khmaladze, R. J. A. Laeven, et al. (2015). Asymptotically distribution-free goodness-of-ﬁt testing for tail copulas. The Annals of Statistics 43(2), 878–902. Coles, S. G. and J. A. Tawn (1991). Modelling extreme multivariate events. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 53(2), 377–392. Davison, A. C., S. A. Padoan, and M. Ribatet (2012). Statistical modeling of spatial extremes. Statistical Science 27(2), 161–186. 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: N

text: We

text: can

text: represent

text: (I

text: P

text: )Z

text: as

text: V

text: with

text: Y

text: I

text: The

text: limiting

text: random

text: variable

text: q

text: q

text: q

text: in

text: is

text: then

text: given

text: by

text: Y

text: T

text: T

text: V

text: D

text: T

text: V

text: s

text: s

text: s

text: Since

text: V

text: is

text: an

text: orthogonal

text: matrix,

text: this

text: expression

text: simpliﬁes

text: to

text: s

text: Y

text: which

text: has

text: the

text: stated

text: j

text: χ2

text: distribution.

text: s

text: Proof

text: of

text: Remark

text: Inspection

text: of

text: the

text: proofs

text: of

text: Corollaries

text: and

text: shows

text: that

text: the

text: dif-

text: ference

text: between

text: the

text: two

text: test

text: statistics

text: converges

text: in

text: distribution

text: to

text: the

text: random

text: variable

text: ZT

text: R(θ

text: Z,

text: where

text: Z

text: is

text: a

text: certain

text: q-variate

text: normal

text: random

text: vector

text: and

text: where

text: R(θ

text: I

text: P

text: (θ

text: T

text: (Σ(θ

text: A(θ

text: I

text: P

text: (θ

text: q

text: q

text: The

text: matrix

text: R(θ

text: can

text: be

text: shown

text: to

text: be

text: equal

text: to

text: zero,

text: proving

text: the

text: claim

text: of

text: the

text: remark.

text: To

text: see

text: why

text: R(θ

text: is

text: zero,

text: note

text: ﬁrst

text: that,

text: suppressing

text: θ

text: and

text: writing

text: Q

text: I

text: P

text: we

text: have

text: Q

text: q

text: and

text: ΣQT

text: QΣ

text: QΣQT

text: Recall

text: the

text: eigenvalue

text: equation

text: QΣQT

text: v

text: ν

text: v

text: for

text: j

text: q.

text: j

text: j

text: j

text: Note

text: that

text: ν

text: if

text: j

text: s

text: and

text: ν

text: if

text: j

text: s

text: The

text: eigenvalue

text: equation

text: implies

text: that

text: j

text: j

text: Qv

text: v

text: for

text: j

text: s

text: while

text: QΣv

text: for

text: j

text: s

text: Since

text: the

text: vectors

text: v

text: v

text: are

text: orthogonal,

text: j

text: j

text: j

text: q

text: we

text: ﬁnd

text: that

text: the

text: vectors

text: v

text: v

text: Σv

text: Σv

text: are

text: linearly

text: independent.

text: It

text: then

text: suﬃces

text: s

text: q

text: to

text: show

text: that

text: Rv

text: for

text: all

text: j

text: s

text: and

text: RΣv

text: for

text: all

text: j

text: s

text: The

text: ﬁrst

text: property

text: follows

text: j

text: j

text: from

text: the

text: fact

text: that

text: Σ

text: ν

text: v

text: and

text: Av

text: ν

text: for

text: j

text: s

text: (use

text: the

text: eigenvalue

text: equation

text: j

text: j

text: j

text: j

text: j

text: j

text: again),

text: while

text: the

text: second

text: property

text: follows

text: from

text: QΣv

text: for

text: j

text: s

text: j

text: References

text: Abadir,

text: K.

text: M.

text: and

text: J.

text: R.

text: Magnus

text: Matrix

text: Algebra,

text: Volume

text: Cambridge

text: University

text: Press.

text: Beirlant,

text: J.,

text: M.

text: Escobar-Bach,

text: Y.

text: Goegebeur,

text: and

text: A.

text: Guillou

text: Bias-corrected

text: estimation

text: of

text: the

text: stable

text: tail

text: dependence

text: function.

text: Available

text: at

text: Beirlant,

text: J.,

text: Y.

text: Goegebeur,

text: J.

text: Segers,

text: and

text: J.

text: Teugels

text: Statistics

text: of

text: Extremes:

text: Theory

text: and

text: Applications.

text: Wiley.

text: Bu¨cher,

text: A.

text: and

text: J.

text: Segers

text: Extreme

text: value

text: copula

text: estimation

text: based

text: on

text: block

text: maxima

text: of

text: a

text: multivariate

text: stationary

text: time

text: series.

text: Extremes

text: Can,

text: S.

text: U.,

text: J.

text: H.

text: J.

text: Einmahl,

text: E.

text: V.

text: Khmaladze,

text: R.

text: J.

text: A.

text: Laeven,

text: et

text: al.

text: Asymptotically

text: distribution-free

text: goodness-of-ﬁt

text: testing

text: for

text: tail

text: copulas.

text: The

text: Annals

text: of

text: Statistics

text: Coles,

text: S.

text: G.

text: and

text: J.

text: A.

text: Tawn

text: Modelling

text: extreme

text: multivariate

text: events.

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society:

text: Series

text: B

text: (Statistical

text: Methodology)

text: Davison,

text: A.

text: C.,

text: S.

text: A.

text: Padoan,

text: and

text: M.

text: Ribatet

text: Statistical

text: modeling

text: of

text: spatial

text: extremes.

text: Statistical

text: Science

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Newton Algorithm on Constraint Manifolds and the 5-electron Thomson problem Petre Birtea, Dan Coma˘nescu Department of Mathematics, West University of Timi¸soara Bd. V. Paˆrvan, No 4, 300223 Timi¸soara, Romaˆnia birtea@math.uvt.ro, comanescu@math.uvt.ro Abstract We give a description of numerical Newton algorithm on a constraint manifold using only the ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold. We apply the numerical Newton algorithm on a sphere in order to ﬁnd the critical conﬁgurations of the 5-electron Thomson problem. As a result, we ﬁnd a new critical conﬁguration of a regular pentagonal type. We also make an analytical study of the critical conﬁgurations found previously and determine their nature using Morse-Bott theory. Last section contains an analytical study of critical conﬁgurations for Riesz s-energy of 5-electron on a sphere and their bifurcation behavior is pointed out. MSC: 49M15, 53-XX Keywords: Newton algorithm, constraint manifold, Hessian operator, Morse theory, 5-electron Thomson problem, Riesz s-energy 1 Introduction Newton algorithm is a eﬃcient numerical iterative method that can be applied for ﬁnding the critical points of the cost function. This type of algorithm uses second-order information about the cost function, fact that guarantees super-linear convergence (in cases it converges). Initially it was constructed on vector spaces, but later was adapted to manifolds for prac(cid:101)tical reasons, see [1] for a careful and detailed discussion of the geometry involved in such a context. → R Let (S, g ) be a smooth Riemannian manifold and G : S be a smooth cost function. The iterative scheSme is given by, see [1]: (cid:101) (cid:101) x = R (v ), (1.1) (cid:101) n+1 xn xn (cid:101) → ∈ where R : TS S is a smooth retraction and the tangent vector v T S is the solution of the xn xn (contravariant) Newton equation (cid:101) (cid:101) (cid:101) · −∇ HG(x ) v = G(x ). (1.2) n xn gS(cid:101) n (cid:101) (cid:101) → In the above equation we have the Hessian operator HG : TS TS. The link between the Hessian (cid:16) (cid:17) × → R operator HG and the associated symmetr(cid:101)ic bilinear form H(cid:101)ess G :(cid:101)TS TS is given by: (cid:101) · · HG(x) v = g(cid:93) (x) Hess G(x) v , x x S 1

text: Newton

label: title

text: Algorithm

text: on

text: Constraint

text: Manifolds

text: and

text: the

text: Thomson

text: problem

text: Petre

label: author

text: Birtea,

text: Dan

text: Coma˘nescu

text: Department

text: of

text: Mathematics,

text: West

text: University

text: of

text: Timi¸soara

text: Bd.

text: V.

text: Paˆrvan,

text: No

text: Timi¸soara,

text: Romaˆnia

text: birtea@math.uvt.ro,

text: comanescu@math.uvt.ro

text: Abstract

text: We

label: abstract

text: give

text: a

text: description

text: of

text: numerical

text: Newton

text: algorithm

text: on

text: a

text: constraint

text: manifold

text: using

text: only

text: the

text: ambient

text: coordinates

text: (usually

text: Euclidean

text: coordinates)

text: and

text: the

text: geometry

text: of

text: the

text: constraint

text: manifold.

text: We

text: apply

text: the

text: numerical

text: Newton

text: algorithm

text: on

text: a

text: sphere

text: in

text: order

text: to

text: ﬁnd

text: the

text: critical

text: conﬁgurations

text: of

text: the

text: Thomson

text: problem.

text: As

text: a

text: result,

text: we

text: ﬁnd

text: a

text: new

text: critical

text: conﬁguration

text: of

text: a

text: regular

text: pentagonal

text: type.

text: We

text: also

text: make

text: an

text: analytical

text: study

text: of

text: the

text: critical

text: conﬁgurations

text: found

text: previously

text: and

text: determine

text: their

text: nature

text: using

text: Morse-Bott

text: theory.

text: Last

text: section

text: contains

text: an

text: analytical

text: study

text: of

text: critical

text: conﬁgurations

text: for

text: Riesz

text: s-energy

text: of

text: on

text: a

text: sphere

text: and

text: their

text: bifurcation

text: behavior

text: is

text: pointed

text: out.

text: MSC:

text: Keywords:

text: Newton

text: algorithm,

text: constraint

text: manifold,

text: Hessian

text: operator,

text: Morse

text: theory,

text: Thomson

text: problem,

text: Riesz

text: s-energy

text: Introduction

text: Newton

text: algorithm

text: is

text: a

text: eﬃcient

text: numerical

text: iterative

text: method

text: that

text: can

text: be

text: applied

text: for

text: ﬁnding

text: the

text: critical

text: points

text: of

text: the

text: cost

text: function.

text: This

text: type

text: of

text: algorithm

text: uses

text: second-order

text: information

text: about

text: the

text: cost

text: function,

text: fact

text: that

text: guarantees

text: super-linear

text: convergence

text: (in

text: cases

text: it

text: converges).

text: Initially

text: it

text: was

text: constructed

text: on

text: vector

text: spaces,

text: but

text: later

text: was

text: adapted

text: to

text: manifolds

text: for

text: reasons,

text: see

text: for

text: a

text: careful

text: and

text: detailed

text: discussion

text: of

text: the

text: geometry

text: involved

text: in

text: such

text: a

text: context.

text: R

text: Let

text: (S,

text: g

text: be

text: a

text: smooth

text: Riemannian

text: manifold

text: and

text: G

text: S

text: be

text: a

text: smooth

text: cost

text: function.

text: The

text: iterative

text: scheSme

text: is

text: given

text: by,

text: see

text: x

text: R

text: (v

text: xn

text: xn

text: where

text: R

text: TS

text: S

text: is

text: a

text: smooth

text: retraction

text: and

text: the

text: tangent

text: vector

text: v

text: T

text: S

text: is

text: the

text: solution

text: of

text: the

text: xn

text: xn

text: (contravariant)

text: Newton

text: equation

text: HG(x

text: v

text: G(x

text: n

text: xn

text: n

text: In

text: the

text: above

text: equation

text: we

text: have

text: the

text: Hessian

text: operator

text: HG

text: TS

text: TS.

text: The

text: link

text: between

text: the

text: Hessian

text: R

text: operator

text: HG

text: and

text: the

text: associated

text: bilinear

text: form

text: G

text: TS

text: is

text: given

text: by:

text: HG(x)

text: v

text: (x)

text: Hess

text: G(x)

text: v

text: x

text: x

text: S


text: ∗ → (cid:101) (cid:101) (cid:101) where g(cid:93) (x) : T S T S is the sharp operator in Riemannian geometry. The (contravariant) Newton S x x · − · ∇ equation (1.2) can be rewritten as Hess G(x ) v = g(cid:91) (x ) G(x ), or equivalently as the (covariant) Newton equation: (cid:101) n (cid:101)xn (cid:101)S n gS n · − Hess G(x ) v = dG(x ). (1.3) n xn n To solve the above equation on the manifold S one needs to put local charts on the manifold in order to make the computations and to have a good knowledge on the Riemannian geometry of the manifold. Another way is to embed the manifold S in a larger manifold (usually an Euclidean space) and work with local charts (Euclidean coordinates) on this ambient space. The details of these constructions in the case when S is described by constraint functions are given in the Section 2. We end this section presenting a new form of the Newton algorithm on constraint manifolds that we call Embedded Newton algorithm. The construction relies on the formula, given in [6], for the Hessian of a cost function restricted to a manifold deﬁned by constraint functions, formula which involves only ambient coordinates. In Section 3 we apply the Embedded Newton algorithm to the 5-electron Thomson problem. We ﬁnd numerically the following critical conﬁgurations: bi-pyramidal conﬁguration, square right pyramid conﬁguration and a new family of saddle critical points in the conﬁguration of a regular pentagon. We also carry out the analytical study of the three critical conﬁgurations discovered numerically. Each type of critical conﬁguration comes as diﬀerential curves of critical points for the embedded gradient vector ﬁeld introduced in [5] and [6]. We prove that all this three critical conﬁgurations are non- degenerate in the sense of Morse-Bott theory. In Section we proceed with an analytical study of the Riesz s-energy of 5-points on a sphere problem which is a generalization of 5-electron Thomson problem. First we prove that the three types of critical conﬁgurations of the 5-electron Thomson problem discussed in Section 3 remain critical conﬁgurations for the Riesz s-energy problem. Various bifurcation phenomena do appear when s varies and we ﬁnd a new bifurcating value for s = 13.5204990011... when the square right pyramid goes from a saddle critical point into a local minima. Some aspects of this bifurcation behavior have been previously pointed out in [21] and [22]. Analyzing the nature of the three critical conﬁgurations for s in the interval (13.5204990011..., 21.1471229401...), Mountain Pass type theorem suggest that there should be another critical conﬁguration with Morse index one. Indeed, we ﬁnd the critical conﬁguration of a double-tetrahedron and prove that it has Morse index one. This conﬁguration was previously discovered in [22] for various values of s in this interval. We also prove that this conﬁguration persists also for s > 21.1471229401.... 2 Embedded Newton algorithm We want to rewrite the (covariant) Newton equation (1.3) on the manifold S in terms of the Rie- mannian geometry of an ambient space M. Assume that there exists an ambient smooth Riemannian → R − manifold (M, g) and a smooth map F : M k such that the manifold S can be written as S = F 1(c), ∈ R where c k (cid:101)is a regular value of the map F. Furthermore, suppose that the induced metric on S by → R the ambient metric g is the metric g , i.e. g = g . Let G : M be a smooth prolongation of the | S S S cost function G. { ∇ ∇ } We ﬁx an adapted frame b , ..., b , F , ..., F on the regular foliation induced by F. More 1 s 1 k precisely, for i = 1, s we have that the vector ﬁeld b restricted to the submanifold S is a tangent vector ∈ ∇ i ∇ ﬁeld to this submanifold, i.e. b X(S) and F , ..., F generate the normal subspace to the i|S 1|S k|S submanifold S. { } Starting with the above adapted frame we consider a co-base Θ , ...Θ , dF , ..., dF for the module 1 s 1 k ∇ of 1-forms Λ1(M), where for i, j = 1, s we have Θ (b ) = δ and for α = 1, k we have Θ ( F ) = 0. i j ij i α On the manifold S the eq(cid:101)uation(cid:101)(1.3)(cid:101)is equivale(cid:101)nt with(cid:101) (cid:101) − · ∀ ∈ Hess G(x )(v , w ) = dG(x ) w , w T S. (2.1) n xn xn n xn xn xn 2 ##LTLine## ##LTLine## ##LTLine##

text: where

text: (x)

text: T

text: S

text: T

text: S

text: is

text: the

text: sharp

text: operator

text: in

text: Riemannian

text: geometry.

text: The

text: (contravariant)

text: Newton

text: S

text: x

text: x

text: equation

text: can

text: be

text: rewritten

text: as

text: Hess

text: G(x

text: v

text: (x

text: G(x

text: or

text: equivalently

text: as

text: the

text: (covariant)

text: Newton

text: equation:

label: equation

text: n

text: n

text: gS

text: n

text: Hess

text: G(x

text: v

text: dG(x

text: n

text: xn

text: n

text: To

text: solve

text: the

text: above

text: equation

text: on

text: the

text: manifold

text: S

text: one

text: needs

text: to

text: put

text: local

text: charts

text: on

text: the

text: manifold

text: in

text: order

text: to

text: make

text: the

text: computations

text: and

text: to

text: have

text: a

text: good

text: knowledge

text: on

text: the

text: Riemannian

text: geometry

text: of

text: the

text: manifold.

text: Another

text: way

text: is

text: to

text: embed

text: the

text: manifold

text: S

text: in

text: a

text: larger

text: manifold

text: (usually

text: an

text: Euclidean

text: space)

text: and

text: work

text: with

text: local

text: charts

text: (Euclidean

text: coordinates)

text: on

text: this

text: ambient

text: space.

text: The

text: details

text: of

text: these

text: constructions

text: in

text: the

text: case

text: when

text: S

text: is

text: described

text: by

text: constraint

text: functions

text: are

text: given

text: in

text: the

text: Section

text: We

text: end

text: this

text: section

text: presenting

text: a

text: new

text: form

text: of

text: the

text: Newton

text: algorithm

text: on

text: constraint

text: manifolds

text: that

text: we

text: call

text: Embedded

text: Newton

text: algorithm.

text: The

text: construction

text: relies

text: on

text: the

text: formula,

text: given

text: in

text: for

text: the

text: Hessian

text: of

text: a

text: cost

text: function

text: restricted

text: to

text: a

text: manifold

text: deﬁned

text: by

text: constraint

text: functions,

text: formula

text: which

text: involves

text: only

text: ambient

text: coordinates.

text: In

text: Section

text: we

text: apply

text: the

text: Embedded

text: Newton

text: algorithm

text: to

text: the

text: Thomson

text: problem.

text: We

text: ﬁnd

text: numerically

text: the

text: following

text: critical

text: conﬁgurations:

text: bi-pyramidal

text: conﬁguration,

text: square

text: right

text: pyramid

text: conﬁguration

text: and

text: a

text: new

text: family

text: of

text: saddle

text: critical

text: points

text: in

text: the

text: conﬁguration

text: of

text: a

text: regular

text: pentagon.

text: We

text: also

text: carry

text: out

text: the

text: analytical

text: study

text: of

text: the

text: three

text: critical

text: conﬁgurations

text: discovered

text: numerically.

text: Each

text: type

text: of

text: critical

text: conﬁguration

text: comes

text: as

text: diﬀerential

text: curves

text: of

text: critical

text: points

text: for

text: the

text: embedded

text: gradient

text: vector

text: ﬁeld

text: introduced

text: in

text: and

text: We

text: prove

text: that

text: all

text: this

text: three

text: critical

text: conﬁgurations

text: are

text: non-

text: degenerate

text: in

text: the

text: sense

text: of

text: Morse-Bott

text: theory.

text: In

text: Section

text: we

text: proceed

text: with

text: an

text: analytical

text: study

text: of

text: the

text: Riesz

text: s-energy

text: of

text: on

text: a

text: sphere

text: problem

text: which

text: is

text: a

text: generalization

text: of

text: Thomson

text: problem.

text: First

text: we

text: prove

text: that

text: the

text: three

text: types

text: of

text: critical

text: conﬁgurations

text: of

text: the

text: Thomson

text: problem

text: discussed

text: in

text: Section

text: remain

text: critical

text: conﬁgurations

text: for

text: the

text: Riesz

text: s-energy

text: problem.

text: Various

text: bifurcation

text: phenomena

text: do

text: appear

text: when

text: s

text: varies

text: and

text: we

text: ﬁnd

text: a

text: new

text: bifurcating

text: value

text: for

text: s

text: when

text: the

text: square

text: right

text: pyramid

text: goes

text: from

text: a

text: saddle

text: critical

text: point

text: into

text: a

text: local

text: minima.

text: Some

text: aspects

text: of

text: this

text: bifurcation

text: behavior

text: have

text: been

text: previously

text: pointed

text: out

text: in

text: and

text: Analyzing

text: the

text: nature

text: of

text: the

text: three

text: critical

text: conﬁgurations

text: for

text: s

text: in

text: the

text: interval

text: Mountain

text: Pass

text: type

text: theorem

text: suggest

text: that

text: there

text: should

text: be

text: another

text: critical

text: conﬁguration

text: with

text: Morse

text: index

text: one.

text: Indeed,

text: we

text: ﬁnd

text: the

text: critical

text: conﬁguration

text: of

text: a

text: double-tetrahedron

text: and

text: prove

text: that

text: it

text: has

text: Morse

text: index

text: one.

text: This

text: conﬁguration

text: was

text: previously

text: discovered

text: in

text: for

text: various

text: values

text: of

text: s

text: in

text: this

text: interval.

text: We

text: also

text: prove

text: that

text: this

text: conﬁguration

text: persists

text: also

text: for

text: s

text: Embedded

text: Newton

text: algorithm

text: We

text: want

text: to

text: rewrite

text: the

text: (covariant)

text: Newton

text: equation

text: on

text: the

text: manifold

text: S

text: in

text: terms

text: of

text: the

text: Rie-

text: mannian

text: geometry

text: of

text: an

text: ambient

text: space

text: M.

text: Assume

text: that

text: there

text: exists

text: an

text: ambient

text: smooth

text: Riemannian

text: R

text: manifold

text: (M,

text: g)

text: and

text: a

text: smooth

text: map

text: F

text: M

text: k

text: such

text: that

text: the

text: manifold

text: S

text: can

text: be

text: written

text: as

text: S

text: F

text: R

text: where

text: c

text: k

text: a

text: regular

text: value

text: of

text: the

text: map

text: F.

text: Furthermore,

text: suppose

text: that

text: the

text: induced

text: metric

text: on

text: S

text: by

text: R

text: the

text: ambient

text: metric

text: g

text: is

text: the

text: metric

text: g

text: i.e.

text: g

text: g

text: Let

text: G

text: M

text: be

text: a

text: smooth

text: prolongation

text: of

text: the

text: S

text: S

text: S

text: cost

text: function

text: G.

text: We

text: ﬁx

text: an

text: adapted

text: frame

text: b

text: b

text: F

text: F

text: on

text: the

text: regular

text: foliation

text: induced

text: by

text: F.

text: More

text: s

text: k

text: precisely,

text: for

text: i

text: s

text: we

text: have

text: that

text: the

text: vector

text: ﬁeld

text: b

text: restricted

text: to

text: the

text: submanifold

text: S

text: is

text: a

text: tangent

text: vector

text: i

text: ﬁeld

text: to

text: this

text: submanifold,

text: i.e.

text: b

text: X(S)

text: and

text: F

text: F

text: generate

text: the

text: normal

text: subspace

text: to

text: the

text: i|S

text: k|S

text: submanifold

text: S.

text: Starting

text: with

text: the

text: above

text: adapted

text: frame

text: we

text: consider

text: a

text: co-base

text: Θ

text: ...Θ

text: dF

text: dF

text: for

text: the

text: module

text: s

text: k

text: of

text: where

text: for

text: i,

text: j

text: s

text: we

text: have

text: Θ

text: (b

text: δ

text: and

text: for

text: α

text: k

text: we

text: have

text: Θ

text: F

text: i

text: j

text: ij

text: i

text: α

text: On

text: the

text: manifold

text: S

text: the

text: Hess

text: G(x

text: )(v

text: w

text: dG(x

text: w

text: w

text: T

text: S.

text: n

text: xn

text: xn

text: n

text: xn

text: xn

text: xn

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:101) R \{ | ∃ ≤ ≤ } where G is the natural extension of G on M = 12 p (i, j), 1 i < j 5, s.t. p = p . s s (cid:101) i j 1. Bi-pyramid conﬁguration. By direct computation we obtain that the bi-pyramid conﬁguration (3.5) veriﬁes equation (4.2) and c(cid:101)onsequently, it remains a critical conﬁguration of G for any s > 0 s with √3 √6 1 (cid:101) G (bi-pyramid) = + + . s 3s 2s 2s Computing the Hessian of G in this bi-pyramid conﬁguration we obtain that for s < 21.1471229401... s we have non-degenerate local minima in the sense of Morse-Bott. This has been previously noticed and discussed in [21] and [22]. For s > 21.1471229401... we obtain non-degenerate saddle points in the sense of Morse-Bott with two strictly negative eigenvalues, which shows that a bifurcation occurs in this type of critical conﬁguration. 2. Square right pyramid conﬁguration. As in the case of Thomson problem s = 1 discussed in the previous section we obtain that the equation (4.2) is equivalent for this conﬁguration with the following equation: ⇔ ∂G (c (λ)) = 0 T (α) = 0, (4.3) s α s (cid:0) (cid:1) (cid:0) (cid:1) where Ts(α) = (2 − 2 α)−2s + 2 2 − 2 α2 −2s α + α 4 − 4 α2 −2s + (2 − 2 α)−2s α. We observe that we have T (α) = Q(α), where Q(α) has been deﬁned in the previous section. For every ≥ 1 − s 1 we obtain that the equation (4.3) has a solution α in the interval [ 0.2432010309..., 0). When ∈ s become bigger the base of the pyramid approaches the equator. For s (0, 1] the solution α of the − − equation (4.3) is in the interval ( 0.25, 0.2432010309...] meaning that when s becomes very small the base of the pyramid is at the distance close to 0.25 from the equatorial plane. The value of the Riesz s-energy on this(cid:101)critical conﬁguration is given by √ 2 √ 4 √ 4 G (square right pyramid) = − + − + − , s ( 4 4α2)s ( 2 2α2)s ( 2 2α)s where α is the solution of the equation (4.3). The study of the Hessian matrix shows that the right square pyramid conﬁguration undergoes various bifurcation phenomena. More precisely for s < 13.5204990011... we have non-degenerate saddle points in the sense of Morse-Bott having one eigenvalue strictly negative. For s > 13.5204990011... the right square pyramid conﬁgurations are non-degenerate local minima in the sense of Morse-Bott. For s between 13.5204990011... and 15.048077392... we have that both bi-pyramidal conﬁguration (cid:101) (cid:101) and square right pyramid conﬁguration are local minima with G (bi-pyramid) < G (square right pyramid). s s For s between 15.048077392... and 21.1471229401... we have that both bi-pyramidal conﬁguration and (cid:101) (cid:101) square right pyramid conﬁguration are local minima with G (bi-pyramid) > G (square right pyramid), s s which is a phenomenon discovered in [21]. For s > 21.1471229401... the above inequality remains true. 3. Regular pentagon. The regular pentagon conﬁguration discussed in the previous section is a solution for the equation (4.2) and consequently, remains critical conﬁguration of Riesz s-energy for any s > 0. The regular pentagon conﬁguration remains non-degenerate saddle in the sense of Morse-Bott 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: where

text: G

text: is

text: the

text: natural

text: extension

text: of

text: G

text: on

text: M

text: p

text: (i,

text: j),

text: i

text: j

text: s.t.

text: p

text: p

text: s

text: s

text: i

text: j

text: Bi-pyramid

text: conﬁguration.

text: By

text: direct

text: computation

text: we

text: obtain

text: that

text: the

text: bi-pyramid

text: conﬁguration

text: veriﬁes

text: equation

text: and

text: it

text: remains

text: a

text: critical

text: conﬁguration

text: of

text: G

text: for

text: any

text: s

text: s

text: with

text: √3

text: √6

text: G

text: (bi-pyramid)

text: s

text: Computing

text: the

text: Hessian

text: of

text: G

text: in

text: this

text: bi-pyramid

text: conﬁguration

text: we

text: obtain

text: that

text: for

text: s

text: s

text: we

text: have

text: non-degenerate

text: local

text: minima

text: in

text: the

text: sense

text: of

text: Morse-Bott.

text: This

text: has

text: been

text: previously

text: noticed

text: and

text: discussed

text: in

text: and

text: For

text: s

text: we

text: obtain

text: non-degenerate

text: saddle

text: points

text: in

text: the

text: sense

text: of

text: Morse-Bott

text: with

text: two

text: strictly

text: negative

text: eigenvalues,

text: which

text: shows

text: that

text: a

text: bifurcation

text: occurs

text: in

text: this

text: type

text: of

text: critical

text: conﬁguration.

text: Square

text: right

text: pyramid

text: conﬁguration.

text: As

text: in

text: the

text: case

text: of

text: Thomson

text: problem

text: s

text: discussed

text: in

text: the

text: previous

text: section

text: we

text: obtain

text: that

text: the

text: equation

text: is

text: equivalent

text: for

text: this

text: conﬁguration

text: with

text: the

text: following

text: equation:

text: ∂G

text: (c

text: (λ))

text: T

text: (α)

text: s

text: α

text: s

text: where

text: Ts(α)

text: α2

text: α

text: α

text: α2

text: α.

text: We

text: observe

text: that

text: we

text: have

text: T

text: (α)

text: Q(α),

text: where

text: Q(α)

text: has

text: been

text: deﬁned

text: in

text: the

text: previous

text: section.

text: For

text: every

text: s

text: we

text: obtain

text: that

text: the

text: equation

text: has

text: a

text: solution

text: α

text: in

text: the

text: interval

text: When

text: s

text: become

text: bigger

text: the

text: base

text: of

text: the

text: pyramid

text: approaches

text: the

text: equator.

text: For

text: s

text: the

text: solution

text: α

text: of

text: the

text: equation

text: is

text: in

text: the

text: interval

text: meaning

text: that

text: when

text: s

text: becomes

text: very

text: small

text: the

text: base

text: of

text: the

text: pyramid

text: is

text: at

text: the

text: distance

text: close

text: to

text: from

text: the

text: equatorial

text: plane.

text: The

text: value

text: of

text: the

text: Riesz

text: s-energy

text: on

text: conﬁguration

text: is

text: given

text: by

text: G

text: (square

text: right

text: pyramid)

text: s

text: where

text: α

text: is

text: the

text: solution

text: of

text: the

text: equation

text: The

text: study

text: of

text: the

text: Hessian

text: matrix

text: shows

text: that

text: the

text: right

text: square

text: pyramid

text: conﬁguration

text: undergoes

text: various

text: bifurcation

text: phenomena.

text: More

text: precisely

text: for

text: s

text: we

text: have

text: non-degenerate

text: saddle

text: points

text: in

text: the

text: sense

text: of

text: Morse-Bott

text: having

text: one

text: eigenvalue

text: strictly

text: negative.

text: For

text: s

text: the

text: right

text: square

text: pyramid

text: conﬁgurations

text: are

text: non-degenerate

text: local

text: minima

text: in

text: the

text: sense

text: of

text: Morse-Bott.

text: For

text: s

text: between

text: and

text: we

text: have

text: that

text: both

text: bi-pyramidal

text: conﬁguration

text: and

text: square

text: right

text: pyramid

text: conﬁguration

text: are

text: local

text: minima

text: with

text: G

text: (bi-pyramid)

text: G

text: (square

text: right

text: pyramid).

text: s

text: s

text: For

text: s

text: between

text: and

text: we

text: have

text: that

text: both

text: bi-pyramidal

text: conﬁguration

text: and

text: square

text: right

text: pyramid

text: conﬁguration

text: are

text: local

text: minima

text: with

text: G

text: (bi-pyramid)

text: G

text: (square

text: right

text: pyramid),

text: s

text: s

text: which

text: is

text: a

text: phenomenon

text: discovered

text: in

text: For

text: s

text: the

text: above

text: inequality

text: remains

text: true.

text: Regular

text: pentagon.

text: The

text: regular

text: pentagon

text: conﬁguration

text: discussed

text: in

text: the

text: previous

text: section

text: is

text: a

text: solution

text: for

text: the

text: equation

text: and

text: consequently,

text: remains

text: critical

text: conﬁguration

text: of

text: Riesz

text: s-energy

text: for

text: any

text: s

text: The

text: regular

text: pentagon

text: conﬁguration

text: remains

text: non-degenerate

text: saddle

text: in

text: the

text: sense

text: of

text: Morse-Bott

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: References [1] P.A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Prince- ton University Press, 2008. R [2] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on n, Birkh¨auser, 2006. [3] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and appli- cations, Jour. Funct. Anal., Vol. 14, (1973), pp. 349-381. [4] B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly, A. Schuermann, Ex- perimental study of energy-minimizing point conﬁgurations on spheres, Experimental Mathematics, Vol. 18 (2009), pp. 257-283. [5] P. Birtea, D. Com˘anescu, Geometric Dissipation for dynamical systems, Comm. Math. Phys., Vol. 316, Issue 2 (2012), pp. 375-394. [6] P. Birtea, D. Com˘anescu, Hessian Operators on Constraint Manifolds, J. Nonlinear Science, Vol. 25, Issue 6 (2015), pp 1285-1305. [7] P. Birtea, D. Com˘anescu, C.A. Popa , Averaging on Manifolds by Embedding Algorithm, J. Math. Imaging Vis., Vol. 49, Issue 2 (2014), pp. 454-466. [8] A. V. Bondarenko, D. P. Hardin, E. B. Saﬀ, Mesh ratios for best-packing and limits of minimal energy conﬁgurations, Acta Math. Hungarica, Vol. 142, Issue 1 (2014), pp. 118-131. [9] R. Bott, Nondegenerate critical manifolds, Ann. of Math., Vol. 60, (1954), pp. 248-261. [10] H. Cohn, Order and disorder in energy minimization, Proceedings of the international congress of mathematicians (ICM 2010), pp. 2416-2443, 2010. [11] H. Cohn, A. Kumar,Universally optimal distribution of points on spheres, Journal of the Amer- ican Mathematical Society, Vol. 20 (2007), pp. 99-148. [12] P.D. Dragnev, D.A. Legg, D.W. Townsend, Discrete logarithmic energy on the sphere, Paciﬁc Journal of Math., Vol. 207, Issue 2 (2002), pp. 345-358. [13] R.J. Gillespie, Fifty years of the VSEPR model, Coordination Chemistry Reviews, Vol. 252, Issue 12-14 (2008), pp. 1315-1327. [14] R.J. Gillespie, I. Hargittai, The VSEPR Model of Molecular Geometry, Allyn and Bacon, Boston, 1991. [15] I. Hargittai, B. Chamberland, The VSEPR Model of Molecular Geometry, Comp. & Maths. with Appls., Vol. 12B (1986), pp. 1021-1038. [16] Xiaorong Huo, Junwei Shao, Spherical Distribution of 5 Points with Maximal Distance Sum, Discrete Comput Geom, Vol. 46, (2011), pp. 156-174. [17] T. W. Melnyk, O. Knop, W. R. Smith, Extremal arrangements of points and unit charges on a sphere: equilibrium conﬁgurations revisited, Can. J. Chem., Vol. 55, (1976), pp. 1745-1761. [18] T. Mu¨ller, J. Frauendiener, Charged particles constrained to a curved surface, European Journal of Physics, Vol. 34, Issue 1 (2013), pp. 147-160. [19] P.K. Newton, T. Sakajo, Point vortex equilibria and optimal packings of circles on a sphere, Proc. R. Soc. A, Vol. 467 (2011), pp. 1468-1490. [20] L. Nicolaescu, An Invitation to Morse Theory, Universitext, Springer, Second Edition 2011. 16

text: References

label: reference

text: P.A.

text: Absil,

text: R.

text: Mahony,

text: R.

text: Sepulchre,

text: Optimization

text: Algorithms

text: on

text: Matrix

text: Manifolds,

text: Prince-

text: ton

text: University

text: Press,

text: R

text: A.

text: Ambrosetti,

text: A.

text: Malchiodi,

text: Perturbation

text: Methods

text: and

text: Semilinear

text: Elliptic

text: Problems

text: on

text: n,

text: Birkh¨auser,

text: A.

text: Ambrosetti,

text: P.H.

text: Rabinowitz,

text: Dual

text: variational

text: methods

text: in

text: critical

text: point

text: theory

text: and

text: appli-

text: cations,

text: Jour.

text: Funct.

text: Anal.,

text: Vol.

text: pp.

text: B.

text: Ballinger,

text: G.

text: Blekherman,

text: H.

text: Cohn,

text: N.

text: Giansiracusa,

text: E.

text: Kelly,

text: A.

text: Schuermann,

text: Ex-

text: perimental

text: study

text: of

text: energy-minimizing

text: point

text: conﬁgurations

text: on

text: spheres,

text: Experimental

text: Mathematics,

text: Vol.

text: pp.

text: P.

text: Birtea,

text: D.

text: Com˘anescu,

text: Geometric

text: Dissipation

text: for

text: dynamical

text: systems,

text: Comm.

text: Math.

text: Phys.,

text: Vol.

text: Issue

text: pp.

text: P.

text: Birtea,

text: D.

text: Com˘anescu,

text: Hessian

text: Operators

text: on

text: Constraint

text: Manifolds,

text: J.

text: Nonlinear

text: Science,

text: Vol.

text: Issue

text: pp

text: P.

text: Birtea,

text: D.

text: Com˘anescu,

text: C.A.

text: Popa

text: Averaging

text: on

text: Manifolds

text: by

text: Embedding

text: Algorithm,

text: J.

text: Math.

text: Imaging

text: Vis.,

text: Vol.

text: Issue

text: pp.

text: A.

text: V.

text: Bondarenko,

text: D.

text: P.

text: Hardin,

text: E.

text: B.

text: Saﬀ,

text: Mesh

text: ratios

text: for

text: best-packing

text: and

text: limits

text: of

text: minimal

text: energy

text: conﬁgurations,

text: Acta

text: Math.

text: Hungarica,

text: Vol.

text: Issue

text: pp.

text: R.

text: Bott,

text: Nondegenerate

text: critical

text: manifolds,

text: Ann.

text: of

text: Math.,

text: Vol.

text: pp.

text: H.

text: Cohn,

text: Order

text: and

text: disorder

text: in

text: energy

text: minimization,

text: Proceedings

text: of

text: the

text: international

text: congress

text: of

text: mathematicians

text: (ICM

text: pp.

text: H.

text: Cohn,

text: A.

text: Kumar,Universally

text: optimal

text: distribution

text: of

text: points

text: on

text: spheres,

text: Journal

text: of

text: the

text: Amer-

text: ican

text: Mathematical

text: Society,

text: Vol.

text: pp.

text: P.D.

text: Dragnev,

text: D.A.

text: Legg,

text: D.W.

text: Townsend,

text: Discrete

text: logarithmic

text: energy

text: on

text: the

text: sphere,

text: Paciﬁc

text: Journal

text: of

text: Math.,

text: Vol.

text: Issue

text: pp.

text: R.J.

text: Gillespie,

text: Fifty

text: years

text: of

text: the

text: VSEPR

text: model,

text: Coordination

text: Chemistry

text: Reviews,

text: Vol.

text: Issue

text: pp.

text: R.J.

text: Gillespie,

text: I.

text: Hargittai,

text: The

text: VSEPR

text: Model

text: of

text: Molecular

text: Geometry,

text: Allyn

text: and

text: Bacon,

text: Boston,

text: I.

text: Hargittai,

text: B.

text: Chamberland,

text: The

text: VSEPR

text: Model

text: of

text: Molecular

text: Geometry,

text: Comp.

text: Maths.

text: with

text: Appls.,

text: Vol.

text: pp.

text: Xiaorong

text: Huo,

text: Junwei

text: Shao,

text: Spherical

text: Distribution

text: of

text: Points

text: with

text: Maximal

text: Distance

text: Sum,

text: Discrete

text: Comput

text: Geom,

text: Vol.

text: pp.

text: T.

text: W.

text: Melnyk,

text: O.

text: Knop,

text: W.

text: R.

text: Smith,

text: Extremal

text: arrangements

text: of

text: points

text: and

text: unit

text: charges

text: on

text: a

text: sphere:

text: equilibrium

text: conﬁgurations

text: revisited,

text: Can.

text: J.

text: Chem.,

text: Vol.

text: pp.

text: T.

text: Mu¨ller,

text: J.

text: Frauendiener,

text: Charged

text: particles

text: constrained

text: to

text: a

text: curved

text: surface,

text: European

text: Journal

text: of

text: Physics,

text: Vol.

text: Issue

text: pp.

text: P.K.

text: Newton,

text: T.

text: Sakajo,

text: Point

text: vortex

text: equilibria

text: and

text: optimal

text: packings

text: of

text: circles

text: on

text: a

text: sphere,

text: Proc.

text: R.

text: Soc.

text: A,

text: Vol.

text: pp.

text: L.

text: Nicolaescu,

text: An

text: Invitation

text: to

text: Morse

text: Theory,

text: Universitext,

text: Springer,

text: Second

text: Edition


text: where p is the position vector of the electron P on the unit sphere. i i Without minimizing the generality of the problem we can suppose that the electron P is ﬁxed in 5 the North pole, i.e. p := (0, 0, 1). The phase space of the problem is the manifold 5 × × × \{ | ∃ ≤ ≤ } S := S2 (cid:101) S2 S2 S2 p := (p , p , p , p ) (i, j), 1 i < j 5, s.t. p = p . 1 2 3 4 i j → R (cid:88) The cost function G : S is a Coul(cid:101)omb potential and it is given by 1 G(p) := || − ||. ≤ ≤ pi pj 1 i<j 5 The aim is to ﬁnd all critical conﬁgurations of the above Coulomb potential. First we will pro- ceed by constructing Embedded Newton algorithm for the constraint conﬁguration space S. This will give us numerically three types of critical conﬁgurations. We also make an analytical study of this conﬁgurations. 3.1 Numerical study: Embedded Newton algorithm R We embed this optimization problem into 12 using the constraint functions F := (F , F , F , F ) : → R || || 1 2 3 4 M 4, where the components are F (p) := 1 p 2, i = 1, 4 and the ambient space M is the open set R \{ | ∃ ≤ ≤ i } 2 (cid:88)i → R 12 p (i, j), 1 i < j 5, s.t. p = p . The extended cost function G : M is given by i j 1 G(p) := || − ||. ≤ ≤ pi pj 1 i<j 5 ⊂ R On the unit sphere S2 3 we can construct a local frame using the North pole stereographic − − − − − − − − projection: e (x, y, z) := (1 z x2, xy, x(1 z)), e (x, y, z) := ( xy, 1 z y2, y(1 z)), see 1 2 Appendix 3.. A local frame on the constraint manifold S is given by: b (p) := (e (p ), 0, 0, 0), b (p) := (e (p ), 0, 0, 0), 1 1 1 2 2 1 b (p) := (0, e (p ), 0, 0), b (p) := (0, e (p ), 0, 0), 3 1 2 4 2 2 b (p) := (0, 0, e (p ), 0), b (p) := (0, 0, e (p ), 0), 5 1 3 6 2 3 b (p) := (0, 0, 0, e (p )), b (p) := (0, 0, 0, e (p )). 7 1 4 8 2 4 The eight coordinate functions are given by: ∂G ∂G g (p) =< (p), e (p ) >, g (p) =< (p), e (p ) >, 1 ∂p 1 1 2 ∂p 2 1 1 1 ∂G ∂G g (p) =< (p), e (p ) >, g (p) =< (p), e (p ) >, 3 ∂p 1 2 4 ∂p 2 2 2 2 ∂G ∂G g (p) =< (p), e (p ) >, g (p) =< (p), e (p ) >, 5 ∂p 1 3 6 ∂p 2 3 3 3 ∂G ∂G g (p) =< (p), e (p ) >, g (p) =< (p), e (p ) > . 7 ∂p 1 4 8 ∂p 2 4 4 4 By a straightforward computation we obtain the Lagrange multipliers functions: ∂G ∂G σ (p) :=< (p), p ) >, σ (p) :=< (p), p ) >, 1 ∂p 1 2 ∂p 2 1 2 ∂G ∂G σ (p) :=< (p), p ) >, σ (p) :=< (p), p ) > . 3 ∂p 3 4 ∂p 4 3 4 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: p

text: is

text: the

text: position

text: vector

text: of

text: the

text: electron

text: P

text: on

text: the

text: unit

text: sphere.

text: i

text: i

text: Without

text: minimizing

text: the

text: generality

text: of

text: the

text: problem

text: we

text: can

text: suppose

text: that

text: the

text: electron

text: P

text: is

text: ﬁxed

text: in

text: the

text: North

text: pole,

text: i.e.

text: p

text: The

text: phase

text: space

text: of

text: the

text: problem

text: is

text: the

text: manifold

text: S

text: p

text: (p

text: p

text: p

text: p

text: (i,

text: j),

text: i

text: j

text: s.t.

text: p

text: p

text: i

text: j

text: R

text: The

text: cost

text: function

text: G

text: S

text: is

text: a

text: potential

text: and

text: it

text: is

text: given

text: by

text: G(p)

text: pi

text: pj

text: i<j

text: The

text: aim

text: is

text: to

text: ﬁnd

text: all

text: critical

text: conﬁgurations

text: of

text: the

text: above

text: Coulomb

text: potential.

text: First

text: we

text: will

text: pro-

text: ceed

text: by

text: constructing

text: Embedded

text: Newton

text: algorithm

text: for

text: the

text: constraint

text: conﬁguration

text: space

text: S.

text: This

text: will

text: give

text: us

text: numerically

text: three

text: types

text: of

text: critical

text: conﬁgurations.

text: We

text: also

text: make

text: an

text: analytical

text: study

text: of

text: this

text: conﬁgurations.

text: Numerical

text: study:

text: Embedded

text: Newton

text: algorithm

text: R

text: We

text: embed

text: this

text: optimization

text: problem

text: into

text: using

text: the

text: constraint

text: functions

text: F

text: (F

text: F

text: F

text: F

text: R

text: M

text: where

text: the

text: components

text: are

text: F

text: (p)

text: p

text: i

text: and

text: the

text: ambient

text: space

text: M

text: is

text: the

text: open

text: set

text: R

text: i

text: R

text: p

text: (i,

text: j),

text: i

text: j

text: s.t.

text: p

text: p

text: The

text: extended

text: cost

text: function

text: G

text: M

text: is

text: given

text: by

text: i

text: j

text: G(p)

text: pi

text: pj

text: i<j

text: R

text: On

text: the

text: unit

text: sphere

text: we

text: can

text: construct

text: a

text: local

text: frame

text: using

text: the

text: North

text: pole

text: stereographic

text: projection:

text: e

text: (x,

text: y,

text: z)

text: z

text: xy,

text: z)),

text: e

text: (x,

text: y,

text: z)

text: xy,

text: z

text: z)),

text: see

text: Appendix

text: A

text: local

text: frame

text: on

text: the

text: constraint

text: manifold

text: S

text: is

text: given

text: by:

text: b

text: (p)

text: (e

text: (p

text: b

text: (p)

text: (e

text: (p

text: b

text: (p)

text: e

text: (p

text: b

text: (p)

text: e

text: (p

text: b

text: (p)

text: e

text: (p

text: b

text: (p)

text: e

text: (p

text: b

text: (p)

text: e

text: (p

text: b

text: (p)

text: e

text: (p

text: The

text: eight

text: coordinate

text: functions

text: are

text: given

text: by:

text: ∂G

text: ∂G

text: g

text: (p)

text: (p),

text: e

text: (p

text: g

text: (p)

text: (p),

text: e

text: (p

text: ∂p

text: ∂p

text: ∂G

text: ∂G

text: g

text: (p)

text: (p),

text: e

text: (p

text: g

text: (p)

text: (p),

text: e

text: (p

text: ∂p

text: ∂p

text: ∂G

text: ∂G

text: g

text: (p)

text: (p),

text: e

text: (p

text: g

text: (p)

text: (p),

text: e

text: (p

text: ∂p

text: ∂p

text: ∂G

text: ∂G

text: g

text: (p)

text: (p),

text: e

text: (p

text: g

text: (p)

text: (p),

text: e

text: (p

text: ∂p

text: ∂p

text: By

text: a

text: straightforward

text: computation

text: we

text: obtain

text: the

text: Lagrange

text: multipliers

text: functions:

text: ∂G

text: ∂G

text: σ

text: (p)

text: (p),

text: p

text: σ

text: (p)

text: (p),

text: p

text: ∂p

text: ∂p

text: ∂G

text: ∂G

text: σ

text: (p)

text: (p),

text: p

text: σ

text: (p)

text: (p),

text: p

text: ∂p

text: ∂p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A&A proofs: manuscript no. alCirArchiv multaneous multi-colour photometry of α Cir in the Walraven the Constellation during the α Cir run. As a consequence of this system. The goal was pulsation mode identiﬁcation (attempted dynamic, instructional operational environment, the α Cir data also by Kurtz & Balona (1984)) based on frequency multiplets set is naturally somewhat complicated. and amplitude changes. The data cover 25 March – 8 August 2014: 146 days, corre- The ﬁrst simultaneous time-resolved photometry and spec- sponding to nearly 33 stellar rotational cycles (see the next Sec. troscopy of α Cir were obtained at ESO by Schneider & Weiss for details). (1989). This yielded only marginal evidence for radial velocity α Cir was observed in the blue by BAb for 131 days and BLb variations for two Ca I lines and the authors remarked on the for 26 days; in the red by UBr for 145 days, and BTr for six very uncertain value on the rotation period for this star (rang- days. The light curves consist of nearly 44 000 blue exposures ing in the literature from one to 12 days). Five years later, Kurtz and 69 000 red exposures, each of duration 1 s in both colours. et al. (1994) provided a robust value of the rotation period (P The sampling time was 21.3 s, which is set by the amount of (cid:39) 4.46 d). Another 15 years elapsed before the ﬁrst light curve processing time needed between consecutive exposures and the sampling the full rotation cycle of α Cir was published based on maximum amount of data that could be stored on board owing WIRE space photometry (Bruntt et al. 2009). to limited download capacity at the time. The quantity and quality of data, and the sophistication of α Cir was observed during individual BRITE orbits for 5 to models, has improved dramatically over the subsequent years 30 min, depending on target visibility, the Earth Exclusion Angle (e.g. Kurtz et al. (1994); Audard et al. (1998); Balona & Laney (angular distance between the illuminated limb of the Earth and (2003); Kochukhov et al. (2007); and many references therein). the star-tracker ﬁeld of view), the required warm-up period for The most complete and recent summary of our knowledge about the star trackers after reacquisition of the target, and passages of the pulsation of α Cir is presented by Bruntt et al. (2008, 2009) the satellites through the South Atlantic Anomaly (when higher and of the complex atmospheric structure and spectral signatures ﬂuxes of cosmic rays prevent useful signal from the star track- of pulsation by Ryabchikova et al. (2007) and Kochukhov et ers). There were also brief interruptions of science data collec- al. (2009). tion when data were being transmitted to ground stations. α Cir was included in proposals submitted in 2009 and 2010 The daily data transfer rate to ground stations in Europe was by several authors of this paper, responding to an Announce- less than planned because of heavy interference with unidentiﬁed ment of Opportunity for targets to be observed by BRITE- transmitters on the ground, which is particularly serious during Constellation. Proposal #05 (WW) focused on roAp stars, #19 the Centaurus ﬁeld observations. For example, data transfer from (DH) on 50 bright stars with the most accurately known Hip- BAb to the station in Graz, Austria, was limited to a maximum parcos parallaxes, and #25 (TL) on the rotation of chemically of 15 min per orbit. peculiar stars (in particular, possible changes in rotation period). 2.2. Reduction: BRITE standard pipeline 2. BRITE-Constellation photometry During the commissioning of BRITE-Constellation, the particular challenges of on-orbit photometry were assessed. To The BRITE-Constellation is a coordinated mission of ﬁve develop an eﬃcient pipeline for fast and optimum extraction of nanosatellites located in low Earth orbits, each hosting an op- the photometry, a Photometry Tiger Team (PHOTT) was estab- tical telescope of 3 cm aperture feeding an uncooled CCD, and ◦ lished, consisting of three groups working independently and observing selected targets in a 24 ﬁeld of view (Weiss et al. pursuing diﬀerent processing and reduction strategies. 2014). Each nanosat is equipped with a single ﬁlter; three have ∼ Issues investigated included: CCD radiation defects (warm a red ﬁlter ( 620 nm) and two have a blue ﬁlter (central wave- ∼ columns, hot and cold pixels), CCD temperature variations, length ( 420 nm). The satellites have overlapping coverage of charge-transfer ineﬃciencies, pixel-to-pixel sensitivity varia- the target ﬁelds to provide two-colour, time-resolved photome- tions and intra-pixel inhomogenities, pointing errors, complex try. ◦ × point spread functions (PSF) varying over the 24 ﬁeld of view The detector is a KAI-11002M CCD with 11 million 9 9 µm − and during long observing runs, identiﬁcation of defective im- pixels, a 14-bit A/D converter, gain of about 3.3 e /ADU, and a − ◦ ages (e.g. incomplete PSF in a sub-raster). Competing reduction dark current of 1e /s at +20 C. The saturation limit of the pixels ∼ methods employed PSF ﬁtting and aperture photometry as well at this temperature is 13 000 ADU, and the response is linear (cid:48)(cid:48) as various threshold criteria for elimination of bad data points up to about 8 000 ADU. The pixel size is 27.3 , projected at the (outliers), various ﬁt-quality parameters, median vs. mean, etc. sky. The methods will be described in detail by Popowicz et al. (in The ﬁve nanosats are designated BRITE-Austria (BAb), prep.). UniBRITE (UBr), BRITE-Lem (BLb), BRITE-Hewelius (BHr), This investigation of α Cir is based on the reduction pipeline and BRITE-Toronto (BTr), where the last letter indicates the ﬁl- developed by PHOTT member Adam Popowicz via aperture ter on board the satellite ("b" = blue, "r" = red). A sixth nanosat, photometry. Figure 1 illustrates the impact of two optimisation BRITE-Montréal (BMb), did not detach from its launch vehicle. parameters on the light-curve quality. The parameters are the The nanosats Bb and UBr were launched ﬁrst, followed by BLb, threshold employed for bad-pixel identiﬁcation in the mean dark BTr, and BHr. frame (T ) and the aperture radius. To assess the light-curve 2.1. α Cir photometry quality, wbeadcomputed the median absolute deviations (MAD) of results obtained for individual satellite orbits, as a robust mea- α Cir was one of the targets in the Centaurus ﬁeld, the sec- sure of scatter in the presence of outliers. The ﬁnal quality mea- ond long-term pointing of BRITE-Constellation after the Orion sure (Q ) is th(cid:40)e median(cid:41) of MADs deﬁned as follows: ﬁeld. The early data were obtained during science commission- M ing of the mission, which meant interruptions in science data collection for engineering tests. Even so, signiﬁcant gaps in the M√AD time series were infrequent, caused mainly by accidental space- QM = median i , (1) craft mode changes from ﬁne-to-coarse pointing (or tumbling) Ni with recovery times shorter than 48 hours. New nanosats joined Article number, page 2 of 10 ##LTLine## ##LTLine##

text: A&A

text: proofs:

text: manuscript

text: no.

text: alCirArchiv

text: multaneous

text: multi-colour

text: photometry

text: of

text: α

font: YVDQFS+rtxmi

text: Cir

text: in

text: the

text: Walraven

text: the

text: Constellation

text: during

text: the

text: α

text: Cir

text: run.

text: As

text: a

text: consequence

text: of

text: this

text: system.

text: The

text: goal

text: was

text: pulsation

text: mode

text: identiﬁcation

text: (attempted

text: dynamic,

text: instructional

text: operational

text: environment,

text: the

text: α

text: Cir

text: data

text: also

text: by

text: Kurtz

text: Balona

text: based

text: on

text: frequency

text: multiplets

text: set

text: is

text: naturally

text: somewhat

text: complicated.

text: and

text: amplitude

text: changes.

text: The

text: data

text: cover

text: March

text: August

text: days,

text: corre-

text: The

text: ﬁrst

text: simultaneous

text: time-resolved

text: photometry

text: and

text: spec-

text: sponding

text: to

text: nearly

text: stellar

text: rotational

text: cycles

text: (see

text: the

text: next

text: Sec.

text: troscopy

text: of

text: α

text: Cir

text: were

text: obtained

text: at

text: ESO

text: by

text: Schneider

text: Weiss

text: for

text: details).

text: This

text: yielded

text: only

text: marginal

text: evidence

text: for

text: radial

text: velocity

text: α

text: Cir

text: was

text: observed

text: in

text: the

text: blue

text: by

text: BAb

text: for

text: days

text: and

text: BLb

text: variations

text: for

text: two

text: Ca

text: I

text: lines

text: and

text: the

text: authors

text: remarked

text: on

text: the

text: for

text: days;

text: in

text: the

text: red

text: by

text: UBr

text: for

text: days,

text: and

text: BTr

text: for

text: six

text: very

text: uncertain

text: value

text: on

text: the

text: rotation

text: period

text: for

text: this

text: star

text: (rang-

text: days.

text: The

text: light

text: curves

text: consist

text: of

text: nearly

text: blue

text: exposures

text: ing

text: in

text: the

text: literature

text: from

text: one

text: to

text: days).

text: Five

text: years

text: later,

text: Kurtz

text: and

text: red

text: exposures,

text: each

text: of

text: duration

text: s

text: in

text: both

text: colours.

text: et

text: al.

text: provided

text: a

text: robust

text: value

text: of

text: the

text: rotation

text: period

text: (P

text: The

text: sampling

text: time

text: was

text: s,

text: which

text: is

text: set

text: by

text: the

text: amount

text: of

text: d).

text: Another

text: years

text: elapsed

text: before

text: the

text: ﬁrst

text: light

text: curve

text: processing

text: time

text: needed

text: between

text: consecutive

text: exposures

text: and

text: the

text: sampling

text: the

text: full

text: rotation

text: cycle

text: of

text: α

text: Cir

text: was

text: published

text: based

text: on

text: maximum

text: amount

text: of

text: data

text: that

text: could

text: be

text: stored

text: on

text: board

text: owing

text: WIRE

text: space

text: photometry

text: (Bruntt

text: et

text: al.

text: to

text: limited

text: download

text: capacity

text: at

text: the

text: time.

text: The

text: quantity

text: and

text: quality

text: of

text: data,

text: and

text: the

text: sophistication

text: of

text: α

text: Cir

text: was

text: observed

text: during

text: individual

text: BRITE

text: orbits

text: for

text: to

text: models,

text: has

text: improved

text: dramatically

text: over

text: the

text: subsequent

text: years

text: min,

text: depending

text: on

text: target

text: visibility,

text: the

text: Earth

text: Exclusion

text: Angle

text: (e.g.

text: Kurtz

text: et

text: al.

text: Audard

text: et

text: al.

text: Balona

text: Laney

text: (angular

text: distance

text: between

text: the

text: illuminated

text: limb

text: of

text: the

text: Earth

text: and

text: Kochukhov

text: et

text: al.

text: and

text: many

text: references

text: therein).

text: the

text: star-tracker

text: ﬁeld

text: of

text: view),

text: the

text: required

text: warm-up

text: period

text: for

text: The

text: most

text: complete

text: and

text: recent

text: summary

text: of

text: our

text: knowledge

text: about

text: the

text: star

text: trackers

text: after

text: reacquisition

text: of

text: the

text: target,

text: and

text: passages

text: of

text: the

text: pulsation

text: of

text: α

text: Cir

text: is

text: presented

text: by

text: Bruntt

text: et

text: al.

text: the

text: satellites

text: through

text: the

text: South

text: Atlantic

text: Anomaly

text: (when

text: higher

text: and

text: of

text: the

text: complex

text: atmospheric

text: structure

text: and

text: spectral

text: signatures

text: ﬂuxes

text: of

text: cosmic

text: rays

text: prevent

text: useful

text: signal

text: from

text: the

text: star

text: track-

text: of

text: pulsation

text: by

text: Ryabchikova

text: et

text: al.

text: and

text: Kochukhov

text: et

text: ers).

text: There

text: were

text: also

text: brief

text: interruptions

text: of

text: science

text: data

text: collec-

text: al.

text: tion

text: when

text: data

text: were

text: being

text: transmitted

text: to

text: ground

text: stations.

text: α

text: Cir

text: was

text: included

text: in

text: proposals

text: submitted

text: in

text: and

text: The

text: daily

text: data

text: transfer

text: rate

text: to

text: ground

text: stations

text: in

text: Europe

text: was

text: by

text: several

text: authors

text: of

text: this

text: paper,

text: responding

text: to

text: an

text: Announce-

text: less

text: than

text: planned

text: because

text: of

text: heavy

text: interference

text: with

text: unidentiﬁed

text: ment

text: of

text: Opportunity

text: for

text: targets

text: to

text: be

text: observed

text: by

text: BRITE-

text: transmitters

text: on

text: the

text: ground,

text: which

text: is

text: particularly

text: serious

text: during

text: Constellation.

text: Proposal

text: (WW)

text: focused

text: on

text: roAp

text: stars,

text: the

text: Centaurus

text: ﬁeld

text: observations.

text: For

text: example,

text: data

text: transfer

text: from

text: (DH)

text: on

text: bright

text: stars

text: with

text: the

text: most

text: accurately

text: known

text: Hip-

text: BAb

text: to

text: the

text: station

text: in

text: Graz,

text: Austria,

text: was

text: limited

text: to

text: a

text: maximum

text: parcos

text: parallaxes,

text: and

text: (TL)

text: on

text: the

text: rotation

text: of

text: chemically

text: of

text: min

text: per

text: orbit.

text: peculiar

text: stars

text: (in

text: particular,

text: possible

text: changes

text: in

text: rotation

text: period).

text: Reduction:

text: BRITE

text: standard

text: pipeline

text: BRITE-Constellation

text: photometry

text: During

text: the

text: commissioning

text: of

text: BRITE-Constellation,

text: the

text: particular

text: challenges

text: of

text: on-orbit

text: photometry

text: were

text: assessed.

text: To

text: The

text: BRITE-Constellation

text: is

text: a

text: coordinated

text: mission

text: of

text: ﬁve

text: develop

text: an

text: eﬃcient

text: pipeline

text: for

text: fast

text: and

text: optimum

text: extraction

text: of

text: nanosatellites

text: located

text: in

text: low

text: Earth

text: orbits,

text: each

text: hosting

text: an

text: op-

text: the

text: photometry,

text: a

text: Photometry

text: Tiger

text: Team

text: (PHOTT)

text: was

text: estab-

text: tical

text: telescope

text: of

text: cm

text: aperture

text: feeding

text: an

text: uncooled

text: CCD,

text: and

text: lished,

text: consisting

text: of

text: three

text: groups

text: working

text: independently

text: and

text: observing

text: selected

text: targets

text: in

text: a

text: ﬁeld

text: of

text: view

text: (Weiss

text: et

text: al.

text: pursuing

text: diﬀerent

text: processing

text: and

text: reduction

text: strategies.

text: Each

text: nanosat

text: is

text: equipped

text: with

text: a

text: single

text: ﬁlter;

text: three

text: have

text: Issues

text: investigated

text: included:

text: CCD

text: radiation

text: defects

text: (warm

text: a

text: red

text: ﬁlter

text: nm)

text: and

text: two

text: have

text: a

text: blue

text: ﬁlter

text: (central

text: wave-

text: columns,

text: hot

text: and

text: cold

text: pixels),

text: CCD

text: temperature

text: variations,

text: length

text: nm).

text: The

text: satellites

text: have

text: overlapping

text: coverage

text: of

text: charge-transfer

text: ineﬃciencies,

text: pixel-to-pixel

text: sensitivity

text: varia-

text: the

text: target

text: ﬁelds

text: to

text: provide

text: two-colour,

text: time-resolved

text: photome-

text: tions

text: and

text: intra-pixel

text: inhomogenities,

text: pointing

text: errors,

text: complex

text: try.

text: point

text: spread

text: functions

text: (PSF)

text: varying

text: over

text: the

text: ﬁeld

text: of

text: view

text: The

text: detector

text: is

text: a

text: CCD

text: with

text: million

text: µm

text: and

text: during

text: long

text: observing

text: runs,

text: identiﬁcation

text: of

text: defective

text: im-

text: pixels,

text: a

text: A/D

text: converter,

text: gain

text: of

text: about

text: e

text: /ADU,

text: and

text: a

text: ages

text: (e.g.

text: incomplete

text: PSF

text: in

text: a

text: sub-raster).

text: Competing

text: reduction

text: dark

text: current

text: of

text: /s

text: at

text: C.

text: The

text: saturation

text: limit

text: of

text: the

text: pixels

text: methods

text: employed

text: PSF

text: ﬁtting

text: and

text: aperture

text: photometry

text: as

text: well

text: at

text: this

text: temperature

text: is

text: ADU,

text: and

text: the

text: response

text: is

text: linear

text: as

text: various

text: threshold

text: criteria

text: for

text: elimination

text: of

text: bad

text: data

text: points

text: up

text: to

text: about

text: ADU.

text: The

text: pixel

text: size

text: is

text: projected

text: at

text: the

text: (outliers),

text: various

text: ﬁt-quality

text: parameters,

text: median

text: vs.

text: mean,

text: etc.

text: sky.

text: The

text: methods

text: will

text: be

text: described

text: in

text: detail

text: by

text: Popowicz

text: et

text: al.

text: (in

text: The

text: ﬁve

text: nanosats

text: are

text: designated

text: BRITE-Austria

text: (BAb),

text: prep.).

text: UniBRITE

text: (UBr),

text: BRITE-Lem

text: (BLb),

text: BRITE-Hewelius

text: (BHr),

text: This

text: investigation

text: of

text: α

text: Cir

text: is

text: based

text: on

text: the

text: reduction

text: pipeline

text: and

text: BRITE-Toronto

text: (BTr),

text: where

text: the

text: last

text: letter

text: indicates

text: the

text: ﬁl-

text: developed

text: by

text: PHOTT

text: member

text: Adam

text: Popowicz

text: via

text: aperture

text: ter

text: on

text: board

text: the

text: satellite

text: ("b"

text: blue,

text: "r"

text: red).

text: A

text: sixth

text: nanosat,

text: photometry.

text: Figure

text: illustrates

text: the

text: impact

text: of

text: two

text: optimisation

text: BRITE-Montréal

text: (BMb),

text: did

text: not

text: detach

text: from

text: its

text: launch

text: vehicle.

text: parameters

text: on

text: the

text: light-curve

text: quality.

text: The

text: parameters

text: are

text: the

text: The

text: nanosats

text: Bb

text: and

text: UBr

text: were

text: launched

text: ﬁrst,

text: followed

text: by

text: BLb,

text: threshold

text: employed

text: for

text: bad-pixel

text: identiﬁcation

text: in

text: the

text: mean

text: dark

text: BTr,

text: and

text: BHr.

text: frame

text: (T

text: and

text: the

text: aperture

text: radius.

text: To

text: assess

text: the

text: light-curve

text: α

text: Cir

text: photometry

text: quality,

text: wbeadcomputed

text: the

text: median

text: absolute

text: deviations

text: (MAD)

text: of

text: results

text: obtained

text: for

text: individual

text: satellite

text: orbits,

text: as

text: a

text: robust

text: mea-

text: α

text: Cir

text: was

text: one

text: of

text: the

text: targets

text: in

text: the

text: Centaurus

text: ﬁeld,

text: the

text: sec-

text: sure

text: of

text: scatter

text: in

text: the

text: presence

text: of

text: outliers.

text: The

text: ﬁnal

text: quality

text: mea-

text: ond

text: long-term

text: pointing

text: of

text: BRITE-Constellation

text: after

text: the

text: Orion

text: sure

text: (Q

text: is

text: of

text: MADs

text: deﬁned

text: as

text: follows:

text: ﬁeld.

text: The

text: early

text: data

text: were

text: obtained

text: during

text: science

text: commission-

text: M

text: ing

text: of

text: the

text: mission,

text: which

text: meant

text: interruptions

text: in

text: science

text: data

text: collection

text: for

text: engineering

text: tests.

text: Even

text: so,

text: signiﬁcant

text: gaps

text: in

text: the

text: M√AD

text: time

text: series

text: were

text: infrequent,

text: caused

text: mainly

text: by

text: accidental

text: space-

text: QM

text: median

text: i

text: craft

text: mode

text: changes

text: from

text: ﬁne-to-coarse

text: pointing

text: (or

text: tumbling)

text: Ni

text: with

text: recovery

text: times

text: shorter

text: than

text: hours.

text: New

text: nanosats

text: joined

text: Article

text: number,

text: page

text: of

text: ##LTLine##

text: ##LTLine##