text: 6. On Hankel non-negative deﬁnite extendable sequences S T (b) κj=2k R(sj) ⊆ R(s2k) and N (s2k) ⊆ κj=2k N (sj) for all k ∈ N0 with 2k ≤ κ. Lemma 6.2 (cf. [14, Lemmn a 2.7]). Let ∈ N. Then o n Hq≥,2,en = (sj)2j=n0 ∈ Hq≥,2n : R(yn+1,2n) ⊆ R(Hn−1) . Remark 6.3. For each n ∈ N0 we see from [14, Proposition 2.24] that H>2,e = H>2 . q, n q, n Proposition 6.4 (cf. [14, Prnoposition 2.13]). Let ∈ N. Then o n Hq≥,2,en = (sj)2j=n0 ∈ Hq≥,2n : N (Ln−1) ⊆ N (Ln) . Proposition 6.4 indicates an essential diﬀerence between the set H≥,e and the set of 2 ﬁnite non-negative deﬁnite sequences from Cq×q. Proposition 6.4 tells qu,snthat there is an inclusion of the null spaces of the consecutive associated Schur complements introduced in Notation 3.4(c), whereas [16, Lemma 6] shows that the corresponding consecutive Schur complements associated with a (Toeplitz-)non-negative deﬁnite sequence from Cq×q are even decreasing with respect to the Löwener semi-ordering in the set Cq×q. Clearly, this H implies the inclusion for the corresponding null spaces. LCReq×mq.maTh6e.n5 (⊆(scjfR.)2j[=n1+041, L∈emHmq≥,2,aen+21.16if])a.nLdetonnly∈ifN0(sajn)2jd=n0let∈(sHj)q≥,2j2=nn+0,1sb∗2en+a1 s=equse2nn+ce1,fraonmd ( ) ( ). y +1 2 +1 H n , n n RP∗erompaor=skiti6o.n6.2L..2e4t] nwe∈sNee0 tahnadt l(estj)(2js=nj+0)2j1=n+∈01 Hbq>e,2,ean+s1eqifueanncde ofrnolmy iCf q(×sjq).2j=nT0h∈enHfrq>o,2mn a[1n4d, s2n+1 s2n+1 Propositionn 6.7 (cf. [14, Proposition 2.22(a)]o). Lent n ∈ N0 and let (sj)2j=no+01 ∈ Hq≥,2,en+1. Then s2n+2 ∈ Cq×q : (sj)2j=n+02 ∈ Hq≥,2n+2 = X ∈ CqH×q : X ≥ Θn+1 . Propositionn 6.8. Let n ∈ N0 and let (sj)2j=n+01o∈ Hnq>,2,en+1. Then o s2n+2 ∈ Cq×q : (sj)2j=n+02 ∈ Hq>,2n+2 = X ∈ CqH×q : X > Θn+1 . Proof. This follows from Remark 6.6 and [14, Proposition 2.24]. S T WeRdeno⊆te Rby Dp×q,κNthe s⊆et of aNll sequences (sj)κj=0 from Cp×q which fulﬁll κj=0 (sj) (s0) and (s0) κj=0 (sj). Proposition 6.9 ( [22, Proposition 4.24]). Let m ∈ N0. Then H≥,e ⊆ D × . q,m q q,m 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: On

text: Hankel

text: non-negative

text: deﬁnite

text: extendable

text: sequences

text: S

text: T

text: (b)

text: R(sj)

text: and

text: N

text: N

text: (sj)

text: for

text: all

text: k

text: with

text: κ.

text: Lemma

text: (cf.

text: Lemmn

text: a

text: Let

text: N.

text: Then

text: o

text: n

text: Remark

text: For

text: each

text: n

text: we

text: see

text: from

text: Proposition

text: that

text: q,

text: n

text: q,

text: n

text: Proposition

text: (cf.

text: Prnoposition

text: Let

text: N.

text: Then

text: o

text: n

text: N

text: N

text: (Ln)

text: Proposition

text: indicates

text: an

text: essential

text: diﬀerence

text: between

text: the

text: set

text: H≥,e

text: and

text: the

text: set

text: of

text: ﬁnite

text: non-negative

text: deﬁnite

text: sequences

text: from

text: Cq×q.

text: Proposition

text: tells

text: qu,snthat

text: there

text: is

text: an

text: inclusion

text: of

text: the

text: spaces

text: of

text: the

text: consecutive

text: associated

text: Schur

text: complements

text: introduced

text: in

text: Notation

text: whereas

text: Lemma

text: shows

text: that

text: the

text: corresponding

text: consecutive

text: Schur

text: complements

text: associated

text: with

text: a

text: (Toeplitz-)non-negative

text: deﬁnite

text: sequence

text: from

text: Cq×q

text: are

text: even

text: decreasing

text: with

text: respect

text: to

text: the

text: Löwener

text: semi-ordering

text: in

text: the

text: set

text: Cq×q.

text: Clearly,

text: this

text: H

text: implies

text: the

text: inclusion

text: for

text: the

text: corresponding

text: spaces.

text: y

text: H

text: n

text: n

text: n

text: tahnadt

text: ofrnolmy

text: iCf

text: Propositionn

text: (cf.

text: Proposition

text: Lent

text: n

text: and

text: let

text: Then

text: Cq×q

text: X

text: CqH×q

text: X

text: Propositionn

text: Let

text: n

text: and

text: let

text: Then

text: o

text: Cq×q

text: X

text: CqH×q

text: X

text: Proof.

text: This

text: follows

text: from

text: Remark

text: and

text: Proposition

text: S

text: T

text: WeRdeno⊆te

text: Rby

text: Dp×q,κNthe

text: s⊆et

text: of

text: aNll

text: sequences

text: from

text: Cp×q

text: which

text: fulﬁll

text: (sj)

text: and

text: (sj).

text: Proposition

text: Proposition

text: Let

text: m

text: Then

text: H≥,e

text: D

text: q,m

text: q

text: q,m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7. Algebraic approach to Proposition 4.5 on the sections of [ ]-Hausdorﬀ non-negative α, β deﬁnite sequences Proof. Using Remark 7.3, we conclude ∆∗ ∆ = , ∆∗ ∇ = , ∇∗ ∆ = , and ∇∗ p,n+1H∇n+1 q,=n+1 . HFnor eapc,nh+1H∈n+C1anqd,n+ea1ch Kn∈ C,pt,nh+is1Himnp+l1iesq,(n7+.51) anKdn(7.6). p,n+1Hn+1 q,n+1 Gn α β Lemma 7.5. Let α ∈ R, β ∈ (α, ∞), and n ∈ N0. Then Fq≥,2n+1,α,β ⊆ Hq≥,2,en+1. Proof. This can be seen from [13, Lemma 2.21]. Remark 7.6. Let ∈ R, let ∈ ( ∞), and let ∈ N . Then, we see from Lemma 7.5 α β α, n 0 F≥ ⊆ D and Proposition 6.9 that q,2n+1,α,β q×q,2n+1. PfrormopCosq×itqi.onTh7e.n7:. Let α ∈ R, let β ∈ (α, ∞), let m ∈ N0, and let (sj)mj=0 be a sequence (a) If (sj)mj=0 ∈ Fq≥,m,α,β, then (sj)‘j=0 ∈ Fq≥,‘,α,β for each ‘ ∈ Z0,m. (b) If (sj)mj=0 ∈ Fq>,m,α,β, then (sj)‘j=0 ∈ Fq>,‘,α,β for each ‘ ∈ Z0,m. ≥ ∈ F≥ (PFsrjo)omj=f.0 T∈hFe q>c,mas,αe,βm). =It0isissutﬃrivciieanl.t Atosspurmoveetthhaatt m(sj)mj=1−01anbdelo(snjg)smj=t0o Fq≥,mq−,m1,,αα,,ββ ((rreesspp.. > −1 ). qF,mirst,αw,βe suppose that there is an integer ∈ N such that = 2 +1. Then Lemma 4.15 n 0 m n yields that the matrix is non-negative Hermitian (resp. positive Hermitian). If = 0, H n ∈ F≥ n ∈ F tNhoewn (wsej)a2j=nss0ume qt,h2na,tα,β ≥(re1s.p.In(svj)ie2j=nw0of Leq>m,2nm,αa,β4).i1m7,mwedeihataevley fo∗ll=ows ffroormalDl eﬁ∈nitZion 4.2.. n s s j 0 2 +1 R j j ⊆ R , n The combination of Lemmata 7.5 and 6.5 provides us ( ) ( ) and y +1 2 +1 H N ( ) = N ( ∗) = R( )⊥ ⊆ R( )⊥ = N ( ∗ n) =, Nn ( ).n From LemHmna 7.2 weHgnet thenHn yn+1,2n+1 h yn+1,2n+i1 zn+1,2n+1 − ∗ ± ( H ) −1 = (β α)∆ ( H) (H ) ∆ . (7.7) α β n q,n α n β n q,n Because of (sj)2j=n+01 ∈ Fq≥,2n+1,α,β (resp. (sj)2j=n+01 ∈ Fq>,2n+1,α,β), Deﬁnition 4.2(a) shows that the matrices ( ) and ( ) are both non-negative Hermitian (resp. positive H H α n β n ± Hermitian). Thus, Lemma B.2 (resp. Remark B.1) implies that the matrix ( ) ( ) H H α− n β n is non-negative Hermitian (resp. positive Hermitian) as well. Because of 0 and β α > rank ∆ = nq, then (7.7) und Remark A.2 yield that the matrix ( H ) −1 is non- q,n α β n negative Hermitian (resp. positive Hermitian). Consequently, in view of Deﬁnition 4.2(b), F≥ F≥ weItsereemthaaints(stjo)2jc=no0nsbideleorntghsetcoaseq,t2hna,αt,β (r=es2p. hqo,l2dns,αt,βru).e with some ∈ N. By virtue of m n n ∈ F≥ ∈ F (sj)2j=n0 q,2n,α,β (resp. (sj)2j=n0 q>,2n,α,β) and Deﬁnition 4.2(b), the matrix (αHβ)n−1 is non-negative Hermitian (resp. positive Hermitian). Lemma 7.4 and yield β > α ( H) −1 ≥ (β − α)−1( H ) −1 and (H ) −1 ≥ (β − α)−1( H ) −1. Hence, the matrices α n α β n β n α β n ( H) −1 and (H ) −1 are both non-negative Hermitian (resp. positive Hermitian). Thus, Dαeﬁnnition 4.2(a)βimn plies (sj)2j=n−01 ∈ Fq≥,2n−1,α,β (resp. (sj)2j=n−01 ∈ Fq>,2n−1,α,β). 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Algebraic

text: approach

text: to

text: Proposition

text: on

text: the

text: sections

text: of

text: ]-Hausdorﬀ

text: non-negative

text: α,

text: β

text: deﬁnite

text: sequences

text: Proof.

text: Using

text: Remark

text: we

text: conclude

text: and

text: HFnor

text: Kn∈

text: Gn

text: α

text: β

text: Lemma

text: Let

text: α

text: R,

text: β

text: (α,

text: and

text: n

text: Then

text: Proof.

text: This

text: can

text: be

text: seen

text: from

text: Lemma

text: Remark

text: Let

text: R,

text: let

text: and

text: let

text: N

text: Then,

text: we

text: see

text: from

text: Lemma

text: α

text: β

text: α,

text: n

text: F≥

text: D

text: and

text: Proposition

text: that

text: Let

text: α

text: R,

text: let

text: β

text: (α,

text: let

text: m

text: and

text: let

text: be

text: a

text: sequence

text: (a)

text: If

text: Fq≥,m,α,β,

text: then

text: Fq≥,‘,α,β

text: for

text: each

text: (b)

text: If

text: Fq>,m,α,β,

text: then

text: Fq>,‘,α,β

text: for

text: each

text: F≥

text: T∈hFe

text: q>c,mas,αe,βm).

text: Atosspurmoveetthhaatt

text: ((rreesspp..

text: −1

text: qF,mirst,αw,βe

text: suppose

text: that

text: there

text: is

text: an

text: integer

text: N

text: such

text: that

text: Then

text: Lemma

text: n

text: m

text: n

text: yields

text: that

text: the

text: matrix

text: is

text: non-negative

text: Hermitian

text: (resp.

text: positive

text: Hermitian).

text: If

text: H

text: n

text: F≥

text: n

text: F

text: tNhoewn

text: fo∗ll=ows

text: ffroormalDl

text: eﬁ∈nitZion

text: n

text: s

text: s

text: j

text: R

text: j

text: j

text: R

text: n

text: The

text: combination

text: of

text: Lemmata

text: and

text: provides

text: us

text: and

text: y

text: H

text: N

text: N

text: R(

text: R(

text: N

text: n)

text: Nn

text: ).n

text: From

text: LemHmna

text: weHgnet

text: thenHn

text: h

text: H

text: −1

text: (β

text: α)∆

text: H)

text: (H

text: α

text: β

text: n

text: q,n

text: α

text: n

text: β

text: n

text: q,n

text: Because

text: of

text: (resp.

text: Deﬁnition

text: shows

text: that

text: the

text: matrices

text: and

text: are

text: both

text: non-negative

text: Hermitian

text: (resp.

text: positive

text: H

text: H

text: α

text: n

text: β

text: n

text: Hermitian).

text: Thus,

text: Lemma

text: (resp.

text: Remark

text: implies

text: that

text: the

text: matrix

text: H

text: H

text: α−

text: n

text: β

text: n

text: is

text: non-negative

text: Hermitian

text: (resp.

text: positive

text: Hermitian)

text: as

text: well.

text: Because

text: of

text: and

text: β

text: α

text: rank

text: nq,

text: then

text: und

text: Remark

text: yield

text: that

text: the

text: matrix

text: H

text: −1

text: is

text: non-

text: q,n

text: α

text: β

text: n

text: negative

text: Hermitian

text: (resp.

text: positive

text: Hermitian).

text: Consequently,

text: in

text: view

text: of

text: Deﬁnition

text: F≥

text: F≥

text: with

text: some

text: N.

text: By

text: virtue

text: of

text: m

text: n

text: n

text: F≥

text: F

text: (resp.

text: and

text: Deﬁnition

text: the

text: matrix

text: is

text: non-negative

text: Hermitian

text: (resp.

text: positive

text: Hermitian).

text: Lemma

text: and

text: yield

text: β

text: α

text: H)

text: −1

text: (β

text: H

text: −1

text: and

text: (H

text: −1

text: (β

text: H

text: Hence,

text: the

text: matrices

text: α

text: n

text: α

text: β

text: n

text: β

text: n

text: α

text: β

text: n

text: H)

text: −1

text: and

text: (H

text: −1

text: are

text: both

text: non-negative

text: Hermitian

text: (resp.

text: positive

text: Hermitian).

text: Thus,

text: Dαeﬁnnition

text: plies

text: (resp.

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: 7. Algebraic approach to Proposition 4.5 on the sections of [ ]-Hausdorﬀ non-negative α, β deﬁnite sequences sCeqoureonllcaerfyrom7.8C. qL×eq.t Tαh∈en:R, let β ∈ (α, ∞), let κ ∈ N0 ∪ {∞}, and let (sj)κj=0 be a (a) I{f( (Ls)j)κj,=(L0 ∈) F}q≥⊆,κ,αC,βq≥,×qthfeonr Lalnl ∈n C∈q≥N×q0 fwoirthal2lnn+∈1N≤0 κw,itahn2dn(≤Lκ), fu∈rtCheq≥×rmq oforer allα ∈n N βwinth 2 + 2 ≤ . α β n n 0 n κ (b) I{f( (Ls)j)κj,=(L0 ∈) F}q>⊆,κ,αC,βq,×qthfeonr Lalnl ∈n C∈q>N×q0 fwoirthal2lnn+∈1N≤0 κw,itahn2dn(≤Lκ), fu∈rtCheq×rmq oforer allα ∈n N βwinth 2 +>2 ≤ . α β n > n 0 n κ Proof. Use Proposition 7.7, Deﬁnitions 4.2 and 4.7, and Remark 3.6. Corollary 7.9. Let α ∈ R, let β ∈ (α, ∞), and let m ∈ N0. Then F≥,e ⊆ F≥ and F>,e ⊆ F> . q,m,α,β q,m,α,β q,m,α,β q,m,α,β Proof. Use Proposition 7.7. Proposition 7.10. Let α ∈ R, let β ∈ (α, ∞), and let m ∈ N0. Then F≥ ⊆ H≥,e F ⊆ H q,m,α,β q,m and > >,e . q,m,α,β q,m Proof. We ﬁrst show F≥ ⊆ H≥,e . Because of H≥0,e = H≥0, the case m = 0 is trivial. If = 2 + 1 for someq,m∈,α,Nβ , thqe,nmLemma 7.5 yielqd, s the aq,ssertion. m n n 0 F≥Now let m = 2n with some n ∈ N. We consider an arbitrary sequen∈ceH(s≥j)2j=n0 ∈ {(q,2Hn,α),β.−1,TGhe−n1}De⊆ﬁniCtin≥oqn×n4q..2(bT)haunsd, RReemmaarrkk A7..31 pyrioevldidseRu(sG(s−j1)2j=+n0 ( H )q,2−n1)an=d Rto(αGFnq≥β,−2n1n−)+1,αR,β(.n(αCHoβn)sne−q1u).enAtlcyc,orDdeinﬁgnittoioPnro4p.2o(saiRt)ioynie7l.d7s(a{)(,αtHhe)sne−q1u,ne(nHcβe)(ns−Rj)1α2j}=n−0⊆β1 nbCen≥lqo×nngqs. Hence, using Remark A.1 again, we conclude (( H) −1 + (H ) −1) = (( H) −1) + R((H ) −1). Since Remark 4.14 shows that H α−1 =n −1 [( βHn) −1 + (H α) −1n] and β n n β α α n β n G −1 + ( H ) −1 = −1 [β2( H(cid:16) ) −1 + α2(cid:17)(H ) −1], we get then n α β n β α α n β n (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) R ⊆ R R (G −1) (G −1) + ( H ) −1 n n α β n (cid:16) (cid:17) = R β2( H) −1 + α2(H ) −1 ⊆ R β2( H) −1 + R α2(H ) −1 α n β n α n β n ⊆ R R R (( H) −1) + (H ) −1 = (H −1). α n β n n × Taking into account that y +1 2 is the last nq q block column of G −1, we infer R(Nyno+w1,2wne) ⊆shoRw(HFn>−1). Th⊆unsH, L>,e,nem. mIaf 6m.2 =yie2ldns w(sijth)2j=ns0om∈eHnq≥,2,e∈n. N0, thisnfollows from Remark 6.3 and Deqﬁ,mn,iαt,iβon 4.2(qb,m). If = 2 + 1 with some ∈ N , this follows from m n n 0 Remark 6.6 in view Lemmata 4.15 and 4.17. 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algebraic

text: approach

text: to

text: Proposition

text: on

text: the

text: sections

text: of

text: ]-Hausdorﬀ

text: non-negative

text: α,

text: β

text: deﬁnite

text: sequences

text: qL×eq.t

text: Tαh∈en:R,

text: let

text: β

text: (α,

text: let

text: κ

text: and

text: let

text: be

text: a

text: (a)

text: I{f(

text: F}q≥⊆,κ,αC,βq≥,×qthfeonr

text: Lalnl

text: ∈n

text: fu∈rtCheq≥×rmq

text: oforer

text: allα

text: ∈n

text: N

text: βwinth

text: α

text: β

text: n

text: n

text: n

text: κ

text: (b)

text: I{f(

text: F}q>⊆,κ,αC,βq,×qthfeonr

text: Lalnl

text: ∈n

text: fu∈rtCheq×rmq

text: oforer

text: allα

text: ∈n

text: N

text: βwinth

text: α

text: β

text: n

text: n

text: n

text: κ

text: Proof.

text: Use

text: Proposition

text: Deﬁnitions

text: and

text: and

text: Remark

text: Corollary

text: Let

text: α

text: R,

text: let

text: β

text: (α,

text: and

text: let

text: m

text: Then

text: F≥,e

text: F≥

text: and

text: F>,e

text: F>

text: q,m,α,β

text: q,m,α,β

text: q,m,α,β

text: q,m,α,β

text: Proof.

text: Use

text: Proposition

text: Proposition

text: Let

text: α

text: R,

text: let

text: β

text: (α,

text: and

text: let

text: m

text: Then

text: F≥

text: H≥,e

text: F

text: H

text: q,m,α,β

text: q,m

text: and

text: >,e

text: q,m,α,β

text: q,m

text: Proof.

text: We

text: ﬁrst

text: show

text: F≥

text: H≥,e

text: Because

text: of

text: the

text: case

text: m

text: is

text: trivial.

text: If

text: for

text: someq,m∈,α,Nβ

text: thqe,nmLemma

text: yielqd,

text: s

text: the

text: aq,ssertion.

text: m

text: n

text: n

text: F≥Now

text: let

text: m

text: with

text: some

text: n

text: N.

text: We

text: consider

text: an

text: arbitrary

text: RReemmaarrkk

text: H

text: nbCen≥lqo×nngqs.

text: Hence,

text: using

text: Remark

text: again,

text: we

text: conclude

text: H)

text: −1

text: (H

text: H)

text: R((H

text: Since

text: Remark

text: shows

text: that

text: H

text: α−1

text: =n

text: −1

text: βHn)

text: −1

text: (H

text: α)

text: and

text: β

text: n

text: n

text: β

text: α

text: α

text: n

text: β

text: n

text: G

text: −1

text: H

text: −1

text: −1

text: −1

text: we

text: get

text: then

text: n

text: α

text: β

text: n

text: β

text: α

text: α

text: n

text: β

text: n

text: R

text: R

text: R

text: (G

text: (G

text: H

text: −1

text: n

text: n

text: α

text: β

text: n

text: R

text: H)

text: −1

text: −1

text: R

text: H)

text: −1

text: R

text: −1

text: α

text: n

text: β

text: n

text: α

text: n

text: β

text: n

text: R

text: R

text: R

text: H)

text: (H

text: −1

text: (H

text: α

text: n

text: β

text: n

text: n

text: Taking

text: into

text: account

text: that

text: y

text: is

text: the

text: last

text: nq

text: q

text: block

text: column

text: of

text: G

text: we

text: infer

text: Th⊆unsH,

text: L>,e,nem.

text: mIaf

text: thisnfollows

text: from

text: Remark

text: and

text: Deqﬁ,mn,iαt,iβon

text: If

text: with

text: some

text: N

text: this

text: follows

text: from

text: m

text: n

text: n

text: Remark

text: in

text: view

text: Lemmata

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 8. On Stieltjes non-negative deﬁnite sequences 8. On Stieltjes non-negative deﬁnite sequences × In this section, we discuss two dual classes of ﬁnite sequences of complex matrices q q F≥ which turn out to be intimately connected with the class . The ﬁrst and second q,m,α,β types of this classes will be associated with the left endpoint and the right endpoint α β of the interval [ ], resp. These classes were already studied by the authors in earlier α, β work (see [14,20,21]). First we consider the class of sequences connected with the left endpoint of [ ]. α α, β Deﬁnition 8.1. Let ∈ R and let ∈ N . α n 0 (a) Let (sj)2j=n+01 be a sequence∞from Cq×q. Then (sj)2j=n+01 is called [α, ∞)-Stieltjes non- negative deﬁnite (resp. [α, )-Stieltjes positive deﬁnite) if both sequences (sj)2j=n0 and ((αs)j)2j=n0 are Hankel non-negative deﬁnite (resp. Hankel positive deﬁnite). (b) Let (sj)2j=n0 be a sequence ∞from Cq×q. Then (sj)2j=n0 is called [α, ∞)-Stieltjes non- neg≥at1iv,eadlseoﬁn(i(te ()re)2s(pn.−[1α), is )H-Satnikeletljensopno-snietigvaetidveeﬁndietﬁen)iitfe(s(rje)s2j=np0. Hanadn,kienl tphoesictaivsee n αs j j=0 deﬁnite). If m ∈ N0, then, K≥ (resp. K> ) stands for the set of all [α, ∞)-Stieltjes non-negative deﬁnite (resp. [α, ∞q,m)-,αStieltjes pqo,msi,tαive deﬁnite) sequences (sj)mj=0 from Cq×q. L∞et α ∈ R, let m ∈ N0, and let (sj)mj=0 be a sequence from∞Cq×q. Then (sj)mj=0 is called [ )-Stieltjes non-negative deﬁnite extendable (resp. [ )-Stieltjes positive deﬁnite α, α, (esxjt)emjn=+d0a1b∈leK) q>if,mt+h1e,αre).eWxiestdseanomteabtryixKq≥s∞,mm,e+,α1(r∈esCp.q×Kqq>,sm,eu,αch) tthheastet(sojf)amj=ll+0[1α∈, ∞K)q≥-S,mti+e1lt,αjes(rneosnp-. negative deﬁnite extendable (resp. [ )-Stieltjes positive deﬁnite extendable) sequences α, (sj)mj=0 from Cq×q. The importance of the class Kq≥,m,e,α in the context of moment problems is caused by the following observation: TfrohmeoCreqm×q.8T.2he(n[1M3, q≥T[h[αe,o∞re)m; (1sj.3)mj])=.0,L=et] 6=α ∈∅ iRf ,amnd∈onNly0,iaf n(dsjl)emjt=0(s∈j)Kmj=q≥0,m,eb,eα.a sequence T(Lαhesem)nj m∈sjaC∈qH8×C.3qqH×f(oq[r2fo0ar,llaLljelm∈jm∈Za0Z,2m0.,−9m(1aa)an–nd(dbs)(2]α)k.s∈)L2ekCtq≥∈α×Cq∈fq≥o×Rrq,aflloeltrkma∈ll∈NkN0∈w0,Nitah0nw2dkilteh≤t 2(mskj.+)mjF=1u0r≤t∈hemKrm.q≥,mor,αe,. Remark 8.4. Let α ∈ R and let m ∈ N0. Then K≥,e ⊆ K≥ and, in the case m ∈ N, furthermore K≥,e 6= K≥ . q,m,α q,m,α q,m,α q,m,α Ralelm‘a∈rkZ08.5−. 1L. et α ∈ R, let m ∈ N, and let (sj)mj=0 ∈ Kq≥,m,α. Then (sj)‘j=0 ∈ Kq≥,‘,e,α for ,m (Rsejm)‘j=ar0k∈8K.6q≥.,‘,e,Lαe(treαsp∈. KRq>,,‘,lαe)t fmor ∈allN‘0∈, aZn0d,κ.let (sj)mj=0 ∈ Kq≥,m,e,α (resp. Kq>,m,α). Then 21

text: On

text: Stieltjes

text: non-negative

text: deﬁnite

text: sequences

text: On

text: Stieltjes

text: non-negative

text: deﬁnite

text: sequences

text: In

text: this

text: section,

text: we

text: discuss

text: two

text: dual

text: classes

text: of

text: ﬁnite

text: sequences

text: of

text: complex

text: matrices

text: q

text: q

text: F≥

text: which

text: turn

text: out

text: to

text: be

text: intimately

text: connected

text: with

text: the

text: class

text: The

text: ﬁrst

text: and

text: second

text: q,m,α,β

text: types

text: of

text: this

text: classes

text: will

text: be

text: associated

text: with

text: the

text: left

text: endpoint

text: and

text: the

text: right

text: endpoint

text: α

text: β

text: of

text: the

text: interval

text: resp.

text: These

text: classes

text: were

text: already

text: studied

text: by

text: the

text: authors

text: in

text: earlier

text: α,

text: β

text: work

text: (see

text: First

text: we

text: consider

text: the

text: class

text: of

text: sequences

text: connected

text: with

text: the

text: left

text: endpoint

text: of

text: α

text: α,

text: β

text: Deﬁnition

text: Let

text: R

text: and

text: let

text: N

text: α

text: n

text: (a)

text: Let

text: be

text: a

text: sequence∞from

text: Cq×q.

text: Then

text: is

text: called

text: [α,

text: ∞)-Stieltjes

text: non-

text: negative

text: deﬁnite

text: (resp.

text: [α,

text: )-Stieltjes

text: positive

text: deﬁnite)

text: if

text: both

text: sequences

text: and

text: are

text: Hankel

text: non-negative

text: deﬁnite

text: (resp.

text: Hankel

text: positive

text: deﬁnite).

text: (b)

text: Let

text: be

text: a

text: sequence

text: ∞from

text: Cq×q.

text: Then

text: is

text: called

text: [α,

text: ∞)-Stieltjes

text: non-

text: is

text: Hanadn,kienl

text: tphoesictaivsee

text: n

text: αs

text: j

text: deﬁnite).

text: If

text: m

text: then,

text: K≥

text: (resp.

text: K>

text: stands

text: for

text: the

text: set

text: of

text: all

text: [α,

text: ∞)-Stieltjes

text: non-negative

text: deﬁnite

text: (resp.

text: [α,

text: ∞q,m)-,αStieltjes

text: pqo,msi,tαive

text: deﬁnite)

text: sequences

text: from

text: Cq×q.

text: L∞et

text: α

text: R,

text: let

text: m

text: and

text: let

text: be

text: a

text: sequence

text: from∞Cq×q.

text: Then

text: is

text: called

text: )-Stieltjes

text: non-negative

text: deﬁnite

text: extendable

text: (resp.

text: )-Stieltjes

text: positive

text: deﬁnite

text: α,

text: α,

text: negative

text: deﬁnite

text: extendable

text: (resp.

text: )-Stieltjes

text: positive

text: deﬁnite

text: extendable)

text: sequences

text: α,

text: from

text: Cq×q.

text: The

text: importance

text: of

text: the

text: class

text: Kq≥,m,e,α

text: in

text: the

text: context

text: of

text: moment

text: problems

text: is

text: caused

text: by

text: the

text: following

text: observation:

text: q≥T[h[αe,o∞re)m;

text: iRf

text: sequence

text: T(Lαhesem)nj

text: Remark

text: Let

text: α

text: R

text: and

text: let

text: m

text: Then

text: K≥,e

text: K≥

text: and,

text: in

text: the

text: case

text: m

text: N,

text: furthermore

text: K≥,e

text: K≥

text: q,m,α

text: q,m,α

text: q,m,α

text: q,m,α

text: et

text: α

text: R,

text: let

text: m

text: N,

text: and

text: let

text: Kq≥,m,α.

text: Then

text: Kq≥,‘,e,α

text: for

text: ,m

text: KRq>,,‘,lαe)t

text: fmor

text: Kq≥,m,e,α

text: (resp.

text: Kq>,m,α).

text: Then


text: 10. On the structure of [ ]-Hausdorﬀ non-negative deﬁnite sequences α, β closed matricial subinterval of Cq×q the endpoints of which are completely determined H by the preceding elements of the sequence. First we introduce two types of matricial intervals: Let A, B ∈ CqH×q. Then we set [A, B] := {X ∈ CqH×q : A ≤ X ≤ B} and (A, B) := {X ∈ CqH×q : A < X < B}. The matrix (resp. ) is called the matricial left (resp. right) endpoint of [ ] or A B A, B ( ). The matrix := 1( + ) (resp. := − ) is called the matricial midpoint A, B C 2 A B D B A (resp. length) of [ ] or ( ). A, B A, B The following result indicates that closed and open matricial intervals are intimately related to closed and open matricial matrix balls. Lemma 10.1. Let A, B ∈ CqH×q and let D := B − A. Then: (a) [A, B] 6= ∅ if and only if D ∈nCq≥×q. In this case, o √ √ ∈ [A, B] = A + DK D : K [0 × , I ] . q q q (b) (A, B) 6= ∅ if and only if D ∈nCq×q. In this case, o > √ √ ∈ (A, B) = A + DK D : K (0 × , I ) . q q q PInropoaf.rt(iacu) laIfr,[AA, B≤] B6=, ∅i,. et.h, eDn t∈heCreq≥×eqx.istCsoanvmeras√terlyix, aX√ssu∈mCe qHD×q ∈wiCthq≥×Aq. ≤ObXvio≤usBly,. 6 ∅ ∈ [0 × , I ] = . Let K [0 × , I ]. The√n X :√= A + DK D is Hermitian. √By vir√tue of q q q q q −q ≥ − − R√emark A.2,√furthermore X A = DK D 0 × and B X = D DK D = − ≥ ∈ q q 6 ∅ D(I K) D 0 × . Hence, X [A, B].√ In particul√ar, [A, B] = . Now let ∈ q q q † − † [ ]. In view of Remark A.8, then := ( ) is Hermitian and we X A, B K D X A D have ≤ − ≤ − 0 × X A B A = D. (10.1) q q R − ⊆ R N ⊆ N − Hence, ( ) ( ) and ( )√ ( )√by virtue of Remark A.11. According to X A D D X A R − ⊆ R N ⊆ N − Remark A.10, thus ( √) √( ) and ( ) ( ). U√sing R√emarks A.5 X A D D X A − and A.6, we obtain then DK D = X A. Henc√e, X =√ A + DK D. With ≤ ≤ † † Remark A.2, we conclude from (10.1) that 0 × K D D D . Using Remarks A.8 q q and A.3 and Proposition A.4, we have √ √ √ √ √ √ √ √ √ √ √ √ † † = † † = † † = † = √ ≤ D D D D D D D D D D D D D PR( D) Iq. ≤ ≤ ∈ Thus, 0 × K I , i. e., K [0 × , I ]. q q q q q q (b) Since every positive Hermitian matrix is invertible, this can be shown more easy in a similar manner. From Lemma 10.1 we conclude the following result: 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: On

text: the

text: structure

text: of

text: ]-Hausdorﬀ

text: non-negative

text: deﬁnite

text: sequences

text: α,

text: β

text: closed

text: matricial

text: subinterval

text: of

text: Cq×q

text: the

text: endpoints

text: of

text: which

text: are

text: completely

text: determined

text: H

text: by

text: the

text: preceding

text: elements

text: of

text: the

text: sequence.

text: First

text: we

text: introduce

text: two

text: types

text: of

text: matricial

text: intervals:

text: Let

text: A,

text: B

text: CqH×q.

text: Then

text: we

text: set

text: [A,

text: B]

text: {X

text: CqH×q

text: A

text: X

text: B}

text: and

text: (A,

text: B)

text: {X

text: CqH×q

text: A

text: X

text: B}.

text: The

text: matrix

text: (resp.

text: is

text: called

text: the

text: matricial

text: left

text: (resp.

text: right)

text: endpoint

text: of

text: or

text: A

text: B

text: A,

text: B

text: The

text: matrix

text: (resp.

text: is

text: called

text: the

text: matricial

text: midpoint

text: A,

text: B

text: C

text: A

text: B

text: D

text: B

text: A

text: (resp.

text: length)

text: of

text: or

text: A,

text: B

text: A,

text: B

text: The

text: following

text: result

text: indicates

text: that

text: closed

text: and

text: open

text: matricial

text: intervals

text: are

text: intimately

text: related

text: to

text: closed

text: and

text: open

text: matricial

text: matrix

text: balls.

text: Lemma

text: Let

text: A,

text: B

text: CqH×q

text: and

text: let

text: D

text: B

text: A.

text: Then:

text: (a)

text: [A,

text: B]

text: if

text: and

text: only

text: if

text: D

text: ∈nCq≥×q.

text: In

text: this

text: case,

text: o

text: [A,

text: B]

text: A

text: DK

text: D

text: K

text: I

text: q

text: q

text: q

text: (b)

text: (A,

text: B)

text: if

text: and

text: only

text: if

text: D

text: ∈nCq×q.

text: In

text: this

text: case,

text: o

text: (A,

text: B)

text: A

text: DK

text: D

text: K

text: I

text: q

text: q

text: q

text: PInropoaf.rt(iacu)

text: laIfr,[AA,

text: B≤]

text: ∅i,.

text: et.h,

text: eDn

text: t∈heCreq≥×eqx.istCsoanvmeras√terlyix,

text: aX√ssu∈mCe

text: qHD×q

text: ∈wiCthq≥×Aq.

text: ≤ObXvio≤usBly,.

text: I

text: Let

text: K

text: I

text: The√n

text: X

text: A

text: DK

text: D

text: is

text: Hermitian.

text: √By

text: vir√tue

text: of

text: q

text: q

text: q

text: q

text: q

text: −q

text: R√emark

text: X

text: A

text: DK

text: D

text: and

text: B

text: X

text: D

text: DK

text: D

text: q

text: q

text: D(I

text: K)

text: D

text: Hence,

text: X

text: [A,

text: B].√

text: In

text: particul√ar,

text: [A,

text: B]

text: Now

text: let

text: q

text: q

text: q

text: In

text: view

text: of

text: Remark

text: then

text: is

text: Hermitian

text: and

text: we

text: X

text: A,

text: B

text: K

text: D

text: X

text: A

text: D

text: have

text: X

text: A

text: B

text: A

text: D.

text: q

text: q

text: R

text: R

text: N

text: N

text: Hence,

text: and

text: )√by

text: virtue

text: of

text: Remark

text: According

text: to

text: X

text: A

text: D

text: D

text: X

text: A

text: R

text: R

text: N

text: N

text: Remark

text: thus

text: and

text: U√sing

text: R√emarks

text: X

text: A

text: D

text: D

text: X

text: A

text: and

text: we

text: obtain

text: then

text: DK

text: D

text: X

text: A.

text: Henc√e,

text: X

text: A

text: DK

text: D.

text: With

text: Remark

text: we

text: conclude

text: from

text: that

text: K

text: D

text: D

text: D

text: Using

text: Remarks

text: q

text: q

text: and

text: and

text: Proposition

text: we

text: have

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: D

text: PR(

text: D)

text: Iq.

text: Thus,

text: K

text: I

text: i.

text: e.,

text: K

text: I

text: q

text: q

text: q

text: q

text: q

text: q

text: (b)

text: Since

text: every

text: positive

text: Hermitian

text: matrix

text: is

text: invertible,

text: this

text: can

text: be

text: shown

text: more

text: easy

text: in

text: a

text: similar

text: manner.

text: From

text: Lemma

text: we

text: conclude

text: the

text: following

text: result:

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: 2. On the matrix version of the truncated [ ]-Hausdorﬀ moment problem α, β R R the indicator function of the set . Then the integral of (Φ Ψ) with respect to over M , µ is deﬁned by Φd Ψ∗ := (1 Φ) 0 (1 Ψ)∗d . An -measurable complex-valued M M µ Ω M µτ M τ A function on Ω is said to be integrable with respect to a non-negative Hermitian measure f on (Ω ) if the pair ( ) is integrable with respect to . In this case the integral µ , A fI , I µ q q Z Z of with respect to is deﬁned by f µ ∗ d := ( )d f µ fI µI Ω Ω q q R R ∈ and for any , the function 1 is integrable with respect to . Then the integral M A f µ of with respect to over is Mdeﬁned by d := ( )d ∗. We denote by f µ M f µ fI µI L1(Ω ; C) the set of all -measurable compMlex-valued Mfunctqions q on Ω which are , A, µ A f integrable with respect to a non-negative Hermitian measure on (Ω ). µ , A Let (resp. ) be the -algebra of all Borel subsets of R (resp. C). For all BR BC σ ∈ \ {∅} M Ω BR , let BΩ be the σ-algebra of all Borel subsets of Ω and let q≥(Ω) be the set of all non-negative Hermitian × measures on (Ω ). For all ∈ N ∪ {∞}, let q q , BΩ κ 0 Mq≥ (Ω) be the set of all σ ∈ Mq≥(Ω) such that for each j ∈ Z0 the function f : Ω → C deﬁn,κed by f (x) := xj belongs to L1(Ω, BZΩ, σ; C). If σ ∈ Mq≥ ,κ(Ω), then let j j ,κ s[σ] := xjσ(dx) (2.1) j Ω ffoorr eavllerjy∈chZo0ic,κe. oOf bΩv∈iously,\M{∅}q≥(aΩn)d⊆ ∈MNq≥,k. (FΩu)rt⊆heMrmq≥o,rke+,1i(fΩΩ) i⊆s aMnoq≥n,∞-e(mΩp)tyhobldousntdruede BR k 0 Borel subset of R, then one can easily see that Mq≥(Ω) = Mq≥ ∞(Ω). In particular, for all ∈ R and ∈ ( ∞) we have , α β α, M M q≥([α, β]) = q≥ ∞([α, β]). (2.2) , Let Mq≥,mol(Ω) be the set of all σ ∈ Mq≥(Ω) for which there exists a ﬁnite subset B of Ω satisfying σ(Ω \ B) = 0q×q. The elements of Mq≥,mol(Ω)Pare called molecular. Obviously, Mq≥,mol(Ω) is the set of all σ ∈ Mq≥(Ω) for which there exist an m ∈ N and siseqtuheenDceisra(cξ‘m)m‘=ea0saunred o(nM‘)m‘=0wfirtohmunΩitanmdasCsq≥a×tq su,ch ∈thaZt σ .=In m‘p=a1rδtξic‘Mula‘,r,wwheerheaδvξe‘ BΩ ξ ‘ 1 Mq≥,mol(Ω) ⊆ Mq≥ ∞(Ω). ‘ ,m , The following matricial moment problem lies in the background of our considerations: M[Ω; (sj)κj=0, =] Let Ω ∈ BR \{∅} and let (sj)κj=0 be a sequence from Cq×q. Parametrize the set Mq≥[Ω; (sj)κj=0, =] of all σ ∈ Mq≥,κ(Ω) such that s[jσ] = sj for all j ∈ Z0,κ. In this paper, we often use the procedure of reﬂecting measures on the real axis. For this reason, we introduce some terminology. Let : R → R be deﬁned by 7→ − . R x x Then is a continuous involution of R and consequently - -measurable. In view of R BR BR Ω ∈ \{∅}, the set Ω∨ := −1(Ω) belongs to \{∅} and the mapping := Rstr BR R BR RΩ Ω R is - -measurable, whereas the mapping := Rstr is - -measurable. BΩ BΩ∨ RΩ∨ Ω∨ R BΩ∨ BΩ 4

text: On

text: the

text: matrix

text: version

text: of

text: the

text: truncated

text: ]-Hausdorﬀ

text: moment

text: problem

text: α,

text: β

text: R

text: R

text: the

text: indicator

text: function

text: of

text: the

text: set

text: Then

text: the

text: integral

text: of

text: (Φ

text: Ψ)

text: with

text: respect

text: to

text: over

text: M

text: µ

text: is

text: deﬁned

text: by

text: Φd

text: Ψ∗

text: Φ)

text: Ψ)∗d

text: An

text: -measurable

text: complex-valued

text: M

text: M

text: µ

text: Ω

text: M

text: µτ

text: M

text: τ

text: A

text: function

text: on

text: Ω

text: is

text: said

text: to

text: be

text: integrable

text: with

text: respect

text: to

text: a

text: non-negative

text: Hermitian

text: measure

text: f

text: on

text: (Ω

text: if

text: the

text: pair

text: is

text: integrable

text: with

text: respect

text: to

text: In

text: this

text: case

text: the

text: integral

text: µ

text: A

text: fI

text: I

text: µ

text: q

text: q

text: Z

text: Z

text: of

text: with

text: respect

text: to

text: is

text: deﬁned

text: by

text: f

text: µ

text: d

text: )d

text: f

text: µ

text: fI

text: µI

text: Ω

text: Ω

text: q

text: q

text: R

text: R

text: and

text: for

text: any

text: the

text: function

text: is

text: integrable

text: with

text: respect

text: to

text: Then

text: the

text: integral

text: M

text: A

text: f

text: µ

text: of

text: with

text: respect

text: to

text: over

text: is

text: Mdeﬁned

text: by

text: d

text: )d

text: We

text: denote

text: by

text: f

text: µ

text: M

text: f

text: µ

text: fI

text: µI

text: C)

text: the

text: set

text: of

text: all

text: -measurable

text: compMlex-valued

text: Mfunctqions

text: q

text: on

text: Ω

text: which

text: are

text: A,

text: µ

text: A

text: f

text: integrable

text: with

text: respect

text: to

text: a

text: non-negative

text: Hermitian

text: measure

text: on

text: (Ω

text: µ

text: A

text: Let

text: (resp.

text: be

text: the

text: -algebra

text: of

text: all

text: Borel

text: subsets

text: of

text: R

text: (resp.

text: C).

text: For

text: all

text: BR

text: BC

text: σ

text: M

text: Ω

text: BR

text: let

text: BΩ

text: be

text: the

text: σ-algebra

text: of

text: all

text: Borel

text: subsets

text: of

text: Ω

text: and

text: let

text: q≥(Ω)

text: be

text: the

text: set

text: of

text: all

text: non-negative

text: Hermitian

text: measures

text: on

text: (Ω

text: For

text: all

text: N

text: let

text: q

text: q

text: BΩ

text: κ

text: Mq≥

text: (Ω)

text: be

text: the

text: set

text: of

text: all

text: σ

text: Mq≥(Ω)

text: such

text: that

text: for

text: each

text: j

text: the

text: function

text: f

text: Ω

text: C

text: deﬁn,κed

text: by

text: f

text: (x)

text: xj

text: belongs

text: to

text: BZΩ,

text: σ;

text: C).

text: If

text: σ

text: Mq≥

text: ,κ(Ω),

text: then

text: let

text: j

text: j

text: ,κ

text: s[σ]

text: xjσ(dx)

text: j

text: Ω

text: ffoorr

text: oOf

text: bΩv∈iously,\M{∅}q≥(aΩn)d⊆

text: ∈MNq≥,k.

text: i⊆s

text: aMnoq≥n,∞-e(mΩp)tyhobldousntdruede

text: BR

text: k

text: Borel

text: subset

text: of

text: R,

text: then

text: one

text: can

text: easily

text: see

text: that

text: Mq≥(Ω)

text: Mq≥

text: ∞(Ω).

text: In

text: particular,

text: for

text: all

text: R

text: and

text: we

text: have

text: α

text: β

text: α,

text: M

text: M

text: q≥([α,

text: β])

text: q≥

text: ∞([α,

text: β]).

text: Let

text: Mq≥,mol(Ω)

text: be

text: the

text: set

text: of

text: all

text: σ

text: Mq≥(Ω)

text: for

text: which

text: there

text: exists

text: a

text: ﬁnite

text: subset

text: B

text: of

text: Ω

text: satisfying

text: σ(Ω

text: B)

text: The

text: elements

text: of

text: Mq≥,mol(Ω)Pare

text: called

text: molecular.

text: Obviously,

text: Mq≥,mol(Ω)

text: is

text: the

text: set

text: of

text: all

text: σ

text: Mq≥(Ω)

text: for

text: which

text: there

text: exist

text: an

text: m

text: N

text: and

text: su,ch

text: ∈thaZt

text: σ

text: .=In

text: BΩ

text: ξ

text: Mq≥,mol(Ω)

text: Mq≥

text: ∞(Ω).

text: ,m

text: The

text: following

text: matricial

text: moment

text: problem

text: lies

text: in

text: the

text: background

text: of

text: our

text: considerations:

text: M[Ω;

text: Let

text: Ω

text: BR

text: and

text: let

text: be

text: a

text: sequence

text: from

text: Cq×q.

text: Parametrize

text: the

text: set

text: Mq≥[Ω;

text: of

text: all

text: σ

text: Mq≥,κ(Ω)

text: such

text: that

text: s[jσ]

text: sj

text: for

text: all

text: j

text: In

text: this

text: paper,

text: we

text: often

text: use

text: the

text: procedure

text: of

text: reﬂecting

text: measures

text: on

text: the

text: real

text: axis.

text: For

text: this

text: reason,

text: we

text: introduce

text: some

text: terminology.

text: Let

text: R

text: R

text: be

text: deﬁned

text: by

text: 7→

text: R

text: x

text: x

text: Then

text: is

text: a

text: continuous

text: involution

text: of

text: R

text: and

text: consequently

text: -measurable.

text: In

text: view

text: of

text: R

text: BR

text: BR

text: Ω

text: the

text: set

text: Ω∨

text: belongs

text: to

text: and

text: the

text: mapping

text: Rstr

text: BR

text: R

text: BR

text: RΩ

text: Ω

text: R

text: is

text: -measurable,

text: whereas

text: the

text: mapping

text: Rstr

text: is

text: -measurable.

text: BΩ

text: BΩ∨

text: RΩ∨

text: Ω∨

text: R

text: BΩ∨

text: BΩ


text: 10. On the structure of [ ]-Hausdorﬀ non-negative deﬁnite sequences α, β h i † † L [( L) −1] ( L) −1 = L and L (L ) −1 (L ) −1 = L . (10.11) n α n α n n n β n β n n Remark 4.12 and Notation 3.4 yield − − ( ) = ( ) = ( ) ( ) = ( ) ( ) + ( ) (10.12) L 0 s 0 β s 0 s 1 β L 0 s 1 M 0 α β α β α α α α α and − − − ( ) = ( ) = ( ) + ( ) = ( ) + ( ) ( ) (10.13) L 0 s 0 α s 0 s 1 α L 0 s 1 M 0. α β α β β β β β β † Using (10.11) for = 1, from (10.12) and (10.13) we obtain then ( ) ( ) = n L1 L 0 L 0 † − † † − α − α β ( ) [( ) ( ) ]+ and ( ) ( ) = ( ) [( ) ( ) ] . This L1 L 0 M 0 s 1 βL1 L1 L 0 L 0 L1 L 0 s 1 M 0 αL1 α α α β α β β β β implies h i † † ( ) ( ) + ( ) ( ) L1 L 0 L 0 L1 L 0 L 0 α α β β α β † − † − − = ( ) [( ) ( ) ] + + ( ) ( ) ( ) (10.14) L1 L 0 M 0 s 1 βL1 L1 L 0 s 1 M 0 αL1. α α α β β β − ± Comparing (10.9) and (10.10) for = 1 with (10.14), we conclude = ( )( ). n d2 β α A2 B2 lFatohnqn≥eNd,2gsnos(e−w(qst2uo,βαwe),njβeF)c.q2j≥es=,nCsu2−0np(o−2(pnα1∈os,sαes)K,eqβju).q≥t2j,eh2,=nenna−H0t−t2le2yn,na,αcnb.e≥de,IcnoPa2fvu.rLioseeeFpwmrooomsfRmitPaeimor8Pon.ar2pro29okp(.s1ao4it().sia1iott)2oni,oittn9mhh.e2pe7,l.sai7ewep(qseapu)l{ehi(ncwa(acvαeteesisok)((njn(()αos2joswβ=nf)−)0Ljj2t))e,h2j2jm=(n=na(−−0t0ms22β(a)∈spjj8r))Lo.2j2j2v=n=≥qn2i−,0−d02,(e2nb1e}−s)b2ut⊆,eβos- h (cid:16) (cid:17)i (( Θ) ) −1 α β n − h (cid:16) † h − i(cid:17)i = β(αs)2n−2 (αM )n−1 + (αL)n−1 βIq + ((αL)β)n−2[((αs)β)2n−3 ((αM)β)n−2] − † − = β(αs)2n−2 (αM )n−1 + (αL)n−1 βIq + (αLβ)n−2 (αsβ)2n−3 (αMβ)n−2 an(cid:0)d, anal(cid:1)ogously, (cid:16) h i(cid:17) − α(Θβ) n−1 = α(sβ)2n−2 + (Mβ)n−1 † − + (Lβ)n−1 αIq + (αLβ)n−2 (αsβ)2n−3 (αMβ)n−2 . By virtue of Remark 4.12, we get (cid:16) h i(cid:17) − − ( L ) −1 = (( L) ) −1 = β( s)2 −2 ( s)2 −1 (( Θ) ) −1 α β n α β n α n α n α β n − † − = (αM )n−1 (αs)2n−1 + (αL)n−1 βIq + (αLβ)n−2 (αsβ)2n−3 (αMβ)n−2 35 ##LTLine##

text: On

text: the

text: structure

text: of

text: ]-Hausdorﬀ

text: non-negative

text: deﬁnite

text: sequences

text: α,

text: β

text: h

label: equation

text: i

text: L

text: L)

text: L)

text: −1

text: L

text: and

text: L

text: (L

text: −1

text: (L

text: −1

text: L

text: n

text: α

text: n

text: α

text: n

text: n

text: n

text: β

text: n

text: β

text: n

text: n

text: Remark

text: and

text: Notation

text: yield

text: L

text: s

text: β

text: s

text: s

text: β

text: L

text: s

text: M

text: α

text: β

text: α

text: β

text: α

text: α

text: α

text: α

text: α

text: and

text: L

text: s

text: α

text: s

text: s

text: α

text: L

text: s

text: M

text: α

text: β

text: α

text: β

text: β

text: β

text: β

text: β

text: β

text: Using

text: for

text: from

text: and

text: we

text: obtain

text: then

text: n

text: L

text: L

text: α

text: α

text: β

text: and

text: This

text: L

text: M

text: s

text: L

text: L

text: L

text: s

text: M

text: α

text: α

text: α

text: β

text: α

text: β

text: β

text: β

text: β

text: implies

text: h

text: i

text: L

text: L

text: L

text: L

text: α

text: α

text: β

text: β

text: α

text: β

text: L

text: M

text: s

text: L

text: s

text: M

text: α

text: α

text: α

text: β

text: β

text: β

text: Comparing

text: and

text: for

text: with

text: we

text: conclude

text: n

text: β

text: α

text: h

text: Θ)

text: −1

text: α

text: β

text: n

text: h

text: h

text: (αM

text: βIq

text: (αM

text: βIq

text: h

text: α(Θβ)

text: αIq

text: By

text: virtue

text: of

text: Remark

text: we

text: get

text: h

text: L

text: −1

text: L)

text: −1

text: β(

text: −2

text: −1

text: Θ)

text: −1

text: α

text: β

text: n

text: α

text: β

text: n

text: α

text: n

text: α

text: n

text: α

text: β

text: n

text: (αM

text: βIq

text: ##LTLine##


text: 4. On [α, β]-Hausdorﬀ non-negative deﬁnite sequences from Cq×q Using Theorem 4.3, we derive now several algebraic results on the matrix sequences introduced in Deﬁnitions 4.2 and 4.4, resp. TPhreonp(ossji)t‘ji=o0n∈4F.5q≥.,‘,Lα,eβt fαor∈alRl ,‘l∈etZβ0,m∈.(α, ∞), let m ∈ N0, and let (sj)mj=0 ∈ Fq≥,m,α,β. PMroof. In view of The⊆oreMm 4.3 we have Mq≥[[α, β]; (sj)Mmj=0, =] 6= ∅. Let ‘ ∈ Z6 0,m∅. Then q≥[[α, β]; (sj)mj=0, =] q≥[[α, β]; (sj)‘j=0, =∈].FH≥ence q≥[[α, β]; (sj)‘j=0, =] = . Thus, again applying Theorem 4.3, we get (sj)‘j=0 q,‘,α,β. Corollary 4.6. Let α ∈ R, β ∈ (α, ∞), and m ∈ N0. Then F≥,e ⊆ F≥ . q,m,α,β q,m,α,β Proof. Use Proposition 4.5. Proposition 4.5 leads us to the following notions: TDheeﬁnn(iti)o∞n 4is.7c.alLleedt[α ∈]-RH,aluestdβorﬀ∈ n(oαn,-∞ne)g,aatinvde dleetﬁn(sitje)∞j(=re0spb.e[ a se]q-HueanucsedofrrﬀomposCitqi×vqe. deﬁnites)j ijf=f0or all m ∈αN, β0 the sequence (sj)mj=0 is [α, βF]-≥Hausdorﬀα,nβoFn-negative deﬁnite (resp. [α, β]-Hausdorﬀ positive deﬁnite). The notation ∞ (resp. >∞ ) stands for q, ,α,β q, ,α,β set of all [ ]-Hausdorﬀ non-negative deﬁnite (resp. [ ]-Hausdorﬀ positive deﬁnite) α, β α, β sequences (sj)∞j=0 from Cq×q. Lemma 4.8. Let α ∈ R, let β ∈ (α, ∞), and let σ ∈ Mq≥([α, β]). Then σ ∈ Mq≥ ∞([α, β]) and the sequence (s[σ])∞=0Tgiven via (2.1) belongs to F≥∞ . , j j q, ,α,β ∈ ∞ M Proof. By the choice of σ we have σ =0 q≥ ([α, β]). Thus, Theorem 4.3 implies m ,m (s[σ])m=0 ∈ F≥ for each m ∈ N0. Hence (s[σ])∞=0 ∈ F≥∞ . j j q,m,α,β j j q, ,α,β TPhreonptohseirteioins a4.s9eq.uLeentceα(s∈j)R∞j=, mle+t1βfro∈m(αC,q∞×q),sluecthmtha∈t N(s0j,)∞ja=n0d∈leFt q≥(,∞sj,)αmj,=β.0 ∈ Fq≥,m,α,β. M 6 ∅ ∈ PMroof. In view of Theorem 4.3 we have q≥[[α, β]; (sj)mj=0, =] =∈ M. Let σ q≥[[α, β]; (sj)mj=0, =]. In view of Lemma 4.8, we have then σ q≥,∞([α, β]), (s[jσ])∞j=0 ∈ Fq≥,∞,α,β, and (s[jσ])mj=0 = (sj)mj=0. Theorem 4.10. Let α ∈ R, let β ∈ (α, ∞), and let m ∈ N0. Then F≥,e = F≥ . q,m,α,β q,m,α,β Proof. Combine Corollary 4.6 and Propositions 4.9 and 4.5. Comparing Theorem 4.10 with Remark 3.9, we see that both statements are completely diﬀerent. Against to the background of Theorems 4.3 and 3.3 we can now immediately M see the reason for this phenomenon, namely, in view of (2.2), we have q≥([α, β]) = Mq≥ ∞([α, β]), whereas on the other hand it can be easily checked that, for each k ∈ N, the p, roper inclusion Mq≥ +1(R) ⊂ Mq≥ (R) is satisﬁed. ,k ,k In view of Lemma 4.8, we will consider the following problem: 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: On

text: [α,

text: β]-Hausdorﬀ

text: non-negative

text: deﬁnite

text: sequences

text: from

text: Cq×q

text: Using

text: Theorem

text: we

text: derive

text: now

text: several

text: algebraic

text: results

text: on

text: the

text: matrix

text: sequences

text: introduced

text: in

text: Deﬁnitions

text: and

text: resp.

text: fαor∈alRl

text: let

text: m

text: and

text: let

text: Fq≥,m,α,β.

text: PMroof.

text: In

text: view

text: of

text: The⊆oreMm

text: we

text: have

text: Mq≥[[α,

text: β];

text: Let

text: Then

text: q≥[[α,

text: β];

text: q≥[[α,

text: β];

text: =∈].FH≥ence

text: q≥[[α,

text: β];

text: Thus,

text: again

text: applying

text: Theorem

text: we

text: get

text: q,‘,α,β.

text: Corollary

text: Let

text: α

text: R,

text: β

text: (α,

text: and

text: m

text: Then

text: F≥,e

text: F≥

text: q,m,α,β

text: q,m,α,β

text: Proof.

text: Use

text: Proposition

text: Proposition

text: leads

text: us

text: to

text: the

text: following

text: notions:

text: TDheeﬁnn(iti)o∞n

text: ∈]-RH,aluestdβorﬀ∈

text: n(oαn,-∞ne)g,aatinvde

text: a

text: se]q-HueanucsedofrrﬀomposCitqi×vqe.

text: deﬁnites)j

text: all

text: m

text: ∈αN,

text: β0

text: the

text: sequence

text: is

text: [α,

text: βF]-≥Hausdorﬀα,nβoFn-negative

text: deﬁnite

text: (resp.

text: [α,

text: β]-Hausdorﬀ

text: positive

text: deﬁnite).

text: The

text: notation

text: (resp.

text: stands

text: for

text: q,

text: ,α,β

text: q,

text: ,α,β

text: set

text: of

text: all

text: ]-Hausdorﬀ

text: non-negative

text: deﬁnite

text: (resp.

text: ]-Hausdorﬀ

text: positive

text: deﬁnite)

text: α,

text: β

text: α,

text: β

text: sequences

text: from

text: Cq×q.

text: Lemma

text: Let

text: α

text: R,

text: let

text: β

text: (α,

text: and

text: let

text: σ

text: Mq≥([α,

text: β]).

text: Then

text: σ

text: Mq≥

text: ∞([α,

text: β])

text: and

text: the

text: sequence

text: via

text: belongs

text: to

text: F≥∞

text: j

text: j

text: q,

text: ,α,β

text: M

text: Proof.

text: By

text: the

text: choice

text: of

text: σ

text: we

text: have

text: σ

text: q≥

text: ([α,

text: β]).

text: Thus,

text: Theorem

text: implies

text: m

text: ,m

text: F≥

text: for

text: each

text: m

text: Hence

text: F≥∞

text: j

text: j

text: q,m,α,β

text: j

text: j

text: q,

text: ,α,β

text: TPhreonptohseirteioins

text: Fq≥,m,α,β.

text: M

text: PMroof.

text: In

text: view

text: of

text: Theorem

text: we

text: have

text: q≥[[α,

text: β];

text: M.

text: Let

text: σ

text: q≥[[α,

text: β];

text: In

text: view

text: of

text: Lemma

text: we

text: have

text: then

text: σ

text: q≥,∞([α,

text: β]),

text: Fq≥,∞,α,β,

text: and

text: Theorem

text: Let

text: α

text: R,

text: let

text: β

text: (α,

text: and

text: let

text: m

text: Then

text: F≥,e

text: F≥

text: q,m,α,β

text: q,m,α,β

text: Proof.

text: Combine

text: Corollary

text: and

text: Propositions

text: and

text: Comparing

text: Theorem

text: with

text: Remark

text: we

text: see

text: that

text: both

text: statements

text: are

text: completely

text: diﬀerent.

text: Against

text: to

text: the

text: background

text: of

text: Theorems

text: and

text: we

text: can

text: now

text: immediately

text: M

text: see

text: the

text: reason

text: for

text: this

text: phenomenon,

text: namely,

text: in

text: view

text: of

text: we

text: have

text: q≥([α,

text: β])

text: Mq≥

text: ∞([α,

text: β]),

text: whereas

text: on

text: the

text: other

text: hand

text: it

text: can

text: be

text: easily

text: checked

text: that,

text: for

text: each

text: k

text: N,

text: the

text: p,

text: roper

text: inclusion

text: Mq≥

text: Mq≥

text: (R)

text: is

text: satisﬁed.

text: ,k

text: ,k

text: In

text: view

text: of

text: Lemma

text: we

text: will

text: consider

text: the

text: following

text: problem:

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text: 14 l index x y z 0 (m,n,p) 0 0 H 1 (m+1,n,p) 2c 0 H − 3d2 c2 − H2 2 (m+2,n,p-1) 2c + c d 0 c2 + d2 c2 + d2 3 (m+2,n+1,p-1) 2c + c d 0 4 (m+1,n+1,p-1) c d 0 − c2 d2 4c2d 5 (m-1,n+1,p) 2c H c2 +−d2 c2 + d2 − 3d2 c2 H2 6 (m-1,n,p-1) c d 0 c−2 + d2 c2−+ d2 7 (m-1,n-1,p-1) c d 0 − 8 (m+1,n-1,p-1) c d 0 − − c2 d2 − 4c2d 9 (m+2,n-1,p) 2c 2c H c2 + d2 c2 + d2 particles dx dy dz 1,0 2c 0 0 4,0 c d -H − c2 d2 4c2d 5,0 2c 0 c2 +−d2 c2 + d2 − 3d2 c2 H2 6,0 c d -H c−2 + d2 c2−+ d2 7,0 c d -H − 8,0 c d -H − 3d2 c2 − H2 2,1 c d -H c2 + d2 c2 + d2 3,1 c d -H − 4,1 c d -H − − 8,1 c d -H − − c2 d2 − 4c2d 9,1 2c 0 c2 + d2 c2 + d2 TABLE S2. Indices and positions of the particles 0 and 1 and their eight neighbours l = 2, ..., 9 (left) and distances between the neighbouring particles and 0 and 1 particles (right) in the unit cell of the maximally zigzagging stripe conﬁguration. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: l

label: table

text: index

text: x

text: y

text: z

text: (m,n,p)

text: H

text: H

text: c

text: d

text: c

text: d

text: c

text: d

text: H

text: c

text: d

text: c

text: d

text: c

text: d

text: H

text: particles

text: dx

text: dy

text: dz

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: c

text: d

text: -H

text: TABLE

text: Indices

text: and

text: positions

text: of

text: the

text: particles

text: and

text: and

text: their

text: eight

text: neighbours

text: l

text: (left)

text: and

text: distances

text: between

text: the

text: neighbouring

text: particles

text: and

text: and

text: particles

text: (right)

text: in

text: the

text: unit

text: cell

text: of

text: the

text: maximally

text: zigzagging

text: stripe

text: conﬁguration.

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text: 3 (considering smaller values of δ did not change the re- sult) with the same number of frustrated bonds n in the fb straight and bent conﬁguration. We ﬁnd that bent stripes are thermodynamically stable, which is the opposite from the Ising antiferromagnet result shown in Fig. 2a. In- creasing the number of free spheres, the entropy diﬀer- ence increases, especially for large angles. From existing experiments on colloidal monolayers [10, 21, 33] it is hard to conclude if the ground state has a FIG. 3. Straight (a-c) and bent (d-f) conﬁgurations. (a,d) yel- preference for straight or bent stripes because the sys- low (red) spheres are in contact with the bottom (top) plane. tem has to be annealed very slowly [20, 22]. From the Phase-space volume of one hard sphere free to move given experimental interparticle potential of NIPA colloids [26] by the 3-dimensional volume V , considering curved (b,e) and 1 it is possible to see that a simple correction to the hard- ﬂat (c,f) surfaces. Here α = 70o and δ/c = 0.08 to emphasize sphere potential can be described by a decreasing expo- the diﬀerence between curved and ﬂat surfaces. nential. However, this potential is in the same “quasi- universality” class with hard spheres [34], hence no qual- − to move while the centers of the other N n remain ﬁxed itative change in the results is expected. Softer interac- in their close-packed positions. tions such as the hard-core soft-shoulder potential does → − The shrinking of spheres, c c δ, implies the scaling not change the preference for bent over straight stripes ∼ V δ3n. V is the phase space volume associated to the [27]. n n n spheres free to move, described by a 3n-dimensional Attraction: Elasticity in the Ising model includes both volume delimited by curved surfaces. Considering one attraction and repulsion which contribute equally, while sphere free to move, the surface of the free volume V is hard- or soft-sphere interactions used to model colloids 1 described by rolling the free sphere in all possible ways are purely repulsive. Adding attraction to repulsive col- over its six neighbor spheres and over one conﬁning wall loids can induce a clustering phase [35], a solid-solid (cid:28) (Fig. 3b,e). For δ c it is possible to neglect the curva- transition [36], a glass-glass transition [37] and many ture of the surfaces [23–25], thus obtaining linear restric- other phenomena [16]. To investigate the role of at- tions (planes) (Fig. 3c,f), and then to compute V as a traction in the entropy selection of the ground state, we 1 3-dimensional polyhedron, and for n free spheres to sim- ﬁrst consider a system of particles in the same straight ilarly compute V as a 3n-dimensional polytope [27, 28]. and bent positions as the buckled colloidal system with n V is given by the Voronoi cell associated to the center of particles interacting either through a harmonic potential 1 tahnedfbreeentspchoenrﬁe(cid:18)gsu(cid:113)craalteidonbsya2tδt,hains(cid:19)dletvheel ceonitnrcoipdye o(af sstfroarigfhcct Ute(nζt)ia=l UUr0(ζζ2) o=r tUh0roζu2g·h−θ(aζ)puwrehleyrereθp(uζl)sivisethhaermHoenavicispidoe- and hcp [25]), but here on√ly up to a deformation angle of step function, ζ = (dr0 dr)/(2δ) and U0 sets the energy α∗ = 2 arctan 8/(7 + 33) (cid:39) 77o [27], after which s{cale. d}r0 = 2c and dr2 = dr02 + du2 + dv2 + dw2 with u, v, w the displacement around the equilibrium posi- bent-stripe entropy is higher than straight-stripe entropy, tion. The entropy of straight and bent conﬁgurations for ∗ see Fig. 2b. For bent stripes with α > α when the free particles interacting through U(ζ) can be exactly calcu- sphere is close to the top wall the condition coming from lated [27], while for Ur(ζ) we use the canonical partition one of the bottom spheres becomes irrelevant [27]. There- function obtained by numerical integration [27]. Fig. 4a,b fore, in conﬁnement already with one free sphere, there shows that the harmonic potential U(ζ) gives a result are local conﬁgurations of a collectively jammed state [29] qualitatively similar to that of the Ising antiferromagnet with the same close-packed density which diﬀer in their (Fig. 2a) with a preference for straight stripes. On the local stability for slightly decreasing density. We will show other hand, considering only the repulsive part of the har- that this is valid also for larger n. monic potential, using Ur(ζ), changes the preference to For n = 2, the free volume can be computed through a bent stripes, as we found for hard spheres (Fig. 2b). 6-dimensional integral [25]. Because the correlated mo- Con·sidering a generalized repulsive potential Urγ(ζ) = tion of the free spheres, to ﬁnd the free volume for n > 2 Ur0ζγ θ(ζ), which reduces to the repulsive harmonic po- requires more sophisticated tools. For that we use a mod- tent−ial for γ →=∞2 and gives hard spheres of diameter iﬁed version of the Lasserre method [30, 31] implemented 2(c δ) for γ 2, for increasing γ, ∆s for one particle in the VINCI code [32]. We calculated the phase-space free to move slowly approaches the hard-sphere result and volumes V for straight and bent conﬁgurations up to n n = 6 spheres that are free to move (see Fig. 1c,d) cor- responding to 18-dimensional polytopes [27]. Figure 2b 2 Urγ diﬀers by a numerical prefactor from the commonly-used tun- shows the entropy diﬀerence per particle for diﬀerent sets able soft repulsive potential [42, 49, 50], with dr0→= σij and of n contiguous spheres free to move with δ/c = 0.001 dr = rij, and the hard-sphere limit obtained for α 0 [49]. ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: (considering

text: smaller

text: values

text: of

text: δ

text: did

text: not

text: change

text: the

text: re-

text: sult)

text: with

text: the

text: same

text: number

text: of

text: frustrated

text: bonds

text: n

text: in

text: the

text: fb

text: straight

text: and

text: bent

text: conﬁguration.

text: We

text: ﬁnd

text: that

text: bent

text: stripes

text: are

text: thermodynamically

text: stable,

text: which

text: is

text: the

text: opposite

text: from

text: the

text: Ising

text: antiferromagnet

text: result

text: shown

text: in

text: Fig.

text: In-

text: creasing

text: the

text: number

text: of

text: free

text: spheres,

text: the

text: entropy

text: diﬀer-

text: ence

text: increases,

text: especially

text: for

text: large

text: angles.

text: From

text: existing

text: experiments

text: on

text: colloidal

text: monolayers

text: it

text: is

text: hard

text: to

text: conclude

text: if

text: the

text: ground

text: state

text: has

text: a

text: FIG.

text: Straight

label: caption

text: (a-c)

text: and

text: bent

text: (d-f)

text: conﬁgurations.

text: (a,d)

text: yel-

text: preference

text: for

text: straight

text: or

text: bent

text: stripes

text: because

text: the

text: sys-

text: low

text: (red)

text: spheres

text: are

text: in

text: contact

text: with

text: the

text: bottom

text: (top)

text: plane.

text: tem

text: has

text: to

text: be

text: annealed

text: very

text: slowly

text: From

text: the

text: Phase-space

text: volume

text: of

text: one

text: hard

text: sphere

text: free

text: to

text: move

text: given

text: experimental

text: interparticle

text: potential

text: of

text: NIPA

text: colloids

text: by

text: the

text: volume

text: V

text: considering

text: curved

text: (b,e)

text: and

text: it

text: is

text: possible

text: to

text: see

text: that

text: a

text: simple

text: correction

text: to

text: the

text: hard-

text: ﬂat

text: (c,f)

text: surfaces.

text: Here

text: α

text: and

text: δ/c

text: to

text: emphasize

text: sphere

text: potential

text: can

text: be

text: described

text: by

text: a

text: decreasing

text: expo-

text: the

text: diﬀerence

text: between

text: curved

text: and

text: ﬂat

text: surfaces.

text: nential.

text: However,

text: this

text: potential

text: is

text: in

text: the

text: same

text: “quasi-

text: universality”

text: class

text: with

text: hard

text: spheres

text: hence

text: no

text: qual-

text: to

text: move

text: while

text: the

text: centers

text: of

text: the

text: other

text: N

text: n

text: remain

text: ﬁxed

text: itative

text: change

text: in

text: the

text: results

text: is

text: expected.

text: Softer

text: interac-

text: in

text: their

text: close-packed

text: positions.

text: tions

text: such

text: as

text: the

text: hard-core

text: soft-shoulder

text: potential

text: does

text: The

text: shrinking

text: of

text: spheres,

text: c

text: c

text: δ,

text: implies

text: the

text: scaling

text: not

text: change

text: the

text: preference

text: for

text: bent

text: over

text: straight

text: stripes

text: V

text: V

text: is

text: the

text: phase

text: space

text: volume

text: associated

text: to

text: the

text: n

text: n

text: n

text: spheres

text: free

text: to

text: move,

text: described

text: by

text: a

text: Attraction:

text: Elasticity

text: in

text: the

text: Ising

text: model

text: includes

text: both

text: volume

text: delimited

text: by

text: curved

text: surfaces.

text: Considering

text: one

text: attraction

text: and

text: repulsion

text: which

text: contribute

text: equally,

text: while

text: sphere

text: free

text: to

text: move,

text: the

text: surface

text: of

text: the

text: free

text: volume

text: V

text: is

text: hard-

text: or

text: soft-sphere

text: interactions

text: used

text: to

text: model

text: colloids

text: described

text: by

text: rolling

text: the

text: free

text: sphere

text: in

text: all

text: possible

text: ways

text: are

text: purely

text: repulsive.

text: Adding

text: attraction

text: to

text: repulsive

text: col-

text: over

text: its

text: six

text: neighbor

text: spheres

text: and

text: over

text: one

text: conﬁning

text: wall

text: loids

text: can

text: induce

text: a

text: clustering

text: phase

text: a

text: solid-solid

text: (Fig.

text: For

text: δ

text: c

text: it

text: is

text: possible

text: to

text: neglect

text: the

text: curva-

text: transition

text: a

text: glass-glass

text: transition

text: and

text: many

text: ture

text: of

text: the

text: surfaces

text: thus

text: obtaining

text: linear

text: restric-

text: other

text: phenomena

text: To

text: investigate

text: the

text: role

text: of

text: at-

text: tions

text: (planes)

text: (Fig.

text: and

text: then

text: to

text: compute

text: V

text: as

text: a

text: traction

text: in

text: the

text: entropy

text: selection

text: of

text: the

text: ground

text: state,

text: we

text: polyhedron,

text: and

text: for

text: n

text: free

text: spheres

text: to

text: sim-

text: ﬁrst

text: consider

text: a

text: system

text: of

text: particles

text: in

text: the

text: same

text: straight

text: ilarly

text: compute

text: V

text: as

text: a

text: polytope

text: and

text: bent

text: positions

text: as

text: the

text: buckled

text: colloidal

text: system

text: with

text: n

text: V

text: is

text: given

text: by

text: the

text: Voronoi

text: cell

text: associated

text: to

text: the

text: center

text: of

text: particles

text: interacting

text: either

text: through

text: a

text: harmonic

text: potential

text: ceonitnrcoipdye

text: o(af

text: sstfroarigfhcct

text: Ute(nζt)ia=l

text: o=r

text: and

text: hcp

text: but

text: here

text: on√ly

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text: such a description will likely overtrain substantially. One way to counteract this is to use a classiﬁcation algorithm with overtraining prevention mechanisms – e.g., a convnet, as it utilizes weight sharing. Another way is to directly transform the initial, “raw” voxel features into some more appropriate “aggregated” features. The available grid-like structure of the raw voxel representation is essential for some aggregation strategies. One example of such aggregation is a wavelet transformation of the voxel data. We consider the simplest case of Haar wavelets [11]. Suppose that a scalar voxel feature is evaluated at a resolution N = 2n, so that the total feature space can be written as ⊗ ⊗ W W W, where W is the 2n-dimensional space of feature values along one coordinate. ⊗ ⊗ × The Haartransform can then be written as H H H, where H is the orthogonal 2n 2n matrix with the entries   − 2 n/2, k = 1,   − − ≤ − − − ≤ − − − 2 (n m)/2, 2m < k 2m+1, (k 2m 1)2n m < s (k 2m 1/2)2n m, Hk,s = − − − ≤ − − − ≤ − − 2 (n m)/2, 2m < k 2m+1, (k 2m 1/2)2n m < s (k 2m)2n m, 0, otherwise, − with m = 0, . . . , n 1 characterizing the length scale. As the Haar transform is linear and orthogonal, the transformed features contain the same information as the original raw features, but this information is now redistributed over several diﬀerent length scales according to the value of m. This explicit exposure of the length scale hierarchy can help reduce overtraining and is an alternative to generating raw voxel features at several resolutions. Another example of feature aggregation is to compute a histogram or some other statistical measure of feature values over the non-empty voxels. In the experiments below we consider a few feature percentiles. In particular, the 0’th percentile corresponds to the minimal value, the 100’th to the maximum value, and the 50’th to the median value. In the case of vector-valued features we perform feature aggregation separately for each component. 4 Experiments We have performed a series of classiﬁcation experiments with the ESB (Engineering Shape Benchmark, [15]) collection of CAD models. This collection consists of 866 mechan- ical parts divided into 3 “superclasses” (Flat-thin wall components, Rectangular cubic prisms, and Solids of revolution). Each superclass is further divided into ﬁner classes (like Clips, Handles, Bolt like parts, etc.); there are 45 ﬁne classes in total (hereafter referred to simply as classes). In particular, each superclass contains a Miscellaneous class for objects that cannot be easily assigned to other classes of this superclass. See Fig. 4 for some examples of ESB objects. We have divided the collection into a training set of 675 objects and a test set of 191 objects, to be used in all the experiments. A classiﬁer’s prediction for a given object is a 45-dimensional vector of class probabilities. We measure the total prediction error simply as the fraction of test objects where the true class is diﬀerent from the one with the highest predicted probability. All the ESB models have consistent, watertight meshes, so all the features from Table 1 are applicable. 7

text: such

text: a

text: description

text: will

text: likely

text: overtrain

text: substantially.

text: One

text: way

text: to

text: counteract

text: this

text: is

text: to

text: use

text: a

text: classiﬁcation

text: algorithm

text: with

text: overtraining

text: prevention

text: mechanisms

text: e.g.,

text: a

text: convnet,

text: as

text: it

text: utilizes

text: weight

text: sharing.

text: Another

text: way

text: is

text: to

text: directly

text: transform

text: the

text: initial,

text: “raw”

text: voxel

text: features

text: into

text: some

text: more

text: appropriate

text: “aggregated”

text: features.

text: The

text: available

text: grid-like

text: structure

text: of

text: the

text: raw

text: voxel

text: representation

text: is

text: essential

text: for

text: some

text: aggregation

text: strategies.

text: One

text: example

text: of

text: such

text: aggregation

text: is

text: a

text: wavelet

text: transformation

text: of

text: the

text: voxel

text: data.

text: We

text: consider

text: the

text: simplest

text: case

text: of

text: Haar

text: wavelets

text: Suppose

text: that

text: a

text: scalar

text: voxel

text: feature

text: is

text: evaluated

text: at

text: a

text: resolution

text: N

text: so

text: that

text: the

text: total

text: feature

text: space

text: can

text: be

text: written

text: as

text: W

text: W

text: W,

text: where

text: W

text: is

text: the

text: space

text: of

text: feature

text: values

text: along

text: one

text: coordinate.

text: The

text: Haartransform

text: can

text: then

text: be

text: written

text: as

text: H

text: H

text: H,

text: where

text: H

text: is

text: the

text: orthogonal

text: matrix

text: with

text: the

text: entries

text: k

text: 2

text: (n

text: k

text: (k

text: m

text: s

text: (k

text: m,

text: Hk,s

text: (n

text: k

text: (k

text: m

text: s

text: (k

text: m,

text: otherwise,

text: with

text: m

text: n

text: characterizing

text: the

text: length

text: scale.

text: As

text: the

text: Haar

text: transform

text: is

text: linear

text: and

text: orthogonal,

text: the

text: transformed

text: features

text: contain

text: the

text: same

text: information

text: as

text: the

text: original

text: raw

text: features,

text: but

text: this

text: information

text: is

text: now

text: redistributed

text: over

text: several

text: diﬀerent

text: length

text: scales

text: according

text: to

text: the

text: value

text: of

text: m.

text: This

text: explicit

text: exposure

text: of

text: the

text: length

text: scale

text: hierarchy

text: can

text: help

text: reduce

text: overtraining

text: and

text: is

text: an

text: alternative

text: to

text: generating

text: raw

text: voxel

text: features

text: at

text: several

text: resolutions.

text: Another

text: example

text: of

text: feature

text: aggregation

text: is

text: to

text: compute

text: a

text: histogram

text: or

text: some

text: other

text: statistical

text: measure

text: of

text: feature

text: values

text: over

text: the

text: non-empty

text: voxels.

text: In

text: the

text: experiments

text: below

text: we

text: consider

text: a

text: few

text: feature

text: percentiles.

text: In

text: particular,

text: the

text: percentile

text: corresponds

text: to

text: the

text: minimal

text: value,

text: the

text: to

text: the

text: maximum

text: value,

text: and

text: the

text: to

text: the

text: median

text: value.

text: In

text: the

text: case

text: of

text: vector-valued

text: features

text: we

text: perform

text: feature

text: aggregation

text: separately

text: for

text: each

text: component.

text: Experiments

text: We

text: have

text: performed

text: a

text: series

text: of

text: classiﬁcation

text: experiments

text: with

text: the

text: ESB

text: (Engineering

text: Shape

text: Benchmark,

text: collection

text: of

text: CAD

text: models.

text: This

text: collection

text: consists

text: of

text: mechan-

text: ical

text: parts

text: divided

text: into

text: “superclasses”

text: (Flat-thin

text: wall

text: components,

text: Rectangular

text: cubic

text: prisms,

text: and

text: Solids

text: of

text: revolution).

text: Each

text: superclass

text: is

text: further

text: divided

text: into

text: ﬁner

text: classes

text: (like

text: Clips,

text: Handles,

text: Bolt

text: like

text: parts,

text: etc.);

text: there

text: are

text: ﬁne

text: classes

text: in

text: total

text: (hereafter

text: referred

text: to

text: simply

text: as

text: classes).

text: In

text: particular,

text: each

text: superclass

text: contains

text: a

text: Miscellaneous

text: class

text: for

text: objects

text: that

text: cannot

text: be

text: easily

text: assigned

text: to

text: other

text: classes

text: of

text: this

text: superclass.

text: See

text: Fig.

text: for

text: some

text: examples

text: of

text: ESB

text: objects.

text: We

text: have

text: divided

text: the

text: collection

text: into

text: a

text: training

text: set

text: of

text: objects

text: and

text: a

text: test

text: set

text: of

text: objects,

text: to

text: be

text: used

text: in

text: all

text: the

text: experiments.

text: A

text: classiﬁer’s

text: prediction

text: for

text: a

text: given

text: object

text: is

text: a

text: vector

text: of

text: class

text: probabilities.

text: We

text: measure

text: the

text: total

text: prediction

text: error

text: simply

text: as

text: the

text: fraction

text: of

text: test

text: objects

text: where

text: the

text: true

text: class

text: is

text: diﬀerent

text: from

text: the

text: one

text: with

text: the

text: highest

text: predicted

text: probability.

text: All

text: the

text: ESB

text: models

text: have

text: consistent,

text: watertight

text: meshes,

text: so

text: all

text: the

text: features

text: from

text: Table

text: are

text: applicable.


text: 1.0 Bool 0.9 SA AN 0.8 QF EV 0.7 VAD 0.6 EAD 0.38 VE 0.36 Raw EV 0.34 0.5 Haar EV 0.32 0.30 0.4 0.28 0.26 0.3 0.24 0.22 0.2 0.20 1 2 4 8 16 32 1 2 4 8 16 Resolution Resolution (a) (b) Figure 2: (a) Relative eﬃciency of diﬀerent features and resolutions for classiﬁcation on ESB. The classiﬁer is trained by XGBoost, as an ensemble of depth-6 decision trees. The classiﬁcation · features are the raw voxel features (e.g., 3 43 = 192 features in the case of EV at resolution 4). (b) Relative eﬃciency of the raw and Haar-aggregated EV features at diﬀerent resolutions. Finally, we construct a classiﬁer using multiple local raw and statistical (percentile) features evaluated at multiple length scales. Speciﬁcally, we include in the set of features all raw non-Bool features from Section at resolutions 1, 2 and 4, and all (non-Bool) { } statistical features at the resolutions 2n 7 . Note that we can consider statistical voxel n=1 features at high resolutions thanks to our sparse representation of voxel features. Our statistical features include the percentiles 0, 25, 50, 75 and 100. It was determined through preliminary experiments that with these features the optimal results are achieved with very shallow, depth-2 trees. In Fig. 3a we show the error convergence history of the trained classiﬁer. The test error is shown separately for the O(3)-symmetrized and non-symmetrized classiﬁers. We observe, in particular, that with the non-symmetrized classiﬁer it takes about 10 (re- spectively, 30) depth-2 trees per class to reach the classiﬁcation errors 0.3 (0.2), while with the symmetrized one it takes 5 (15) trees. At the end of training the test error is approximately 0.15. In Fig. 3b we show the top 20 features (both raw and aggregated) in terms of the number of their occurrences in the trees of the ﬁnal classiﬁer. We observe that the top features include raw features at the trivial resolution 1 as well as statistical features at large resolutions. While the earlier experiment with individual features in Fig. 2a suggests that raw features can be useful at resolutions higher than 1 with deeper trees, in the present experiment they are clearly outperformed by statistical features. We also observe that the classiﬁer especially favors the VAD-, EAD-, and EV-based features and does not favor the SA-, QF- and AN-based ones. Interestingly, neither the total area ([1][SA]) nor the total volume ([1][VE]) is among the top-20 features, though some 9 rorre noitacifissalC rorre noitacifissalC ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Bool

text: SA

text: AN

text: QF

text: EV

text: VAD

text: EAD

text: VE

text: Raw

text: EV

text: Haar

text: EV

text: Resolution

text: Resolution

text: (a)

text: (b)

text: Figure

text: (a)

text: Relative

text: eﬃciency

text: of

text: diﬀerent

text: features

text: and

text: resolutions

text: for

text: classiﬁcation

text: on

text: ESB.

text: The

text: classiﬁer

text: is

text: trained

text: by

text: XGBoost,

text: as

text: an

text: ensemble

text: of

text: decision

text: trees.

text: The

text: classiﬁcation

text: features

text: are

text: the

text: raw

text: voxel

text: features

text: (e.g.,

text: features

text: in

text: the

text: case

text: of

text: EV

text: at

text: resolution

text: (b)

text: Relative

text: eﬃciency

text: of

text: the

text: raw

text: and

text: Haar-aggregated

text: EV

text: features

text: at

text: diﬀerent

text: resolutions.

text: Finally,

text: we

text: construct

text: a

text: classiﬁer

text: using

text: multiple

text: local

text: raw

text: and

text: statistical

text: (percentile)

text: features

text: evaluated

text: at

text: multiple

text: length

text: scales.

text: Speciﬁcally,

text: we

text: include

text: in

text: the

text: set

text: of

text: features

text: all

text: raw

text: non-Bool

text: features

text: from

text: Section

text: at

text: resolutions

text: and

text: and

text: all

text: (non-Bool)

text: statistical

text: features

text: at

text: the

text: resolutions

text: Note

text: that

text: we

text: can

text: consider

text: statistical

text: voxel

text: features

text: at

text: high

text: resolutions

text: thanks

text: to

text: our

text: sparse

text: representation

text: of

text: voxel

text: features.

text: Our

text: statistical

text: features

text: include

text: the

text: percentiles

text: and

text: It

text: was

text: determined

text: through

text: preliminary

text: experiments

text: that

text: with

text: these

text: features

text: the

text: optimal

text: results

text: are

text: achieved

text: with

text: very

text: shallow,

text: trees.

text: In

text: Fig.

text: we

text: show

text: the

text: error

text: convergence

text: history

text: of

text: the

text: trained

text: classiﬁer.

text: The

text: test

text: error

text: is

text: shown

text: separately

text: for

text: the

text: and

text: non-symmetrized

text: classiﬁers.

text: We

text: observe,

text: in

text: particular,

text: that

text: with

text: the

text: non-symmetrized

text: classiﬁer

text: it

text: takes

text: about

text: (re-

text: spectively,

text: trees

text: per

text: class

text: to

text: reach

text: the

text: classiﬁcation

text: errors

text: while

text: with

text: the

text: symmetrized

text: one

text: it

text: takes

text: trees.

text: At

text: the

text: end

text: of

text: training

text: the

text: test

text: error

text: is

text: approximately

text: In

text: Fig.

text: we

text: show

text: the

text: top

text: features

text: (both

text: raw

text: and

text: aggregated)

text: in

text: terms

text: of

text: the

text: number

text: of

text: their

text: occurrences

text: in

text: the

text: trees

text: of

text: the

text: ﬁnal

text: classiﬁer.

text: We

text: observe

text: that

text: the

text: top

text: features

text: include

text: raw

text: features

text: at

text: the

text: trivial

text: resolution

text: as

text: well

text: as

text: statistical

text: features

text: at

text: large

text: resolutions.

text: While

text: the

text: earlier

text: experiment

text: with

text: individual

text: features

text: in

text: Fig.

text: suggests

text: that

text: raw

text: features

text: can

text: be

text: useful

text: at

text: resolutions

text: higher

text: than

text: with

text: deeper

text: trees,

text: in

text: the

text: present

text: experiment

text: they

text: are

text: clearly

text: outperformed

text: by

text: statistical

text: features.

text: We

text: also

text: observe

text: that

text: the

text: classiﬁer

text: especially

text: favors

text: the

text: VAD-,

text: EAD-,

text: and

text: EV-based

text: features

text: and

text: does

text: not

text: favor

text: the

text: SA-,

text: QF-

text: and

text: AN-based

text: ones.

text: Interestingly,

text: neither

text: the

text: total

text: area

text: nor

text: the

text: total

text: volume

text: is

text: among

text: the

text: features,

text: though

text: some

text: rorre

text: noitacifissalC

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