text: sound-meaning associations are preferred by culturally, historically, and geo- graphically diverse human groups (Blasi, Wichmann, Hammarström, Stadler, & Christiansen, 2016). The framework above is agnostic concerning the type of association between sound and meaning. By doing that, we are borrowing the abstract perspective of network theory, that is a priori neutral concerning the nature or the origins of the edges (Newman, 2010; Barthélemy, 2011). Our framework could be generalized to accommodate Peirce’s classic types of refer- ence, i.e., iconic, indexical and symbolic (Deacon, 1997), or the state-of-the-art on the iconicity/systematicity distinction (Dingemanse, Blasi, Lupyan, Chris- tiansen, & Monaghan, 2015). An crucial reason to remain neutral is that the distinctions above were not made when deﬁning the laws of meaning that are the target of this article. The framework allows one to model lexical ambiguity: a lexically ambiguous word is a word such that its degree is greater than one. Although the model starts from a ﬂat hierarchy of concepts (by default all concepts have the same degree of generality), a word with an abstract meaning could be approximated either as a word linked to a single abstract concept or as a word linked to the multiple speciﬁc meanings it covers (Ferrer-i-Cancho, 2016c). As for the latter approach, the word for vehicle would be linked to the meanings for car, bike, ship, airplane,... Suppose that p(s , r ) is the joint probability of the unordered pair formed i j by s and r . By deﬁnition, (cid:88) (cid:88) i j n m p(s , r ) = 1. (10) i j i=1 j=1 The probability of s is (cid:88) i m p(s ) = p(s , r ) (11) i i j j=1 and the probability of r is (cid:88) i n p(r ) = p(s , r ). (12) j i j i=1 Our model shares the assumptions of distributional semantics that the mean- ing of a word is represented as a vector of the weights of diﬀerent concepts for that word (Lund & Burgess, 1996). In our framework, the meaning of the word s , is represented by the m-dimensional vector i { } p(s , r ), ..., p(s , r ), ..., p(s , r ) i 1 i j i m The joint probabilities p(s , r ) for all words and meanings deﬁnes a weighted i j matrix of the same size of A. In the coming sections, we will derive the meaning frequency-law deﬁning p(s , r ) as a function of A. Put diﬀerently, we will i j 5

text: sound-meaning

text: associations

text: are

text: preferred

text: by

text: culturally,

text: historically,

text: and

text: geo-

text: graphically

text: diverse

text: human

text: groups

text: (Blasi,

text: Wichmann,

text: Hammarström,

text: Stadler,

text: Christiansen,

text: The

text: framework

text: above

text: is

text: agnostic

text: concerning

text: the

text: type

text: of

text: association

text: between

text: sound

text: and

text: meaning.

text: By

text: doing

text: that,

text: we

text: are

text: borrowing

text: the

text: abstract

text: perspective

text: of

text: network

text: theory,

text: that

text: is

text: a

text: priori

text: neutral

text: concerning

text: the

text: nature

text: or

text: the

text: origins

text: of

text: the

text: edges

text: (Newman,

text: Barthélemy,

text: Our

text: framework

text: could

text: be

text: generalized

text: to

text: accommodate

text: Peirce’s

text: classic

text: types

text: of

text: refer-

text: ence,

text: i.e.,

text: iconic,

text: indexical

text: and

text: symbolic

text: (Deacon,

text: or

text: the

text: state-of-the-art

text: on

text: the

text: iconicity/systematicity

text: distinction

text: (Dingemanse,

text: Blasi,

text: Lupyan,

text: Chris-

text: tiansen,

text: Monaghan,

text: An

text: crucial

text: reason

text: to

text: remain

text: neutral

text: is

text: that

text: the

text: distinctions

text: above

text: were

text: not

text: made

text: when

text: deﬁning

text: the

text: laws

text: of

text: meaning

text: that

text: are

text: the

text: target

text: of

text: this

text: article.

text: The

text: framework

text: allows

text: one

text: to

text: model

text: lexical

text: ambiguity:

text: a

text: lexically

text: ambiguous

text: word

text: is

text: a

text: word

text: such

text: that

text: its

text: degree

text: is

text: greater

text: than

text: one.

text: Although

text: the

text: model

text: starts

text: from

text: a

text: ﬂat

text: hierarchy

text: of

text: concepts

text: (by

text: default

text: all

text: concepts

text: have

text: the

text: same

text: degree

text: of

text: generality),

text: a

text: word

text: with

text: an

text: abstract

text: meaning

text: could

text: be

text: approximated

text: either

text: as

text: a

text: word

text: linked

text: to

text: a

text: single

text: abstract

text: concept

text: or

text: as

text: a

text: word

text: linked

text: to

text: the

text: multiple

text: speciﬁc

text: meanings

text: it

text: covers

text: (Ferrer-i-Cancho,

text: As

text: for

text: the

text: latter

text: approach,

text: the

text: word

text: for

text: vehicle

text: would

text: be

text: linked

text: to

text: the

text: meanings

text: for

text: car,

text: bike,

text: ship,

text: airplane,...

text: Suppose

text: that

text: p(s

text: r

text: is

text: the

text: joint

text: probability

text: of

text: the

text: unordered

text: pair

text: formed

text: i

text: j

text: by

text: s

text: and

text: r

text: By

text: deﬁnition,

text: i

text: j

text: n

text: m

text: p(s

text: r

text: i

text: j

text: The

text: probability

text: of

text: s

text: is

text: i

text: m

text: p(s

text: p(s

text: r

text: i

text: i

text: j

text: and

text: the

text: probability

text: of

text: r

text: is

text: i

text: n

text: p(r

text: p(s

text: r

text: j

text: i

text: j

text: Our

text: model

text: shares

text: the

text: assumptions

text: of

text: distributional

text: semantics

text: that

text: the

text: mean-

text: ing

text: of

text: a

text: word

text: is

text: represented

text: as

text: a

text: vector

text: of

text: the

text: weights

text: of

text: diﬀerent

text: concepts

text: for

text: that

text: word

text: (Lund

text: Burgess,

text: In

text: our

text: framework,

text: the

text: meaning

text: of

text: the

text: word

text: s

text: is

text: represented

text: by

text: the

text: m-dimensional

text: vector

text: i

text: p(s

text: r

text: p(s

text: r

text: p(s

text: r

text: i

text: i

text: j

text: i

text: m

text: The

text: joint

text: probabilities

text: p(s

text: r

text: for

text: all

text: words

text: and

text: meanings

text: deﬁnes

text: a

text: weighted

text: i

text: j

text: matrix

text: of

text: the

text: same

text: size

text: of

text: A.

text: In

text: the

text: coming

text: sections,

text: we

text: will

text: derive

text: the

text: meaning

text: frequency-law

text: deﬁning

text: p(s

text: r

text: as

text: a

text: function

text: of

text: A.

text: Put

text: diﬀerently,

text: we

text: will

text: i

text: j


text: (cid:48) − µ(cid:126)e = µ(cid:126)e (cid:126)1 i i (cid:48) − ω(cid:126)e = ω(cid:126)e (cid:126)1. i i ∈ { } (cid:48) ∈ { } The condition ω 0, 1 is equivalent to ω(cid:126)e = (cid:126)0. ω 0, 1 leads to (cid:48) (cid:48) j (cid:48) i j µ(cid:126)eω(cid:126)e = 0 but trivially for being ω(cid:126)e null. i i ∈i { } The assumption φ = 1 and ω 0, 1 (orthogonality of row vectors of A), j transform Eq. 18 into Eq. 13 because a = 0 and ω = 0 are equivalent when ij j ω does not exceed 1. In general, Eq. 18 combined with the principle of contrast j gives (cid:88) m p(s ) = cµφ a i i ij j=1 = cµφ+1 i and ﬁnally Eq. 1 with 1 δ = . φ + 1 When φ = 1, we get δ = 1/2 again. Interestingly, the principle of contrast follows from the principle of mutual information maximization, a more fundamental principle that allows one to predict vocabulary learning in children and that can be combined with the principle of entropy minimization to predict Zipf’s law for word frequencies (Ferrer-i-Cancho, 2016c). With Eq. 15, we follow Bunge (Bunge, 2013, pp. 32-33) preventing scientiﬁc knowledge from becoming “an aggregation of disconnected information” and aspiring to build a “system of ideas that are logically connected among themselves”. It is possible to obtain a relaxed meaning frequency-law under more general conditions. In particular, we would like to get rid of the heavy constraint that meaning degrees cannot exceed one. Suppose that d is a constant such that ≤ ∈ { } 0 < d n. Some obvious but not very general conditions are ω 0, d for j all j or ω = d for all j. It is easy to see that they lead again to Eq. 13 when j φ = 1. A more general condition can be deﬁned as follows. First, we deﬁne | E[ωφ µ] as the conditional expectation of ωφ given µ for an edge. Here µ and ω are the degrees at both ends of an edge. Then suppose that A is given and that | | the E[ωφ µ] = E[ωφ] ωφ is mean independent of µ, namely E[ωφ µ] = E[ωφ] for each value of µ (Kolmogorov, 1956; Poirier, 1995), then the expectation of p(s ) i (as deﬁned in Eq. 18) knowing µ is i 8 ##LTLine##

label: equation

text: i

text: i

text: i

text: i

label: paragraph

text: The

text: condition

text: ω

text: is

text: equivalent

text: to

text: ω

text: leads

text: to

text: j

text: i

text: j

text: but

text: trivially

text: for

text: being

text: null.

text: i

text: i

text: ∈i

text: The

text: assumption

text: φ

text: and

text: ω

text: (orthogonality

text: of

text: row

text: vectors

text: of

text: A),

text: j

text: transform

text: Eq.

text: into

text: Eq.

text: because

text: a

text: and

text: ω

text: are

text: equivalent

text: when

text: ij

text: j

text: ω

text: does

text: not

text: exceed

text: In

text: general,

text: Eq.

text: combined

text: with

text: the

text: principle

text: of

text: contrast

text: j

text: gives

text: m

text: p(s

text: cµφ

text: a

text: i

text: i

text: ij

text: i

text: and

text: ﬁnally

text: Eq.

text: with

text: δ

text: φ

text: When

text: φ

text: we

text: get

text: δ

text: again.

text: Interestingly,

text: the

text: principle

text: of

text: contrast

text: follows

text: from

text: the

text: principle

text: of

text: mutual

text: information

text: maximization,

text: a

text: more

text: fundamental

text: principle

text: that

text: allows

text: one

text: to

text: predict

text: vocabulary

text: learning

text: in

text: children

text: and

text: that

text: can

text: be

text: combined

text: with

text: the

text: principle

text: of

text: entropy

text: minimization

text: to

text: predict

text: Zipf’s

text: law

text: for

text: word

text: frequencies

text: (Ferrer-i-Cancho,

text: With

text: Eq.

text: we

text: follow

text: Bunge

text: (Bunge,

text: pp.

text: preventing

text: scientiﬁc

text: knowledge

text: from

text: becoming

text: “an

text: aggregation

text: of

text: disconnected

text: information”

text: and

text: aspiring

text: to

text: build

text: a

text: “system

text: of

text: ideas

text: that

text: are

text: logically

text: connected

text: among

text: themselves”.

text: It

text: is

text: possible

text: to

text: obtain

text: a

text: relaxed

text: meaning

text: frequency-law

text: under

text: more

text: general

text: conditions.

text: In

text: particular,

text: we

text: would

text: like

text: to

text: get

text: rid

text: of

text: the

text: heavy

text: constraint

text: that

text: meaning

text: degrees

text: cannot

text: exceed

text: one.

text: Suppose

text: that

text: d

text: is

text: a

text: constant

text: such

text: that

text: d

text: n.

text: Some

text: obvious

text: but

text: not

text: very

text: general

text: conditions

text: are

text: ω

text: d

text: for

text: j

text: all

text: j

text: or

text: ω

text: d

text: for

text: all

text: j.

text: It

text: is

text: easy

text: to

text: see

text: that

text: they

text: lead

text: again

text: to

text: Eq.

text: when

text: j

text: φ

text: A

text: more

text: general

text: condition

text: can

text: be

text: deﬁned

text: as

text: follows.

text: First,

text: we

text: deﬁne

text: E[ωφ

text: µ]

text: as

text: the

text: conditional

text: expectation

text: of

text: ωφ

text: given

text: µ

text: for

text: an

text: edge.

text: Here

text: µ

text: and

text: ω

text: are

text: the

text: degrees

text: at

text: both

text: ends

text: of

text: an

text: edge.

text: Then

text: suppose

text: that

text: A

text: is

text: given

text: and

text: that

text: the

text: E[ωφ

text: µ]

text: E[ωφ]

text: ωφ

text: is

text: mean

text: independent

text: of

text: µ,

text: namely

text: E[ωφ

text: µ]

text: E[ωφ]

text: for

text: each

text: value

text: of

text: µ

text: (Kolmogorov,

text: Poirier,

text: then

text: the

text: expectation

text: of

text: p(s

text: i

text: (as

text: deﬁned

text: in

text: Eq.

text: knowing

text: µ

text: is

text: i

text: ##LTLine##


text: one has − − − yˆ = l + Ls 1[θ(cid:126) − ] y + Ls 1[θ(cid:126) − ] y + . . . + Ls−1[θ(cid:126) − ] y − . s s 0 [s 1,T] 0 1 [s 1,T] 1 s 1 [s 1,T] s 1 − (cid:3) Given that v = y yˆ (23) follows and therefore the induction is completed proving (ii). t t t It should be noted that the equivalence between the aﬃne output-based and aﬃne puriﬁed-output- based policies assumes that both policies can depend, in a non-anticipative manner, on the entire sequence of the regime switching sequence. In order to avoid exponential exploding in the number of parameters specifying the policies as N grows, we, in the sequel, will restrict the “switching memory” T of the policies, by restricting their parameters to depend on (at most) a ﬁxed moderate number of the latest values of the switching signal. With this restriction, the two classes of aﬃne policies in question are, in general, not equivalent and the respective families of achievable state-control trajectories are in a “general position” to each other. 5 Convex Design In this section, we develop a computationally tractable approach for designing an aﬃne puriﬁed- output-based policy. To this end in Section 5.1 we provide a semideﬁnite program specifying, when feasible, policy parameters ensuring the validity of the design speciﬁcations. Furthermore, in Section 5.2 we show that the coeﬃcients of our explicit convex program, can be computed in polynomial in N, M arithmetic operations, validating the computational tractability of the approach. 5.1 Convex Programming Formulation The aﬃne puriﬁed-output-based policy (10) has the favorable feature that it results in a state- control trajectory which for a ﬁxed realization of the switching signal is bi-aﬃne in the initial con- { } dition and disturbances ζ(cid:126) as well as the policy parameters h , Ht ≤ ≤ ≤ − . The computational t j j t,0 t N 1 tractability of our proposed policy design is owed to this fact that leads to a convex AARC via appropriate subsequent use of the inhomogeneous S-lemma. Theorem 2 Consider a switching signal trajectory θ(cid:126) ∈ ON. For a given disturbance vector ζ(cid:126) ∈ DN { } the state-control trajectory w(cid:126) is aﬃne in the policy parameters h , Ht ≤ ≤ ≤ − . Similarly ﬁx { } t j j t,0 t N 1 the values of the policy parameters h , Ht ≤ ≤ ≤ − then the state-control trajectory w(cid:126) is aﬃne t j j t,0 t N 1 in the disturbance vector ζ(cid:126). (cid:18) (cid:19) (cid:18) (cid:19) − Proof: By resorting (cid:89)to the system eq(cid:88)uation(cid:89)s (1), one has for k = 0, 1, . . . , N 1, − − − k 1 k 1 k 1 x = A [θ ] z + A [θ ] (R [θ ] d + B [θ ] u ). (24) k i i i (cid:81)i j j j j j j (cid:2) i=0 (cid:3) j=0 i=j+1 − − In the above expression we employ the convention k 1 A[θ ] = I , where k = 0, 1, . . . , N 1. i=k i nx Let ζ(cid:126)T = dT dT . . . dT− denote the exogenous disturbance trajectory, (24) is expressed in d 0 1 N 1 matrix notation as (cid:126)x = A z + B (cid:126)u + R ζ(cid:126) . (25) d The individual matrices are given explicitely in Appendix A1. 12 ##LTLine## ##LTLine## ##LTLine##

text: one

text: has

text: yˆ

text: l

text: Ls

text: y

text: Ls

text: y

text: y

text: s

text: s

text: [s

text: [s

text: s

text: [s

text: s

text: Given

text: that

text: v

text: y

text: yˆ

text: follows

text: and

text: therefore

text: the

text: induction

text: is

text: completed

text: proving

text: (ii).

text: t

text: t

text: t

text: It

text: should

text: be

text: noted

text: that

text: the

text: equivalence

text: between

text: the

text: aﬃne

text: output-based

text: and

text: aﬃne

text: puriﬁed-output-

text: based

text: policies

text: assumes

text: that

text: both

text: policies

text: can

text: depend,

text: in

text: a

text: non-anticipative

text: manner,

text: on

text: the

text: entire

text: sequence

text: of

text: the

text: regime

text: switching

text: sequence.

text: In

text: order

text: to

text: avoid

text: exponential

text: exploding

text: in

text: the

text: number

text: of

text: parameters

text: specifying

text: the

text: policies

text: as

text: N

text: grows,

text: we,

text: in

text: the

text: sequel,

text: will

text: restrict

text: the

text: “switching

text: memory”

text: T

text: of

text: the

text: policies,

text: by

text: restricting

text: their

text: parameters

text: to

text: depend

text: on

text: (at

text: most)

text: a

text: ﬁxed

text: moderate

text: number

text: of

text: the

text: latest

text: values

text: of

text: the

text: switching

text: signal.

text: With

text: this

text: restriction,

text: the

text: two

text: classes

text: of

text: aﬃne

text: policies

text: in

text: question

text: are,

text: in

text: general,

text: not

text: equivalent

text: and

text: the

text: respective

text: families

text: of

text: achievable

text: state-control

text: trajectories

text: are

text: in

text: a

text: “general

text: position”

text: to

text: each

text: other.

text: Convex

text: Design

text: In

text: this

text: section,

text: we

text: develop

text: a

text: computationally

text: tractable

text: approach

text: for

text: designing

text: an

text: aﬃne

text: puriﬁed-

text: output-based

text: policy.

text: To

text: this

text: end

text: in

text: Section

text: we

text: provide

text: a

text: semideﬁnite

text: program

text: specifying,

text: when

text: feasible,

text: policy

text: parameters

text: ensuring

text: the

text: validity

text: of

text: the

text: design

text: speciﬁcations.

text: Furthermore,

text: in

text: Section

text: we

text: show

text: that

text: the

text: coeﬃcients

text: of

text: our

text: explicit

text: convex

text: program,

text: can

text: be

text: computed

text: in

text: polynomial

text: in

text: N,

text: M

text: arithmetic

text: operations,

text: validating

text: the

text: computational

text: tractability

text: of

text: the

text: approach.

text: Convex

text: Programming

text: Formulation

text: The

text: aﬃne

text: puriﬁed-output-based

text: policy

text: has

text: the

text: favorable

text: feature

text: that

text: it

text: results

text: in

text: a

text: state-

text: control

text: trajectory

text: which

text: for

text: a

text: ﬁxed

text: realization

text: of

text: the

text: switching

text: signal

text: is

text: bi-aﬃne

text: in

text: the

text: initial

text: con-

text: dition

text: and

text: disturbances

text: as

text: well

text: as

text: the

text: policy

text: parameters

text: h

text: Ht

text: The

text: computational

text: t

text: j

text: j

text: t

text: N

text: tractability

text: of

text: our

text: proposed

text: policy

text: design

text: is

text: owed

text: to

text: this

text: fact

text: that

text: leads

text: to

text: a

text: convex

text: AARC

text: via

text: appropriate

text: subsequent

text: use

text: of

text: the

text: inhomogeneous

text: S-lemma.

text: Theorem

text: Consider

text: a

text: switching

text: signal

text: trajectory

text: ON.

text: For

text: a

text: given

text: disturbance

text: vector

text: DN

text: the

text: state-control

text: trajectory

text: is

text: aﬃne

text: in

text: the

text: policy

text: parameters

text: h

text: Ht

text: Similarly

text: ﬁx

text: t

text: j

text: j

text: t

text: N

text: the

text: values

text: of

text: the

text: policy

text: parameters

text: h

text: Ht

text: then

text: the

text: state-control

text: trajectory

text: is

text: aﬃne

text: t

text: j

text: j

text: t

text: N

text: in

text: the

text: disturbance

text: vector

text: Proof:

text: By

text: resorting

text: the

text: system

text: one

text: has

text: for

text: k

text: N

text: k

text: k

text: k

text: x

text: A

text: [θ

text: z

text: A

text: [θ

text: (R

text: [θ

text: d

text: B

text: [θ

text: u

text: k

text: i

text: i

text: i

text: j

text: j

text: j

text: j

text: j

text: j

text: In

text: the

text: above

text: expression

text: we

text: employ

text: the

text: convention

text: k

text: A[θ

text: I

text: where

text: k

text: N

text: i=k

text: i

text: nx

text: Let

text: dT

text: dT

text: dT−

text: denote

text: the

text: exogenous

text: disturbance

text: trajectory,

text: is

text: expressed

text: in

text: d

text: N

text: matrix

text: notation

text: as

text: A

text: z

text: B

text: R

text: d

text: The

text: individual

text: matrices

text: are

text: given

text: explicitely

text: in

text: Appendix

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Bertsimas et al. (2011). The degree of suboptimality of aﬃne policies was investigated via duality theory in Kuhn et al. (2011). Instances where aﬃne policies are provably optimal also include the linear quadratic Gaussian problem (LQG) (Kwakernaak and Sivan 1972, Anderson and Moore 1990), as well as its risk sensitive version with an exponential cost criterion (LEQG) (Whittle 1990), both problems admitting a dynamic programming solution. Further applications of aﬃne policies can be found in stochastic programming settings (Nemirovski and Shapiro 2005, Kuhn et al. 2011, Chen et al. 2008), and in the area of robust model predictive and receding horizon control (L¨ofberg 2003, Goulart et al. 2006, Goulart and Kerrigan 2007, Skaf and Boyd 2010). From a historical perspective some of the earliest accounts on the use of aﬃne policies in stochastic programming include (Charnes et al. 1958, Gartska and Wets 1974). The paper is organized as follows. Section 2 contains some common notation that will be used in the subsequent discussion. The formal problem statement is given in Section 3 along with two examples suggesting its relevance in operations research applications. Our contributions are stated in Section 3 as well. Section 4 introduces the concept of puriﬁed-output-based aﬃne policies for regime-switching models. The main theorems of the paper describing our tractable approach are in Section 5. We provide a numerical illustration of our methodology in Section 6, and we conclude with a summary and thoughts for future directions in Section 7. 2 Notation Z N The set of positive integers is denoted by , the set of non-negative integers by , the set of real R ∈ Z R + R × numbers by . For n, m let n denote the Euclidean n-space and n m the space of n-by-m + real matrices. Typically, vectors and matrices will be written with boldface letters. The transpose of ∈ R ∈ Z the column vector x n is denoted by xT and for i , [x] refers to the i-th entry of x. Similarly ∈ Z + i ∈ R × R for i, j , [A] refers to the (i, j)-th entry of the matrix A n m. The vector of 1’s in n + i,j R is denoted by 1 = [1, 1, . . . , 1]T, the vector of 0’s in n is 0 = [0, 0, . . . , 0]T. The identity/zero R × n × n ∈ R (cid:104) (cid:105) matrix in n n is I / O , the zero matrix in Rn m is O × . For x, y n, x, y denotes the n n × n m inner product of the two vectors. A sum (product) of n n matrices over an empty set of indices is × the n n zero (identity) matrix. 3 Problem Description, Examples, and Contributions In this section we ﬁrst provide the formal problem statement. Subsequently we give two examples suggesting its relevance to operations research applications. We conclude with a game-theoretic interpretation of our formulation, as well as by stating our contributions. 3.1 Problem statement We consider the state space description of a partially observed linear regime-switching model: x = z, 0 x = A [θ ] x + B [θ ] u + R [θ ] d , (1) t+1 t t t t t t t t t − y = C [θ ] x + D [θ ] d , t = 0, 1, . . . , N 1, t t t t t t t ∈ R ∈ R ∈ R ∈ R where xt nx, ut nu, yt ny, dt nd are the state, control, output, and exogeneous disturbance at time t, respectively, while N is the ﬁnite time horizon. At each instant of time t 3

text: Bertsimas

text: et

text: al.

text: The

text: degree

text: of

text: suboptimality

text: of

text: aﬃne

text: policies

text: was

text: investigated

text: via

text: duality

text: theory

text: in

text: Kuhn

text: et

text: al.

text: Instances

text: where

text: aﬃne

text: policies

text: are

text: provably

text: optimal

text: also

text: include

text: the

text: linear

text: quadratic

text: Gaussian

text: problem

text: (LQG)

text: (Kwakernaak

text: and

text: Sivan

text: Anderson

text: and

text: Moore

text: as

text: well

text: as

text: its

text: risk

text: sensitive

text: version

text: with

text: an

text: exponential

text: cost

text: criterion

text: (LEQG)

text: (Whittle

text: both

text: problems

text: admitting

text: a

text: dynamic

text: programming

text: solution.

text: Further

text: applications

text: of

text: aﬃne

text: policies

text: can

text: be

text: found

text: in

text: stochastic

text: programming

text: settings

text: (Nemirovski

text: and

text: Shapiro

text: Kuhn

text: et

text: al.

text: Chen

text: et

text: al.

text: and

text: in

text: the

text: area

text: of

text: robust

text: model

text: predictive

text: and

text: receding

text: horizon

text: control

text: (L¨ofberg

text: Goulart

text: et

text: al.

text: Goulart

text: and

text: Kerrigan

text: Skaf

text: and

text: Boyd

text: From

text: a

text: historical

text: perspective

text: some

text: of

text: the

text: earliest

text: accounts

text: on

text: the

text: use

text: of

text: aﬃne

text: policies

text: in

text: stochastic

text: programming

text: include

text: (Charnes

text: et

text: al.

text: Gartska

text: and

text: Wets

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: contains

text: some

text: common

text: notation

text: that

text: will

text: be

text: used

text: in

text: the

text: subsequent

text: discussion.

text: The

text: formal

text: problem

text: statement

text: is

text: given

text: in

text: Section

text: along

text: with

text: two

text: examples

text: suggesting

text: its

text: relevance

text: in

text: operations

text: research

text: applications.

text: Our

text: contributions

text: are

text: stated

text: in

text: Section

text: as

text: well.

text: Section

text: introduces

text: the

text: concept

text: of

text: puriﬁed-output-based

text: aﬃne

text: policies

text: for

text: regime-switching

text: models.

text: The

text: main

text: theorems

text: of

text: the

text: paper

text: describing

text: our

text: tractable

text: approach

text: are

text: in

text: Section

text: We

text: provide

text: a

text: numerical

text: illustration

text: of

text: our

text: methodology

text: in

text: Section

text: and

text: we

text: conclude

text: with

text: a

text: summary

text: and

text: thoughts

text: for

text: future

text: directions

text: in

text: Section

text: Notation

text: Z

text: N

text: The

text: set

text: of

text: positive

text: integers

text: is

text: denoted

text: by

text: the

text: set

text: of

text: non-negative

text: integers

text: by

text: the

text: set

text: of

text: real

text: R

text: Z

text: R

text: R

text: numbers

text: by

text: For

text: n,

text: m

text: let

text: n

text: denote

text: the

text: Euclidean

text: n-space

text: and

text: n

text: m

text: the

text: space

text: of

text: n-by-m

text: real

text: matrices.

text: Typically,

text: vectors

text: and

text: matrices

text: will

text: be

text: written

text: with

text: boldface

text: letters.

text: The

text: transpose

text: of

text: R

text: Z

text: the

text: column

text: vector

text: x

text: n

text: is

text: denoted

text: by

text: xT

text: and

text: for

text: i

text: [x]

text: refers

text: to

text: the

text: i-th

text: entry

text: of

text: x.

text: Similarly

text: Z

text: i

text: R

text: R

text: for

text: i,

text: j

text: [A]

text: refers

text: to

text: the

text: (i,

text: j)-th

text: entry

text: of

text: the

text: matrix

text: A

text: n

text: m.

text: The

text: vector

text: of

text: in

text: n

text: i,j

text: R

text: is

text: denoted

text: by

text: the

text: vector

text: of

text: in

text: n

text: is

text: The

text: identity/zero

text: R

text: n

text: n

text: R

text: matrix

text: in

text: n

text: n

text: is

text: I

text: O

text: the

text: zero

text: matrix

text: in

text: Rn

text: m

text: is

text: O

text: For

text: x,

text: y

text: n,

text: x,

text: y

text: denotes

text: the

text: n

text: n

text: n

text: m

text: inner

text: product

text: of

text: the

text: two

text: vectors.

text: A

text: sum

text: (product)

text: of

text: n

text: n

text: matrices

text: over

text: an

text: empty

text: set

text: of

text: indices

text: is

text: the

text: n

text: n

text: zero

text: (identity)

text: matrix.

text: Problem

text: Description,

text: Examples,

text: and

text: Contributions

text: In

text: this

text: section

text: we

text: ﬁrst

text: provide

text: the

text: formal

text: problem

text: statement.

text: Subsequently

text: we

text: give

text: two

text: examples

text: suggesting

text: its

text: relevance

text: to

text: operations

text: research

text: applications.

text: We

text: conclude

text: with

text: a

text: game-theoretic

text: interpretation

text: of

text: our

text: formulation,

text: as

text: well

text: as

text: by

text: stating

text: our

text: contributions.

text: Problem

text: statement

text: We

text: consider

text: the

text: state

text: space

text: description

text: of

text: a

text: partially

text: observed

text: linear

text: regime-switching

text: model:

text: x

text: z,

text: x

text: A

text: [θ

text: x

text: B

text: [θ

text: u

text: R

text: [θ

text: d

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: y

text: C

text: [θ

text: x

text: D

text: [θ

text: d

text: t

text: N

text: t

text: t

text: t

text: t

text: t

text: t

text: t

text: R

text: R

text: R

text: R

text: where

text: xt

text: nx,

text: ut

text: nu,

text: yt

text: ny,

text: dt

text: nd

text: are

text: the

text: state,

text: control,

text: output,

text: and

text: exogeneous

text: disturbance

text: at

text: time

text: t,

text: respectively,

text: while

text: N

text: is

text: the

text: ﬁnite

text: time

text: horizon.

text: At

text: each

text: instant

text: of

text: time

text: t


text: THIS PAPER IS ACCEPTED BY CSEE JOURNAL OF POWER AND ENERGY SYSTEM, DOI WILL BE PROVIDED WHEN IT IS AVAILABLE. 7 x 105 1.8 w/o TCSC (Peak) w/t TCSC (Peak) 1.7 w/o TCSC (Normal) w/t TCSC (Normal) 1.6 1.5 1.4 1.3 1.2 1B.a1se 60−61 8−545−495−114−1115−1947−6915−1730−1738−3747−4917−3148−4926−3023−2523−2425−278−3038−6523−13124−11519−3427−3217−1834−3620−2143−4428−2937−3949−66 Operating State Fig. 3. Hourly generation cost for peak and normal load level of IEEE 118-bus system. TABLE II load is provided for this test system in [39], we treat that PLANNING CRITERION FOR THE TEST SYSTEM load as the normal load level. The peak load level is 20% − higher than the normal load level and the low load level is Base N 1 Contingency ≤ ≤ ≤ ≤ 20% less than the normal load level. The computer used for Voltage (p.u.) 0.94 V 1.06 0.9 V 1.1 Thermal Limits Smax 1.1Smax all the simulations has an Inter Core(TM) i5-2400M CPU @ k k 2.30 GHz with 4.00 GB of RAM. The complete model is programmed in GAMS [40]. The MILP master problem is A. IEEE 118-Bus System solved by CPLEX [41] and the NLP subproblems are solved by IPOPT [42]. The IEEE 118-bus system has 118 buses, 177 transmission We consider the allocation strategy for the TCSC in this lines, 9 transformers and 19 generators. The total active and paper. The compensation rate for the TCSC is allowed to reactive load for the peak load level are 4930 MW and 1661 vary from -70% to 20% of its corresponding reactance [20]. MVar. The active and reactive power generation capacity are According to [15], the annual investment cost of TCSC is 6466 MW and 6325 MVar. The thermal limits for the transmis- converted from its total investment cost with the yearly interest sion lines are decreased artiﬁcially to create congestions. We rate and life span using equation (58). In this study, the interest consider 30 contingencies based on the congestion severity rate d is assumed to be 5% and the life span LT is selected [44] so the number of operating states is 93. Moreover, we to be 5 years [20]. leverage the sensitivity approach [29] to select 30 candidate locations to install TCSC. The complete procedures are given · d(1 + d)LT below: A˜I = AI − (58) k k (1 + d)LT 1 1) Run an OPF for each operating state without TCSC using IPOPT as the solver. Note that the coupling constraints between the base state and contingency state are not The budget Amax is assumed to be 3 M$. The algorithm considered. In addition, each line reactance is treated as tolerance (cid:15) is set to be 0.2%. The constant α in (39) is − down an optimization variable and constraint (59) is included in selected to be 1010 through a trial-and-error process. The the OPF model, i.e., ﬁx the line reactance to its original duration of peak, normal and low load level are assumed to value: be 2190 h, 4380 h and 2190 h, respectively. In addition, based on the typical values of line outage rate provided in [43], we assume that the forced outage rate for the contingency x˜ = x (59) kct k branch is 0.1%. Table I gives the number of operating hours in each state. Finally, the planning criterion for the base and contingency operating condition is provided in Table II. 2) Obtain the sensitivity (λkct) of the operation cost with respect to the change of line reactance in each state, i.e., TABLE I value of the dual variable associated with constraint (59). DURATION OF EACH OPERATING STATE 3) Compute the weighted sensitivity (λ¯ ) of branch k by (cid:88) (cid:88) k equation (60): Peak Normal Low # of hours for base state 2124.3 4248.6 2124.3 | | # of hours for each contingency state 2.19 4.38 2.19 λ¯ = π λ x (60) k ct kct k ∈ ∈ t ΩT c Ωc ]h/$[ tsoC noitareneG ##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## ##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: THIS

text: PAPER

text: IS

text: ACCEPTED

text: BY

text: CSEE

text: JOURNAL

text: OF

text: POWER

text: AND

text: ENERGY

text: SYSTEM,

text: DOI

text: WILL

text: BE

text: PROVIDED

text: WHEN

text: IT

text: IS

text: AVAILABLE.

text: x

font: default

text: w/o

text: TCSC

text: (Peak)

text: w/t

text: TCSC

text: (Peak)

text: w/o

text: TCSC

text: (Normal)

text: w/t

text: TCSC

text: (Normal)

text: Operating

text: State

text: Fig.

text: Hourly

label: caption

text: generation

text: cost

text: for

text: peak

text: and

text: normal

text: load

text: level

text: of

text: IEEE

text: system.

text: TABLE

text: II

text: load

text: is

text: provided

text: for

text: this

text: test

text: system

text: in

text: we

text: treat

text: that

text: PLANNING

text: CRITERION

text: FOR

text: THE

text: TEST

text: SYSTEM

text: load

text: as

text: the

text: normal

text: load

text: level.

text: The

text: peak

text: load

text: level

text: is

text: higher

text: than

text: the

text: normal

text: load

text: level

text: and

text: the

text: low

text: load

text: level

text: is

text: Base

text: N

text: Contingency

text: less

text: than

text: the

text: normal

text: load

text: level.

text: The

text: computer

text: used

text: for

text: Voltage

text: (p.u.)

text: V

text: V

text: Thermal

text: Limits

text: Smax

text: all

text: the

text: simulations

text: has

text: an

text: Inter

text: Core(TM)

text: CPU

text: k

text: k

text: GHz

text: with

text: GB

text: of

text: RAM.

text: The

text: complete

text: model

text: is

text: programmed

text: in

text: GAMS

text: The

text: MILP

text: master

text: problem

text: is

text: A.

text: IEEE

text: System

text: solved

text: by

text: CPLEX

text: and

text: the

text: NLP

text: subproblems

text: are

text: solved

text: by

text: IPOPT

text: The

text: IEEE

text: system

text: has

text: buses,

text: transmission

text: We

text: consider

text: the

text: allocation

text: strategy

text: for

text: the

text: TCSC

text: in

text: this

text: lines,

text: transformers

text: and

text: generators.

text: The

text: total

text: active

text: and

text: paper.

text: The

text: compensation

text: rate

text: for

text: the

text: TCSC

text: is

text: allowed

text: to

text: reactive

text: load

text: for

text: the

text: peak

text: load

text: level

text: are

text: MW

text: and

text: vary

text: from

text: to

text: of

text: its

text: corresponding

text: reactance

text: MVar.

text: The

text: active

text: and

text: reactive

text: power

text: generation

text: capacity

text: are

text: According

text: to

text: the

text: annual

text: investment

text: cost

text: of

text: TCSC

text: is

text: MW

text: and

text: MVar.

text: The

text: thermal

text: limits

text: for

text: the

text: transmis-

text: converted

text: from

text: its

text: total

text: investment

text: cost

text: with

text: the

text: yearly

text: interest

text: sion

text: lines

text: are

text: decreased

text: artiﬁcially

text: to

text: create

text: congestions.

text: We

text: rate

text: and

text: life

text: span

text: using

text: equation

text: In

text: this

text: study,

text: the

text: interest

text: consider

text: contingencies

text: based

text: on

text: the

text: congestion

text: severity

text: rate

text: d

text: is

text: assumed

text: to

text: be

text: and

text: the

text: life

text: span

text: LT

text: is

text: selected

text: so

text: the

text: number

text: of

text: operating

text: states

text: is

text: Moreover,

text: we

text: to

text: be

text: years

text: leverage

text: the

text: sensitivity

text: approach

text: to

text: select

text: candidate

text: locations

text: to

text: install

text: TCSC.

text: The

text: complete

text: procedures

text: are

text: given

text: d)LT

text: below:

text: A˜I

text: AI

text: k

text: k

text: d)LT

text: Run

text: an

text: OPF

text: for

text: each

text: operating

text: state

text: without

text: TCSC

text: using

text: IPOPT

text: as

text: the

text: solver.

text: Note

text: that

text: the

text: coupling

text: constraints

text: between

text: the

text: base

text: state

text: and

text: contingency

text: state

text: are

text: not

text: The

text: budget

text: Amax

text: is

text: assumed

text: to

text: be

text: M$.

text: The

text: algorithm

text: considered.

text: In

text: addition,

text: each

text: line

text: reactance

text: is

text: treated

text: as

text: tolerance

text: is

text: set

text: to

text: be

text: The

text: constant

text: α

text: in

text: is

text: down

text: an

text: optimization

text: variable

text: and

text: constraint

text: is

text: included

text: in

text: selected

text: to

text: be

text: through

text: a

text: trial-and-error

text: process.

text: The

text: the

text: OPF

text: model,

text: i.e.,

text: ﬁx

text: the

text: line

text: reactance

text: to

text: its

text: original

text: duration

text: of

text: peak,

text: normal

text: and

text: low

text: load

text: level

text: are

text: assumed

text: to

text: value:

text: be

text: h,

text: h

text: and

text: h,

text: respectively.

text: In

text: addition,

text: based

text: on

text: the

text: typical

text: values

text: of

text: line

text: outage

text: rate

text: provided

text: in

text: we

text: assume

text: that

text: the

text: forced

text: outage

text: rate

text: for

text: the

text: contingency

text: x˜

text: x

text: kct

text: k

text: branch

text: is

text: Table

text: I

text: gives

text: the

text: number

text: of

text: operating

text: hours

text: in

text: each

text: state.

text: Finally,

text: the

text: planning

text: criterion

text: for

text: the

text: base

text: and

text: contingency

text: operating

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text: 6 (cid:20) (cid:21) (cid:20) (cid:21) situation, and write an expansion of the functional g , f given by eq. (32), with η = 0, as a power series in the scalar order parameter, 9 1 − 2 1 b = β 1 + n T δ2 + − = 0 (43) 4 c 2 s c 1 + 2S 1 S c c g = g (0) + A S + A S2 + A S3 + ...., (47) f f 1 2 3 and where the coeﬃcients A , A , ..., depend on the ther- 1 2 −1 4 1 1 modynamic ﬁeld variables, T and σ, and on the model c = + − = 0. (44) 2 (1 + 2S )2 2 (1 S )2 parameters. It is straightforward to write the expressions c c of these coeﬃcients. For exam(cid:0)ple, (cid:1)A(cid:0) is given b(cid:1)y 1 From eq. (44), we have (cid:0) (cid:1) − 3 − 3 1 + exp 3δσ 1 + 3µ δ2 A = µ δ2 4 4 s . (48) 1 1 8 s 2 2 + exp 3βδσ S = . (45) 4 c 4 From this equation, we see that A = 0 for σ = 0, 1 regardless of the value of the distortion, which conﬁrms We then use eq. (43), and write the critical tempera- the existence of a disordered phase at suﬃciently high ture, (cid:20) (cid:21) temperatures. It is not diﬃcult to relate the coeﬃcients of the Landau expansion (47) with the corresponding co- − (cid:54) 27 − 27 1 eﬃcients of eq. (36). In particular, A = 0 indicates T = 1 n δ2 . (46) 3 c 32 64 s the characteristic ﬁrst-order nature of the nematic tran- sition. Also, we can use the Landau expansion to check − and conﬁrm the location of the critical point in the σ T This critical temperature increases with the parameter phase diagram. ω, given by B. Biaxial transitions ω = n δ2, s If the external stress is suﬃciently large, and negative, which gauges the strength of the coupling, and diverges with the usual positive strain anisotropy, δ > 0, the ne- → − for ω 64/27 = 2.3703..., which is an indication that matic directors tend to be parallel to the x y planes, − the contact with the experimental situation is restricted which gives rise to biaxial ordering. In the σ T phase to relatively small (cid:20)values of(cid:21)ω. We can also write an diagram, we then predict the existence of a line of second expression for the stress at the critical point, order transitions to a low-temperature biaxial phase. − This second-order line in the σ T plane can be located 4 − 2 from an analysis of the general equations of state for small δσ = T ln 2 = (0.03531...) T , c 3 c 3 c values of the order parameter η. Taking into account equations (29), (30), and (31), we w(cid:0)rit(cid:1)e the expansions which is a linear function of T . We remark that this is a c positive and relatively small stress ﬁeld, and that it does not make physical sense for ω > 64/27. S = S + S η2 + O η4 , (49) 0 2 As it is pointed out by Warner and Terentjev [1], the engineering shear modulus of polymeric chains, at room temperatures, is of the order of 104 to 106 GPa, which is and (cid:0) (cid:1) at least ﬁve orders of magnitude smaller than the corre- sponding shear modulus of usual solids. Therefore, tak- 1 = E + E η2 + O η4 , (50) ing into account that 0 < δ < 1, at least for prolate 1 2 elastomers, we claim that physically realistic and acces- sible results will be restricted to quite small values of the where the coeﬃcients S , S , E , E , ...., depend on parameter ω = n δ2. In some recent numerical simula- 0 2 1 2 s stress, temperature, and the parameters of the model sys- tions for a uniaxial nematic elastomer on a lattice, Pasini ≈ tem. and coworkers [22] assumed that n 0.3. In the next (cid:54) s The line of stability of the biaxial solution, η2 = 0, Section, we use a typical small value, ω = 0.2, to draw comes from the equation some graphs to illustrate our main ﬁndings. It is certainly interesting to make contact with the phe- nomenological expansions of the free energy, which have 1 = E , (51) 1 been written by several authors, as Selinger, Jeon, and Ratna [7, 8]. Let us then consider the special uniaxial ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: situation,

text: and

text: write

text: an

text: expansion

text: of

text: the

text: functional

text: g

text: f

text: given

text: by

text: eq.

text: with

text: η

text: as

text: a

text: power

text: series

text: in

text: the

text: scalar

text: order

text: parameter,

text: b

text: β

text: n

text: T

text: δ2

text: c

text: s

text: c

text: S

text: c

text: c

text: g

text: g

text: A

text: S

text: A

text: A

text: f

text: f

text: and

text: where

text: the

text: coeﬃcients

text: A

text: A

text: depend

text: on

text: the

text: ther-

text: −1

text: modynamic

text: ﬁeld

text: variables,

text: T

text: and

text: σ,

text: and

text: on

text: the

text: model

text: c

text: S

text: parameters.

text: It

text: is

text: straightforward

text: to

text: write

text: the

text: expressions

text: c

text: c

text: of

text: these

text: coeﬃcients.

text: For

text: is

text: given

text: From

text: eq.

text: we

text: have

text: exp

text: 3δσ

text: 3µ

text: δ2

text: A

text: µ

text: δ2

text: s

text: s

text: exp

text: 3βδσ

text: S

text: c

text: From

text: this

text: equation,

text: we

text: see

text: that

text: A

text: for

text: σ

text: regardless

text: of

text: the

text: value

text: of

text: the

text: distortion,

text: which

text: conﬁrms

text: We

text: then

text: use

text: eq.

text: and

text: write

text: the

text: critical

text: tempera-

text: the

text: existence

text: of

text: a

text: disordered

text: phase

text: at

text: suﬃciently

text: high

text: ture,

text: temperatures.

text: It

text: is

text: not

text: diﬃcult

text: to

text: relate

text: the

text: coeﬃcients

text: of

text: the

text: Landau

text: expansion

text: with

text: the

text: corresponding

text: co-

text: eﬃcients

text: of

text: eq.

text: In

text: particular,

text: A

text: indicates

text: T

text: n

text: δ2

text: c

text: s

text: the

text: characteristic

text: ﬁrst-order

text: nature

text: of

text: the

text: nematic

text: tran-

text: sition.

text: Also,

text: we

text: can

text: use

text: the

text: Landau

text: expansion

text: to

text: check

text: and

text: conﬁrm

text: the

text: location

text: of

text: the

text: critical

text: point

text: in

text: the

text: σ

text: T

text: This

text: critical

text: temperature

text: increases

text: with

text: the

text: parameter

text: phase

text: diagram.

text: ω,

text: given

text: by

text: B.

text: Biaxial

text: transitions

text: ω

text: n

text: s

text: If

text: the

text: external

text: stress

text: is

text: suﬃciently

text: large,

text: and

text: negative,

text: which

text: gauges

text: the

text: strength

text: of

text: the

text: coupling,

text: and

text: diverges

text: with

text: the

text: usual

text: positive

text: strain

text: anisotropy,

text: δ

text: the

text: ne-

text: for

text: ω

text: which

text: is

text: an

text: indication

text: that

text: matic

text: directors

text: tend

text: to

text: be

text: parallel

text: to

text: the

text: x

text: y

text: planes,

text: the

text: contact

text: with

text: the

text: experimental

text: situation

text: is

text: restricted

text: which

text: gives

text: rise

text: to

text: biaxial

text: ordering.

text: In

text: the

text: σ

text: T

text: phase

text: to

text: relatively

text: small

text: We

text: can

text: also

text: write

text: an

text: diagram,

text: we

text: then

text: predict

text: the

text: existence

text: of

text: a

text: line

text: of

text: second

text: expression

text: for

text: the

text: stress

text: at

text: the

text: critical

text: point,

text: order

text: transitions

text: to

text: a

text: low-temperature

text: biaxial

text: phase.

text: This

text: second-order

text: line

text: in

text: the

text: σ

text: T

text: plane

text: can

text: be

text: located

text: from

text: an

text: analysis

text: of

text: the

text: general

text: equations

text: of

text: state

text: for

text: small

text: δσ

text: T

text: ln

text: T

text: c

text: c

text: c

text: values

text: of

text: the

text: order

text: parameter

text: η.

text: Taking

text: into

text: account

text: equations

text: and

text: we

text: the

text: expansions

text: which

text: is

text: a

text: linear

text: function

text: of

text: T

text: We

text: remark

text: that

text: this

text: is

text: a

text: c

text: positive

text: and

text: relatively

text: small

text: stress

text: ﬁeld,

text: and

text: that

text: it

text: does

text: not

text: make

text: physical

text: sense

text: for

text: ω

text: S

text: S

text: S

text: η2

text: O

text: η4

text: As

text: it

text: is

text: pointed

text: out

text: by

text: Warner

text: and

text: Terentjev

text: the

text: engineering

text: shear

text: modulus

text: of

text: polymeric

text: chains,

text: at

text: room

text: temperatures,

text: is

text: of

text: the

text: order

text: of

text: to

text: GPa,

text: which

text: is

text: and

text: at

text: least

text: ﬁve

text: orders

text: of

text: magnitude

text: smaller

text: than

text: the

text: corre-

text: sponding

text: shear

text: modulus

text: of

text: usual

text: solids.

text: Therefore,

text: tak-

text: E

text: E

text: η2

text: O

text: η4

text: ing

text: into

text: account

text: that

text: δ

text: at

text: least

text: for

text: prolate

text: elastomers,

text: we

text: claim

text: that

text: physically

text: realistic

text: and

text: acces-

text: sible

text: results

text: will

text: be

text: restricted

text: to

text: quite

text: small

text: values

text: of

text: the

text: where

text: the

text: coeﬃcients

text: S

text: S

text: E

text: E

text: depend

text: on

text: parameter

text: ω

text: n

text: In

text: some

text: recent

text: numerical

text: simula-

text: s

text: stress,

text: temperature,

text: and

text: the

text: parameters

text: of

text: the

text: model

text: sys-

text: tions

text: for

text: a

text: uniaxial

text: nematic

text: elastomer

text: on

text: a

text: lattice,

text: Pasini

text: tem.

text: and

text: coworkers

text: assumed

text: that

text: n

text: In

text: the

text: next

text: s

text: The

text: line

text: of

text: stability

text: of

text: the

text: biaxial

text: solution,

text: η2

text: Section,

text: we

text: use

text: a

text: typical

text: small

text: value,

text: ω

text: to

text: draw

text: comes

text: from

text: the

text: equation

text: some

text: graphs

text: to

text: illustrate

text: our

text: main

text: ﬁndings.

text: It

text: is

text: certainly

text: interesting

text: to

text: make

text: contact

text: with

text: the

text: phe-

text: nomenological

text: expansions

text: of

text: the

text: free

text: energy,

text: which

text: have

text: E

text: been

text: written

text: by

text: several

text: authors,

text: as

text: Selinger,

text: Jeon,

text: and

text: Ratna

text: Let

text: us

text: then

text: consider

text: the

text: special

text: uniaxial

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text: Bulge eﬀect on bar instability 11 t=0 t=9.78 GYr along bar major axis t=9.78 GYr along bar minor axis 4 MA00 MA00 MA00 2 0 −2 −4 4 MA01 MA01 MA01 2 0 −2 −4 4 MA02 MA02 MA02 2 0 −2 −4 4 MA03 MA03 MA03 2 0 −2 −4 −20−15−10 −5 0 5 10 15 20 −20−15−10 −5 0 5 10 15 20 −20−15−10 −5 0 5 10 15 20 Y(kpc) Y(kpc) Y(kpc) Figure 18. First column shows edge on(YZ cross section) of MA models at t=0; second and third column shows 2 diﬀerent edge on view at t=9.78 Gyr t=0 t=9.78 GYr along bar major axis t=9.78 GYr along bar minor axis 4 2 MB00 MB00 MB00 0 −2 −4 4 2 MB01 MB01 MB01 0 −2 −4 4 2 MB02 MB02 MB02 0 −2 −4 4 2 MB03 MB03 MB03 0 −2 −4 4 2 MB04 MB04 MB04 0 −2 −4 4 2 MB05 MB05 MB05 0 −2 −4 4 2 MB06 MB06 MB06 0 −2 −4 −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 Y(kpc) Y(kpc) Y(kpc) Figure 19. First column shows edge on(YZ cross section) of MB models at t=0; second and third column shows 2 diﬀerent edge on view at t=9.78 Gyr MNRAS 000, 1–12 (2016) )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z )cpk(Z ##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## 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text: 12 Kataria & Das REFERENCES Ostriker J. P., Peebles P. J. E., 1973, ApJ, 186, 467 Pfenniger D., Norman C., 1990, ApJ, 363, 391 Aguerri J. A. L., Balcells M., Peletier R. F., 2001, A&A, 367, 428 Raha N., Sellwood J. A., James R. A., Kahn F. D., 1991, Nature, 352, 411 Athanassoula E., 2002, ApJ, 569, L83 Romano-Díaz E., Shlosman I., Heller C., Hoﬀman Y., 2008, ApJ, 687, L13 Athanassoula E., 2003, MNRAS, 341, 1179 Saha K., 2015, ApJ, 806, L29 Athanassoula E., Misiriotis A., 2002, MNRAS, 330, 35 Saha K., Naab T., 2013, MNRAS, 434, 1287 Barazza F. D., Jogee S., Marinova I., 2008, 396, 351 Saha K., Martinez-Valpuesta I., Gerhard O., 2012, MNRAS, 421, 333 Barnes J. E., Hernquist L. E., 1991, ApJ, 370, L65 Saha K., Gerhard O., Martinez-Valpuesta I., 2016, A&A, 588, A42 Barnes J., Hut P., 1986, Nature, 324, 446 Salo H., 1991, A&A, 243, 118 Barway S., Saha K., Vaghmare K., Kembhavi A. K., 2016, MNRAS, 463, Scannapieco C., Athanassoula E., 2012, MNRAS, 425, L10 L41 Sellwood J. A., 1980a, A&A, 89, 296 Baugh C. M., Cole S., Frenk C. S., 1996, MNRAS, 283, 1361 Sellwood J. A., 1980b, A&A, 89, 296 Binney J., Merriﬁeld M., 1998, Galactic Astronomy Sellwood J. A., Evans N. W., 2001, ApJ, 546, 176 BinneyJ.,TremaineS.,2008,GalacticDynamics:SecondEdition.Princeton Sheth K., et al., 2008a, ApJ, 675, 1141 University Press Sheth K., et al., 2008b, ApJ, 675, 1141 Bournaud F., Jog C. J., Combes F., 2007, A&A, 476, 1179 Simmons B. D., et al., 2014, MNRAS, 445, 3466 Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., Kravtsov A. V., Klypin Skibba R. A., et al., 2012, MNRAS, 423, 1485 A. A., Primack J. R., Dekel A., 2001, MNRAS, 321, 559 Springel V., 2005, MNRAS, 364, 1105 Buta R. J., et al., 2015, ApJS, 217, 32 Springel V., Yoshida N., White S. D. M., 2001, New Astron., 6, 79 Combes F., Sanders R. H., 1981, A&A, 96, 164 Springel V., Di Matteo T., Hernquist L., 2005, MNRAS, 361, 776 Combes F., Debbasch F., Friedli D., Pfenniger D., 1990, A&A, 233, 82 ToomreA.,1981,Thestructureandevolutionofnormalgalaxies.Cambridge Das M., Teuben P. J., Vogel S. N., Regan M. W., Sheth K., Harris A. I., University Press, Cambridge Jeﬀerys W. H., 2003, ApJ, 582, 190 Tremaine S., Weinberg M. D., 1984, MNRAS, 209, 729 Das M., Laurikainen E., Salo H., Buta R., 2008, Ap&SS, 317, 163 Valenzuela O., Klypin A., 2003, MNRAS, 345, 406 Debattista V. P., Sellwood J. A., 2000, The Astrophysical Journal, 543, 704 Weinberg M. D., Katz N., 2007, MNRAS, 375, 460 Dubinski J., Berentzen I., Shlosman I., 2009, ApJ, 697, 293 Widrow L. M., Pym B., Dubinski J., 2008, ApJ, 679, 1239 Efstathiou G., Lake G., Negroponte J., 1982a, MNRAS, 199, 1069 Wyse R. F. G., Gilmore G., Franx M., 1997, ARA&A, 35, 637 Efstathiou G., Lake G., Negroponte J., 1982b, MNRAS, 199, 1069 Yurin D., Springel V., 2014, MNRAS, 444, 62 Eggen O. J., Lynden-Bell D., Sandage A. R., 1962, ApJ, 136, 748 Elmegreen B. G., Elmegreen D. M., 1985, ApJ, 288, 438 This paper has been typeset from a TEX/LATEX ﬁle prepared by the author. Elmegreen B. G., Elmegreen D. M., Hirst A. C., 2004, The Astrophysical Journal, 612, 191 Erwin P., 2005, MNRAS, 364, 283 Eskridge P. B., et al., 2000, AJ, 119, 536 Fisher D. B., Drory N., 2008, AJ, 136, 773 Gadotti D. A., 2009, MNRAS, 393, 1531 Gerin M., Combes F., Athanassoula E., 1990, A&A, 230, 37 Hohl F., 1971, ApJ, 168, 343 Hohl F., 1976, AJ, 81, 30 Honey M., Das M., Ninan J. P., Manoj P., 2016, MNRAS, 462, 2099 Hopkins P. F., Cox T. J., Younger J. D., Hernquist L., 2009, ApJ, 691, 1168 Hopkins P. F., et al., 2010, ApJ, 715, 202 Hoyle B., et al., 2011, MNRAS, 415, 3627 Jogee S., et al., 2004, ApJ, 615, L105 Kalnajs A. J., 1972, ApJ, 175, 63 Kauﬀmann G., White S. D. M., Guiderdoni B., 1993, MNRAS, 264, 201 Kim T., Gadotti D. A., Sheth K., Lee M., S4G Team 2014, in American Astronomical Society Meeting Abstracts #223. p. 205.03 Klypin A., Valenzuela O., Colín P., Quinn T., 2009, MNRAS, 398, 1027 Knapen J. H., Shlosman I., Peletier R. F., 2000, ApJ, 529, 93 Kormendy J., Kennicutt Jr. R. C., 2004, ARA&A, 42, 603 Long S., Shlosman I., Heller C., 2014, ApJ, 783, L18 Lynden-Bell D., 1979, MNRAS, 187, 101 Lynden-Bell D., Kalnajs A. J., 1972, MNRAS, 157, 1 MachadoR.E.G.,AthanassoulaE.,RodionovS.,2012,inCapuzzo-Dolcetta R., Limongi M., Tornambè A., eds, Astronomical Society of the Paciﬁc Conference Series Vol. 453, Advances in Computational Astrophysics: Methods, Tools, and Outcome. p. 363 ( ) arXiv:1207.3452 Marinova I., Jogee S., 2007, ApJ, 659, 1176 Martinez-Valpuesta I., Shlosman I., Heller C., 2006, ApJ, 637, 214 Mayer L., Wadsley J., 2004, MNRAS, 347, 277 Melvin T., et al., 2014, MNRAS, 438, 2882 Menéndez-Delmestre K., Sheth K., Schinnerer E., Jarrett T. H., Scoville N. Z., 2007, ApJ, 657, 790 Naab T., et al., 2014, MNRAS, 444, 3357 Nair P. B., Abraham R. G., 2010, ApJ, 714, L260 Noguchi M., 1987, MNRAS, 228, 635 Norman C. A., Sellwood J. A., Hasan H., 1996, ApJ, 462, 114 MNRAS 000, 1–12 (2016)

text: Kataria

text: Das

text: REFERENCES

font: AXZFNX+TeXGyreTermes-Bold

text: Ostriker

text: J.

text: P.,

text: Peebles

text: P.

text: J.

text: E.,

text: ApJ,

text: Pfenniger

text: D.,

text: Norman

text: C.,

text: ApJ,

text: Aguerri

text: J.

text: A.

text: L.,

text: Balcells

text: M.,

text: Peletier

text: R.

text: F.,

text: A&A,

text: Raha

text: N.,

text: Sellwood

text: J.

text: A.,

text: James

text: R.

text: A.,

text: Kahn

text: F.

text: D.,

text: Nature,

text: Athanassoula

text: E.,

text: ApJ,

text: Romano-Díaz

text: E.,

text: Shlosman

text: I.,

text: Heller

text: C.,

text: Hoﬀman

text: Y.,

text: ApJ,

text: Athanassoula

text: E.,

text: MNRAS,

text: Saha

text: K.,

text: ApJ,

text: Athanassoula

text: E.,

text: Misiriotis

text: A.,

text: MNRAS,

text: Saha

text: K.,

text: Naab

text: T.,

text: MNRAS,

text: Barazza

text: F.

text: D.,

text: Jogee

text: S.,

text: Marinova

text: I.,

text: Saha

text: K.,

text: Martinez-Valpuesta

text: I.,

text: Gerhard

text: O.,

text: MNRAS,

text: Barnes

text: J.

text: E.,

text: Hernquist

text: L.

text: E.,

text: ApJ,

text: Saha

text: K.,

text: Gerhard

text: O.,

text: Martinez-Valpuesta

text: I.,

text: A&A,

text: Barnes

text: J.,

text: Hut

text: P.,

text: Nature,

text: Salo

text: H.,

text: A&A,

text: Barway

text: S.,

text: Saha

text: K.,

text: Vaghmare

text: K.,

text: Kembhavi

text: A.

text: K.,

text: MNRAS,

text: Scannapieco

text: C.,

text: Athanassoula

text: E.,

text: MNRAS,

text: Sellwood

text: J.

text: A.,

text: A&A,

text: Baugh

text: C.

text: M.,

text: Cole

text: S.,

text: Frenk

text: C.

text: S.,

text: MNRAS,

text: Sellwood

text: J.

text: A.,

text: A&A,

text: Binney

text: J.,

text: Merriﬁeld

text: M.,

text: Galactic

text: Astronomy

text: Sellwood

text: J.

text: A.,

text: Evans

text: N.

text: W.,

text: ApJ,

text: Sheth

text: K.,

text: et

text: al.,

text: ApJ,

text: University

text: Press

text: Sheth

text: K.,

text: et

text: al.,

text: ApJ,

text: Bournaud

text: F.,

text: Jog

text: C.

text: J.,

text: Combes

text: F.,

text: A&A,

text: Simmons

text: B.

text: D.,

text: et

text: al.,

text: MNRAS,

text: Bullock

text: J.

text: S.,

text: Kolatt

text: T.

text: S.,

text: Sigad

text: Y.,

text: Somerville

text: R.

text: S.,

text: Kravtsov

text: A.

text: V.,

text: Klypin

text: Skibba

text: R.

text: A.,

text: et

text: al.,

text: MNRAS,

text: A.

text: A.,

text: Primack

text: J.

text: R.,

text: Dekel

text: A.,

text: MNRAS,

text: Springel

text: V.,

text: MNRAS,

text: Buta

text: R.

text: J.,

text: et

text: al.,

text: ApJS,

text: Springel

text: V.,

text: Yoshida

text: N.,

text: White

text: S.

text: D.

text: M.,

text: New

text: Astron.,

text: Combes

text: F.,

text: Sanders

text: R.

text: H.,

text: A&A,

text: Springel

text: V.,

text: Di

text: Matteo

text: T.,

text: Hernquist

text: L.,

text: MNRAS,

text: Combes

text: F.,

text: Debbasch

text: F.,

text: Friedli

text: D.,

text: Pfenniger

text: D.,

text: A&A,

text: Das

text: M.,

text: Teuben

text: P.

text: J.,

text: Vogel

text: S.

text: N.,

text: Regan

text: M.

text: W.,

text: Sheth

text: K.,

text: Harris

text: A.

text: I.,

text: University

text: Press,

text: Cambridge

text: Jeﬀerys

text: W.

text: H.,

text: ApJ,

text: Tremaine

text: S.,

text: Weinberg

text: M.

text: D.,

text: MNRAS,

text: Das

text: M.,

text: Laurikainen

text: E.,

text: Salo

text: H.,

text: Buta

text: R.,

text: Ap&SS,

text: Valenzuela

text: O.,

text: Klypin

text: A.,

text: MNRAS,

text: Debattista

text: V.

text: P.,

text: Sellwood

text: J.

text: A.,

text: The

text: Astrophysical

text: Journal,

text: Weinberg

text: M.

text: D.,

text: Katz

text: N.,

text: MNRAS,

text: Dubinski

text: J.,

text: Berentzen

text: I.,

text: Shlosman

text: I.,

text: ApJ,

text: Widrow

text: L.

text: M.,

text: Pym

text: B.,

text: Dubinski

text: J.,

text: ApJ,

text: Efstathiou

text: G.,

text: Lake

text: G.,

text: Negroponte

text: J.,

text: MNRAS,

text: Wyse

text: R.

text: F.

text: G.,

text: Gilmore

text: G.,

text: Franx

text: M.,

text: ARA&A,

text: Efstathiou

text: G.,

text: Lake

text: G.,

text: Negroponte

text: J.,

text: MNRAS,

text: Yurin

text: D.,

text: Springel

text: V.,

text: MNRAS,

text: Eggen

text: O.

text: J.,

text: Lynden-Bell

text: D.,

text: Sandage

text: A.

text: R.,

text: ApJ,

text: Elmegreen

text: B.

text: G.,

text: Elmegreen

text: D.

text: M.,

text: ApJ,

text: This

text: paper

text: has

text: been

text: typeset

text: from

text: a

text: TEX/LATEX

text: ﬁle

text: prepared

text: by

text: the

text: author.

text: Elmegreen

text: B.

text: G.,

text: Elmegreen

text: D.

text: M.,

text: Hirst

text: A.

text: C.,

text: The

text: Astrophysical

text: Journal,

text: Erwin

text: P.,

text: MNRAS,

text: Eskridge

text: P.

text: B.,

text: et

text: al.,

text: AJ,

text: Fisher

text: D.

text: B.,

text: Drory

text: N.,

text: AJ,

text: Gadotti

text: D.

text: A.,

text: MNRAS,

text: Gerin

text: M.,

text: Combes

text: F.,

text: Athanassoula

text: E.,

text: A&A,

text: Hohl

text: F.,

text: ApJ,

text: Hohl

text: F.,

text: AJ,

text: Honey

text: M.,

text: Das

text: M.,

text: Ninan

text: J.

text: P.,

text: Manoj

text: P.,

text: MNRAS,

text: Hopkins

text: P.

text: F.,

text: Cox

text: T.

text: J.,

text: Younger

text: J.

text: D.,

text: Hernquist

text: L.,

text: ApJ,

text: Hopkins

text: P.

text: F.,

text: et

text: al.,

text: ApJ,

text: Hoyle

text: B.,

text: et

text: al.,

text: MNRAS,

text: Jogee

text: S.,

text: et

text: al.,

text: ApJ,

text: Kalnajs

text: A.

text: J.,

text: ApJ,

text: Kauﬀmann

text: G.,

text: White

text: S.

text: D.

text: M.,

text: Guiderdoni

text: B.,

text: MNRAS,

text: Kim

text: T.,

text: Gadotti

text: D.

text: A.,

text: Sheth

text: K.,

text: Lee

text: M.,

text: Team

text: in

text: American

text: Astronomical

text: Society

text: Meeting

text: Abstracts

text: p.

text: Klypin

text: A.,

text: Valenzuela

text: O.,

text: Colín

text: P.,

text: Quinn

text: T.,

text: MNRAS,

text: Knapen

text: J.

text: H.,

text: Shlosman

text: I.,

text: Peletier

text: R.

text: F.,

text: ApJ,

text: Kormendy

text: J.,

text: Kennicutt

text: Jr.

text: R.

text: C.,

text: ARA&A,

text: Long

text: S.,

text: Shlosman

text: I.,

text: Heller

text: C.,

text: ApJ,

text: Lynden-Bell

text: D.,

text: MNRAS,

text: Lynden-Bell

text: D.,

text: Kalnajs

text: A.

text: J.,

text: MNRAS,

text: R.,

text: Limongi

text: M.,

text: Tornambè

text: A.,

text: eds,

text: Astronomical

text: Society

text: of

text: the

text: Paciﬁc

text: Conference

text: Series

text: Vol.

text: Advances

text: in

text: Computational

text: Astrophysics:

text: Methods,

text: Tools,

text: and

text: Outcome.

text: p.

text: Marinova

text: I.,

text: Jogee

text: S.,

text: ApJ,

text: Martinez-Valpuesta

text: I.,

text: Shlosman

text: I.,

text: Heller

text: C.,

text: ApJ,

text: Mayer

text: L.,

text: Wadsley

text: J.,

text: MNRAS,

text: Melvin

text: T.,

text: et

text: al.,

text: MNRAS,

text: Menéndez-Delmestre

text: K.,

text: Sheth

text: K.,

text: Schinnerer

text: E.,

text: Jarrett

text: T.

text: H.,

text: Scoville

text: N.

text: Z.,

text: ApJ,

text: Naab

text: T.,

text: et

text: al.,

text: MNRAS,

text: Nair

text: P.

text: B.,

text: Abraham

text: R.

text: G.,

text: ApJ,

text: Noguchi

text: M.,

text: MNRAS,

text: Norman

text: C.

text: A.,

text: Sellwood

text: J.

text: A.,

text: Hasan

text: H.,

text: ApJ,

text: MNRAS