text: Ciollaro, Genovese, Lei and Wasserman Trevor Hastie and Werner Stuetzle. Principal curves. Journal of the American Statistical Association, 84(406):502–516, 1989. Lajos Horv´ath and Piotr Kokoszka. Inference for functional data with applications. Springer, 2012. Julien Jacques and Cristian Preda. Functional data clustering: a survey. Advances in Data Analysis and Classiﬁcation, pages 1–25, 2013. Jussi Klemel¨a. Adaptive estimation of the mode of a multivariate density. Journal of Nonparametric Statistics, 17(1):83–105, 2005. Jing Lei, Alessandro Rinaldo, and Larry Wasserman. A conformal prediction approach to explore functional data. Annals of Mathematics and Artiﬁcial Intelligence, pages 1–15, 2013. Jia Li, Surajit Ray, and Bruce G. Lindsay. A nonparametric statistical approach to clus- tering via mode identiﬁcation. Journal of Machine Learning Research, 8(8):1687–1723, 2007a. Xiangru Li, Zhanyi Hu, and Fuchao Wu. A note on the convergence of the mean shift. Pattern Recognition, 40(6):1756 – 1762, 2007b. Byeong U. Park and James S. Marron. Comparison of data-driven bandwidth selectors. Journal of the American Statistical Association, 85(409):66–72, 1990. Emanuel Parzen. On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33(3):1065–1076, 1962. James O. Ramsay and Bernard W. Silverman. Functional Data Analysis. Springer, 2005. Alessandro Rinaldo and Larry Wasserman. Generalized density clustering. The Annals of Statistics, 38(5):2678–2722, 2010. Alessandro Rinaldo, Aarti Singh, Rebecca Nugent, and Larry Wasserman. Stability of density-based clustering. Journal of Machine Learning Research, 13(1):905–948, 2012. Joseph P. Romano. On weak convergence and optimality of kernel density estimates of the mode. The Annals of Statistics, 16(2):629–647, 1988. Simon J. Sheather and Michael C. Jones. A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53(3): 683–690, 1991. Bernard W. Silverman. Using kernel density estimates to investigate multimodality. Journal of the Royal Statistical Society, Series B, pages 97–99, 1981. Bernard W. Silverman. Density estimation for statistics and data analysis. Chapman & Hall, 1986. 38

text: Ciollaro,

text: Genovese,

text: Lei

text: and

text: Wasserman

text: Trevor

text: Hastie

text: and

text: Werner

text: Stuetzle.

text: Principal

text: curves.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Lajos

text: Horv´ath

text: and

text: Piotr

text: Kokoszka.

text: Inference

text: for

text: functional

text: data

text: with

text: applications.

text: Springer,

text: Julien

text: Jacques

text: and

text: Cristian

text: Preda.

text: Functional

text: data

text: clustering:

text: a

text: survey.

text: Advances

text: in

text: Data

text: Analysis

text: and

text: Classiﬁcation,

text: pages

text: Jussi

text: Klemel¨a.

text: Adaptive

text: estimation

text: of

text: the

text: mode

text: of

text: a

text: multivariate

text: density.

text: Journal

text: of

text: Nonparametric

text: Statistics,

text: Jing

text: Lei,

text: Alessandro

text: Rinaldo,

text: and

text: Larry

text: Wasserman.

text: A

text: conformal

text: prediction

text: approach

text: to

text: explore

text: functional

text: data.

text: Annals

text: of

text: Mathematics

text: and

text: Artiﬁcial

text: Intelligence,

text: pages

text: Jia

text: Li,

text: Surajit

text: Ray,

text: and

text: Bruce

text: G.

text: Lindsay.

text: A

text: nonparametric

text: statistical

text: approach

text: to

text: clus-

text: tering

text: via

text: mode

text: identiﬁcation.

text: Journal

text: of

text: Machine

text: Learning

text: Research,

text: Xiangru

text: Li,

text: Zhanyi

text: Hu,

text: and

text: Fuchao

text: Wu.

text: A

text: note

text: on

text: the

text: convergence

text: of

text: the

text: mean

text: shift.

text: Pattern

text: Recognition,

text: Byeong

text: U.

text: Park

text: and

text: James

text: S.

text: Marron.

text: Comparison

text: of

text: data-driven

text: bandwidth

text: selectors.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Emanuel

text: Parzen.

text: On

text: estimation

text: of

text: a

text: probability

text: density

text: function

text: and

text: mode.

text: The

text: Annals

text: of

text: Mathematical

text: Statistics,

text: James

text: O.

text: Ramsay

text: and

text: Bernard

text: W.

text: Silverman.

text: Functional

text: Data

text: Analysis.

text: Springer,

text: Alessandro

text: Rinaldo

text: and

text: Larry

text: Wasserman.

text: Generalized

text: density

text: clustering.

text: The

text: Annals

text: of

text: Statistics,

text: Alessandro

text: Rinaldo,

text: Aarti

text: Singh,

text: Rebecca

text: Nugent,

text: and

text: Larry

text: Wasserman.

text: Stability

text: of

text: density-based

text: clustering.

text: Journal

text: of

text: Machine

text: Learning

text: Research,

text: Joseph

text: P.

text: Romano.

text: On

text: weak

text: convergence

text: and

text: optimality

text: of

text: kernel

text: density

text: estimates

text: of

text: the

text: mode.

text: The

text: Annals

text: of

text: Statistics,

text: Simon

text: J.

text: Sheather

text: and

text: Michael

text: C.

text: Jones.

text: A

text: reliable

text: data-based

text: bandwidth

text: selection

text: method

text: for

text: kernel

text: density

text: estimation.

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society,

text: Series

text: B,

text: Bernard

text: W.

text: Silverman.

text: Using

text: kernel

text: density

text: estimates

text: to

text: investigate

text: multimodality.

text: Journal

text: of

text: the

text: Royal

text: Statistical

text: Society,

text: Series

text: B,

text: pages

text: Bernard

text: W.

text: Silverman.

text: Density

text: estimation

text: for

text: statistics

text: and

text: data

text: analysis.

text: Chapman

text: Hall,


text: 4 (cid:104) (cid:105) − (cid:104) (cid:105) Due to the fact that merging two layers of WSN mod- that M = M( k 2)/ k . This means that each node w els destroys the Granovetterian structure, we investigated has two strong links while the rest are weak. This can how the correlation between layers aﬀects the properties be justiﬁed by considering the eﬀect of cyclic closure: of the network. We created the second layer by copying The cyclic closure is the most frequent interaction which the ﬁrst layer and then shuﬄed the fraction p of the nodes strengthens two links of a focal node with positive feed- in the second layer. Shuﬄing nodes i and j means that back of their weights. Assuming a random network for all original links (i, k) become (j, k) and vice versa. This the groups at the percolation threshold, one should get (cid:104) (cid:105) ≡ is just a relabeling of the nodes in that layer, meaning M (fa) = N /2. This gives k 2M /N = 3.26 for g c g (cid:104)g (cid:105) g g that the topology remains the same, i.e., both layers cor- the single-layer model, where k is the average degree g respond to single-layer WSN models but with increasing of the connections between the groups, i.e., that of the p the correlations between them decrease. This is called supernodes. Snapshots of the single-layer model as in the “copy-and-shuﬄe” model (see Fig. 2). Fig. 2(a) justify the low contact number for the groups. When p = 0, the aggregate network is equivalent to the The shuﬄing of agents creates new connections from single-layer network whose link weights are doubled. For one group to another. These connections increase the p = 1, it is the same as the double-layer model; the Gra- intergroup connection degree by novetterian structure gets entirely destroyed by random- ization. By controlling p between 0 and 1, a transient (cid:104) (cid:105) ≈ (cid:104) (cid:105) (cid:104) (cid:105) behavior is observed. Figure 3 shows how the thresh- kg (p) kg (p = 0) + ps k . (2) old values fa get closer to fd as p is increased and for → c → c p 1 we get ∆f 0. The reason is that strong links c Thus there is a linear increase of the average degree of in the second layer connect the communities more ran- the groups with p. domly since the correlation between the ﬁrst and the sec- Now we can use Eq. (1) to go in the reverse direction, ond layer diminishes. namely that knowing M /N we can get the percolation The randomization has the consequence that the per- g g threshold (note that M and M also depend on p in a colation threshold fa gets closer to that of the corre- w c trivial way): (cid:18) (cid:19) sponding Erd˝os-R´enyi random network. However, for a reasonably large range of p, we can clearly diﬀerentiate fthcaeaGnrdanfcodvtehttuesritahnesstirmuciltaurritey. between the layer assures fca = MMw((pp)) (cid:18)1 − 2(M +NgpN(cid:104)k(cid:105)) (cid:19) g The gradual transition can be understood in the follow- (cid:104) (cid:105) − k 2 − 1 ing way. Let us make the assumptions that the original = (cid:104) (cid:105) 1 (cid:104) (cid:105) (cid:104) (cid:105) . (3) network (ﬁrst layer) is composed of strongly connected k kg (p = 0) + 2s k p groups interconnected by weak links [see Fig. 2(a)] and the average size of these groups is s which is small and The average size of the groups for the single layer can independent of the total number of agents N. Eventually be obtained by an infomap analysis [19] and was found the number of groups is N = N/s. In the link perco- to be s = 15.1. The resulting curve is shown in Fig. 3 as g lation analysis starting from the weak links (ascending a dashed line and is compared to the empirical threshold order) we can consider the groups as “supernodes” and values. The calculated line ﬁts the initial part very well, we have to solve the percolation problem for the links where the above picture is expected to work. connecting them. The copy-and-shuﬄe model produces a region of p, Not only the intergroup links turn out to be weak but where a multilayer Granovetterian structure exists. Now also some intragroup ones. Let us denote the number of we have to check whether our construction has lead to links by M, the total number of weak links by M and enhancement of the overlapping of the communities, too. w the intergroup weak links by M which are a subset of We have analyzed the aggregate networks by the method g M . Let us remove the fa fraction of the total links in of Ahn et al. [22] and calculated c/c , the ratio of the w 0 ascending order. In this case the number of weak links average numbers of communities a node belongs to at − gets M (fa) = M Mfa since we only removed weak parameter value p and p = 0 3. We expect that c/c w w 0 links. Thus it is clear that this approximation will not should increase as shuﬄing goes on. Figure 4 shows the work for fa > M /M. The intergroup links are removed dependence of this quantity on p. The overlap starts w by the same rate as weak links, so the number of inter- to increase only when the Granovetterian correlation be- group links after removing the fa part of the total links tween link weight and topology is already wiped away. (cid:18) (cid:19) in ascending order is In the next section we make another attempt to produce − M M (fa) = M 1 fa . (1) g g M w 3 We get even for p = 0 a value larger than 1 (c0 = 2.96. This is due to the fact that the method of [22] is a partition of the In order to have quantitative results we need the num- link graph producing a large number of single links (diads) as ber of strong links. This can be estimated if we assume communities, which enhance the overlap. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Due

text: to

text: the

text: fact

text: that

text: merging

text: two

text: layers

text: of

text: WSN

text: mod-

text: that

text: M

text: M(

text: k

text: k

text: This

text: means

text: that

text: each

text: node

text: w

text: els

text: destroys

text: the

text: Granovetterian

text: structure,

text: we

text: investigated

text: has

text: two

text: strong

text: links

text: while

text: the

text: rest

text: are

text: weak.

text: This

text: can

text: how

text: the

text: correlation

text: between

text: layers

text: aﬀects

text: the

text: properties

text: be

text: justiﬁed

text: by

text: considering

text: the

text: eﬀect

text: of

text: cyclic

text: closure:

text: of

text: the

text: network.

text: We

text: created

text: the

text: second

text: layer

text: by

text: copying

text: The

text: cyclic

text: closure

text: is

text: the

text: most

text: frequent

text: interaction

text: which

text: the

text: ﬁrst

text: layer

text: and

text: then

text: shuﬄed

text: the

text: fraction

text: p

text: of

text: the

text: nodes

text: strengthens

text: two

text: links

text: of

text: a

text: focal

text: node

text: with

text: positive

text: feed-

text: in

text: the

text: second

text: layer.

text: Shuﬄing

text: nodes

text: i

text: and

text: j

text: means

text: that

text: back

text: of

text: their

text: weights.

text: Assuming

text: a

text: random

text: network

text: for

text: all

text: original

text: links

text: (i,

text: k)

text: become

text: (j,

text: k)

text: and

text: vice

text: versa.

text: This

text: the

text: groups

text: at

text: the

text: percolation

text: threshold,

text: one

text: should

text: get

text: is

text: just

text: a

text: relabeling

text: of

text: the

text: nodes

text: in

text: that

text: layer,

text: meaning

text: M

text: (fa)

text: N

text: This

text: gives

text: k

text: /N

text: for

text: g

text: c

text: g

text: g

text: g

text: that

text: the

text: topology

text: remains

text: the

text: same,

text: i.e.,

text: both

text: layers

text: cor-

text: the

text: single-layer

text: model,

text: where

text: k

text: is

text: the

text: average

text: degree

text: g

text: respond

text: to

text: single-layer

text: WSN

text: models

text: but

text: with

text: increasing

text: of

text: the

text: connections

text: between

text: the

text: groups,

text: i.e.,

text: that

text: of

text: the

text: p

text: the

text: correlations

text: between

text: them

text: decrease.

text: This

text: is

text: called

text: supernodes.

text: Snapshots

text: of

text: the

text: single-layer

text: model

text: as

text: in

text: the

text: “copy-and-shuﬄe”

text: model

text: (see

text: Fig.

text: Fig.

text: justify

text: the

text: low

text: contact

text: number

text: for

text: the

text: groups.

text: When

text: p

text: the

text: aggregate

text: network

text: is

text: equivalent

text: to

text: the

text: The

text: shuﬄing

text: of

text: agents

text: creates

text: new

text: connections

text: from

text: single-layer

text: network

text: whose

text: link

text: weights

text: are

text: doubled.

text: For

text: one

text: group

text: to

text: another.

text: These

text: connections

text: increase

text: the

text: p

text: it

text: is

text: the

text: same

text: as

text: the

text: double-layer

text: model;

text: the

text: Gra-

text: intergroup

text: connection

text: degree

text: by

text: novetterian

text: structure

text: gets

text: entirely

text: destroyed

text: by

text: random-

text: ization.

text: By

text: controlling

text: p

text: between

text: and

text: a

text: transient

text: behavior

text: is

text: observed.

text: Figure

text: shows

text: how

text: the

text: thresh-

text: kg

text: (p)

text: kg

text: (p

text: ps

text: k

text: old

text: values

text: fa

text: get

text: closer

text: to

text: fd

text: as

text: p

text: is

text: increased

text: and

text: for

text: c

text: c

text: p

text: we

text: get

text: ∆f

text: The

text: reason

text: is

text: that

text: strong

text: links

text: c

text: Thus

text: there

text: is

text: a

text: linear

text: increase

text: of

text: the

text: average

text: degree

text: of

text: in

text: the

text: second

text: layer

text: connect

text: the

text: communities

text: more

text: ran-

text: the

text: groups

text: with

text: p.

text: domly

text: since

text: the

text: correlation

text: between

text: the

text: ﬁrst

text: and

text: the

text: sec-

text: Now

text: we

text: can

text: use

text: Eq.

text: to

text: go

text: in

text: the

text: reverse

text: direction,

text: ond

text: layer

text: diminishes.

text: namely

text: that

text: knowing

text: M

text: /N

text: we

text: can

text: get

text: the

text: percolation

text: The

text: randomization

text: has

text: the

text: consequence

text: that

text: the

text: per-

text: g

text: g

text: threshold

text: (note

text: that

text: M

text: and

text: M

text: also

text: depend

text: on

text: p

text: in

text: a

text: colation

text: threshold

text: fa

text: gets

text: closer

text: to

text: that

text: of

text: the

text: corre-

text: w

text: c

text: trivial

text: way):

text: sponding

text: Erd˝os-R´enyi

text: random

text: network.

text: However,

text: for

text: a

text: reasonably

text: large

text: range

text: of

text: p,

text: we

text: can

text: clearly

text: diﬀerentiate

text: between

text: the

text: layer

text: assures

text: fca

text: MMw((pp))

text: g

text: The

text: gradual

text: transition

text: can

text: be

text: understood

text: in

text: the

text: follow-

text: k

text: ing

text: way.

text: Let

text: us

text: make

text: the

text: assumptions

text: that

text: the

text: original

text: network

text: (ﬁrst

text: layer)

text: is

text: composed

text: of

text: strongly

text: connected

text: k

text: kg

text: (p

text: k

text: p

text: groups

text: interconnected

text: by

text: weak

text: links

text: [see

text: Fig.

text: and

text: the

text: average

text: size

text: of

text: these

text: groups

text: is

text: s

text: which

text: is

text: small

text: and

text: The

text: average

text: size

text: of

text: the

text: groups

text: for

text: the

text: single

text: layer

text: can

text: independent

text: of

text: the

text: total

text: number

text: of

text: agents

text: N.

text: Eventually

text: be

text: obtained

text: by

text: an

text: infomap

text: analysis

text: and

text: was

text: found

text: the

text: number

text: of

text: groups

text: is

text: N

text: N/s.

text: In

text: the

text: link

text: perco-

text: to

text: be

text: s

text: The

text: resulting

text: curve

text: is

text: shown

text: in

text: Fig.

text: as

text: g

text: lation

text: analysis

text: starting

text: from

text: the

text: weak

text: links

text: (ascending

text: a

text: dashed

text: line

text: and

text: is

text: compared

text: to

text: the

text: empirical

text: threshold

text: order)

text: we

text: can

text: consider

text: the

text: groups

text: as

text: “supernodes”

text: and

text: values.

text: The

text: calculated

text: line

text: ﬁts

text: the

text: initial

text: part

text: very

text: well,

text: we

text: have

text: to

text: solve

text: the

text: percolation

text: problem

text: for

text: the

text: links

text: where

text: the

text: above

text: picture

text: is

text: expected

text: to

text: work.

text: connecting

text: them.

text: The

text: copy-and-shuﬄe

text: model

text: produces

text: a

text: region

text: of

text: p,

text: Not

text: only

text: the

text: intergroup

text: links

text: turn

text: out

text: to

text: be

text: weak

text: but

text: where

text: a

text: multilayer

text: Granovetterian

text: structure

text: exists.

text: Now

text: also

text: some

text: intragroup

text: ones.

text: Let

text: us

text: denote

text: the

text: number

text: of

text: we

text: have

text: to

text: check

text: whether

text: our

text: construction

text: has

text: lead

text: to

text: links

text: by

text: M,

text: the

text: total

text: number

text: of

text: weak

text: links

text: by

text: M

text: and

text: enhancement

text: of

text: the

text: overlapping

text: of

text: the

text: communities,

text: too.

text: w

text: the

text: intergroup

text: weak

text: links

text: by

text: M

text: which

text: are

text: a

text: subset

text: of

text: We

text: have

text: analyzed

text: the

text: aggregate

text: networks

text: by

text: the

text: method

text: g

text: M

text: Let

text: us

text: remove

text: the

text: fa

text: fraction

text: of

text: the

text: total

text: links

text: in

text: of

text: Ahn

text: et

text: al.

text: and

text: calculated

text: c/c

text: the

text: ratio

text: of

text: the

text: w

text: ascending

text: order.

text: In

text: this

text: case

text: the

text: number

text: of

text: weak

text: links

text: average

text: numbers

text: of

text: communities

text: a

text: node

text: belongs

text: to

text: at

text: gets

text: M

text: (fa)

text: M

text: Mfa

text: since

text: we

text: only

text: removed

text: weak

text: parameter

text: value

text: p

text: and

text: p

text: We

text: expect

text: that

text: c/c

text: w

text: w

text: links.

text: Thus

text: it

text: is

text: clear

text: that

text: this

text: approximation

text: will

text: not

text: should

text: increase

text: as

text: shuﬄing

text: goes

text: on.

text: Figure

text: shows

text: the

text: work

text: for

text: fa

text: M

text: /M.

text: The

text: intergroup

text: links

text: are

text: removed

text: dependence

text: of

text: this

text: quantity

text: on

text: p.

text: The

text: overlap

text: starts

text: w

text: by

text: the

text: same

text: rate

text: as

text: weak

text: links,

text: so

text: the

text: number

text: of

text: inter-

text: to

text: increase

text: only

text: when

text: the

text: Granovetterian

text: correlation

text: be-

text: group

text: links

text: after

text: removing

text: the

text: fa

text: part

text: of

text: the

text: total

text: links

text: tween

text: link

text: weight

text: and

text: topology

text: is

text: already

text: wiped

text: away.

text: in

text: ascending

text: order

text: is

text: In

text: the

text: next

text: section

text: we

text: make

text: another

text: attempt

text: to

text: produce

text: M

text: M

text: (fa)

text: M

text: fa

text: g

text: g

text: M

text: w

text: We

text: get

text: even

text: for

text: p

text: a

text: value

text: larger

text: than

text: This

text: is

text: due

text: to

text: the

text: fact

text: that

text: the

text: method

text: of

text: is

text: a

text: partition

text: of

text: the

text: In

text: order

text: to

text: have

text: quantitative

text: results

text: we

text: need

text: the

text: num-

text: link

text: graph

text: producing

text: a

text: large

text: number

text: of

text: single

text: links

text: (diads)

text: as

text: ber

text: of

text: strong

text: links.

text: This

text: can

text: be

text: estimated

text: if

text: we

text: assume

text: communities,

text: which

text: enhance

text: the

text: overlap.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: The contribution of this paper is to ﬁll this gap, by proposing a novel Bayesian spatio-temporal model for cluster detection and risk estimation. The model is able to detect clusters dynamically, so that both cluster membership and their average risk levels can evolve over time. Inference is based on Markov chain Monte Carlo (McMC) simulation, and unlike most existing models for spatio- temporal disease mapping, our methodology can be implemented in freely available software, making this research reproducible. This software is the R package CARBayesST, which was developed in conjunction with this paper. The methodology developed here is motivated by two case studies, and the motivating questions for both analyses focus on the relationship between public health policy and disease clustering. The background and motivation for these case studies are outlined in Section 2, together with a brief review of spatio-temporal disease mapping. Section 3 presents our methodological contribution, and details the software that has been developed. Section 4 quantiﬁes the performance of our methodology by simulation, while the results from the two case studies are presented in Section 5. Finally, Section 6 concludes the paper. 2 Background and motivation 2.1 Spatio-temporal disease mapping Disease data (y , e ) denote the observed and expected numbers of disease cases across i = 1, . . . , N it it contiguous areal units during t = 1, . . . , T time periods. The latter, e , is computed using internal it or external standardisation, and accounts for the heterogeneous population sizes and demographic structures across the N areal units. The same set of age and sex speciﬁc disease rates are used in the standardisation for all time periods, because any estimated temporal trend in disease risk is | ∼ then relative to a common baseline. The Poisson model y e , θ Poisson(e θ ) is commonly it it it it it assumed for the disease counts y in the spatio-temporal disease mapping literature, where θ is the it it risk of disease in areal unit i during time period t relative to e . Log-linear models are typically it speciﬁed for θ , and the ﬁrst model to be proposed in the literature was Bernardinelli et al. (1995), it { } who represented θ by a set of spatially varying linear time trends. However, linear trends are not it always appropriate, and the most popular model in the literature is the main eﬀects and interaction decomposition proposed by Knorr-Held (2000): 3

text: The

text: contribution

text: of

text: this

text: paper

text: is

text: to

text: ﬁll

text: this

text: gap,

text: by

text: proposing

text: a

text: novel

text: Bayesian

text: spatio-temporal

text: model

text: for

text: cluster

text: detection

text: and

text: risk

text: estimation.

text: The

text: model

text: is

text: able

text: to

text: detect

text: clusters

text: dynamically,

text: so

text: that

text: both

text: cluster

text: membership

text: and

text: their

text: average

text: risk

text: levels

text: can

text: evolve

text: over

text: time.

text: Inference

text: is

text: based

text: on

text: Markov

text: chain

text: Monte

text: Carlo

text: (McMC)

text: simulation,

text: and

text: unlike

text: most

text: existing

text: models

text: for

text: spatio-

text: temporal

text: disease

text: mapping,

text: our

text: methodology

text: can

text: be

text: implemented

text: in

text: freely

text: available

text: software,

text: making

text: this

text: research

text: reproducible.

text: This

text: software

text: is

text: the

text: R

text: package

text: CARBayesST,

text: which

text: was

text: developed

text: in

text: conjunction

text: with

text: this

text: paper.

text: The

text: methodology

text: developed

text: here

text: is

text: motivated

text: by

text: two

text: case

text: studies,

text: and

text: the

text: motivating

text: questions

text: for

text: both

text: analyses

text: focus

text: on

text: the

text: relationship

text: between

text: public

text: health

text: policy

text: and

text: disease

text: clustering.

text: The

text: background

text: and

text: motivation

text: for

text: these

text: case

text: studies

text: are

text: outlined

text: in

text: Section

text: together

text: with

text: a

text: brief

text: review

text: of

text: spatio-temporal

text: disease

text: mapping.

text: Section

text: presents

text: our

text: methodological

text: contribution,

text: and

text: details

text: the

text: software

text: that

text: has

text: been

text: developed.

text: Section

text: quantiﬁes

text: the

text: performance

text: of

text: our

text: methodology

text: by

text: simulation,

text: while

text: the

text: results

text: from

text: the

text: two

text: case

text: studies

text: are

text: presented

text: in

text: Section

text: Finally,

text: Section

text: concludes

text: the

text: paper.

text: Background

text: and

text: motivation

text: Spatio-temporal

text: disease

text: mapping

text: Disease

text: data

text: (y

text: e

text: denote

text: the

text: observed

text: and

text: expected

text: numbers

text: of

text: disease

text: cases

text: across

text: i

text: N

text: it

text: it

text: contiguous

text: areal

text: units

text: during

text: t

text: T

text: time

text: periods.

text: The

text: latter,

text: e

text: is

text: computed

text: using

text: internal

text: it

text: or

text: external

text: standardisation,

text: and

text: accounts

text: for

text: the

text: heterogeneous

text: population

text: sizes

text: and

text: demographic

text: structures

text: across

text: the

text: N

text: areal

text: units.

text: The

text: same

text: set

text: of

text: age

text: and

text: sex

text: speciﬁc

text: disease

text: rates

text: are

text: used

text: in

text: the

text: standardisation

text: for

text: all

text: time

text: periods,

text: because

text: any

text: estimated

text: temporal

text: trend

text: in

text: disease

text: risk

text: is

text: then

text: relative

text: to

text: a

text: common

text: baseline.

text: The

text: Poisson

text: model

text: y

text: e

text: θ

text: Poisson(e

text: θ

text: is

text: commonly

text: it

text: it

text: it

text: it

text: it

text: assumed

text: for

text: the

text: disease

text: counts

text: y

text: in

text: the

text: spatio-temporal

text: disease

text: mapping

text: literature,

text: where

text: θ

text: is

text: the

text: it

text: it

text: risk

text: of

text: disease

text: in

text: areal

text: unit

text: i

text: during

text: time

text: period

text: t

text: relative

text: to

text: e

text: Log-linear

text: models

text: are

text: typically

text: it

text: speciﬁed

text: for

text: θ

text: and

text: the

text: ﬁrst

text: model

text: to

text: be

text: proposed

text: in

text: the

text: literature

text: was

text: Bernardinelli

text: et

text: al.

text: it

text: who

text: represented

text: θ

text: by

text: a

text: set

text: of

text: spatially

text: varying

text: linear

text: time

text: trends.

text: However,

text: linear

text: trends

text: are

text: not

text: it

text: always

text: appropriate,

text: and

text: the

text: most

text: popular

text: model

text: in

text: the

text: literature

text: is

text: the

text: main

text: eﬀects

text: and

text: interaction

text: decomposition

text: proposed

text: by

text: Knorr-Held


text: Figure 3: The maps in the top row display the estimated risks for 2001 from Model-KH (left panel (a)) and Model-2 (right panel (b)), while the bottom panel (panel (c)) displays boxplots of estimated risks from Model-2 for all years by PHD. 24 ##LTFigure## ##LTFigure##

text: Figure

text: The

label: caption

text: maps

text: in

text: the

text: top

text: row

text: display

text: the

text: estimated

text: risks

text: for

text: from

text: Model-KH

text: (left

text: panel

text: (a))

text: and

text: (right

text: panel

text: (b)),

text: while

text: the

text: bottom

text: panel

text: (panel

text: (c))

text: displays

text: boxplots

text: of

text: estimated

text: risks

text: from

text: for

text: all

text: years

text: by

text: PHD.

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##


text: (cid:32)(cid:80) (cid:33) ln(θit) = β + φ(cid:80)i + θi + αt +(cid:80)δt + γit, (1) φ |φ− , W, τ2 ∼ N Nj=1 wijφj , τφ2 , i i φ N w N w j=1 ij j=1 ij | ∼ θ τ2 N(0, τ2), i θ θ | ∼ α α − , τ2 N(α − , τ2), t t 1 α t 1 α | ∼ δ τ2 N(0, τ2), t δ δ | ∼ γ γ− , τ2 N(m , τ2v ). it it γ it γ it The spatial (φ + θ ) and temporal (α + δ ) main eﬀects are a convolution of independent (θ , δ ) i i t t i t and autocorrelated (φ , α ) processes. Spatial autocorrelation is modelled by an intrinsic conditional i t × autoregressive (ICAR, Besag et al., 1991) model with a binary N N neighbourhood matrix W , where w = 1 if areal units (i, j) share a common border and is zero otherwise. Thus the conditional expec- ij tation is the mean of the random eﬀects in neighbouring areal units, inducing spatial smoothness into φ . Temporal autocorrelation is modelled by a ﬁrst order random walk, where the conditional expecta- i tion of α is α − . Four diﬀerent models were proposed by Knorr-Held (2000) for the interaction term t t 1 γ , which respectively assume independence, purely spatial autocorrelation, purely temporal autocor- it relation and spatio-temporal autocorrelation. The ﬁrst of these has (m = 0, v = 1), and further it it details are given by Knorr-Held (2000). Weakly informative Inverse-gamma and Gaussian priors can be speciﬁed for the variance parameters (τ2, τ2, τ2, τ2, τ2) and intercept term (β) respectively. φ θ α δ γ Another popular approach is to allow the spatially smoothed log-risk surface to evolve over time via an autoregressive process, and a number of models of this type have been proposed (see for example Mugglin et al. (2002), Ugarte et al. (2012) and Rushworth et al. (2014)). Here we outline the model of Rushworth et al. (2014), which is given by: 4 ##LTLine## ##LTLine##

label: equation

text: ln(θit)

text: β

text: θi

text: αt

text: γit,

text: φ

text: |φ−

text: W,

text: τ2

text: N

text: wijφj

text: τφ2

text: i

text: i

text: φ

text: N

text: w

text: N

text: w

text: ij

text: ij

text: θ

text: τ2

text: i

text: θ

text: θ

text: α

text: α

text: τ2

text: N(α

text: t

text: t

text: α

text: t

text: α

text: δ

text: τ2

text: t

text: δ

text: δ

text: γ

text: γ−

text: τ2

text: N(m

text: it

text: it

text: γ

text: it

text: γ

text: it

text: The

text: spatial

text: (φ

text: θ

text: and

text: temporal

text: (α

text: δ

text: main

text: eﬀects

text: are

text: a

text: convolution

text: of

text: independent

text: (θ

text: δ

text: i

text: i

text: t

text: t

text: i

text: t

text: and

text: autocorrelated

text: (φ

text: α

text: processes.

text: Spatial

text: autocorrelation

text: is

text: modelled

text: by

text: an

text: intrinsic

text: conditional

text: i

text: t

text: autoregressive

text: (ICAR,

text: Besag

text: et

text: al.,

text: model

text: with

text: a

text: binary

text: N

text: N

text: neighbourhood

text: matrix

text: W

text: where

text: w

text: if

text: areal

text: units

text: (i,

text: j)

text: share

text: a

text: common

text: border

text: and

text: is

text: zero

text: otherwise.

text: Thus

text: the

text: conditional

text: expec-

text: ij

text: tation

text: is

text: the

text: mean

text: of

text: the

text: random

text: eﬀects

text: in

text: neighbouring

text: areal

text: units,

text: inducing

text: spatial

text: smoothness

text: into

text: φ

text: Temporal

text: autocorrelation

text: is

text: modelled

text: by

text: a

text: ﬁrst

text: order

text: random

text: walk,

text: where

text: the

text: conditional

text: expecta-

text: i

text: tion

text: of

text: α

text: is

text: α

text: Four

text: diﬀerent

text: models

text: were

text: proposed

text: by

text: Knorr-Held

text: for

text: the

text: interaction

text: term

text: t

text: t

text: γ

text: which

text: respectively

text: assume

text: independence,

text: purely

text: spatial

text: autocorrelation,

text: purely

text: temporal

text: autocor-

text: it

text: relation

text: and

text: spatio-temporal

text: autocorrelation.

text: The

text: ﬁrst

text: of

text: these

text: has

text: (m

text: v

text: and

text: further

text: it

text: it

text: details

text: are

text: given

text: by

text: Knorr-Held

text: Weakly

text: informative

text: Inverse-gamma

text: and

text: Gaussian

text: priors

text: can

text: be

text: speciﬁed

text: for

text: the

text: variance

text: parameters

text: and

text: intercept

text: term

text: (β)

text: respectively.

text: φ

text: θ

text: α

text: δ

text: γ

text: Another

text: popular

text: approach

text: is

text: to

text: allow

text: the

text: spatially

text: smoothed

text: log-risk

text: surface

text: to

text: evolve

text: over

text: time

text: via

text: an

text: autoregressive

text: process,

text: and

text: a

text: number

text: of

text: models

text: of

text: this

text: type

text: have

text: been

text: proposed

text: (see

text: for

text: example

text: Mugglin

text: et

text: al.

text: Ugarte

text: et

text: al.

text: and

text: Rushworth

text: et

text: al.

text: Here

text: we

text: outline

text: the

text: model

text: of

text: Rushworth

text: et

text: al.

text: which

text: is

text: given

text: by:

text: ##LTLine##

text: ##LTLine##


text: 2 LEXING YING accuracy. If we assume that both the diameter and the boundary length of Ω are of order Θ(1), then the boundary ∂Ω shall be discretized with a set P of n = Θ(1/λ) = Θ(ω) points. Discretizing (1) with an appropriate numerical quadrature scheme results in the following discrete summation problem: for ∈ (cid:88) each x P , evaluate (2) u(x) = G(x, y)f(y). ∈ y P Direct evaluation of (2) takes Θ(n2) steps. This can be computationally intensive for high frequency problems where ω and n are large. Hence, there is a practical need for algorithms that can evaluate (2) rapidly and accurately. In the past thirty years, a lot of research has been devoted to this prob- lem and several successful algorithms have been proposed. The ﬁrst, and probably the most well-known approach is the high frequency fast mul- tipole method initiated by Rokhlin in [18, 19] and further developed in [6,8,12,20,21]. This method follows the structure of the classical fast mul- tipole method for the Laplace equation and uses fast spherical harmonic transforms and diagonal forms to construct the multipole and local expan- sions eﬃciently. This method has an Θ(n log n) complexity but is rather technical. The second method is the butterﬂy algorithm proposed by Michielssen and Boag in [15], where the main observation is that the interaction be- tween two regions under certain geometric conﬁguration is numerically low- rank even for high frequency scattering problems. The complexity of this algorithm is Θ(n log2 n) with a relatively large prefactor. A third method is the directional FMM-type algorithm proposed by En- gquist and Ying in [10, 11], where the concept of a directional parabolic separation condition is introduced to construct directional low-rank approx- imations between certain square regions and wedges with narrow opening angle. This algorithm has a Θ(n log n) complexity and the advantage of being kernel-independent. Other algorithms that rely on this directional low-rank idea include [2,14]. While all the above methods are based on hierarchical partitioning of the boundary, methods based on the fast Fourier transform [4, 5] are also used widely. These methods are relatively easy to implement, but have super- linear complexity like Θ(nα log n) with α = 1.5 for example. In this paper, we introduce a new algorithm for 2D problems. This algo- rithm leverages the directional idea of [10, 11] and uses oscillatory Cheby- shev interpolation and local FFTs to speed up the computation. This new method has a Θ(n log2 n) complexity and is conceptually much simpler than the previous hierarchical methods. The rest of this paper is organized as follows. Section 2 explained the algorithm in detail. Section 3 provides numerical results and discussions are given in Section 4.

text: LEXING

text: YING

text: accuracy.

text: If

text: we

text: assume

text: that

text: both

text: the

text: diameter

text: and

text: the

text: boundary

text: length

text: of

text: Ω

text: are

text: of

text: order

text: then

text: the

text: boundary

text: ∂Ω

text: shall

text: be

text: discretized

text: with

text: a

text: set

text: P

text: of

text: n

text: Θ(ω)

text: points.

text: Discretizing

text: with

text: an

text: appropriate

text: numerical

text: quadrature

text: scheme

text: results

text: in

text: the

text: following

text: discrete

text: summation

text: problem:

text: for

text: each

text: x

text: P

text: evaluate

text: u(x)

text: G(x,

text: y)f(y).

text: y

text: P

text: Direct

text: evaluation

text: of

text: takes

text: steps.

text: This

text: can

text: be

text: computationally

text: intensive

text: for

text: high

text: frequency

text: problems

text: where

text: ω

text: and

text: n

text: are

text: large.

text: Hence,

text: there

text: is

text: a

text: practical

text: need

text: for

text: algorithms

text: that

text: can

text: evaluate

text: rapidly

text: and

text: accurately.

text: In

text: the

text: past

text: thirty

text: years,

text: a

text: lot

text: of

text: research

text: has

text: been

text: devoted

text: to

text: this

text: prob-

text: lem

text: and

text: several

text: successful

text: algorithms

text: have

text: been

text: proposed.

text: The

text: ﬁrst,

text: and

text: probably

text: the

text: most

text: well-known

text: approach

text: is

text: the

text: high

text: frequency

text: fast

text: mul-

text: tipole

text: method

text: initiated

text: by

text: Rokhlin

text: in

text: and

text: further

text: developed

text: in

text: This

text: method

text: follows

text: the

text: structure

text: of

text: the

text: classical

text: fast

text: mul-

text: tipole

text: method

text: for

text: the

text: Laplace

text: equation

text: and

text: uses

text: fast

text: spherical

text: harmonic

text: transforms

text: and

text: diagonal

text: forms

text: to

text: construct

text: the

text: multipole

text: and

text: local

text: expan-

text: sions

text: eﬃciently.

text: This

text: method

text: has

text: an

text: Θ(n

text: log

text: n)

text: complexity

text: but

text: is

text: rather

text: technical.

text: The

text: second

text: method

text: is

text: the

text: butterﬂy

text: algorithm

text: proposed

text: by

text: Michielssen

text: and

text: Boag

text: in

text: where

text: the

text: main

text: observation

text: is

text: that

text: the

text: interaction

text: be-

text: tween

text: two

text: regions

text: under

text: certain

text: geometric

text: conﬁguration

text: is

text: numerically

text: low-

text: rank

text: even

text: for

text: high

text: frequency

text: scattering

text: problems.

text: The

text: complexity

text: of

text: this

text: algorithm

text: is

text: Θ(n

text: n)

text: with

text: a

text: relatively

text: large

text: prefactor.

text: A

text: third

text: method

text: is

text: the

text: directional

text: FMM-type

text: algorithm

text: proposed

text: by

text: En-

text: gquist

text: and

text: Ying

text: in

text: where

text: the

text: concept

text: of

text: a

text: directional

text: parabolic

text: separation

text: condition

text: is

text: introduced

text: to

text: construct

text: directional

text: low-rank

text: approx-

text: imations

text: between

text: certain

text: square

text: regions

text: and

text: wedges

text: with

text: narrow

text: opening

text: angle.

text: This

text: algorithm

text: has

text: a

text: Θ(n

text: log

text: n)

text: complexity

text: and

text: the

text: advantage

text: of

text: being

text: kernel-independent.

text: Other

text: algorithms

text: that

text: rely

text: on

text: this

text: directional

text: low-rank

text: idea

text: include

text: While

text: all

text: the

text: above

text: methods

text: are

text: based

text: on

text: hierarchical

text: partitioning

text: of

text: the

text: boundary,

text: methods

text: based

text: on

text: the

text: fast

text: Fourier

text: transform

text: are

text: also

text: used

text: widely.

text: These

text: methods

text: are

text: relatively

text: easy

text: to

text: implement,

text: but

text: have

text: super-

text: linear

text: complexity

text: like

text: Θ(nα

text: log

text: n)

text: with

text: α

text: for

text: example.

text: In

text: this

text: paper,

text: we

text: introduce

text: a

text: new

text: algorithm

text: for

text: problems.

text: This

text: algo-

text: rithm

text: leverages

text: the

text: directional

text: idea

text: of

text: and

text: uses

text: oscillatory

text: Cheby-

text: shev

text: interpolation

text: and

text: local

text: FFTs

text: to

text: speed

text: up

text: the

text: computation.

text: This

text: new

text: method

text: has

text: a

text: Θ(n

text: n)

text: complexity

text: and

text: is

text: conceptually

text: much

text: simpler

text: than

text: the

text: previous

text: hierarchical

text: methods.

text: The

text: rest

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: explained

text: the

text: algorithm

text: in

text: detail.

text: Section

text: provides

text: numerical

text: results

text: and

text: discussions

text: are

text: given

text: in

text: Section


text: DIRECTIONAL COMPUTATION OF HIGH FREQUENCY BOUNDARY INTEGRALS11 · where the superscript m means that the exponential modulation is done only for S. Similarly if T is almost-planar, we construct the representation · · G(T, S) = diag(F (:, kT )) Gm (T, S) T TS TS and the approximation ≈ · · · G(T, S) diag(F (:, kT )) I Gm (R , S), T TS T TS T · where the superscript m means that the exponential modulation is done only for T . Since Ω has a C2 boundary, the curvature is uniformly bounded. Let us G ﬁrst estimate the number of segments in . From the setup algorith√m, it is G clear that the√length of a segment T in is bounded by min(2qλ, 2qλ/ κ ) = T Θ(2qλ) = Θ( ωλ). As the total length√of the boun√dary is Θ(1), the number G of segments in is bounded by Θ(1/( ωλ)) = Θ( ω). F Next, let us consider the number of pairs in . Since that the boundary F is C2, for almost all pairs (T, S) in the segments T and S have the same length. Consider ﬁrst the segments of length 2qλ. As there are 2q segments of this length, the total number of pairs is bounded by 22q = Θ(ω). For length equal to smaller values, a segment T only appears together with a constant number of segments S in its neighborhood. Therefore, the total F number of pairs in is again Θ(ω). The complexity of the setup algorithm contains three parts: • F The generation of the set . It takes O(ω log ω) steps. • The evaluation of the dense pairs. Clearly there are at most Θ(ω) dense pairs. Since each leaf segment contains at most O(1) points, the dense matrix G(T, S) has O(1) entries. Therefore, the evaluation cost of all dense pairs is O(ω). • The evaluation of the low-rank pairs. Again, the number of low-rank pairs is at most Θ(ω). In all cases of the low-rank pairs, the matrices · · Gmm, G m , and Gm have O(1) entries. Therefore, the evaluation T,S T,S T,S cost of all low-rank pairs is also O(ω). Summing these contributions together gives a Θ(ω log ω) = Θ(n log n) com- plexity for the setup algorithm. (cid:88) Now we are ready to describe how to evaluate the sums ∈ u(x) = G(x, y)f(y), x P ∈ y P eﬃciently. In the following discussion, we slightly abuse notation by using u to denote the vector (u(x)) ∈ and f to denote the vector (f(y)) ∈ . For x P y P a segment T , u(T ) stands for the subvector obtained by restricting u to the points in T . Similarly, f(S) is the restriction of f to the points in S. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DIRECTIONAL

text: COMPUTATION

text: OF

text: HIGH

text: FREQUENCY

text: BOUNDARY

text: where

text: the

text: superscript

text: m

text: means

text: that

text: the

text: exponential

text: modulation

text: is

text: done

text: only

text: for

text: S.

text: Similarly

text: if

text: T

text: is

text: almost-planar,

text: we

text: construct

text: the

text: representation

text: G(T,

text: S)

text: diag(F

text: kT

text: Gm

text: (T,

text: S)

text: T

text: TS

text: TS

text: and

text: the

text: approximation

text: G(T,

text: S)

text: diag(F

text: kT

text: I

text: Gm

text: (R

text: S),

text: T

text: TS

text: T

text: TS

text: T

text: where

text: the

text: superscript

text: m

text: means

text: that

text: the

text: exponential

text: modulation

text: is

text: done

text: only

text: for

text: T

text: Since

text: Ω

text: has

text: a

text: boundary,

text: the

text: curvature

text: is

text: uniformly

text: bounded.

text: Let

text: us

text: G

text: ﬁrst

text: estimate

text: the

text: number

text: of

text: segments

text: in

text: From

text: the

text: setup

text: algorith√m,

text: it

text: is

text: G

text: clear

text: that

text: the√length

text: of

text: a

text: segment

text: T

text: in

text: is

text: bounded

text: by

text: κ

text: T

text: Θ(

text: ωλ).

text: As

text: the

text: total

text: length√of

text: the

text: boun√dary

text: is

text: the

text: number

text: G

text: of

text: segments

text: in

text: is

text: bounded

text: by

text: ωλ))

text: Θ(

text: ω).

text: F

text: Next,

text: let

text: us

text: consider

text: the

text: number

text: of

text: pairs

text: in

text: Since

text: that

text: the

text: boundary

text: F

text: is

text: for

text: almost

text: all

text: pairs

text: (T,

text: S)

text: in

text: the

text: segments

text: T

text: and

text: S

text: have

text: the

text: same

text: length.

text: Consider

text: ﬁrst

text: the

text: segments

text: of

text: length

text: As

text: there

text: are

text: segments

text: of

text: this

text: length,

text: the

text: total

text: number

text: of

text: pairs

text: is

text: bounded

text: by

text: Θ(ω).

text: For

text: length

text: equal

text: to

text: smaller

text: values,

text: a

text: segment

text: T

text: only

text: appears

text: together

text: with

text: a

text: constant

text: number

text: of

text: segments

text: S

text: in

text: its

text: neighborhood.

text: Therefore,

text: the

text: total

text: F

text: number

text: of

text: pairs

text: in

text: is

text: again

text: Θ(ω).

text: The

text: complexity

text: of

text: the

text: setup

text: algorithm

text: contains

text: three

text: parts:

text: F

text: The

text: generation

text: of

text: the

text: set

text: It

text: takes

text: O(ω

text: log

text: ω)

text: steps.

text: The

text: evaluation

text: of

text: the

text: dense

text: pairs.

text: Clearly

text: there

text: are

text: at

text: most

text: Θ(ω)

text: dense

text: pairs.

text: Since

text: each

text: leaf

text: segment

text: contains

text: at

text: most

text: points,

text: the

text: dense

text: matrix

text: G(T,

text: S)

text: has

text: entries.

text: Therefore,

text: the

text: evaluation

text: cost

text: of

text: all

text: dense

text: pairs

text: is

text: O(ω).

text: The

text: evaluation

text: of

text: the

text: low-rank

text: pairs.

text: Again,

text: the

text: number

text: of

text: low-rank

text: pairs

text: is

text: at

text: most

text: Θ(ω).

text: In

text: all

text: cases

text: of

text: the

text: low-rank

text: pairs,

text: the

text: matrices

text: Gmm,

text: G

text: m

text: and

text: Gm

text: have

text: entries.

text: Therefore,

text: the

text: evaluation

text: T,S

text: T,S

text: T,S

text: cost

text: of

text: all

text: low-rank

text: pairs

text: is

text: also

text: O(ω).

text: Summing

text: these

text: contributions

text: together

text: gives

text: a

text: Θ(ω

text: log

text: ω)

text: Θ(n

text: log

text: n)

text: com-

text: plexity

text: for

text: the

text: setup

text: algorithm.

text: Now

text: we

text: are

text: ready

text: to

text: describe

text: how

text: to

text: evaluate

text: the

text: sums

text: u(x)

text: G(x,

text: y)f(y),

text: x

text: P

text: y

text: P

text: eﬃciently.

text: In

text: the

text: following

text: discussion,

text: we

text: slightly

text: abuse

text: notation

text: by

text: using

text: u

text: to

text: denote

text: the

text: vector

text: (u(x))

text: and

text: f

text: to

text: denote

text: the

text: vector

text: (f(y))

text: For

text: x

text: P

text: y

text: P

text: a

text: segment

text: T

text: u(T

text: stands

text: for

text: the

text: subvector

text: obtained

text: by

text: restricting

text: u

text: to

text: the

text: points

text: in

text: T

text: Similarly,

text: f(S)

text: is

text: the

text: restriction

text: of

text: f

text: to

text: the

text: points

text: in

text: S.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DIRECTIONAL COMPUTATION OF HIGH FREQUENCY BOUNDARY INTEGRALS21

text: DIRECTIONAL

text: COMPUTATION

text: OF

text: HIGH

text: FREQUENCY

text: BOUNDARY


text: 2 I. INTRODUCTION Short-range repulsive interactions are those mainly responsible for the structure of classical particle ﬂuid systems; this is what originally conferred worthiness to hard–body particle models.[1] These have actually proven to be a very good representation of colloidal particle systems, with a very good agreement between the theoretical phase diagram of hard spheres and the experimental phase behaviour of colloidal spheres.[2] Today, ever-increasing importance of colloids and advances in the synthesis of colloidal particles of non-spherical symmetry [3–5] are depriving theoretical studies of any hard particle system of what is remaining of a mere academic aura, demonstrating that these studies may be crucial for the design of new colloidal materials.[6] Helical particles are especially worth investigating as Nature has conferred to the helix a rather prominent role. Helical polynucleotides and polypeptides function at large enough densities that the details of their shape start to be relevant.[7] The desire of mimicking Nature in reproducing the functions carried out by helical biopolymers has, in turn, led to a very active area in polymer research – the synthesis and characterisation of helical polymers, aiming at exploiting the inherent chirality of the helical structure to produce new functional materials to be used especially in asymmetric catalysis and enantiomeric separation. [8, 9] This material interest merges with its inherent biological interest in the currently pursued attempt to employ DNA, perhaps the most emblematic of all helical biomolecular systems, as a building-block for new materials.[10, 11] Rather surprisingly, in spite of this wealth of inspiration sources, helices appear to have been mostly overlooked in past theoretical studies on hard-body non-spherical particle systems, focussing mostly on rod- or disc-like particles,[12, 13] possibly due to the tacit assumption that helices, as elongated objects, can be assimilated to rods. To ﬁll this gap, we have undertaken a systematic investigation of the phase behaviour of hard helices, using numerical simulation and density functional theory. We have found how ﬁnely the isotropic–nematic phase boundaries depend on the structural parameters deﬁning a helical particle, with a dependence not simply rooted in its aspect ratio, thus making a mapping onto an eﬀective rod rather loose.[14] More importantly, we have also provided evidence for the existence of a new chiral nematic phase, named screw-nematic, the helix twofold symmetry axes spiral around the main phase director.[16] This was the phase observed in experiments on systems of colloidal helical ﬁlaments,[17] but our results on such a basic model suggest this screw-nematic phase to be a general feature of any helical particle system, including DNA suspensions at suﬃciently high densities.[18] In the present work, we build upon past work by extending it in several respects. (i) We present a complete phase diagram in the density-pressure plane, with a special emphasis on the smectic phases occurring at densities higher than those typical of the conventional and screw-nematic phases, and discuss how the screw-nematic order merges with a tendency to layering to produce new chiral, screw-like, smectic phases. (ii) We perform a detailed study of the isotropic–nematic coexistence; (iii) We extend the second-virial theory for the nematic–to–screw-nematic phase transition [16] by adding the third-virial contribution, and validate it against numerical simulations. In the next section, we provide details on the various theoretical and computational methods used. We ﬁrst describe the Monte Carlo simulation technique and then the density functional theory at diﬀerent levels of approximation. Section III presents and discusses the results, sub-divided in several parts. In the ﬁrst, phase diagrams, as obtained from isobaric Monte Carlo simulations, are shown and the structure of the smectic phases occurring in the higher density regions is described by means of positional and orientational order parameters and pair correlation functions. In the second part, attention is paid to the isotropic–nematic phase transition, with the aim of a proper location of the coexisting densities and pressure. The third part presents the theoretical results for the nematic–to–screw-nematic phase transition. Finally, Section IV concludes this work by giving a brief summary and possible outlooks. II. MODEL AND METHODS Helices were simply modelled as a line of 15 fused hard spherical beads of diameter D rigidly arranged into a helicoidal shape, with a contour length ﬁxed to L = 10D.[14] On changing the radius r and pitch p at ﬁxed L, the shape of the helical particle can be tuned from a straight rod to very wound coils (Fig.1). We focus on increasingly ∈ ∈ twisted helical shapes in the range r/D [0.2; 0.4] and p/D [2; 8]. Such particles have a suﬃciently large eﬀective aspect ratio to display a rich polymorphic liquid-crystal phase behaviour, and yet they have an intermediate degree O of ”curliness” (L/p = (1)), so that phase sequences and phase structures are expected to depend sensitively on the overall set of parameters deﬁning the particle shape. Hereafter, all lengths will be expressed in units of D. To trace the full phase diagrams of such objects, we resorted to Monte Carlo (MC) numerical simulations in the isobaric(-isothermal) ensemble (MC-NPT) [19, 20]. These calculations were preceded by the construction of the initial compact conﬁgurations. Additional MC simulations in the canonical ensemble (MC-NVT) [19, 20] were performed in one speciﬁc case to identify the precise value of the volume fractions and pressure at the coexistence. Onsager theory, in diﬀerent forms,[21–23] was used for studying the isotropic-to-nematic and nematic–screw-nematic phase transitions. The remainder of this section provides details on the various methods used to investigate the phase behaviour.

text: I.

label: section

text: INTRODUCTION

text: Short-range

text: repulsive

text: interactions

text: are

text: those

text: mainly

text: responsible

text: for

text: the

text: structure

text: of

text: classical

text: particle

text: ﬂuid

text: systems;

text: this

text: is

text: what

text: originally

text: conferred

text: worthiness

text: to

text: hard–body

text: particle

text: These

text: have

text: actually

text: proven

text: to

text: be

text: a

text: very

text: good

text: representation

text: of

text: colloidal

text: particle

text: systems,

text: with

text: a

text: very

text: good

text: agreement

text: between

text: the

text: theoretical

text: phase

text: diagram

text: of

text: hard

text: spheres

text: and

text: the

text: experimental

text: phase

text: behaviour

text: of

text: colloidal

text: Today,

text: ever-increasing

text: importance

text: of

text: colloids

text: and

text: advances

text: in

text: the

text: synthesis

text: of

text: colloidal

text: particles

text: of

text: non-spherical

text: symmetry

text: are

text: depriving

text: theoretical

text: studies

text: of

text: any

text: hard

text: particle

text: system

text: of

text: what

text: is

text: remaining

text: of

text: a

text: mere

text: academic

text: aura,

text: demonstrating

text: that

text: these

text: studies

text: may

text: be

text: crucial

text: for

text: the

text: design

text: of

text: new

text: colloidal

text: Helical

text: particles

text: are

text: especially

text: worth

text: investigating

text: as

text: Nature

text: has

text: conferred

text: to

text: the

text: helix

text: a

text: rather

text: prominent

text: role.

text: Helical

text: polynucleotides

text: and

text: polypeptides

text: function

text: at

text: large

text: enough

text: densities

text: that

text: the

text: details

text: of

text: their

text: shape

text: start

text: to

text: be

text: The

text: desire

text: of

text: mimicking

text: Nature

text: in

text: reproducing

text: the

text: functions

text: carried

text: out

text: by

text: helical

text: biopolymers

text: has,

text: in

text: turn,

text: led

text: to

text: a

text: very

text: active

text: area

text: in

text: polymer

text: research

text: the

text: synthesis

text: and

text: characterisation

text: of

text: helical

text: polymers,

text: aiming

text: at

text: exploiting

text: the

text: inherent

text: chirality

text: of

text: the

text: helical

text: structure

text: to

text: produce

text: new

text: functional

text: materials

text: to

text: be

text: used

text: especially

text: in

text: asymmetric

text: catalysis

text: and

text: enantiomeric

text: separation.

text: This

text: material

text: interest

text: merges

text: with

text: its

text: inherent

text: biological

text: interest

text: in

text: the

text: currently

text: pursued

text: attempt

text: to

text: employ

text: DNA,

text: perhaps

text: the

text: most

text: emblematic

text: of

text: all

text: helical

text: biomolecular

text: systems,

text: as

text: a

text: building-block

text: for

text: new

text: Rather

text: surprisingly,

text: in

text: spite

text: of

text: this

text: wealth

text: of

text: inspiration

text: sources,

text: helices

text: appear

text: to

text: have

text: been

text: mostly

text: overlooked

text: in

text: past

text: theoretical

text: studies

text: on

text: hard-body

text: non-spherical

text: particle

text: systems,

text: focussing

text: mostly

text: on

text: rod-

text: or

text: disc-like

text: possibly

text: due

text: to

text: the

text: tacit

text: assumption

text: that

text: helices,

text: as

text: elongated

text: objects,

text: can

text: be

text: assimilated

text: to

text: rods.

text: To

text: ﬁll

text: this

text: gap,

text: we

text: have

text: undertaken

text: a

text: systematic

text: investigation

text: of

text: the

text: phase

text: behaviour

text: of

text: hard

text: helices,

text: using

text: numerical

text: simulation

text: and

text: density

text: functional

text: theory.

text: We

text: have

text: found

text: how

text: ﬁnely

text: the

text: isotropic–nematic

text: phase

text: boundaries

text: depend

text: on

text: the

text: structural

text: parameters

text: deﬁning

text: a

text: helical

text: particle,

text: with

text: a

text: dependence

text: not

text: simply

text: rooted

text: in

text: its

text: aspect

text: ratio,

text: thus

text: making

text: a

text: mapping

text: onto

text: an

text: eﬀective

text: rod

text: rather

text: More

text: importantly,

text: we

text: have

text: also

text: provided

text: evidence

text: for

text: the

text: existence

text: of

text: a

text: new

text: chiral

text: nematic

text: phase,

text: named

text: screw-nematic,

text: the

text: helix

text: twofold

text: symmetry

text: axes

text: spiral

text: around

text: the

text: main

text: phase

text: This

text: was

text: the

text: phase

text: observed

text: in

text: experiments

text: on

text: systems

text: of

text: colloidal

text: helical

text: but

text: our

text: results

text: on

text: such

text: a

text: basic

text: model

text: suggest

text: this

text: screw-nematic

text: phase

text: to

text: be

text: a

text: general

text: feature

text: of

text: any

text: helical

text: particle

text: system,

text: including

text: DNA

text: suspensions

text: at

text: suﬃciently

text: high

text: In

text: the

text: present

text: work,

text: we

text: build

text: upon

text: past

text: work

text: by

text: extending

text: it

text: in

text: several

text: respects.

text: (i)

text: We

text: present

text: a

text: complete

text: phase

text: diagram

text: in

text: the

text: density-pressure

text: plane,

text: with

text: a

text: special

text: emphasis

text: on

text: the

text: smectic

text: phases

text: occurring

text: at

text: densities

text: higher

text: than

text: those

text: typical

text: of

text: the

text: conventional

text: and

text: screw-nematic

text: phases,

text: and

text: discuss

text: how

text: the

text: screw-nematic

text: order

text: merges

text: with

text: a

text: tendency

text: to

text: layering

text: to

text: produce

text: new

text: chiral,

text: screw-like,

text: smectic

text: phases.

text: (ii)

text: We

text: perform

text: a

text: detailed

text: study

text: of

text: the

text: isotropic–nematic

text: coexistence;

text: (iii)

text: We

text: extend

text: the

text: second-virial

text: theory

text: for

text: the

text: nematic–to–screw-nematic

text: phase

text: transition

text: by

text: adding

text: the

text: third-virial

text: contribution,

text: and

text: validate

text: it

text: against

text: numerical

text: simulations.

text: In

text: the

text: next

text: section,

text: we

text: provide

text: details

text: on

text: the

text: various

text: theoretical

text: and

text: computational

text: methods

text: used.

text: We

text: ﬁrst

text: describe

text: the

text: Monte

text: Carlo

text: simulation

text: technique

text: and

text: then

text: the

text: density

text: functional

text: theory

text: at

text: diﬀerent

text: levels

text: of

text: approximation.

text: Section

text: III

text: presents

text: and

text: discusses

text: the

text: results,

text: sub-divided

text: in

text: several

text: parts.

text: In

text: the

text: ﬁrst,

text: phase

text: diagrams,

text: as

text: obtained

text: from

text: isobaric

text: Monte

text: Carlo

text: simulations,

text: are

text: shown

text: and

text: the

text: structure

text: of

text: the

text: smectic

text: phases

text: occurring

text: in

text: the

text: higher

text: density

text: regions

text: is

text: described

text: by

text: means

text: of

text: positional

text: and

text: orientational

text: order

text: parameters

text: and

text: pair

text: correlation

text: functions.

text: In

text: the

text: second

text: part,

text: attention

text: is

text: paid

text: to

text: the

text: isotropic–nematic

text: phase

text: transition,

text: with

text: the

text: aim

text: of

text: a

text: proper

text: location

text: of

text: the

text: coexisting

text: densities

text: and

text: pressure.

text: The

text: third

text: part

text: presents

text: the

text: theoretical

text: results

text: for

text: the

text: nematic–to–screw-nematic

text: phase

text: transition.

text: Finally,

text: Section

text: IV

text: concludes

text: this

text: work

text: by

text: giving

text: a

text: brief

text: summary

text: and

text: possible

text: outlooks.

text: II.

text: MODEL

text: AND

text: METHODS

text: Helices

text: were

text: simply

text: modelled

text: as

text: a

text: line

text: of

text: fused

text: hard

text: spherical

text: beads

text: of

text: diameter

text: D

text: rigidly

text: arranged

text: into

text: a

text: helicoidal

text: shape,

text: with

text: a

text: contour

text: length

text: ﬁxed

text: to

text: L

text: On

text: changing

text: the

text: radius

text: r

text: and

text: pitch

text: p

text: at

text: ﬁxed

text: L,

text: the

text: shape

text: of

text: the

text: helical

text: particle

text: can

text: be

text: tuned

text: from

text: a

text: straight

text: rod

text: to

text: very

text: wound

text: coils

text: We

text: focus

text: on

text: increasingly

text: twisted

text: helical

text: shapes

text: in

text: the

text: range

text: r/D

text: and

text: p/D

text: Such

text: particles

text: have

text: a

text: suﬃciently

text: large

text: eﬀective

text: aspect

text: ratio

text: to

text: display

text: a

text: rich

text: polymorphic

text: liquid-crystal

text: phase

text: behaviour,

text: and

text: yet

text: they

text: have

text: an

text: intermediate

text: degree

text: O

text: of

text: ”curliness”

text: (L/p

text: so

text: that

text: phase

text: sequences

text: and

text: phase

text: structures

text: are

text: expected

text: to

text: depend

text: sensitively

text: on

text: the

text: overall

text: set

text: of

text: parameters

text: deﬁning

text: the

text: particle

text: shape.

text: Hereafter,

text: all

text: lengths

text: will

text: be

text: expressed

text: in

text: units

text: of

text: D.

text: To

text: trace

text: the

text: full

text: phase

text: diagrams

text: of

text: such

text: objects,

text: we

text: resorted

text: to

text: Monte

text: Carlo

text: (MC)

text: numerical

text: simulations

text: in

text: the

text: isobaric(-isothermal)

text: ensemble

text: (MC-NPT)

text: These

text: calculations

text: were

text: preceded

text: by

text: the

text: construction

text: of

text: the

text: initial

text: compact

text: conﬁgurations.

text: Additional

text: MC

text: simulations

text: in

text: the

text: canonical

text: ensemble

text: (MC-NVT)

text: were

text: performed

text: in

text: one

text: speciﬁc

text: case

text: to

text: identify

text: the

text: precise

text: value

text: of

text: the

text: volume

text: fractions

text: and

text: pressure

text: at

text: the

text: coexistence.

text: Onsager

text: theory,

text: in

text: diﬀerent

text: was

text: used

text: for

text: studying

text: the

text: isotropic-to-nematic

text: and

text: nematic–screw-nematic

text: phase

text: transitions.

text: The

text: remainder

text: of

text: this

text: section

text: provides

text: details

text: on

text: the

text: various

text: methods

text: used

text: to

text: investigate

text: the

text: phase

text: behaviour.


text: 11 (cid:98) ∗ FIG. 7. The g(cid:107)(R(cid:107)) and gw(cid:107)(R(cid:107)) correlation functions for helices with r = 0.2 and p = 8, calculated for P = 0.9 in the N ∗ 1, ∗ phase (left) and P = 1.0 in the N phase (right). Also depicted are two corresponding snapshots, colour coded according to S the local tangent as explained in the text. (cid:98) ∗ ∗ FIG. 8. The functions g(cid:107)(R(cid:107)) (top left) and gw(cid:107)(R(cid:107)) (top right) at P = 1.1, for helices with r = 0.2 and p = 8 (Sm phase.. 1, A,S Also depicted is g⊥(R⊥) and a corresponding snapshot colour coded according to the local tangent to helices (right). ∗ corresponding to the N state point (P = 0.9), highlight the diﬀerent organization occurring at the two pressures. ∗ In Fig. 7 one can further notice a small amplitude oscillation in function g(cid:107)(R(cid:107)) at P = 1, that is absent in the ∗ corresponding lower pressure case P = 0.9. This is indicative of an incipient smectic order, which sets in at the ∗ slightly higher pressure P = 1.1, as conﬁrmed by the solid-like behaviour of g(cid:107)(R(cid:107)) shown in Fig.8 (top left). Here ∼ (cid:98) we can recognize a periodicity of 12, only slightly longer than the eﬀective length of the helices, which is equal to 10.88. This is diﬀerent from the periodicity of gw(cid:107)(R(cid:107)) (top right), which corresponds to the helix pitch p, here 1, equal to 8. Thus the screw-like order has combined with layer ordering to give rise to a new chiral smectic phase. The presence of two diﬀerent periodicities is evident in the snapshot in Fig. 8. The correlation function g⊥(R⊥) (cid:104) (cid:105) (Fig. 8, bottom left) does not provide indication of translational order within each single layer, and ψ is found 6 to be very small. This screw-smectic phase, globally uniaxial with the main director perpendicular to the layers, is ∗ ∗ ≈ of type A and labelled as Sm . At higher pressure, P = 1.3 (η 0.52), hexatic order sets in within each single A,S layer, as sho√wn by the behavior of g⊥(R⊥) that exhibits well developed characteristic double peak structure, with maxima at 3σ and 2σ (Fig. 9, left panel), σ being the position of the main, nearest-neighbour peak. The presence (cid:104) (cid:105) (cid:98) of hexatic order is further conﬁrmed by the high value of ψ (Fig.10 left panel) and by the fact that the average (cid:104) (cid:105) ≈ 6 number of nearest-neighbours n tends to 6 at η 0.52. The plot of gw(cid:107)(R(cid:107)) (Fig.9, top right panel) shows a clear 1, ∗ in-plane azimuthal correlation, but the absence of the helical periodicity that was present in the Sm phase. The A,S ∗ correlation now is only within layers, which are uncorrelated from each other. This diﬀerence from the Sm phase A,S ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: FIG.

text: The

text: and

text: correlation

text: functions

text: for

text: helices

text: with

text: r

text: and

text: p

text: calculated

text: for

text: P

text: in

text: the

text: N

text: phase

text: (left)

text: and

text: P

text: in

text: the

text: N

text: phase

text: (right).

text: Also

text: depicted

text: are

text: two

text: corresponding

text: snapshots,

text: colour

text: coded

text: according

text: to

text: S

text: the

text: local

text: tangent

text: as

text: explained

text: in

text: the

text: text.

text: FIG.

text: The

text: functions

text: (top

text: left)

text: and

text: (top

text: right)

text: at

text: P

text: for

text: helices

text: with

text: r

text: and

text: p

text: (Sm

text: phase..

text: A,S

text: Also

text: depicted

text: is

text: g⊥(R⊥)

text: and

text: a

text: corresponding

text: snapshot

text: colour

text: coded

text: according

text: to

text: the

text: local

text: tangent

text: to

text: helices

text: (right).

text: corresponding

text: to

text: the

text: N

text: state

text: point

text: (P

text: highlight

text: the

text: diﬀerent

text: organization

text: occurring

text: at

text: the

text: two

text: pressures.

text: In

text: Fig.

text: one

text: can

text: further

text: notice

text: a

text: small

text: amplitude

text: oscillation

text: in

text: function

text: at

text: P

text: that

text: is

text: absent

text: in

text: the

text: corresponding

text: lower

text: pressure

text: case

text: P

text: This

text: is

text: indicative

text: of

text: an

text: incipient

text: smectic

text: order,

text: which

text: sets

text: in

text: at

text: the

text: slightly

text: higher

text: pressure

text: P

text: as

text: conﬁrmed

text: by

text: the

text: solid-like

text: behaviour

text: of

text: shown

text: in

text: (top

text: left).

text: Here

text: we

text: can

text: recognize

text: a

text: periodicity

text: of

text: only

text: slightly

text: longer

text: than

text: the

text: eﬀective

text: length

text: of

text: the

text: helices,

text: which

text: is

text: equal

text: to

text: This

text: is

text: diﬀerent

text: from

text: the

text: periodicity

text: of

text: (top

text: right),

text: which

text: corresponds

text: to

text: the

text: helix

text: pitch

text: p,

text: here

text: equal

text: to

text: Thus

text: the

text: screw-like

text: order

text: has

text: combined

text: with

text: layer

text: ordering

text: to

text: give

text: rise

text: to

text: a

text: new

text: chiral

text: smectic

text: phase.

text: The

text: presence

text: of

text: two

text: diﬀerent

text: periodicities

text: is

text: evident

text: in

text: the

text: snapshot

text: in

text: Fig.

text: The

text: correlation

text: function

text: g⊥(R⊥)

text: (Fig.

text: bottom

text: left)

text: does

text: not

text: provide

text: indication

text: of

text: translational

text: order

text: within

text: each

text: single

text: layer,

text: and

text: ψ

text: is

text: found

text: to

text: be

text: very

text: small.

text: This

text: screw-smectic

text: phase,

text: globally

text: uniaxial

text: with

text: the

text: main

text: director

text: perpendicular

text: to

text: the

text: layers,

text: is

text: of

text: type

text: A

text: and

text: labelled

text: as

text: Sm

text: At

text: higher

text: pressure,

text: P

text: (η

text: hexatic

text: order

text: sets

text: in

text: within

text: each

text: single

text: A,S

text: layer,

text: as

text: sho√wn

text: by

text: the

text: behavior

text: of

text: g⊥(R⊥)

text: that

text: exhibits

text: well

text: developed

text: characteristic

text: double

text: peak

text: structure,

text: with

text: maxima

text: at

text: 3σ

text: and

text: 2σ

text: (Fig.

text: left

text: panel),

text: σ

text: being

text: the

text: position

text: of

text: the

text: main,

text: nearest-neighbour

text: peak.

text: The

text: presence

text: of

text: hexatic

text: order

text: is

text: further

text: conﬁrmed

text: by

text: the

text: high

text: value

text: of

text: ψ

text: left

text: panel)

text: and

text: by

text: the

text: fact

text: that

text: the

text: average

text: number

text: of

text: nearest-neighbours

text: n

text: tends

text: to

text: at

text: η

text: The

text: plot

text: of

text: top

text: right

text: panel)

text: shows

text: a

text: clear

text: in-plane

text: azimuthal

text: correlation,

text: but

text: the

text: absence

text: of

text: the

text: helical

text: periodicity

text: that

text: was

text: present

text: in

text: the

text: Sm

text: phase.

text: The

text: A,S

text: correlation

text: now

text: is

text: only

text: within

text: layers,

text: which

text: are

text: uncorrelated

text: from

text: each

text: other.

text: This

text: diﬀerence

text: from

text: the

text: Sm

text: phase

text: A,S

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 19 with respect to next layers, to gain translational entropy. Under these conditions, layering is very strong a(cid:98)s testiﬁed by the solid-like peaks in the observed g(cid:107)(R(cid:107)). This situation, depicted in the cartoon of Fig.14 (right panel), diﬀers from the conventional smectic B phase for the presence of the in-plane (polar) correlation between the w axes of helices, and hence the phase was denoted as Sm . B,p The(cid:98)alternative scenario stems from the possibility that tips of helices protrud(cid:98)ing out from a layer are still able to propagate the ordering to the neighboring layers. Thus, helices belonging to diﬀerent layers stack on top of each other along n to form parallel, ”inﬁnitely” long, (cid:98)helices. The alignement of the the w axes t(cid:98)hen translates into a screw- like ordering propagating across the layers. Under these conditions, clearly diﬀerent layers are strongly correlated with each other. The layer structure along n is preserved, but positional ordering along n is less eﬀective, due to the presence of protruding helices, as testiﬁed by the reduced peaked structure of g(cid:107)(R(cid:107)). This is the situation represented ∗ in the left panel of Fig.14. We denoted this phase as Sm , because it couples layering with hexatic positional order B,S and screw-like azimuthal correlations. The entropic advantage of this scheme is to form a set of ”inﬁnite” parallel helices, which allows the favourable screw-like motion to be still operative. One may envisage the additional presence of a columnar phase in the case of helices with suﬃciently long contour lengths, well beyond those considered here. This is a subject that deserves a dedicated study. V. CONCLUSIONS AND OUTLOOK Phase Code Organisation type Conventional nematic N (a) ∗ Screw- nematic N (b) S∗ Screw-smectic A Sm (c) A,S Polar smectic B Sm (d) B,p ∗ Screw-smectic B Sm (cid:98) (e) (cid:98) B,S (cid:98) (cid:98) (cid:98) (cid:98) (cid:98) TABLE I. Sum(cid:98)mary of the diﬀerent phases exhibited by hard helices. (a)u axes oriented along n; (b) as in N with azimuthal coupling of w axes along c that in turn spirals about n; (c)as in Sm with azimuthal coupling of w axes along c that in turn A ∗ spirals about n; (d) as in Sm with additional in-plane polar and hexatic order; (e) as in Sm with additional hexatic order. A A,S In this work, we have studied the self-assembly properties of systems of hard helices as a function of helix morphology. Helical particles have been modelled as a set of fused hard spheres properly arranged to form a rigid helix of a ﬁxed contour length. Using a combination of numerical simulations and density functional theories, we have analyzed the sequence of diﬀerent liquid crystal phases appearing at increasing density, using a set of suitable order parameters and correlation functions. The rich and unconventional polymorphism that we found is in striking contrast with the conventional wisdom of assimilating the phase behaviour of helical particles to that of rods, an assumption commonly adopted also in the analysis of experiments on helical (bio)polymers. Table I summarizes the distinctive features of all phases discussed in this work. (cid:98) ∗ (cid:98) The ﬁrst novel phase encountered with increasing density is the screw-nematic (N ). As neighboring helices tend to S lock into an in-phase nematic conﬁguration by an azimuthal correlation of the helix w axes a(cid:98)long a common direction c, there must be a(cid:98)concomitant gain in translational entropy counterbalancing that loss of rotational entropy for this new phase to be stable. This is achieved through a translational-rotational coupling where c spirals around the main nematic director n, with a periodicity equal to the pitch of the single helix. We have also implemented a density ∗ functional theory with increasing degrees of accuracy, for the screw-nematic N phase, under the assumption of S perfectly aligned helices, and tested its accuracy with numerical simulations on the same system. We ﬁnd the results of the most accurate versions of the theory in reasonably good quantitative agreement with numerical simulations. ∗ ∗ With increasing density a smectic A phase with screw-like order (Sm ) can appear, which(cid:98) diﬀers from the N A,S S for the presence of layers. However helices laying in the interlayer regions provide a bridge between adjacent layers, which allows to keep the screw-like organization. As density increases, positional ordering along n also increases, while in-p(cid:98)lane hexatic order tends to set in. This leads to the formation of either a polar Sm phase, characterized by the B,p fact that diﬀerent layers can rotate and translate independently of each other with no coupling between orientations ∗ of w axes in diﬀerent layers, or of a Sm , with screw-like coupling between adjacent layers, Our results indicate B,S ∗ that a Sm phase is more favoured for slender helices, with a gradual transition to screw-smectic Sm phases B,p B,S for curlier particles. At even higher densities, a very compact phase, that we generally labelled as C, is achieved. This phase is likely to display some regular crystal structure, as indicated by the regular peaks in several correlation functions the we have monitored. A detailed study of this phase will be discussed elsewhere. The results presented here call for experimental veriﬁcation. To this purpose, an important distinctive feature of ##LTLine## ##LTLine## ##LTLine##

text: with

text: respect

text: to

text: next

text: layers,

text: to

text: gain

text: translational

text: entropy.

text: Under

text: these

text: conditions,

text: layering

text: is

text: very

text: strong

text: testiﬁed

text: by

text: the

text: solid-like

text: peaks

text: in

text: the

text: observed

text: This

text: situation,

text: depicted

text: in

text: the

text: cartoon

text: of

text: (right

text: panel),

text: diﬀers

text: from

text: the

text: conventional

text: smectic

text: B

text: phase

text: for

text: the

text: presence

text: of

text: the

text: in-plane

text: (polar)

text: correlation

text: between

text: the

text: w

text: axes

text: of

text: helices,

text: and

text: hence

text: the

text: phase

text: was

text: denoted

text: as

text: Sm

text: B,p

text: scenario

text: stems

text: from

text: the

text: possibility

text: that

text: tips

text: of

text: helices

text: out

text: from

text: a

text: layer

text: are

text: still

text: able

text: to

text: propagate

text: the

text: ordering

text: to

text: the

text: neighboring

text: layers.

text: Thus,

text: helices

text: belonging

text: to

text: diﬀerent

text: layers

text: stack

text: on

text: top

text: of

text: each

text: other

text: along

text: n

text: to

text: form

text: parallel,

text: ”inﬁnitely”

text: long,

text: The

text: alignement

text: of

text: the

text: the

text: w

text: axes

text: translates

text: into

text: a

text: screw-

text: like

text: ordering

text: propagating

text: across

text: the

text: layers.

text: Under

text: these

text: conditions,

text: clearly

text: diﬀerent

text: layers

text: are

text: strongly

text: correlated

text: with

text: each

text: other.

text: The

text: layer

text: structure

text: along

text: n

text: is

text: preserved,

text: but

text: positional

text: ordering

text: along

text: n

text: is

text: less

text: eﬀective,

text: due

text: to

text: the

text: presence

text: of

text: protruding

text: helices,

text: as

text: testiﬁed

text: by

text: the

text: reduced

text: peaked

text: structure

text: of

text: This

text: is

text: the

text: situation

text: represented

text: in

text: the

text: left

text: panel

text: of

text: We

text: denoted

text: this

text: phase

text: as

text: Sm

text: because

text: it

text: couples

text: layering

text: with

text: hexatic

text: positional

text: order

text: B,S

text: and

text: screw-like

text: azimuthal

text: correlations.

text: The

text: entropic

text: advantage

text: of

text: this

text: scheme

text: is

text: to

text: form

text: a

text: set

text: of

text: ”inﬁnite”

text: parallel

text: helices,

text: which

text: allows

text: the

text: favourable

text: screw-like

text: motion

text: to

text: be

text: still

text: operative.

text: One

text: may

text: envisage

text: the

text: additional

text: presence

text: of

text: a

text: columnar

text: phase

text: in

text: the

text: case

text: of

text: helices

text: with

text: suﬃciently

text: long

text: contour

text: lengths,

text: well

text: beyond

text: those

text: considered

text: here.

text: This

text: is

text: a

text: subject

text: that

text: deserves

text: a

text: dedicated

text: study.

text: V.

text: CONCLUSIONS

text: AND

text: OUTLOOK

text: Phase

text: Code

text: Organisation

text: type

text: Conventional

text: nematic

text: N

text: (a)

text: Screw-

text: nematic

text: N

text: (b)

text: S∗

text: Screw-smectic

text: A

text: Sm

text: (c)

text: A,S

text: Polar

text: smectic

text: B

text: Sm

text: (d)

text: B,p

text: Screw-smectic

text: B

text: Sm

text: (e)

text: B,S

text: TABLE

text: I.

text: of

text: the

text: diﬀerent

text: phases

text: exhibited

text: by

text: hard

text: helices.

text: (a)u

text: axes

text: oriented

text: along

text: n;

text: (b)

text: as

text: in

text: N

text: with

text: azimuthal

text: coupling

text: of

text: w

text: axes

text: along

text: c

text: that

text: in

text: turn

text: spirals

text: about

text: n;

text: (c)as

text: in

text: Sm

text: with

text: azimuthal

text: coupling

text: of

text: w

text: axes

text: along

text: c

text: that

text: in

text: turn

text: A

text: spirals

text: about

text: n;

text: (d)

text: as

text: in

text: Sm

text: with

text: additional

text: in-plane

text: polar

text: and

text: hexatic

text: order;

text: (e)

text: as

text: in

text: Sm

text: with

text: additional

text: hexatic

text: order.

text: A

text: A,S

text: In

text: this

text: work,

text: we

text: have

text: studied

text: the

text: self-assembly

text: properties

text: of

text: systems

text: of

text: hard

text: helices

text: as

text: a

text: function

text: of

text: helix

text: morphology.

text: Helical

text: particles

text: have

text: been

text: modelled

text: as

text: a

text: set

text: of

text: fused

text: hard

text: spheres

text: properly

text: arranged

text: to

text: form

text: a

text: rigid

text: helix

text: of

text: a

text: ﬁxed

text: contour

text: length.

text: Using

text: a

text: combination

text: of

text: numerical

text: simulations

text: and

text: density

text: functional

text: theories,

text: we

text: have

text: analyzed

text: the

text: sequence

text: of

text: diﬀerent

text: liquid

text: crystal

text: phases

text: appearing

text: at

text: increasing

text: density,

text: using

text: a

text: set

text: of

text: suitable

text: order

text: parameters

text: and

text: correlation

text: functions.

text: The

text: rich

text: and

text: unconventional

text: polymorphism

text: that

text: we

text: found

text: is

text: in

text: striking

text: contrast

text: with

text: the

text: conventional

text: wisdom

text: of

text: assimilating

text: the

text: phase

text: behaviour

text: of

text: helical

text: particles

text: to

text: that

text: of

text: rods,

text: an

text: assumption

text: commonly

text: adopted

text: also

text: in

text: the

text: analysis

text: of

text: experiments

text: on

text: helical

text: (bio)polymers.

text: Table

text: I

text: summarizes

text: the

text: distinctive

text: features

text: of

text: all

text: phases

text: discussed

text: in

text: this

text: work.

text: The

text: ﬁrst

text: novel

text: phase

text: encountered

text: with

text: increasing

text: density

text: is

text: the

text: screw-nematic

text: (N

text: As

text: neighboring

text: helices

text: tend

text: to

text: S

text: lock

text: into

text: an

text: in-phase

text: nematic

text: conﬁguration

text: by

text: an

text: azimuthal

text: correlation

text: of

text: the

text: helix

text: w

text: axes

text: a

text: common

text: direction

text: c,

text: there

text: must

text: be

text: gain

text: in

text: translational

text: entropy

text: counterbalancing

text: that

text: loss

text: of

text: rotational

text: entropy

text: for

text: this

text: new

text: phase

text: to

text: be

text: stable.

text: This

text: is

text: achieved

text: through

text: a

text: translational-rotational

text: coupling

text: where

text: c

text: spirals

text: around

text: the

text: main

text: nematic

text: director

text: n,

text: with

text: a

text: periodicity

text: equal

text: to

text: the

text: pitch

text: of

text: the

text: single

text: helix.

text: We

text: have

text: also

text: implemented

text: a

text: density

text: functional

text: theory

text: with

text: increasing

text: degrees

text: of

text: accuracy,

text: for

text: the

text: screw-nematic

text: N

text: phase,

text: under

text: the

text: assumption

text: of

text: S

text: perfectly

text: aligned

text: helices,

text: and

text: tested

text: its

text: accuracy

text: with

text: numerical

text: simulations

text: on

text: the

text: same

text: system.

text: We

text: ﬁnd

text: the

text: results

text: of

text: the

text: most

text: accurate

text: versions

text: of

text: the

text: theory

text: in

text: reasonably

text: good

text: quantitative

text: agreement

text: with

text: numerical

text: simulations.

text: With

text: increasing

text: density

text: a

text: smectic

text: A

text: phase

text: with

text: screw-like

text: order

text: (Sm

text: can

text: appear,

text: diﬀers

text: from

text: the

text: N

text: A,S

text: S

text: for

text: the

text: presence

text: of

text: layers.

text: However

text: helices

text: laying

text: in

text: the

text: interlayer

text: regions

text: provide

text: a

text: bridge

text: between

text: adjacent

text: layers,

text: which

text: allows

text: to

text: keep

text: the

text: screw-like

text: organization.

text: As

text: density

text: increases,

text: positional

text: ordering

text: along

text: n

text: also

text: increases,

text: while

text: hexatic

text: order

text: tends

text: to

text: set

text: in.

text: This

text: leads

text: to

text: the

text: formation

text: of

text: either

text: a

text: polar

text: Sm

text: phase,

text: characterized

text: by

text: the

text: B,p

text: fact

text: that

text: diﬀerent

text: layers

text: can

text: rotate

text: and

text: translate

text: independently

text: of

text: each

text: other

text: with

text: no

text: coupling

text: between

text: orientations

text: of

text: w

text: axes

text: in

text: diﬀerent

text: layers,

text: or

text: of

text: a

text: Sm

text: with

text: screw-like

text: coupling

text: between

text: adjacent

text: layers,

text: Our

text: results

text: indicate

text: B,S

text: that

text: a

text: Sm

text: phase

text: is

text: more

text: favoured

text: for

text: slender

text: helices,

text: with

text: a

text: gradual

text: transition

text: to

text: screw-smectic

text: Sm

text: phases

text: B,p

text: B,S

text: for

text: curlier

text: particles.

text: At

text: even

text: higher

text: densities,

text: a

text: very

text: compact

text: phase,

text: that

text: we

text: generally

text: labelled

text: as

text: C,

text: is

text: achieved.

text: This

text: phase

text: is

text: likely

text: to

text: display

text: some

text: regular

text: crystal

text: structure,

text: as

text: indicated

text: by

text: the

text: regular

text: peaks

text: in

text: several

text: correlation

text: functions

text: the

text: we

text: have

text: monitored.

text: A

text: detailed

text: study

text: of

text: this

text: phase

text: will

text: be

text: discussed

text: elsewhere.

text: The

text: results

text: presented

text: here

text: call

text: for

text: experimental

text: veriﬁcation.

text: To

text: this

text: purpose,

text: an

text: important

text: distinctive

text: feature

text: of

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text: ##LTLine##

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