text: straints and microscopical properties of the chromatin, for instance [25, 46]. Additionally, some of these elements [25, 44, 46, 47], as well as strong steric [43] and hydrodynamic [64] eﬀects, can alter the scaling behaviors of equation (9). The two experiments reproduced in Suppl. ﬁgure 1 ([29, 37]) display a robust scaling, but are not made in identical controlled environments. As such, we decided to ﬁx the simulation spatial unit through a deﬁnition of the maximum extension of linkers between three nucleosomes b = 24nm and, after, to ﬁt the temporal scaling behavior of MSD as in Suppl. ﬁgure 1B for the deﬁnition of the unit of time. To obtain Suppl. ﬁgure 1, the end-to-end distance as a function of the genomic distance, we measured the equilibrium conformation along time of 64 parallel polymer simulations, then calculated the mean distance in function of genomic distance and log-binned the data though a geometric progression of ratio of 2. To obtain the MSD as a function of time, we collected for the same 64 parallel equilibrium simulations the trajectories of the bead placed in the center of the polymer chain; then we subtracted from each trajectory the position at time zero and we calculated the square of each three-dimensional vector and log-binned the resulting curve through a geometric progression or ratio of 2. 4. Irreversible crosslinking process The simulation implements Rouse dynamics with the addition of crosslinks, namely ir- reversible and conﬁguration-dependent changes in the chain topology. In detail, after each accepted MC move, in case a couple of beads are found at a distance lower than r = b/64 int units, a crosslink is introduced between them with a given probability p that we consider as a proxy for the crosslinker concentration c and the couple is added to the list of adjacent beads (in the 3D space). This section reconsiders this proxy connecting the parameter p to physico-chemical quantities. The problem of irreversible reactions can be restated as a ﬁrst passage-time problem: in the most basic examples (e.g. the decay radioactive atomic nuclei), the on-rate of transition from state A to state B is a constant in time, which is called K . As such, considering on a ﬁnite (continuous) amount of time ∆t, the probability p of passing from A to B in that amount of time can be calculated from the rate of doing the transition integrated over the elapsed time 18

text: straints

text: and

text: microscopical

text: properties

text: of

text: the

text: chromatin,

text: for

text: instance

text: Additionally,

text: some

text: of

text: these

text: elements

text: as

text: well

text: as

text: strong

text: steric

text: and

text: hydrodynamic

text: eﬀects,

text: can

text: alter

text: the

text: scaling

text: behaviors

text: of

text: equation

text: The

text: two

text: experiments

text: reproduced

text: in

text: Suppl.

text: ﬁgure

text: display

text: a

text: robust

text: scaling,

text: but

text: are

text: not

text: made

text: in

text: identical

text: controlled

text: environments.

text: As

text: such,

text: we

text: decided

text: to

text: ﬁx

text: the

text: simulation

text: spatial

text: unit

text: through

text: a

text: deﬁnition

text: of

text: the

text: maximum

text: extension

text: of

text: linkers

text: between

text: three

text: nucleosomes

text: b

text: and,

text: after,

text: to

text: ﬁt

text: the

text: temporal

text: scaling

text: behavior

text: of

text: MSD

text: as

text: in

text: Suppl.

text: ﬁgure

text: for

text: the

text: deﬁnition

text: of

text: the

text: unit

text: of

text: time.

text: To

text: obtain

text: Suppl.

text: ﬁgure

text: the

text: end-to-end

text: distance

text: as

text: a

text: function

text: of

text: the

text: genomic

text: distance,

text: we

text: measured

text: the

text: equilibrium

text: conformation

text: along

text: time

text: of

text: parallel

text: polymer

text: simulations,

text: then

text: calculated

text: the

text: mean

text: distance

text: in

text: function

text: of

text: genomic

text: distance

text: and

text: log-binned

text: the

text: data

text: though

text: a

text: geometric

text: progression

text: of

text: ratio

text: of

text: To

text: obtain

text: the

text: MSD

text: as

text: a

text: function

text: of

text: time,

text: we

text: collected

text: for

text: the

text: same

text: parallel

text: equilibrium

text: simulations

text: the

text: trajectories

text: of

text: the

text: bead

text: placed

text: in

text: the

text: center

text: of

text: the

text: polymer

text: chain;

text: then

text: we

text: subtracted

text: from

text: each

text: trajectory

text: the

text: position

text: at

text: time

text: zero

text: and

text: we

text: calculated

text: the

text: square

text: of

text: each

text: three-dimensional

text: vector

text: and

text: log-binned

text: the

text: resulting

text: curve

text: through

text: a

text: geometric

text: progression

text: or

text: ratio

text: of

text: Irreversible

text: crosslinking

text: process

text: The

text: simulation

text: implements

text: Rouse

text: dynamics

text: with

text: the

text: addition

text: of

text: crosslinks,

text: namely

text: ir-

text: reversible

text: and

text: conﬁguration-dependent

text: changes

text: in

text: the

text: chain

text: topology.

text: In

text: detail,

text: after

text: each

text: accepted

text: MC

text: move,

text: in

text: case

text: a

text: couple

text: of

text: beads

text: are

text: found

text: at

text: a

text: distance

text: lower

text: than

text: r

text: int

text: units,

text: a

text: crosslink

text: is

text: introduced

text: between

text: them

text: with

text: a

text: given

text: probability

text: p

text: that

text: we

text: consider

text: as

text: a

text: proxy

text: for

text: the

text: crosslinker

text: concentration

text: c

text: and

text: the

text: couple

text: is

text: added

text: to

text: the

text: list

text: of

text: adjacent

text: beads

text: (in

text: the

text: space).

text: This

text: section

text: reconsiders

text: this

text: proxy

text: connecting

text: the

text: parameter

text: p

text: to

text: physico-chemical

text: quantities.

text: The

text: problem

text: of

text: irreversible

text: reactions

text: can

text: be

text: restated

text: as

text: a

text: ﬁrst

text: passage-time

text: problem:

text: in

text: the

text: most

text: basic

text: examples

text: (e.g.

text: the

text: decay

text: radioactive

text: atomic

text: nuclei),

text: the

text: on-rate

text: of

text: transition

text: from

text: state

text: A

text: to

text: state

text: B

text: is

text: a

text: constant

text: in

text: time,

text: which

text: is

text: called

text: K

text: As

text: such,

text: considering

text: on

text: a

text: ﬁnite

text: (continuous)

text: amount

text: of

text: time

text: ∆t,

text: the

text: probability

text: p

text: of

text: passing

text: from

text: A

text: to

text: B

text: in

text: that

text: amount

text: of

text: time

text: can

text: be

text: calculated

text: from

text: the

text: rate

text: of

text: doing

text: the

text: transition

text: integrated

text: over

text: the

text: elapsed

text: time


text: a rescaling, are similar to the results of the simpler algorithm (Supplementary ﬁgure 2A, squares). Finally, we speculate that taking into account the transition from diﬀusion-limited to ∗ reaction-limited regimes, the expressions for pearling size ξ and time t will be (denoting ξ cr ∗ and t the pearling size and time for p = 1): cr ∗ − ∗ t t − ∗ cr = p 2δ, and t −cr ξ ξ − ∗ cr = D1/2p δ, with (11) t 1/2 R cr · ∗ ξ2 = D t and δ = 0.4; cr R cr which model the crossover. Equation (11) resumes equations 5, 6, 7 and 8 from the main text. They display explicitly the dependency on the crosslink probability p, introduce a ∗ soft crossover toward the critical value p = 1 and the bounds ξ and t to be determined cr cr experimentally. B. Simulation data analysis 1. Generation of contact matrices For each set of parameters, simulations has been launched in batches of 2000 identical runs on the TARS cluster of “Pasteur Institut”, powered by 3500 identical CPUs. Time and information about the couple of involved beads have been recorded for each crosslinking event for each parallel simulation in distinct data-ﬁles. Sparse matrices have been generated, merging all batches, summing the number of crosslinking events happening for each couple of beads up to a speciﬁed time. Time has been sampled according to a geometric progression of initial value and common ratio equal to 1.1, discarding all matrices with less than 512 contacts. Contact matrices for ideal polymer models at very long time (after the transition) at various crosslinking probabilities look like the one reported in Suppl. ﬁgure 3. 2. Calculation of P (s) in the simulation In the simulation, sparse matrices have been coarse-grained to bead-level dense matrices. Binning of the contact probability P (s) at bead level has been made by calculating the mean value of secondary diagonals for s varying in the range from 1 to 2048 beads. In order to 21 ##LTLine## ##LTLine##

text: a

label: paragraph

text: rescaling,

text: are

text: similar

text: to

text: the

text: results

text: of

text: the

text: simpler

text: algorithm

text: (Supplementary

text: ﬁgure

text: squares).

text: Finally,

text: we

text: speculate

text: that

text: taking

text: into

text: account

text: the

text: transition

text: from

text: diﬀusion-limited

text: to

text: reaction-limited

text: regimes,

text: the

text: expressions

text: for

text: pearling

text: size

text: ξ

text: and

text: time

text: t

text: will

text: be

text: (denoting

text: ξ

text: cr

text: and

text: t

text: the

text: pearling

text: size

text: and

text: time

text: for

text: p

text: cr

label: equation

text: t

text: t

text: cr

text: p

text: and

text: t

text: −cr

text: ξ

text: ξ

text: cr

text: δ,

text: with

text: t

text: R

text: cr

text: ξ2

text: D

text: t

text: and

text: δ

text: cr

text: R

text: cr

text: which

text: model

text: the

text: crossover.

text: Equation

text: resumes

text: equations

text: and

text: from

text: the

text: main

text: text.

text: They

text: display

text: explicitly

text: the

text: dependency

text: on

text: the

text: crosslink

text: probability

text: p,

text: introduce

text: a

text: soft

text: crossover

text: toward

text: the

text: critical

text: value

text: p

text: and

text: the

text: bounds

text: ξ

text: and

text: t

text: to

text: be

text: determined

text: cr

text: cr

text: experimentally.

text: B.

text: Simulation

text: data

text: analysis

text: Generation

text: of

text: contact

text: matrices

text: For

text: each

text: set

text: of

text: parameters,

text: simulations

text: has

text: been

text: launched

text: in

text: batches

text: of

text: identical

text: runs

text: on

text: the

text: TARS

text: cluster

text: of

text: “Pasteur

text: Institut”,

text: powered

text: by

text: identical

text: CPUs.

text: Time

text: and

text: information

text: about

text: the

text: couple

text: of

text: involved

text: beads

text: have

text: been

text: recorded

text: for

text: each

text: crosslinking

text: event

text: for

text: each

text: parallel

text: simulation

text: in

text: distinct

text: data-ﬁles.

text: Sparse

text: matrices

text: have

text: been

text: generated,

text: merging

text: all

text: batches,

text: summing

text: the

text: number

text: of

text: crosslinking

text: events

text: happening

text: for

text: each

text: couple

text: of

text: beads

text: up

text: to

text: a

text: speciﬁed

text: time.

text: Time

text: has

text: been

text: sampled

text: according

text: to

text: a

text: geometric

text: progression

text: of

text: initial

text: value

text: and

text: common

text: ratio

text: equal

text: to

text: discarding

text: all

text: matrices

text: with

text: less

text: than

text: contacts.

text: Contact

text: matrices

text: for

text: ideal

text: polymer

text: models

text: at

text: very

text: long

text: time

text: (after

text: the

text: transition)

text: at

text: various

text: crosslinking

text: probabilities

text: look

text: like

text: the

text: one

text: reported

text: in

text: Suppl.

text: ﬁgure

text: Calculation

text: of

text: P

text: (s)

text: in

text: the

text: simulation

text: In

text: the

text: simulation,

text: sparse

text: matrices

text: have

text: been

text: coarse-grained

text: to

text: bead-level

text: dense

text: matrices.

text: Binning

text: of

text: the

text: contact

text: probability

text: P

text: (s)

text: at

text: bead

text: level

text: has

text: been

text: made

text: by

text: calculating

text: the

text: mean

text: value

text: of

text: secondary

text: diagonals

text: for

text: s

text: varying

text: in

text: the

text: range

text: from

text: to

text: beads.

text: In

text: order

text: to

text: ##LTLine##

text: ##LTLine##


text: in the transition of the asymptotic exponent γ∞(s) from short-distance to large-distance values (Fig. 2A, lower panel). This length ξ corresponds to the average length of the polymer segments captured in the pearls, and will hereafter be referred as the pearling length. The characteristic length ξ could also be recovered from the mean squared distance between monomers as a function of the chemical distance s (data not shown). Individual pearls were identiﬁed by clustering together monomers on the contact graph [66] using the Louvain algorithm [67], and their size was computed in order to conﬁrm that ξ indeed reﬂects the average number of monomers in pearls. (see Suppl. Fig. 5). For s > ξ, γ∞(s) = 3/2, consistent with the initial equilibrium state of the polymer, whereas γ∞(s) tends inside the pearls to a limiting value γ < 1 at small enough s. lim ∝ − The length ξ scales with the crosslink probability p as ξ p δ, with δ = 0.4 (Fig. 3A), indicating that the extent along the chain of the crosslink-induced collapse is paradoxically more prominent for small p, i.e. low crosslinking rate. Indeed, conformation changes of polymer loops of size greater than ξ are diﬀusion-limited, while for smaller loops, Rouse FIG. 1. Kinetics of the pearling transition (simulation). (A) Time evolution of the contact proba- bility curve P (s) at ﬁxed crosslink probability p = 0.1, displayed as a superposition of semitrans- t parent plots obtained at increasing simulation time t (black arrow); the resulting color density is given in the inset. A crossover at a length ξ arises at large enough times. Error bars are smaller than the thickness of the line. (B) Evolution of γ , the short-distance slope of the log-log plot of 0 the P (s), as a function of the rescaled time variable φ, for diﬀerent values of p. Inset: snapshots t of the time evolution of the polymer conformation (p = 0.1). 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: in

text: the

text: transition

text: of

text: the

text: asymptotic

text: exponent

text: γ∞(s)

text: from

text: short-distance

text: to

text: large-distance

text: values

text: (Fig.

text: lower

text: panel).

text: This

text: length

text: ξ

text: corresponds

text: to

text: the

text: average

text: length

text: of

text: the

text: polymer

text: segments

text: captured

text: in

text: the

text: pearls,

text: and

text: will

text: hereafter

text: be

text: referred

text: as

text: the

text: pearling

text: length.

text: The

text: characteristic

text: length

text: ξ

text: could

text: also

text: be

text: recovered

text: from

text: the

text: mean

text: squared

text: distance

text: between

text: monomers

text: as

text: a

text: function

text: of

text: the

text: chemical

text: distance

text: s

text: (data

text: not

text: shown).

text: Individual

text: pearls

text: were

text: identiﬁed

text: by

text: clustering

text: together

text: monomers

text: on

text: the

text: contact

text: graph

text: using

text: the

text: Louvain

text: algorithm

text: and

text: their

text: size

text: was

text: computed

text: in

text: order

text: to

text: conﬁrm

text: that

text: ξ

text: indeed

text: reﬂects

text: the

text: average

text: number

text: of

text: monomers

text: in

text: pearls.

text: (see

text: Suppl.

text: Fig.

text: For

text: s

text: ξ,

text: γ∞(s)

text: consistent

text: with

text: the

text: initial

text: equilibrium

text: state

text: of

text: the

text: polymer,

text: whereas

text: γ∞(s)

text: tends

text: inside

text: the

text: pearls

text: to

text: a

text: limiting

text: value

text: γ

text: at

text: small

text: enough

text: s.

text: lim

text: The

text: length

text: ξ

text: scales

text: with

text: the

text: crosslink

text: probability

text: p

text: as

text: ξ

text: p

text: δ,

text: with

text: δ

text: (Fig.

text: indicating

text: that

text: the

text: extent

text: along

text: the

text: chain

text: of

text: the

text: crosslink-induced

text: collapse

text: is

text: paradoxically

text: more

text: prominent

text: for

text: small

text: p,

text: i.e.

text: low

text: crosslinking

text: rate.

text: Indeed,

text: conformation

text: changes

text: of

text: polymer

text: loops

text: of

text: size

text: greater

text: than

text: ξ

text: are

text: diﬀusion-limited,

text: while

text: for

text: smaller

text: loops,

text: Rouse

text: FIG.

text: Kinetics

text: of

text: the

text: pearling

text: transition

text: (simulation).

text: (A)

text: Time

text: evolution

text: of

text: the

text: contact

text: proba-

text: bility

text: curve

text: P

text: (s)

text: at

text: ﬁxed

text: crosslink

text: probability

text: p

text: displayed

text: as

text: a

text: superposition

text: of

text: semitrans-

text: t

text: parent

text: plots

text: obtained

text: at

text: increasing

text: simulation

text: time

text: t

text: (black

text: arrow);

text: the

text: resulting

text: color

text: density

text: is

text: given

text: in

text: the

text: inset.

text: A

text: crossover

text: at

text: a

text: length

text: ξ

text: arises

text: at

text: large

text: enough

text: times.

text: Error

text: bars

text: are

text: smaller

text: than

text: the

text: thickness

text: of

text: the

text: line.

text: (B)

text: Evolution

text: of

text: γ

text: the

text: short-distance

text: slope

text: of

text: the

text: log-log

text: plot

text: of

text: the

text: P

text: (s),

text: as

text: a

text: function

text: of

text: the

text: rescaled

text: time

text: variable

text: φ,

text: for

text: diﬀerent

text: values

text: of

text: p.

text: Inset:

text: snapshots

text: t

text: of

text: the

text: time

text: evolution

text: of

text: the

text: polymer

text: conformation

text: (p

text: ##LTFigure##

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text: (cid:90) (cid:90) 3k T L 1 − 3k T L 1 − = B dp B dp e p2t/τ = 4π(cid:90)3K R (cid:16) p2 4π3K(cid:17) (cid:114)R p2 3k T L 1 − − 1 · = B dp 1 e p2t/τ = 3k T L t1/2 (21) 4π3K R p2 Kπ3ζ B (cid:112) by integrating by parts. It corresponds to Rouse sub-diﬀusion, whose coeﬃcient in this model is equal to C = 3k T L/ Kπ3ζ and is a measurable quantity. t B (cid:28) (cid:29) (cid:12) (cid:12) (cid:68) (cid:69) (cid:12) (cid:12) (cid:10)(cid:12) (cid:12)(cid:11) At ﬁx(cid:12)ed time, we can calcula(cid:12)te the po(cid:12)int-to-poi(cid:12)nt mean(cid:12)square distance: (cid:12) (cid:12) (cid:12) − 2 − R(cid:126) (s , 0) R(cid:126) (s + s , 0) (cid:90) = 6 Ri(s , 0) 2 (cid:90) 6 Ri(s , 0)Ri(s + s , 0) = 0 0 0 0 0 3k T L 1 − 3k T L 1 = B dp B dp ei2πps = 4π3K R p2 4π3K R p2 3k T L · | | B = s (22) 2Kπ by integrating by parts and through Residue theorem. This equation reproduces the well known random-walk behavior, whose coeﬃcient C = 3k T L/(2Kπ) is a measurable s B quantity. From a micro-rheological point-of-view, we can determine model parameters by ﬁtting the scaling of the Rouse sub-diﬀusion and the scaling of the point-to-point mean square distance: (cid:18) (cid:19) 6 C 3 1 1 C 2 s s ζ = Lk T, and K = Lk T, implying τ = ; (23) π2 C2 B 2π C B π3 C t s t Notice that the scaling of the parameters ζ (and K) in terms of the arbitrary length L, emerging from purely algebraic considerations, is reminiscent of the scaling of viscosity as a ∝ function of the persistence length of a worm-like-chain ζ = 6πηb (with b L). We can ﬁnally write the relaxation time equation (19) for each mode p in the following terms: (cid:18) (cid:19) 1 C 2 · 1 s τ (p) = (24) R π3 C p2 t 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

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text: Rouse

text: sub-diﬀusion,

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text: Residue

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text: whose

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text: From

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text: ﬁtting

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text: scaling

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text: FIG. 2. Quantitative features of the pearling transition (simulation). (A) (Upper panel) asymptotic curve P∞(s) and (Lower panel) its local slope γ∞(s) and pearling length ξ, for diﬀerent crosslink probabilities p. Inset: example of a pearled state (p = 0.1). (B) (Upper panel) asymptotic curve P∞(s) and (Lower panel) its local slope γ∞(s) for diﬀerent polymer dynamics, parameterized by the Rouse coeﬃcient D . Inset: monomer mean square displacement (MSD) as a function of time, R whose intercept yields a measurement of C , see Eq. 9. t diﬀusion is faster than the crosslinking reaction. In this latter reaction-limited regime, many conformational ﬂuctuations and contacts can occur and be ﬁxed by crosslinks, producing pearls of mean size ξ. Based on this qualitative picture, we propose a mean-ﬁeld calculation of the dependence of ξ in p. The relaxation time for a ﬁxed loop of size s scales as: (cid:18) (cid:19) − · π3 C 2 τ (s) = D 1 s2, with D = t , (3) R R R 4 C s § (detailed derivation in SM II.E.4) while the average duration τ needed to crosslink cross contacting beads is inversely proportional to the crosslink probability: 5 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: Quantitative

label: caption

text: features

text: of

text: the

text: pearling

text: transition

text: (simulation).

text: (A)

text: (Upper

text: panel)

text: asymptotic

text: curve

text: P∞(s)

text: and

text: (Lower

text: panel)

text: its

text: local

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text: γ∞(s)

text: and

text: pearling

text: length

text: ξ,

text: for

text: diﬀerent

text: crosslink

text: probabilities

text: p.

text: Inset:

text: example

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text: a

text: pearled

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text: (p

text: (B)

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text: panel)

text: asymptotic

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text: P∞(s)

text: and

text: (Lower

text: panel)

text: its

text: local

text: slope

text: γ∞(s)

text: for

text: diﬀerent

text: polymer

text: dynamics,

text: parameterized

text: by

text: the

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text: D

text: Inset:

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text: mean

text: square

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text: (MSD)

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text: a

text: function

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text: intercept

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text: measurement

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text: Eq.

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text: faster

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text: crosslinking

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text: In

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text: latter

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text: conformational

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text: and

text: contacts

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text: by

text: crosslinks,

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text: pearls

text: of

text: mean

text: size

text: ξ.

text: Based

text: on

text: this

text: qualitative

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text: we

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text: a

text: mean-ﬁeld

text: calculation

text: of

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text: dependence

text: of

text: ξ

text: in

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text: The

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text: s

text: scales

text: as:

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text: τ

text: (s)

text: D

text: with

text: D

text: t

text: R

text: R

text: R

text: C

text: s

text: (detailed

text: derivation

text: in

text: SM

text: while

text: the

text: average

text: duration

text: τ

text: needed

text: to

text: crosslink

text: cross

text: contacting

text: beads

text: is

text: inversely

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text: the

text: crosslink

text: probability:

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text: ∝ − ξ(p) p δ, (5) with δ = 1/2 correctly recapitulating the decrease of ξ at increasing p. We here assumed that the dynamics is consistent with Rouse diﬀusion during the pearling formation and col- lapse. However, Rouse diﬀusion is not expected to apply to the mesh into what the initially linear polymer is transformed after enough crosslinks, which may explain the diﬀerent value δ = 0.4 measured in the simulations (Fig. 3A). With the same argument we also predict that ξ varies with the dynamical properties of the polymer. Simulations actually show that variation of the Rouse diﬀusion coeﬃcient D has a dramatic eﬀect on ξ (Fig. 2B). For R small D , ξ is small and crosslinking has mostly a local eﬀect. When D increases, longer R R polymer segments can reach their equilibrium conformation between two crosslink events so that ξ becomes larger. In the line of the above calculation, we expect a scaling ∝ ξ(D ) D1/2, (6) R R which is well reproduced in the simulations (Fig. 3B). Our simulation moreover shows that the collapse happens abruptly. The short-distance ∗ exponent γ presents a sharp decrease at a time t , which we call the pearling time. Before 0 (cid:28) ∗ this transition (t t ), γ coincides with the exponent at long distances, 3/2, as expected 0 for an equilibrium state. Only after the transition a smaller exponent is observed, with a ∗ limiting value γ < 1 depending on the kinetic parameters. t depends on the crosslink lim ∗ ∝ − · probability with a scaling t p 0.8 prompting to deﬁne a re-scaled variable φ = p0.8 t. The evolution of γ as a function of φ re-scales at any p into a single transition curve (Fig. 1B). 0 ∗ ∗ The scaling of t can also be explained with the above mean-ﬁeld argument: as t emerges from pearling (see polymer snapshots along the transition curve in Fig. 1B), it is equal to ∗ the relaxation time of pearls of mean size ξ: t = τ (ξ). From Eq. 27, R ∗ ∝ − t p 2δ, (7) 7

text: ξ(p)

text: p

text: δ,

text: with

text: δ

text: correctly

text: recapitulating

text: the

text: decrease

text: of

text: ξ

text: at

text: increasing

text: p.

text: We

text: here

text: assumed

text: that

text: the

text: dynamics

text: is

text: consistent

text: with

text: Rouse

text: diﬀusion

text: during

text: the

text: pearling

text: formation

text: and

text: col-

text: lapse.

text: However,

text: Rouse

text: diﬀusion

text: is

text: not

text: expected

text: to

text: apply

text: to

text: the

text: mesh

text: into

text: what

text: the

text: initially

text: linear

text: polymer

text: is

text: transformed

text: after

text: enough

text: crosslinks,

text: which

text: may

text: explain

text: the

text: diﬀerent

text: value

text: δ

text: measured

text: in

text: the

text: simulations

text: (Fig.

text: With

text: the

text: same

text: argument

text: we

text: also

text: predict

text: that

text: ξ

text: varies

text: with

text: the

text: dynamical

text: properties

text: of

text: the

text: polymer.

text: Simulations

text: actually

text: show

text: that

text: variation

text: of

text: the

text: Rouse

text: diﬀusion

text: coeﬃcient

text: D

text: has

text: a

text: dramatic

text: eﬀect

text: on

text: ξ

text: (Fig.

text: For

text: R

text: small

text: D

text: ξ

text: is

text: small

text: and

text: crosslinking

text: has

text: mostly

text: a

text: local

text: eﬀect.

text: When

text: D

text: increases,

text: longer

text: R

text: R

text: polymer

text: segments

text: can

text: reach

text: their

text: equilibrium

text: conformation

text: between

text: two

text: crosslink

text: events

text: so

text: that

text: ξ

text: becomes

text: larger.

text: In

text: the

text: line

text: of

text: the

text: above

text: calculation,

text: we

text: expect

text: a

text: scaling

text: ξ(D

text: R

text: R

text: which

text: is

text: well

text: reproduced

text: in

text: the

text: simulations

text: (Fig.

text: Our

text: simulation

text: moreover

text: shows

text: that

text: the

text: collapse

text: happens

text: abruptly.

text: The

text: short-distance

text: exponent

text: γ

text: presents

text: a

text: sharp

text: decrease

text: at

text: a

text: time

text: t

text: which

text: we

text: call

text: the

text: pearling

text: time.

text: Before

text: this

text: transition

text: (t

text: t

text: γ

text: coincides

text: with

text: the

text: exponent

text: at

text: long

text: distances,

text: as

text: expected

text: for

text: an

text: equilibrium

text: state.

text: Only

text: after

text: the

text: transition

text: a

text: smaller

text: exponent

text: is

text: observed,

text: with

text: a

text: limiting

text: value

text: γ

text: depending

text: on

text: the

text: kinetic

text: parameters.

text: t

text: depends

text: on

text: the

text: crosslink

text: lim

text: probability

text: with

text: a

text: scaling

text: t

text: p

text: prompting

text: to

text: deﬁne

text: a

text: re-scaled

text: variable

text: φ

text: t.

text: The

text: evolution

text: of

text: γ

text: as

text: a

text: function

text: of

text: φ

text: re-scales

text: at

text: any

text: p

text: into

text: a

text: single

text: transition

text: curve

text: (Fig.

text: The

text: scaling

text: of

text: t

text: can

text: also

text: be

text: explained

text: with

text: the

text: above

text: mean-ﬁeld

text: argument:

text: as

text: t

text: emerges

text: from

text: pearling

text: (see

text: polymer

text: snapshots

text: along

text: the

text: transition

text: curve

text: in

text: Fig.

text: it

text: is

text: equal

text: to

text: the

text: relaxation

text: time

text: of

text: pearls

text: of

text: mean

text: size

text: ξ:

text: t

text: τ

text: (ξ).

text: From

text: Eq.

text: R

text: t

text: p


text: – 2 – 1Space Sciences Laboratory, 7 Gauss way, University of California, Berkeley, CA, 94720, USA 2Department of Astronomy, University of California, Berkeley, 501 Campbell Hall #3411, Berkeley, CA, 94720, USA 3Radboud University, Nijmegen, Comeniuslaan 4, 6525 HP Nijmegen, The Netherlands 4SETI Institute, Mountain View, California 5Institute of Space Sciences and Astronomy, University of Malta 6Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia 7Radio Astronomy Lab, University of California, Berkeley, CA, 94720, USA 8ASTRON, Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The Netherlands 9Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands 10Cornell Center for Astrophysics and Planetary Science and Department of Astronomy, Cornell University, Ithaca, NY 14853, USA 11Academia Sinica Institute of Astronomy and Astrophysics, 645 N. A’ohoku Place, Hilo, HI 96720, USA 12Physics Department, University of Vermont, Burlington, VT 05401, USA 13Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, USA 14Center for Gravitational Waves and Cosmology, Chestnut Ridge Research Building, Mor- gantown, WV 26505, USA 15University of Oxford, Sub-Department of Astrophysics, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom 16Hellenic Open University, School of Science and Technology, Parodos Aristotelous 18, ##LTLine##

text: Sciences

text: Laboratory,

text: Gauss

text: way,

text: University

text: of

text: California,

text: Berkeley,

text: CA,

text: USA

text: of

text: Astronomy,

text: University

text: of

text: California,

text: Berkeley,

text: Campbell

text: Hall

text: Berkeley,

text: CA,

text: USA

text: University,

text: Nijmegen,

text: Comeniuslaan

text: HP

text: Nijmegen,

text: The

text: Netherlands

text: Institute,

text: Mountain

text: View,

text: California

text: of

text: Space

text: Sciences

text: and

text: Astronomy,

text: University

text: of

text: Malta

text: for

text: Astrophysics

text: Supercomputing,

text: Swinburne

text: University

text: of

text: Technology,

text: Hawthorn,

text: VIC

text: Australia

text: Astronomy

text: Lab,

text: University

text: of

text: California,

text: Berkeley,

text: CA,

text: USA

text: Netherlands

text: Institute

text: for

text: Radio

text: Astronomy,

text: Postbus

text: AA,

text: Dwingeloo,

text: The

text: Netherlands

text: Pannekoek

text: Institute

text: for

text: Astronomy,

text: University

text: of

text: Amsterdam,

text: Science

text: Park

text: XH

text: Amsterdam,

text: The

text: Netherlands

text: Center

text: for

text: Astrophysics

text: and

text: Planetary

text: Science

text: and

text: Department

text: of

text: Astronomy,

text: Cornell

text: University,

text: Ithaca,

text: NY

text: USA

text: Sinica

text: Institute

text: of

text: Astronomy

text: and

text: Astrophysics,

text: N.

text: A’ohoku

text: Place,

text: Hilo,

text: HI

text: USA

text: Department,

text: University

text: of

text: Vermont,

text: Burlington,

text: VT

text: USA

text: of

text: Physics

text: and

text: Astronomy,

text: West

text: Virginia

text: University,

text: Morgantown,

text: WV

text: USA

text: for

text: Gravitational

text: Waves

text: and

text: Cosmology,

text: Chestnut

text: Ridge

text: Research

text: Building,

text: Mor-

text: gantown,

text: WV

text: USA

text: of

text: Oxford,

text: Sub-Department

text: of

text: Astrophysics,

text: Denys

text: Wilkinson

text: Building,

text: Keble

text: Road,

text: Oxford,

text: United

text: Kingdom

text: Open

text: University,

text: School

text: of

text: Science

text: and

text: Technology,

text: Parodos

text: Aristotelous

text: ##LTLine##


text: – 14 – We used the NE2001 model (Cordes & Lazio 2002), and estimated a Galactic scattering − timescale τ = 20 µs ν α towards the direction of FRB 121102 (ν is the observing frequency s in GHz while α is the scaling parameter). Assuming a frequency scaling for a Kolmogorov spectrum with scaling parameter between 4 to 4.4 (Cordes & Lazio 2002; Bhat et al. 2004), we estimate the expected diﬀractive interstellar scintillation bandwidth as (Lambert & Rickett 1999) 1.16 1.16ν[4,4.4] ∆f = = MHz. (2) DISS 2πτ 40π s For the observed band (4.5 to 8 GHz), the predicted ∆f varies from 7 MHz to DISS 87 MHz (assuming a scaling parameter of 4.4). Thus, the large-scale frequency structures ∼ ( GHz wide) are likely to be intrinsic to the source and/or due to propagation eﬀects in the source’s local environment. To compare these predicted values for ∆f with ﬁne-scale frequency structures DISS from the detected bursts, we obtained spectra for each burst by selecting an appropriate on-pulse window. Following the procedure of Cordes et al. (1985) and Cordes (1986), we then computed auto-correlation functions (ACFs) for these spectra (Figure 5). As the spectral extent of a few bursts were signiﬁcantly broad—spanning over 3 GHz—the ACFs over the entire spectra produce a multi-featured ACF with multiple widths. To measure the characteristic bandwidth for a given burst, we divided each spectrum into two or three parts, depending upon the spectral extent. We then ﬁtted a Gaussian function to the ACF to obtain the half width at half maximum (HWHM) as a characteristic bandwidth for the given sub-spectra for each burst. We found that the ACFs showed diﬀerent characteristic bandwidths in diﬀerent sub-spectra in the 4.5 to 8 GHz band. Table 2 shows these characteristic bandwidths ##LTLine## ##LTLine##

text: We

text: used

text: the

text: model

text: (Cordes

text: Lazio

text: and

text: estimated

text: a

text: Galactic

text: scattering

text: timescale

text: τ

text: µs

text: ν

text: α

text: towards

text: the

text: direction

text: of

text: FRB

text: (ν

text: is

text: the

text: observing

text: frequency

text: s

text: in

text: GHz

text: while

text: α

text: is

text: the

text: scaling

text: parameter).

text: Assuming

text: a

text: frequency

text: scaling

text: for

text: a

text: Kolmogorov

text: spectrum

text: with

text: scaling

text: parameter

text: between

text: to

text: (Cordes

text: Lazio

text: Bhat

text: et

text: al.

text: we

text: estimate

text: the

text: expected

text: diﬀractive

text: interstellar

text: scintillation

text: bandwidth

text: as

text: (Lambert

text: Rickett

text: ∆f

text: MHz.

text: DISS

text: 2πτ

text: s

text: For

text: the

text: observed

text: band

text: to

text: GHz),

text: the

text: predicted

text: ∆f

text: varies

text: from

text: MHz

text: to

text: DISS

text: MHz

text: (assuming

text: a

text: scaling

text: parameter

text: of

text: Thus,

text: the

text: large-scale

text: frequency

text: structures

text: GHz

text: wide)

text: are

text: likely

text: to

text: be

text: intrinsic

text: to

text: the

text: source

text: and/or

text: due

text: to

text: propagation

text: eﬀects

text: in

text: the

text: source’s

text: local

text: environment.

text: To

text: compare

text: these

text: predicted

text: values

text: for

text: ∆f

text: with

text: ﬁne-scale

text: frequency

text: structures

text: DISS

text: from

text: the

text: detected

text: bursts,

text: we

text: obtained

text: spectra

text: for

text: each

text: burst

text: by

text: selecting

text: an

text: appropriate

text: on-pulse

text: window.

text: Following

text: the

text: procedure

text: of

text: Cordes

text: et

text: al.

text: and

text: Cordes

text: we

text: then

text: computed

text: auto-correlation

text: functions

text: (ACFs)

text: for

text: these

text: spectra

text: (Figure

text: As

text: the

text: spectral

text: extent

text: of

text: a

text: few

text: bursts

text: were

text: signiﬁcantly

text: broad—spanning

text: over

text: GHz—the

text: ACFs

text: over

text: the

text: entire

text: spectra

text: produce

text: a

text: multi-featured

text: ACF

text: with

text: multiple

text: widths.

text: To

text: measure

text: the

text: characteristic

text: bandwidth

text: for

text: a

text: given

text: burst,

text: we

text: divided

text: each

text: spectrum

text: into

text: two

text: or

text: three

text: parts,

text: depending

text: upon

text: the

text: spectral

text: extent.

text: We

text: then

text: ﬁtted

text: a

text: Gaussian

text: function

text: to

text: the

text: ACF

text: to

text: obtain

text: the

text: half

text: width

text: at

text: half

text: maximum

text: (HWHM)

text: as

text: a

text: characteristic

text: bandwidth

text: for

text: the

text: given

text: sub-spectra

text: for

text: each

text: burst.

text: We

text: found

text: that

text: the

text: ACFs

text: showed

text: diﬀerent

text: characteristic

text: bandwidths

text: in

text: diﬀerent

text: sub-spectra

text: in

text: the

text: to

text: GHz

text: band.

text: Table

text: shows

text: these

text: characteristic

text: bandwidths

text: ##LTLine##

text: ##LTLine##


text: – 15 – from 15 stronger bursts. The measured characteristic bandwidth follows the expected ∆f across the 4.5 to 8 GHz band (see Figure 6) suggesting that the ﬁne-scale frequency DISS structures seen in each of these bursts are likely due to Galactic interstellar scintillation, with a few exceptions. For example, burst 11H does not appear to show characteristic bandwidth matching with the ∆f at higher frequencies. However, this could be a DISS statistical ﬂuctuation given the small number of scintles in the burst. 4. Discussion 4.1. Instantaneous Burst Rate Our detection of 21 bursts within 60 minutes represents the highest number of bursts detected within a short interval (i.e. hour timescale), with 18 bursts occurring in the ﬁrst 30 minutes. We did not detect any bursts during the following 4 hours of observations, which supports the idea that FRB 121102’s bursting behavior is episodic. This could be due to intrinsic changes in the emission conditions or due to more favorable ‘plasma lensing’ conditions (Cordes et al. 2017) during the ﬁrst hour, potentially enhancing observed burst energies by an order of magnitude. Our observations highlight the advantages aﬀorded by wider instantaneous bandwidth. If our 4 GHz instantaneous bandwidth were halved, we would only have detected 10 to 12 bursts, mainly as the band-limited spectral structures they exhibit would have fallen out of band (Figure 2). Law et al. (2017) have also pointed out similar limitations of narrow bandwidth observations. The spectral peak variation of each burst and apparent band-limited characteristics indicate that further bursts may be detected by searching for pulses over subbands, particularly in observations over large fractional bandwidths. The existence of spectral structure may also impact the performance of single-pulse search pipelines, which are

text: from

text: stronger

text: bursts.

text: The

text: measured

text: characteristic

text: bandwidth

text: follows

text: the

text: expected

text: ∆f

text: across

text: the

text: to

text: GHz

text: band

text: (see

text: Figure

text: suggesting

text: that

text: the

text: ﬁne-scale

text: frequency

text: DISS

text: structures

text: seen

text: in

text: each

text: of

text: these

text: bursts

text: are

text: likely

text: due

text: to

text: Galactic

text: interstellar

text: scintillation,

text: with

text: a

text: few

text: exceptions.

text: For

text: example,

text: burst

text: does

text: not

text: appear

text: to

text: show

text: characteristic

text: bandwidth

text: matching

text: with

text: the

text: ∆f

text: at

text: higher

text: frequencies.

text: However,

text: this

text: could

text: be

text: a

text: DISS

text: statistical

text: ﬂuctuation

text: given

text: the

text: small

text: number

text: of

text: scintles

text: in

text: the

text: burst.

label: section

text: Discussion

text: Instantaneous

text: Burst

text: Rate

text: Our

text: detection

text: of

text: bursts

text: within

text: minutes

text: represents

text: the

text: highest

text: number

text: of

text: bursts

text: detected

text: within

text: a

text: short

text: interval

text: (i.e.

text: hour

text: timescale),

text: with

text: bursts

text: occurring

text: in

text: the

text: ﬁrst

text: minutes.

text: We

text: did

text: not

text: detect

text: any

text: bursts

text: during

text: the

text: following

text: hours

text: of

text: observations,

text: which

text: supports

text: the

text: idea

text: that

text: FRB

text: bursting

text: behavior

text: is

text: episodic.

text: This

text: could

text: be

text: due

text: to

text: intrinsic

text: changes

text: in

text: the

text: emission

text: conditions

text: or

text: due

text: to

text: more

text: favorable

text: ‘plasma

text: lensing’

text: conditions

text: (Cordes

text: et

text: al.

text: during

text: the

text: ﬁrst

text: hour,

text: potentially

text: enhancing

text: observed

text: burst

text: energies

text: by

text: an

text: order

text: of

text: magnitude.

text: Our

text: observations

text: highlight

text: the

text: advantages

text: aﬀorded

text: by

text: wider

text: instantaneous

text: bandwidth.

text: If

text: our

text: GHz

text: instantaneous

text: bandwidth

text: were

text: halved,

text: we

text: would

text: only

text: have

text: detected

text: to

text: bursts,

text: mainly

text: as

text: the

text: band-limited

text: spectral

text: structures

text: they

text: exhibit

text: would

text: have

text: fallen

text: out

text: of

text: band

text: (Figure

text: Law

text: et

text: al.

text: have

text: also

text: pointed

text: out

text: similar

text: limitations

text: of

text: narrow

text: bandwidth

text: observations.

text: The

text: spectral

text: peak

text: variation

text: of

text: each

text: burst

text: and

text: apparent

text: band-limited

text: characteristics

text: indicate

text: that

text: further

text: bursts

text: may

text: be

text: detected

text: by

text: searching

text: for

text: pulses

text: over

text: subbands,

text: particularly

text: in

text: observations

text: over

text: large

text: fractional

text: bandwidths.

text: The

text: existence

text: of

text: spectral

text: structure

text: may

text: also

text: impact

text: the

text: performance

text: of

text: single-pulse

text: search

text: pipelines,

text: which

text: are


text: – 19 – frequency the apparently ﬂat spectral index transitions and how the event rate varies as a function of frequency. Acknowledgements VG thanks A. Noutsos for useful comments on the draft. We would like to thank our referee, J. I. Katz for a critical review of the paper and for suggesting several improvements to the manuscript. VG would like to acknowledge NSF grant 1407804 and the Marilyn and Watson Alberts SETI Chair funds. CJL is supported by the NSF award 1611606. DM and JWTH acknowledge support from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement nr. 337062. JWTH also acknowledges funding from an NWO Vidi fellowship. L.G.S. acknowledges ﬁnancial support from the ERC Starting Grant BEACON under contract number 279702, as well as the Max Planck Society. MAM and SBS are supported by NSF awards NSF-OIA award number 1458952. SBS, SC, JMC, TJWL, MAM, SMR, and AS are members of the NANOGrav Physics Frontiers Center supported by NSF award PHY-1430284. NG is supported by HOU for her current sabbatical leave. Breakthrough Listen is managed by the Breakthrough Initiatives, sponsored by the Breakthrough Prize Foundation (breakthroughinitiatives.org). The Green Bank Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

text: frequency

text: the

text: apparently

text: ﬂat

text: spectral

text: index

text: transitions

text: and

text: how

text: the

text: event

text: rate

text: varies

text: as

text: a

text: function

text: of

text: frequency.

text: Acknowledgements

text: VG

text: thanks

text: A.

text: Noutsos

text: for

text: useful

text: comments

text: on

text: the

text: draft.

text: We

text: would

text: like

text: to

text: thank

text: our

text: referee,

text: J.

text: I.

text: Katz

text: for

text: a

text: critical

text: review

text: of

text: the

text: paper

text: and

text: for

text: suggesting

text: several

text: improvements

text: to

text: the

text: manuscript.

text: VG

text: would

text: like

text: to

text: acknowledge

text: NSF

text: grant

text: and

text: the

text: Marilyn

text: and

text: Watson

text: Alberts

text: SETI

text: Chair

text: funds.

text: CJL

text: is

text: supported

text: by

text: the

text: NSF

text: award

text: DM

text: and

text: JWTH

text: acknowledge

text: support

text: from

text: the

text: European

text: Research

text: Council

text: under

text: the

text: European

text: Union’s

text: Seventh

text: Framework

text: Programme

text: ERC

text: Grant

text: Agreement

text: nr.

text: JWTH

text: also

text: acknowledges

text: funding

text: from

text: an

text: NWO

text: Vidi

text: fellowship.

text: L.G.S.

text: acknowledges

text: ﬁnancial

text: support

text: from

text: the

text: ERC

text: Starting

text: Grant

text: BEACON

text: under

text: contract

text: number

text: as

text: well

text: as

text: the

text: Max

text: Planck

text: Society.

text: MAM

text: and

text: SBS

text: are

text: supported

text: by

text: NSF

text: awards

text: NSF-OIA

text: award

text: number

text: SBS,

text: SC,

text: JMC,

text: TJWL,

text: MAM,

text: SMR,

text: and

text: AS

text: are

text: members

text: of

text: the

text: NANOGrav

text: Physics

text: Frontiers

text: Center

text: supported

text: by

text: NSF

text: award

text: NG

text: is

text: supported

text: by

text: HOU

text: for

text: her

text: current

text: sabbatical

text: leave.

text: Breakthrough

text: Listen

text: is

text: managed

text: by

text: the

text: Breakthrough

text: Initiatives,

text: sponsored

text: by

text: the

text: Breakthrough

text: Prize

text: Foundation

text: (breakthroughinitiatives.org).

text: The

text: Green

text: Bank

text: Observatory

text: is

text: a

text: facility

text: of

text: the

text: National

text: Science

text: Foundation

text: operated

text: under

text: cooperative

text: agreement

text: by

text: Associated

text: Universities,

text: Inc.


text: – 26 – Lawrence, E., Vander Wiel, S., Law, C., Burke Spolaor, S., & Bower, G. C. 2017, AJ, 154, 117 Lingam, M., & Loeb, A. 2017, ApJL, 837, L23. http://stacks.iop.org/2041-8205/837/ i=2/a=L23 Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777 MacMahon, D. H. E., Price, D. C., Lebofsky, M., et al. 2017, ArXiv e-prints, arXiv:1707.06024 Macquart, J.-P., & Johnston, S. 2015, MNRAS, 451, 3278 Marcote, B., Paragi, Z., Hessels, J. W. T., et al. 2017, ApJ, 834, L8 Masui, K., Lin, H.-H., Sievers, J., et al. 2015, Nature, 528, 523 Meyers, B. W., Tremblay, S. E., Bhat, N. D. R., et al. 2017, ApJ, 851, 20 Michilli, D., Seymour, A., Hessels, J. W. T., et al. 2018, Nature, 553, 182. http: //www.nature.com/doifinder/10.1038/nature25149 Piro, A. L. 2016, ApJ, 824, L32 Price, D. C., Gajjar, V., Rosenthal, L., et al. 2018, ArXiv e-prints, arXiv:1802.04446 Scholz, P., Spitler, L. G., Hessels, J. W. T., et al. 2016, ApJ, 833, 177 Scholz, P., Bogdanov, S., Hessels, J. W. T., et al. 2017, ApJ, 846, 80 Spitler, L. G., Cordes, J. M., Hessels, J. W. T., et al. 2014, ApJ, 790, 101 Spitler, L. G., Scholz, P., Hessels, J. W. T., et al. 2016, Nature, 531, 202

text: Lawrence,

text: E.,

text: Vander

text: Wiel,

text: S.,

text: Law,

text: C.,

text: Burke

text: Spolaor,

text: S.,

text: Bower,

text: G.

text: C.

text: AJ,

text: Lingam,

text: M.,

text: Loeb,

text: A.

text: ApJL,

text: Lorimer,

text: D.

text: R.,

text: Bailes,

text: M.,

text: McLaughlin,

text: M.

text: A.,

text: Narkevic,

text: D.

text: J.,

text: Crawford,

text: F.

text: Science,

text: MacMahon,

text: D.

text: H.

text: E.,

text: Price,

text: D.

text: C.,

text: Lebofsky,

text: M.,

text: et

text: al.

text: ArXiv

text: e-prints,

text: Macquart,

text: J.-P.,

text: Johnston,

text: S.

text: MNRAS,

text: Marcote,

text: B.,

text: Paragi,

text: Z.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: ApJ,

text: Masui,

text: K.,

text: Lin,

text: H.-H.,

text: Sievers,

text: J.,

text: et

text: al.

text: Nature,

text: Meyers,

text: B.

text: W.,

text: Tremblay,

text: S.

text: E.,

text: Bhat,

text: N.

text: D.

text: R.,

text: et

text: al.

text: ApJ,

text: Michilli,

text: D.,

text: Seymour,

text: A.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: Nature,

text: http:

text: Piro,

text: A.

text: L.

text: ApJ,

text: Price,

text: D.

text: C.,

text: Gajjar,

text: V.,

text: Rosenthal,

text: L.,

text: et

text: al.

text: ArXiv

text: e-prints,

text: Scholz,

text: P.,

text: Spitler,

text: L.

text: G.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: ApJ,

text: Scholz,

text: P.,

text: Bogdanov,

text: S.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: ApJ,

text: Spitler,

text: L.

text: G.,

text: Cordes,

text: J.

text: M.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: ApJ,

text: Spitler,

text: L.

text: G.,

text: Scholz,

text: P.,

text: Hessels,

text: J.

text: W.

text: T.,

text: et

text: al.

text: Nature,


text: – 8 – channelized to 512 ‘coarse’ frequency channels, requantized to 8 bits, and then distributed to a cluster of compute nodes that recorded these data to disk. Directly after observations, the coarsely-channelized voltage data were further channelized (to 366 kHz resolution), integrated (with a sampling time of 350 µsec) using a custom GPU-accelerated spectroscopy code, and ﬁnally written toSIGPROC3 ﬁlterbank ﬁles as total intensity (Stokes I) dynamic spectra with 4096 spectral channels. These dynamic spectra were searched using the Heimdall package (Barsdell et al. − 2012) for dispersed pulses within the DM range of 500 to 700 pc cm 3, using a DM interval − of 0.1 pc cm 3. We detected 21 bursts above a threshold signal-to-noise ratio (S/N) of six. A section of raw voltages (of total 1.5 seconds) around each detected burst was extracted − for further processing and coherently dedispersed to a DM of 565.0 pc cm 3 (see Section ) using the DSPSR package (van Straten & Bailes 2011). The coherently dedispersed PSRFITS data products have temporal and spectral resolutions of 10.1 µs and 183 kHz, respectively4. We have discarded lower part of the frequency band (4 to 4.5 GHz) in further analysis due to spurious radio frequency interferences. 3. Analysis PSRCHIVE (van Straten et al. 2012) and custom Python scripts were used to measure the properties of the detected bursts. Table 1 lists various parameters determined for each burst. Among the ten recorded scans, all reported detections occurred in the ﬁrst hour, i.e. from scans 11 and 12. We assigned the bursts identiﬁers 11A through 11R, and 12A 3http://sigproc.sourceforge.net 4PSRFITS data for all these bursts are publicly available to download at seti.berkeley. edu/frb121102/ ##LTLine##

text: channelized

text: to

text: ‘coarse’

text: frequency

text: channels,

text: requantized

text: to

text: bits,

text: and

text: then

text: distributed

text: to

text: a

text: cluster

text: of

text: compute

text: nodes

text: that

text: recorded

text: these

text: data

text: to

text: disk.

text: Directly

text: after

text: observations,

text: the

text: coarsely-channelized

text: voltage

text: data

text: were

text: further

text: channelized

text: (to

text: kHz

text: resolution),

text: integrated

text: (with

text: a

text: sampling

text: time

text: of

text: µsec)

text: using

text: a

text: custom

text: GPU-accelerated

text: spectroscopy

text: code,

text: and

text: ﬁnally

text: written

text: ﬁlterbank

text: ﬁles

text: as

text: total

text: intensity

text: (Stokes

text: I)

text: dynamic

text: spectra

text: with

text: spectral

text: channels.

text: These

text: dynamic

text: spectra

text: were

text: searched

text: using

text: the

text: Heimdall

text: package

text: (Barsdell

text: et

text: al.

text: for

text: dispersed

text: pulses

text: within

text: the

text: DM

text: range

text: of

text: to

text: pc

text: cm

text: using

text: a

text: DM

text: interval

text: of

text: pc

text: cm

text: We

text: detected

text: bursts

text: above

text: a

text: threshold

text: signal-to-noise

text: ratio

text: (S/N)

text: of

text: six.

text: A

text: section

text: of

text: raw

text: voltages

text: (of

text: total

text: seconds)

text: around

text: each

text: detected

text: burst

text: was

text: extracted

text: for

text: further

text: processing

text: and

text: coherently

text: dedispersed

text: to

text: a

text: DM

text: of

text: pc

text: cm

text: (see

text: Section

text: using

text: the

text: DSPSR

text: package

text: (van

text: Straten

text: Bailes

text: The

text: coherently

text: dedispersed

text: PSRFITS

text: data

text: products

text: have

text: temporal

text: and

text: spectral

text: resolutions

text: of

text: µs

text: and

text: kHz,

text: We

text: have

text: discarded

text: lower

text: part

text: of

text: the

text: frequency

text: band

text: to

text: GHz)

text: in

text: further

text: analysis

text: due

text: to

text: spurious

text: radio

text: frequency

text: interferences.

text: Analysis

text: PSRCHIVE

text: (van

text: Straten

text: et

text: al.

text: and

text: custom

text: Python

text: scripts

text: were

text: used

text: to

text: measure

text: the

text: properties

text: of

text: the

text: detected

text: bursts.

text: Table

text: lists

text: various

text: parameters

text: determined

text: for

text: each

text: burst.

text: Among

text: the

text: ten

text: recorded

text: scans,

text: all

text: reported

text: detections

text: occurred

text: in

text: the

text: ﬁrst

text: hour,

text: i.e.

text: from

text: scans

text: and

text: We

text: assigned

text: the

text: bursts

text: identiﬁers

text: through

text: and

text: data

text: for

text: all

text: these

text: bursts

text: are

text: publicly

text: available

text: to

text: download

text: at

text: seti.berkeley.

text: ##LTLine##


text: ON SKEW BRACES AND THEIR IDEALS 15 Proof. Clearly every solvable skew brace is a Baer radical skew brace since every n-sequence will reach zero. Suppose now that A is a Baer radical skew brace, ⊆ so A B(A). We will prove that A is solvable by induction on the number of elements in A. If A has only one element then A is a trivial brace and the result holds. Suppose the result holds for all skew braces of cardinality smaller than i, (cid:54) (cid:54) and suppose that A has cardinality i + 1. Since B(A) = 0 we get W(A) = 0 (by Theorem 4.28). By Lemma 5.5, W(A) is a solvable ideal in A. Since A is Baer radical it follows that A/W(A) is Baer radical. By the inductive assumption A/W(A) is solvable. By Lemma 5.4 applied for I = W(A) we get that A is (cid:3) solvable. Corollary 5.7. Let A be a ﬁnite skew brace and I be an ideal in A. Then I is solvable if and only if I is Baer radical. In particular, the Baer radical of A equals the largest solvable ideal. Moreover, if A has a nonzero solvable ideal then A has a ∗ non-zero ideal I such that I I = 0. Proof. Every Baer radical ideal is solvable, hence B(A) is solvable. On the other hand every solvable ideal is Baer radical by Theorem 5.6. The rest follows from (cid:3) Lemma 4.20. Corollary 5.8. There exists an inﬁnite brace such that its Baer radical is not solvable. Proof. It follows from the fact that there are Baer radical rings which are not (cid:3) nilpotent. Remark 5.9. Question 4.5 asks for two-sided braces that are prime. Note that there exists an inﬁnite prime and not simple Jacobson radical ring. Thus, by Lemma 4.30, there exists a prime and not simple inﬁnite two-sided brace. The results in this section allow us to answer the following question: Is it true that a product of any number of non-zero ideals in A (in any order) is nonzero? Lemma 5.10. Let A be a prime skew brace and let I and J be non-zero ideals in ∩ A. Then I J is a non-zero ideal in A. Moreover, the intersection of any ﬁnite number of ideals in A is a non-zero ideal in A. ∗ ⊆ ∩ Proof. The intersection of any two ideals is an ideal. Notice that I J I J, ∩ (cid:54) therefore I J = 0. The last assertion can be proved by induction on the number (cid:3) of ideals.

text: ON

text: SKEW

text: BRACES

text: AND

text: THEIR

text: IDEALS

text: Proof.

text: Clearly

text: every

text: solvable

text: skew

text: brace

text: is

text: a

text: Baer

text: radical

text: skew

text: brace

text: since

text: every

text: n-sequence

text: will

text: reach

text: zero.

text: Suppose

text: now

text: that

text: A

text: is

text: a

text: Baer

text: radical

text: skew

text: brace,

text: so

text: A

text: B(A).

text: We

text: will

text: prove

text: that

text: A

text: is

text: solvable

text: by

text: induction

text: on

text: the

text: number

text: of

text: elements

text: in

text: A.

text: If

text: A

text: has

text: only

text: one

text: element

text: then

text: A

text: is

text: a

text: trivial

text: brace

text: and

text: the

text: result

text: holds.

text: Suppose

text: the

text: result

text: holds

text: for

text: all

text: skew

text: braces

text: of

text: cardinality

text: smaller

text: than

text: i,

font: default

text: and

text: suppose

text: that

text: A

text: has

text: cardinality

text: i

text: Since

text: B(A)

text: we

text: get

text: W(A)

text: (by

text: Theorem

text: By

text: Lemma

text: W(A)

text: is

text: a

text: solvable

text: ideal

text: in

text: A.

text: Since

text: A

text: is

text: Baer

text: radical

text: it

text: follows

text: that

text: A/W(A)

text: is

text: Baer

text: radical.

text: By

text: the

text: inductive

text: assumption

text: A/W(A)

text: is

text: solvable.

text: By

text: Lemma

text: applied

text: for

text: I

text: W(A)

text: we

text: get

text: that

text: A

text: is

text: solvable.

text: Corollary

text: Let

text: A

text: be

text: a

text: ﬁnite

text: skew

text: brace

text: and

text: I

text: be

text: an

text: ideal

text: in

text: A.

text: Then

text: I

text: is

text: solvable

text: if

text: and

text: only

text: if

text: I

text: is

text: Baer

text: radical.

text: In

text: particular,

text: the

text: Baer

text: radical

text: of

text: A

text: equals

text: the

text: largest

text: solvable

text: ideal.

text: Moreover,

text: if

text: A

text: has

text: a

text: nonzero

text: solvable

text: ideal

text: then

text: A

text: has

text: a

text: non-zero

text: ideal

text: I

text: such

text: that

text: I

text: I

text: Proof.

text: Every

text: Baer

text: radical

text: ideal

text: is

text: solvable,

text: hence

text: B(A)

text: is

text: solvable.

text: On

text: the

text: other

text: hand

text: every

text: solvable

text: ideal

text: is

text: Baer

text: radical

text: by

text: Theorem

text: The

text: rest

text: follows

text: from

text: Lemma

text: Corollary

text: There

text: exists

text: an

text: inﬁnite

text: brace

text: such

text: that

text: its

text: Baer

text: radical

text: is

text: not

text: solvable.

text: Proof.

text: It

text: follows

text: from

text: the

text: fact

text: that

text: there

text: are

text: Baer

text: radical

text: rings

text: which

text: are

text: not

text: nilpotent.

text: Remark

text: Question

text: asks

text: for

text: two-sided

text: braces

text: that

text: are

text: prime.

text: Note

text: that

text: there

text: exists

text: an

text: inﬁnite

text: prime

text: and

text: not

text: simple

text: Jacobson

text: radical

text: ring.

text: Thus,

text: by

text: Lemma

text: there

text: exists

text: a

text: prime

text: and

text: not

text: simple

text: inﬁnite

text: two-sided

text: brace.

text: The

text: results

text: in

text: this

text: section

text: allow

text: us

text: to

text: answer

text: the

text: following

text: question:

text: Is

text: it

text: true

text: that

text: a

text: product

text: of

text: any

text: number

text: of

text: non-zero

text: ideals

text: in

text: A

text: (in

text: any

text: order)

text: is

text: nonzero?

text: Lemma

text: Let

text: A

text: be

text: a

text: prime

text: skew

text: brace

text: and

text: let

text: I

text: and

text: J

text: be

text: non-zero

text: ideals

text: in

text: A.

text: Then

text: I

text: J

text: is

text: a

text: non-zero

text: ideal

text: in

text: A.

text: Moreover,

text: the

text: intersection

text: of

text: any

text: ﬁnite

text: number

text: of

text: ideals

text: in

text: A

text: is

text: a

text: non-zero

text: ideal

text: in

text: A.

text: Proof.

text: The

text: intersection

text: of

text: any

text: two

text: ideals

text: is

text: an

text: ideal.

text: Notice

text: that

text: I

text: J

text: I

text: J,

text: therefore

text: I

text: J

text: The

text: last

text: assertion

text: can

text: be

text: proved

text: by

text: induction

text: on

text: the

text: number

text: of

text: ideals.


text: ON SKEW BRACES AND THEIR IDEALS 7 3. Series of ideals Following Rump [30], one deﬁnes the left series of a skew brace A recursively by ∗ ≥ A1 = A and An+1 = A An for n 1. Each An is a left ideal of A. The following example shows that in general An is not a normal subgroup of the additive group of A: (cid:39) Example 3.1. Let A = S . The left series of A is A1 = A, A2 S and (cid:39) 36,191 18,22 A3 B . Then the additive group A3 is not normal in the additive group of A. 3,1 Indeed, the additive group of A contains no normal subgroup of order three. ∗ Similarly the right series of A is deﬁned by A(1) = A and A(n+1) = A(n) A ≥ for n 1. Each A(n) is an ideal of A. A skew brace A is said to be left nilpotent ∈ N (resp. right nilpotent) if An = 0 (resp. A(n) = 0) for some n . See [9] or [30] for examples. Example 3.2. Let A = B . Up to isomorphism, the ideals of A are 16,73 0, B , B , B , B , B , B , B , B . 2,1 4,1 4,2 4,3 8,10 8,13 8,19 16,73 ∗ Let I be the ideal isomorphic to B . Then I I is a subset of size two which is 8,10 not an ideal of A. { } Recall that a skew brace A is said to be simple if its only ideals are 0 and A. Simple skew braces are intensively studied, in particular simple classical braces [7, 9]. (cid:39) A ◦ (cid:39) (cid:111) Example 3.3. Let A = S . Then (A, +) and (A, ) C C . The skew 12,22 4 ∈ N 3 4 brace A is a simple skew brace and A = An = A(n) for all n . (cid:39) ◦ (cid:39) (cid:111) Example 3.4. Let A = S . Then (A, +) SL (3) and (A, ) C C . 24,50 ∈ N 2 3 8 Furthermore A = An = A(n) for all n . This skew brace is not simple since for (cid:39) example Soc(A) B . 2,1 Let us count how many simple classical braces appear in our database. It is known that classical braces of prime-power size are not simple. Computer calcula- tions show the following results: ≤ ≤ Proposition 3.5. Let A be a simple brace of order n, where 1 n 127 and (cid:54) n = 96. Then A is isomorphic to B or B . 24,94 72,475 For skew braces we can prove the following proposition: ≤ ≤ Pr(cid:54)∈o{position 3}.6. Let A be a simple skew brace of order n, whe∼re 1 n 63 and n 32, 48, 54 . Then A is isomorphic to S , S , S = B or to one ≤ ≤ 12,22 12,23 24,853 24,94 of the skew braces S , where 145 k 152, which are the only skew braces with 60,k A additive group isomorphic to . 5

text: ON

text: SKEW

text: BRACES

text: AND

text: THEIR

text: IDEALS

text: Series

text: of

text: ideals

text: Following

text: Rump

text: one

text: deﬁnes

text: the

text: left

text: series

text: of

text: a

text: skew

text: brace

text: A

text: recursively

text: by

text: A

text: and

text: A

text: An

text: for

text: n

text: Each

text: An

text: is

text: a

text: left

text: ideal

text: of

text: A.

text: The

text: following

text: example

text: shows

text: that

text: in

text: general

text: An

text: is

text: not

text: a

text: normal

text: subgroup

text: of

text: the

text: additive

text: group

text: of

text: A:

text: Example

text: Let

text: A

text: S

text: The

text: left

text: series

text: of

text: A

text: is

text: A,

text: S

text: and

text: B

text: Then

text: the

text: additive

text: group

text: is

text: not

text: normal

text: in

text: the

text: additive

text: group

text: of

text: A.

text: Indeed,

text: the

text: additive

text: group

text: of

text: A

text: contains

text: no

text: normal

text: subgroup

text: of

text: order

text: three.

text: Similarly

text: the

text: right

text: series

text: of

text: A

text: is

text: deﬁned

text: by

text: A

text: and

text: A(n)

text: A

text: for

text: n

text: Each

text: A(n)

text: is

text: an

text: ideal

text: of

text: A.

text: A

text: skew

text: brace

text: A

text: is

text: said

text: to

text: be

text: left

text: nilpotent

text: N

text: (resp.

text: right

text: nilpotent)

text: if

text: An

text: (resp.

text: A(n)

text: for

text: some

text: n

text: See

text: or

text: for

text: examples.

text: Example

text: Let

text: A

text: B

text: Up

text: to

text: isomorphism,

text: the

text: ideals

text: of

text: A

text: are

text: B

text: B

text: B

text: B

text: B

text: B

text: B

text: B

text: Let

text: I

text: be

text: the

text: ideal

text: isomorphic

text: to

text: B

text: Then

text: I

text: I

text: is

text: a

text: subset

text: of

text: size

text: two

text: which

text: is

text: not

text: an

text: ideal

text: of

text: A.

text: Recall

text: that

text: a

text: skew

text: brace

text: A

text: is

text: said

text: to

text: be

text: simple

text: if

text: its

text: only

text: ideals

text: are

text: and

text: A.

text: Simple

text: skew

text: braces

text: are

text: intensively

text: studied,

text: in

text: particular

text: simple

text: classical

text: braces

text: A

text: Example

text: Let

text: A

text: S

text: Then

text: (A,

text: and

text: (A,

text: C

text: C

text: The

text: skew

text: N

text: brace

text: A

text: is

text: a

text: simple

text: skew

text: brace

text: and

text: A

text: An

text: A(n)

text: for

text: all

text: n

text: Example

text: Let

text: A

text: S

text: Then

text: (A,

text: SL

text: and

text: (A,

text: C

text: C

text: N

text: Furthermore

text: A

text: An

text: A(n)

text: for

text: all

text: n

text: This

text: skew

text: brace

text: is

text: not

text: simple

text: since

text: for

text: example

text: Soc(A)

text: B

text: Let

text: us

text: count

text: how

text: many

text: simple

text: classical

text: braces

text: appear

text: in

text: our

text: database.

text: It

text: is

text: known

text: that

text: classical

text: braces

text: of

text: prime-power

text: size

text: are

text: not

text: simple.

text: Computer

text: calcula-

text: tions

text: show

text: the

text: following

text: results:

text: Proposition

text: Let

text: A

text: be

text: a

text: simple

text: brace

text: of

text: order

text: n,

text: where

text: n

text: and

text: n

text: Then

text: A

text: is

text: isomorphic

text: to

text: B

text: or

text: B

text: For

text: skew

text: braces

text: we

text: can

text: prove

text: the

text: following

text: proposition:

text: Let

text: A

text: be

text: a

text: simple

text: skew

text: brace

text: of

text: order

text: n,

text: whe∼re

text: n

text: and

text: n

text: Then

text: A

text: is

text: isomorphic

text: to

text: S

text: S

text: S

text: B

text: or

text: to

text: one

text: of

text: the

text: skew

text: braces

text: S

text: where

text: k

text: which

text: are

text: the

text: only

text: skew

text: braces

text: with

text: A

text: additive

text: group

text: isomorphic

text: to