text: 2 come and go, being swapped in or out with other mem- fer. Here we brieﬂy review the main points about hori- bers of the community via horizontal transfer. But be- zontal gene transfer relevant to the mathematical model cause biochemical innovations produced by any member developed below. of the community were available to all, evolution at this While reproduction implies a vertical transfer of genes time was rapid—probably more rapid than at any time from one entity to the next in the phylogenetic tree, there since. Selection acted on whole communities, not on in- are also processes in which possibly unrelated individuals dividual progenotes. Those communities that were bet- exchange genetic material during their lifetimes, i.e., hor- ter at sharing their biochemical breakthroughs ﬂourished. izontally in the sense of the tree. This transfer of genes Out of this cauldron of evolutionary innovation, the uni- within one generation is consequently termed horizontal versal genetic code and its translational machinery co- gene transfer (HGT) [24–27] or lateral gene transfer. It evolved, in response to the selective pressures favoring is now widely accepted that HGT is a fundamental driv- eﬃcient sharing and interoperability. ing force of evolution [25, 28–33], and that its existence Vetsigian, Woese, and Goldenfeld [19] confronted and raises profound theoretical problems for evolutionary bi- constrained these speculations with mathematical anal- ology. For example, the longstanding problem of deﬁn- ysis and computer simulations. Going beyond Woese’s ing bacterial species [30, 34–38] is due, in part, to the conjectures, they probed early evolution scientiﬁcally by promiscuous use of HGT by bacteria. A recent primer on interpreting the available data on the genetic code. Their horizontal gene transfer and its potential for evolutionary dynamical model for the evolution of the genetic code [19] processes in general is given in [39]. showed that a collective state of life is required to obtain As discussed above, if HGT was rampant in the early the observed [20, 21] statistical properties of the code, in stages of evolution, the last universal common ancestor particular, its simultaneous universality and optimality. was a community, not a single organism [16, 17, 19, 40]. A later study by Goldenfeld and colleagues [22] provided In this collective state, individuals could not yet be dis- further evidence that only a collective state of life could tinguished, as each progenote’s genes were frequently ex- have created the highly optimized code used by all life changed through HGT. In terms of the model to be devel- today [21]. oped below, the total pool of genotypes in the collective C. The Darwinian transition state would be spread out and thus broadly distributed in the state space of all theoretically possible genotypes. How did the era of collective evolution come to an end? Conversely, a genotype distribution that is highly local- Woese speculated that as the translation process began ized in state space, being concentrated on just one or a to improve, and as progenotic subsystems became in- few genotypes, would be the model’s version of a well- creasingly complex and specialized, it would have become deﬁned species. harder to ﬁnd foreign parts compatible with them. Thus, Woese postulated that as the collective state of the horizontal gene transfer would have become increasingly progenote population slowly evolved toward higher com- diﬃcult. The only possible modiﬁcations at this point plexity, its rate of HGT slowly decreased [17]. At some would have come from within the progenote’s lineage it- point the system crossed the “Darwinian threshold” [17]. self, through mutation and gene duplication. It was in Then natural selection instead of HGT started to dom- this way that the progenotes would have made the Dar- inate the dynamics. The ﬁtter individuals were selected winian transition [17, 18] to become “genotes,” i.e., life for and the ﬁrst species emerged from the distributed forms with a tight coupling between their nucleic acid state. In the colorful language of Dyson [41]: genotypes and their protein phenotypes, and that could But then, one evil day, a cell resembling a therefore evolve through the familiar Darwinian process primitive bacterium happened to ﬁnd itself of vertical descent. one jump ahead of its neighbors in eﬃciency. The model considered below is an attempt to explore, That cell, anticipating Bill Gates by three bil- in mathematical terms, how the Darwinian transition lion years, separated itself from the commu- from the collective state to the modern era of individ- nity and refused to share. Its oﬀspring be- uality might have taken place. Our approach shares with came the ﬁrst species of bacteria—and the Ref. [19] the outlook that a dynamical systems calcula- ﬁrst species of any kind—reserving their in- tion should be devised to support or refute the hypothe- tellectual property for their own private use. ses considered. Our results lend support to the proposed With their superior eﬃciency, the bacteria collective state of life [15–19] by providing a potential continued to prosper and to evolve separately, mechanism for the exit from that state. while the rest of the community continued its D. Horizontal gene transfer communal life. Some millions of years later, another cell separated itself from the commu- Over the last decades more and more evidence has ac- nity and became the ancestor of the archaea. cumulated that, besides selection, mutation, and drift [23], Some time after that, a third cell separated another process drives evolution: horizontal gene trans- itself and became the ancestor of the eukary-

text: come

text: and

text: go,

text: being

text: swapped

text: in

text: or

text: out

text: with

text: other

text: mem-

text: fer.

text: Here

text: we

text: brieﬂy

text: review

text: the

text: main

text: points

text: about

text: hori-

text: bers

text: of

text: the

text: community

text: via

text: horizontal

text: transfer.

text: But

text: be-

text: zontal

text: gene

text: transfer

text: relevant

text: to

text: the

text: mathematical

text: model

text: cause

text: biochemical

text: innovations

text: produced

text: by

text: any

text: member

text: developed

text: below.

text: of

text: the

text: community

text: were

text: available

text: to

text: all,

text: evolution

text: at

text: this

text: While

text: reproduction

text: implies

text: a

text: vertical

text: transfer

text: of

text: genes

text: time

text: was

text: rapid—probably

text: more

text: rapid

text: than

text: at

text: any

text: time

text: from

text: one

text: entity

text: to

text: the

text: next

text: in

text: the

text: phylogenetic

text: tree,

text: there

text: since.

text: Selection

text: acted

text: on

text: whole

text: communities,

text: not

text: on

text: in-

text: are

text: also

text: processes

text: in

text: which

text: possibly

text: unrelated

text: individuals

text: dividual

text: progenotes.

text: Those

text: communities

text: that

text: were

text: bet-

text: exchange

text: genetic

text: material

text: during

text: their

text: lifetimes,

text: i.e.,

text: hor-

text: ter

text: at

text: sharing

text: their

text: biochemical

text: breakthroughs

text: ﬂourished.

text: izontally

text: in

text: the

text: sense

text: of

text: the

text: tree.

text: This

text: transfer

text: of

text: genes

text: Out

text: of

text: this

text: cauldron

text: of

text: evolutionary

text: innovation,

text: the

text: uni-

text: within

text: one

text: generation

text: is

text: consequently

text: termed

text: horizontal

text: versal

text: genetic

text: code

text: and

text: its

text: translational

text: machinery

text: co-

text: gene

text: transfer

text: (HGT)

text: or

text: lateral

text: gene

text: transfer.

text: It

text: evolved,

text: in

text: response

text: to

text: the

text: selective

text: pressures

text: favoring

text: is

text: now

text: widely

text: accepted

text: that

text: HGT

text: is

text: a

text: fundamental

text: driv-

text: eﬃcient

text: sharing

text: and

text: interoperability.

text: ing

text: force

text: of

text: evolution

text: and

text: that

text: its

text: existence

text: Vetsigian,

text: Woese,

text: and

text: Goldenfeld

text: confronted

text: and

text: raises

text: profound

text: theoretical

text: problems

text: for

text: evolutionary

text: bi-

text: constrained

text: these

text: speculations

text: with

text: mathematical

text: anal-

text: ology.

text: For

text: example,

text: the

text: longstanding

text: problem

text: of

text: deﬁn-

text: ysis

text: and

text: computer

text: simulations.

text: Going

text: beyond

text: Woese’s

text: ing

text: bacterial

text: species

text: is

text: due,

text: in

text: part,

text: to

text: the

text: conjectures,

text: they

text: probed

text: early

text: evolution

text: scientiﬁcally

text: by

text: promiscuous

text: use

text: of

text: HGT

text: by

text: bacteria.

text: A

text: recent

text: primer

text: on

text: interpreting

text: the

text: available

text: data

text: on

text: the

text: genetic

text: code.

text: Their

text: horizontal

text: gene

text: transfer

text: and

text: its

text: potential

text: for

text: evolutionary

text: dynamical

text: model

text: for

text: the

text: evolution

text: of

text: the

text: genetic

text: code

text: processes

text: in

text: general

text: is

text: given

text: in

text: showed

text: that

text: a

text: collective

text: state

text: of

text: life

text: is

text: required

text: to

text: obtain

text: As

text: discussed

text: above,

text: if

text: HGT

text: was

text: rampant

text: in

text: the

text: early

text: the

text: observed

text: statistical

text: properties

text: of

text: the

text: code,

text: in

text: stages

text: of

text: evolution,

text: the

text: last

text: universal

text: common

text: ancestor

text: particular,

text: its

text: simultaneous

text: universality

text: and

text: optimality.

text: was

text: a

text: community,

text: not

text: a

text: single

text: organism

text: A

text: later

text: study

text: by

text: Goldenfeld

text: and

text: colleagues

text: provided

text: In

text: this

text: collective

text: state,

text: individuals

text: could

text: not

text: yet

text: be

text: dis-

text: further

text: evidence

text: that

text: only

text: a

text: collective

text: state

text: of

text: life

text: could

text: tinguished,

text: as

text: each

text: progenote’s

text: genes

text: were

text: frequently

text: ex-

text: have

text: created

text: the

text: highly

text: optimized

text: code

text: used

text: by

text: all

text: life

text: changed

text: through

text: HGT.

text: In

text: terms

text: of

text: the

text: model

text: to

text: be

text: devel-

text: today

text: oped

text: below,

text: the

text: total

text: pool

text: of

text: genotypes

text: in

text: the

text: collective

text: C.

text: The

text: Darwinian

text: transition

text: state

text: would

text: be

text: spread

text: out

text: and

text: thus

text: broadly

text: distributed

text: in

text: the

text: state

text: space

text: of

text: all

text: theoretically

text: possible

text: genotypes.

text: How

text: did

text: the

text: era

text: of

text: collective

text: evolution

text: come

text: to

text: an

text: end?

text: Conversely,

text: a

text: genotype

text: distribution

text: that

text: is

text: highly

text: local-

text: Woese

text: speculated

text: that

text: as

text: the

text: translation

text: process

text: began

text: ized

text: in

text: state

text: space,

text: being

text: concentrated

text: on

text: just

text: one

text: or

text: a

text: to

text: improve,

text: and

text: as

text: progenotic

text: subsystems

text: became

text: in-

text: few

text: genotypes,

text: would

text: be

text: the

text: model’s

text: version

text: of

text: a

text: well-

text: creasingly

text: complex

text: and

text: specialized,

text: it

text: would

text: have

text: become

text: deﬁned

text: species.

text: harder

text: to

text: ﬁnd

text: foreign

text: parts

text: compatible

text: with

text: them.

text: Thus,

text: Woese

text: postulated

text: that

text: as

text: the

text: collective

text: state

text: of

text: the

text: horizontal

text: gene

text: transfer

text: would

text: have

text: become

text: increasingly

text: progenote

text: population

text: slowly

text: evolved

text: toward

text: higher

text: com-

text: diﬃcult.

text: The

text: only

text: possible

text: modiﬁcations

text: at

text: this

text: point

text: plexity,

text: its

text: rate

text: of

text: HGT

text: slowly

text: decreased

text: At

text: some

text: would

text: have

text: come

text: from

text: within

text: the

text: progenote’s

text: lineage

text: it-

text: point

text: the

text: system

text: crossed

text: the

text: “Darwinian

text: threshold”

text: self,

text: through

text: mutation

text: and

text: gene

text: duplication.

text: It

text: was

text: in

text: Then

text: natural

text: selection

text: instead

text: of

text: HGT

text: started

text: to

text: dom-

text: this

text: way

text: that

text: the

text: progenotes

text: would

text: have

text: made

text: the

text: Dar-

text: inate

text: the

text: dynamics.

text: The

text: ﬁtter

text: individuals

text: were

text: selected

text: winian

text: transition

text: to

text: become

text: “genotes,”

text: i.e.,

text: life

text: for

text: and

text: the

text: ﬁrst

text: species

text: emerged

text: from

text: the

text: distributed

text: forms

text: with

text: a

text: tight

text: coupling

text: between

text: their

text: nucleic

text: acid

text: state.

text: In

text: the

text: colorful

text: language

text: of

text: Dyson

text: genotypes

text: and

text: their

text: protein

text: phenotypes,

text: and

text: that

text: could

text: But

text: then,

text: one

text: evil

text: day,

text: a

text: cell

text: resembling

text: a

text: therefore

text: evolve

text: through

text: the

text: familiar

text: Darwinian

text: process

text: primitive

text: bacterium

text: happened

text: to

text: ﬁnd

text: itself

text: of

text: vertical

text: descent.

text: one

text: jump

text: ahead

text: of

text: its

text: neighbors

text: in

text: eﬃciency.

text: The

text: model

text: considered

text: below

text: is

text: an

text: attempt

text: to

text: explore,

text: That

text: cell,

text: anticipating

text: Bill

text: Gates

text: by

text: three

text: bil-

text: in

text: mathematical

text: terms,

text: how

text: the

text: Darwinian

text: transition

text: lion

text: years,

text: separated

text: itself

text: from

text: the

text: commu-

text: from

text: the

text: collective

text: state

text: to

text: the

text: modern

text: era

text: of

text: individ-

text: nity

text: and

text: refused

text: to

text: share.

text: Its

text: oﬀspring

text: be-

text: uality

text: might

text: have

text: taken

text: place.

text: Our

text: approach

text: shares

text: with

text: came

text: the

text: ﬁrst

text: species

text: of

text: bacteria—and

text: the

text: Ref.

text: the

text: outlook

text: that

text: a

text: dynamical

text: systems

text: calcula-

text: ﬁrst

text: species

text: of

text: any

text: kind—reserving

text: their

text: in-

text: tion

text: should

text: be

text: devised

text: to

text: support

text: or

text: refute

text: the

text: hypothe-

text: tellectual

text: property

text: for

text: their

text: own

text: private

text: use.

text: ses

text: considered.

text: Our

text: results

text: lend

text: support

text: to

text: the

text: proposed

text: With

text: their

text: superior

text: eﬃciency,

text: the

text: bacteria

text: collective

text: state

text: of

text: life

text: by

text: providing

text: a

text: potential

text: continued

text: to

text: prosper

text: and

text: to

text: evolve

text: separately,

text: mechanism

text: for

text: the

text: exit

text: from

text: that

text: state.

text: while

text: the

text: rest

text: of

text: the

text: community

text: continued

text: its

text: D.

text: Horizontal

text: gene

text: transfer

text: communal

text: life.

text: Some

text: millions

text: of

text: years

text: later,

text: another

text: cell

text: separated

text: itself

text: from

text: the

text: commu-

text: Over

text: the

text: last

text: decades

text: more

text: and

text: more

text: evidence

text: has

text: ac-

text: nity

text: and

text: became

text: the

text: ancestor

text: of

text: the

text: archaea.

text: cumulated

text: that,

text: besides

text: selection,

text: mutation,

text: and

text: drift

text: Some

text: time

text: after

text: that,

text: a

text: third

text: cell

text: separated

text: another

text: process

text: drives

text: evolution:

text: horizontal

text: gene

text: trans-

text: itself

text: and

text: became

text: the

text: ancestor

text: of

text: the

text: eukary-


text: 3 otes. on a ﬁtness landscape [23, 53] in the presence of selec- tion, mutation, drift, and HGT. Each progenote carries After making a Darwinian transition, evolution pro- a genome of length l composed of a sequence of the bases ceeds in the familiar vertical manner, being driven by se- 0 and 1. The genome of progenote i determines its ﬁt- lection, mutation, and drift [23], with HGT playing only ness f . The progenotes reproduce by the Moran pro- i a minor role. Such Darwinian dynamics have, of course, cess [23, 54], i.e., each progenote reproduces randomly in been studied extensively in both experimental and model time, with its reproduction rate given by its ﬁtness f . i settings [23, 42–46]. Compared to the dynamics of HGT Whenever a progenote of genotype i reproduces, an oﬀ- their properties are relatively well understood. Recently, spring is added to the population which is either identical potential inﬂuences of HGT on such evolutionary dynam- to genotype i or a mutant of genotype j with probabil- ics have been investigated [16, 19, 47–51]. Some mathe- ity µ . Instantaneously after such a reproduction event, ij matical models of HGT have focused on how it can in- one progenote in the population is chosen randomly to die crease a population’s ﬁtness in Darwinian evolution [52]. and is hence removed from the population. We assume Keep in mind, however, that the hypothesized HGT as- that one mutation event will only aﬀect one of the bases sociated with progenotes and the Darwinian transition, of the genome, so that the Hamming distance between being associated with ribosomal genes and the rest of genotypes i and j is 1. the core machinery of the cell, would have been of far Hence, our ﬁtness landscape may be represented by a greater evolutionary signiﬁcance than the HGT of, say, network where the diﬀerent genotypes are the nodes of antibiotic resistance genes seen in bacterial communities the network and the possible mutations form the links. today. In Woese’s scenario, the ancient form of HGT was Assigning two diﬀerent bases, 0 and 1, and given the rampant, pervasive, and tremendously disruptive and in- structure of the mutations, the resulting network is an novative. It was the prime mover in shaping the fabric l-dimensional hypercube (Fig. 1). of the cell [18]. The ﬁtness landscape underlying our model is assumed We would like to understand what such a Darwinian to be a Mount Fuji landscape [23]: The highest ﬁtness is transition would look like, mathematically. The model assigned to one single genotype, the peak. Other geno- described in the next section is deliberately minimal. It types are assigned lower ﬁtness: the farther away from leaves out all the biology of ribosomes, proteins, genetic the peak in genotype space, the lower the ﬁtness. Thus, codes, and the like. What remains is an attempt to cap- a single-peaked mountain landscape is created on geno- ture the essence of Woese’s speculations. In place of type space, and a population evolving purely through the a community of progenotes, we consider a community processes of selection and mutation should converge to of abstract genomes, represented by binary sequences. this peak. Note that the Moran process described above They interact via HGT, and are subject to mutation, is a random process. It thereby constitutes a minimal selection, and drift on a ﬁtness landscape. Our work model intrinsically including the eﬀects of selection, mu- suggests that HGT-dominated dynamics may have been tation and genetic drift [23]. The latter is induced by the intermittently interrupted by selection-driven processes stochastic selection in the combined process of reproduc- during which genotypes became ﬁtter and decreased their tion, mutation and death and has the eﬀect of randomly inclination toward HGT. Such stochastic switching in the walking the population around in genotype space even if nonlinear population dynamics may have destabilized the no ﬁtness diﬀerences were present. HGT-dominated state and thus led to a Darwinian tran- To reveal the potential impact of HGT we incorpo- sition and the emergence of the ﬁrst species in early evo- rate its basic features into the stochastic evolution model. lution. Two progenotes A and B may meet and a subsequence On a side note, an interesting mathematical aspect of s of progenote B’s genome may be inserted into A’s the model is that it necessarily involves three-point in- genome. As a result of this horizontal gene transfer event, teractions, since HGT transforms one genotype into a the genotype of progenote A will transform into another second by importing pieces of a third. Thus the model genotype C, determined by its original genotype and the provides a natural biological example of a complex hy- subsequence s. This process is illustrated in Figure 1. pernetwork. Until now, most models in evolutionary dy- To model this process we add HGT-hyperlinks to the namics and population biology did not need to go beyond hypercube network representing the ﬁtness landscape. ordinary network structure, with two-point interactions One such hyperlink symbolizes a three-genotype inter- between nodes connected by links. action and is deﬁned through the following process. We choose two genotypes A and B randomly as well as a random subsequence of genome B with length between II. EVOLUTIONARY MODEL x = 2 and x = l − 2 bases. This subsequence is inserted at a random position of genome A. The remaining x To explore the consequences of HGT on evolution, we bases at the end of A’s sequence are cut oﬀ, keeping the consider a model community of N progenotes evolving sequence length of A constant. The new sequence deter-

label: paragraph

text: otes.

text: on

text: a

text: ﬁtness

text: landscape

text: in

text: the

text: presence

text: of

text: selec-

text: tion,

text: mutation,

text: drift,

text: and

text: HGT.

text: Each

text: progenote

text: carries

text: After

text: making

text: a

text: Darwinian

text: transition,

text: evolution

text: pro-

text: a

text: genome

text: of

text: length

text: l

text: composed

text: of

text: a

text: sequence

text: of

text: the

text: bases

text: ceeds

text: in

text: the

text: familiar

text: vertical

text: manner,

text: being

text: driven

text: by

text: se-

text: and

text: The

text: genome

text: of

text: progenote

text: i

text: determines

text: its

text: ﬁt-

text: lection,

text: mutation,

text: and

text: drift

text: with

text: HGT

text: playing

text: only

text: ness

text: f

text: The

text: progenotes

text: reproduce

text: by

text: the

text: Moran

text: pro-

text: i

text: a

text: minor

text: role.

text: Such

text: Darwinian

text: dynamics

text: have,

text: of

text: course,

text: cess

text: i.e.,

text: each

text: progenote

text: reproduces

text: randomly

text: in

text: been

text: studied

text: extensively

text: in

text: both

text: experimental

text: and

text: model

text: time,

text: with

text: its

text: reproduction

text: rate

text: given

text: by

text: its

text: ﬁtness

text: f

text: i

text: settings

text: Compared

text: to

text: the

text: dynamics

text: of

text: HGT

text: Whenever

text: a

text: progenote

text: of

text: genotype

text: i

text: reproduces,

text: an

text: oﬀ-

text: their

text: properties

text: are

text: relatively

text: well

text: understood.

text: Recently,

text: spring

text: is

text: added

text: to

text: the

text: population

text: which

text: is

text: either

text: identical

text: potential

text: inﬂuences

text: of

text: HGT

text: on

text: such

text: evolutionary

text: dynam-

text: to

text: genotype

text: i

text: or

text: a

text: mutant

text: of

text: genotype

text: j

text: with

text: probabil-

text: ics

text: have

text: been

text: investigated

text: Some

text: mathe-

text: ity

text: µ

text: Instantaneously

text: after

text: such

text: a

text: reproduction

text: event,

text: ij

text: matical

text: models

text: of

text: HGT

text: have

text: focused

text: on

text: how

text: it

text: can

text: in-

text: one

text: progenote

text: in

text: the

text: population

text: is

text: chosen

text: randomly

text: to

text: die

text: crease

text: a

text: population’s

text: ﬁtness

text: in

text: Darwinian

text: evolution

text: and

text: is

text: hence

text: removed

text: from

text: the

text: population.

text: We

text: assume

text: Keep

text: in

text: mind,

text: however,

text: that

text: the

text: hypothesized

text: HGT

text: as-

text: that

text: one

text: mutation

text: event

text: will

text: only

text: aﬀect

text: one

text: of

text: the

text: bases

text: sociated

text: with

text: progenotes

text: and

text: the

text: Darwinian

text: transition,

text: of

text: the

text: genome,

text: so

text: that

text: the

text: Hamming

text: distance

text: between

text: being

text: associated

text: with

text: ribosomal

text: genes

text: and

text: the

text: rest

text: of

text: genotypes

text: i

text: and

text: j

text: is

text: the

text: core

text: machinery

text: of

text: the

text: cell,

text: would

text: have

text: been

text: of

text: far

text: Hence,

text: our

text: ﬁtness

text: landscape

text: may

text: be

text: represented

text: by

text: a

text: greater

text: evolutionary

text: signiﬁcance

text: than

text: the

text: HGT

text: of,

text: say,

text: network

text: where

text: the

text: diﬀerent

text: genotypes

text: are

text: the

text: nodes

text: of

text: antibiotic

text: resistance

text: genes

text: seen

text: in

text: bacterial

text: communities

text: the

text: network

text: and

text: the

text: possible

text: mutations

text: form

text: the

text: links.

text: today.

text: In

text: Woese’s

text: scenario,

text: the

text: ancient

text: form

text: of

text: HGT

text: was

text: Assigning

text: two

text: diﬀerent

text: bases,

text: and

text: and

text: given

text: the

text: rampant,

text: pervasive,

text: and

text: tremendously

text: disruptive

text: and

text: in-

text: structure

text: of

text: the

text: mutations,

text: the

text: resulting

text: network

text: is

text: an

text: novative.

text: It

text: was

text: the

text: prime

text: mover

text: in

text: shaping

text: the

text: fabric

text: l-dimensional

text: hypercube

text: (Fig.

text: of

text: the

text: cell

text: The

text: ﬁtness

text: landscape

text: underlying

text: our

text: model

text: is

text: assumed

text: We

text: would

text: like

text: to

text: understand

text: what

text: such

text: a

text: Darwinian

text: to

text: be

text: a

text: Mount

text: Fuji

text: landscape

text: The

text: highest

text: ﬁtness

text: is

text: transition

text: would

text: look

text: like,

text: mathematically.

text: The

text: model

text: assigned

text: to

text: one

text: single

text: genotype,

text: the

text: peak.

text: Other

text: geno-

text: described

text: in

text: the

text: next

text: section

text: is

text: deliberately

text: minimal.

text: It

text: types

text: are

text: assigned

text: lower

text: ﬁtness:

text: the

text: farther

text: away

text: from

text: leaves

text: out

text: all

text: the

text: biology

text: of

text: ribosomes,

text: proteins,

text: genetic

text: the

text: peak

text: in

text: genotype

text: space,

text: the

text: lower

text: the

text: ﬁtness.

text: Thus,

text: codes,

text: and

text: the

text: like.

text: What

text: remains

text: is

text: an

text: attempt

text: to

text: cap-

text: a

text: single-peaked

text: mountain

text: landscape

text: is

text: created

text: on

text: geno-

text: ture

text: the

text: essence

text: of

text: Woese’s

text: speculations.

text: In

text: place

text: of

text: type

text: space,

text: and

text: a

text: population

text: evolving

text: purely

text: through

text: the

text: a

text: community

text: of

text: progenotes,

text: we

text: consider

text: a

text: community

text: processes

text: of

text: selection

text: and

text: mutation

text: should

text: converge

text: to

text: of

text: abstract

text: genomes,

text: represented

text: by

text: binary

text: sequences.

text: this

text: peak.

text: Note

text: that

text: the

text: Moran

text: process

text: described

text: above

text: They

text: interact

text: via

text: HGT,

text: and

text: are

text: subject

text: to

text: mutation,

text: is

text: a

text: random

text: process.

text: It

text: thereby

text: constitutes

text: a

text: minimal

text: selection,

text: and

text: drift

text: on

text: a

text: ﬁtness

text: landscape.

text: Our

text: work

text: model

text: intrinsically

text: including

text: the

text: eﬀects

text: of

text: selection,

text: mu-

text: suggests

text: that

text: HGT-dominated

text: dynamics

text: may

text: have

text: been

text: tation

text: and

text: genetic

text: drift

text: The

text: latter

text: is

text: induced

text: by

text: the

text: intermittently

text: interrupted

text: by

text: selection-driven

text: processes

text: stochastic

text: selection

text: in

text: the

text: combined

text: process

text: of

text: reproduc-

text: during

text: which

text: genotypes

text: became

text: ﬁtter

text: and

text: decreased

text: their

text: tion,

text: mutation

text: and

text: death

text: and

text: has

text: the

text: eﬀect

text: of

text: randomly

text: inclination

text: toward

text: HGT.

text: Such

text: stochastic

text: switching

text: in

text: the

text: walking

text: the

text: population

text: around

text: in

text: genotype

text: space

text: even

text: if

text: nonlinear

text: population

text: dynamics

text: may

text: have

text: destabilized

text: the

text: no

text: ﬁtness

text: diﬀerences

text: were

text: present.

text: HGT-dominated

text: state

text: and

text: thus

text: led

text: to

text: a

text: Darwinian

text: tran-

text: To

text: reveal

text: the

text: potential

text: impact

text: of

text: HGT

text: we

text: incorpo-

text: sition

text: and

text: the

text: emergence

text: of

text: the

text: ﬁrst

text: species

text: in

text: early

text: evo-

text: rate

text: its

text: basic

text: features

text: into

text: the

text: stochastic

text: evolution

text: model.

text: lution.

text: Two

text: progenotes

text: A

text: and

text: B

text: may

text: meet

text: and

text: a

text: subsequence

text: On

text: a

text: side

text: note,

text: an

text: interesting

text: mathematical

text: aspect

text: of

text: s

text: of

text: progenote

text: B’s

text: genome

text: may

text: be

text: inserted

text: into

text: A’s

text: the

text: model

text: is

text: that

text: it

text: necessarily

text: involves

text: three-point

text: in-

text: genome.

text: As

text: a

text: result

text: of

text: this

text: horizontal

text: gene

text: transfer

text: event,

text: teractions,

text: since

text: HGT

text: transforms

text: one

text: genotype

text: into

text: a

text: the

text: genotype

text: of

text: progenote

text: A

text: will

text: transform

text: into

text: another

text: second

text: by

text: importing

text: pieces

text: of

text: a

text: third.

text: Thus

text: the

text: model

text: genotype

text: C,

text: determined

text: by

text: its

text: original

text: genotype

text: and

text: the

text: provides

text: a

text: natural

text: biological

text: example

text: of

text: a

text: complex

text: hy-

text: subsequence

text: s.

text: This

text: process

text: is

text: illustrated

text: in

text: Figure

text: pernetwork.

text: Until

text: now,

text: most

text: models

text: in

text: evolutionary

text: dy-

text: To

text: model

text: this

text: process

text: we

text: add

text: HGT-hyperlinks

text: to

text: the

text: namics

text: and

text: population

text: biology

text: did

text: not

text: need

text: to

text: go

text: beyond

text: hypercube

text: network

text: representing

text: the

text: ﬁtness

text: landscape.

text: ordinary

text: network

text: structure,

text: with

text: two-point

text: interactions

text: One

text: such

text: hyperlink

text: symbolizes

text: a

text: three-genotype

text: inter-

text: between

text: nodes

text: connected

text: by

text: links.

text: action

text: and

text: is

text: deﬁned

text: through

text: the

text: following

text: process.

text: We

text: choose

text: two

text: genotypes

text: A

text: and

text: B

text: randomly

text: as

text: well

text: as

text: a

text: random

text: subsequence

text: of

text: genome

text: B

text: with

text: length

text: between

text: II.

text: EVOLUTIONARY

text: MODEL

text: x

text: and

text: x

text: l

text: bases.

text: This

text: subsequence

text: is

text: inserted

text: at

text: a

text: random

text: position

text: of

text: genome

text: A.

text: The

text: remaining

text: x

text: To

text: explore

text: the

text: consequences

text: of

text: HGT

text: on

text: evolution,

text: we

text: bases

text: at

text: the

text: end

text: of

text: A’s

text: sequence

text: are

text: cut

text: oﬀ,

text: keeping

text: the

text: consider

text: a

text: model

text: community

text: of

text: N

text: progenotes

text: evolving

text: sequence

text: length

text: of

text: A

text: constant.

text: The

text: new

text: sequence

text: deter-


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text: on

text: an

text: idealized

text: model

text: now

text: demonstrate

text: how

text: stochastic

label: reference

text: D.

text: L.

text: Theobald,

text: “A

text: formal

text: test

text: of

text: the

text: theory

text: of

text: universal

text: switching

text: and

text: ﬁtness-dependent

text: competence

text: may

text: com-

text: common

text: ancestry,”

text: Nature

text: bine

text: to

text: create

text: a

text: transition

text: from

text: a

text: bistable

text: state

text: to

text: a

text: Tal

text: Dagan,

text: Mayo

text: Roettger,

text: David

text: Bryant,

text: and

text: William

text: Martin,

text: “Genome

text: networks

text: root

text: the

text: tree

text: of

text: life

text: between

text: speciated-only

text: state.

text: They

text: in

text: particular

text: also

text: suggest

text: that

text: prokaryotic

text: domains,”

text: Genome

text: Biology

text: and

text: Evolution

text: HGT

text: may

text: be

text: present

text: at

text: similar

text: competence

text: levels

text: be-

text: fore

text: and

text: after

text: the

text: emergence

text: of

text: the

text: ﬁrst

text: species.

text: From

text: a

text: Tom

text: A

text: Williams,

text: Peter

text: G

text: Foster,

text: Cymon

text: J

text: Cox,

text: and

text: complementary

text: perspective,

text: whereas

text: one

text: or

text: a

text: few

text: species

text: T

text: Martin

text: Embley,

text: “An

text: archaeal

text: origin

text: of

text: eukaryotes

text: sup-

text: may

text: already

text: have

text: existed,

text: other

text: population

text: parts

text: may

text: ports

text: only

text: two

text: primary

text: domains

text: of

text: life,”

text: Nature

text: still

text: have

text: been

text: mixed

text: without

text: any

text: clear

text: species.

text: So

text: the

text: very

text: ﬁrst

text: species

text: may

text: only

text: have

text: marked

text: the

text: beginning

text: Nicolas

text: Glansdorﬀ,

text: Ying

text: Xu,

text: Bernard

text: Labedan,

text: et

text: al.,

text: “The

text: last

text: universal

text: common

text: ancestor:

text: emergence,

text: consti-

text: of

text: the

text: decline

text: of

text: genuinely

text: non-speciﬁc

text: life,

text: with

text: other

text: tution

text: and

text: genetic

text: legacy

text: of

text: an

text: elusive

text: forerunner,”

text: Biol

text: Darwinian

text: transitions

text: to

text: follow.

text: Direct

text: Acknowlegdements:

text: We

text: thank

text: Nigel

text: Goldenfeld,

text: Ste-

text: C.

text: R.

text: Woese

text: and

text: G.

text: E.

text: Fox,

text: “The

text: concept

text: of

text: cellular

text: evolution,”

text: J.

text: Mol.

text: Evol.

text: fan

text: Grosskinsky,

text: Oskar

text: Hallatschek

text: and

text: Arne

text: Traulsen

text: C.

text: R.

text: Woese,

text: “The

text: universal

text: ancestor,”

text: Proc.

text: Natl.

text: Acad.

text: for

text: valuable

text: comments

text: and

text: discussions.

text: Supported

text: by

text: a

text: Sci.

text: USA

text: grant

text: of

text: the

text: Max

text: Planck

text: Society

text: to

text: MT.

text: C.

text: R.

text: Woese,

text: “On

text: the

text: evolution

text: of

text: cells,”

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: USA

text: C.

text: R.

text: Woese,

text: “Evolving

text: biological

text: organization,”

text: in

text: Mi-

text: crobial

text: Phylogeny

text: and

text: Evolution:

text: Concepts

text: and

text: Contro-

text: versies,

text: edited

text: by

text: J.

text: Sapp

text: (Oxford

text: University

text: Press,

text: Ox-

text: ford,

text: UK,

text: pp.

text: K.

text: Vetsigian,

text: C.Woese,

text: and

text: N.

text: Goldenfeld,

text: “Collective

text: evolution

text: and

text: the

text: genetic

text: code,”

text: Proc.

text: Natl.

text: Acad.

text: Sci.

text: C.

text: R.

text: Darwin,

text: On

text: the

text: Origin

text: of

text: Species

text: (John

text: Murray,

text: USA

text: London,

text: UK,

text: David

text: Haig

text: and

text: Laurence

text: D

text: Hurst,

text: “A

text: quantitative

text: mea-

text: MW

text: Nirenberg,

text: OW

text: Jones,

text: P

text: Leder,

text: BFC

text: Clark,

text: WS

text: Sly,

text: sure

text: of

text: error

text: minimization

text: in

text: the

text: genetic

text: code,”

text: Journal

text: and

text: S

text: Pestka,

text: “On

text: the

text: coding

text: of

text: genetic

text: information,”

text: in

text: of

text: Molecular

text: Evolution

text: Cold

text: Spring

text: Harbor

text: Symposia

text: on

text: Quantitative

text: Biology,

text: Stephen

text: J

text: Freeland

text: and

text: Laurence

text: D

text: Hurst,

text: “The

text: genetic

text: Vol.

text: (Cold

text: Spring

text: Harbor

text: Laboratory

text: Press,

text: pp.

text: code

text: is

text: one

text: in

text: a

text: million,”

text: Journal

text: of

text: Molecular

text: Evolution

text: C.

text: R.

text: Woese,

text: The

text: Genetic

text: Code

text: (Harper

text: and

text: Row,

text: New

text: Thomas

text: Butler,

text: Nigel

text: Goldenfeld,

text: Damien

text: Mathew,

text: and

text: York,

text: Zaida

text: Luthey-Schulten,

text: “Extreme

text: genetic

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text: optimal-

text: Robin

text: D

text: Knight,

text: Stephen

text: J

text: Freeland,

text: and

text: Laura

text: F

text: ity

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text: a

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text: Landweber,

text: “Rewiring

text: the

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text: acid

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text: requirement,”

text: Physical

text: Review

text: E

text: genetic

text: code,”

text: Nature

text: Reviews

text: Genetics

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text: Emma

text: Lehmer.

text: “On

text: Congruences

text: involving

text: Bernoulli

text: numbers

text: and

text: the

text: quo-

text: tients

text: of

text: Fermat

text: and

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text: of

text: Math.

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text: Lerch,

text: “Zur

text: Theorie

text: des

text: Fermatschen

text: Quotienten.

text: Math.

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text: Petr

text: Ležák,

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text: Wieferichových

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text: Mirimanoﬀ,

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text: de

text: Fermat,”

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text: R.

text: Acad.

text: Sci.

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text: angew.

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text: orem,”

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text: Math.

text: Mathew

text: D.

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text: har-

text: p

text: monic

text: series,”

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text: Schwindt,

text: “Three

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text: Math.

text: Comp.

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text: Bi-

text: quarterly

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text: author’s

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text: Discrete

text: Math.


text: 2 The Divisibility of Harmonic Numbers It may be helpful to distinguish the purpose of the present study with that of the more general problem of the divisibility of Harmonic numbers. Eswarathasan and Levine [8] note that all primes greater than 3 divide the Harmonic numbers − − − of indices 1, ( 1), and 2 1, and Wolstenholme’s theorem states that p p p p | for a prime p > 3, p2 Hp−1. These results may be inverted to give three general rules for the divisibility of Harmonic numbers: 1. is divisible by ( + 1)2 if + 1 is a prime 3; H n n > n √ 2. Hn is divisible by √1+ 24n+1 if the latter is a prime > 3; 3. is divisible by + 1 if the latter is a prime 3. H n > n Eswarathasan and Levine deﬁne harmonic primes as primes that divide only the three aforementioned Harmonic numbers (OEIS A092101), and anharmonic primes as those that divide additional Harmonic numbers (OEIS A092102). | From the fact that we consider only cases (1) where with 1, p Hb c N > p/N it will be evident that any such is anharmonic, and that we are seeking a p subset of anharmonic primes that divide Harmonic numbers of relatively small index; for example, from Table 5 we see that = 2 is solved by = 1093, N p | implying that 1093 , while = 46 is solved by = 11731, implying that H546 N p | 11731 . In contrast, the work of Boyd [1] and Rogers [23] entails, in part, H255 ﬁnding Harmonic numbers of large index divisible by relatively small primes, | such as the case 11 . The overlap between our H1011849771855214912968404217247 results and theirs is slight. Indeed, our results entail only a minority of the anharmonic primes, because − they require that divide an Harmonic number with 1, and thus p H n < p n belong to a subset of the anharmonic primes which has sometimes been called the Harmonic irregular primes (see OEIS A092194 and the Wikipedia entry for “Regular prime”). Although the standard deﬁnition of such a prime is that p − it divide an Harmonic number with 1, it may be noted that any H n < p n prime satisfying this criterion must in fact divide some Harmonic number H − ≡ n of smaller index n < (p 1)/2, since by symmetry Hp−1−n −Hn mod p. The inequality sign in this condition is strict because if = ( 1) 2, then is n p / p a Wieferich prime and by (2) and (4) must likewise divide the smaller . Hbp/4c But this inequality also gives a sharp upper bound on the index of the least Harmonic number divisible by an Harmonic irregular prime , since among the p − ﬁrst 999 primes we have = ( 3) 2 for = 29 37 3373 (see OEIS A125854). n p / p , , In general, there does not seem to be any way of recognizing an Harmonic irregular prime other than by exhaustively testing it as a divisor of Harmonic ≤ − numbers with ( 3) 2 (though one would of course not attempt to do H n p / n this by actual division except for very small ). p In turn, our results entail only a minority of the Harmonic irregular primes, b c because we must have = for some . Nonetheless, for some small m p/N N primes this relationship is satisﬁed by more than one value of ; for example, N | b c b c b c b c b c with = 137, , and 23 = 24 = 25 = 26 = 27 = 5. p p H5 p/ p/ p/ p/ p/ 6 ##LTLine## ##LTLine## ##LTLine##

label: section

text: The

text: Divisibility

text: of

text: Harmonic

text: Numbers

text: It

text: may

text: be

text: helpful

text: to

text: distinguish

text: the

text: purpose

text: of

text: the

text: present

text: study

text: with

text: that

text: of

text: the

text: more

text: general

text: problem

text: of

text: the

text: divisibility

text: of

text: Harmonic

text: numbers.

text: Eswarathasan

text: and

text: Levine

text: note

text: that

text: all

text: primes

text: greater

text: than

text: divide

text: the

text: Harmonic

text: numbers

text: of

text: indices

text: and

text: and

text: Wolstenholme’s

text: theorem

text: states

text: that

text: p

text: p

text: p

text: p

text: for

text: a

text: prime

text: p

text: These

text: results

text: may

text: be

text: inverted

text: to

text: give

text: three

text: general

text: rules

text: for

text: the

text: divisibility

text: of

text: Harmonic

text: numbers:

label: list

text: is

text: divisible

text: by

text: if

text: is

text: a

text: prime

text: H

text: n

text: n

text: n

text: Hn

text: is

text: divisible

text: by

text: if

text: the

text: latter

text: is

text: a

text: prime

text: is

text: divisible

text: by

text: if

text: the

text: latter

text: is

text: a

text: prime

text: H

text: n

text: n

text: Eswarathasan

text: and

text: Levine

text: deﬁne

text: harmonic

text: primes

text: as

text: primes

text: that

text: divide

text: only

text: the

text: three

text: aforementioned

text: Harmonic

text: numbers

text: (OEIS

text: and

text: anharmonic

text: primes

text: as

text: those

text: that

text: divide

text: additional

text: Harmonic

text: numbers

text: (OEIS

text: From

text: the

text: fact

text: that

text: we

text: consider

text: only

text: cases

text: where

text: with

text: p

text: Hb

text: c

text: N

text: p/N

text: it

text: will

text: be

text: evident

text: that

text: any

text: such

text: is

text: anharmonic,

text: and

text: that

text: we

text: are

text: seeking

text: a

text: p

text: subset

text: of

text: anharmonic

text: primes

text: that

text: divide

text: Harmonic

text: numbers

text: of

text: relatively

text: small

text: index;

text: for

text: example,

text: from

text: Table

text: we

text: see

text: that

text: is

text: solved

text: by

text: N

text: p

text: implying

text: that

text: while

text: is

text: solved

text: by

text: implying

text: that

text: N

text: p

text: In

text: contrast,

text: the

text: work

text: of

text: Boyd

text: and

text: Rogers

text: entails,

text: in

text: part,

text: ﬁnding

text: Harmonic

text: numbers

text: of

text: large

text: index

text: divisible

text: by

text: relatively

text: small

text: primes,

text: such

text: as

text: the

text: case

text: The

text: overlap

text: between

text: our

text: results

text: and

text: theirs

text: is

text: slight.

text: Indeed,

text: our

text: results

text: entail

text: only

text: a

text: minority

text: of

text: the

text: anharmonic

text: primes,

text: because

text: they

text: require

text: that

text: divide

text: an

text: Harmonic

text: number

text: with

text: and

text: thus

text: p

text: H

text: n

text: p

text: n

text: belong

text: to

text: a

text: subset

text: of

text: the

text: anharmonic

text: primes

text: which

text: has

text: sometimes

text: been

text: called

text: the

text: Harmonic

text: irregular

text: primes

text: (see

text: OEIS

text: and

text: the

text: Wikipedia

text: entry

text: for

text: “Regular

text: prime”).

text: Although

text: the

text: standard

text: deﬁnition

text: of

text: such

text: a

text: prime

text: is

text: that

text: p

text: it

text: divide

text: an

text: Harmonic

text: number

text: with

text: it

text: may

text: be

text: noted

text: that

text: any

text: H

text: n

text: p

text: n

text: prime

text: satisfying

text: this

text: criterion

text: must

text: in

text: fact

text: divide

text: some

text: Harmonic

text: number

text: H

text: n

text: of

text: smaller

text: index

text: n

text: (p

text: since

text: by

text: symmetry

text: −Hn

text: mod

text: p.

text: The

text: inequality

text: sign

text: in

text: this

text: condition

text: is

text: strict

text: because

text: if

text: then

text: is

text: n

text: p

text: p

text: a

text: Wieferich

text: prime

text: and

text: by

text: and

text: must

text: likewise

text: divide

text: the

text: smaller

text: But

text: this

text: inequality

text: also

text: gives

text: a

text: sharp

text: upper

text: bound

text: on

text: the

text: index

text: of

text: the

text: least

text: Harmonic

text: number

text: divisible

text: by

text: an

text: Harmonic

text: irregular

text: prime

text: since

text: among

text: the

text: p

text: ﬁrst

text: primes

text: we

text: have

text: for

text: (see

text: OEIS

text: n

text: p

text: p

text: In

text: general,

text: there

text: does

text: not

text: seem

text: to

text: be

text: any

text: way

text: of

text: recognizing

text: an

text: Harmonic

text: irregular

text: prime

text: other

text: than

text: by

text: exhaustively

text: testing

text: it

text: as

text: a

text: divisor

text: of

text: Harmonic

text: numbers

text: with

text: (though

text: one

text: would

text: of

text: course

text: not

text: attempt

text: to

text: do

text: H

text: n

text: p

text: n

text: this

text: by

text: actual

text: division

text: except

text: for

text: very

text: small

text: p

text: In

text: turn,

text: our

text: results

text: entail

text: only

text: a

text: minority

text: of

text: the

text: Harmonic

text: irregular

text: primes,

text: b

text: c

text: because

text: we

text: must

text: have

text: for

text: some

text: Nonetheless,

text: for

text: some

text: small

text: m

text: p/N

text: N

text: primes

text: this

text: relationship

text: is

text: satisﬁed

text: by

text: more

text: than

text: one

text: value

text: of

text: for

text: example,

text: N

text: b

text: c

text: b

text: c

text: b

text: c

text: b

text: c

text: b

text: c

text: with

text: and

text: p

text: p

text: p/

text: p/

text: p/

text: p/

text: p/

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 15 10 5 5 10 15 20 25 30 35 Figure 1: Least divisors 319 900 000 of Harmonic numbers for p < , , Hb c p/N = 2 through 46, arranged in ascending order of ; horizontal N p axis = order of discovery; vertical axis = log . p for the sake of brevity we focus mainly on the results of successful searches. In ≡ [4], we noted that a statement in pp. 389–390 of [3], implying that 0 Hb c p/N mod has a solution 2 000 for every between 2 and 46 other than 5, is p p < , N incorrect (at least if we require to ensure that is not a vacuous p > N Hb c p/N sum). Indeed, as reported below, for = 17 18 20 29 31 43 there are no N , , , , , solutions with 319 900 000, and for = 5 12 there are no solutions with p < , , N , 9 520 000 000 000. p < , , , , The divisors of and are the Wieferich primes (OEIS A001220). p Hbp/2c Hbp/4c The ﬁrst of these, 1093, was found in 1913 by Meissner, and the second, 3511, in 1922 by Beeger. The Wieferich primes have inspired some of the most intensive numerical searches ever conducted, but no further instances have been found. The current search record doubtless belongs to the ongoing test by PrimeGrid [20], which had reached 484 357 700 000 000 000 as of 6 October 2015. p < , , , , , The divisors of are the Mirimanoﬀ primes (OEIS A014127). The p Hbp/3c ﬁrst of these, 11, was announced by Jacobi in 1828 [12], the second, 1006003, by Kloss in 1965 [13]. No further instances have been found. The current search record appears to be 970 453 984 500 000, held by Dorais and Klyve [6]. p < , , , , We have found no computational literature on apart from the negative Hbp/5c result in [3] that it does not vanish modulo for 2 000, though inevitably, p p < , reports of negative results are more diﬃcult to locate. One wonders why such results were not reported by Schwindt [24], since he could have obtained them − as a byproduct of his study of the interval . In any case, we Hbp/5c Hbp/6c have tested to a much greater limit than that attainable by Schwindt’s Hbp/5c method, without ﬁnding any instance where it vanishes modulo . p The divisors of (OEIS A238201, but ignoring solutions with 6) Hbp/6c p < were investigated in 1983 by Schwindt [24], whose study employed a rather na´’ive 10 ##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##

font: default

text: Figure

text: Least

label: caption

text: divisors

text: of

text: Harmonic

text: numbers

text: for

text: p

text: Hb

text: c

text: p/N

text: through

text: arranged

text: in

text: ascending

text: order

text: of

text: horizontal

text: N

text: p

text: axis

text: order

text: of

text: discovery;

text: vertical

text: axis

text: log

text: p

text: for

text: the

text: sake

text: of

text: brevity

text: we

text: focus

text: mainly

text: on

text: the

text: results

text: of

text: successful

text: searches.

text: In

text: we

text: noted

text: that

text: a

text: statement

text: in

text: pp.

text: of

text: implying

text: that

text: Hb

text: c

text: p/N

text: mod

text: has

text: a

text: solution

text: for

text: every

text: between

text: and

text: other

text: than

text: is

text: p

text: p

text: N

text: incorrect

text: (at

text: least

text: if

text: we

text: require

text: to

text: ensure

text: that

text: is

text: not

text: a

text: vacuous

text: p

text: N

text: Hb

text: c

text: p/N

text: sum).

text: Indeed,

text: as

text: reported

text: below,

text: for

text: there

text: are

text: no

text: N

text: solutions

text: with

text: and

text: for

text: there

text: are

text: no

text: solutions

text: with

text: p

text: N

text: p

text: The

text: divisors

text: of

text: and

text: are

text: the

text: Wieferich

text: primes

text: (OEIS

text: p

text: The

text: ﬁrst

text: of

text: these,

text: was

text: found

text: in

text: by

text: Meissner,

text: and

text: the

text: second,

text: in

text: by

text: Beeger.

text: The

text: Wieferich

text: primes

text: have

text: inspired

text: some

text: of

text: the

text: most

text: intensive

text: numerical

text: searches

text: ever

text: conducted,

text: but

text: no

text: further

text: instances

text: have

text: been

text: found.

text: The

text: current

text: search

text: record

text: doubtless

text: belongs

text: to

text: the

text: ongoing

text: test

text: by

text: PrimeGrid

text: which

text: had

text: reached

text: as

text: of

text: October

text: p

text: The

text: divisors

text: of

text: are

text: the

text: Mirimanoﬀ

text: primes

text: (OEIS

text: The

text: p

text: ﬁrst

text: of

text: these,

text: was

text: announced

text: by

text: Jacobi

text: in

text: the

text: second,

text: by

text: Kloss

text: in

text: No

text: further

text: instances

text: have

text: been

text: found.

text: The

text: current

text: search

text: record

text: appears

text: to

text: be

text: held

text: by

text: Dorais

text: and

text: Klyve

text: p

text: We

text: have

text: found

text: no

text: computational

text: literature

text: on

text: apart

text: from

text: the

text: negative

text: result

text: in

text: that

text: it

text: does

text: not

text: vanish

text: modulo

text: for

text: though

text: inevitably,

text: p

text: p

text: reports

text: of

text: negative

text: results

text: are

text: more

text: diﬃcult

text: to

text: locate.

text: One

text: wonders

text: why

text: such

text: results

text: were

text: not

text: reported

text: by

text: Schwindt

text: since

text: he

text: could

text: have

text: obtained

text: them

text: as

text: a

text: byproduct

text: of

text: his

text: study

text: of

text: the

text: interval

text: In

text: any

text: case,

text: we

text: have

text: tested

text: to

text: a

text: much

text: greater

text: limit

text: than

text: that

text: attainable

text: by

text: Schwindt’s

text: method,

text: without

text: ﬁnding

text: any

text: instance

text: where

text: it

text: vanishes

text: modulo

text: p

text: The

text: divisors

text: of

text: (OEIS

text: but

text: ignoring

text: solutions

text: with

text: p

text: were

text: investigated

text: in

text: by

text: Schwindt

text: whose

text: study

text: employed

text: a

text: rather

text: na´’ive

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text: Improving the Description of Nonmagnetic and Magnetic Molecular Crystals ∗ via the van der Waals Density Functional Masao Obata, Makoto Nakamura, and Ikutaro Hamada † ‡ Tatsuki Oda † and ‡ 1 We have derived and implemented a stress tensor formulation for the van der Waals density functional (vdW-DF) with spin-polarization-dependent gradient correction (GC) recently proposed by the authors [J. Phys. Soc. Jpn. 82, 093701 (2013)] and applied it to non- magnetic and magnetic molecular crystals under ambient condition. We found that the cell parameters of the molecular crystals obtained with vdW-DF show an overall improvement compared with those obtained using local density and generalized gradient approximations. In particular, the original vdW-DF with GC gives the equilibrium structural parameters of solid oxygen in the α-phase, which are in good agreement with the experiment.

text: Improving

label: title

text: the

text: Description

text: of

text: Nonmagnetic

text: and

text: Magnetic

text: Molecular

text: Crystals

text: via

text: the

text: van

text: der

text: Waals

text: Density

text: Functional

text: Masao

label: author

text: Obata,

text: Makoto

text: Nakamura,

text: and

text: Ikutaro

text: Hamada

text: Tatsuki

text: Oda

text: and

text: We

label: abstract

text: have

text: derived

text: and

text: implemented

text: a

text: stress

text: tensor

text: formulation

text: for

text: the

text: van

text: der

text: Waals

text: density

text: functional

text: (vdW-DF)

text: with

text: spin-polarization-dependent

text: gradient

text: correction

text: (GC)

text: recently

text: proposed

text: by

text: the

text: authors

text: [J.

text: Phys.

text: Soc.

text: Jpn.

text: and

text: applied

text: it

text: to

text: non-

text: magnetic

text: and

text: magnetic

text: molecular

text: crystals

text: under

text: ambient

text: condition.

text: We

text: found

text: that

text: the

text: cell

text: parameters

text: of

text: the

text: molecular

text: crystals

text: obtained

text: with

text: vdW-DF

text: show

text: an

text: overall

text: improvement

text: compared

text: with

text: those

text: obtained

text: using

text: local

text: density

text: and

text: generalized

text: gradient

text: approximations.

text: In

text: particular,

text: the

text: original

text: vdW-DF

text: with

text: GC

text: gives

text: the

text: equilibrium

text: structural

text: parameters

text: of

text: solid

text: oxygen

text: in

text: the

text: α-phase,

text: which

text: are

text: in

text: good

text: agreement

text: with

text: the

text: experiment.


text: 0q fo noitubirtsiD )mota/Ve( ygrene noitalerroc lacolnoN 20 vdW−DF 00 .0.12.525 aa== 150.2.598ÅÅ DD((qqαα)) εεccnnll 011..82 0.6 0.1 0.4 0.05 0.2 0 0 0 1 2 3 4 5 6 7 8 vdW−DF2 00 .0.12.525 aa== 150.2.598ÅÅ DD((qqαα)) εεccnnll 011..82 0.6 0.1 0.4 0.05 0.2 0 0 0 1 2 3 4 5 6 7 8 q α FIG. 5. (Color online) Distributions of q function (solid curves) and εnl(α) (dotted curves) for fcc argon 0 c calculated using vdW-DF (upper panel) and vdW-DF2 (lower panel). The red open and blue ﬁlled symbols are for the lattice parameters (a) of 5.29 and 10.58 ˚A, respectively. (cid:88)   Enl = εnl(α), c (cid:88)c (cid:88) α   Ω | | ∗ εnl(α) = η (G) 2φ (G) + 2 Re(η (G)η (G))φ (G) , (C1) c 2 α αα α β αβ G β(<α) where εnl(α) is a positive value. εnl(α)’s for fcc argon calculated using vdW-DF and vdW-DF2 c c are shown in Fig. 5. There is almost no contribution of the nonlocal correlation energy at the smallest q , although the samples of D(q ) exist there. Note that the samples at the smallest q α α α come from diluted electron densities. In the case of a = 5.29 ˚A, εnl(α)’s have a peak at q = 2.4 c and are damped rapidly as q increases. In the case of a = 10.58 ˚A, the shrunken electron cloud α around atomic cores causes the distributions of D(q ) at larger q ’s(> 5). However, there is no α α energy contribution from the Enl term. This is presumably because the vdW interaction is both c canceled out in the atom and fully damped at the atomic distances. From the above result, it is conﬁrmed that the q mesh deﬁned in the present work covers the energy range in the system. α ∗ † refercolor ##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##

text: fo

text: noitubirtsiD

text: )mota/Ve(

text: ygrene

text: noitalerroc

text: lacolnoN

text: vdW−DF

text: aa==

text: DD((qqαα))

text: εεccnnll

text: aa==

text: DD((qqαα))

text: εεccnnll

text: q

text: α

text: FIG.

text: (Color

text: online)

text: Distributions

text: of

text: q

text: function

text: (solid

text: curves)

text: and

text: εnl(α)

text: (dotted

text: curves)

text: for

text: fcc

text: argon

text: c

text: calculated

text: using

text: vdW-DF

text: (upper

text: panel)

text: and

text: (lower

text: panel).

text: The

text: red

text: open

text: and

text: blue

text: ﬁlled

text: symbols

text: are

text: for

text: the

text: lattice

text: parameters

text: (a)

text: of

text: and

text: ˚A,

text: respectively.

text: Enl

text: εnl(α),

text: c

text: α

text: Ω

text: εnl(α)

text: η

text: (G)

text: 2φ

text: (G)

text: Re(η

text: (G)η

text: (G))φ

text: (G)

text: c

text: α

text: αα

text: α

text: β

text: αβ

text: G

text: β(<α)

text: where

text: εnl(α)

text: is

text: a

text: positive

text: value.

text: εnl(α)’s

text: for

text: fcc

text: argon

text: calculated

text: using

text: vdW-DF

text: and

text: c

text: c

text: are

text: shown

text: in

text: Fig.

text: There

text: is

text: almost

text: no

text: contribution

text: of

text: the

text: nonlocal

text: correlation

text: energy

text: at

text: the

text: smallest

text: q

text: although

text: the

text: samples

text: of

text: D(q

text: exist

text: there.

text: Note

text: that

text: the

text: samples

text: at

text: the

text: smallest

text: q

text: α

text: α

text: α

text: come

text: from

text: diluted

text: electron

text: densities.

text: In

text: the

text: case

text: of

text: a

text: ˚A,

text: εnl(α)’s

text: have

text: a

text: peak

text: at

text: q

text: c

text: and

text: are

text: damped

text: rapidly

text: as

text: q

text: increases.

text: In

text: the

text: case

text: of

text: a

text: ˚A,

text: the

text: shrunken

text: electron

text: cloud

text: α

text: around

text: atomic

text: cores

text: causes

text: the

text: distributions

text: of

text: D(q

text: at

text: larger

text: q

text: ’s(>

text: However,

text: there

text: is

text: no

text: α

text: α

text: energy

text: contribution

text: from

text: the

text: Enl

text: term.

text: This

text: is

text: presumably

text: because

text: the

text: vdW

text: interaction

text: is

text: both

text: c

text: canceled

text: out

text: in

text: the

text: atom

text: and

text: fully

text: damped

text: at

text: the

text: atomic

text: distances.

text: From

text: the

text: above

text: result,

text: it

text: is

text: conﬁrmed

text: that

text: the

text: q

text: mesh

text: deﬁned

text: in

text: the

text: present

text: work

text: covers

text: the

text: energy

text: range

text: in

text: the

text: system.

text: α

text: refercolor

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text: 23 64 H. E. Swanson, N. T. Gilfrich, and G. M. Ugrinic , National Bureau of Standerds Circular 539, Vol. 5 (U. S. Government Printing Oﬃce, Washington, D. C., 1955) p. 54. 65 P. Cherin and P. Unger, Inorg. Chem. 6, 1589 (1967). 66 A. D. Corso and R. Resta, Phys. Rev. B 50, 4327 (1994). 67 G. Kresse, J. Furthmu¨ller, and J. Hafner, Phys. Rev. B 50, 13181 (1994). 68 T. Buˇcko, J. Hafner, S. Leb`egue, and J. G. A´ngy´an, J. Phys. Chem. A 114, 11814 (2010). 69 O. Sode, M. Kec¸, K. Yagi, and S. Hirata, J. Chem. Phys. 138, 074501 (2013). 70 A. Simon and K. Peters, Acta Cryst. B 36, 2750 (1980). 71 A. Otero-de-la-Roza and E. R. Johnson, J. Chem. Phys. 137, 054103 (2012). 72 S. A. Bonev, F. Gygi, T. Ogitsu, and G. Galli, Phys. Rev. Lett. 91, 065501 (2003). 73 I. Hamada, private communication. 74 F. Corsetti, E. Artacho, J. M. Soler, S. S. Alexandre, and M.-V. Fernandez-Serra, J. Chem. Phys. 139, 194502 (2013).

text: H.

text: E.

text: Swanson,

text: N.

text: T.

text: Gilfrich,

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text: M.

text: Ugrinic

text: National

text: Bureau

text: of

text: Standerds

text: Circular

text: Vol.

text: (U.

text: S.

text: Government

text: Printing

text: Oﬃce,

text: Washington,

text: D.

text: C.,

text: p.

text: P.

text: Cherin

text: and

text: P.

text: Unger,

text: Inorg.

text: Chem.

text: A.

text: D.

text: Corso

text: and

text: R.

text: Resta,

text: Phys.

text: Rev.

text: B

text: G.

text: Kresse,

text: J.

text: Furthmu¨ller,

text: and

text: J.

text: Hafner,

text: Phys.

text: Rev.

text: B

text: T.

text: Buˇcko,

text: J.

text: Hafner,

text: S.

text: Leb`egue,

text: and

text: J.

text: G.

text: A´ngy´an,

text: J.

text: Phys.

text: Chem.

text: A

text: O.

text: Sode,

text: M.

text: Kec¸,

text: K.

text: Yagi,

text: and

text: S.

text: Hirata,

text: J.

text: Chem.

text: Phys.

text: A.

text: Simon

text: and

text: K.

text: Peters,

text: Acta

text: Cryst.

text: B

text: A.

text: Otero-de-la-Roza

text: and

text: E.

text: R.

text: Johnson,

text: J.

text: Chem.

text: Phys.

text: S.

text: A.

text: Bonev,

text: F.

text: Gygi,

text: T.

text: Ogitsu,

text: and

text: G.

text: Galli,

text: Phys.

text: Rev.

text: Lett.

text: I.

text: Hamada,

text: private

text: communication.

text: F.

text: Corsetti,

text: E.

text: Artacho,

text: J.

text: M.

text: Soler,

text: S.

text: S.

text: Alexandre,

text: and

text: M.-V.

text: Fernandez-Serra,

text: J.

text: Chem.

text: Phys.


text: ENTROPY AND DRIFT IN WORD HYPERBOLIC GROUPS SÉBASTIEN GOUËZEL, FRÉDÉRIC MATHÉUS, FRANÇOIS MAUCOURANT Abstract. The fundamental inequality of Guivarc’h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary hyperbolic group which is not virtually free, endowed with a word distance, the fundamental inequality is strict for symmetric measures with ﬁnite support, uniformly for measures with a given support. This answers a conjecture of S. Lalley. For admissible measures, this is proved using previous results of Ancona and Blachère-Haïssinsky-Mathieu. For non-admissible measures, this follows from a counting result, interesting in its own right: we show that, in any inﬁnite index subgroup, the number of non-distorted points is exponentially small. The uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup. 1. Main results Let Γ be a ﬁnitely generated inﬁnite group. Although the following discussion makes sense in a much broader context, we will assume that Γ is hyperbolic since all results of this article are devoted to this setting. There are two natural ways to construct random elements in Γ: • Let d be a proper left-invariant distance on Γ (for instance a word distance). For large n, one can pick an element at random with respect to the uniform measure ρ n on the ball B = B(e, n) (where e denotes the identity of Γ). • n Let µ be a probability measure on Γ. For large n, one can pick an element at ∗ random with respect to the measure µ n (the n-th convolution of the measure µ). Equivalently, let g , g , . . . be a sequence of random elements of Γ that are distributed 1 2 · · · independently according to µ. Form the random walk X = g g . Then the ∗ n 1 n distribution of X is µ n. n From a theoretical point of view, these methods share a lot of properties. From a compu- tational point of view, the second method is much easier to implement in general groups since it does not require the computation of the ball B (note however that, in hyperbolic n groups, simulating the uniform measure is very easy thanks to the automatic structure of the group). It is therefore of interest to ﬁnd probability measures µ such that these two methods give equivalent results, in a sense that will be made precise below. This is the main question of Vershik in [Ver00]. In free groups (with the word distance coming from the usual set of generators), everything can be computed: if µ is the uniform measure on the ∗ generators, then µ n and ρ behave essentially in the same way. The situation is the same n in free products of ﬁnite groups, again thanks to the underlying tree structure. However, in more complicated groups, explicit computations are essentially impossible, and it is expected Date: January 20, 2015. 1 ##LTLine##

text: ENTROPY

text: AND

text: DRIFT

text: IN

text: WORD

text: HYPERBOLIC

text: GROUPS

text: SÉBASTIEN

text: GOUËZEL,

text: FRÉDÉRIC

text: MATHÉUS,

text: FRANÇOIS

text: MAUCOURANT

text: Abstract.

text: The

text: fundamental

text: inequality

text: of

text: Guivarc’h

text: relates

text: the

text: entropy

text: and

text: the

text: drift

text: of

text: random

text: walks

text: on

text: groups.

text: It

text: is

text: strict

text: if

text: and

text: only

text: if

text: the

text: random

text: walk

text: does

text: not

text: behave

text: like

text: the

text: uniform

text: measure

text: on

text: balls.

text: We

text: prove

text: that,

text: in

text: any

text: nonelementary

text: hyperbolic

text: group

text: which

text: is

text: not

text: virtually

text: free,

text: endowed

text: with

text: a

text: word

text: distance,

text: the

text: fundamental

text: inequality

text: is

text: strict

text: for

text: symmetric

text: measures

text: with

text: ﬁnite

text: support,

text: uniformly

text: for

text: measures

text: with

text: a

text: given

text: support.

text: This

text: answers

text: a

text: conjecture

text: of

text: S.

text: Lalley.

text: For

text: admissible

text: measures,

text: this

text: is

text: proved

text: using

text: previous

text: results

text: of

text: Ancona

text: and

text: Blachère-Haïssinsky-Mathieu.

text: For

text: non-admissible

text: measures,

text: this

text: follows

text: from

text: a

text: counting

text: result,

text: interesting

text: in

text: its

text: own

text: right:

text: we

text: show

text: that,

text: in

text: any

text: inﬁnite

text: index

text: subgroup,

text: the

text: number

text: of

text: non-distorted

text: points

text: is

text: exponentially

text: small.

text: The

text: uniformity

text: is

text: obtained

text: by

text: studying

text: the

text: behavior

text: of

text: measures

text: that

text: degenerate

text: towards

text: a

text: measure

text: supported

text: on

text: an

text: elementary

text: subgroup.

text: Main

text: results

text: Let

text: Γ

text: be

text: a

text: ﬁnitely

text: generated

text: inﬁnite

text: group.

text: Although

text: the

text: following

text: discussion

text: makes

text: sense

text: in

text: a

text: much

text: broader

text: context,

text: we

text: will

text: assume

text: that

text: Γ

text: is

text: hyperbolic

text: since

text: all

text: results

text: of

text: this

text: article

text: are

text: devoted

text: to

text: this

text: setting.

text: There

text: are

text: two

text: natural

text: ways

text: to

text: construct

text: random

text: elements

text: in

text: Γ:

text: Let

text: d

text: be

text: a

text: proper

text: left-invariant

text: distance

text: on

text: Γ

text: (for

text: instance

text: a

text: word

text: distance).

text: For

text: large

text: n,

text: one

text: can

text: pick

text: an

text: element

text: at

text: random

text: with

text: respect

text: to

text: the

text: uniform

text: measure

text: ρ

text: n

text: on

text: the

text: ball

text: B

text: B(e,

text: n)

text: (where

text: e

text: denotes

text: the

text: identity

text: of

text: Γ).

text: n

text: Let

text: µ

text: be

text: a

text: probability

text: measure

text: on

text: Γ.

text: For

text: large

text: n,

text: one

text: can

text: pick

text: an

text: element

text: at

text: random

text: with

text: respect

text: to

text: the

text: measure

text: µ

text: n

text: (the

text: n-th

text: convolution

text: of

text: the

text: measure

text: µ).

text: Equivalently,

text: let

text: g

text: g

text: be

text: a

text: sequence

text: of

text: random

text: elements

text: of

text: Γ

text: that

text: are

text: distributed

text: independently

text: according

text: to

text: µ.

text: Form

text: the

text: random

text: walk

text: X

text: g

text: g

text: Then

text: the

text: n

text: n

text: distribution

text: of

text: X

text: is

text: µ

text: n.

text: n

text: From

text: a

text: theoretical

text: point

text: of

text: view,

text: these

text: methods

text: share

text: a

text: lot

text: of

text: properties.

text: From

text: a

text: compu-

text: tational

text: point

text: of

text: view,

text: the

text: second

text: method

text: is

text: much

text: easier

text: to

text: implement

text: in

text: general

text: groups

text: since

text: it

text: does

text: not

text: require

text: the

text: computation

text: of

text: the

text: ball

text: B

text: (note

text: however

text: that,

text: in

text: hyperbolic

text: n

text: groups,

text: simulating

text: the

text: uniform

text: measure

text: is

text: very

text: easy

text: thanks

text: to

text: the

text: automatic

text: structure

text: of

text: the

text: group).

text: It

text: is

text: therefore

text: of

text: interest

text: to

text: ﬁnd

text: probability

text: measures

text: µ

text: such

text: that

text: these

text: two

text: methods

text: give

text: equivalent

text: results,

text: in

text: a

text: sense

text: that

text: will

text: be

text: made

text: precise

text: below.

text: This

text: is

text: the

text: main

text: question

text: of

text: Vershik

text: in

text: In

text: free

text: groups

text: (with

text: the

text: word

text: distance

text: coming

text: from

text: the

text: usual

text: set

text: of

text: generators),

text: everything

text: can

text: be

text: computed:

text: if

text: µ

text: is

text: the

text: uniform

text: measure

text: on

text: the

text: generators,

text: then

text: µ

text: n

text: and

text: ρ

text: behave

text: essentially

text: in

text: the

text: same

text: way.

text: The

text: situation

text: is

text: the

text: same

text: n

text: in

text: free

text: products

text: of

text: ﬁnite

text: groups,

text: again

text: thanks

text: to

text: the

text: underlying

text: tree

text: structure.

text: However,

text: in

text: more

text: complicated

text: groups,

text: explicit

text: computations

text: are

text: essentially

text: impossible,

text: and

text: it

text: is

text: expected

text: Date:

text: January

text: ##LTLine##