text: VIRIAL EXPANSION BOUNDS THROUGH TREE PARTITION SCHEMES 7 5. Penrose is the same as Groeneveld R − In [11], Groeneveld obtains, for stable potentials, the bound > α(u)C(β) 1, − Vir where α(u) is the smallest positive root of α(u)e α(u) = 1 . His method relies (1+u)e upon ﬁnding recursion relations for graphs, expressing these as a system of diﬀeren- tial equations in three variables for multivariable generating functions and solving these equations in order to obtain an expression for the virial coeﬃcients. Here we indicate how our result actually gives the same. In order to show that the two bounds are equivalent, we use the fact that the (cid:55)→ − rooted tree generating function is the inverse of s se s , which can be found in [7]. The function T (z) as deﬁned in (7.12) is shown to be equal to T (z) in 1 Pen,1 section and we use in this section two identities proved in section for T (z), Pen,1 but we write these in terms of T (z) to maintain consistency with the notation of 1 the previous section. − Lemma 5.1. The derivative of the generating function T (z), evaluated at se s 1 satisﬁes the following identity: (cid:48) − 1 T (se s) = − . (5.1) 1 1 s (cid:48) (cid:55)→ − Proof. We use the relation (7.30): T (z) = − 1• . If we change variables z se s • − 1 1 T (z) and use that T (se s) = s, then we easily obtain: (cid:48) − 1 T (se s) = − . (5.2) 1 1 s (cid:3) − Lemma 5.2. The generating function T (z) evaluated at se s satisﬁes: 1 − − − T (se s) = 1 e s. (5.3) 1 − (cid:55)→ Pr−oof. We know from (7.28) that T1(z) = 1 T•z(z), making the substitution z se s, we immediately get: − − − T (se s) = 1 e s. (5.4) 1 (cid:3) Proposition 5.3. The lower bound of the radius of convergence of the virial ex- pansion: R ≥ − c(u) C(β) 1 (5.5) Vir 1 + 1 T (uc(u)) u 1 (cid:48) − where c(u) is the smallest positive solution to uc(u)T (uc(u)) T (c(u)) = 1 is equal − 1 1 − to α(u)C(β) 1, where α(u) is the smallest positive solution to α(u)e α(u) = 1 . (1+u)e ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: VIRIAL

text: EXPANSION

text: BOUNDS

text: THROUGH

text: TREE

text: PARTITION

text: SCHEMES

text: Penrose

text: is

text: the

text: same

text: as

text: Groeneveld

text: R

text: In

text: Groeneveld

text: obtains,

text: for

text: stable

text: potentials,

text: the

text: bound

text: α(u)C(β)

text: Vir

text: where

text: α(u)

text: is

text: the

text: smallest

text: positive

text: root

text: of

text: α(u)e

text: α(u)

text: His

text: method

text: relies

text: upon

text: ﬁnding

text: recursion

text: relations

text: for

text: graphs,

text: expressing

text: these

text: as

text: a

text: system

text: of

text: diﬀeren-

text: tial

text: equations

text: in

text: three

text: variables

text: for

text: multivariable

text: generating

text: functions

text: and

text: solving

text: these

text: equations

text: in

text: order

text: to

text: obtain

text: an

text: expression

text: for

text: the

text: virial

text: coeﬃcients.

text: Here

text: we

text: indicate

text: how

text: our

text: result

text: actually

text: gives

text: the

text: same.

text: In

text: order

text: to

text: show

text: that

text: the

text: two

text: bounds

text: are

text: equivalent,

text: we

text: use

text: the

text: fact

text: that

text: the

text: rooted

text: tree

text: generating

text: function

text: is

text: the

text: inverse

text: of

text: s

text: se

text: s

text: which

text: can

text: be

text: found

text: in

text: The

text: function

text: T

text: (z)

text: as

text: deﬁned

text: in

text: is

text: shown

text: to

text: be

text: equal

text: to

text: T

text: (z)

text: in

text: section

text: and

text: we

text: use

text: in

text: this

text: section

text: two

text: identities

text: proved

text: in

text: section

text: for

text: T

text: (z),

text: but

text: we

text: write

text: these

text: in

text: terms

text: of

text: T

text: (z)

text: to

text: maintain

text: consistency

text: with

text: the

text: notation

text: of

text: the

text: previous

text: section.

text: Lemma

text: The

text: derivative

text: of

text: the

text: generating

text: function

text: T

text: (z),

text: evaluated

text: at

text: se

text: s

text: satisﬁes

text: the

text: following

text: identity:

text: T

text: (se

text: s)

text: s

text: Proof.

text: We

text: use

text: the

text: relation

text: T

text: (z)

text: 1•

text: If

text: we

text: change

text: variables

text: z

text: se

text: s

text: T

text: (z)

text: and

text: use

text: that

text: T

text: (se

text: s)

text: s,

text: then

text: we

text: easily

text: obtain:

text: T

text: (se

text: s)

text: s

text: Lemma

text: The

text: generating

text: function

text: T

text: (z)

text: evaluated

text: at

text: se

text: s

text: satisﬁes:

text: T

text: (se

text: s)

text: e

text: s.

text: Pr−oof.

text: We

text: know

text: from

text: that

text: T•z(z),

text: making

text: the

text: substitution

text: z

text: se

text: s,

text: we

text: immediately

text: get:

text: T

text: (se

text: s)

text: e

text: s.

text: Proposition

text: The

text: lower

text: bound

text: of

text: the

text: radius

text: of

text: convergence

text: of

text: the

text: virial

text: ex-

text: pansion:

text: R

text: c(u)

text: C(β)

text: Vir

text: T

text: (uc(u))

text: u

text: where

text: c(u)

text: is

text: the

text: smallest

text: positive

text: solution

text: to

text: uc(u)T

text: (uc(u))

text: T

text: (c(u))

text: is

text: equal

text: to

text: α(u)C(β)

text: where

text: α(u)

text: is

text: the

text: smallest

text: positive

text: solution

text: to

text: α(u)e

text: α(u)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: | Figure 14. The directional dependence of kSZ-galaxy cross correlation f(θ (cid:96)). The panels are for e [0.01, 0.3] (bottom left), , [0.9, 1.1] (bottom right), [1.9, 2.1] (top left) and [3.9, 4.1] (top right). In each panel, there are four curves for (cid:96) = 1000 (solid), 3000 (dot), 9000 (short dash) and the approximation (cid:54) Eq. 4.2 (long dash), respectively. All the curves with A = 0 and B = 0 are normalized to unity at ◦ (cid:54) − ◦ θ = 0 . For clarity, curves with A = 0 and B = 0 are normalized to 1 at θ = 0 . One exception is e ◦ (cid:39) e ◦ the redshift bin [0.9, 1.1]. It has ∆T(θ = 0 ) 0. So we normalize with the value at θ = 45 instead. e e Curves of diﬀerent (cid:96) overlap over each other and are barely distinguishable against each other. This | | (cid:39) shows that (1) f(θ (cid:96)) has negligible (cid:96) dependence (f(θ (cid:96)) f(θ )) and (2) the approximation Eq. e e e 4.2 describes f(θ ) to excellent accuracy. e the noise term. We discuss two limiting cases. One is the optimistic case that all large scale structure related components (other than kSZ) in the CMB maps have been removed without b(cid:104)ias (cid:105)by combining multiple frequency bands. I(cid:88)n this limit, the expectation value of Ni is zero ( N = 0). The r.m.s error of the weighted correlation is i (cid:104) (cid:105) NW,2 = W 2σ2 . (4.4) i i i ≡ (cid:104) (cid:105) Here, σ2 N2 . The normalization of W is irrele(cid:88)vant. However it is convenient to ﬁx it in i order to ﬁnd the optimal weighting. We choose the normalization ⇒ CW ((cid:96)) = C ((cid:96)) W f = 1 . (4.5) pivot i i i (cid:80) The optimal weighting is then f /σ2 W = i i . (4.6) i f2/σ2 i i The second case is more conservative, and perhaps more realistic. In this case, there are residual large scale structure related components other than kSZ in the CMB maps. Therefore – 19 – ##LTFigure## ##LTLine##

label: caption

text: Figure

label: paragraph

text: The

text: directional

text: dependence

text: of

text: kSZ-galaxy

text: cross

text: correlation

text: f(θ

text: The

text: panels

text: are

text: for

text: e

text: (bottom

text: left),

text: (bottom

text: right),

text: (top

text: left)

text: and

text: (top

text: right).

text: In

text: each

text: panel,

text: there

text: are

text: four

text: curves

text: for

text: (solid),

text: (dot),

text: (short

text: dash)

text: and

text: the

text: approximation

font: default

text: Eq.

text: (long

text: dash),

text: respectively.

text: All

text: the

text: curves

text: with

text: A

text: and

text: B

text: are

text: normalized

text: to

text: unity

text: at

text: θ

text: For

text: clarity,

text: curves

text: with

text: A

text: and

text: B

text: are

text: normalized

text: to

text: at

text: θ

text: One

text: exception

text: is

text: e

text: e

text: the

text: redshift

text: bin

text: It

text: has

text: ∆T(θ

text: So

text: we

text: normalize

text: with

text: the

text: value

text: at

text: θ

text: instead.

text: e

text: e

text: Curves

text: of

text: diﬀerent

text: overlap

text: over

text: each

text: other

text: and

text: are

text: barely

text: distinguishable

text: against

text: each

text: other.

text: This

text: shows

text: that

text: f(θ

text: has

text: negligible

text: dependence

text: (f(θ

text: f(θ

text: and

text: the

text: approximation

text: Eq.

text: e

text: e

text: e

text: describes

text: f(θ

text: to

text: excellent

text: accuracy.

text: e

text: the

text: noise

text: term.

text: We

text: discuss

text: two

text: limiting

text: cases.

text: One

text: is

text: the

text: optimistic

text: case

text: that

text: all

text: large

text: scale

text: structure

text: related

text: components

text: (other

text: than

text: kSZ)

text: in

text: the

text: CMB

text: maps

text: have

text: been

text: removed

text: without

text: combining

text: multiple

text: frequency

text: bands.

text: this

text: limit,

text: the

text: expectation

text: value

text: of

text: Ni

text: is

text: zero

text: N

text: The

text: r.m.s

text: error

text: of

text: the

text: weighted

text: correlation

text: is

text: i

text: W

text: i

text: i

text: i

text: Here,

text: σ2

text: The

text: normalization

text: of

text: W

text: is

text: However

text: it

text: is

text: convenient

text: to

text: ﬁx

text: it

text: in

text: i

text: order

text: to

text: ﬁnd

text: the

text: optimal

text: weighting.

text: We

text: choose

text: the

text: normalization

text: CW

text: C

text: W

text: f

text: pivot

text: i

text: i

text: i

text: The

text: optimal

text: weighting

text: is

text: then

text: f

text: W

text: i

text: i

text: i

text: i

text: i

text: The

text: second

text: case

text: is

text: more

text: conservative,

text: and

text: perhaps

text: more

text: realistic.

text: In

text: this

text: case,

text: there

text: are

text: residual

text: large

text: scale

text: structure

text: related

text: components

text: other

text: than

text: kSZ

text: in

text: the

text: CMB

text: maps.

text: Therefore

text: ##LTFigure##

text: ##LTLine##


text: Figure 15. The forecasted constrain on bubble collision. For the future CMB experiment, we adopt (cid:48) a FWHM=2 and a noise level of 1µK per square arcminute. The overlapping sky area is taken as (cid:54) f = 0.5. Points with ﬁlled circles are for the case of A = 0 and B = 0. Points with open square are sky (cid:54) for the case of A = 0 and B = 0. For the angular resolution of Planck, primary CMB overwhelms. So Planck in combination with LSST has limited capability to detect the kSZ generated by bubble collision. A future CMB experiment with better resolution can improve the measurement by orders of magnitude. We estimate that a PRISM-like experiment has the capability of measure A and B with − ≤ r.m.s error of 10 5. Constraints on B are systematically better than constraints on A when z 4, c simply reﬂecting the fact that when A = B, the case B has on the average larger curvature potential inside of our horizon. The two constraints at z = 4 are roughly the same, for the reason that they c have roughly the same mean curvature perturbation inside of our horizon. kSZ eﬀect including patchy reionization. CFor is the power spectrum of residual foregrounds, which includes all large scale structure related contaminants such as CIB, the relativistic ther- mal SZ eﬀect and the cosmic radio background. Since this relies heavily on multi frequency information, whose eﬀectiveness depends on exposure time, this may also vary with direction. Variation of the gray-body power index and the temperature of CIB over the sky can add more directional variation. Given uncertainties in the modelling of the conventional kSZ eﬀect and foregrounds, we will adopt a ﬂat spectrum (cid:96)2(CkSZ +CFor)/(2π)(2.73K)2 = 10µK2. C is g the galaxy auto power spectrum in the given patch of the sky and the given redshift bin. CN g is the associated (statistical) noise. We only consider shot noise so that CN = 4πf /N . g sky,i g,i f is the fractional sky coverage. N is the total number of galaxies in this patch of sky,i g,i the sky in the given redshift bin. The last approximation of Eq. 4.13 holds since the kSZ eﬀect induced by bubble collision is always subdominant/negligible to sum of other CMB components/contaminations. Given that the bubble collision induced kSZ signal is weak, we require both advanced CMB experiments and galaxy surveys in order to be sensitive to bubble collisions. Fig. 15 – 21 – ##LTFigure##

text: Figure

text: The

text: forecasted

text: constrain

text: on

text: bubble

text: collision.

text: For

text: the

text: future

text: CMB

text: experiment,

text: we

text: adopt

text: a

text: and

text: a

text: noise

text: level

text: of

text: per

text: square

text: arcminute.

text: The

text: overlapping

text: sky

text: area

text: is

text: taken

text: as

text: f

text: Points

text: with

text: ﬁlled

text: circles

text: are

text: for

text: the

text: case

text: of

text: A

text: and

text: B

text: Points

text: with

text: open

text: square

text: are

text: sky

text: for

text: the

text: case

text: of

text: A

text: and

text: B

text: For

text: the

text: angular

text: resolution

text: of

text: Planck,

text: primary

text: CMB

text: overwhelms.

text: So

text: Planck

text: in

text: combination

text: with

text: LSST

text: has

text: limited

text: capability

text: to

text: detect

text: the

text: kSZ

text: generated

text: by

text: bubble

text: collision.

text: A

text: future

text: CMB

text: experiment

text: with

text: better

text: resolution

text: can

text: improve

text: the

text: measurement

text: by

text: orders

text: of

text: magnitude.

text: We

text: estimate

text: that

text: a

text: PRISM-like

text: experiment

text: has

text: the

text: capability

text: of

text: measure

text: A

text: and

text: B

text: with

text: r.m.s

text: error

text: of

text: Constraints

text: on

text: B

text: are

text: systematically

text: better

text: than

text: constraints

text: on

text: A

text: when

text: z

text: c

text: simply

text: reﬂecting

text: the

text: fact

text: that

text: when

text: A

text: B,

text: the

text: case

text: B

text: has

text: on

text: the

text: average

text: larger

text: curvature

text: potential

text: inside

text: of

text: our

text: horizon.

text: The

text: two

text: constraints

text: at

text: z

text: are

text: roughly

text: the

text: same,

text: for

text: the

text: reason

text: that

text: they

text: c

text: have

text: roughly

text: the

text: same

text: mean

text: curvature

text: perturbation

text: inside

text: of

text: our

text: horizon.

text: kSZ

text: eﬀect

text: including

text: patchy

text: reionization.

text: CFor

text: is

text: the

text: power

text: spectrum

text: of

text: residual

text: foregrounds,

text: which

text: includes

text: all

text: large

text: scale

text: structure

text: related

text: contaminants

text: such

text: as

text: CIB,

text: the

text: relativistic

text: ther-

text: mal

text: SZ

text: eﬀect

text: and

text: the

text: cosmic

text: radio

text: background.

text: Since

text: this

text: relies

text: heavily

text: on

text: multi

text: frequency

text: information,

text: whose

text: eﬀectiveness

text: depends

text: on

text: exposure

text: time,

text: this

text: may

text: also

text: vary

text: with

text: direction.

text: Variation

text: of

text: the

text: gray-body

text: power

text: index

text: and

text: the

text: temperature

text: of

text: CIB

text: over

text: the

text: sky

text: can

text: add

text: more

text: directional

text: variation.

text: Given

text: uncertainties

text: in

text: the

text: modelling

text: of

text: the

text: conventional

text: kSZ

text: eﬀect

text: and

text: foregrounds,

text: we

text: will

text: adopt

text: a

text: ﬂat

text: spectrum

text: C

text: is

text: g

text: the

text: galaxy

text: auto

text: power

text: spectrum

text: in

text: the

text: given

text: patch

text: of

text: the

text: sky

text: and

text: the

text: given

text: redshift

text: bin.

text: CN

text: g

text: is

text: the

text: associated

text: (statistical)

text: noise.

text: We

text: only

text: consider

text: shot

text: noise

text: so

text: that

text: CN

text: /N

text: g

text: sky,i

text: g,i

text: f

text: is

text: the

text: fractional

text: sky

text: coverage.

text: N

text: is

text: the

text: total

text: number

text: of

text: galaxies

text: in

text: this

text: patch

text: of

text: sky,i

text: g,i

text: the

text: sky

text: in

text: the

text: given

text: redshift

text: bin.

text: The

text: last

text: approximation

text: of

text: Eq.

text: holds

text: since

text: the

text: kSZ

text: eﬀect

text: induced

text: by

text: bubble

text: collision

text: is

text: always

text: subdominant/negligible

text: to

text: sum

text: of

text: other

text: CMB

text: components/contaminations.

text: Given

text: that

text: the

text: bubble

text: collision

text: induced

text: kSZ

text: signal

text: is

text: weak,

text: we

text: require

text: both

text: advanced

text: CMB

text: experiments

text: and

text: galaxy

text: surveys

text: in

text: order

text: to

text: be

text: sensitive

text: to

text: bubble

text: collisions.

text: Fig.

text: ##LTFigure##


text: Discrete curvature and abelian groups ∗ † ‡ ‡ Bo’az Klartag, Gady Kozma, Peter Ralli and Prasad Tetali August 27, 2015 Abstract We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called Γ -calculus (of Bakry-E´mery) seems to be that it 2 is fairly straightforward to compute this notion of curvature parameter for several speciﬁc graphs of interest – particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (`a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs – a result of independent interest. 1 Introduction For several decades now it has been a fruitful endeavour to translate notions from Riemannian geometry to graph theory. It is now clear what are the graph analogs of the laplacian, Poincar´e inequality, Harnack inequality, and many related notions. The graph point of view led to generalizations which would have been less natural in Riemannian geometry, such as β-parabolic Harnack inequalities (see, e.g., [5]), and to some counterexamples [4, 13, 22]. Despite all this progress, the graph analog of the notion of curvature remained elusive. In ´ their 1985 paper, Bakry and Emery [2] suggested a notion analogous to curvature that would work in the very general framework of a Markov semigroup (which, of course, incorporates both continuous diﬀusions and random walks on graphs). The condition was based on the ∞ Bochner formula and was denoted by CD(K, ) (for curvature-dimension) where K is a ∞ curvature parameter. A semigroup satisfying CD(K, ) is a generalization of Brownian ≥ ∞ motion on a manifold with Ricci curvature K and hence the condition CD(K, ) is ∗ Research supported in part by the European Research Council. † Research supported in part by the Israel Science Foundation and the Jesselson Foundation. ‡ Research supported in part by the NSF grant DMS-1407657. 1 ##LTLine##

text: Discrete

label: title

text: curvature

text: and

text: abelian

text: groups

label: author

text: Bo’az

text: Klartag,

text: Gady

text: Kozma,

text: Peter

text: Ralli

text: and

text: Prasad

text: Tetali

text: August

label: date

text: Abstract

text: We

label: abstract

text: study

text: a

text: natural

text: discrete

text: Bochner-type

text: inequality

text: on

text: graphs,

text: and

text: explore

text: its

text: merit

text: as

text: a

text: notion

text: of

text: “curvature”

text: in

text: discrete

text: spaces.

text: An

text: appealing

text: feature

text: of

text: this

text: discrete

text: version

text: of

text: the

text: so-called

text: Γ

text: -calculus

text: (of

text: Bakry-E´mery)

text: seems

text: to

text: be

text: that

text: it

text: is

text: fairly

text: straightforward

text: to

text: compute

text: this

text: notion

text: of

text: curvature

text: parameter

text: for

text: several

text: speciﬁc

text: graphs

text: of

text: interest

text: particularly,

text: abelian

text: groups,

text: slices

text: of

text: the

text: hypercube,

text: and

text: the

text: symmetric

text: group

text: under

text: various

text: sets

text: of

text: generators.

text: We

text: further

text: develop

text: this

text: notion

text: by

text: deriving

text: Buser-type

text: inequalities

text: (`a

text: la

text: Ledoux),

text: relating

text: functional

text: and

text: isoperimetric

text: constants

text: associated

text: with

text: a

text: graph.

text: Our

text: derivations

text: provide

text: a

text: tight

text: bound

text: on

text: the

text: Cheeger

text: constant

text: (i.e.,

text: the

text: edge-isoperimetric

text: constant)

text: in

text: terms

text: of

text: the

text: spectral

text: gap,

text: for

text: graphs

text: with

text: nonnegative

text: curvature,

text: particularly,

text: the

text: class

text: of

text: abelian

text: Cayley

text: graphs

text: a

text: result

text: of

text: independent

text: interest.

text: Introduction

text: For

text: several

text: decades

text: now

text: it

text: has

text: been

text: a

text: fruitful

text: endeavour

text: to

text: translate

text: notions

text: from

text: Riemannian

text: geometry

text: to

text: graph

text: theory.

text: It

text: is

text: now

text: clear

text: what

text: are

text: the

text: graph

text: analogs

text: of

text: the

text: laplacian,

text: Poincar´e

text: inequality,

text: Harnack

text: inequality,

text: and

text: many

text: related

text: notions.

text: The

text: graph

text: point

text: of

text: view

text: led

text: to

text: generalizations

text: which

text: would

text: have

text: been

text: less

text: natural

text: in

text: Riemannian

text: geometry,

text: such

text: as

text: β-parabolic

text: Harnack

text: inequalities

text: (see,

text: e.g.,

text: and

text: to

text: some

text: counterexamples

text: Despite

text: all

text: this

text: progress,

text: the

text: graph

text: analog

text: of

text: the

text: notion

text: of

text: curvature

text: remained

text: elusive.

text: In

text: their

text: paper,

text: Bakry

text: and

text: Emery

text: suggested

text: a

text: notion

text: analogous

text: to

text: curvature

text: that

text: would

text: work

text: in

text: the

text: very

text: general

text: framework

text: of

text: a

text: Markov

text: semigroup

text: (which,

text: of

text: course,

text: incorporates

text: both

text: continuous

text: diﬀusions

text: and

text: random

text: walks

text: on

text: graphs).

text: The

text: condition

text: was

text: based

text: on

text: the

text: Bochner

text: formula

text: and

text: was

text: denoted

text: by

text: CD(K,

text: (for

text: curvature-dimension)

text: where

text: K

text: is

text: a

text: curvature

text: parameter.

text: A

text: semigroup

text: satisfying

text: CD(K,

text: is

text: a

text: generalization

text: of

text: Brownian

text: motion

text: on

text: a

text: manifold

text: with

text: Ricci

text: curvature

text: K

text: and

text: hence

text: the

text: condition

text: CD(K,

text: is

text: Research

text: supported

text: in

text: part

text: by

text: the

text: European

text: Research

text: Council.

text: Research

text: supported

text: in

text: part

text: by

text: the

text: Israel

text: Science

text: Foundation

text: and

text: the

text: Jesselson

text: Foundation.

text: Research

text: supported

text: in

text: part

text: by

text: the

text: NSF

text: grant

text: ##LTLine##


text: ∼ (cid:54)∼ ∼ as for each pair v, w x with v w, there is exactly one u with v, w u and d(x, u) = 2. (cid:54) (Here we use the notation ∆(x, v, w) to denote the set of unordered pairs (v, w) of distinct (cid:54)∼ neighbors of x for which v w.) ∼ { } − Also notice that any v x has t( x, v ) = n 2: if v = s x, the vertices that make a (cid:54) ij (cid:54) triangle with x and v are s x when l = i and x = x , and s x when m = j and x = x . lj l i im m j Now we may compute (cid:32) (cid:33) (cid:18) (cid:19) (cid:88) (cid:88) (cid:88) 2Γ (f)(x) 2 2 − ≥ − − n 2 (f(cid:88)(v) f(w))2 + f(v) + 2 d + f(v)2 ∼ ∼ 2 ∼ (cid:54) v,w x v x v x ∆(xvw) − + 2 (f(v) f(w(cid:32)))2 (cid:33) (cid:88) (cid:88) (cid:16) (cid:17) (cid:88) ∆(vwx) 2 ≥ (cid:88)− (cid:88) (cid:88)− n (cid:88) (f(v) f(w))2 + f(v) + 1 d + f(v)2 ∼ ∼ 2 ∼ v,w x v x v x − − (cid:88) = (d 1) f(v)2 2 f(v)f(w) + f(v)2 + 2 f(v)f(w) ∼ ∼ ∼ ∼ v x v,w x v x v,w x (cid:16) −(cid:17) (cid:88) n + (1 d + ) f(v)2 2 ∼ v x n = 1 + f(v)2 . 2 ∼ v x ≥ So Ric(G) 1 + n. Together with Theorem 1.2 we get that Ric = 1 + n. 2 2 (cid:80) 2.5.2 Middle slice with adjacent transpositions { ∈ {− } } ∼ ⇐⇒ − We now consider G with V = x 1, 1 2n : x = 0 , where x y supp(x y) i i Z consists of 2 consecutive elements. Alternately, V is the set of paths in 2 that move from − (0, 0) to (2n, 0) with steps of (+1, +1) and (+1, 1), and paths x and y are neighbors if y − can be achieved by transposing an adjacent (+1, +1) and (+1, 1) in x. ≥ − − Theorem 2.8. Ric(G) 1. Further, lim Ric(G) = 1. →∞ n ∈ { ∈ { − } (cid:54) } ∈ Proof. Let x V . Let I(x) = i 1, . . . , 2n 1 : x = x , so i I if and only if we ∈ i i+1 are allowed to switch segments i and i + 1. If i I(x), denote by a x the vertex obtained | | i by making this switch. Observe I(x) = deg(x). ∈ | − | The neighbors of a x are: a (a x) = x, a (a x) for any j I(x) with i j > 1, and ∈ i |i −i | j i (cid:54) a (a x) for any j / I(x) with i j = 1 and j = 0, 2n. We calculate that deg(a x) = j i − { ∈ | − | } − − i deg(x) + 2 2# j I(x) : i j = 1 1 1 − . i=1 i=2n∈1 | − | We observe that a neighbor of the form a (a x) if j I(x) and i j > 1 will be identical j i to a (a x), and have d(x, a a x) = 2. i j j i 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: as

text: for

text: each

text: pair

text: v,

text: w

text: x

text: with

text: v

text: w,

text: there

text: is

text: exactly

text: one

text: u

text: with

text: v,

text: w

text: u

text: and

text: d(x,

text: u)

text: (Here

text: we

text: use

text: the

text: notation

text: ∆(x,

text: v,

text: w)

text: to

text: denote

text: the

text: set

text: of

text: unordered

text: pairs

text: (v,

text: w)

text: of

text: distinct

text: neighbors

text: of

text: x

text: for

text: which

text: v

text: w.)

text: Also

text: notice

text: that

text: any

text: v

text: x

text: has

text: t(

text: x,

text: v

text: n

text: if

text: v

text: s

text: x,

text: the

text: vertices

text: that

text: make

text: a

text: ij

text: triangle

text: with

text: x

text: and

text: v

text: are

text: s

text: x

text: when

text: l

text: i

text: and

text: x

text: x

text: and

text: s

text: x

text: when

text: m

text: j

text: and

text: x

text: x

text: lj

text: l

text: i

text: im

text: m

text: j

text: Now

text: we

text: may

text: compute

text: 2Γ

text: (f)(x)

text: n

text: f(v)

text: d

text: v,w

text: x

text: v

text: x

text: v

text: x

text: ∆(xvw)

text: (f(v)

text: ∆(vwx)

text: n

text: (f(v)

text: f(v)

text: d

text: v,w

text: x

text: v

text: x

text: v

text: x

text: (d

text: f(v)f(w)

text: f(v)f(w)

text: v

text: x

text: v,w

text: x

text: v

text: x

text: v,w

text: x

text: n

text: d

text: v

text: x

text: n

text: v

text: x

text: So

text: Ric(G)

text: n.

text: Together

text: with

text: Theorem

text: we

text: get

text: that

text: Ric

text: n.

text: Middle

text: slice

text: with

text: adjacent

text: transpositions

text: We

text: now

text: consider

text: G

text: with

text: V

text: x

text: x

text: where

text: x

text: y

text: supp(x

text: y)

text: i

text: i

text: Z

text: consists

text: of

text: consecutive

text: elements.

text: Alternately,

text: V

text: is

text: the

text: set

text: of

text: paths

text: in

text: that

text: move

text: from

text: to

text: with

text: steps

text: of

text: and

text: and

text: paths

text: x

text: and

text: y

text: are

text: neighbors

text: if

text: y

text: can

text: be

text: achieved

text: by

text: transposing

text: an

text: adjacent

text: and

text: in

text: x.

text: Theorem

text: Ric(G)

text: Further,

text: lim

text: Ric(G)

text: n

text: Proof.

text: Let

text: x

text: V

text: Let

text: I(x)

text: i

text: x

text: x

text: so

text: i

text: I

text: if

text: and

text: only

text: if

text: we

text: i

text: are

text: allowed

text: to

text: switch

text: segments

text: i

text: and

text: i

text: If

text: i

text: I(x),

text: denote

text: by

text: a

text: x

text: the

text: vertex

text: obtained

text: i

text: by

text: making

text: this

text: switch.

text: Observe

text: I(x)

text: deg(x).

text: The

text: neighbors

text: of

text: a

text: x

text: are:

text: a

text: (a

text: x)

text: x,

text: a

text: (a

text: x)

text: for

text: any

text: j

text: I(x)

text: with

text: i

text: j

text: and

text: i

text: |i

text: −i

text: j

text: i

text: a

text: (a

text: x)

text: for

text: any

text: j

text: I(x)

text: with

text: i

text: j

text: and

text: j

text: We

text: calculate

text: that

text: deg(a

text: x)

text: j

text: i

text: i

text: deg(x)

text: j

text: I(x)

text: i

text: j

text: We

text: observe

text: that

text: a

text: neighbor

text: of

text: the

text: form

text: a

text: (a

text: x)

text: if

text: j

text: I(x)

text: and

text: i

text: j

text: will

text: be

text: identical

text: j

text: i

text: to

text: a

text: (a

text: x),

text: and

text: have

text: d(x,

text: a

text: a

text: x)

text: i

text: j

text: j

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:80) where in the| ﬁ|rst equality, we used the ide(cid:88)ntity (1) and the fact that for any g, ≡∆g = 0. Now let V = n, and recall the Dirichlet form (with respect to the measure π 1), E − (f, f) = (f(y) f(x))2 ∼ x y (cid:88) (cid:16) (cid:88) (cid:17) (cid:88) and that E − − − ( ∆f, f) = ∆f(x) (f(x) f(y)) = (∆f(x))2 . ∼ x y x x Plugging into the above inequality gives E − − E ≥ 2 ( ∆f, f) 2K (f, f) 0, and so E ≤ E − K (f, f) ( ∆f, f) , ≥ resulting in λ K . 4 Buser-type Inequalities § The proofs in this section are a straightforward discrete version of 5 of Ledoux’s paper [23]. First we derive a key gradient estimate on the heat kernel associated with a graph, which will then be used in deriving a Buser inequality for graphs, as mentioned in the introduction. 4.1 Gradient estimates ≥ (cid:52) For t 0, we write P = exp(t ) for the heat kernel associated with the graph G. Then t R P is a positive deﬁnite matrix on V , with P being the identity matrix. Note that P t (cid:52) 0 (cid:80) (cid:52) (cid:52) t commutes with and with P , and that ∂P /∂t = P = P . Finally, the matrix P s t t t t has non-negative entries. So if f has non-negative entries, then also P (f) has non-negative → R (cid:107) (cid:107) | | t entries. For a vector f : V we write f = ( f(v) p)1/p. ≥ p (cid:112)v ∈ R → R Lemma 4.1. Suppose G has Ric(G) K for some K . Then, for any f : V and ≤ ≤ | | √ any 0 t 1/ 2K , (cid:107) − (cid:107) ≤ (cid:107) (cid:107) f P f 2 t Γ(f) . t 1 1 ≥ Note that the restriction on t applies only when K is negative: if K > 0 then Ric K ≥ implies Ric 0 and the lemma holds with no restriction on t. Proof. The proof is in three steps. Step 1. We ﬁrst prove that ≤ − Γ(P f) e 2KtP (Γ(f)) , t t 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: in

text: the|

text: ﬁ|rst

text: equality,

text: we

text: used

text: the

text: and

text: the

text: fact

text: that

text: for

text: any

text: g,

text: ≡∆g

text: Now

text: let

text: V

text: n,

text: and

text: recall

text: the

text: Dirichlet

text: form

text: (with

text: respect

text: to

text: the

text: measure

text: π

text: E

text: (f,

text: f)

text: (f(y)

text: x

text: y

text: and

text: that

text: E

text: ∆f,

text: f)

text: ∆f(x)

text: (f(x)

text: f(y))

text: x

text: y

text: x

text: x

text: Plugging

text: into

text: the

text: above

text: inequality

text: gives

text: E

text: E

text: ∆f,

text: f)

text: (f,

text: f)

text: and

text: so

text: E

text: E

text: K

text: (f,

text: f)

text: ∆f,

text: f)

text: resulting

text: in

text: λ

text: K

text: Buser-type

text: Inequalities

text: The

text: proofs

text: in

text: this

text: section

text: are

text: a

text: straightforward

text: discrete

text: version

text: of

text: of

text: Ledoux’s

text: paper

text: First

text: we

text: derive

text: a

text: key

text: gradient

text: estimate

text: on

text: the

text: heat

text: kernel

text: associated

text: with

text: a

text: graph,

text: which

text: will

text: then

text: be

text: used

text: in

text: deriving

text: a

text: Buser

text: inequality

text: for

text: graphs,

text: as

text: mentioned

text: in

text: the

text: introduction.

text: Gradient

text: estimates

text: For

text: t

text: we

text: write

text: P

text: exp(t

text: for

text: the

text: heat

text: kernel

text: associated

text: with

text: the

text: graph

text: G.

text: Then

text: t

text: R

text: P

text: is

text: a

text: positive

text: deﬁnite

text: matrix

text: on

text: V

text: with

text: P

text: being

text: the

text: identity

text: matrix.

text: Note

text: that

text: P

text: t

text: t

text: commutes

text: with

text: and

text: with

text: P

text: and

text: that

text: ∂P

text: /∂t

text: P

text: P

text: Finally,

text: the

text: matrix

text: P

text: s

text: t

text: t

text: t

text: t

text: has

text: non-negative

text: entries.

text: So

text: if

text: f

text: has

text: non-negative

text: entries,

text: then

text: also

text: P

text: (f)

text: has

text: non-negative

text: R

text: t

text: entries.

text: For

text: a

text: vector

text: f

text: V

text: we

text: write

text: f

text: f(v)

text: p

text: R

text: R

text: Lemma

text: Suppose

text: G

text: has

text: Ric(G)

text: K

text: for

text: some

text: K

text: Then,

text: for

text: any

text: f

text: V

text: and

text: any

text: t

text: f

text: P

text: f

text: t

text: Γ(f)

text: t

text: Note

text: that

text: the

text: restriction

text: on

text: t

text: applies

text: only

text: when

text: K

text: is

text: negative:

text: if

text: K

text: then

text: Ric

text: K

text: implies

text: Ric

text: and

text: the

text: lemma

text: holds

text: with

text: no

text: restriction

text: on

text: t.

text: Proof.

text: The

text: proof

text: is

text: in

text: three

text: steps.

text: Step

text: We

text: ﬁrst

text: prove

text: that

text: Γ(P

text: f)

text: e

text: (Γ(f))

text: t

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the Markov chain version. A result of the above type is also recently derived independently by Erbar and by Oveis-Gharan and Trevisan (private communications). An earlier, weaker result, λ(M) = O(d2 h2(M)) follows from the work in [6], which uses a diﬀerent notion of curvature (and a diﬀerent argument of Ledoux), starting from a ﬁnite-dimensional curvature- dimension CD(K, n) inequality for graphs. Recently there have been several attempts to modify the CD(K, n) criterion in order to allow certain results involving the heat equation [6, 21, 29]. A recent result of Mu¨nch (cid:48) ´ [29] is that the CDE (K, n) criterion of [21] implies the CD(K, n) criterion of Bakry-Emery. These criteria are often useful; for example, it is known that Ricci-ﬂat graphs satisfy both ∞ (cid:48) ∞ the CDE(0, ) criterion of [6] and the CDE (0, ) criterion. In the remainder of this section, we introduce Bochner’s Γ -type curvature for graphs 2 along with various notations and deﬁnitions. In Section , we bound the curvature for several examples, including slices of the discrete cube, symmetric group with adjacent as well as all transpositions as the generating sets; and nonnegativity of curvature for Cayley graphs of abelian groups. In Section , we show that the spectral gap can be bounded from below by curvature. In Section , we derive the above-mentioned Buser-type inequalities. 1.1 Preliminaries We ﬁrst recall some basic deﬁnitions and fairly standard notions. Let G = (V, E) be an undirected and locally ﬁnite graph. Throughout, we will assume that G has no isolated − − vertices. The graph Laplacian ∆ = ∆(G) = (D(G) A(G)), where D(G) is the diagonal matrix of the degrees of the vertices, and A(cid:88)(G) is the adjacency matrix of G. As an operator, → R its action on an f : V can be described as: − ∆f(x) = (f(y) f(x)) . ∼ y x ∼ where here and below the notation y x means that y is a neighbour of x in the graph. The sum is of course only over the y. Note that ∆ is a negative semi-deﬁnite matrix. − The spectral gap λ(G) is the least non-zero eigenvalue of ∆. We deﬁne the Cheeger constant | | ∂A h(G) = min | | , 0<|A|≤|V |/2 A | | − where ∂A denotes the num→berRof edg(cid:88)es from A to V A. Given functions f, g : V , we also deﬁne: 1 − − Γ(f, g)(x) = (f(x) f(y))(g(x) g(y)) . 2 ∼ y x When f = g, the abo∈ve becomes the more co(cid:88)mmonly denoted (square of the l2-type) discrete gradient: for each x V , 1 − |∇ | Γ(f)(x) := Γ(f, f)(x) = (f(x) f(y))2 =: f(x) 2 . 2 ∼ y x 3 ##LTLine## ##LTLine## ##LTLine##

text: the

text: Markov

text: chain

text: version.

text: A

text: result

text: of

text: the

text: above

text: type

text: is

text: also

text: recently

text: derived

text: independently

text: by

text: Erbar

text: and

text: by

text: Oveis-Gharan

text: and

text: Trevisan

text: (private

text: communications).

text: An

text: earlier,

text: weaker

text: result,

text: λ(M)

text: follows

text: from

text: the

text: work

text: in

text: which

text: uses

text: a

text: diﬀerent

text: notion

text: of

text: curvature

text: (and

text: a

text: diﬀerent

text: argument

text: of

text: Ledoux),

text: starting

text: from

text: a

text: ﬁnite-dimensional

text: curvature-

text: dimension

text: CD(K,

text: n)

text: inequality

text: for

text: graphs.

text: Recently

text: there

text: have

text: been

text: several

text: attempts

text: to

text: modify

text: the

text: CD(K,

text: n)

text: criterion

text: in

text: order

text: to

text: allow

text: certain

text: results

text: involving

text: the

text: heat

text: equation

text: A

text: recent

text: result

text: of

text: Mu¨nch

text: is

text: that

text: the

text: CDE

text: (K,

text: n)

text: criterion

text: of

text: implies

text: the

text: CD(K,

text: n)

text: criterion

text: of

text: Bakry-Emery.

text: These

text: criteria

text: are

text: often

text: useful;

text: for

text: example,

text: it

text: is

text: known

text: that

text: Ricci-ﬂat

text: graphs

text: satisfy

text: both

text: the

text: criterion

text: of

text: and

text: the

text: CDE

text: criterion.

text: In

text: the

text: remainder

text: of

text: this

text: section,

text: we

text: introduce

text: Bochner’s

text: Γ

text: -type

text: curvature

text: for

text: graphs

text: along

text: with

text: various

text: notations

text: and

text: deﬁnitions.

text: In

text: Section

text: we

text: bound

text: the

text: curvature

text: for

text: several

text: examples,

text: including

text: slices

text: of

text: the

text: discrete

text: cube,

text: symmetric

text: group

text: with

text: adjacent

text: as

text: well

text: as

text: all

text: transpositions

text: as

text: the

text: generating

text: sets;

text: and

text: nonnegativity

text: of

text: curvature

text: for

text: Cayley

text: graphs

text: of

text: abelian

text: groups.

text: In

text: Section

text: we

text: show

text: that

text: the

text: spectral

text: gap

text: can

text: be

text: bounded

text: from

text: below

text: by

text: curvature.

text: In

text: Section

text: we

text: derive

text: the

text: above-mentioned

text: Buser-type

text: inequalities.

text: Preliminaries

text: We

text: ﬁrst

text: recall

text: some

text: basic

text: deﬁnitions

text: and

text: fairly

text: standard

text: notions.

text: Let

text: G

text: (V,

text: E)

text: be

text: an

text: undirected

text: and

text: locally

text: ﬁnite

text: graph.

text: Throughout,

text: we

text: will

text: assume

text: that

text: G

text: has

text: no

text: isolated

text: vertices.

text: The

text: graph

text: Laplacian

text: ∆(G)

text: (D(G)

text: A(G)),

text: where

text: D(G)

text: is

text: the

text: diagonal

text: matrix

text: of

text: the

text: degrees

text: of

text: the

text: vertices,

text: and

text: is

text: the

text: adjacency

text: matrix

text: of

text: G.

text: As

text: an

text: operator,

text: R

text: its

text: action

text: on

text: an

text: f

text: V

text: can

text: be

text: described

text: as:

text: ∆f(x)

text: (f(y)

text: f(x))

text: y

text: x

text: where

text: here

text: and

text: below

text: the

text: notation

text: y

text: x

text: means

text: that

text: y

text: is

text: a

text: neighbour

text: of

text: x

text: in

text: the

text: graph.

text: The

text: sum

text: is

text: of

text: course

text: only

text: over

text: the

text: y.

text: Note

text: that

text: is

text: a

text: negative

text: semi-deﬁnite

text: matrix.

text: The

text: spectral

text: gap

text: λ(G)

text: is

text: the

text: least

text: non-zero

text: eigenvalue

text: of

text: We

text: deﬁne

text: the

text: Cheeger

text: constant

text: ∂A

text: h(G)

text: min

text: A

text: where

text: ∂A

text: denotes

text: the

text: num→berRof

text: from

text: A

text: to

text: V

text: A.

text: Given

text: functions

text: f,

text: g

text: V

text: we

text: also

text: deﬁne:

text: Γ(f,

text: g)(x)

text: (f(x)

text: f(y))(g(x)

text: g(y))

text: y

text: x

text: When

text: f

text: g,

text: the

text: abo∈ve

text: becomes

text: the

text: more

text: denoted

text: (square

text: of

text: the

text: discrete

text: gradient:

text: for

text: each

text: x

text: V

text: Γ(f)(x)

text: Γ(f,

text: f)(x)

text: (f(x)

text: f(x)

text: y

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [28] R. Montenegro and P. Tetali. Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci., 1(3):x+121, 2006. [29] F. Mu¨nch. Remarks on curvature dimension conditions on graphs. arXiv:1501.05839. [30] Y. Ollivier. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal., 256(3):810–864, 2009. [31] P. Papasoglu. An algorithm detecting hyperbolicity. In Geometric and computational perspectives on inﬁnite groups (Minneapolis, MN and New Brunswick, NJ, 1994), vol- ume 25 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 193–200. Amer. Math. Soc., Providence, RI, 1996. [32] M.D. Sammer. Aspects of mass transportation in discrete concentration inequalities. PhD thesis, Georgia institute of technology, 2005. [33] M. Schmuckenschla¨ger. Curvature of nonlocal Markov generators. In Convex geometric analysis (Berkeley, CA, 1996), volume 34 of Math. Sci. Res. Inst. Publ., pages 189–197. Cambridge Univ. Press, Cambridge, 1999. [34] K.-Th. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196(1):65–131, 2006. 22

label: reference

text: R.

text: Montenegro

text: and

text: P.

text: Tetali.

text: Mathematical

text: aspects

text: of

text: mixing

text: times

text: in

text: Markov

text: chains.

text: Found.

text: Trends

text: Theor.

text: Comput.

text: Sci.,

text: F.

text: Mu¨nch.

text: Remarks

text: on

text: curvature

text: dimension

text: conditions

text: on

text: graphs.

text: Y.

text: Ollivier.

text: Ricci

text: curvature

text: of

text: Markov

text: chains

text: on

text: metric

text: spaces.

text: J.

text: Funct.

text: Anal.,

text: P.

text: Papasoglu.

text: An

text: algorithm

text: detecting

text: hyperbolicity.

text: In

text: Geometric

text: and

text: computational

text: perspectives

text: on

text: inﬁnite

text: groups

text: (Minneapolis,

text: MN

text: and

text: New

text: Brunswick,

text: NJ,

text: vol-

text: ume

text: of

text: DIMACS

text: Ser.

text: Discrete

text: Math.

text: Theoret.

text: Comput.

text: Sci.,

text: pages

text: Amer.

text: Math.

text: Soc.,

text: Providence,

text: RI,

text: M.D.

text: Sammer.

text: Aspects

text: of

text: mass

text: transportation

text: in

text: discrete

text: concentration

text: inequalities.

text: PhD

text: thesis,

text: Georgia

text: institute

text: of

text: technology,

text: M.

text: Schmuckenschla¨ger.

text: Curvature

text: of

text: nonlocal

text: Markov

text: generators.

text: In

text: Convex

text: geometric

text: analysis

text: (Berkeley,

text: CA,

text: volume

text: of

text: Math.

text: Sci.

text: Res.

text: Inst.

text: Publ.,

text: pages

text: Cambridge

text: Univ.

text: Press,

text: Cambridge,

text: K.-Th.

text: Sturm.

text: On

text: the

text: geometry

text: of

text: metric

text: measure

text: spaces.

text: I.

text: Acta

text: Math.,


text: (cid:88) observing that | | 1 ≤ dT ∆(x, v, u) = t(x, v) 2 ∼ 2 v x and that Γ(f)(x) = 1d. Any value of K > 2 + T will not satisfy (3) for the function f at 2 ≤ 2 vertex x, thus Ric(G) 2 + T . 2 2 Examples In this section we provide bounds on the curvature for several graphs of general interest. 2.1 The hypercube H n Let H represent the n-dimensional hypercube, where vertices are adjacent if their Ham- n ming distance is one. While the following result also follows from the tensorization result of [33], we provide here a direct proof. ≥ Theorem 2.1. Ric(H ) = 2 if n 1. n (cid:32) (cid:33) ∈ Proof. For any vertex x(cid:88) H (cid:88), and for any f with f(x(cid:88)) = 0, we get from (5(cid:88)) n 2 1 − − 2Γ (f)(x) = (f(u) 2f(v))2 + f(v) + (2 n) f2(v). 2 2 ∼ ∼ u: ∼v:∼ v x v x d(x,u)=2 x v u Let u be a vertex of distance 2 from x, and let v and w be the two distinct vertices so ∼ ∼ ∼ ∼ ∼ that u v x w (cid:88)u. Then for ﬁxed values of f(v) where v x, according to (6) Γ (f)(x) is minimized by f(u) = f(v) + f(w). With this value, 2 − − (f(u) 2f(v))2 = 2 (f(v) f(w))2 . ∼ ∼ v:u v x (cid:32) (cid:33) ∼ ∼ As for every pair v, w x(cid:88)there is a unique verte(cid:88)x u with u v, w and(cid:88)d(x, u) = 2, 2 ≥ − − 2Γ (f)(x) (f(v) f(w))2 + f(v) + (2 n) f2(v) , 2 (cid:54) ∼ ∼ v=∼w v x v x v,w x w(cid:88)here the ﬁrst sum is ove(cid:88)r all unordered p(cid:88)airs (v, w) of(cid:88)distinct neighbors of x. W(cid:88)e use this convention throughout the paper. Expanding the above gives − − (cid:88)(f2(v) + f2(w)) 2f(v)f(w) + f2(v) + 2f(v)f(w) + (2 n) f2(v) (cid:54) (cid:54) ∼ (cid:54) ∼ v=∼w v=∼w v x v=∼w v x v,w x v,w x v,w x = 2 f2(v) = 4Γ(f)(x) . ∼ v x ≥ So Ric 2, and by Theorem 1.2 we may conclude that Ric = 2. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: observing

text: that

text: dT

text: ∆(x,

text: v,

text: u)

text: t(x,

text: v)

text: v

text: x

text: and

text: that

text: Γ(f)(x)

text: Any

text: value

text: of

text: K

text: T

text: will

text: not

text: satisfy

text: for

text: the

text: function

text: f

text: at

text: vertex

text: x,

text: thus

text: Ric(G)

text: T

label: section

text: Examples

text: In

text: this

text: section

text: we

text: provide

text: bounds

text: on

text: the

text: curvature

text: for

text: several

text: graphs

text: of

text: general

text: interest.

text: The

text: hypercube

text: H

text: n

text: Let

text: H

text: represent

text: the

text: n-dimensional

text: hypercube,

text: where

text: vertices

text: are

text: adjacent

text: if

text: their

text: Ham-

text: n

text: ming

text: distance

text: is

text: one.

text: While

text: the

text: following

text: result

text: also

text: follows

text: from

text: the

text: tensorization

text: result

text: of

text: we

text: provide

text: here

text: a

text: direct

text: proof.

text: Theorem

text: Ric(H

text: if

text: n

text: n

text: Proof.

text: For

text: any

text: vertex

text: H

text: and

text: for

text: any

text: f

text: with

text: we

text: get

text: from

text: n

text: 2Γ

text: (f)(x)

text: (f(u)

text: f(v)

text: n)

text: u:

text: ∼v:∼

text: v

text: x

text: v

text: x

text: x

text: v

text: u

text: Let

text: u

text: be

text: a

text: vertex

text: of

text: distance

text: from

text: x,

text: and

text: let

text: v

text: and

text: w

text: be

text: the

text: two

text: distinct

text: vertices

text: so

text: that

text: u

text: v

text: x

text: w

text: Then

text: for

text: ﬁxed

text: values

text: of

text: f(v)

text: where

text: v

text: x,

text: according

text: to

text: Γ

text: (f)(x)

text: is

text: minimized

text: by

text: f(u)

text: f(v)

text: f(w).

text: With

text: this

text: value,

text: (f(u)

text: (f(v)

text: v:u

text: v

text: x

text: As

text: for

text: every

text: pair

text: v,

text: w

text: is

text: a

text: unique

text: u

text: with

text: u

text: v,

text: w

text: u)

text: 2Γ

text: (f)(x)

text: (f(v)

text: f(v)

text: n)

text: v=∼w

text: v

text: x

text: v

text: x

text: v,w

text: x

text: the

text: ﬁrst

text: sum

text: is

text: all

text: unordered

text: (v,

text: w)

text: neighbors

text: of

text: x.

text: use

text: this

text: convention

text: throughout

text: the

text: paper.

text: Expanding

text: the

text: above

text: gives

text: n)

text: v=∼w

text: v=∼w

text: v

text: x

text: v=∼w

text: v

text: x

text: v,w

text: x

text: v,w

text: x

text: v,w

text: x

text: v

text: x

text: So

text: Ric

text: and

text: by

text: Theorem

text: we

text: may

text: conclude

text: that

text: Ric

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In the following, we compute the curvature of the complete graph. With the tensorization result of [26], this provide another proof of the fact that the hypercube has curvature 2. 2.2 The complete graph K n ≥ Theorem 2.2. Ric(K ) = 1 + n if n 2. n 2 ∈ → R Proof. For the complete graph on n vertices, we have, for every x V and every f : V such that f(x) = 0, from (5), (cid:16) (cid:88) (cid:17) (cid:88) (cid:88) (cid:16) (cid:0) (cid:1) (cid:17) 2Γ (f)(x) = 2 2 − − 1 f(v) + (3 n) f2(v) + 2 (f(v) f(u))2 + f(u)2 + f(v)2 . ∼ ∼ ∼ 2 v x v x u,v(cid:54) x u=v (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) Expanding the above gives − 5 − f2(v) + 2f(u)f(v) + (3 n) f2(v) + (f2(v) + f2(u)) 4f(u)f(v) (cid:88) (cid:88) (cid:88) ∼ ∼ ∼ 2 ∼ ∼ v x u,v(cid:54) x v x u,v(cid:54) x u,v(cid:54) x u=v u=v u=v − 5 − − = (cid:18)(4 n) (cid:19) f2(v) + (n 2) f2(v) 2 f(u)f(v) (cid:88) (cid:88) (cid:88) (cid:16) (cid:88) (cid:17) ∼ 2 ∼ ∼ v x v x u,v(cid:54) x u=v 3n − − 3n − 2 = 1 f2(v) 2 f(u)f(v) = f2(v) f(v) . 2 ∼ ∼(cid:80) 2 ∼ (cid:80)∼ v x u,v(cid:54) x v x v x (cid:80) u=v ≤ |{ ∼ }| − By the Cauchy-Schwarz inequality, ( ∼ f(v))2 v : v x ∼ f2(v) = (n (cid:32) v (cid:33)x v x 1) ∼ f2(v), so (cid:88) (cid:88) (cid:16) (cid:17) (cid:88) v x 2 3n − ≥ n f2(v) f(v) 1 + f2(v) . 2 ∼ ∼ 2 ∼ v x v x v x ≥ Thus Ric 1 + n, once again by Theorem 1.2, we conclude that Ric = 1 + n. 2 2 2.3 Finite abelian Cayley graphs A ﬁnite abelian group is of course a product of cyclic groups and hence one might think that the curvature of the graph can be deduced from the tensorization result of [26]. However, a Cayley graph is determined by an underlying group and a generating set for that group. Here we show that a ﬁnitely generated abelian group with any set of generators has positive Ricci curvature - not only with the generating set inherited from a decomposition into cyclic 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: following,

text: we

text: compute

text: the

text: curvature

text: of

text: the

text: complete

text: graph.

text: With

text: the

text: tensorization

text: result

text: of

text: this

text: provide

text: another

text: proof

text: of

text: the

text: fact

text: that

text: the

text: hypercube

text: has

text: curvature

text: The

text: complete

text: graph

text: K

text: n

text: Theorem

text: Ric(K

text: n

text: if

text: n

text: n

text: R

text: Proof.

text: For

text: the

text: complete

text: graph

text: on

text: n

text: vertices,

text: we

text: have,

text: for

text: every

text: x

text: V

text: and

text: every

text: f

text: V

text: such

text: that

text: f(x)

text: from

text: 2Γ

text: (f)(x)

text: f(v)

text: n)

text: (f(v)

text: v

text: x

text: v

text: x

text: x

text: u=v

text: Expanding

text: the

text: above

text: gives

text: n)

text: v

text: x

text: x

text: v

text: x

text: x

text: x

text: u=v

text: u=v

text: u=v

text: n)

text: (n

text: f(u)f(v)

text: v

text: x

text: v

text: x

text: x

text: u=v

text: f(u)f(v)

text: f(v)

text: v

text: x

text: x

text: v

text: x

text: v

text: x

text: u=v

text: By

text: the

text: Cauchy-Schwarz

text: inequality,

text: v

text: v

text: x

text: (n

text: v

text: v

text: x

text: so

text: v

text: x

text: n

text: f(v)

text: v

text: x

text: v

text: x

text: v

text: x

text: Thus

text: Ric

text: n,

text: once

text: again

text: by

text: Theorem

text: we

text: conclude

text: that

text: Ric

text: n.

text: Finite

text: abelian

text: Cayley

text: graphs

text: A

text: ﬁnite

text: abelian

text: group

text: is

text: of

text: course

text: a

text: product

text: of

text: cyclic

text: groups

text: and

text: hence

text: one

text: might

text: think

text: that

text: the

text: curvature

text: of

text: the

text: graph

text: can

text: be

text: deduced

text: from

text: the

text: tensorization

text: result

text: of

text: However,

text: a

text: Cayley

text: graph

text: is

text: determined

text: by

text: an

text: underlying

text: group

text: and

text: a

text: generating

text: set

text: for

text: that

text: group.

text: Here

text: we

text: show

text: that

text: a

text: ﬁnitely

text: generated

text: abelian

text: group

text: with

text: any

text: set

text: of

text: generators

text: has

text: positive

text: Ricci

text: curvature

text: not

text: only

text: with

text: the

text: generating

text: set

text: inherited

text: from

text: a

text: decomposition

text: into

text: cyclic

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 1. Identically cropped images of LMC SNR 0505-67.9 (DEM L71); on the left is the combined g’r’i’+Hα Gemini image, and on the right is the Chandra color image, which is a combination of 0.3-0.7 keV (red), 0.7-1.1 keV (green), and 1.1-2.4 keV (blue) observations. The Gemini images were taken with the GMOS imager on the 8.1m Gemini South Telescope. (The dark horizontal line running across the bottom of the Gemini image is an artifact due to a nearby bright star.) The Chandra image on the right shows the complicated X-ray structure of the remnant. To measure the remnant center, we used four diﬀerent gas regions visible in X-ray: the extreme outer edge, the rim of the outer shell, the edge of the inner region, and the central minimum. This al(cid:48)l(cid:48)ows us to get a more accurate measurement of the center as well as a better understanding of the errors on that measurement. The 15.8 99.73% containment central region is marked on both images. The error circle is very large, because the remnant is both old (4360 years) and asymmetric, and therefore it contains a number of possible ex-companion stars. Stars 19 and 32 are marked; star 19 is the nearest red giant to the center, and star 32 is the nearest subgiant. (The numbers correspond to their identiﬁcations in Table 2.) Future spectroscopy may allow us to identify which of the central stars are likely to be the actual ex-companion, if any. (X-ray image source: The Chandra Supernova Remnant Catalog (http://hea-www.harvard.edu/ChandraSNR/), used with permission) SD and DD progenitors are still possibilities. Further hanced heavy elements, is likely no longer detectable due observations of the potential ex-companion stars within to normal mixing processes in the outer atmosphere of the central region, namely high resolution spectra, are any potential ex-companion. In general, the method of needed to identify any likely ex-companions. It may be searching for ex-companion stars in historical SNRs is diﬃcult, however, to ever obtain any conclusive answers best applied to younger remnants; it is likely that all for this remnant, due to its age. To do complete follow- older SNRs will have these same diﬃculties. It is not, up on the large number of stars within the central region however, impossible to obtain an answer, so investiga- woDurldafttavkeerscioornreMsapyon2d7,in2g0l2y0 large amounts of telescope tion and follow-up should be continue to be attempted anPdreaprninatlytyspisesettimusein,gaLAnTdEXasttylleeaesmtuloatneeapoj fv.t1h2e/16id/1e1ntifying for all known nearby Ia SNRs. characteristics of an ex-companion, the presence of en- INVESTIGATION OF THE PROGENITORS OF THE TYPE IA SUPERNOVAE ASSOCIATED WITH THE LMC SUPERNOVA REMNANTS 0505-67.9 AND 0509-68.7 Ashley Pagnotta 0 Department of Astrophysics, American Museum of Natural History, New York, NY 10024 Bradley E. Schaefer 0 Department of Astrophysics, American Museum of Natural History, New York, NY 10024 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 Keywords: supernova remnants ##LTFigure##

text: Figure

text: Identically

text: cropped

text: images

text: of

text: LMC

text: SNR

text: (DEM

text: on

text: the

text: left

text: is

text: the

text: combined

text: g’r’i’+Hα

text: Gemini

text: image,

text: and

text: on

text: the

text: right

text: is

text: the

text: Chandra

text: color

text: image,

text: which

text: is

text: a

text: combination

text: of

text: keV

text: (red),

text: keV

text: (green),

text: and

text: keV

text: (blue)

text: observations.

text: The

text: Gemini

text: images

text: were

text: taken

text: with

text: the

text: GMOS

text: imager

text: on

text: the

text: Gemini

text: South

text: Telescope.

text: (The

text: dark

text: horizontal

text: line

text: running

text: across

text: the

text: bottom

text: of

text: the

text: Gemini

text: image

text: is

text: an

text: artifact

text: due

text: to

text: a

text: nearby

text: bright

text: star.)

text: The

text: Chandra

text: image

text: on

text: the

text: right

text: shows

text: the

text: complicated

text: X-ray

text: structure

text: of

text: the

text: remnant.

text: To

text: measure

text: the

text: remnant

text: center,

text: we

text: used

text: four

text: diﬀerent

text: gas

text: regions

text: visible

text: in

text: X-ray:

text: the

text: extreme

text: outer

text: edge,

text: the

text: rim

text: of

text: the

text: outer

text: shell,

text: the

text: edge

text: of

text: the

text: inner

text: region,

text: and

text: the

text: central

text: minimum.

text: This

text: us

text: to

text: get

text: a

text: more

text: accurate

text: measurement

text: of

text: the

text: center

text: as

text: well

text: as

text: a

text: better

text: understanding

text: of

text: the

text: errors

text: on

text: that

text: measurement.

text: The

text: containment

text: central

text: region

text: is

text: marked

text: on

text: both

text: images.

text: The

text: error

text: circle

text: is

text: very

text: large,

text: because

text: the

text: remnant

text: is

text: both

text: old

text: years)

text: and

text: asymmetric,

text: and

text: therefore

text: it

text: contains

text: a

text: number

text: of

text: possible

text: ex-companion

text: stars.

text: Stars

text: and

text: are

text: marked;

text: star

text: is

text: the

text: nearest

text: red

text: giant

text: to

text: the

text: center,

text: and

text: star

text: is

text: the

text: nearest

text: subgiant.

text: (The

text: numbers

text: correspond

text: to

text: their

text: identiﬁcations

text: in

text: Table

text: Future

text: spectroscopy

text: may

text: allow

text: us

text: to

text: identify

text: which

text: of

text: the

text: central

text: stars

text: are

text: likely

text: to

text: be

text: the

text: actual

text: ex-companion,

text: if

text: any.

text: (X-ray

text: image

text: source:

text: The

text: Chandra

text: Supernova

text: Remnant

text: Catalog

text: used

text: with

text: permission)

text: SD

text: and

text: DD

text: progenitors

text: are

text: still

text: possibilities.

text: Further

text: hanced

text: heavy

text: elements,

text: is

text: likely

text: no

text: longer

text: detectable

text: due

text: observations

text: of

text: the

text: potential

text: ex-companion

text: stars

text: within

text: to

text: normal

text: mixing

text: processes

text: in

text: the

text: outer

text: atmosphere

text: of

text: the

text: central

text: region,

text: namely

text: high

text: resolution

text: spectra,

text: are

text: any

text: potential

text: ex-companion.

text: In

text: general,

text: the

text: method

text: of

text: needed

text: to

text: identify

text: any

text: likely

text: ex-companions.

text: It

text: may

text: be

text: searching

text: for

text: ex-companion

text: stars

text: in

text: historical

text: SNRs

text: is

text: diﬃcult,

text: however,

text: to

text: ever

text: obtain

text: any

text: conclusive

text: answers

text: best

text: applied

text: to

text: younger

text: remnants;

text: it

text: is

text: likely

text: that

text: all

text: for

text: this

text: remnant,

text: due

text: to

text: its

text: age.

text: To

text: do

text: complete

text: follow-

text: older

text: SNRs

text: will

text: have

text: these

text: same

text: diﬃculties.

text: It

text: is

text: not,

text: up

text: on

text: the

text: large

text: number

text: of

text: stars

text: within

text: the

text: central

text: region

text: however,

text: impossible

text: to

text: obtain

text: an

text: answer,

text: so

text: investiga-

text: large

text: amounts

text: of

text: telescope

text: tion

text: and

text: follow-up

text: should

text: be

text: continue

text: to

text: be

text: attempted

text: for

text: all

text: known

text: nearby

text: Ia

text: SNRs.

text: characteristics

text: of

text: an

text: ex-companion,

text: the

text: presence

text: of

text: en-

text: INVESTIGATION

text: OF

text: THE

text: PROGENITORS

text: OF

text: THE

text: TYPE

text: IA

text: SUPERNOVAE

text: ASSOCIATED

text: WITH

text: THE

text: LMC

text: SUPERNOVA

text: REMNANTS

text: AND

text: Ashley

text: Pagnotta

text: Department

text: of

text: Astrophysics,

text: American

text: Museum

text: of

text: Natural

text: History,

text: New

text: York,

text: NY

text: Bradley

text: E.

text: Schaefer

text: Department

text: of

text: Astrophysics,

text: American

text: Museum

text: of

text: Natural

text: History,

text: New

text: York,

text: NY

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: Louisiana

text: State

text: University,

text: Baton

text: Rouge,

text: LA

text: Keywords:

text: supernova

text: remnants

text: ##LTFigure##


text: |)x( V|,|)x( V| C 2 1 ) (mI (cid:104) |)x( W|,|)x( W| |)x(c|,|)x(a| 2 1 |)0,x(u| |)5.6,x(u| |)61,x(u| |)0,x(v| |)5.6,x(v| |)61,x(v| 12 0 5 0.1 2 2 4 0 0 3 −50 0 50 −50 0 50 4 x x 2 500 6 1 0 2 8 0 5 −500 0 D −2 0 2 −50 0 50 Re( ) x (cid:104) 10 0.4 2 5 0.2 1 0 0 0 −50 0 50 −50 0 50 −40 −20 0 20 40 x x x FIG. 4: This ﬁgure shows stability analysis and time propagation plots for the bright-bright soliton case with p = q = 1. The placement of the ﬁgures follows the same pattern as that of Figure 1. Here the parameters common to all plots are d = d = 1, ≈ 1 2 r = 1 and K = 0.1. The point C = 0.5, D = 3 from the upper left contour plot corresponds to max(Re(λ)) 1.7055 and more details regarding these speciﬁc amplitude parameters are shown in the other seven plots. Here we ﬁnd that as time progresses the solution loses its soliton shape with turbulent, symmetry-breaking behaviour occurring ﬁrst nearby the central peak and then leaking outward across the x axis. D in (29) increase. Here, it is clear that the instability results in the breaking of the parity symmetry, leading to a symmetry-breaking pattern. 5. Bright and Dark-in-Bright solitary Wave − For the solutions presented in Section in equations (32)-(35) we choose G(y) = Ky(1 y2) and obtain − − − ω = d r2p2, V = Byq + d r2y2(p + 1)(p + 2) + d K2y2(1 y2)2 1 1 1 1 − −A2 − − − σ = d r2q2, V = y2p q(1 y2) + d r2y2q(q + 1) + 4d K2y2(1 y2)2 2 2 B 2 2 − W = Krd sech(rx) tanh(rx)(1 + 2p (5 + 2p)y2) 1 1 − W = 2Krd sech(rx) tanh(rx)(1 + 2q (3 + 2q)y2) 2 2 by choosing ω, σ as usual. The stability analysis is presented in Figure 5. Similar to the quadratic case, the strength of instability increases as A increases and as B increases. The propagation plots in Figure 5 show that the peak destabilizes and the amplitude spreads out over the x axis while maintaining some comparative concentration at the center of the axis. Furthermore, a symmetry-breaking feature appears once again to be amplifed and be distinctly observable at the end of the simulation’s reporting horizon. B. Oscillatory Solutions (k = 0) 1. Quadratic oscillatory solution For the solutions in Section in equations (27)-(28) we take G(y) = KF (y) similar to the dark-dark soliton case, 1 obtaining the following solutions − − − ω = AC + d K2 + 4d r2, V = 2AC y2 + d K2(1 2y2)2 d K2 (48) 0 1 1 1 0 1 1 C − C − − σ = 0 + 4d K2 + 4d r2, V = 2 0 y2 + 4d K2(1 2y2)2 4d K2 A 2 2 2 A 2 2 − W = 12jrd K cos(rx) sin(rx) j j ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: |)x(

text: V|,|)x(

text: V|

text: C

text: (mI

text: |)x(

text: W|,|)x(

text: W|

text: |)x(c|,|)x(a|

text: x

text: x

text: D

text: −2

text: Re(

text: x

text: x

text: x

text: x

text: FIG.

text: This

text: ﬁgure

text: shows

text: stability

text: analysis

text: and

text: time

text: propagation

text: plots

text: for

text: the

text: bright-bright

text: soliton

text: case

text: with

text: p

text: q

text: The

text: placement

text: of

text: the

text: ﬁgures

text: follows

text: the

text: same

text: pattern

text: as

text: that

text: of

text: Figure

text: Here

text: the

text: parameters

text: common

text: to

text: all

text: plots

text: are

text: d

text: d

text: r

text: and

text: K

text: The

text: point

text: C

text: D

text: from

text: the

text: upper

text: left

text: contour

text: plot

text: corresponds

text: to

text: max(Re(λ))

text: and

text: more

text: details

text: regarding

text: these

text: speciﬁc

text: amplitude

text: parameters

text: are

text: shown

text: in

text: the

text: other

text: seven

text: plots.

text: Here

text: we

text: ﬁnd

text: that

text: as

text: time

text: progresses

text: the

text: solution

text: loses

text: its

text: soliton

text: shape

text: with

text: turbulent,

text: symmetry-breaking

text: behaviour

text: occurring

text: ﬁrst

text: nearby

text: the

text: central

text: peak

text: and

text: then

text: leaking

text: outward

text: across

text: the

text: x

text: axis.

text: D

text: in

text: increase.

text: Here,

text: it

text: is

text: clear

text: that

text: the

text: instability

text: results

text: in

text: the

text: breaking

text: of

text: the

text: parity

text: symmetry,

text: leading

text: to

text: a

text: symmetry-breaking

text: pattern.

text: Bright

text: and

text: Dark-in-Bright

text: solitary

text: Wave

text: For

text: the

text: solutions

text: presented

text: in

text: Section

text: in

text: equations

text: we

text: choose

text: G(y)

text: and

text: obtain

text: ω

text: d

text: V

text: Byq

text: d

text: d

text: σ

text: d

text: V

text: d

text: B

text: W

text: Krd

text: sech(rx)

text: W

text: sech(rx)

text: by

text: choosing

text: ω,

text: σ

text: as

text: usual.

text: The

text: stability

text: analysis

text: is

text: presented

text: in

text: Figure

text: Similar

text: to

text: the

text: quadratic

text: case,

text: the

text: strength

text: of

text: instability

text: increases

text: as

text: A

text: increases

text: and

text: as

text: B

text: increases.

text: The

text: propagation

text: plots

text: in

text: Figure

text: show

text: that

text: the

text: peak

text: destabilizes

text: and

text: the

text: amplitude

text: spreads

text: out

text: over

text: the

text: x

text: axis

text: while

text: maintaining

text: some

text: comparative

text: concentration

text: at

text: the

text: center

text: of

text: the

text: axis.

text: Furthermore,

text: a

text: symmetry-breaking

text: feature

text: appears

text: once

text: again

text: to

text: be

text: amplifed

text: and

text: be

text: distinctly

text: observable

text: at

text: the

text: end

text: of

text: the

text: simulation’s

text: reporting

text: horizon.

text: B.

text: Oscillatory

text: Solutions

text: (k

text: Quadratic

text: oscillatory

text: solution

text: For

text: the

text: solutions

text: in

text: Section

text: in

text: equations

text: we

text: take

text: G(y)

text: KF

text: (y)

text: similar

text: to

text: the

text: dark-dark

text: soliton

text: case,

text: obtaining

text: the

text: following

text: solutions

text: ω

text: AC

text: d

text: V

text: d

text: d

text: C

text: C

text: σ

text: V

text: A

text: A

text: W

text: K

text: cos(rx)

text: sin(rx)

text: j

text: j

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: |)x( V|,|)x( V| |)x( V|,|)x( V| C 2 1 C 2 1 0 |)x( W|,|)x( W| |)x(c|,|)x(a| )(cid:104)(mI |)x(2W|,|)x(1W| |)x(c|,|)x(a| )(cid:104)(mI 2 1 |)71,x(u| |)0,x(u| |)5.8,x(u| |)0,x(u| |)9.4,x(u| |)6,x(u| |)71,x(v| |)0,x(v| |)5.8,x(v| |)0,x(v| |)9.4,x(v| |)6,x(v| 14 0 0.1 2 6 2 0 0 4 −20 0 20 −20 0 20 4 x x 50 6 2 0 2 8 0 5 −50 0 AC −5 0 5 −20 0 20 0 Re( ) x (cid:104) 10 0.04 4 5 0.02 2 0 0 0 −20 0 20 −20 0 20 −20 0 20 x x x FIG. 6: This ﬁgure shows stability analysis and time propagation plots for the quadratic oscillatory case. The placement of the ﬁgures follows the same pattern as that of Figure 1. Here the parameters common to all plots are d = d = 0.1, r = 0.5 and ≈1 2 K = 0.05. The point C = 0.5, D = 3 from the upper left contour plot corresponds to max(Re(λ)) 2.3602 and more details regarding these speciﬁc amplitude parameters are shown in the other seven plots. Here we ﬁnd that as time progresses the instability of the solution appears across the x axis at near regular intervals, close to the periodicity of the stationary solution. However, the solution over time does not remain truly periodic. 4 0 0.1 6 0.05 2 2 0 0 −50 0 50 −50 0 50 4 4 x x 50 4 6 2 0 2 8 0 5 −50 0 D −5 0 5 −50 0 50 Re( ) x (cid:104) 4 0.1 10 2 0.05 5 0 0 0 −50 0 50 −50 0 50 −50 0 50 x x x FIG. 7: This ﬁgure shows stability analysis and time propagation plots for the linear oscillatory case. The placement of the ﬁgures follows the same pattern as that of Figure 1. Here the parameters common to all plots are d = d = 0.35, r = 0.25 and 1≈ 2 K = 0.1. The point C = 0.5, D = 3 from the upper left contour plot corresponds to max(Re(λ)) 2.6630 and more details regarding these speciﬁc amplitude parameters are shown in the other seven plots. Similar to the quadratic oscillatory case, here we ﬁnd that as time progresses the solution does not remain truly periodic. The stability graph on the left of Figure 8 shows that the strength of instability increases as B increases, yet it appears to be roughly independent of A. The propagation panels show the distortion of the original solution occurring over time at points across the x axis corresponding to an eigenvector that is also spread across x. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: |)x(

text: V|,|)x(

text: V|

text: |)x(

text: V|,|)x(

text: V|

text: C

text: C

text: |)x(

text: W|,|)x(

text: W|

text: |)x(c|,|)x(a|

text: |)x(c|,|)x(a|

text: x

text: x

text: AC

text: −5

text: Re(

text: x

text: x

text: x

text: x

text: FIG.

text: This

text: ﬁgure

text: shows

text: stability

text: analysis

text: and

text: time

text: propagation

text: plots

text: for

text: the

text: quadratic

text: oscillatory

text: case.

text: The

text: placement

text: of

text: the

text: ﬁgures

text: follows

text: the

text: same

text: pattern

text: as

text: that

text: of

text: Figure

text: Here

text: the

text: parameters

text: common

text: to

text: all

text: plots

text: are

text: d

text: d

text: r

text: and

text: ≈1

text: K

text: The

text: point

text: C

text: D

text: from

text: the

text: upper

text: left

text: contour

text: plot

text: corresponds

text: to

text: max(Re(λ))

text: and

text: more

text: details

text: regarding

text: these

text: speciﬁc

text: amplitude

text: parameters

text: are

text: shown

text: in

text: the

text: other

text: seven

text: plots.

text: Here

text: we

text: ﬁnd

text: that

text: as

text: time

text: progresses

text: the

text: instability

text: of

text: the

text: solution

text: appears

text: across

text: the

text: x

text: axis

text: at

text: near

text: regular

text: intervals,

text: close

text: to

text: the

text: periodicity

text: of

text: the

text: stationary

text: solution.

text: However,

text: the

text: solution

text: over

text: time

text: does

text: not

text: remain

text: truly

text: periodic.

text: x

text: x

text: D

text: −5

text: Re(

text: x

text: x

text: x

text: x

text: FIG.

text: This

text: ﬁgure

text: shows

text: stability

text: analysis

text: and

text: time

text: propagation

text: plots

text: for

text: the

text: linear

text: oscillatory

text: case.

text: The

text: placement

text: of

text: the

text: ﬁgures

text: follows

text: the

text: same

text: pattern

text: as

text: that

text: of

text: Figure

text: Here

text: the

text: parameters

text: common

text: to

text: all

text: plots

text: are

text: d

text: d

text: r

text: and

text: 1≈

text: K

text: The

text: point

text: C

text: D

text: from

text: the

text: upper

text: left

text: contour

text: plot

text: corresponds

text: to

text: max(Re(λ))

text: and

text: more

text: details

text: regarding

text: these

text: speciﬁc

text: amplitude

text: parameters

text: are

text: shown

text: in

text: the

text: other

text: seven

text: plots.

text: Similar

text: to

text: the

text: quadratic

text: oscillatory

text: case,

text: here

text: we

text: ﬁnd

text: that

text: as

text: time

text: progresses

text: the

text: solution

text: does

text: not

text: remain

text: truly

text: periodic.

text: The

text: stability

text: graph

text: on

text: the

text: left

text: of

text: Figure

text: shows

text: that

text: the

text: strength

text: of

text: instability

text: increases

text: as

text: B

text: increases,

text: yet

text: it

text: appears

text: to

text: be

text: roughly

text: independent

text: of

text: A.

text: The

text: propagation

text: panels

text: show

text: the

text: distortion

text: of

text: the

text: original

text: solution

text: occurring

text: over

text: time

text: at

text: points

text: across

text: the

text: x

text: axis

text: corresponding

text: to

text: an

text: eigenvector

text: that

text: is

text: also

text: spread

text: across

text: x.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 7 ∈ { } for j 1, 2 . Equations (27)-(28) give us a solution that we call the quadratic oscillatory case. In Section we show the shape of the solution, analyze the stability, and explore its dynamics over the evolution variable (z). B. Power Functions Next we take F (y) and F (y) to be power functions 1 2 { } F (y) = i 0,1 Cyp1, F (y) = Dyp2 (29) 1 2 ≥ (cid:54) for p , p 0 and C, D = 0. This special case considerably simpliﬁes the relevant compatibility conditions. In 1 2 ≥ particular, substituting (29) into (6) and examining conditions (7)-(8) we require that either k = 1 and 2p p , or 1 2 p = p = 1. 1 2 1. Cnoidal parameter k = 1 and y = sech(rx) In the case of k = 1 we can have non-integer p and p ; this is in contrast to the polynomial case. For this case, we 1 2 apply (6) and ﬁnd ± − − − − − − V (x) = Dyp2 d r2p (1 2y2) d r2p (p 1)(1 y2) + d G2(y) ω (30) 1 1 1 1 1 1 1 ±C2 − − − − − − − V (x) = y2p1 p2 d r2p (1 2y2) d r2p (p 1)(1 y2) + 4d G2(y) σ 2 D 2 2 2 2 2 2 (cid:48) W (x) = jrd tanh(rx) (2p G(y) + yG (y)) j j j ∈ { } (cid:48) for j 1, 2 . As long as (2p G(y) + yG (y)) are even functions in x, which is easy to choose since y = sech(rx) j PT is an even function of x, then W and W are odd and compatible with the symmetry criterion. We refer to the 1 2 solutions given in (29) and (30) as the bright-bright soliton case. In section we show the wave’s shape, analyze its stability and explore its direct numerical evolution. Note that the case p = p = 2 corresponds to solitonic solutions 1 2 found by Karamzin-Sukhorukov in [41], and 2p = p = 2 to solitonic solutions found by Menyuk et al. [44]. 1 2 2. Powers p = p = 1 1 2 In the case p = p = 1, we have 1 2 ± − − − − V (x) = Dy d r2(2k2 1 2k2y2) + d G2(y) ω (31) 1 1 1 ±C2 − − − − V (x) = y d r2(2k2 1 2k2y2) + 4d G2(y) σ 2 D 2 2 (cid:48) W (x) = jrd dc(rx, k)sn(rx, k) (2G(y) + yG (y)) j j ∈ { } (cid:48) where j 1, 2 . Similar to the previous section, we only need to choose a function G(y) so that (2G(y) + yG (y)) PT is an even function of x in order to satisfy the symmetry criterion. Equations (29) and (31) give us a solution we call the linear oscillatory case. More details are included in Section . C. Other solutions Here, we introduce a possibility which is distinct from the previous ones as follows. We introduce F (y) and F (y) 1 2 in the form − F = iAyp(1 y2)1/2, F = Byq (32) 1 2 ≥ for p, q 0 i.e., a non-polynomial form. By (6) and examining conditions (7)-(8) we ﬁnd that we must have − − − − ≤ ≤ p(p 1)(1 k2) = q(q 1)(1 k2) = 0 and q 2p. That is, we require that either k = 1 with q 2p, or p = q = 1, or p = q = 0. We will focus on the former two cases. As for W in (6) and condition (9), we ﬁnd that G(y) must be 1,2 in the form ##LTLine## ##LTLine##

text: for

text: j

text: Equations

text: give

text: us

text: a

text: solution

text: that

text: we

text: call

text: the

text: quadratic

text: oscillatory

text: case.

text: In

text: Section

text: we

text: show

text: the

text: shape

text: of

text: the

text: solution,

text: analyze

text: the

text: stability,

text: and

text: explore

text: its

text: dynamics

text: over

text: the

text: evolution

text: variable

text: (z).

text: B.

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