text: are called vertices and the elements of A(G) are called arcs. When referring to an arc (u, v), we say that u is adjacent to v and v is adjacent from u. The −→ ordered pair (u, v) will often be denoted by uv. Sometimes we use u v −→ ∈ G or simply u v to represent an arc (u, v) A(G). The vertex u is the start-vertex and v is the end-vertex of a given arc (u, v). An arc of the form (u, u), or equivalently uu is called a loop. A mixed graph cannot contain multiple arcs, that is, it cannot contain the arc (u, v) more than once. A mixed graph G is called bipartite if there is a partition of V(G) into two sets X and Y , which we call colour classes, such that for each arc uv of { } G the set u, v intersects both X and Y . A set S of arcs is self-paired if, ∈ { } whenever (u, v) S, (v, u) is also in S. If S = (u, v), (v, u) , then we con- { } sider S to be the unordered pair u, v ; this unordered pair is called an edge. It is useful to consider two special cases of mixed graphs. A graph is a mixed graph without loops whose arc-set is self-paired. The edge set of a graph is denoted by E(G). A digraph is a mixed graph with no loops in (cid:48) which no set of arcs is self-paired. The inverse G of a mixed graph G is (cid:48) obtained from G by reversing all its arcs, that is V(G ) =V(G) and (v, u) is (cid:48) an arc of G if and only if (u, v) is an arc of G. A digraph G may therefore be (cid:48) characterised as a mixed graph for which A(G) and A(G ) are disjoint. Given ∈ a mixed graph G and a vertex v V(G), we deﬁne the in-neighbourhood { ∈ | ∈ } N (v) by N (v) = x V(G) (x, v) A(G) . Similarly we deﬁne the in in { ∈ | ∈ } out-neighbourhood N (v) by N (v) = x V(G) (v, x) A(G) . The in- out out | | degree ρ (v) of a vertex v is deﬁned by ρ (v) = N (v) and the out-degree in in | in| ρ (v) of a vertex v is deﬁned by ρ (v) = N (v) . When G is a graph, out out out these notions reduce to the usual neighbourhood N(v) = N (v) = N (v) in out and degree ρ(v) = ρ (v) = ρ (v). in out ∈ Let G be a graph and let v V(G). Let N(v) be the neighbourhood of (cid:54) v. We say that G is vertex-determining if N(x) = N(y) for any two distinct vertices x and y of G [7]. A set P of arcs is called an alternating trail or Z-trail if its elements · · · can be ordered in a sequence a , a , , a such that if a = (u, v) or (cid:54) 1 2 k (cid:54) i a = (u, w), w = v or a = (w, v), w = u. Note that in a Z-trail i+1 i+1 vertices may be repeated but arcs may not. If a = (p, q) and a = (r, s) 1 k we then say that the Z-trail joins p and s. A Z-trail can be open (when the ﬁrst and last vertex are diﬀerent), semi-closed (when the ﬁrst arc is uv and the last arc is wu) or closed (when the ﬁrst arc is uv and the last arc is uw or the ﬁrst arc is vu and the last arc is wu). Note that the ordering of arcs described in the deﬁnition is not unique, but it is basically unique for open and semi-closed Z-trails (one just chooses the beginning or the end), while for closed Z-trails all vertices can be taken as the ﬁrst vertex. Whether a Z-trail is open, semi-closed or closed is determined by the ﬁrst and last 2

text: are

text: called

text: vertices

text: and

text: the

text: elements

text: of

text: A(G)

text: are

text: called

text: arcs.

text: When

text: referring

text: to

text: an

text: arc

text: (u,

text: v),

text: we

text: say

text: that

text: u

text: is

text: adjacent

text: to

text: v

text: and

text: v

text: is

text: adjacent

text: from

text: u.

text: The

text: ordered

text: pair

text: (u,

text: v)

text: will

text: often

text: be

text: denoted

text: by

text: uv.

text: Sometimes

text: we

text: use

text: u

text: v

text: G

text: or

text: simply

text: u

text: v

text: to

text: represent

text: an

text: arc

text: (u,

text: v)

text: A(G).

text: The

text: vertex

text: u

text: is

text: the

text: start-vertex

text: and

text: v

text: is

text: the

text: end-vertex

text: of

text: a

text: given

text: arc

text: (u,

text: v).

text: An

text: arc

text: of

text: the

text: form

text: (u,

text: u),

text: or

text: equivalently

text: uu

text: is

text: called

text: a

text: loop.

text: A

text: mixed

text: graph

text: cannot

text: contain

text: multiple

text: arcs,

text: that

text: is,

text: it

text: cannot

text: contain

text: the

text: arc

text: (u,

text: v)

text: more

text: than

text: once.

text: A

text: mixed

text: graph

text: G

text: is

text: called

text: bipartite

text: if

text: there

text: is

text: a

text: partition

text: of

text: V(G)

text: into

text: two

text: sets

text: X

text: and

text: Y

text: which

text: we

text: call

text: colour

text: classes,

text: such

text: that

text: for

text: each

text: arc

text: uv

text: of

text: G

text: the

text: set

text: u,

text: v

text: intersects

text: both

text: X

text: and

text: Y

text: A

text: set

text: S

text: of

text: arcs

text: is

text: self-paired

text: if,

text: whenever

text: (u,

text: v)

text: S,

text: (v,

text: u)

text: is

text: also

text: in

text: S.

text: If

text: S

text: (u,

text: v),

text: (v,

text: u)

text: then

text: we

text: con-

text: sider

text: S

text: to

text: be

text: the

text: unordered

text: pair

text: u,

text: v

text: this

text: unordered

text: pair

text: is

text: called

text: an

text: edge.

text: It

text: is

text: useful

text: to

text: consider

text: two

text: special

text: cases

text: of

text: mixed

text: graphs.

text: A

text: graph

text: is

text: a

text: mixed

text: graph

text: without

text: loops

text: whose

text: arc-set

text: is

text: self-paired.

text: The

text: edge

text: set

text: of

text: a

text: graph

text: is

text: denoted

text: by

text: E(G).

text: A

text: digraph

text: is

text: a

text: mixed

text: graph

text: with

text: no

text: loops

text: in

text: which

text: no

text: set

text: of

text: arcs

text: is

text: self-paired.

text: The

text: inverse

text: G

text: of

text: a

text: mixed

text: graph

text: G

text: is

text: obtained

text: from

text: G

text: by

text: reversing

text: all

text: its

text: arcs,

text: that

text: is

text: V(G

text: =V(G)

text: and

text: (v,

text: u)

text: is

text: an

text: arc

text: of

text: G

text: if

text: and

text: only

text: if

text: (u,

text: v)

text: is

text: an

text: arc

text: of

text: G.

text: A

text: digraph

text: G

text: may

text: therefore

text: be

text: characterised

text: as

text: a

text: mixed

text: graph

text: for

text: which

text: A(G)

text: and

text: A(G

text: are

text: disjoint.

text: Given

text: a

text: mixed

text: graph

text: G

text: and

text: a

text: vertex

text: v

text: V(G),

text: we

text: deﬁne

text: the

text: in-neighbourhood

text: N

text: (v)

text: by

text: N

text: (v)

text: x

text: V(G)

text: (x,

text: v)

text: A(G)

text: Similarly

text: we

text: deﬁne

text: the

text: in

text: in

text: out-neighbourhood

text: N

text: (v)

text: by

text: N

text: (v)

text: x

text: V(G)

text: (v,

text: x)

text: A(G)

text: The

text: in-

text: out

text: out

text: degree

text: ρ

text: (v)

text: of

text: a

text: vertex

text: v

text: is

text: deﬁned

text: by

text: ρ

text: (v)

text: N

text: (v)

text: and

text: the

text: out-degree

text: in

text: in

text: in|

text: ρ

text: (v)

text: of

text: a

text: vertex

text: v

text: is

text: deﬁned

text: by

text: ρ

text: (v)

text: N

text: (v)

text: When

text: G

text: is

text: a

text: graph,

text: out

text: out

text: out

text: these

text: notions

text: reduce

text: to

text: the

text: usual

text: neighbourhood

text: N(v)

text: N

text: (v)

text: N

text: (v)

text: in

text: out

text: and

text: degree

text: ρ(v)

text: ρ

text: (v)

text: ρ

text: (v).

text: in

text: out

text: Let

text: G

text: be

text: a

text: graph

text: and

text: let

text: v

text: V(G).

text: Let

text: N(v)

text: be

text: the

text: neighbourhood

text: of

text: v.

text: We

text: say

text: that

text: G

text: is

text: vertex-determining

text: if

text: N(x)

text: N(y)

text: for

text: any

text: two

text: distinct

text: vertices

text: x

text: and

text: y

text: of

text: G

text: A

text: set

text: P

text: of

text: arcs

text: is

text: called

text: an

text: alternating

text: trail

text: or

text: Z-trail

text: if

text: its

text: elements

text: can

text: be

text: ordered

text: in

text: a

text: sequence

text: a

text: a

text: a

text: such

text: that

text: if

text: a

text: (u,

text: v)

text: or

text: k

text: i

text: a

text: (u,

text: w),

text: w

text: v

text: or

text: a

text: (w,

text: v),

text: w

text: u.

text: Note

text: that

text: in

text: a

text: Z-trail

text: vertices

text: may

text: be

text: repeated

text: but

text: arcs

text: may

text: not.

text: If

text: a

text: (p,

text: q)

text: and

text: a

text: (r,

text: s)

text: k

text: we

text: then

text: say

text: that

text: the

text: Z-trail

text: joins

text: p

text: and

text: s.

text: A

text: Z-trail

text: can

text: be

text: open

text: (when

text: the

text: ﬁrst

text: and

text: last

text: vertex

text: are

text: diﬀerent),

text: semi-closed

text: (when

text: the

text: ﬁrst

text: arc

text: is

text: uv

text: and

text: the

text: last

text: arc

text: is

text: wu)

text: or

text: closed

text: (when

text: the

text: ﬁrst

text: arc

text: is

text: uv

text: and

text: the

text: last

text: arc

text: is

text: uw

text: or

text: the

text: ﬁrst

text: arc

text: is

text: vu

text: and

text: the

text: last

text: arc

text: is

text: wu).

text: Note

text: that

text: the

text: ordering

text: of

text: arcs

text: described

text: in

text: the

text: deﬁnition

text: is

text: not

text: unique,

text: but

text: it

text: is

text: basically

text: unique

text: for

text: open

text: and

text: semi-closed

text: Z-trails

text: (one

text: just

text: chooses

text: the

text: beginning

text: or

text: the

text: end),

text: while

text: for

text: closed

text: Z-trails

text: all

text: vertices

text: can

text: be

text: taken

text: as

text: the

text: ﬁrst

text: vertex.

text: Whether

text: a

text: Z-trail

text: is

text: open,

text: semi-closed

text: or

text: closed

text: is

text: determined

text: by

text: the

text: ﬁrst

text: and

text: last


text: bipartite graphs, that is, the characterisation of those graphs G for which × (cid:39) × (cid:39) G C H C implies G H, whenever C is a bipartite graph (see [2] Chapter 9). The second part of Theorem 3.2 could be rephrased as follows: “G is unstable if and only if it has an anti-automorphism of order diﬀerent from 2”. Note that the existence of an anti-automorphisms of order 2 does not imply instability since such a map corresponds to a trivial TF- automorphisms. (cid:111) Z Theorem 3.2. Let G be a graph. Then Aut B(G) = AutTFG . Fur- 2 thermore, G is unstable if and only if it has a non-trivial TF-automorphism. Proof. From Lemma 3.1, f(Σ) = AutTFG which must have index 2 in (cid:55)→ Aut B(G). The permutation δ(v, ε) (v, ε + 1) is an automorphism of (cid:54)∈ B(G) and δ f(Σ). Then AutB(G) is generated by f(Σ) and δ. Further- ∩ (cid:104) (cid:105) (cid:67) more, f(Σ) δ = id and f(Σ) Aut B(G) being of index 2. (cid:111) Z Since Aut B(G) = AutTFG , G is stable if and only if AutTFG = 2 (cid:3) Aut G. Figure 2: A stable and unstable graph which are TF-isomorphic. It is natural to ask whether it can happen that a stable graph is TF- isomorphic to an unstable one. The answer is yes and an example is shown in Figure 2 with the Petersen graph being stable and the other graph which is TF-isomorphic to it being unstable. Both graphs have the same bipartite canonical double cover since they are TF-isomorphic. The reader may refer to [9] to learn more about graphs having the same canonical double cover. We should point out here (as noted by Surowski in [10]) that if a graph G is (cid:111) Z stable, that is, AutB(G) = AutG , then the semi-direct product must 2 be a direct product because AutG is normal in AutB(G) since it has index Z 2 and also is normal since its generator commutes with every element 2 of AutB(G), by stability. In fact, as Surowski comments, the stability of Z G is equivalent to the centrality of in AutB(G), which is the lift of the 2 5 ##LTFigure##

text: bipartite

label: paragraph

text: graphs,

text: that

text: is,

text: the

text: characterisation

text: of

text: those

text: graphs

text: G

text: for

text: which

text: G

text: C

text: H

text: C

text: implies

text: G

text: H,

text: whenever

text: C

text: is

text: a

text: bipartite

text: graph

text: (see

text: Chapter

text: The

text: second

text: part

text: of

text: Theorem

text: could

text: be

text: rephrased

text: as

text: follows:

text: “G

text: is

text: unstable

text: if

text: and

text: only

text: if

text: it

text: has

text: an

text: anti-automorphism

text: of

text: order

text: diﬀerent

text: from

text: Note

text: that

text: the

text: existence

text: of

text: an

text: anti-automorphisms

text: of

text: order

text: does

text: not

text: imply

text: instability

text: since

text: such

text: a

text: map

text: corresponds

text: to

text: a

text: trivial

text: TF-

text: automorphisms.

text: Z

text: Theorem

text: Let

text: G

text: be

text: a

text: graph.

text: Then

text: Aut

text: B(G)

text: AutTFG

text: Fur-

text: thermore,

text: G

text: is

text: unstable

text: if

text: and

text: only

text: if

text: it

text: has

text: a

text: non-trivial

text: TF-automorphism.

text: Proof.

text: From

text: Lemma

text: f(Σ)

text: AutTFG

text: which

text: must

text: have

text: index

text: in

text: Aut

text: B(G).

text: The

text: permutation

text: δ(v,

text: ε)

text: (v,

text: ε

text: is

text: an

text: automorphism

text: of

text: B(G)

text: and

text: δ

text: f(Σ).

text: Then

text: AutB(G)

text: is

text: generated

text: by

text: f(Σ)

text: and

text: δ.

text: Further-

text: more,

text: f(Σ)

text: δ

text: id

text: and

text: f(Σ)

text: Aut

text: B(G)

text: being

text: of

text: index

text: Z

text: Since

text: Aut

text: B(G)

text: AutTFG

text: G

text: is

text: stable

text: if

text: and

text: only

text: if

text: AutTFG

text: Aut

text: G.

text: Figure

text: A

text: stable

text: and

text: unstable

text: graph

text: which

text: are

text: TF-isomorphic.

text: It

text: is

text: natural

text: to

text: ask

text: whether

text: it

text: can

text: happen

text: that

text: a

text: stable

text: graph

text: is

text: TF-

text: isomorphic

text: to

text: an

text: unstable

text: one.

text: The

text: answer

text: is

text: yes

text: and

text: an

text: example

text: is

text: shown

text: in

text: Figure

text: with

text: the

text: Petersen

text: graph

text: being

text: stable

text: and

text: the

text: other

text: graph

text: which

text: is

text: TF-isomorphic

text: to

text: it

text: being

text: unstable.

text: Both

text: graphs

text: have

text: the

text: same

text: bipartite

text: canonical

text: double

text: cover

text: since

text: they

text: are

text: TF-isomorphic.

text: The

text: reader

text: may

text: refer

text: to

text: to

text: learn

text: more

text: about

text: graphs

text: having

text: the

text: same

text: canonical

text: double

text: cover.

text: We

text: should

text: point

text: out

text: here

text: (as

text: noted

text: by

text: Surowski

text: in

text: that

text: if

text: a

text: graph

text: G

text: is

text: Z

text: stable,

text: that

text: is,

text: AutB(G)

text: AutG

text: then

text: the

text: semi-direct

text: product

text: must

text: be

text: a

text: direct

text: product

text: because

text: AutG

text: is

text: normal

text: in

text: AutB(G)

text: since

text: it

text: has

text: index

text: Z

text: and

text: also

text: is

text: normal

text: since

text: its

text: generator

text: commutes

text: with

text: every

text: element

text: of

text: AutB(G),

text: by

text: stability.

text: In

text: fact,

text: as

text: Surowski

text: comments,

text: the

text: stability

text: of

text: Z

text: G

text: is

text: equivalent

text: to

text: the

text: centrality

text: of

text: in

text: AutB(G),

text: which

text: is

text: the

text: lift

text: of

text: the

text: ##LTFigure##


text: identity in AutG. (a,0) (a,1) (b,0) (b,1) a c e (c,0) (c,1) b d G (d,0) (d,1) (e,0) (e,1) B(G) Figure 3: An example used to show how a TF-automorphism (α, β) of G can be obtained from an automorphism σ of B(G). It is worth noting that the ideas explored in the proof of Lemma 3.1 may be used to extract TF-automorphisms of a graph G from automorphisms of B(G) which ﬁx the colour classes. In fact, let σ be such an automor- phism. Deﬁne the permutations α and β of V(G) as follows: α(x) = y if and only if σ(x, 0) = (y, 0) and β(x) = y if and only if σ(x, 1) = (y, 1). Then (α, β) is a TF-automorphism of G. We remark that α and β are not necessarily automorphisms of G as we shall show in the example shown in Figure 3. The automorphism σ is chosen so that it ﬁxes one compo- nent of B(G) whilst being an automorphism of the other component. In order to have a more concise representation, we denote vertices of B(G) of the form (u, 0), that is, elements of the colour class V by u and sim- 0 0 ilarly denote vertices of the form (u, 1) in V by u . Using this notation, 1 1 σ = (a )(b )(c )(d )(e)(a e )(b d )(c ). The permutations α and β of G 0 1 0 1 1 1 0 0 1 are extracted from σ as described in the proof of Lemma 3.1. For instance, to obtain α, we restrict the action of σ to the elements of V , that is, those 0 vertices of the form (v, 0) or v when using the new notation and then drop 0 the subscript. Similarly, the permutation β is obtained from the action of σ restricted to V . Therefore, α = (a)(c)(e)(b d) and β = (b)(d)(a e)(c). 1 Note that neither α nor β is an automorphism of G, but (α, β) is a TF- automorphism of G which in turn can be lifted to the unexpected automor- phism σ of B(G). This example illustrates Lemma 3.1 since the graph G is unstable and has a non-trivial TF-automorphism. The result of Lemma 3.1 and the subsequent example lead us to other questions regarding the nature of the permutations α and β which, as dis- 6 ##LTFigure##

text: identity

text: in

text: AutG.

font: default

text: a

text: c

text: e

text: b

text: d

text: G

text: B(G)

text: Figure

text: An

label: caption

text: example

text: used

text: to

text: show

text: how

text: a

text: TF-automorphism

text: (α,

text: β)

text: of

text: G

text: can

text: be

text: obtained

text: from

text: an

text: automorphism

text: σ

text: of

text: B(G).

text: It

text: is

text: worth

text: noting

text: that

text: the

text: ideas

text: explored

text: in

text: the

text: proof

text: of

text: Lemma

text: may

text: be

text: used

text: to

text: extract

text: TF-automorphisms

text: of

text: a

text: graph

text: G

text: from

text: automorphisms

text: of

text: B(G)

text: which

text: ﬁx

text: the

text: colour

text: classes.

text: In

text: fact,

text: let

text: σ

text: be

text: such

text: an

text: automor-

text: phism.

text: Deﬁne

text: the

text: permutations

text: α

text: and

text: β

text: of

text: V(G)

text: as

text: follows:

text: α(x)

text: y

text: if

text: and

text: only

text: if

text: σ(x,

text: (y,

text: and

text: β(x)

text: y

text: if

text: and

text: only

text: if

text: σ(x,

text: (y,

text: Then

text: (α,

text: β)

text: is

text: a

text: TF-automorphism

text: of

text: G.

text: We

text: remark

text: that

text: α

text: and

text: β

text: are

text: not

text: necessarily

text: automorphisms

text: of

text: G

text: as

text: we

text: shall

text: show

text: in

text: the

text: example

text: shown

text: in

text: Figure

text: The

text: automorphism

text: σ

text: is

text: chosen

text: so

text: that

text: it

text: ﬁxes

text: one

text: compo-

text: nent

text: of

text: B(G)

text: whilst

text: being

text: an

text: automorphism

text: of

text: the

text: other

text: component.

text: In

text: order

text: to

text: have

text: a

text: more

text: concise

text: representation,

text: we

text: denote

text: vertices

text: of

text: B(G)

text: of

text: the

text: form

text: (u,

text: that

text: is,

text: elements

text: of

text: the

text: colour

text: class

text: V

text: by

text: u

text: and

text: sim-

text: ilarly

text: denote

text: vertices

text: of

text: the

text: form

text: (u,

text: in

text: V

text: by

text: u

text: Using

text: this

text: notation,

text: σ

text: (a

text: )(b

text: )(c

text: )(d

text: )(e)(a

text: e

text: )(b

text: d

text: )(c

text: The

text: permutations

text: α

text: and

text: β

text: of

text: G

text: are

text: extracted

text: from

text: σ

text: as

text: described

text: in

text: the

text: proof

text: of

text: Lemma

text: For

text: instance,

text: to

text: obtain

text: α,

text: we

text: restrict

text: the

text: action

text: of

text: σ

text: to

text: the

text: elements

text: of

text: V

text: that

text: is,

text: those

text: vertices

text: of

text: the

text: form

text: (v,

text: or

text: v

text: when

text: using

text: the

text: new

text: notation

text: and

text: then

text: drop

text: the

text: subscript.

text: Similarly,

text: the

text: permutation

text: β

text: is

text: obtained

text: from

text: the

text: action

text: of

text: σ

text: restricted

text: to

text: V

text: Therefore,

text: α

text: (a)(c)(e)(b

text: d)

text: and

text: β

text: (b)(d)(a

text: e)(c).

text: Note

text: that

text: neither

text: α

text: nor

text: β

text: is

text: an

text: automorphism

text: of

text: G,

text: but

text: (α,

text: β)

text: is

text: a

text: TF-

text: automorphism

text: of

text: G

text: which

text: in

text: turn

text: can

text: be

text: lifted

text: to

text: the

text: unexpected

text: automor-

text: phism

text: σ

text: of

text: B(G).

text: This

text: example

text: illustrates

text: Lemma

text: since

text: the

text: graph

text: G

text: is

text: unstable

text: and

text: has

text: a

text: non-trivial

text: TF-automorphism.

text: The

text: result

text: of

text: Lemma

text: and

text: the

text: subsequent

text: example

text: lead

text: us

text: to

text: other

text: questions

text: regarding

text: the

text: nature

text: of

text: the

text: permutations

text: α

text: and

text: β

text: which,

text: as

text: dis-

text: ##LTFigure##


text: 9 is observed to transition from ω4 below the boson peak Our approach predicts that the vibrational properties frequency to ω2 above it [2, 3]. Our theory is fully con- of amorphous solids depend on their excess-coordination sistent with these results, if ω ≈ ω . Below we will δz = z−z and compressive strain e (keeping in mind that BP 0 c argue that silica corresponds to δz = 0, e < 0 for which the presence of weak interactions can be incorporated by this property is satisﬁed, as illustrated in Figure 1b. lowering the value of e). In the (δz, e) plane there is a forbidden region where no mechanically stable glasses are possible, represented in white in Fig. 2. Purely repulsive 100 particles such as elastic spheres and colloidal glasses must lie in the red-blue region, which is stable despite e > 0. In network glasses at small pressure, such as chalcogenides, 10−5 z−z can be monitored by changing the valence, allowing c to explore the blue part of the phase diagram of Fig. 2. In this case e < 0 due to the presence of weak Van der ) (ω 10−10 Waals interactions that can stabilize the system even if Γ z − z < 0. We shall recall below why silica corresponds c 1−e/ec = 5 × 10−1 to δz = 0 and e < 0. We now discuss the consequence of 1−e/ec = 5 × 10−2 this classiﬁcation in each case. 10−15 1−e/ec = 5 × 10−3 Repulsive short-range particles: In granular materials, 1−e/ec = 5 × 10−4 emulsions, and hard sphere colloidal glasses the particle interaction is repulsive and short-range, implying that 10−20 e > 0. Thus these systems lie in the upper right corner 10−2 10−1 100 101 102 ω/ω of the phase diagram of Fig. 2. Considerable attention ∗ has been given in the “jamming” literature to the case of FIG. 10. Sound attenuation Γ(ω) as strain approaches its frictionless spheres interacting via a ﬁnite range poten- critical value, at δz = 0.012. tial [33], because vibrational properties display critical properties as compression vanishes. To some extent, this scaling behavior can be experimentally observed [33], in The interpretation of S in terms of a resonant peak in particular in emulsions (see e.g. [66]). Numerically, scal- is not appropriate as soon as the scattering length of ing exponents can be extracted precisely, which enables phonons is equal to half their wavelength. This deﬁnes stringent testing of theories of elasticity and transport in the Ioﬀe-Regel frequency ω at which πΓ = Ω [64]. We IR amorphous materials. predict Here we consider particles interacting via a one-sided (cid:113) harmonic potential (extension to other potentials, e.g. ω ≈ ω2 + ω2/π2 (22) IR 0 ∗ Hertzian, is straightforward). The contact compression is simply proportional to the increase of packing fraction It has been suggested that the boson peak frequency is e ∝ φ − φ , where φ is the point at which pressure van- c c equal to ω . As illustrated in Figure 1b, we ﬁnd that IR ishes. Another useful system to consider can be made by it depends on the microscopic structure of the glass. For replacing particles by points, and contacts by harmonic sphere packings e → e (see below) and ω and ω c IR BP springs at rest [40]. This essentially removes the pre- are predicted to diﬀer. For silica and well-coordinated stress term in the expansion of δE, see Eq. (1), and cor- covalent networks e < 0 and we predict ω ≈ ω , in IR BP responds to setting e = 0 in our formalism. It was found agreement with observations [2]. previously that (i) the shear modulus satisﬁes µ ∼ δz Above ω , modes no longer resemble plane waves. IR independently of pre-stress [37, 51], and the prefactor of This is apparent in Figure 6, which shows that ν (cid:54)= ν res this relation decreases as the compressive pre-stress in- √ above ω : the phase velocity of the total response is IR creases [67, 68]. (ii) δz ∼ e [37]. (iii) The vibrational not characterized at all by ν . Similarly, the scattering res spectrum displays one frequency scale ω ∼ δz above length of the resonant wavenumber, (cid:96) (ω) = 2ν Γ−1, ∗ res res which D(ω) displays a plateau [38, 40]. This is true in- diﬀers from (cid:96) above ω , as shown in Figure 7. The dif- s IR dependent of the pre-stress, but with pre-stress D(ω) is ference can be dramatic: we see that for ω ∼ 30ω , the to- ∗ much larger for ω < ω [40], and presents non-plane- ∗ tal response persists for nearly 2 decades longer than the wave-like modes up to very small frequencies. (iv) The contribution from the resonant wavenumber. These com- diﬀusivity is essentially independent of frequency in pack- ments underline the relevance of ν and (cid:96) as the physical s ings, but presents a behaviour consistent with Rayleigh velocity and length scales in the entire frequency range scattering below ω when pre-stress is removed [5, 6]. ∗ (below the localization transition). All these behaviours follow precisely our predictions, if we suppose that packings lie very close to the stabil- ity boundary in Figure 1a. Such marginal stability was VII. COMPARISON WITH SPECIFIC GLASSES proposed in [40] to rationalize the structure of packings; proposed explanations for this behaviour can be found ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: observed

text: to

text: transition

text: from

text: ω4

text: below

text: the

text: boson

text: peak

text: Our

text: approach

text: predicts

text: that

text: the

text: vibrational

text: properties

text: frequency

text: to

text: ω2

text: above

text: it

text: Our

text: theory

text: is

text: fully

text: con-

text: of

text: amorphous

text: solids

text: depend

text: on

text: their

text: excess-coordination

text: sistent

text: with

text: these

text: results,

text: if

text: ω

text: ω

text: Below

text: we

text: will

text: δz

text: z−z

text: and

text: compressive

text: strain

text: e

text: (keeping

text: in

text: mind

text: that

text: BP

text: c

text: argue

text: that

text: silica

text: corresponds

text: to

text: δz

text: e

text: for

text: which

text: the

text: presence

text: of

text: weak

text: interactions

text: can

text: be

text: incorporated

text: by

text: this

text: property

text: is

text: satisﬁed,

text: as

text: illustrated

text: in

text: Figure

text: lowering

text: the

text: value

text: of

text: e).

text: In

text: the

text: (δz,

text: e)

text: plane

text: there

text: is

text: a

text: forbidden

text: region

text: where

text: no

text: mechanically

text: stable

text: glasses

text: are

text: possible,

text: represented

text: in

text: white

text: in

text: Fig.

text: Purely

text: repulsive

text: particles

text: such

text: as

text: elastic

text: spheres

text: and

text: colloidal

text: glasses

text: must

text: lie

text: in

text: the

text: red-blue

text: region,

text: which

text: is

text: stable

text: despite

text: e

text: In

text: network

text: glasses

text: at

text: small

text: pressure,

text: such

text: as

text: chalcogenides,

text: z−z

text: can

text: be

text: monitored

text: by

text: changing

text: the

text: valence,

text: allowing

text: c

text: to

text: explore

text: the

text: blue

text: part

text: of

text: the

text: phase

text: diagram

text: of

text: Fig.

text: In

text: this

text: case

text: e

text: due

text: to

text: the

text: presence

text: of

text: weak

text: Van

text: der

text: (ω

text: Waals

text: interactions

text: that

text: can

text: stabilize

text: the

text: system

text: even

text: if

text: Γ

text: z

text: z

text: We

text: shall

text: recall

text: below

text: why

text: silica

text: corresponds

text: c

text: to

text: δz

text: and

text: e

text: We

text: now

text: discuss

text: the

text: consequence

text: of

text: this

text: classiﬁcation

text: in

text: each

text: case.

text: Repulsive

text: short-range

text: particles:

text: In

text: granular

text: materials,

text: emulsions,

text: and

text: hard

text: sphere

text: colloidal

text: glasses

text: the

text: particle

text: interaction

text: is

text: repulsive

text: and

text: short-range,

text: implying

text: that

text: e

text: Thus

text: these

text: systems

text: lie

text: in

text: the

text: upper

text: right

text: corner

text: ω/ω

text: of

text: the

text: phase

text: diagram

text: of

text: Fig.

text: Considerable

text: attention

text: has

text: been

text: given

text: in

text: the

text: “jamming”

text: literature

text: to

text: the

text: case

text: of

text: FIG.

text: Sound

text: attenuation

text: Γ(ω)

text: as

text: strain

text: approaches

text: its

text: frictionless

text: spheres

text: interacting

text: via

text: a

text: ﬁnite

text: range

text: poten-

text: critical

text: value,

text: at

text: δz

text: tial

text: because

text: vibrational

text: properties

text: display

text: critical

text: properties

text: as

text: compression

text: vanishes.

text: To

text: some

text: extent,

text: this

text: scaling

text: behavior

text: can

text: be

text: experimentally

text: observed

text: in

text: The

text: interpretation

text: of

text: S

text: in

text: terms

text: of

text: a

text: resonant

text: peak

text: in

text: particular

text: in

text: emulsions

text: (see

text: e.g.

text: Numerically,

text: scal-

text: is

text: not

text: appropriate

text: as

text: soon

text: as

text: the

text: scattering

text: length

text: of

text: ing

text: exponents

text: can

text: be

text: extracted

text: precisely,

text: which

text: enables

text: phonons

text: is

text: equal

text: to

text: half

text: their

text: wavelength.

text: This

text: deﬁnes

text: stringent

text: testing

text: of

text: theories

text: of

text: elasticity

text: and

text: transport

text: in

text: the

text: Ioﬀe-Regel

text: frequency

text: ω

text: at

text: which

text: πΓ

text: Ω

text: We

text: IR

text: amorphous

text: materials.

text: predict

text: Here

text: we

text: consider

text: particles

text: interacting

text: via

text: a

text: one-sided

text: harmonic

text: potential

text: (extension

text: to

text: other

text: potentials,

text: e.g.

text: ω

text: ω2

text: IR

text: Hertzian,

text: is

text: straightforward).

text: The

text: contact

text: compression

text: is

text: simply

text: proportional

text: to

text: the

text: increase

text: of

text: packing

text: fraction

text: It

text: has

text: been

text: suggested

text: that

text: the

text: boson

text: peak

text: frequency

text: is

text: e

text: φ

text: φ

text: where

text: φ

text: is

text: the

text: point

text: at

text: which

text: pressure

text: van-

text: c

text: c

text: equal

text: to

text: ω

text: As

text: illustrated

text: in

text: Figure

text: we

text: ﬁnd

text: that

text: IR

text: ishes.

text: Another

text: useful

text: system

text: to

text: consider

text: can

text: be

text: made

text: by

text: it

text: depends

text: on

text: the

text: microscopic

text: structure

text: of

text: the

text: glass.

text: For

text: replacing

text: particles

text: by

text: points,

text: and

text: contacts

text: by

text: harmonic

text: sphere

text: packings

text: e

text: e

text: (see

text: below)

text: and

text: ω

text: and

text: ω

text: c

text: IR

text: BP

text: springs

text: at

text: rest

text: This

text: essentially

text: removes

text: the

text: pre-

text: are

text: predicted

text: to

text: diﬀer.

text: For

text: silica

text: and

text: well-coordinated

text: stress

text: term

text: in

text: the

text: expansion

text: of

text: δE,

text: see

text: Eq.

text: and

text: cor-

text: covalent

text: networks

text: e

text: and

text: we

text: predict

text: ω

text: ω

text: in

text: IR

text: BP

text: responds

text: to

text: setting

text: e

text: in

text: our

text: formalism.

text: It

text: was

text: found

text: agreement

text: with

text: observations

text: previously

text: that

text: (i)

text: the

text: shear

text: modulus

text: satisﬁes

text: µ

text: δz

text: Above

text: ω

text: modes

text: no

text: longer

text: resemble

text: plane

text: waves.

text: IR

text: independently

text: of

text: pre-stress

text: and

text: the

text: prefactor

text: of

text: This

text: is

text: apparent

text: in

text: Figure

text: which

text: shows

text: that

text: ν

text: ν

text: res

text: this

text: relation

text: decreases

text: as

text: the

text: compressive

text: pre-stress

text: in-

text: above

text: ω

text: the

text: phase

text: velocity

text: of

text: the

text: total

text: response

text: is

text: IR

text: creases

text: (ii)

text: δz

text: e

text: (iii)

text: The

text: vibrational

text: not

text: characterized

text: at

text: all

text: by

text: ν

text: Similarly,

text: the

text: scattering

text: res

text: spectrum

text: displays

text: one

text: frequency

text: scale

text: ω

text: δz

text: above

text: length

text: of

text: the

text: resonant

text: wavenumber,

text: (ω)

text: 2ν

text: Γ−1,

text: res

text: res

text: which

text: D(ω)

text: displays

text: a

text: plateau

text: This

text: is

text: true

text: in-

text: diﬀers

text: from

text: above

text: ω

text: as

text: shown

text: in

text: Figure

text: The

text: dif-

text: s

text: IR

text: dependent

text: of

text: the

text: pre-stress,

text: but

text: with

text: pre-stress

text: D(ω)

text: is

text: ference

text: can

text: be

text: dramatic:

text: we

text: see

text: that

text: for

text: ω

text: the

text: to-

text: much

text: larger

text: for

text: ω

text: ω

text: and

text: presents

text: non-plane-

text: tal

text: response

text: persists

text: for

text: nearly

text: decades

text: longer

text: than

text: the

text: wave-like

text: modes

text: up

text: to

text: very

text: small

text: frequencies.

text: (iv)

text: The

text: contribution

text: from

text: the

text: resonant

text: wavenumber.

text: These

text: com-

text: diﬀusivity

text: is

text: essentially

text: independent

text: of

text: frequency

text: in

text: pack-

text: ments

text: underline

text: the

text: relevance

text: of

text: ν

text: and

text: as

text: the

text: physical

text: s

text: ings,

text: but

text: presents

text: a

text: behaviour

text: consistent

text: with

text: Rayleigh

text: velocity

text: and

text: length

text: scales

text: in

text: the

text: entire

text: frequency

text: range

text: scattering

text: below

text: ω

text: when

text: pre-stress

text: is

text: removed

text: (below

text: the

text: localization

text: transition).

text: All

text: these

text: behaviours

text: follow

text: precisely

text: our

text: predictions,

text: if

text: we

text: suppose

text: that

text: packings

text: lie

text: very

text: close

text: to

text: the

text: stabil-

text: ity

text: boundary

text: in

text: Figure

text: Such

text: marginal

text: stability

text: was

text: VII.

text: COMPARISON

text: WITH

text: SPECIFIC

text: GLASSES

text: proposed

text: in

text: to

text: rationalize

text: the

text: structure

text: of

text: packings;

text: proposed

text: explanations

text: for

text: this

text: behaviour

text: can

text: be

text: found

text: ##LTFigure##

text: ##LTLine##

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text: → → ¯ ¯ ¯ b B (bc¯)) is more suppressed compared with a single heavy meson (for example b B (bd)) by c d − ∼ − a factor of 10 4 10 3 [31]. The case of the SUSY hadron production is similar, i.e. production of mesino t˜ ¯b(c¯) is more suppressed than t˜ q¯ (q = u, d, s). 1 1 A theoretical estimate shows that the so called stoponium can be formed, and the binding energy is about 1-3 GeV [32] which is much smaller than the mass of stop and does not aﬀect the phase space of the production. Next we calculate the production rate of e+e− → X˜X¯˜ near its threshold at ILC, whose low background makes it more advantageous over hadron colliders. B. Estimating the SUSY mesino production rate To predict the production rate of the SUSY mesinos near threshold, one could associate it with the B-meson production near threshold. Indeed, we wish to use the data of the B-factory to predict the production rates as σtheor(e+e− → X˜X¯˜) ∼ σtheor(e+e− → BB¯) σ(e+e− → X˜X¯˜) σexp(e+e− → BB¯) , (7) − → where the superscript ”theor” means the theoretically predicted value and σexp(e+e BB¯) is the measured value at B-factories and σ(e+e− → X˜X¯˜) is what we expect. The ratio of σtheor(e+e− → X˜X¯˜) − → σtheor(e+e BB¯) can be obtained in terms of the superﬂavor symmetry, so that one can eventually obtain σ(e+e− → X˜X¯˜). In fact, by the superﬂavor symmetry we can relate the matrix element < X˜X¯˜|Jµ|0 > to the matrix element < BB¯|J(cid:48)µ|0 >, where Jµ and J(cid:48)µ are vector currents corresponding to squark-anti-sqaurk and quark-anti-quark productions respectively. However, there is a serious problem that all the available data about the B-meson productions − → → are not exactly what we need, because the available data are from e+e Υ(4s)/Υ(5s)/Υ(6s) ¯ BB, namely via the Υ resonances. Instead, we need the data on the direct production of − → e+e BB¯, i.e the contribution of the continuum spectrum near the threshold. The total −→ σ(e+e hadrons) spectrum on Rb (deﬁned as Rb = σ(e+e−→µ+µ−) ) provided by experimentalists [23] which is close to 0.3 also cannot be used either.1 1 For this point, we thank Dr. C.Z. Yuan of IHEP who told us that there are no such data about the continuum 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: b

text: B

text: (bc¯))

text: is

text: more

text: suppressed

text: compared

text: with

text: a

text: single

text: heavy

text: meson

text: (for

text: example

text: b

text: B

text: (bd))

text: by

text: c

text: d

text: a

text: factor

text: of

text: The

text: case

text: of

text: the

text: SUSY

text: hadron

text: production

text: is

text: similar,

text: i.e.

text: production

text: of

text: mesino

text: t˜

text: ¯b(c¯)

text: is

text: more

text: suppressed

text: than

text: t˜

text: q¯

text: (q

text: u,

text: d,

text: s).

text: A

text: theoretical

text: estimate

text: shows

text: that

text: the

text: so

text: called

text: stoponium

text: can

text: be

text: formed,

text: and

text: the

text: binding

text: energy

text: is

text: about

text: GeV

text: which

text: is

text: much

text: smaller

text: than

text: the

text: mass

text: of

text: stop

text: and

text: does

text: not

text: aﬀect

text: the

text: phase

text: space

text: of

text: the

text: production.

text: Next

text: we

text: calculate

text: the

text: production

text: rate

text: of

text: e+e−

text: X˜X¯˜

text: near

text: its

text: threshold

text: at

text: ILC,

text: whose

text: low

text: background

text: makes

text: it

text: more

text: advantageous

text: over

text: hadron

text: colliders.

text: B.

text: Estimating

text: the

text: SUSY

text: mesino

text: production

text: rate

text: To

text: predict

text: the

text: production

text: rate

text: of

text: the

text: SUSY

text: mesinos

text: near

text: threshold,

text: one

text: could

text: associate

text: it

text: with

text: the

text: B-meson

text: production

text: near

text: threshold.

text: Indeed,

text: we

text: wish

text: to

text: use

text: the

text: data

text: of

text: the

text: B-factory

text: to

text: predict

text: the

text: production

text: rates

text: as

text: σtheor(e+e−

text: X˜X¯˜)

text: σtheor(e+e−

text: BB¯)

text: σ(e+e−

text: X˜X¯˜)

text: σexp(e+e−

text: BB¯)

text: where

text: the

text: superscript

text: ”theor”

text: means

text: the

text: theoretically

text: predicted

text: value

text: and

text: σexp(e+e

text: BB¯)

text: is

text: the

text: measured

text: value

text: at

text: B-factories

text: and

text: σ(e+e−

text: X˜X¯˜)

text: is

text: what

text: we

text: expect.

text: The

text: ratio

text: of

text: σtheor(e+e−

text: X˜X¯˜)

text: σtheor(e+e

text: BB¯)

text: can

text: be

text: obtained

text: in

text: terms

text: of

text: the

text: superﬂavor

text: symmetry,

text: so

text: that

text: one

text: can

text: eventually

text: obtain

text: σ(e+e−

text: X˜X¯˜).

text: In

text: fact,

text: by

text: the

text: superﬂavor

text: symmetry

text: we

text: can

text: relate

text: the

text: matrix

text: element

text: to

text: the

text: matrix

text: element

text: where

text: Jµ

text: and

text: are

text: vector

text: currents

text: corresponding

text: to

text: squark-anti-sqaurk

text: and

text: quark-anti-quark

text: productions

text: respectively.

text: However,

text: there

text: is

text: a

text: serious

text: problem

text: that

text: all

text: the

text: available

text: data

text: about

text: the

text: B-meson

text: productions

text: are

text: not

text: exactly

text: what

text: we

text: need,

text: because

text: the

text: available

text: data

text: are

text: from

text: e+e

text: BB,

text: namely

text: via

text: the

text: Υ

text: resonances.

text: Instead,

text: we

text: need

text: the

text: data

text: on

text: the

text: direct

text: production

text: of

text: e+e

text: BB¯,

text: i.e

text: the

text: contribution

text: of

text: the

text: continuum

text: spectrum

text: near

text: the

text: threshold.

text: The

text: total

text: σ(e+e

text: hadrons)

text: spectrum

text: on

text: Rb

text: (deﬁned

text: as

text: Rb

text: σ(e+e−→µ+µ−)

text: provided

text: by

text: experimentalists

text: which

text: is

text: close

text: to

text: also

text: cannot

text: be

text: used

text: For

text: this

text: point,

text: we

text: thank

text: Dr.

text: C.Z.

text: Yuan

text: of

text: IHEP

text: who

text: told

text: us

text: that

text: there

text: are

text: no

text: such

text: data

text: about

text: the

text: continuum

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: k1 b p2 t˜1 p2 k1 γ γ,Z0 k2 ¯b p1 k2 t˜¯1 p1 FIG. 1: The process of e+e− → BB¯. FIG. 2: The process of e+e− → X˜X¯˜. − M − − 1 − i − i = ξ( ω)( i e(p p ) ) v¯(k )( ieγµ)u(k ) , (10) B 2 1 µ 2 1 3 s 1 and for mesinos are M − − − − i = ξ( ω)[u¯(p )i2e(p2 p1)µ v(p ) iv¯(k )( ieγµ)u(k ) X˜ 2 −3 2mt˜1 − 1 s2 2 1 (11) +u¯(p )g (p2 p1)µ v(p ) −i v¯(k )γ g u(k )] . 2 tz 2mt˜1 1 s2 m2Z 2 µ ez 1 · (cid:48) − Here ω = v v = s 1, k and k are the momenta of the incoming electron and positron, p 2m2 1 2 1 and p are the momenta of the outgoing anti-hadron and hadron. It is noted that the hadronic 2 matrix elements are determined according to the superﬂavor symmetry as shown in Eqs.(5) and (6). Thus we obtain the cross section for pair productions as (cid:82) (cid:88) − − |M| σ = 1 d3p1 1 d3p2 1 (2π)4δ4(p + p k k )1 2. (12) 2s (2π)3 2E1 (2π)3 2E2 1 2 1 2 4 spin | − | The ﬁnal expression includes the Isgur-Wise function ξ( ω) 2 which determines the hadronic matrix elements and manifests the non-perturbative QCD eﬀects in the hadronization. As mentioned above, we cannot use the data to ﬁx the parameters, so generally we will obtain the values of the Isgur-Wise function for certain ω by employing some phenomenological models. III. NUMERICAL ANALYSIS 8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: b

text: γ

text: ¯b

text: t˜¯1

text: FIG.

text: The

text: process

text: of

text: e+e−

text: BB¯.

text: FIG.

text: The

text: process

text: of

text: e+e−

text: X˜X¯˜.

label: equation

text: M

text: i

text: i

text: ξ(

text: ω)(

text: i

text: e(p

text: p

text: v¯(k

text: ieγµ)u(k

text: B

text: µ

text: s

text: and

text: for

text: mesinos

text: are

text: M

text: i

text: ξ(

text: ω)[u¯(p

text: v(p

text: iv¯(k

text: ieγµ)u(k

text: X˜

text: −3

text: +u¯(p

text: )g

text: v(p

text: −i

text: v¯(k

text: )γ

text: g

text: u(k

text: tz

text: µ

text: ez

text: Here

text: ω

text: v

text: v

text: s

text: k

text: and

text: k

text: are

text: the

text: momenta

text: of

text: the

text: incoming

text: electron

text: and

text: positron,

text: p

text: and

text: p

text: are

text: the

text: momenta

text: of

text: the

text: outgoing

text: anti-hadron

text: and

text: hadron.

text: It

text: is

text: noted

text: that

text: the

text: hadronic

text: matrix

text: elements

text: are

text: determined

text: according

text: to

text: the

text: superﬂavor

text: symmetry

text: as

text: shown

text: in

text: and

text: Thus

text: we

text: obtain

text: the

text: cross

text: section

text: for

text: pair

text: productions

text: as

text: |M|

text: σ

text: p

text: k

text: k

text: spin

text: The

text: ﬁnal

text: expression

text: includes

text: the

text: Isgur-Wise

text: function

text: ξ(

text: ω)

text: which

text: determines

text: the

text: hadronic

text: matrix

text: elements

text: and

text: manifests

text: the

text: non-perturbative

text: QCD

text: eﬀects

text: in

text: the

text: hadronization.

text: As

text: mentioned

text: above,

text: we

text: cannot

text: use

text: the

text: data

text: to

text: ﬁx

text: the

text: parameters,

text: so

text: generally

text: we

text: will

text: obtain

text: the

text: values

text: of

text: the

text: Isgur-Wise

text: function

text: for

text: certain

text: ω

text: by

text: employing

text: some

text: phenomenological

text: models.

text: III.

text: NUMERICAL

text: ANALYSIS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: BLOW-UP FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM KATRIN GRUNERT Abstract. Following conservative solutions of the two-component Camassa– − − − Holm system ut utxx+3uux 2uxuxx uuxxx+ρρx = 0, ρt+(uρ)x = 0 along characteristics, we determine if wave breaking occurs in the nearby future or ∈ R ∈ R not, for initial data u0 H1( ) and ρ0 L2( ). 1. Introduction Over the last few years a lot of generalizations of the well-studied Camassa–Holm (CH) equation, [2], − − − (1.1) u u + 3uu 2u u uu = 0 t txx x x xx xxx have been introduced. Among them the two-component Camassa–Holm (2CH) system, − − − (1.2) u u + 3uu 2u u uu + ρρ = 0 t txx x x xx xxx x ρ + (uρ) = 0, t x which we want to consider here. It has been derived as a modell for shallow water by Constantin and Ivanov [4] and has been studied intensively over the last few years, see e.g. [7, 8, 12, 13, 14] and the references therein. The great interest in the CH equation and its generalizations, see e.g. [3, 5, 6, 10, 11, 18, 19] relies on the fact that even smooth initial data might lead to classical solutions which only exist locally. In the case of the 2CH system, this is due to the fact that wave breaking can occur within ﬁnite time, that is u (cid:107) (cid:107) (cid:107) (cid:107) x becomes unbounded from below pointwise while u and ρ remain bounded, HR1 L2R see e.g. [12, 13, 14, 15, 20, 21] and the references therein. More precisely the function u2 +ρ2 turns into a positive, ﬁnite Radon measure at breaking time, which x means that energy concentrates on sets of measure zero. Thus in order to describe ∈ R weak solutions, one does not only consider pairs (u, ρ), such that u H1( ) and ∈ R ∈ D ρ L2( ), but triplets (u, ρ, µ) , where µ is a positive, ﬁnite Radon measure with µ = (u2 + ρ2)dx, see Deﬁnition 2.1. ac x However, one is still facing the problem, what would be a natural way to prolong the solution beyond wave breaking. One possibility is to reformulate the 2CH 2010 Mathematics Subject Classiﬁcation. Primary: 35Q53, 35B35; Secondary: 35B44. Key words and phrases. Two-component Camassa–Holm system, blow up, regularization. Research supported by the Austrian Science Fund (FWF) under Grant No. J3147. 1 ##LTLine##

text: BLOW-UP

text: FOR

text: THE

text: TWO-COMPONENT

text: CAMASSA–HOLM

text: SYSTEM

text: KATRIN

text: GRUNERT

text: Abstract.

text: Following

label: abstract

text: conservative

text: solutions

text: of

text: the

text: two-component

text: Camassa–

text: Holm

text: system

text: ut

text: uuxxx+ρρx

text: ρt+(uρ)x

text: along

text: characteristics,

text: we

text: determine

text: if

text: wave

text: breaking

text: occurs

text: in

text: the

text: nearby

text: future

text: or

text: R

text: R

text: not,

text: for

text: initial

text: data

text: and

text: ρ0

label: section

text: Introduction

text: Over

text: the

text: last

text: few

text: years

text: a

text: lot

text: of

text: generalizations

text: of

text: the

text: well-studied

text: Camassa–Holm

text: (CH)

text: equation,

text: u

text: u

text: u

text: uu

text: t

text: txx

text: x

text: x

text: xx

text: xxx

text: have

text: been

text: introduced.

text: Among

text: them

text: the

text: two-component

text: Camassa–Holm

text: system,

text: u

text: u

text: u

text: uu

text: ρρ

text: t

text: txx

text: x

text: x

text: xx

text: xxx

text: x

text: ρ

text: (uρ)

text: t

text: x

text: which

text: we

text: want

text: to

text: consider

text: here.

text: It

text: has

text: been

text: derived

text: as

text: a

text: modell

text: for

text: shallow

text: water

text: by

text: Constantin

text: and

text: Ivanov

text: and

text: has

text: been

text: studied

text: intensively

text: over

text: the

text: last

text: few

text: years,

text: see

text: e.g.

text: and

text: the

text: references

text: therein.

text: The

text: great

text: interest

text: in

text: the

text: CH

text: equation

text: and

text: its

text: generalizations,

text: see

text: e.g.

text: relies

text: on

text: the

text: fact

text: that

text: even

text: smooth

text: initial

text: data

text: might

text: lead

text: to

text: classical

text: solutions

text: which

text: only

text: exist

text: locally.

text: In

text: the

text: case

text: of

text: the

text: system,

text: this

text: is

text: due

text: to

text: the

text: fact

text: that

text: wave

text: breaking

text: can

text: occur

text: within

text: ﬁnite

text: time,

text: that

text: is

text: u

text: x

text: becomes

text: unbounded

text: from

text: below

text: pointwise

text: while

text: u

text: and

text: ρ

text: remain

text: bounded,

text: see

text: e.g.

text: and

text: the

text: references

text: therein.

text: More

text: precisely

text: the

text: function

text: turns

text: into

text: a

text: positive,

text: ﬁnite

text: Radon

text: measure

text: at

text: breaking

text: time,

text: which

text: x

text: means

text: that

text: energy

text: concentrates

text: on

text: sets

text: of

text: measure

text: zero.

text: Thus

text: in

text: order

text: to

text: describe

text: R

text: weak

text: solutions,

text: one

text: does

text: not

text: only

text: consider

text: pairs

text: (u,

text: ρ),

text: such

text: that

text: u

text: and

text: R

text: D

text: ρ

text: but

text: triplets

text: (u,

text: ρ,

text: µ)

text: where

text: µ

text: is

text: a

text: positive,

text: ﬁnite

text: Radon

text: measure

text: with

text: µ

text: see

text: Deﬁnition

text: ac

text: x

text: However,

text: one

text: is

text: still

text: facing

text: the

text: problem,

text: what

text: would

text: be

text: a

text: natural

text: way

text: to

text: prolong

text: the

text: solution

text: beyond

text: wave

text: breaking.

text: One

text: possibility

text: is

text: to

text: reformulate

text: the

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary:

text: Secondary:

text: Key

text: words

text: and

text: phrases.

text: Two-component

text: Camassa–Holm

text: system,

text: blow

text: up,

text: regularization.

text: Research

text: supported

text: by

text: the

text: Austrian

text: Science

text: Fund

text: (FWF)

text: under

text: Grant

text: No.

text: ##LTLine##


text: BLOW-UP FOR THE 2CH SYSTEM 5 whose time evolution is given through (2.2). However, this ansatz has two draw- backs, on the one hand we do not know yet how µ(t, x) evolves with respect to time, and on the other hand how to combine µ(t, x) and y(t, ξ). The last issue will D be resolved next by introducing a suitable mapping from Eulerian coordinates F to Lagrangian coordinates . D Deﬁnition 2.2. For any (u, ρ, µ) in , let { | −∞ } (2.6a) y(ξ) = sup y µ(( , y)) + y < ξ , − (2.6b) h(ξ) = 1 y (ξ), ξ ◦ (2.6c) U(ξ) = u y(ξ), ◦ (2.6d) r(ξ) = ρ y(ξ)y (ξ). ξ ∈ F D → F Then (y, U, h, r) . We denote by L: the mapping which to any ∈ D ∈ F element (u, ρ, µ) associates X = (y, U, h, r) given by (2.6). Since µ is a positive, ﬁnite Radon measure, we have that the function F(x) = −∞ µ(( , x)) is increasing and lower semi-continuous. We therefore obtain, as an immediate consequence, that y is an increasing function, such that we can ﬁnd to ∈ R ∈ R any x at least one ξ such that y(ξ) = x. In particular, we have that if { } a positive amount of energy concentrates at some point x, i.e. µ( x ) = c > 0, then (2.6a) implies that the point x will be mapped to an interval [ξ , ξ ], such that − ∈ 1 2 ξ ξ = c and y(ξ) = x for all ξ [ξ , ξ ]. Moreover, y is Lipschitz continuous 2 1 1 2 (cid:82) with Lipschitz constant at most one, and hence diﬀerentiable almost everywhere ∈ by Rademacher’s theorem. Thus y (ξ) = 0 for almost every ξ [ξ , ξ ], which is ξ 1 2 ∈ { } equivalent to h(ξ) = 1 for almost every ξ [ξ , ξ ]. Thus µ( x ) = ξ2 h(ξ)dξ. In 1 2 ξ1 general, one has that { ∈ R | } { ∈ R | ∈ } ξ y (ξ) = 0 = ξ y(ξ) supp(µ ) , ξ s R (cid:107) (cid:107) by Besicovitch’s derivation theorem, see e.g. [1, Theorem 2.22], and µ( ) = h . L1R The above considerations show that wave breaking in Lagrangian coordinates means that y (ξ) equals 0, and the concentrated energy can be read of from the ξ { ∈ R | } function h(ξ) integrated over ξ y (ξ) = 0 . In particular, one can view, ξ roughly speaking, h, which is a function, as the relabeled version of the measure µ. ∈ R Therefore, (2.1) implies that hy = U2 + r2 for almost every ξ . Since y and h ξ ξ ξ are bounded, we have that y (ξ) = 0 implies U (ξ) = r(ξ) = 0. ξ ξ Before introducing the set of Lagrangian coordinates, which provides us with a list of all properties inherited by (y, U, h, r) necessary for global conservative solutions to exist, we will characterize relabeling functions. These functions will F not only play a key role when deﬁning , but also when identifying equivalence classes in Lagrangian coordinates, which turn up since we introduced 4 Lagrangian coordinates in contrast to 3 Eulerian coordinates, and hence a degree of freedom, which could cause trouble, when considering questions concerning the uniqueness of solutions. Deﬁnition 2.3 (Relabeling functions). We denote by G the subgroup of the group R R of homeomorphisms from to such that

text: BLOW-UP

text: FOR

text: THE

text: SYSTEM

text: whose

text: time

text: evolution

text: is

text: given

text: through

text: However,

text: this

text: ansatz

text: has

text: two

text: draw-

text: backs,

text: on

text: the

text: one

text: hand

text: we

text: do

text: not

text: know

text: yet

text: how

text: µ(t,

text: x)

text: evolves

text: with

text: respect

text: to

text: time,

text: and

text: on

text: the

text: other

text: hand

text: how

text: to

text: combine

text: µ(t,

text: x)

text: and

text: y(t,

text: ξ).

text: The

text: last

text: issue

text: will

text: D

text: be

text: resolved

text: next

text: by

text: introducing

text: a

text: suitable

text: mapping

text: from

text: Eulerian

text: coordinates

text: F

text: to

text: Lagrangian

text: coordinates

text: D

text: Deﬁnition

text: For

text: any

text: (u,

text: ρ,

text: µ)

text: in

text: let

text: y(ξ)

text: sup

text: y

text: µ((

text: y))

text: y

text: ξ

text: h(ξ)

text: y

text: (ξ),

text: ξ

text: U(ξ)

text: u

text: y(ξ),

text: r(ξ)

text: ρ

text: y(ξ)y

text: (ξ).

text: ξ

text: F

text: D

text: F

text: Then

text: (y,

text: U,

text: h,

text: r)

text: We

text: denote

text: by

text: L:

text: the

text: mapping

text: which

text: to

text: any

text: D

text: F

text: element

text: (u,

text: ρ,

text: µ)

text: associates

text: X

text: (y,

text: U,

text: h,

text: r)

text: given

text: by

text: Since

text: µ

text: is

text: a

text: positive,

text: ﬁnite

text: Radon

text: measure,

text: we

text: have

text: that

text: the

text: function

text: F(x)

text: µ((

text: x))

text: is

text: increasing

text: and

text: lower

text: semi-continuous.

text: We

text: therefore

text: obtain,

text: as

text: an

text: immediate

text: consequence,

text: that

text: y

text: is

text: an

text: increasing

text: function,

text: such

text: that

text: we

text: can

text: ﬁnd

text: to

text: R

text: R

text: any

text: x

text: at

text: least

text: one

text: ξ

text: such

text: that

text: y(ξ)

text: x.

text: In

text: particular,

text: we

text: have

text: that

text: if

text: a

text: positive

text: amount

text: of

text: energy

text: concentrates

text: at

text: some

text: point

text: x,

text: i.e.

text: µ(

text: x

text: c

text: then

text: implies

text: that

text: the

text: point

text: x

text: will

text: be

text: mapped

text: to

text: an

text: interval

text: [ξ

text: ξ

text: such

text: that

text: ξ

text: ξ

text: c

text: and

text: y(ξ)

text: x

text: for

text: all

text: ξ

text: [ξ

text: ξ

text: Moreover,

text: y

text: is

text: Lipschitz

text: continuous

text: with

text: Lipschitz

text: constant

text: at

text: most

text: one,

text: and

text: hence

text: diﬀerentiable

text: almost

text: everywhere

text: by

text: Rademacher’s

text: theorem.

text: Thus

text: y

text: (ξ)

text: for

text: almost

text: every

text: ξ

text: [ξ

text: ξ

text: which

text: is

text: ξ

text: equivalent

text: to

text: h(ξ)

text: for

text: almost

text: every

text: ξ

text: [ξ

text: ξ

text: Thus

text: µ(

text: x

text: ξ2

text: h(ξ)dξ.

text: In

text: ξ1

text: general,

text: one

text: has

text: that

text: R

text: R

text: ξ

text: y

text: (ξ)

text: ξ

text: y(ξ)

text: supp(µ

text: ξ

text: s

text: R

text: by

text: Besicovitch’s

text: derivation

text: theorem,

text: see

text: e.g.

text: Theorem

text: and

text: µ(

text: h

text: The

text: above

text: considerations

text: show

text: that

text: wave

text: breaking

text: in

text: Lagrangian

text: coordinates

text: means

text: that

text: y

text: (ξ)

text: equals

text: and

text: the

text: concentrated

text: energy

text: can

text: be

text: read

text: of

text: from

text: the

text: ξ

text: R

text: function

text: h(ξ)

text: integrated

text: over

text: ξ

text: y

text: (ξ)

text: In

text: particular,

text: one

text: can

text: view,

text: ξ

text: roughly

text: speaking,

text: h,

text: which

text: is

text: a

text: function,

text: as

text: the

text: relabeled

text: version

text: of

text: the

text: measure

text: µ.

text: R

text: Therefore,

text: implies

text: that

text: hy

text: for

text: almost

text: every

text: ξ

text: Since

text: y

text: and

text: h

text: ξ

text: ξ

text: ξ

text: are

text: bounded,

text: we

text: have

text: that

text: y

text: (ξ)

text: implies

text: U

text: (ξ)

text: r(ξ)

text: ξ

text: ξ

text: Before

text: introducing

text: the

text: set

text: of

text: Lagrangian

text: coordinates,

text: which

text: provides

text: us

text: with

text: a

text: list

text: of

text: all

text: properties

text: inherited

text: by

text: (y,

text: U,

text: h,

text: r)

text: necessary

text: for

text: global

text: conservative

text: solutions

text: to

text: exist,

text: we

text: will

text: characterize

text: relabeling

text: functions.

text: These

text: functions

text: will

text: F

text: not

text: only

text: play

text: a

text: key

text: role

text: when

text: deﬁning

text: but

text: also

text: when

text: identifying

text: equivalence

text: classes

text: in

text: Lagrangian

text: coordinates,

text: which

text: turn

text: up

text: since

text: we

text: introduced

text: Lagrangian

text: coordinates

text: in

text: contrast

text: to

text: Eulerian

text: coordinates,

text: and

text: hence

text: a

text: degree

text: of

text: freedom,

text: which

text: could

text: cause

text: trouble,

text: when

text: considering

text: questions

text: concerning

text: the

text: uniqueness

text: of

text: solutions.

text: Deﬁnition

text: (Relabeling

text: functions).

text: We

text: denote

text: by

text: G

text: the

text: subgroup

text: of

text: the

text: group

text: R

text: R

text: of

text: homeomorphisms

text: from

text: to

text: such

text: that


text: 10 K. GRUNERT −1 − (3.9) α = α2 + (U2 P). t 2 (cid:113) Thus we see that α (t, ξ) ≤ 0 for α2(t, ξ) ≥ 4((cid:107)u (cid:107)2 + µ (R)). (cid:113) t 0 L2R 0 Assume that α(0, ξ) ≤ −2 (cid:107)u (cid:107)2 + µ (R), then we can ﬁnd to any constant 0 L2R 0 C < α(0, ξ) < −2 (cid:107)u (cid:107)2 + µ (R), a time t , such that α(t , ξ) = C, and α(t, ξ) 0 L2R 0 C C is strictly decreasing on the interval [0, t ]. C ∈ N Let C be a decreasing sequence of constants such that C < α(0, ξ) for all n →n −∞ n and C . Then we can associate to each C , as before, a time t , such that n n n α(t , ξ) = C and α(t, ξ) is strictly decreasing on the interval [0, t ]. Thus n n n → −∞ → ∞ (3.10) α(t , ξ) = C , as n . n n Unfortunately, this argument has one drawback so far, we do not know if the sequence t is convergent. However what we do know, is that the sequence t is n n increasing. This means if we can show that the sequence t is uniformly bounded n then there exists a positive time t such that b ↑ → ∞ (3.11) t t , as n , n b and t is the time where wave breaking occurs, since α(t, ξ) becomes unbounded b from below. Therefore consider the diﬀerential equation −1 (3.12) γ (t) = γ2(t) + C, t 2 where C = 2((cid:107)u (cid:107)2 + µ (R)), with initial data γ(0) = α(0, ξ). Then γ2(0) − 0 L2R 0 2C > 0 and γ(t) is strictly decreasing. Moreover, γ(t) is negative and an upper ≥ bound for α(t, ξ), i.e. α2(t, ξ) γ2(t) > 2C, as long as α(t, ξ) is well-deﬁned. A qualitative analysis of (3.12) shows that γ(t), will blow up. We claim that this happens within ﬁnite time. If so the blow up time of γ(t) provides us with an upper bound for the sequence t . n The solution of (3.12) is given by √ √ √ − − 2Cγ(0) +√2C + ( 2Cγ(√0) 2C) exp√( 2Ct) (3.13) γ(t) = − − − . γ(0) + 2C (γ(0) 2C) exp( 2Ct) A close look reveals that both the numerator, which remains negative for all times, and the denominator, which is positive at initial time, are absolutely bounded. Thus γ2(t) can only blow up within ﬁnite time if the denominator equals zero, i.e. √ √ γ(0) + √2C − (3.14) − = exp( 2Ct). γ(0) 2C ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K.

text: GRUNERT

text: −1

text: α

text: α2

text: P).

text: t

text: Thus

text: we

text: see

text: that

text: α

text: (t,

text: ξ)

text: for

text: ξ)

text: µ

text: (R)).

text: t

text: Assume

text: that

text: ξ)

text: −2

text: µ

text: (R),

text: then

text: we

text: can

text: ﬁnd

text: to

text: any

text: constant

text: C

text: ξ)

text: −2

text: µ

text: (R),

text: a

text: time

text: t

text: such

text: that

text: α(t

text: ξ)

text: C,

text: and

text: α(t,

text: ξ)

text: C

text: C

text: is

text: strictly

text: decreasing

text: on

text: the

text: interval

text: t

text: C

text: N

text: Let

text: C

text: be

text: a

text: decreasing

text: sequence

text: of

text: constants

text: such

text: that

text: C

text: ξ)

text: for

text: all

text: n

text: →n

text: n

text: and

text: C

text: Then

text: we

text: can

text: associate

text: to

text: each

text: C

text: as

text: before,

text: a

text: time

text: t

text: such

text: that

text: n

text: n

text: n

text: α(t

text: ξ)

text: C

text: and

text: α(t,

text: ξ)

text: is

text: strictly

text: decreasing

text: on

text: the

text: interval

text: t

text: Thus

text: n

text: n

text: n

text: α(t

text: ξ)

text: C

text: as

text: n

text: n

text: n

text: Unfortunately,

text: this

text: argument

text: has

text: one

text: drawback

text: so

text: far,

text: we

text: do

text: not

text: know

text: if

text: the

text: sequence

text: t

text: is

text: convergent.

text: However

text: what

text: we

text: do

text: know,

text: is

text: that

text: the

text: sequence

text: t

text: is

text: n

text: n

text: increasing.

text: This

text: means

text: if

text: we

text: can

text: show

text: that

text: the

text: sequence

text: t

text: is

text: uniformly

text: bounded

text: n

text: then

text: there

text: exists

text: a

text: positive

text: time

text: t

text: such

text: that

text: b

text: t

text: t

text: as

text: n

text: n

text: b

text: and

text: t

text: is

text: the

text: time

text: where

text: wave

text: breaking

text: occurs,

text: since

text: α(t,

text: ξ)

text: becomes

text: unbounded

text: b

text: from

text: below.

text: Therefore

text: consider

text: the

text: diﬀerential

text: equation

text: −1

text: γ

text: (t)

text: C,

text: t

text: where

text: C

text: µ

text: (R)),

text: with

text: initial

text: data

text: ξ).

text: Then

text: and

text: γ(t)

text: is

text: strictly

text: decreasing.

text: Moreover,

text: γ(t)

text: is

text: negative

text: and

text: an

text: upper

text: bound

text: for

text: α(t,

text: ξ),

text: i.e.

text: ξ)

text: as

text: long

text: as

text: α(t,

text: ξ)

text: is

text: well-deﬁned.

text: A

text: qualitative

text: analysis

text: of

text: shows

text: that

text: γ(t),

text: will

text: blow

text: up.

text: We

text: claim

text: that

text: this

text: happens

text: within

text: ﬁnite

text: time.

text: If

text: so

text: the

text: blow

text: up

text: time

text: of

text: γ(t)

text: provides

text: us

text: with

text: an

text: upper

text: bound

text: for

text: the

text: sequence

text: t

text: n

text: The

text: solution

text: of

text: is

text: given

text: by

text: exp√(

text: γ(t)

text: exp(

text: A

text: close

text: look

text: reveals

text: that

text: both

text: the

text: numerator,

text: which

text: remains

text: negative

text: for

text: all

text: times,

text: and

text: the

text: denominator,

text: which

text: is

text: positive

text: at

text: initial

text: time,

text: are

text: absolutely

text: bounded.

text: Thus

text: can

text: only

text: blow

text: up

text: within

text: ﬁnite

text: time

text: if

text: the

text: denominator

text: equals

text: zero,

text: i.e.

text: exp(

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: ##LTLine##


text: not depend on an external knowledge source, and eleven different centrality measures and their are therefore more suitable for deployment in on- variations (cf. Section ). line systems. An important point to note here is • Performance evaluation of different central- that PageRank is a centrality measure, and returns ity measures and collocation networks on the most central words in word co-occurrence net- four benchmark datasets, and comparison of works. our method with tf-idf – a state-of-the-art un- Against this background, we ask the following supervised keyphrase extraction baseline (cf. question: Are there other centrality measures that Section ). perform as well as or better than PageRank in keyword/keyphrase extraction task? Furthermore, • Identiﬁcation of centrality measures and net- are these centrality measures simpler than PageR- work types that perform well, and centrality ank? That this might indeed be the case, is ev- measures that perform poorly (cf. Section ). ident from Table 1, where we show the top ten • words selected by Degree, Strength (weighted de- Design of an online graph-based keyword gree), and PageRank on a sample word network of and keyphrase extraction system, based on a newswire document. Note how similar the top- the centrality measures used in this paper.2 ranking words are for these three centrality mea- sures. This calls into question whether we should 2 Related Work use these other simpler centrality measures like In the supervised paradigm, key term extrac- Degree and Strength, and whether they are better tion is formulated as a classiﬁcation problem. than or competitive with PageRank on benchmark Each term is encoded using different feature rep- corpora. The research that we describe in this pa- resentations such as tf-idf (Witten et al., 1999; per speciﬁcally addresses these questions. Zhang et al., 2006), ﬁrst occurrence (Hulth, 2003), We experiment with an array of centrality mea- phrase length and phrase distribution (Jiang et al., sures and their variants (cf. Table 2) on differ- 2009), is-in-title (Litvak and Last, 2008), etc. Sev- ent types of word and noun phrase collocation eral machine learning algorithms have been used networks. Construction of these collocation net- by different groups, e.g., na¨ıve Bayes by Frank works has been described in Section . These net- et al. (1999), and Witten et al. (1999), decision works are then used to extract key terms from four trees by Ercan and Cicekli (2007), support vec- benchmark datasets (cf. Section ). We treat word tor machines by Zhang et al. (2006) and Jiang et and noun phrase networks separately to have an al. (2009), and conditional random ﬁelds by Zhang idea about which centrality measures perform well et al. (2008). on what types of networks. Note also that using In the unsupervised paradigm, the key term ex- noun phrase networks allows us to directly derive traction problem is framed as a ranking problem. a ranking for phrases, as opposed to combining the Dominant ranking strategies include: tf (Barker ranks of constituent words from word networks. and Cornacchia, 2000), tf-idf (Tonella et al., Contributions and Organization. Although 2003), term informativeness and term phrase- centrality measures on word networks have been ness (Tomokiyo and Hurst, 2003), etc. used before, there is no unifying study that looked The performance of key term extraction sys- systematically into different centrality measures tems is usually measured by Precision, Recall, and on different types of collocation networks, in or- F-score. The state-of-the-art performance hovers der to determine what works well and when. This around an F-score of 20%-30% (Kim et al., 2010), is a gap we aim to bridge in our study. Our contri- thereby showing that key term extraction is a hard butions are: • problem in general. Construction and experimentation with four Graph-based methods to key term extraction are different types of collocation networks on inherently unsupervised, where the philosophy is words and noun phrases (cf. Section )1. to build a network of words/phrases, and then rank • Ranking words and noun phrases based on the nodes using some kind of centrality measure. Researchers used variants of PageRank (Mihalcea 1Code and data available at http:// ec2-107-22-235-109.compute-1.amazonaws. 2http://ec2-107-22-235-109.compute-1. com:8000. amazonaws.com:8000. ##LTLine## ##LTLine##

text: not

text: depend

text: on

text: an

text: external

text: knowledge

text: source,

text: and

text: eleven

label: list

text: different

text: centrality

text: measures

text: and

text: their

text: are

text: therefore

text: more

text: suitable

text: for

text: deployment

text: in

text: on-

text: variations

text: (cf.

text: Section

text: line

text: systems.

text: An

text: important

text: point

text: to

text: note

text: here

text: is

text: Performance

text: evaluation

text: of

text: different

text: central-

text: that

text: PageRank

text: is

text: a

text: centrality

text: measure,

text: and

text: returns

text: ity

text: measures

text: and

text: collocation

text: networks

text: on

text: the

text: most

text: central

text: words

text: in

text: word

text: co-occurrence

text: net-

text: four

text: benchmark

text: datasets,

text: and

text: comparison

text: of

text: works.

text: our

text: method

text: with

text: tf-idf

text: a

text: state-of-the-art

text: un-

text: Against

text: this

text: background,

text: we

text: ask

text: the

text: following

text: supervised

text: keyphrase

text: extraction

text: baseline

text: (cf.

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