text: Geometrically, the membranes are disconnected from each other and thus two membrane equations for two tympanic membrane displacements must be solved. The second issue is a consequence of the ∅ ﬁrst issue: By deﬁnition, the boundary ∂Ω of each ﬁber Ω are topologically closed with ∂2Ω = . t { } t t For the pairwisely in ∂Ω unconnected parts Γ ≤ ≤ , this is not the case if N > 1: They will 0 (cid:54) ∅ i 1 i N generically have a boundary ∂Γ = . Thus, we need to specialize boundary conditions to the i boundary vibrations equations. The third issue to address asks why the boundaries ∂Ω should t actually start oscillating. Mathematically, this can be reformulated and we can ask what source term to the boundary vibrations we should specialize. • Source term and boundary conditions: We have to give the boundary vibrations boundary and initial conditions as well as an external source term,say Ψ = Ψ(p, p ). One way around boundary ex ∅ conditions is to impose a periodicity condition on u by arguing that ∂(∂Ω ) = such that an { } 0 integration by parts w.r.t. the coordinate yi ≤ ≤ (cid:54) results in no boundary term. This has been 1 i n,i=s done in [96] for closed bio-membranes such as cellular membranes. However, in [98, 99, 101] tympanic membranes, i.e., membranes which do not cover all of ∂Ω , have been considered. In the set-up the 0 (cid:54) tympanic membranes do not bound all of the interaural cavity, i.e., in our language u = 0 only on R connected components of ∂Ω . By compactness of Ω in n, the boundary manifold ∂Ω is relatively R 0 ∈ N 0 0 compact in n. I.e., a there are N, M such that 0 (cid:93) (cid:93) ≡ N (cid:93) N+M ∂Ω Γ Γ , (184) 0 i i i=1 i=N+1 − − is a decomposition of the (n 2)-connected ∂Ω in (n 2)-connected and retractible components ≤ ≤ ∩ 0 (cid:54) Γ , 1 i N + M such that Vol − (Γ Γ ) = 0 for i = j. I.e., the Γ ’s are allowed to share at i n 1 i j i B → R most a Lebesgue Null-set w.r.t. the Lebesgue-Borel integration measure Vol − : (∂Ω ) + on n 1 M 0 R ×0 the Borel-algebra of the boundary ∂Ω of the ﬁber Ω of the unperturbed bundle = + Ω . ≤ ≤ 0 0 0 0 0 We deﬁne Γ for 1 i N by being a decomposition of ∂Ω in smooth, retractible, bounded − i 0 (n 1)-dimensional sub-manifolds Γ , i (cid:93) (cid:91) (cid:91) N ≡ ≡ { ∈ | (cid:54) } { ∈ |∃ ≥ (cid:54) } Γ supp (u) y ∂Ω : u(t, y) = 0 = y ∂Ω t 0 : u(t, y) = 0 . (185) i ≥ ∂Ω0 ≥ 0 0 i=1 t 0 t 0 The smoothness and manifold properties follow from smoothness of u by smoothness of the ψ → ’s ≥ M M 0 t for t 0 stored in the deﬁnition of the unperturbed bundle and the perturbed bundle . The 0 over-line denotes topological closure in ∂Ω topologized by the relative topology τ (Ω ) where we 0 R ∂Ω0 →0 R recall that Ω has been topologized by the relative topology τ ( n), using that Ω (cid:44) n by the 0 (cid:107) (cid:107) Ω0 0 embedding ι , and the topology is w.r.t. the norm . induced by the metric g extended from Ω R 0 ≤g0 ≤ 0 0 to n by radial extrapolation. In turn Γ for N + 1 i N + M is deﬁned as a decomposition in i≤ Γ ’s of the same properties as the Γ , 1‘i N, i (cid:32) (cid:33) i (cid:32) (cid:33) (cid:93) (cid:93) (cid:93) (cid:123) N+M ≡ N ≡ \ N \ { ∈ |∃ ≥ (cid:54) } Γ Γ ∂Ω Γ = ∂Ω y ∂Ω t 0 : u(t, y) = 0 . (186) i i 0 i 0 0 i=N+1 i=1 i=1 61 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Geometrically,

text: the

text: membranes

text: are

text: disconnected

text: from

text: each

text: other

text: and

text: thus

text: two

text: membrane

text: equations

text: for

text: two

text: tympanic

text: membrane

text: displacements

text: must

text: be

text: solved.

text: The

text: second

text: issue

text: is

text: a

text: consequence

text: of

text: the

text: ﬁrst

text: issue:

text: By

text: deﬁnition,

text: the

text: boundary

text: ∂Ω

text: of

text: each

text: ﬁber

text: Ω

text: are

text: topologically

text: closed

text: with

text: ∂2Ω

text: t

text: t

text: t

text: For

text: the

text: pairwisely

text: in

text: ∂Ω

text: unconnected

text: parts

text: Γ

text: this

text: is

text: not

text: the

text: case

text: if

text: N

text: They

text: will

font: default

text: i

text: i

text: N

text: generically

text: have

text: a

text: boundary

text: ∂Γ

text: Thus,

text: we

text: need

text: to

text: specialize

text: boundary

text: conditions

text: to

text: the

text: i

text: boundary

text: vibrations

text: equations.

text: The

text: third

text: issue

text: to

text: address

text: asks

text: why

text: the

text: boundaries

text: ∂Ω

text: should

text: t

text: actually

text: start

text: oscillating.

text: Mathematically,

text: this

text: can

text: be

text: reformulated

text: and

text: we

text: can

text: ask

text: what

text: source

text: term

text: to

text: the

text: boundary

text: vibrations

text: we

text: should

text: specialize.

text: Source

text: term

text: and

text: boundary

text: conditions:

text: We

text: have

text: to

text: give

text: the

text: boundary

text: vibrations

text: boundary

text: and

text: initial

text: conditions

text: as

text: well

text: as

text: an

text: external

text: source

text: term,say

text: Ψ

text: Ψ(p,

text: p

text: One

text: way

text: around

text: boundary

text: ex

text: conditions

text: is

text: to

text: impose

text: a

text: periodicity

text: condition

text: on

text: u

text: by

text: arguing

text: that

text: ∂(∂Ω

text: such

text: that

text: an

text: integration

text: by

text: parts

text: w.r.t.

text: the

text: coordinate

text: yi

text: results

text: in

text: no

text: boundary

text: term.

text: This

text: has

text: been

text: i

text: n,i=s

text: done

text: in

text: for

text: closed

text: bio-membranes

text: such

text: as

text: cellular

text: membranes.

text: However,

text: in

text: tympanic

text: membranes,

text: i.e.,

text: membranes

text: which

text: do

text: not

text: cover

text: all

text: of

text: ∂Ω

text: have

text: been

text: considered.

text: In

text: the

text: set-up

text: the

text: tympanic

text: membranes

text: do

text: not

text: bound

text: all

text: of

text: the

text: interaural

text: cavity,

text: i.e.,

text: in

text: our

text: language

text: u

text: only

text: on

text: R

text: connected

text: components

text: of

text: ∂Ω

text: By

text: compactness

text: of

text: Ω

text: in

text: n,

text: the

text: boundary

text: manifold

text: ∂Ω

text: is

text: relatively

text: R

text: N

text: compact

text: in

text: n.

text: I.e.,

text: a

text: there

text: are

text: N,

text: M

text: such

text: that

text: N

text: N+M

text: ∂Ω

text: Γ

text: Γ

text: i

text: i

text: is

text: a

text: decomposition

text: of

text: the

text: (n

text: ∂Ω

text: in

text: (n

text: and

text: retractible

text: components

text: Γ

text: i

text: N

text: M

text: such

text: that

text: Vol

text: (Γ

text: Γ

text: for

text: i

text: j.

text: I.e.,

text: the

text: Γ

text: ’s

text: are

text: allowed

text: to

text: share

text: at

text: i

text: n

text: i

text: j

text: i

text: B

text: R

text: most

text: a

text: Lebesgue

text: Null-set

text: w.r.t.

text: the

text: Lebesgue-Borel

text: integration

text: measure

text: Vol

text: (∂Ω

text: on

text: n

text: M

text: R

text: ×0

text: the

text: Borel-algebra

text: of

text: the

text: boundary

text: ∂Ω

text: of

text: the

text: ﬁber

text: Ω

text: of

text: the

text: unperturbed

text: bundle

text: Ω

text: We

text: deﬁne

text: Γ

text: for

text: i

text: N

text: by

text: being

text: a

text: decomposition

text: of

text: ∂Ω

text: in

text: smooth,

text: retractible,

text: bounded

text: i

text: (n

text: sub-manifolds

text: Γ

text: i

text: N

text: Γ

text: supp

text: (u)

text: y

text: ∂Ω

text: u(t,

text: y)

text: y

text: ∂Ω

text: t

text: u(t,

text: y)

text: i

text: ∂Ω0

text: t

text: t

text: The

text: smoothness

text: and

text: manifold

text: properties

text: follow

text: from

text: smoothness

text: of

text: u

text: by

text: smoothness

text: of

text: the

text: ψ

text: ’s

text: M

text: M

text: t

text: for

text: t

text: stored

text: in

text: the

text: deﬁnition

text: of

text: the

text: unperturbed

text: bundle

text: and

text: the

text: perturbed

text: bundle

text: The

text: over-line

text: denotes

text: topological

text: closure

text: in

text: ∂Ω

text: topologized

text: by

text: the

text: relative

text: topology

text: τ

text: (Ω

text: where

text: we

text: R

text: ∂Ω0

text: →0

text: R

text: recall

text: that

text: Ω

text: has

text: been

text: topologized

text: by

text: the

text: relative

text: topology

text: τ

text: n),

text: using

text: that

text: Ω

text: n

text: by

text: the

text: Ω0

text: embedding

text: ι

text: and

text: the

text: topology

text: is

text: w.r.t.

text: the

text: norm

text: induced

text: by

text: the

text: metric

text: g

text: extended

text: from

text: Ω

text: R

text: to

text: n

text: by

text: radial

text: extrapolation.

text: In

text: turn

text: Γ

text: for

text: N

text: i

text: N

text: M

text: is

text: deﬁned

text: as

text: a

text: decomposition

text: in

text: i≤

text: Γ

text: ’s

text: of

text: the

text: same

text: properties

text: as

text: the

text: Γ

text: N,

text: i

text: i

text: N+M

text: N

text: N

text: Γ

text: Γ

text: ∂Ω

text: Γ

text: ∂Ω

text: y

text: ∂Ω

text: t

text: u(t,

text: y)

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:123) (cid:93) The operation denotes complement in ∂Ω and stands as before for disjoint union modulo 0 B → R Lebesgue Null-sets w.r.t. the Lebesgue-Borel integration measure Vol − : (∂Ω ) +. n 1 0 0 In the following, we will summarize our results on the equations we are interested in. As such, we decompose ≡ | the action functional S into N action funct(cid:34)ionals f(cid:35)or the u u , geom,diss (cid:88) (cid:88) i Γi N N S [u] = S u = S [u ], (187) geom,diss geom,diss i geom,diss i i=1 i=1 by pairwise disjointness modulo a Null-set of the (Γ ) ≤ ≤ . For the derivation of the boundary con- ∈ { } i 1 i N ditions, let i 1, ..., N be arbitrary. We will also suppress the index i of the u ’s notationally in order i to avoid confusion with the coordinate index i. Using the previously obtained expressions for the sec- ∇ ∇ tional curvatures K and mean curvature H in terms of the Levi-Civita connection = g0 and the ij Laplace-Beltrami operator ∆ , g0 ∇ ∇ O H = T r [ u] + ((cid:15)2) g0 i j − we can start integration by parts. We choose the normal vector n(t = 0) = ∂ for the boundary term | t ≤ ≤ at t = 0 and the normal vector n = n for the boundary term at ∂Γ for 1 i N. We choose ∂Ω0 ∂Γi ∈ ≤ ≤ i the Dirichlet boundary conditions u (t, y) = 0 if y ∂Γ for all 1 i N. We the ﬁrst and second i i contribution to the geometrical action function by parts once, We integrate twice by parts the third contribution containing the Laplace-Beltrami acting on u squared. The ﬁrst boundary term resulting k from the third contribution to the geometrical action functional, i.e., the ﬁrst integral over ∂Γ containing k only covariant derivatives of u is integrated by parts once more using ∂2Γ = 0. This allows us to obtain k 62

label: list

text: The

text: operation

text: denotes

text: complement

text: in

text: ∂Ω

text: and

text: stands

text: as

text: before

text: for

text: disjoint

text: union

text: modulo

text: B

text: R

text: Lebesgue

text: Null-sets

text: w.r.t.

text: the

text: Lebesgue-Borel

text: integration

text: measure

text: Vol

text: (∂Ω

text: n

text: In

label: paragraph

text: the

text: following,

text: we

text: will

text: summarize

text: our

text: results

text: on

text: the

text: equations

text: we

text: are

text: interested

text: in.

text: As

text: such,

text: we

text: decompose

text: the

text: action

text: functional

text: S

text: into

text: N

text: action

label: equation

text: the

text: u

text: u

text: geom,diss

text: i

text: Γi

text: N

text: N

text: S

text: [u]

text: S

text: u

text: S

text: [u

text: geom,diss

text: geom,diss

text: i

text: geom,diss

text: i

text: by

text: pairwise

text: disjointness

text: modulo

text: a

text: Null-set

text: of

text: the

text: (Γ

text: For

text: the

text: derivation

text: of

text: the

text: boundary

text: con-

text: i

text: i

text: N

text: ditions,

text: let

text: i

text: N

text: be

text: arbitrary.

text: We

text: will

text: also

text: suppress

text: the

text: index

text: i

text: of

text: the

text: u

text: ’s

text: notationally

text: in

text: order

text: i

text: to

text: avoid

text: confusion

text: with

text: the

text: coordinate

text: index

text: i.

text: Using

text: the

text: previously

text: obtained

text: expressions

text: for

text: the

text: sec-

text: tional

text: curvatures

text: K

text: and

text: mean

text: curvature

text: H

text: in

text: terms

text: of

text: the

text: Levi-Civita

text: connection

text: and

text: the

text: ij

text: Laplace-Beltrami

text: operator

text: O

text: H

text: T

text: r

text: u]

text: i

text: j

text: we

text: can

text: start

text: integration

text: by

text: parts.

text: We

text: choose

text: the

text: normal

text: vector

text: n(t

text: for

text: the

text: boundary

text: term

text: t

text: at

text: t

text: and

text: the

text: normal

text: vector

text: n

text: n

text: for

text: the

text: boundary

text: term

text: at

text: ∂Γ

text: for

text: i

text: N.

text: We

text: choose

text: ∂Ω0

text: ∂Γi

text: i

text: the

text: Dirichlet

text: boundary

text: conditions

text: u

text: (t,

text: y)

text: if

text: y

text: ∂Γ

text: for

text: all

text: i

text: N.

text: We

text: the

text: ﬁrst

text: and

text: second

text: i

text: i

text: contribution

text: to

text: the

text: geometrical

text: action

text: function

text: by

text: parts

text: once,

text: We

text: integrate

text: twice

text: by

text: parts

text: the

text: third

text: contribution

text: containing

text: the

text: Laplace-Beltrami

text: acting

text: on

text: u

text: squared.

text: The

text: ﬁrst

text: boundary

text: term

text: resulting

text: k

text: from

text: the

text: third

text: contribution

text: to

text: the

text: geometrical

text: action

text: functional,

text: i.e.,

text: the

text: ﬁrst

text: integral

text: over

text: ∂Γ

text: containing

text: k

text: only

text: covariant

text: derivatives

text: of

text: u

text: is

text: integrated

text: by

text: parts

text: once

text: more

text: using

text: ∂2Γ

text: This

text: allows

text: us

text: to

text: obtain

text: k


text: a boundary term that contains a factor u to be evaluated on ∂Γ without derivatives. Let us calculate, k k (cid:88) (cid:90){ } (cid:113) (cid:16) (cid:17) S [ u ≤ ≤ ] geom,diss. i 1 k N 1 N − −| | = (cid:88) (cid:90) dtdn 1y (cid:113) GΣ σ GΣ,tt(∂ u (cid:88))2 +(cid:90)T (∂ u )2 + ((cid:113)T d2)(∆∂ u )2 2 R+×Γ 0 m 0 t k 0 i k curv g0 k k=1 0 k σ N − −| | σ N − −| | − = m(cid:88) (cid:90) dn 1y (cid:113) GΣ uGtt∂ u + m (cid:88) (cid:90) dn 1y G(cid:113)Σ u Σ 1∂ (Σ∂ u ) 2 {0}×Γ 0 0 t 2 R+×Γ 0 k t t k k=1 k k=1 0 k T N − −| | − T N − −| | + 0 (cid:88) (cid:90) dn 2y GΣ(cid:113)u (GΣ)ij∂ u 0 dn(cid:88)1y(cid:90) GΣ u ∆∂ u(cid:113)) 2 R+×∂Γ 0 k 0 n k 2 R+×Γ 0 k g0 k k=1 0 k k=1 0 k T d2 N − −| | − T d2 N − −| | + cu2rv (cid:88) (cid:90)R+×∂Γ dtdn 2y(cid:112) GΣ0 nj∂juk(∆∂g0uk) cu2rv R+×∂Γ dtdn 2y GΣ0 uknj∂j(∆∂g0uk) k=1 0 k k=1 0 k T d2 N − −| | + cur(cid:88)v (cid:90) dtd(cid:113)n 1y GΣ u (∆∂ )2(cid:88)u (cid:90) (cid:113) 2 R+×Γ o k g0 k k=1 0 k σ N − −| | σ N − −| | − = m(cid:88) (cid:90) dn 1y (cid:113) GΣ uGtt∂ u + m (cid:88) (cid:90) dn 1y G(cid:113)Σ u Σ 1∂ (Σ∂ u ) 2 {0}×Γ 0 0 t 2 R+×Γ 0 k t t k k=1 k k=1 0 k T N − −| | − T N − −| | + 0 (cid:88) (cid:90) dn 2y G(cid:113)Σ u (GΣ)ij∂ u 0 d(cid:88)n 1y(cid:90) GΣ u ∆∂ (cid:113)u ) 2 R+×∂Γ 0 k 0 n k 2 R+×Γ 0 k g0 k k=1 0 k k=1 0 k − N − −| | T d2 N − −| | T (cid:88)d2 (cid:90) dt(cid:113)dn 2y GΣ u n ∂j(∆∂ u ) + curv dtdn 1y GΣ u (∆∂ )2u curv R+×∂Γ 0 k j g0 k 2 R+×Γ 0 k g0 k k=1 0 k k=1 0 k σ N − −| | − = m(cid:88) (cid:90) dn 1y(cid:113) GΣ u Σ 1∂ (Σ∂ u ) 2 R+×Γ 0 k t t k k=1 0 k − T N − −| | 0 (cid:88) (cid:90) dn 1y G(cid:113)Σ u ∆∂ u ) 2 R+×Γ 0 k g0 k k=1 0 k T d2 N − −| | + c(cid:88)u2rv (cid:90) R+×Γ (cid:113)dtdn 1y (cid:0) GΣ0 uk(∆∂g0)2uk (cid:1) k=1 0 k 1 N − −| | − − = dtdn 1y GΣ u σ Σ 1∂ (Σu ) T ∆∂ u + T d2(∆∂ )2u . 2 R+ 0 k m t k 0 g0 k 0 g0 k k=1 0 We have used the homogeneous initial conditions and the homogeneous Dirichlet boundary conditions for ≤ ≤ u for all k, 1 k N to set the boundary terms equal to 0. The functional derivative of the geometrical k { } action function w.r.t. the u ≤ ≤ now is easy. Let k be arbitrary but given. Hamilton’s principle of k 1 k N least action gives us the necessary condition for a minimum, δS geom ! = 0, (188) δu k 63 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: a

text: boundary

text: term

text: that

text: contains

text: a

text: factor

text: u

text: to

text: be

text: evaluated

text: on

text: ∂Γ

text: without

text: derivatives.

text: Let

text: us

text: calculate,

text: k

text: k

text: S

text: u

text: geom,diss.

text: i

text: k

text: N

text: N

text: dtdn

text: GΣ

text: σ

text: GΣ,tt(∂

text: u

text: u

text: u

text: R+×Γ

text: m

text: t

text: k

text: i

text: k

text: curv

text: k

text: k

text: σ

text: N

text: σ

text: N

text: dn

text: GΣ

text: uGtt∂

text: u

text: m

text: dn

text: u

text: Σ

text: 1∂

text: (Σ∂

text: u

text: t

text: R+×Γ

text: k

text: t

text: t

text: k

text: k

text: k

text: T

text: N

text: T

text: N

text: dn

text: (GΣ)ij∂

text: u

text: GΣ

text: u

text: R+×∂Γ

text: k

text: n

text: k

text: R+×Γ

text: k

text: k

text: k

text: k

text: T

text: N

text: T

text: N

text: dtdn

text: R+×∂Γ

text: dtdn

text: k

text: k

text: T

text: N

text: GΣ

text: u

text: R+×Γ

text: o

text: k

text: k

text: k

text: σ

text: N

text: σ

text: N

text: dn

text: GΣ

text: uGtt∂

text: u

text: m

text: dn

text: u

text: Σ

text: 1∂

text: (Σ∂

text: u

text: t

text: R+×Γ

text: k

text: t

text: t

text: k

text: k

text: k

text: T

text: N

text: T

text: N

text: dn

text: u

text: (GΣ)ij∂

text: u

text: GΣ

text: u

text: R+×∂Γ

text: k

text: n

text: k

text: R+×Γ

text: k

text: k

text: k

text: k

text: N

text: T

text: N

text: T

text: GΣ

text: u

text: n

text: ∂j(∆∂

text: u

text: curv

text: dtdn

text: GΣ

text: u

text: curv

text: R+×∂Γ

text: k

text: j

text: k

text: R+×Γ

text: k

text: k

text: k

text: k

text: σ

text: N

text: dn

text: GΣ

text: u

text: Σ

text: 1∂

text: (Σ∂

text: u

text: R+×Γ

text: k

text: t

text: t

text: k

text: k

text: T

text: N

text: dn

text: u

text: u

text: R+×Γ

text: k

text: k

text: k

text: T

text: N

text: R+×Γ

text: k

text: N

text: dtdn

text: GΣ

text: u

text: σ

text: Σ

text: 1∂

text: (Σu

text: T

text: u

text: T

text: R+

text: k

text: m

text: t

text: k

text: k

text: k

text: We

text: have

text: used

text: the

text: homogeneous

text: initial

text: conditions

text: and

text: the

text: homogeneous

text: Dirichlet

text: boundary

text: conditions

text: for

text: u

text: for

text: all

text: k,

text: k

text: N

text: to

text: set

text: the

text: boundary

text: terms

text: equal

text: to

text: The

text: functional

text: derivative

text: of

text: the

text: geometrical

text: k

text: action

text: function

text: w.r.t.

text: the

text: u

text: now

text: is

text: easy.

text: Let

text: k

text: be

text: arbitrary

text: but

text: given.

text: Hamilton’s

text: principle

text: of

text: k

text: k

text: N

text: least

text: action

text: gives

text: us

text: the

text: necessary

text: condition

text: for

text: a

text: minimum,

text: δS

text: geom

text: δu

text: k

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: ##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##

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text: ##LTLine##


text: which is fulﬁlled if − − σ Σ 1∂ (Σu ) T ∆∂ u + T d2(∆∂ )2u = 0. (189) m t k 0 g0 k 0 g0 k Division by σ gives us with the deﬁnitions c2 = T σ and c2 = T /σ of the ﬂexural wave m m 0 m H curv m propagation velocity c and the bending wave propagation velocity C a non-standard damped wave m h equation, − − Σ 1∂ (Σu ) c2 ∆∂ u + c2 d2(∆∂ )2u (190) t k m g0 k H g0 k We restrict ourselves to the cases, where the terminology wave equation is appropriate, i.e., we require (cid:107) (cid:107) (cid:107) (cid:107) c2 ∆∂ → > c2 d2 ∆2 → . Using multiplicativity of the operator norm [103] for a self-adjoint m g0 L2 H2,2 H g0 L2 H2,2 operator squared, (cid:107) (cid:107) c2 1 ∆∂ → < m . g0 L2 H2,2 c2 d2 H (cid:107) (cid:107) ≤ Since ∆ → 1, we obtain g0 L2 H2,2 c2 > c2 d2 (191) m H This means that ﬂexural waves of the boundary vibrations propagate faster than bending waves of the boundary vibrations. Experimentally, the condition can be veriﬁed by claculating the measuring the eigenfrequencies of the elastic structure in question. For the ICE model the condition is fulﬁlled, c.f., − the model [98, 99, 101]. Introducing the polynomial p = c2 x c2 d2x2 and using the Bochner functional m H calculus [104], we can re-write the homogeneous boundary vibrations equation in the more compact form , − − Σ 1∂ (Σ∂ u ) p(∆∂ )u = 0. (192) t t k g0 k M We will be concerned with reducing this non-standard wave equation to a simpler equation on in 0 short. The last issue to address is the source term. Let us re-consider the ICE model [98, 99, 101]. The boundary vibrations there, the membrane displacements, are pressure diﬀerence receivers. They respond M to the diﬀerence of an external acoustic wave hitting the membranes from the outside, i.e., at ∂ + in the M ⊃ M M ⊂ M M embedding space n+1 and from the interior, i.e., at ∂ − . Note that ∂ is the boundary of M M the perturbation bundle and not the unperturbed bundle . In the ICE model, the acoustic pressure ≥ 0 hitting the tympana from the outside of Ω for a given t 0 is prescribed and denoted by p . The idea is t ex that there is a sound source localized outside the lizard’s head such that the tympanic membranes undergo vibrations caused by the external sound stimulus. We will apply this idea as well. Since we prescribe | (cid:54) | p = p M , we introduce a discontinuity, i.e., p M = p M in general. In the context of the ICE model, ex ∂ + ∂ + ∂ − the directional hearing of lizards is then explained as a reaction of the membrane system coupled by the 64 ##LTLine## ##LTLine##

text: which

text: is

text: fulﬁlled

text: if

text: σ

text: Σ

text: 1∂

text: (Σu

text: T

text: u

text: T

text: m

text: t

text: k

text: k

text: k

text: Division

text: by

text: σ

text: gives

text: us

text: with

text: the

text: deﬁnitions

text: T

text: σ

text: and

text: T

text: /σ

text: of

text: the

text: ﬂexural

text: wave

text: m

text: m

text: m

text: H

text: curv

text: m

text: propagation

text: velocity

text: c

text: and

text: the

text: bending

text: wave

text: propagation

text: velocity

text: C

text: a

text: non-standard

text: damped

text: wave

text: m

text: h

text: equation,

text: Σ

text: 1∂

text: (Σu

text: u

text: t

text: k

text: m

text: k

text: H

text: k

text: We

text: restrict

text: ourselves

text: to

text: the

text: cases,

text: where

text: the

text: terminology

text: wave

text: equation

text: is

text: appropriate,

text: i.e.,

text: we

text: require

text: ∆2

text: Using

text: multiplicativity

text: of

text: the

text: operator

text: norm

text: for

text: a

text: self-adjoint

text: m

text: H

text: operator

text: squared,

text: m

text: H

text: Since

text: we

text: obtain

text: m

text: H

text: This

text: means

text: that

text: ﬂexural

text: waves

text: of

text: the

text: boundary

text: vibrations

text: propagate

text: faster

text: than

text: bending

text: waves

text: of

text: the

text: boundary

text: vibrations.

text: Experimentally,

text: the

text: condition

text: can

text: be

text: veriﬁed

text: by

text: claculating

text: the

text: measuring

text: the

text: eigenfrequencies

text: of

text: the

text: elastic

text: structure

text: in

text: question.

text: For

text: the

text: ICE

text: model

text: the

text: condition

text: is

text: fulﬁlled,

text: c.f.,

text: the

text: model

text: Introducing

text: the

text: polynomial

text: p

text: x

text: and

text: using

text: the

text: Bochner

text: functional

text: m

text: H

text: calculus

text: we

text: can

text: re-write

text: the

text: homogeneous

text: boundary

text: vibrations

text: equation

text: in

text: the

text: more

text: compact

text: form

text: Σ

text: 1∂

text: (Σ∂

text: u

text: p(∆∂

text: )u

text: t

text: t

text: k

text: k

text: M

text: We

text: will

text: be

text: concerned

text: with

text: reducing

text: this

text: non-standard

text: wave

text: equation

text: to

text: a

text: simpler

text: equation

text: on

text: in

text: short.

text: The

text: last

text: issue

text: to

text: address

text: is

text: the

text: source

text: term.

text: Let

text: us

text: re-consider

text: the

text: ICE

text: model

text: The

text: boundary

text: vibrations

text: there,

text: the

text: membrane

text: displacements,

text: are

text: pressure

text: diﬀerence

text: receivers.

text: They

text: respond

text: M

text: to

text: the

text: diﬀerence

text: of

text: an

text: external

text: acoustic

text: wave

text: hitting

text: the

text: membranes

text: from

text: the

text: outside,

text: i.e.,

text: at

text: in

text: the

text: M

text: M

text: M

text: M

text: M

text: embedding

text: space

text: and

text: from

text: the

text: interior,

text: i.e.,

text: at

text: Note

text: that

text: is

text: the

text: boundary

text: of

text: M

text: M

text: the

text: perturbation

text: bundle

text: and

text: not

text: the

text: unperturbed

text: bundle

text: In

text: the

text: ICE

text: model,

text: the

text: acoustic

text: pressure

text: hitting

text: the

text: tympana

text: from

text: the

text: outside

text: of

text: Ω

text: for

text: a

text: given

text: t

text: is

text: prescribed

text: and

text: denoted

text: by

text: p

text: The

text: idea

text: is

text: t

text: ex

text: that

text: there

text: is

text: a

text: sound

text: source

text: localized

text: outside

text: the

text: lizard’s

text: head

text: such

text: that

text: the

text: tympanic

text: membranes

text: undergo

text: vibrations

text: caused

text: by

text: the

text: external

text: sound

text: stimulus.

text: We

text: will

text: apply

text: this

text: idea

text: as

text: well.

text: Since

text: we

text: prescribe

text: p

text: p

text: M

text: we

text: introduce

text: a

text: discontinuity,

text: i.e.,

text: p

text: M

text: p

text: M

text: in

text: general.

text: In

text: the

text: context

text: of

text: the

text: ICE

text: model,

text: ex

text: the

text: directional

text: hearing

text: of

text: lizards

text: is

text: then

text: explained

text: as

text: a

text: reaction

text: of

text: the

text: membrane

text: system

text: coupled

text: by

text: the

text: ##LTLine##

text: ##LTLine##


text: { } The decomposition of u in the localized boundary vibrations u ≤ ≤ from the previous sub-paragraph k 1 k N M gives us in total N + 1 equations: An acoustic wave equation on the perturbed bundle for p and N { } equations describing the dynamics of the localized boundary vibrations u ≤ ≤ k 1 k N (cid:88) − N ◦ − | ∈ R × M ∂t2p c2∆g0p = ρ0c2 ∂t2uk Φ0→1 t R+0 ×Γkδ((t, x) +0 Γk) on (200) k=1 − − | R × ⊂ M Σ 1∂ (Σ∂ u ) p(∆∂ )u = Ψ on + Γ ∂ . (201) t t k g0 k Γk 0 k 0 | R × The symbol Ψ denotes restriction of the source term for the boundary vibrations u to + Γ , where ∈ { Γk } 0 k for each k 1, ..., N the boundary vibration u is localized. The initial conditions for both equations k stay unchanged, the Neumann boundary condition to the acoustic wave equation stays unchanged as well, but the Laplace-Beltrami operator ∆∂ is now deﬁned for all Γ ’s individually and breaks the periodic g0 k boundary conditions that we amended to the Laplace-Beltrami operator on ∂Ω . The derivation from 0 the previous paragraph demonstrated that we can take Dirichlet boundary conditions for the localized { } boundary vibrations equations, i.e., the equations for u ≤ ≤ , k 1 k N | R × u = 0 on + ∂Γ . (202) k ∂Γk 0 k M M If we pull back the acoustic wave equation on to the well-understood reference bundle we 0M introduce the perturbation operator W but we arrive at equations that live on the reference bundle M 0 and its boundary ∂ , denoting p˜(Φ → (t, x)) = p(t, x ) also by the symbol p, 0 0 t t (cid:88) − N ∈ R × ∂2p c2∆ p = c2W[u]p + ρ c2 ∂2u δ((t, x) + Γ ) (203) t g0 0 t k 0 k k=1 − − | R × ⊂ M Σ 1∂ (Σ∂ u ) p(∆∂ )u = Ψ on + Γ ∂ . (204) t t k g0 k Γk 0 k 0 M The Neumann boundary conditions for the acoustic pressure now become ∂ p = 0 on ∂ . For illus- n 0 tration purposes, the mutual acoustic feedback between the boundary vibrations and the acoustic pressure is shown in Fig. 13. Let us comment on the diﬀerential equations adopting for a moment a mathematically more precise language. The Eigenfunctions revisited: We investigate the eigenfunctions again. By the assumptions on the ge- ≥ ometry of Ω for all t 0, namely smoothness, compactness, we can apply Lichernowicz’ theorem to t ﬁnd for all individual t eigenfunctions for the Neumann Laplace-Beltrami operator, i.e., we can ﬁnd the { } sequence of pairs (λ (t), Ψ (t)) ∈N consisting of pairwisely orthornormal eigenfunctions Ψ (t) for ∆ n n n n g0 on Ω with Neumann eigenvalues λ (t), t n − ∆ Ψ (t) = λ (t)Ψ (t) on Ω , (205) g0 n n n t 67

text: The

text: decomposition

text: of

text: u

text: in

text: the

text: localized

text: boundary

text: vibrations

text: u

text: from

text: the

text: previous

text: sub-paragraph

text: k

text: k

text: N

text: M

text: gives

text: us

text: in

text: total

text: N

text: equations:

text: An

text: acoustic

text: wave

text: equation

text: on

text: the

text: perturbed

text: bundle

text: for

text: p

text: and

text: N

text: equations

text: describing

text: the

text: dynamics

text: of

text: the

text: localized

text: boundary

text: vibrations

text: u

text: k

text: k

text: N

text: N

text: R

text: M

text: Φ0→1

text: t

text: ×Γkδ((t,

text: x)

text: Γk)

text: on

text: R

text: M

text: Σ

text: 1∂

text: (Σ∂

text: u

text: p(∆∂

text: )u

text: Ψ

text: on

text: Γ

text: t

text: t

text: k

text: k

text: Γk

text: k

text: R

text: The

text: symbol

text: Ψ

text: denotes

text: restriction

text: of

text: the

text: source

text: term

text: for

text: the

text: boundary

text: vibrations

text: u

text: to

text: Γ

text: where

text: Γk

text: k

text: for

text: each

text: k

text: N

text: the

text: boundary

text: vibration

text: u

text: is

text: localized.

text: The

text: initial

text: conditions

text: for

text: both

text: equations

text: k

text: stay

text: unchanged,

text: the

text: Neumann

text: boundary

text: condition

text: to

text: the

text: acoustic

text: wave

text: equation

text: stays

text: unchanged

text: as

text: well,

text: but

text: the

text: Laplace-Beltrami

text: operator

text: is

text: now

text: deﬁned

text: for

text: all

text: Γ

text: ’s

text: individually

text: and

text: breaks

text: the

text: periodic

text: k

text: boundary

text: conditions

text: that

text: we

text: amended

text: to

text: the

text: Laplace-Beltrami

text: operator

text: on

text: ∂Ω

text: The

text: derivation

text: from

text: the

text: previous

text: paragraph

text: demonstrated

text: that

text: we

text: can

text: take

text: Dirichlet

text: boundary

text: conditions

text: for

text: the

text: localized

text: boundary

text: vibrations

text: equations,

text: i.e.,

text: the

text: equations

text: for

text: u

text: k

text: k

text: N

text: R

text: u

text: on

text: ∂Γ

text: k

text: ∂Γk

text: k

text: M

text: M

text: If

text: we

text: pull

text: back

text: the

text: acoustic

text: wave

text: equation

text: on

text: to

text: the

text: well-understood

text: reference

text: bundle

text: we

text: introduce

text: the

text: perturbation

text: operator

text: W

text: but

text: we

text: arrive

text: at

text: equations

text: that

text: live

text: on

text: the

text: reference

text: bundle

text: M

text: and

text: its

text: boundary

text: denoting

text: p˜(Φ

text: (t,

text: x))

text: p(t,

text: x

text: also

text: by

text: the

text: symbol

text: p,

text: t

text: t

text: N

text: R

text: p

text: ρ

text: δ((t,

text: x)

text: Γ

text: t

text: t

text: k

text: k

text: R

text: M

text: Σ

text: 1∂

text: (Σ∂

text: u

text: p(∆∂

text: )u

text: Ψ

text: on

text: Γ

text: t

text: t

text: k

text: k

text: Γk

text: k

text: M

text: The

text: Neumann

text: boundary

text: conditions

text: for

text: the

text: acoustic

text: pressure

text: now

text: become

text: p

text: on

text: For

text: illus-

text: n

text: tration

text: purposes,

text: the

text: mutual

text: acoustic

text: feedback

text: between

text: the

text: boundary

text: vibrations

text: and

text: the

text: acoustic

text: pressure

text: is

text: shown

text: in

text: Fig.

text: Let

text: us

text: comment

text: on

text: the

text: diﬀerential

text: equations

text: adopting

text: for

text: a

text: moment

text: a

text: mathematically

text: more

text: precise

text: language.

text: The

text: Eigenfunctions

text: revisited:

text: We

text: investigate

text: the

text: eigenfunctions

text: again.

text: By

text: the

text: assumptions

text: on

text: the

text: ge-

text: ometry

text: of

text: Ω

text: for

text: all

text: t

text: namely

text: smoothness,

text: compactness,

text: we

text: can

text: apply

text: Lichernowicz’

text: theorem

text: to

text: t

text: ﬁnd

text: for

text: all

text: individual

text: t

text: eigenfunctions

text: for

text: the

text: Neumann

text: Laplace-Beltrami

text: operator,

text: i.e.,

text: we

text: can

text: ﬁnd

text: the

text: sequence

text: of

text: pairs

text: (λ

text: (t),

text: Ψ

text: (t))

text: ∈N

text: consisting

text: of

text: pairwisely

text: orthornormal

text: eigenfunctions

text: Ψ

text: (t)

text: for

text: n

text: n

text: n

text: n

text: on

text: Ω

text: with

text: Neumann

text: eigenvalues

text: λ

text: (t),

text: t

text: n

text: Ψ

text: (t)

text: λ

text: (t)Ψ

text: (t)

text: on

text: Ω

text: n

text: n

text: n

text: t


text: ∈ R Convention: In the following let I = [0, a), a ¯+ be the intersection of the maximal interval of existence of a solution to a system of ﬁrst order ordinary diﬀerential equations, I , intersected with all ∩ R max non-negative reals, I = I +. max 0 Duhamel’s principle - I: Duhamel’s principle is a method to calculate the solution to a liner ﬁrst order diﬀerential equation. We start from the ordinary diﬀerential equations case for the reader’s con- ∈ → R venience. Let g = g(t) C1(I n) be the vector-valued function which is sought for and let ∈ → R ∈ → R A(t) gl(n, C0(I )). Last,let f C0(I n) be a given vector-valued function. Consider the following ordinary diﬀerential equation with initial conditions g(0) = 0, dg (t) = A(t)g(t) + f(t) (220) dt As a ﬁrst step, we assume that A is time-independent and set A(t) = A. Duhamel’s principle states that there is a unique solution to the initial(cid:90)value problem, given by t − g(t) = dτ exp (G(t τ)) f(τ), (221) 0 − · − where G(t τ) = G (t τ) and G = A is referred to as the generator of the smoothly parameterized R (cid:51) ∈ R group GL(n ) exp(tG), t . The deﬁnition uses the correspondence of the exponential map for a Lie-group G and its derivation, i.e., its Lie-algebra g, R × → → exp : + g G, (t, A) exp(tA), (222) 0 where the exponential is for ﬁnite-dimensional Lie-groups G in one of their associated matrix repre- sentation7 just the ordinary matrix exponentiation rewritten in terms of the Laurent series expansion, (cid:88) ∞ 1 exp(tA) = tkAk, (223) k! k=0 R → R where the t could be pulled in front of the powers of A because A deﬁnes a linear mapping n n. Indeed, we have respecting that we(cid:90)also have to diﬀerentiate the integral and not only the integral in the sense of parameter integrals, dg d (cid:90) t − = dτ exp (G(t τ)) f(τ) dt dt 0 t − − = G dτ exp (G(t τ)) f(τ) + exp(G(t t))f(t) 0 = Ag + f. 7One could distinguish the matrix representation of a Lie group and the Lie group itself. However, we will identify the two notions mainly since for most practical purposes physicists resort directly to (the adjoint or fundamental) matrix representation. 73 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: Convention:

text: In

text: the

text: following

text: let

text: I

text: a),

text: a

text: be

text: the

text: intersection

text: of

text: the

text: maximal

text: interval

text: of

text: existence

text: of

text: a

text: solution

text: to

text: a

text: system

text: of

text: ﬁrst

text: order

text: ordinary

text: diﬀerential

text: equations,

text: I

text: intersected

text: with

text: all

text: R

text: max

text: non-negative

text: reals,

text: I

text: I

text: max

text: Duhamel’s

text: principle

text: I:

text: Duhamel’s

text: principle

text: is

text: a

text: method

text: to

text: calculate

text: the

text: solution

text: to

text: a

text: liner

text: ﬁrst

text: order

text: diﬀerential

text: equation.

text: We

text: start

text: from

text: the

text: ordinary

text: diﬀerential

text: equations

text: case

text: for

text: the

text: reader’s

text: con-

text: R

text: venience.

text: Let

text: g

text: g(t)

text: n)

text: be

text: the

text: vector-valued

text: function

text: which

text: is

text: sought

text: for

text: and

text: let

text: R

text: R

text: A(t)

text: gl(n,

text: Last,let

text: f

text: n)

text: be

text: a

text: given

text: vector-valued

text: function.

text: Consider

text: the

text: following

text: ordinary

text: diﬀerential

text: equation

text: with

text: initial

text: conditions

text: dg

text: (t)

text: A(t)g(t)

text: f(t)

text: dt

text: As

text: a

text: ﬁrst

text: step,

text: we

text: assume

text: that

text: A

text: is

text: time-independent

text: and

text: set

text: A(t)

text: A.

text: Duhamel’s

text: principle

text: states

text: that

text: there

text: is

text: a

text: unique

text: solution

text: to

text: the

text: problem,

text: given

text: by

text: t

text: g(t)

text: dτ

text: exp

text: (G(t

text: τ))

text: f(τ),

text: where

text: G(t

text: τ)

text: G

text: (t

text: τ)

text: and

text: G

text: A

text: is

text: referred

text: to

text: as

text: the

text: generator

text: of

text: the

text: smoothly

text: parameterized

text: R

text: R

text: group

text: GL(n

text: exp(tG),

text: t

text: The

text: deﬁnition

text: uses

text: the

text: correspondence

text: of

text: the

text: exponential

text: map

text: for

text: a

text: Lie-group

text: G

text: and

text: its

text: derivation,

text: i.e.,

text: its

text: Lie-algebra

text: g,

text: R

text: exp

text: g

text: G,

text: (t,

text: A)

text: exp(tA),

text: where

text: the

text: exponential

text: is

text: for

text: ﬁnite-dimensional

text: Lie-groups

text: G

text: in

text: one

text: of

text: their

text: associated

text: matrix

text: repre-

text: just

text: the

text: ordinary

text: matrix

text: exponentiation

text: rewritten

text: in

text: terms

text: of

text: the

text: Laurent

text: series

text: expansion,

text: exp(tA)

text: tkAk,

text: k!

text: R

text: R

text: where

text: the

text: t

text: could

text: be

text: pulled

text: in

text: front

text: of

text: the

text: powers

text: of

text: A

text: because

text: A

text: deﬁnes

text: a

text: linear

text: mapping

text: n

text: n.

text: Indeed,

text: we

text: have

text: respecting

text: that

text: have

text: to

text: diﬀerentiate

text: the

text: integral

text: and

text: not

text: only

text: the

text: integral

text: in

text: the

text: sense

text: of

text: parameter

text: integrals,

text: dg

text: d

text: t

text: dτ

text: exp

text: (G(t

text: τ))

text: f(τ)

text: dt

text: dt

text: t

text: G

text: dτ

text: exp

text: (G(t

text: τ))

text: f(τ)

text: exp(G(t

text: t))f(t)

text: Ag

text: f.

text: could

text: distinguish

text: the

text: matrix

text: representation

text: of

text: a

text: Lie

text: group

text: and

text: the

text: Lie

text: group

text: itself.

text: However,

text: we

text: will

text: identify

text: the

text: two

text: notions

text: mainly

text: since

text: for

text: most

text: practical

text: purposes

text: physicists

text: resort

text: directly

text: to

text: (the

text: adjoint

text: or

text: fundamental)

text: matrix

text: representation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where the generator G = G(t)now depends on (cid:90)time and is given by τ (cid:48) (cid:48) G(τ) = dτ A(τ ). (226) 0 Indeed,following the procedure outlined in the Picard-Lindelo¨ﬀ-theorem and symmetrizing the resulting iteration scheme, we can compare with the deﬁnition of the matrix exponential deﬁning here τ = τ− = τ , 1 0 0/0 = 0 and (A(t))0 = 1n, (cid:18)(cid:90) (cid:19) (cid:88) (cid:18)(cid:90) (cid:19) ∞ τ (cid:48) 1 τ (cid:48) (cid:48) k (cid:90) exp dτ A(τ ) = (cid:88) (cid:89) dτ A(τ ) (227) k! 0 0 k=0 ∞ − k τi−1:=τ(1 i/k) = dτ A(τ ) (228) i i 0 k=0 i=0 The last equation is the A-dependent part in the Picard-iteration, i.e., the application of the Banach ﬁxed point theorem in the proof of the Picard-Lindel¨oﬀ theorem. The integrals could be symmetrized − with the result of equal integration domain [0, τ] and a symmetrization factor (k!) 1. Notice that only in the symmetrization procedure we need the commutativity of the t-dependent matrices: In our setting, we ∈ R have for all t , t that [A(t , ), A(t )] = 0. In other words, we could commute the t-dependent matrices 1 2 1 2 just as we do for ordinary functions depending on t. We check that Duhamel’s principle yields a solution (cid:90) to the ordinary diﬀerential equation system under consideration dg (cid:90)d t − = dτ exp(G(t τ)f(τ) dt dt 0 − (cid:90) t dG(t τ(cid:12)(cid:12)) (cid:18)(cid:90) − (cid:19) = dτ (cid:12) exp(G(t τ)f(τ) + f(t) dt (cid:12) 0 − (cid:90) t (cid:18)d(cid:90) (cid:19)t τ (cid:48) (cid:48) − = dτ − dτ A(τ ) exp(G(t τ)f(τ) + f d(t τ) 0 τ=0 0 (cid:90) t d t (cid:48) (cid:48) − = dτ dτ A(τ ) exp(G(t τ)f(τ) + f dt 0 0 t (cid:90) − = dτ A(t) exp(G(t τ)f(τ) + f 0 t − = A(t) dτ exp(G(t τ)f(τ) + f 0 = A(t)g + f, − | − (cid:82) using that t = (t τ) and d(t τ) = dt. Also in this case, Duhamel’s principle works. We have τ=0 ∈ R summarized the geometrical idea of Duhamel’s principle in Fig. 13. A Lie algebra element A gl(n, ) (cid:16) (cid:17) t (cid:82) or in gl /[gl , gl ] can be used to deﬁne a straight line G(t) = dτ A which is a curve in the Lie algebra. c c c 0 ∈ R By means of the exponential map, we can in a neighborhood of 0 gl(n ) or gl /[gl , gl ], and by suitable c c c R t translations of the origin also elsewhere, map G(t) to an element exp dτ A in the Lie group GL(n, ) 0 75 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: the

text: generator

text: G

text: G(t)now

text: depends

text: on

text: and

text: is

text: given

text: by

text: τ

text: G(τ)

text: dτ

text: A(τ

text: Indeed,following

text: the

text: procedure

text: outlined

text: in

text: the

text: Picard-Lindelo¨ﬀ-theorem

text: and

text: symmetrizing

text: the

text: resulting

text: iteration

text: scheme,

text: we

text: can

text: compare

text: with

text: the

text: deﬁnition

text: of

text: the

text: matrix

text: exponential

text: deﬁning

text: here

text: τ

text: τ−

text: τ

text: and

text: τ

text: τ

text: k

text: exp

text: dτ

text: A(τ

text: dτ

text: A(τ

text: k!

text: k

text: i/k)

text: dτ

text: A(τ

text: i

text: i

text: The

text: last

text: equation

text: is

text: the

text: A-dependent

text: part

text: in

text: the

text: Picard-iteration,

text: i.e.,

text: the

text: application

text: of

text: the

text: Banach

text: ﬁxed

text: point

text: theorem

text: in

text: the

text: proof

text: of

text: the

text: Picard-Lindel¨oﬀ

text: theorem.

text: The

text: integrals

text: could

text: be

text: symmetrized

text: with

text: the

text: result

text: of

text: equal

text: integration

text: domain

text: τ]

text: and

text: a

text: symmetrization

text: factor

text: (k!)

text: Notice

text: that

text: only

text: in

text: the

text: symmetrization

text: procedure

text: we

text: need

text: the

text: commutativity

text: of

text: the

text: t-dependent

text: matrices:

text: In

text: our

text: setting,

text: we

text: R

text: have

text: for

text: all

text: t

text: t

text: that

text: [A(t

text: A(t

text: In

text: other

text: words,

text: we

text: could

text: commute

text: the

text: t-dependent

text: matrices

text: just

text: as

text: we

text: do

text: for

text: ordinary

text: functions

text: depending

text: on

text: t.

text: We

text: check

text: that

text: Duhamel’s

text: principle

text: yields

text: a

text: solution

text: to

text: the

text: ordinary

text: diﬀerential

text: equation

text: system

text: under

text: consideration

text: dg

text: t

text: dτ

text: exp(G(t

text: τ)f(τ)

text: dt

text: dt

text: t

text: dG(t

text: dτ

text: exp(G(t

text: τ)f(τ)

text: f(t)

text: dt

text: t

text: τ

text: dτ

text: dτ

text: A(τ

text: exp(G(t

text: τ)f(τ)

text: f

text: d(t

text: τ)

text: t

text: d

text: t

text: dτ

text: dτ

text: A(τ

text: exp(G(t

text: τ)f(τ)

text: f

text: dt

text: t

text: dτ

text: A(t)

text: exp(G(t

text: τ)f(τ)

text: f

text: t

text: A(t)

text: dτ

text: exp(G(t

text: τ)f(τ)

text: f

text: A(t)g

text: f,

text: using

text: that

text: t

text: (t

text: τ)

text: and

text: d(t

text: τ)

text: dt.

text: Also

text: in

text: this

text: case,

text: Duhamel’s

text: principle

text: works.

text: We

text: have

text: R

text: summarized

text: the

text: geometrical

text: idea

text: of

text: Duhamel’s

text: principle

text: in

text: Fig.

text: A

text: Lie

text: algebra

text: element

text: A

text: gl(n,

text: t

text: or

text: in

text: gl

text: /[gl

text: gl

text: can

text: be

text: used

text: to

text: deﬁne

text: a

text: straight

text: line

text: G(t)

text: dτ

text: A

text: which

text: is

text: a

text: curve

text: in

text: the

text: Lie

text: algebra.

text: c

text: c

text: c

text: R

text: By

text: means

text: of

text: the

text: exponential

text: map,

text: we

text: can

text: in

text: a

text: neighborhood

text: of

text: gl(n

text: or

text: gl

text: /[gl

text: gl

text: and

text: by

text: suitable

text: c

text: c

text: c

text: R

text: t

text: translations

text: of

text: the

text: origin

text: also

text: elsewhere,

text: map

text: G(t)

text: to

text: an

text: element

text: exp

text: dτ

text: A

text: in

text: the

text: Lie

text: group

text: GL(n,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R ∈ )) := GL the element A(t) gl . This is achieved by a convenient property of the covariant10 Lie-functor c c L, namely that it respects the lower central series decomposition for the Lie-group and the Lie-algebra. More precisely, (cid:77) (cid:77) ∞ ∞ Adk [GL ] (cid:55)→ adk [gl ] L : GL = GLc c gl = glc c , (260) c Adk+1[GL ] c adk+1[gl ] k=0 GLc c k=0 glc c preserves the individual factors where Ad denotes the action of the GL on itself by conjugation. GLc c Moreover, the operators that we are interested in, namely Laplacians are unbounded operators in the L2 and the H1,2 norm. However, when restricting to H2,2, i.e., another Sobolev-space, which we can also equip with the L2-norm, the Laplacians can be bounded by the H2,2-norms of the functions they are acting on. At the end of this paragraph let us brieﬂy recapitulate what the Magnus series does: In the non-commutative case of the full gl , we cannot use the exponential straight away to map from the Lie c → algebra to the Lie group, c.f., the upper row in Fig. 16. However, we can deﬁne a curve t G(t) in the Lie-algebra such that the exponential maps G(t) to an element S(t) of the Lie-group such that its − (cid:48) diﬀerential logarithm, d exp 1 S(t) = S (t)/S(t) = A(t). It is more useful in the context of diﬀerential ≥ equation solving to interpret G(t) for t 0 as a matrix semigroup, c.f. [58, 8, 2, 82, 28, 47]. The process is depicted in the lower half of Fig. 16. Duhamel IV Duhamel’s principle and its one of the iteration schemes derived thereof - the Magnus series - can be generalized to operator, [58, 103, 104]. The analogy in physics used to give sense to this process is the quantum mechanical imagination of (linear) operators as inﬁnite-dimensional, but else well- behaved matrices mapping from one inﬁnite-dimensional vector space with an inner product, the so-called Hilbert space, to another. The discussion can be extended to Banach spaces as a more general class of function spaces, [49, 103, 104], but we only need Hilbert spaces which have more convenient structures on them. From the more mathematical angle, the analogy is useful but not totally correct. We will be interested in two speciﬁc issues, namely boundedness and compactness of the operators. We will use the notion of an operator as a synonym for an operator between Hilbert spaces and the notion matrix as a synonym for a linear map between ﬁnite-dimensional vector spaces. Recall, that an operator or a matrix, (cid:107) (cid:107) → (cid:107) (cid:107) ∈ R A : (V , . ) (V , . ) is called bounded if for a c +, the relative operator norm of A is bounded by 1 1 2 2 0 c, i.e., (cid:20) (cid:21) (cid:107) (cid:107) (cid:107) (cid:107) Av ∞ A → := sup (cid:107) (cid:107) 2 = c < . (261) 1 2 ∈ v v V1 1 In the ﬁnite dimensional case, i.e., when A is a matrix, we need not distinguish between two diﬀerent (cid:107) (cid:107) | | relative norms . → and . → on HomC(V , V ). The norms are said to be equivalent. In the inﬁnite- 1 2 1 2 1 2 C− dimensional case, i.e., when V , V are Hilbert spaces and the operator spaces is the space of linear 1 2 operators between V and V , LinOpC(V , V ), the distinction becomes crucial. We can have one relative (cid:107) (cid:107) 1 2 1 2 | | operator norm . → on LinOpC(V , V ) where an operator A is bounded, but another norm . → on 1 2 1 2 1 2 LinOpC(V , V ) where it is unbounded. With the deﬁnition of the operator norm in the context of bound- 1 2 edness, an appropriate norm choices can be broken down to an appropriate selection of norms on the domain Hilbert space V of the operator A and an appropriate norm on the target Hilbert space V . The 1 2 10in the functorial sense 81 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: GL

text: the

text: element

text: A(t)

text: gl

text: This

text: is

text: achieved

text: by

text: a

text: convenient

text: property

text: of

text: the

text: Lie-functor

text: c

text: c

text: L,

text: namely

text: that

text: it

text: respects

text: the

text: lower

text: central

text: series

text: decomposition

text: for

text: the

text: Lie-group

text: and

text: the

text: Lie-algebra.

text: More

text: precisely,

text: Adk

text: [GL

text: adk

text: [gl

text: L

text: GL

text: GLc

text: c

text: gl

text: glc

text: c

text: c

text: c

text: GLc

text: c

text: glc

text: c

text: preserves

text: the

text: individual

text: factors

text: where

text: Ad

text: denotes

text: the

text: action

text: of

text: the

text: GL

text: on

text: itself

text: by

text: conjugation.

text: GLc

text: c

text: Moreover,

text: the

text: operators

text: that

text: we

text: are

text: interested

text: in,

text: namely

text: Laplacians

text: are

text: unbounded

text: operators

text: in

text: the

text: and

text: the

text: norm.

text: However,

text: when

text: restricting

text: to

text: i.e.,

text: another

text: Sobolev-space,

text: which

text: we

text: can

text: also

text: equip

text: with

text: the

text: the

text: Laplacians

text: can

text: be

text: bounded

text: by

text: the

text: of

text: the

text: functions

text: they

text: are

text: acting

text: on.

text: At

text: the

text: end

text: of

text: this

text: paragraph

text: let

text: us

text: brieﬂy

text: recapitulate

text: what

text: the

text: Magnus

text: series

text: does:

text: In

text: the

text: non-commutative

text: case

text: of

text: the

text: full

text: gl

text: we

text: cannot

text: use

text: the

text: exponential

text: straight

text: away

text: to

text: map

text: from

text: the

text: Lie

text: c

text: algebra

text: to

text: the

text: Lie

text: group,

text: c.f.,

text: the

text: upper

text: row

text: in

text: Fig.

text: However,

text: we

text: can

text: deﬁne

text: a

text: curve

text: t

text: G(t)

text: in

text: the

text: Lie-algebra

text: such

text: that

text: the

text: exponential

text: maps

text: G(t)

text: to

text: an

text: element

text: S(t)

text: of

text: the

text: Lie-group

text: such

text: that

text: its

text: diﬀerential

text: logarithm,

text: d

text: exp

text: S(t)

text: S

text: (t)/S(t)

text: A(t).

text: It

text: is

text: more

text: useful

text: in

text: the

text: context

text: of

text: diﬀerential

text: equation

text: solving

text: to

text: interpret

text: G(t)

text: for

text: t

text: as

text: a

text: matrix

text: semigroup,

text: c.f.

text: The

text: process

text: is

text: depicted

text: in

text: the

text: lower

text: half

text: of

text: Fig.

text: Duhamel

text: IV

text: Duhamel’s

text: principle

text: and

text: its

text: one

text: of

text: the

text: iteration

text: schemes

text: derived

text: thereof

text: the

text: Magnus

text: series

text: can

text: be

text: generalized

text: to

text: operator,

text: The

text: analogy

text: in

text: physics

text: used

text: to

text: give

text: sense

text: to

text: this

text: process

text: is

text: the

text: quantum

text: mechanical

text: imagination

text: of

text: (linear)

text: operators

text: as

text: inﬁnite-dimensional,

text: but

text: else

text: well-

text: behaved

text: matrices

text: mapping

text: from

text: one

text: inﬁnite-dimensional

text: vector

text: space

text: with

text: an

text: inner

text: product,

text: the

text: so-called

text: Hilbert

text: space,

text: to

text: another.

text: The

text: discussion

text: can

text: be

text: extended

text: to

text: Banach

text: spaces

text: as

text: a

text: more

text: general

text: class

text: of

text: function

text: spaces,

text: but

text: we

text: only

text: need

text: Hilbert

text: spaces

text: which

text: have

text: more

text: convenient

text: structures

text: on

text: them.

text: From

text: the

text: more

text: mathematical

text: angle,

text: the

text: analogy

text: is

text: useful

text: but

text: not

text: totally

text: correct.

text: We

text: will

text: be

text: interested

text: in

text: two

text: speciﬁc

text: issues,

text: namely

text: boundedness

text: and

text: compactness

text: of

text: the

text: operators.

text: We

text: will

text: use

text: the

text: notion

text: of

text: an

text: operator

text: as

text: a

text: synonym

text: for

text: an

text: operator

text: between

text: Hilbert

text: spaces

text: and

text: the

text: notion

text: matrix

text: as

text: a

text: synonym

text: for

text: a

text: linear

text: map

text: between

text: ﬁnite-dimensional

text: vector

text: spaces.

text: Recall,

text: that

text: an

text: operator

text: or

text: a

text: matrix,

text: R

text: A

text: (V

text: (V

text: is

text: called

text: bounded

text: if

text: for

text: a

text: c

text: the

text: relative

text: operator

text: norm

text: of

text: A

text: is

text: bounded

text: by

text: c,

text: i.e.,

text: Av

text: A

text: sup

text: c

text: v

text: v

text: In

text: the

text: ﬁnite

text: dimensional

text: case,

text: i.e.,

text: when

text: A

text: is

text: a

text: matrix,

text: we

text: need

text: not

text: distinguish

text: between

text: two

text: diﬀerent

text: relative

text: norms

text: and

text: on

text: HomC(V

text: V

text: The

text: norms

text: are

text: said

text: to

text: be

text: equivalent.

text: In

text: the

text: inﬁnite-

text: C−

text: dimensional

text: case,

text: i.e.,

text: when

text: V

text: V

text: are

text: Hilbert

text: spaces

text: and

text: the

text: operator

text: spaces

text: is

text: the

text: space

text: of

text: linear

text: operators

text: between

text: V

text: and

text: V

text: LinOpC(V

text: V

text: the

text: distinction

text: becomes

text: crucial.

text: We

text: can

text: have

text: one

text: relative

text: operator

text: norm

text: on

text: LinOpC(V

text: V

text: where

text: an

text: operator

text: A

text: is

text: bounded,

text: but

text: another

text: norm

text: on

text: LinOpC(V

text: V

text: where

text: it

text: is

text: unbounded.

text: With

text: the

text: deﬁnition

text: of

text: the

text: operator

text: norm

text: in

text: the

text: context

text: of

text: bound-

text: edness,

text: an

text: appropriate

text: norm

text: choices

text: can

text: be

text: broken

text: down

text: to

text: an

text: appropriate

text: selection

text: of

text: norms

text: on

text: the

text: domain

text: Hilbert

text: space

text: V

text: of

text: the

text: operator

text: A

text: and

text: an

text: appropriate

text: norm

text: on

text: the

text: target

text: Hilbert

text: space

text: V

text: The

text: the

text: functorial

text: sense

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The subtlety is that we don’t know what S is. The Dyson series expansion uses the perturbation lemma [2, 89, 74, 52] that if S (t, τ) is an evolution family and V is a perturbation, then S(t, τ) is 0 A an evolution as well, given by the integral equation (cid:90) t (cid:48) (cid:48) (cid:48) (cid:48) S(t, τ) = S (t, τ) + dτ S (t, τ )V (τ )S(τ , τ). (276) 0 0 A τ This is Duhamel’s principle applied to the operator equation for the evolution family S(t, τ) treating V S as the inhomogeneity by Banach ﬁxed point theorem. The proof of existence uses Gronwall’s A lemma, [37]. Since the opera(cid:13)tors A(cid:18),(cid:90)A0 are bound(cid:19)e(cid:13)d, Gronwall gives (cid:18)(cid:90) (cid:19) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) ≤ (cid:107) (cid:107) t (cid:48) (cid:48) ≤ (cid:107) (cid:107) t (cid:107) (cid:107) S(t, τ) S (t, τ) exp dτ V (τ ) S (t, τ) exp dτ V (τ) . (277) 0 A 0 A τ τ (cid:107) (cid:107) (cid:28) If V (t) 1, the series converges. Bounds can be found e.g. in [8, 27]. Mathematicians work A (cid:107) (cid:107) (cid:28) on improvement of bounds. In the case we are interested in, we have V 1 such that locally A these series converge. The summation problem that Dyson worked on stays the same as for the Dirac series, except that now the operators S are to be build up from S and V. Dyson found the 0 expression (cid:90) (cid:18)(cid:90) (cid:19) (cid:88) (cid:89) ∞ ∞ ∞ k (cid:18) (cid:18)(cid:90) (cid:19)(cid:19) S(t, τ) = dτ S (t, τ ) dτ V (τ − )S (τ − , τ ) (278) 0 0 0 j A j 1 0 j 1 j 0 0 k=0 j=1 ≡ T t (cid:48) (cid:48) exp dτ A(τ ) . (279) τ T using the time-ordering symbol for symmetrization of the Neumann series again. • Magnus’ method: Magnus [58] proceeded in a diﬀerent way and circumvented the formal time- ordering symbol. The main issue lies in the fundamental theorem of calculus, no longer giving the familiar result for t(cid:16)he scalar qu(cid:17)antities, (cid:82) (cid:18)(cid:90) (cid:19) (cid:18)(cid:90) (cid:19) (cid:48) (cid:48) t d exp τ dτ a(τ ) (cid:48) t (cid:48) (cid:48) t (cid:48) (cid:48) (cid:48) = a (t) exp dτ a(τ ) = exp dτ a(τ ) a (t), (280) dt τ τ in the case of A(t) being an element of a Lie-algebra g or more generally of an associative Banach algebra turned into a Lie-algebra by endowment with a Lie bracket, (cid:82) (cid:18)(cid:90) (cid:19) (cid:18)(cid:90) (cid:19) (cid:48) ∂ exp( t dτ A(τ)) (cid:48) t t (cid:48) τ !=?!? A (t) exp dτ A(τ) !=?!? exp dτ A(τ) A (t). (281) ∂t 0 0 89 ##LTLine## ##LTLine##

text: The

text: subtlety

text: is

text: that

text: we

text: don’t

text: know

text: what

text: S

text: is.

text: The

text: Dyson

text: series

text: expansion

text: uses

text: the

text: perturbation

text: lemma

text: that

text: if

text: S

text: (t,

text: τ)

text: is

text: an

text: evolution

text: family

text: and

text: V

text: is

text: a

text: perturbation,

text: then

text: S(t,

text: τ)

text: is

text: A

text: an

text: evolution

text: as

text: well,

text: given

text: by

text: the

text: integral

text: equation

text: t

text: S(t,

text: τ)

text: S

text: (t,

text: τ)

text: dτ

text: S

text: (t,

text: τ

text: )V

text: (τ

text: )S(τ

text: τ).

text: A

text: τ

text: This

text: is

text: Duhamel’s

text: principle

text: applied

text: to

text: the

text: operator

text: equation

text: for

text: the

text: evolution

text: family

text: S(t,

text: τ)

text: treating

text: V

text: S

text: as

text: the

text: inhomogeneity

text: by

text: Banach

text: ﬁxed

text: point

text: theorem.

text: The

text: proof

text: of

text: existence

text: uses

text: Gronwall’s

text: A

text: lemma,

text: Since

text: the

text: are

text: Gronwall

text: gives

text: t

text: t

text: S(t,

text: τ)

text: S

text: (t,

text: τ)

text: exp

text: dτ

text: V

text: (τ

text: S

text: (t,

text: τ)

text: exp

text: dτ

text: V

text: (τ)

text: A

text: A

text: τ

text: τ

text: If

text: V

text: (t)

text: the

text: series

text: converges.

text: Bounds

text: can

text: be

text: found

text: e.g.

text: in

text: Mathematicians

text: work

text: A

text: on

text: improvement

text: of

text: bounds.

text: In

text: the

text: case

text: we

text: are

text: interested

text: in,

text: we

text: have

text: V

text: such

text: that

text: locally

text: A

text: these

text: series

text: converge.

text: The

text: summation

text: problem

text: that

text: Dyson

text: worked

text: on

text: stays

text: the

text: same

text: as

text: for

text: the

text: Dirac

text: series,

text: except

text: that

text: now

text: the

text: operators

text: S

text: are

text: to

text: be

text: build

text: up

text: from

text: S

text: and

text: V.

text: Dyson

text: found

text: the

text: expression

text: k

text: S(t,

text: τ)

text: dτ

text: S

text: (t,

text: τ

text: dτ

text: V

text: (τ

text: )S

text: (τ

text: τ

text: j

text: A

text: j

text: j

text: j

text: T

text: t

text: exp

text: dτ

text: A(τ

text: τ

text: T

text: using

text: the

text: time-ordering

text: symbol

text: for

text: symmetrization

text: of

text: the

text: Neumann

text: series

text: again.

text: Magnus’

text: method:

text: Magnus

text: proceeded

text: in

text: a

text: diﬀerent

text: way

text: and

text: circumvented

text: the

text: formal

text: time-

text: ordering

text: symbol.

text: The

text: main

text: issue

text: lies

text: in

text: the

text: fundamental

text: theorem

text: of

text: calculus,

text: no

text: longer

text: giving

text: the

text: familiar

text: result

text: for

text: scalar

text: t

text: d

text: exp

text: τ

text: dτ

text: a(τ

text: t

text: t

text: a

text: (t)

text: exp

text: dτ

text: a(τ

text: exp

text: dτ

text: a(τ

text: a

text: (t),

text: dt

text: τ

text: τ

text: in

text: the

text: case

text: of

text: A(t)

text: being

text: an

text: element

text: of

text: a

text: Lie-algebra

text: g

text: or

text: more

text: generally

text: of

text: an

text: associative

text: Banach

text: algebra

text: turned

text: into

text: a

text: Lie-algebra

text: by

text: endowment

text: with

text: a

text: Lie

text: bracket,

text: exp(

text: t

text: dτ

text: A(τ))

text: t

text: t

text: τ

text: A

text: (t)

text: exp

text: dτ

text: A(τ)

text: exp

text: dτ

text: A(τ)

text: A

text: (t).

text: ∂t

text: ##LTLine##

text: ##LTLine##


text: endfor Comment: Now the set T makes the algorithm hear collisions at all the steps when the set F does. Phase 4 (increasing size) for i := 1 to k do | | ∩ (cid:54) ∅ if T < m and Q T = then begin i ⊆ ∩ | ∪ | ≤ ∪ choose X Q F of a maximal size such that T X m ; T := T X i end Comment: In this phase we increase the size of T , by adding edges included in queries, to be as close as possible to m. This phase is productive if the set Q is large enough, otherwise we still have Phase 5 as the last resort. Phase 5 (padding with edges not mentioned) | | − | | If T < m then add m T edges to T that are not in Q. A Comment: This phase is needed in case algorithm has the queries involving only a small set of edges. This completes the ﬁve-phase construction of the graph G. We say that a vertex v is a Y -witness, ∩ { } ≤ for a set Y of edges, if none of the edges (v, w) satisﬁes Q Y = (v, w) , for any number i k i and for any vertex w in V distinct from v. Correctness of the ﬁve-phase construction is formulated as Lemma 4. Lemma 4 The ﬁve-phase construction results in obtaining a set T of size m and with a vertex that is a T -witness. Proof: We examine the ﬁve phases one by one. ∩ { } After Phase 1. Suppose to the contrary that there are no F -witness vertices. We have Q F = e \ { } ∩ { i } holds only if F e has some isolated vertex. An edge (v, w), such that Q F = (v, w) , is not i removed only if otherwise one among the vertices v and w would become isolated in F . Notice also that at least − − · · · − · − − (n 1) + (n 2) + + (n l) = l (2n l 1)/2 edges need to be removed in order to isolate l or more vertices, which follows by induction on l. Removing one edge results in the intersection of at least one query with F being empty. We can remove at most k edges. Should we like to restrain all the n vertices from being F -witnesses we − − would need to isolate at least n/2 vertices, thus needing to remove at least n/2(2n n/2 1)/2 = − − − n(3n 1)/4 = 3n2 n/4 edges from F . Because of the inequality k < n2/8 n/2, at least one 2 8 vertex v has not been prevented from being an F -witness. It follows that all the singleton intersections that contain an edge adjacent to vertex v have | | ≥ | | − been removed from F . The inequality F Q k follows by the fact that an edge is removed from the set F only if at least one query Q becomes disjoint with the updated F . i 12 ##LTLine## ##LTLine##

text: endfor

text: Comment:

text: Now

text: the

text: set

text: T

text: makes

text: the

text: algorithm

text: hear

text: collisions

text: at

text: all

text: the

text: steps

text: when

text: the

text: set

text: F

text: does.

text: Phase

text: (increasing

text: size)

text: for

text: i

text: to

text: k

text: do

text: if

text: T

text: m

text: and

text: Q

text: T

text: then

text: begin

text: i

text: choose

text: X

text: Q

text: F

text: of

text: a

text: maximal

text: size

text: such

text: that

text: T

text: X

text: m

text: T

text: T

text: X

text: i

text: end

text: Comment:

text: In

text: this

text: phase

text: we

text: increase

text: the

text: size

text: of

text: T

text: by

text: adding

text: edges

text: included

text: in

text: queries,

text: to

text: be

text: as

text: close

text: as

text: possible

text: to

text: m.

text: This

text: phase

text: is

text: productive

text: if

text: the

text: set

text: Q

text: is

text: large

text: enough,

text: otherwise

text: we

text: still

text: have

text: Phase

text: as

text: the

text: last

text: resort.

text: Phase

text: (padding

text: with

text: edges

text: not

text: mentioned)

text: If

text: T

text: m

text: then

text: add

text: m

text: T

text: edges

text: to

text: T

text: that

text: are

text: not

text: in

text: Q.

text: A

text: Comment:

text: This

text: phase

text: is

text: needed

text: in

text: case

text: algorithm

text: has

text: the

text: queries

text: involving

text: only

text: a

text: small

text: set

text: of

text: edges.

text: This

text: completes

text: the

text: ﬁve-phase

text: construction

text: of

text: the

text: graph

text: G.

text: We

text: say

text: that

text: a

text: vertex

text: v

text: is

text: a

text: Y

text: -witness,

text: for

text: a

text: set

text: Y

text: of

text: edges,

text: if

text: of

text: the

text: edges

text: (v,

text: w)

text: satisﬁes

text: Q

text: Y

text: (v,

text: w)

text: for

text: any

text: number

text: i

text: k

text: i

text: and

text: for

text: any

text: vertex

text: w

text: in

text: V

text: distinct

text: from

text: v.

text: Correctness

text: of

text: the

text: ﬁve-phase

text: construction

text: is

text: formulated

text: as

text: Lemma

text: Lemma

text: The

text: ﬁve-phase

text: construction

text: results

text: in

text: obtaining

text: a

text: set

text: T

text: of

text: size

text: m

text: and

text: with

text: a

text: vertex

text: that

text: is

text: a

text: T

text: -witness.

text: Proof:

text: We

text: examine

text: the

text: ﬁve

text: phases

text: one

text: by

text: one.

text: After

text: Phase

text: Suppose

text: to

text: the

text: contrary

text: that

text: there

text: are

text: no

text: F

text: -witness

text: vertices.

text: We

text: have

text: Q

text: F

text: e

text: i

text: holds

text: only

text: if

text: F

text: e

text: has

text: some

text: isolated

text: vertex.

text: An

text: edge

text: (v,

text: w),

text: such

text: that

text: Q

text: F

text: (v,

text: w)

text: is

text: not

text: i

text: removed

text: only

text: if

text: otherwise

text: one

text: among

text: the

text: vertices

text: v

text: and

text: w

text: would

text: become

text: isolated

text: in

text: F

text: Notice

text: also

text: that

text: at

text: least

text: (n

text: (n

text: (n

text: l)

text: l

text: l

text: edges

text: need

text: to

text: be

text: removed

text: in

text: order

text: to

text: isolate

text: l

text: or

text: more

text: vertices,

text: which

text: follows

text: by

text: induction

text: on

text: l.

text: Removing

text: one

text: edge

text: results

text: in

text: the

text: intersection

text: of

text: at

text: least

text: one

text: query

text: with

text: F

text: being

text: empty.

text: We

text: can

text: remove

text: at

text: most

text: k

text: edges.

text: Should

text: we

text: like

text: to

text: restrain

text: all

text: the

text: n

text: vertices

text: from

text: being

text: F

text: -witnesses

text: we

text: would

text: need

text: to

text: isolate

text: at

text: least

text: vertices,

text: thus

text: needing

text: to

text: remove

text: at

text: least

text: edges

text: from

text: F

text: Because

text: of

text: the

text: inequality

text: k

text: at

text: least

text: one

text: vertex

text: v

text: has

text: not

text: been

text: prevented

text: from

text: being

text: an

text: F

text: -witness.

text: It

text: follows

text: that

text: all

text: the

text: singleton

text: intersections

text: that

text: contain

text: an

text: edge

text: adjacent

text: to

text: vertex

text: v

text: have

text: been

text: removed

text: from

text: F

text: The

text: inequality

text: F

text: Q

text: k

text: follows

text: by

text: the

text: fact

text: that

text: an

text: edge

text: is

text: removed

text: from

text: the

text: set

text: F

text: only

text: if

text: at

text: least

text: one

text: query

text: Q

text: becomes

text: disjoint

text: with

text: the

text: updated

text: F

text: i

text: ##LTLine##

text: ##LTLine##


text: ∩ { } ≤ After Phase 2. The tree T contains each edge (v, w) such that Q F = (v, w) , for some i k. i This follows from the observation that such an edge (v, w) has not been removed in Phase 1, hence one among the vertices v and w is of degree 1 in F , and the edge (v, w) was put into set T . Clearly, | | ≤ − T n 1, and hence there is a T -witness at this point. − After Phase 3. Set T has at most n 1 + 2k < m edges. This is because at most two edges can ≤ be added to the set T in each step i k. The vertex v that is a T -witness is the same as in the ∩ { } analysis of the previous phases. More precisely, there was no edge (v, w) such that Q T = (v, w) i in the beginning of Phase 3, the set T was not decreased during Phase 3, and no edge such that ∩ ∅ Q F = was added in step i. i | | − After Phase 4. We obtain set T of a size that is between Q k and m. This is because we can \ | | ≥ | | − add to T any number of edges from F T , as long as the inequality F Q k holds. There is a T -witness because during Phase 4 we add only edges in such queries that have at least one element in T . After Phase 5. The property of an existence of a T -witness is maintained because Phase 5 does | | | | − not interfere with intersections of F and Q . Since the size T was between Q k and m, we can i { ∈ (cid:54) } \ − − | | ≥ add suﬃciently many elements from the set (v, w) : v, w V, v = w Q of size n(n 1)/2 Q − | | | | (cid:3) m + k Q to obtain T = m. Lemma 4 contains all the essential ingredients of a lower bound in terms of the number m of edges alone. We show a more general fact formulated in terms of both the numbers of nodes and edges. A Theorem 3 For any deterministic oblivious algorithm ﬁnding spanning forests, and numbers n O and m, where m = (n2), there exists a connected graph GA with n vertices and Θ(m) edges, such A that algorithm requires time Ω(m) to ﬁnd a spanning tree of GA. ≤ ≤ − Proof: If the inequality 2n m n(n 1)/4 holds, then apply the ﬁve-phase construction. It follows from Lemma 4 that if we take the ﬁnal T , obtained by completing the ﬁve-phase construction, A as the set of edges E, then the oblivious algorithm needs more than k = Ω(m) steps to broadcast a spanning tree of the graph G = (V, E). This is because a T -witness has not been heard by the kth step as an endpoint of an edge. In the remaining cases of dependencies between the numbers m and n, we can proceed as − follows. If n(n 1)/4 < m, then we can use the same construction as if the number m were equal − − to n(n 1)/4. If m < 2n, then a simple path of length (cid:96) = min [ n 1 , m ] does the job, because (cid:3) any algorithm needs to broadcast at least (cid:96) edges to reveal the whole path. There is a simple oblivious algorithm that operates in time m + 1: the station i broadcasts its edge at step i. Theorem 3 may be interpreted as follows: listing all the edges systematically is asymptotically optimal among oblivious algorithms. 4 Weighted Graphs In this section we consider graphs that have positive weights assigned to their edges. The input is speciﬁed at the start of an execution, no adversaries are involved. 13

text: After

text: Phase

text: The

text: tree

text: T

text: contains

text: each

text: edge

text: (v,

text: w)

text: such

text: that

text: Q

text: F

text: (v,

text: w)

text: for

text: some

text: i

text: k.

text: i

text: This

text: follows

text: from

text: the

text: observation

text: that

text: such

text: an

text: edge

text: (v,

text: w)

text: has

text: not

text: been

text: removed

text: in

text: Phase

text: hence

text: one

text: among

text: the

text: vertices

text: v

text: and

text: w

text: is

text: of

text: degree

text: in

text: F

text: and

text: the

text: edge

text: (v,

text: w)

text: was

text: put

text: into

text: set

text: T

text: Clearly,

text: T

text: n

text: and

text: hence

text: there

text: is

text: a

text: T

text: -witness

text: at

text: this

text: point.

text: After

text: Phase

text: Set

text: T

text: has

text: at

text: most

text: n

text: m

text: edges.

text: This

text: is

text: because

text: at

text: most

text: two

text: edges

text: can

text: be

text: added

text: to

text: the

text: set

text: T

text: in

text: each

text: step

text: i

text: k.

text: The

text: vertex

text: v

text: that

text: is

text: a

text: T

text: -witness

text: is

text: the

text: same

text: as

text: in

text: the

text: analysis

text: of

text: the

text: previous

text: phases.

text: More

text: precisely,

text: there

text: was

text: no

text: edge

text: (v,

text: w)

text: such

text: that

text: Q

text: T

text: (v,

text: w)

text: i

text: in

text: the

text: beginning

text: of

text: Phase

text: the

text: set

text: T

text: was

text: not

text: decreased

text: during

text: Phase

text: and

text: no

text: edge

text: such

text: that

text: Q

text: F

text: was

text: added

text: in

text: step

text: i.

text: i

text: After

text: Phase

text: We

text: obtain

text: set

text: T

text: of

text: a

text: size

text: that

text: is

text: between

text: Q

text: k

text: and

text: m.

text: This

text: is

text: because

text: we

text: can

text: add

text: to

text: T

text: any

text: number

text: of

text: edges

text: from

text: F

text: T

text: as

text: long

text: as

text: the

text: inequality

text: F

text: Q

text: k

text: holds.

text: There

text: is

text: a

text: T

text: -witness

text: because

text: during

text: Phase

text: we

text: add

text: only

text: edges

text: in

text: such

text: queries

text: that

text: have

text: at

text: least

text: one

text: element

text: in

text: T

text: After

text: Phase

text: The

text: property

text: of

text: an

text: existence

text: of

text: a

text: T

text: -witness

text: is

text: maintained

text: because

text: Phase

text: does

text: not

text: interfere

text: with

text: intersections

text: of

text: F

text: and

text: Q

text: Since

text: the

text: size

text: T

text: was

text: between

text: Q

text: k

text: and

text: m,

text: we

text: can

text: i

text: add

text: suﬃciently

text: many

text: elements

text: from

text: the

text: set

text: (v,

text: w)

text: v,

text: w

text: V,

text: v

text: w

text: Q

text: of

text: size

text: n(n

text: Q

text: m

text: k

text: Q

text: to

text: obtain

text: T

text: m.

text: Lemma

text: contains

text: all

text: the

text: essential

text: ingredients

text: of

text: a

text: lower

text: bound

text: in

text: terms

text: of

text: the

text: number

text: m

text: of

text: edges

text: alone.

text: We

text: show

text: a

text: more

text: general

text: fact

text: formulated

text: in

text: terms

text: of

text: both

text: the

text: numbers

text: of

text: nodes

text: and

text: edges.

text: A

text: Theorem

text: For

text: any

text: deterministic

text: oblivious

text: algorithm

text: ﬁnding

text: spanning

text: forests,

text: and

text: numbers

text: n

text: O

text: and

text: m,

text: where

text: m

text: there

text: exists

text: a

text: connected

text: graph

text: GA

text: with

text: n

text: vertices

text: and

text: Θ(m)

text: edges,

text: such

text: A

text: that

text: algorithm

text: requires

text: time

text: Ω(m)

text: to

text: ﬁnd

text: a

text: spanning

text: tree

text: of

text: GA.

text: Proof:

text: If

text: the

text: inequality

text: m

text: n(n

text: holds,

text: then

text: apply

text: the

text: ﬁve-phase

text: construction.

text: It

text: follows

text: from

text: Lemma

text: that

text: if

text: we

text: take

text: the

text: ﬁnal

text: T

text: obtained

text: by

text: completing

text: the

text: ﬁve-phase

text: construction,

text: A

text: as

text: the

text: set

text: of

text: edges

text: E,

text: then

text: the

text: oblivious

text: algorithm

text: needs

text: more

text: than

text: k

text: Ω(m)

text: steps

text: to

text: broadcast

text: a

text: spanning

text: tree

text: of

text: the

text: graph

text: G

text: (V,

text: E).

text: This

text: is

text: because

text: a

text: T

text: -witness

text: has

text: not

text: been

text: heard

text: by

text: the

text: kth

text: step

text: as

text: an

text: endpoint

text: of

text: an

text: edge.

text: In

text: the

text: remaining

text: cases

text: of

text: dependencies

text: between

text: the

text: numbers

text: m

text: and

text: n,

text: we

text: can

text: proceed

text: as

text: follows.

text: If

text: n(n

text: m,

text: then

text: we

text: can

text: use

text: the

text: same

text: construction

text: as

text: if

text: the

text: number

text: m

text: were

text: equal

text: to

text: n(n

text: If

text: m

text: then

text: a

text: simple

text: path

text: of

text: length

text: min

text: n

text: m

text: does

text: the

text: job,

text: because

text: any

text: algorithm

text: needs

text: to

text: broadcast

text: at

text: least

text: edges

text: to

text: reveal

text: the

text: whole

text: path.

text: There

text: is

text: a

text: simple

text: oblivious

text: algorithm

text: that

text: operates

text: in

text: time

text: m

text: the

text: station

text: i

text: broadcasts

text: its

text: edge

text: at

text: step

text: i.

text: Theorem

text: may

text: be

text: interpreted

text: as

text: follows:

text: listing

text: all

text: the

text: edges

text: systematically

text: is

text: asymptotically

text: optimal

text: among

text: oblivious

text: algorithms.

text: Weighted

text: Graphs

text: In

text: this

text: section

text: we

text: consider

text: graphs

text: that

text: have

text: positive

text: weights

text: assigned

text: to

text: their

text: edges.

text: The

text: input

text: is

text: speciﬁed

text: at

text: the

text: start

text: of

text: an

text: execution,

text: no

text: adversaries

text: are

text: involved.


text: Theorem 4 There is a randomized algorithm that ﬁnds a minimum-weight spanning forest T in O | | expected time ( T + W log m), where W is the number of distinct weights on the edges in T , and O in worst-case time (m). Proof: To guarantee the claimed worst-case performance, we apply a similar stratagem as in algorithm RandSimple for simple graphs. We have all the edges broadcast on the channel in a systematic way, until a forest of minimum weight has been found or all the edges have been exhausted, whatever comes ﬁrst. More precisely, let the algorithm RandWeighted be run in odd-numbered steps, while in even-numbered steps the stations broadcast their edges and weights in order of their IDs. Unlike algorithm RandSimple, the processes should be independent of each other, in the sense that the edges broadcast in the odd-numbered steps are not categorized as revealed, cycle or waiting, until all have been exhausted and the algorithm stops with all the edges broadcasted. This allows us to combine the expected performance of RandWeighted given in (cid:3) Lemma 5 with a worst-case upper bound proportional to the number m of all the edges. 4.2 Lower Bound for Weighted Graphs In this section we prove a lower bound Ω(m) for deterministic adaptive algorithms for weighted graphs. The lower bound is of the same form as in Section , the diﬀerence is that the algorithms are adaptive rather than oblivious, while the graphs are weighted rather than simple. A For each deterministic adaptive algorithm , we construct a certain weighted graph GA of n vertices and m edges. We start with any assignment of edges to the stations at the very beginning N of computation, so that the graph G = (V, E) has no isolated vertices. Let be the set of positive { ∈ N} integers. In the construction we use only weights from set 1/j : j . Our goal is to assign { ∈ N} weights to all the edges. Each station i will have a set A (t) of numbers in 1/j : j assigned i { ∈ N} to it after step t of the construction. Initially we set A (0) = 1/j : j , for every station i. i Since there is a one-to-one correspondence between the edges and the stations, we treat E also as a set of stations. Denote by E(t) the set of stations which have an edge with some weight A assigned to it by step t of algorithm . Each station i from E(t) has just one element in A (t). We ∈ \ i assume the following invariant after step t: for each station i E E(t), the set A (t) is inﬁnite. i A Consider step t + 1 of algorithm , as determined by step t of the construction. Let S (t + 1) A0 denote the set of these stations i that do not broadcast during step t + 1 of algorithm , for an inﬁnite number of possible weights from A (t). Similarly, let S (t + 1) denote the set of stations i 1 that broadcast in step t + 1, for inﬁnitely many possible initial values. Step t + 1 of the construction is broken into the following cases: \ ∅ Case 1: S (t + 1) S (t + 1) = . 1 0 Make set A (t + 1) contain all the weights from A (t), for which station i does not broadcast during i A ∈ i step t + 1 of algorithm , for each station i S (t + 1). Hence E(t + 1) = E(t) and the invariant 0 holds after step t + 1. Moreover, there are no new stations broadcasting on the channel, hence we can determine what happens in step t + 1 by considering only stations in E(t). \ { } Case 2: S (t + 1) S (t + 1) = i . 1 0 { } We set A (t + 1) = 1/(cid:96) , where 1/(cid:96) is the maximum weight in A (t) for which station i broadcasts i A \ i ∩ in step t+1 of algorithm . For the other stations j in E E(t), we set A (t+1) = A (t) (0, 1/(cid:96)). j j 16

text: Theorem

text: There

text: is

text: a

text: randomized

text: algorithm

text: that

text: ﬁnds

text: a

text: minimum-weight

text: spanning

text: forest

text: T

text: in

text: O

text: expected

text: time

text: T

text: W

text: log

text: m),

text: where

text: W

text: is

text: the

text: number

text: of

text: distinct

text: weights

text: on

text: the

text: edges

text: in

text: T

text: and

text: O

text: in

text: worst-case

text: time

text: (m).

text: Proof:

text: To

text: guarantee

text: the

text: claimed

text: worst-case

text: performance,

text: we

text: apply

text: a

text: similar

text: stratagem

text: as

text: in

text: algorithm

text: RandSimple

text: for

text: simple

text: graphs.

text: We

text: have

text: all

text: the

text: edges

text: broadcast

text: on

text: the

text: channel

text: in

text: a

text: systematic

text: way,

text: until

text: a

text: forest

text: of

text: minimum

text: weight

text: has

text: been

text: found

text: or

text: all

text: the

text: edges

text: have

text: been

text: exhausted,

text: whatever

text: comes

text: ﬁrst.

text: More

text: precisely,

text: let

text: the

text: algorithm

text: RandWeighted

text: be

text: run

text: in

text: odd-numbered

text: steps,

text: while

text: in

text: even-numbered

text: steps

text: the

text: stations

text: broadcast

text: their

text: edges

text: and

text: weights

text: in

text: order

text: of

text: their

text: IDs.

text: Unlike

text: algorithm

text: RandSimple,

text: the

text: processes

text: should

text: be

text: independent

text: of

text: each

text: other,

text: in

text: the

text: sense

text: that

text: the

text: edges

text: broadcast

text: in

text: the

text: odd-numbered

text: steps

text: are

text: not

text: categorized

text: as

text: revealed,

text: cycle

text: or

text: waiting,

text: until

text: all

text: have

text: been

text: exhausted

text: and

text: the

text: algorithm

text: stops

text: with

text: all

text: the

text: edges

text: broadcasted.

text: This

text: allows

text: us

text: to

text: combine

text: the

text: expected

text: performance

text: of

text: RandWeighted

text: given

text: in

text: Lemma

text: with

text: a

text: worst-case

text: upper

text: bound

text: proportional

text: to

text: the

text: number

text: m

text: of

text: all

text: the

text: edges.

text: Lower

text: Bound

text: for

text: Weighted

text: Graphs

text: In

text: this

text: section

text: we

text: prove

text: a

text: lower

text: bound

text: Ω(m)

text: for

text: deterministic

text: adaptive

text: algorithms

text: for

text: weighted

text: graphs.

text: The

text: lower

text: bound

text: is

text: of

text: the

text: same

text: form

text: as

text: in

text: Section

text: the

text: diﬀerence

text: is

text: that

text: the

text: algorithms

text: are

text: adaptive

text: rather

text: than

text: oblivious,

text: while

text: the

text: graphs

text: are

text: weighted

text: rather

text: than

text: simple.

text: A

text: For

text: each

text: deterministic

text: adaptive

text: algorithm

text: we

text: construct

text: a

text: certain

text: weighted

text: graph

text: GA

text: of

text: n

text: vertices

text: and

text: m

text: edges.

text: We

text: start

text: with

text: any

text: assignment

text: of

text: edges

text: to

text: the

text: stations

text: at

text: the

text: very

text: beginning

text: N

text: of

text: computation,

text: so

text: that

text: the

text: graph

text: G

text: (V,

text: E)

text: has

text: no

text: isolated

text: vertices.

text: Let

text: be

text: the

text: set

text: of

text: positive

text: N}

text: integers.

text: In

text: the

text: construction

text: we

text: use

text: only

text: weights

text: from

text: set

text: j

text: Our

text: goal

text: is

text: to

text: assign

text: N}

text: weights

text: to

text: all

text: the

text: edges.

text: Each

text: station

text: i

text: will

text: have

text: a

text: set

text: A

text: (t)

text: of

text: numbers

text: in

text: j

text: assigned

text: i

text: N}

text: to

text: it

text: after

text: step

text: t

text: of

text: the

text: construction.

text: Initially

text: we

text: set

text: A

text: j

text: for

text: every

text: station

text: i.

text: i

text: Since

text: there

text: is

text: a

text: one-to-one

text: correspondence

text: between

text: the

text: edges

text: and

text: the

text: stations,

text: we

text: treat

text: E

text: also

text: as

text: a

text: set

text: of

text: stations.

text: Denote

text: by

text: E(t)

text: the

text: set

text: of

text: stations

text: which

text: have

text: an

text: edge

text: with

text: some

text: weight

text: A

text: assigned

text: to

text: it

text: by

text: step

text: t

text: of

text: algorithm

text: Each

text: station

text: i

text: from

text: E(t)

text: has

text: just

text: one

text: element

text: in

text: A

text: (t).

text: We

text: i

text: assume

text: the

text: following

text: invariant

text: after

text: step

text: t:

text: for

text: each

text: station

text: i

text: E

text: E(t),

text: the

text: set

text: A

text: (t)

text: is

text: inﬁnite.

text: i

text: A

text: Consider

text: step

text: t

text: of

text: algorithm

text: as

text: determined

text: by

text: step

text: t

text: of

text: the

text: construction.

text: Let

text: S

text: (t

text: denote

text: the

text: set

text: of

text: these

text: stations

text: i

text: that

text: do

text: not

text: broadcast

text: during

text: step

text: t

text: of

text: algorithm

text: for

text: an

text: inﬁnite

text: number

text: of

text: possible

text: weights

text: from

text: A

text: (t).

text: Similarly,

text: let

text: S

text: (t

text: denote

text: the

text: set

text: of

text: stations

text: i

text: that

text: broadcast

text: in

text: step

text: t

text: for

text: inﬁnitely

text: many

text: possible

text: initial

text: values.

text: Step

text: t

text: of

text: the

text: construction

text: is

text: broken

text: into

text: the

text: following

text: cases:

text: Case

text: S

text: (t

text: S

text: (t

text: Make

text: set

text: A

text: (t

text: contain

text: all

text: the

text: weights

text: from

text: A

text: (t),

text: for

text: which

text: station

text: i

text: does

text: not

text: broadcast

text: during

text: i

text: A

text: i

text: step

text: t

text: of

text: algorithm

text: for

text: each

text: station

text: i

text: S

text: (t

text: Hence

text: E(t

text: E(t)

text: and

text: the

text: invariant

text: holds

text: after

text: step

text: t

text: Moreover,

text: there

text: are

text: no

text: new

text: stations

text: broadcasting

text: on

text: the

text: channel,

text: hence

text: we

text: can

text: determine

text: what

text: happens

text: in

text: step

text: t

text: by

text: considering

text: only

text: stations

text: in

text: E(t).

text: Case

text: S

text: (t

text: S

text: (t

text: i

text: We

text: set

text: A

text: (t

text: where

text: is

text: the

text: maximum

text: weight

text: in

text: A

text: (t)

text: for

text: which

text: station

text: i

text: broadcasts

text: i

text: A

text: i

text: in

text: step

text: of

text: algorithm

text: For

text: the

text: other

text: stations

text: j

text: in

text: E

text: E(t),

text: we

text: set

text: A

text: A

text: (t)

text: j

text: j