text: 0 −0.24 σ (H ) ess κ E(κ) −0.25 −0.5 −0.26 −1 −0.27 −1.5 −0.28 Numerical simulation Leading term −2 √ −0.29 1/ 2 κc 0.1 0.2 0.3 0.4 κ κ (a) (b) Figure 1: In (a) a plot of the ground state energy is given as a function of the impurity charge κ. Figure (b) is a comparison of the leading term of the discrete eigenvalue, and the numerically calculated discrete eigenvalue. Theorem 2.2. Let H be the self-adjoint operator formally described by κ,κ˜ −1 − − − H = ∆ δ(x y) + κ˜δ(x) κδ(y). (2.4) κ,κ˜ 2 on L2(R2). Given any κ˜ > 1 there exists κ such that H has no discrete eigenvalues for all ≥ M κ,κ˜ κ κ . Furthermore, given any 0 < κ˜ < 1 there exists some κ such that H has at least one M ≥ M κ,κ˜ discrete eigenvalue for all κ κ . M As a consequence of the previous two theorems we will also prove the following corollary: Corollary 2.3. Let H be the operator in (2.1). Then there exists a critical charge of the impurity, κ which we will denote κ , such that the discrete spectrum of H is non-empty for all 0 < κ < κ and ≥ c κ c empty for κ κ . c ≈ Using numerical simulations to calculate the smallest discrete eigenvalue of H we see that at κ κ 1.546 the ground state energy hits the essential spectrum. Thus, we expect that the true κ is close to c 1.546. In Figure 2 a numerical calculation of the critical charge κ is plotted against the mass fraction c σ. We see that as the mass of the impurity decreases the critical charge is increased, and thus bound states exists for impurities with larger charges. We have also plotted the coeﬃcient β in (2.3) against the mass fraction, and we see that the coeﬃcient decreases as the mass of the impurity decreases. 3 The Framework In this section we introduce the framework we use to study the discrete spectrum of H . This κ,σ framework has been used in [20, 2], and we refer to those papers for more details. We deﬁne H as the unique self-adjoint operator associated to the sesqui-linear form κ,σ 1(cid:104)∇ ∇ (cid:105) − (cid:104) (cid:105) Q(f, g) = f, A g R f(x, x), g(x, x) R 2 L2( 2) L2( ) (3.1) (cid:104) (cid:105) − (cid:104) (cid:105) + κ f(0, y), g(0, y) R κ f(x, 0), g(x, 0) R , L2( ) L2( ) on H1(R2) × H1(R2), where H1(R2) is the So(cid:20)bolev sp(cid:21)ace of ﬁrst order and A ∈ R2×2 is the matrix 1 σ A = . (3.2) σ 1 3 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: σ

text: (H

text: ess

text: κ

text: E(κ)

text: −1

text: Numerical

text: simulation

text: Leading

text: term

text: −2

text: κc

text: κ

text: κ

text: (a)

text: (b)

text: Figure

text: In

text: (a)

text: a

text: plot

text: of

text: the

text: ground

text: state

text: energy

text: is

text: given

text: as

text: a

text: function

text: of

text: the

text: impurity

text: charge

text: κ.

text: Figure

text: (b)

text: is

text: a

text: comparison

text: of

text: the

text: leading

text: term

text: of

text: the

text: discrete

text: eigenvalue,

text: and

text: the

text: numerically

text: calculated

text: discrete

text: eigenvalue.

text: Theorem

text: Let

text: H

text: be

text: the

text: self-adjoint

text: operator

text: formally

text: described

text: by

text: κ,κ˜

text: −1

text: H

text: δ(x

text: y)

text: κ˜δ(x)

text: κδ(y).

text: κ,κ˜

text: on

text: Given

text: any

text: κ˜

text: there

text: exists

text: κ

text: such

text: that

text: H

text: has

text: no

text: discrete

text: eigenvalues

text: for

text: all

text: M

text: κ,κ˜

text: κ

text: κ

text: Furthermore,

text: given

text: any

text: κ˜

text: there

text: exists

text: some

text: κ

text: such

text: that

text: H

text: has

text: at

text: least

text: one

text: M

text: M

text: κ,κ˜

text: discrete

text: eigenvalue

text: for

text: all

text: κ

text: κ

text: M

text: As

text: a

text: consequence

text: of

text: the

text: previous

text: two

text: theorems

text: we

text: will

text: also

text: prove

text: the

text: following

text: corollary:

text: Corollary

text: Let

text: H

text: be

text: the

text: operator

text: in

text: Then

text: there

text: exists

text: a

text: critical

text: charge

text: of

text: the

text: impurity,

text: κ

text: which

text: we

text: will

text: denote

text: κ

text: such

text: that

text: the

text: discrete

text: spectrum

text: of

text: H

text: is

text: non-empty

text: for

text: all

text: κ

text: κ

text: and

text: c

text: κ

text: c

text: empty

text: for

text: κ

text: κ

text: c

text: Using

text: numerical

text: simulations

text: to

text: calculate

text: the

text: smallest

text: discrete

text: eigenvalue

text: of

text: H

text: we

text: see

text: that

text: at

text: κ

text: κ

text: the

text: ground

text: state

text: energy

text: hits

text: the

text: essential

text: spectrum.

text: Thus,

text: we

text: expect

text: that

text: the

text: true

text: κ

text: is

text: close

text: to

text: c

text: In

text: Figure

text: a

text: numerical

text: calculation

text: of

text: the

text: critical

text: charge

text: κ

text: is

text: plotted

text: against

text: the

text: mass

text: fraction

text: c

text: σ.

text: We

text: see

text: that

text: as

text: the

text: mass

text: of

text: the

text: impurity

text: decreases

text: the

text: critical

text: charge

text: is

text: increased,

text: and

text: thus

text: bound

text: states

text: exists

text: for

text: impurities

text: with

text: larger

text: charges.

text: We

text: have

text: also

text: plotted

text: the

text: coeﬃcient

text: β

text: in

text: against

text: the

text: mass

text: fraction,

text: and

text: we

text: see

text: that

text: the

text: coeﬃcient

text: decreases

text: as

text: the

text: mass

text: of

text: the

text: impurity

text: decreases.

text: The

text: Framework

text: In

text: this

text: section

text: we

text: introduce

text: the

text: framework

text: we

text: use

text: to

text: study

text: the

text: discrete

text: spectrum

text: of

text: H

text: This

text: κ,σ

text: framework

text: has

text: been

text: used

text: in

text: and

text: we

text: refer

text: to

text: those

text: papers

text: for

text: more

text: details.

text: We

text: deﬁne

text: H

text: as

text: the

text: unique

text: self-adjoint

text: operator

text: associated

text: to

text: the

text: sesqui-linear

text: form

text: κ,σ

text: Q(f,

text: g)

text: f,

text: A

text: g

text: R

text: f(x,

text: x),

text: g(x,

text: x)

text: R

text: κ

text: y),

text: y)

text: R

text: κ

text: f(x,

text: g(x,

text: R

text: on

text: where

text: is

text: the

text: of

text: ﬁrst

text: order

text: and

text: A

text: is

text: the

text: matrix

text: σ

text: A

text: σ

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: for some a > 0. Write s = s + is and t = t + it , with s , s , t , t ∈ R, then 1 2 1 2 1 2 1 2 1 1 Tˆ (s, t) = − − − − − − . 2 π t2 t2 + 2it t + (s t )2 (s t )2 + 2i(s t )(s t ) + ε + 1 1 2 1 2 1 1 2 2 1 1 2 2 (cid:26) (cid:27)− This function has no poles for t and s satisfying t2 + (s t )2 < 1, and is analytic on the subset 2 2 2 2 2 ∈ C | | 1 ⊂ C ξ 2 : Im(ξ) < 2. 2 Take η = (s2, t2) ∈ R2 such that |η| < 1/2(cid:90)and deﬁne δ := ε + 1 − t22 − (s2 − t2)2. By the choice of η we get δ > 0, and the norm (cid:107) · (cid:107) ≤ 1 1 1 ∞ Tˆ2( + iη) 2L2(R2) π2 R2 (t21 + (s1 − t1)2 + δ)2 ds1 dt1 = πδ < . Th√us, (cid:107)Tˆ (· + iη)(cid:107) R < ∞ for all such η ∈ R2. Then the Paley-Wiener theorem implies that 2,0 L2( 2) eα x2+y2T (x, y) ∈ L2(R2) for all α < 1/2. This concludes the proof of T (x, y)eα|x| ∈ L2(R2). 2 2 We are now ready to sh(cid:90)ow that C(x, y) is an integral(cid:90)kernel of a Hilbert-Schmidt operator. To do that we use the following inequality ≤ ∗ | | (cid:48) (cid:48) ∗ | (cid:48)| (cid:48) (cid:48) C(x, y) T (x, t) t T (t , y) dt dt + T (x, t) t T (t , y) dt dt , (4.16) R 2 2 R 2 2 2 2 (cid:90) which follows from the deﬁnition of C(x, y) and the inequality | − (cid:48)| √ t t − ≤ | − (cid:48)| ≤ | | | (cid:48)| e εs ds t t t + t . 0 We will show that the last the t(cid:90)erm in (4.16) is in L2(R2) (the proof that the ﬁrst term is also in L2(R2) is identical). Note that the integral is separable and ∗ | (cid:48)| (cid:48) (cid:48) T (x, t) t T (t , y) dt dt =: F(x)G(y). R 2 2 2 We will show that F, G ∈ L2(R). Applying the Cauchy-Schwarz inequality with respect to the t-integral and using Lemma 4.3 we ﬁnd (cid:90) (cid:12)(cid:90) (cid:12) (cid:90) (cid:12)(cid:90) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:107) (cid:107) (cid:90) ∗ 2 ∗ | | − | | 2 F 2 R = T (x, t) dt dx = T (x, t)eα t e α t dt dx L2( ) R R 2 R R 2 ≤ | ∗ | | | ∞ C T (x, t) 2e2α t dt dx < , (4.17) α R 2 2 (cid:90) (cid:12)(cid:90) (cid:12) (cid:90) (cid:12)(cid:90) (cid:12) for α > 0 suﬃciently(cid:12) small. Similarly(cid:12): (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:107) (cid:107) (cid:90) (cid:12)|(cid:90)(cid:48)| (cid:48) (cid:48) 2 ≤ (cid:12) | (cid:48)|(cid:90) (cid:48) (cid:48) 2 G 2L2(R) = R R(cid:12)(cid:12)t T2(t , y) dt dy Cα (cid:12)(cid:12)R R eα2 t T2(t , y) dt dy (cid:12) (cid:12) − | (cid:48)| | (cid:48)| (cid:48) (cid:48) 2 ≤ | (cid:48)|| (cid:48) | (cid:48) ∞ = Cα R R e α2 t eα t T2(t , y) dt dy C˜α R e2α t T2(t , y) 2 dt dy < , 2 again for α > 0 suﬃ(cid:90)ciently small. We conclude that C(x, y) ∈ L2(R2) uniformly in ε > 0. Using the expansion in (4.14) the integral kernel of the Birman-Schwinger operator is − − − (cid:48) (cid:48) (cid:48) − √κ | (cid:105)(cid:104) | 12 κ R V2(x, t)[(1 T3) 1](t, t )V1(t , y) dt dt = 12 ε Ψ Φ + κBε(x, y), (4.18) 2 where B (x, y) is the integral kernel of the uniformly bounded operator for ε > 0 that comes from ε the non-singular terms of (4.14). Also: 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

label: paragraph

text: some

text: a

text: Write

text: s

text: s

text: is

text: and

text: t

text: t

text: it

text: with

text: s

text: s

text: t

text: t

text: R,

text: then

text: Tˆ

text: (s,

text: t)

text: π

text: t

text: (s

text: t

text: (s

text: t

text: t

text: )(s

text: t

text: ε

text: This

text: function

text: has

text: no

text: poles

text: for

text: t

text: and

text: s

text: satisfying

text: (s

text: t

text: and

text: is

text: analytic

text: on

text: the

text: subset

text: C

text: C

text: ξ

text: Im(ξ)

text: Take

text: η

text: such

text: that

text: |η|

text: deﬁne

text: δ

text: ε

text: By

text: the

text: choice

text: of

text: η

text: we

text: get

text: δ

text: and

text: the

text: norm

text: iη)

text: π2

text: πδ

text: Th√us,

text: R

text: for

text: all

text: such

text: η

text: Then

text: the

text: Paley-Wiener

text: theorem

text: implies

text: that

text: eα

text: (x,

text: y)

text: for

text: all

text: α

text: This

text: concludes

text: the

text: proof

text: of

text: T

text: (x,

text: y)eα|x|

text: We

text: are

text: now

text: ready

text: to

text: that

text: C(x,

text: y)

text: is

text: an

text: of

text: a

text: Hilbert-Schmidt

text: operator.

text: To

text: do

text: that

text: we

text: use

text: the

text: following

text: inequality

text: C(x,

text: y)

text: T

text: (x,

text: t)

text: t

text: T

text: (t

text: y)

text: dt

text: dt

text: T

text: (x,

text: t)

text: t

text: T

text: (t

text: y)

text: dt

text: dt

text: R

text: R

text: which

text: follows

text: from

text: the

text: deﬁnition

text: of

text: C(x,

text: y)

text: and

text: the

text: inequality

text: t

text: t

text: e

text: εs

text: ds

text: t

text: t

text: t

text: t

text: We

text: will

text: show

text: that

text: the

text: last

text: the

text: in

text: is

text: in

text: (the

text: proof

text: that

text: the

text: ﬁrst

text: term

text: is

text: also

text: in

text: is

text: identical).

text: Note

text: that

text: the

text: integral

text: is

text: separable

text: and

text: T

text: (x,

text: t)

text: t

text: T

text: (t

text: y)

text: dt

text: dt

text: F(x)G(y).

text: R

text: We

text: will

text: show

text: that

text: F,

text: G

text: Applying

text: the

text: Cauchy-Schwarz

text: inequality

text: with

text: respect

text: to

text: the

text: t-integral

text: and

text: using

text: Lemma

text: we

text: ﬁnd

text: F

text: R

text: T

text: (x,

text: t)

text: dt

text: dx

text: T

text: (x,

text: t)eα

text: t

text: e

text: α

text: t

text: dt

text: dx

text: R

text: R

text: R

text: R

text: C

text: T

text: (x,

text: t)

text: t

text: dt

text: dx

text: α

text: R

text: for

text: α

text: small.

text: G

text: R

text: y)

text: dt

text: dy

text: Cα

text: R

text: t

text: y)

text: dt

text: dy

text: Cα

text: R

text: R

text: e

text: α2

text: t

text: eα

text: t

text: y)

text: dt

text: dy

text: C˜α

text: R

text: t

text: y)

text: dt

text: dy

text: again

text: for

text: α

text: small.

text: We

text: conclude

text: that

text: C(x,

text: y)

text: uniformly

text: in

text: ε

text: Using

text: the

text: expansion

text: in

text: the

text: integral

text: kernel

text: of

text: the

text: Birman-Schwinger

text: operator

text: is

text: √κ

text: κ

text: R

text: t

text: y)

text: dt

text: dt

text: ε

text: Ψ

text: Φ

text: κBε(x,

text: y),

text: where

text: B

text: (x,

text: y)

text: is

text: the

text: integral

text: kernel

text: of

text: the

text: uniformly

text: bounded

text: operator

text: for

text: ε

text: that

text: comes

text: from

text: ε

text: the

text: non-singular

text: terms

text: of

text: Also:

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: (3:2 and 14:13) one can construct related three-body resonances with libration periods of 0.6, 0.8, 2.5 and 14.3 years (compare with Figure 2 and Table 1) by subtracting the corresponding resonant arguments: ϕ = 3λ − 16λ + 13λ (2) 3:16:13,1 Mim Pnd Ble ϕ = 3λ − 16λ + 13λ − (cid:36) + (cid:36) (3) 3:16:13,2 Mim Pnd Ble Mim Pnd ϕ = 3λ − 16λ + 13λ − (cid:36) + (cid:36) (4) 3:16:13,3 Mim Pnd Ble Pnd Ble ϕ = 3λ − 16λ + 13λ − (cid:36) + (cid:36) . (5) 3:16:13,4 Mim Pnd Ble Mim Ble Adding Prometheus, Titan and Tethys, all having inﬂuence on the orbital dynamics of Pandora and Mimas, leads to a non-stationary signal in the Fourier spectrum. This could be caused by the chaotic and strong interactions between Prometheus and Pandora resulting in wandering libration frequencies and changing amplitudes in the orbital dynamics of Bl´eriot. Performing long-time simulations for the test moonlet considering a time span of up to 100 Saturnian years, the 42:40 IVR of Prometheus seems to become more and more important forcing the moonlet to librate with a period of about 4 years with a radial amplitude of 20 meters and an amplitude around 1 km in mean longitude. This signal is clearly visible in the Fourier spectrum and stationary in contrast to the other signals. Although the libration periods of the test moonlet from our simulations agree fairly well with the observational data, the resulting amplitudes are far too small. 3 Moonlet-Propeller Interactions Next, we will consider the gravitational interaction between the embedded moonlet with its created gap region. Imagine a non-symmetric propeller structure so that the moonlet gets acceler- ated by the ring gravity. We will show, that the evolution of the related longitudinal residual can be described by a harmonic oscillator. This harmonic oscillator feels a periodic external forcing caused by the gravity of the outer moons. The amplitudes of the external forced frequencies get the more magniﬁed, the closer the forcing frequency matches the eigenfrequency of the harmonic oscillator (propeller-moonlet system). Consider a moonlet located at (x , y ) as illustrated in Figure 3. Interacting with the sur- m m rounding viscous ring material, the propeller moonlet gravitationally scatters ring particles to larger and smaller orbits creating two gaps in its vicinity in the ring material (regions of reduced surface mass density) and which are decorated by two density enhanced regions pairwise downstream of its orbit (Spahn and Wiebicke, 1989). Viscous diﬀusion of the ring material counteracts this gap- creation, smoothing out the structure with growing azimuthal distance downstream the moonlet (Spahn and Sremˇcevi´c, 2000; Sremˇcevi´c et al., 2002). For simplicity, we assume the gap shape to be a rectangular area with reduced density σ , illustrated by the shaded regions at |x| = x and g g width ∆ in Figure 3. The diﬀusion process deﬁnes the length L ± y of the gaps, while the width (cid:18) (cid:19) m of the propeller-blades is set by the moonlet’s mass m or its Hill radius h 1 ∆ m 3 ∝ h(m) ∼ a , (6) 2 0 3m c and Saturn’s mass is labeled by m . We assume that both gaps are anchored at the position c y following the moonlet’s motion. Further, the ends at L and −L are ﬁxed, because the imprint m of the moonlet motion will be averaged out along the azimuth due to the diﬀusion process and dragged along with the Kepler shear. In the symmetric case (x = 0, y = 0) the torque on the moonlet due to the gravitational m m interaction with the gap is zero because(cid:90)of point (cid:90)symmetry. When the moonlet is leaving its mean position, the gravitational force F(cid:126) of the ring material on the moonlet is g ∞ ∞ (cid:126)r − (cid:126)r F(cid:126) ((cid:126)r ) = −G dx dy σ ((cid:126)r) m , (7) g m −∞ −∞ |(cid:126)r − (cid:126)r|3 m 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: one

text: can

text: construct

text: related

text: three-body

text: resonances

text: with

text: libration

text: periods

text: of

text: and

text: years

text: (compare

text: with

text: Figure

text: and

text: Table

text: by

text: subtracting

text: the

text: corresponding

text: resonant

text: arguments:

text: ϕ

label: equation

text: 3λ

text: Mim

text: Pnd

text: Ble

text: ϕ

text: 3λ

text: Mim

text: Pnd

text: Ble

text: Mim

text: Pnd

text: ϕ

text: 3λ

text: Mim

text: Pnd

text: Ble

text: Pnd

text: Ble

text: ϕ

text: 3λ

text: Mim

text: Pnd

text: Ble

text: Mim

text: Ble

text: Adding

text: Prometheus,

text: Titan

text: and

text: Tethys,

text: all

text: having

text: inﬂuence

text: on

text: the

text: orbital

text: dynamics

text: of

text: Pandora

text: and

text: Mimas,

text: leads

text: to

text: a

text: non-stationary

text: signal

text: in

text: the

text: Fourier

text: spectrum.

text: This

text: could

text: be

text: caused

text: by

text: the

text: chaotic

text: and

text: strong

text: interactions

text: between

text: Prometheus

text: and

text: Pandora

text: resulting

text: in

text: wandering

text: libration

text: frequencies

text: and

text: changing

text: amplitudes

text: in

text: the

text: orbital

text: dynamics

text: of

text: Bl´eriot.

text: Performing

text: long-time

text: simulations

text: for

text: the

text: test

text: moonlet

text: considering

text: a

text: time

text: span

text: of

text: up

text: to

text: Saturnian

text: years,

text: the

text: IVR

text: of

text: Prometheus

text: seems

text: to

text: become

text: more

text: and

text: more

text: important

text: forcing

text: the

text: moonlet

text: to

text: librate

text: with

text: a

text: period

text: of

text: about

text: years

text: with

text: a

text: radial

text: amplitude

text: of

text: meters

text: and

text: an

text: amplitude

text: around

text: km

text: in

text: mean

text: longitude.

text: This

text: signal

text: is

text: clearly

text: visible

text: in

text: the

text: Fourier

text: spectrum

text: and

text: stationary

text: in

text: contrast

text: to

text: the

text: other

text: signals.

text: Although

text: the

text: libration

text: periods

text: of

text: the

text: test

text: moonlet

text: from

text: our

text: simulations

text: agree

text: fairly

text: well

text: with

text: the

text: observational

text: data,

text: the

text: resulting

text: amplitudes

text: are

text: far

text: too

text: small.

text: Moonlet-Propeller

text: Interactions

text: Next,

text: we

text: will

text: consider

text: the

text: gravitational

text: interaction

text: between

text: the

text: embedded

text: moonlet

text: with

text: its

text: created

text: gap

text: region.

text: Imagine

text: a

text: non-symmetric

text: propeller

text: structure

text: so

text: that

text: the

text: moonlet

text: gets

text: acceler-

text: ated

text: by

text: the

text: ring

text: gravity.

text: We

text: will

text: show,

text: that

text: the

text: evolution

text: of

text: the

text: related

text: longitudinal

text: residual

text: can

text: be

text: described

text: by

text: a

text: harmonic

text: oscillator.

text: This

text: harmonic

text: oscillator

text: feels

text: a

text: periodic

text: external

text: forcing

text: caused

text: by

text: the

text: gravity

text: of

text: the

text: outer

text: moons.

text: The

text: amplitudes

text: of

text: the

text: external

text: forced

text: frequencies

text: get

text: the

text: more

text: magniﬁed,

text: the

text: closer

text: the

text: forcing

text: frequency

text: matches

text: the

text: eigenfrequency

text: of

text: the

text: harmonic

text: oscillator

text: (propeller-moonlet

text: system).

text: Consider

text: a

text: moonlet

text: located

text: at

text: (x

text: y

text: as

text: illustrated

text: in

text: Figure

text: Interacting

text: with

text: the

text: sur-

text: m

text: m

text: rounding

text: viscous

text: ring

text: material,

text: the

text: propeller

text: moonlet

text: gravitationally

text: scatters

text: ring

text: particles

text: to

text: larger

text: and

text: smaller

text: orbits

text: creating

text: two

text: gaps

text: in

text: its

text: vicinity

text: in

text: the

text: ring

text: material

text: (regions

text: of

text: reduced

text: surface

text: mass

text: density)

text: and

text: which

text: are

text: decorated

text: by

text: two

text: density

text: enhanced

text: regions

text: pairwise

text: downstream

text: of

text: its

text: orbit

text: (Spahn

text: and

text: Wiebicke,

text: Viscous

text: diﬀusion

text: of

text: the

text: ring

text: material

text: counteracts

text: this

text: gap-

text: creation,

text: smoothing

text: out

text: the

text: structure

text: with

text: growing

text: azimuthal

text: distance

text: downstream

text: the

text: moonlet

text: (Spahn

text: and

text: Sremˇcevi´c,

text: Sremˇcevi´c

text: et

text: al.,

text: For

text: simplicity,

text: we

text: assume

text: the

text: gap

text: shape

text: to

text: be

text: a

text: rectangular

text: area

text: with

text: reduced

text: density

text: σ

text: illustrated

text: by

text: the

text: shaded

text: regions

text: at

text: |x|

text: x

text: and

text: g

text: g

text: width

text: in

text: Figure

text: The

text: diﬀusion

text: process

text: deﬁnes

text: the

text: length

text: L

text: y

text: of

text: the

text: gaps,

text: while

text: the

text: width

text: m

text: of

text: the

text: propeller-blades

text: is

text: set

text: by

text: the

text: moonlet’s

text: mass

text: m

text: or

text: its

text: Hill

text: radius

text: h

text: m

text: h(m)

text: a

text: c

text: and

text: Saturn’s

text: mass

text: is

text: labeled

text: by

text: m

text: We

text: assume

text: that

text: both

text: gaps

text: are

text: anchored

text: at

text: the

text: position

text: c

text: y

text: following

text: the

text: moonlet’s

text: motion.

text: Further,

text: the

text: ends

text: at

text: L

text: and

text: −L

text: are

text: ﬁxed,

text: because

text: the

text: imprint

text: m

text: of

text: the

text: moonlet

text: motion

text: will

text: be

text: averaged

text: out

text: along

text: the

text: azimuth

text: due

text: to

text: the

text: diﬀusion

text: process

text: and

text: dragged

text: along

text: with

text: the

text: Kepler

text: shear.

text: In

text: the

text: symmetric

text: case

text: (x

text: y

text: the

text: torque

text: on

text: the

text: moonlet

text: due

text: to

text: the

text: gravitational

text: m

text: m

text: interaction

text: with

text: the

text: gap

text: is

text: zero

text: point

text: When

text: the

text: moonlet

text: is

text: leaving

text: its

text: mean

text: position,

text: the

text: gravitational

text: force

text: of

text: the

text: ring

text: material

text: on

text: the

text: moonlet

text: is

text: g

text: −G

text: dx

text: dy

text: σ

text: m

text: g

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In order to estimate the eigenlibration period for the moonlet, we use values, which have been inspired by theoretical and observational data (Tiscareno et al., 2007; Spahn and Sremˇcevi´c, 2000). We can estimate the length of the gaps from the analytical model derived by Sremˇcevi´c et al. (cid:90) (2002). The relative mass moved out of the gaps along the azimuth is L σ − σ g 0 dy = −2.1aK , σ 0 0 which can be obtained from a direct numerical integration. Assuming δσ/σ = −0.5 for the 0 rectangular simpliﬁcation of the gaps, the gap length is L = 4.2aK and thus, for Bl´eriot L ≈ 600km, (cid:34) (cid:35) (cid:20) (cid:21) (cid:20) (cid:21) where the scaling length aK can be calculated by 400cm2 h 3 Ω aK = 160km s − ν 500m 10 4 1 s with ν, h and Ω denoting the viscosity, the Hill radius and the Kepler frequency. The value of ν ﬁts to extrapolated values from observations of Tiscareno et al. (2007) and simulations of Daisaka et al. (2001). Further, the width of the gap ∆ = 2h ≈ 1km. (cid:34) (cid:18) (cid:19) (cid:35) As a result, we estimate a eigenlibration period of that ring-moonlet harmonic oscillator: (cid:112) − 1 2πL L |δσ| ∆ 2 T = ≈ 13yr . 6G |δσ| ∆ 600km 20 g 1km cm2 The Hill radius h of the propeller moonlet has a strong inﬂuence on the gap length L ∝ h3 ∼ m and thus on the libration period. With h = 480m ± 50m the libration period is T = (13 ± 4)yr. If the propeller moonlet is placed in the Saturnian ring system with all the other moons, this harmonic oscillator will be forced with a zoo of frequencies due to the Saturnian moons. When a forcing frequency matches the eigenfrequency close enough, a large gain in the resulting amplitude is possible. For example, if the period of the resonant interaction and the eigenperiod of the oscillator diﬀer by a third of a year, the amplitude is ampliﬁed by two orders of magnitude. The closer the forcing frequency is to the eigenfrequency, the larger the resulting amplitudes. Formally, to get to the observed amplitudes, which are about three orders of magnitude larger than the values from Table 1, the diﬀerence of the periods has to be smaller than 0.03 years. The longest period from Table 1 falls well into the uncertainty interval of the estimated libration period T, so that our model can indeed induce much larger libration amplitudes. One should note, that our simple model is generally not fully consistent, because the libration amplitude y is not much smaller than the used gap length L. Thus, the condition to approximate m the force for small amplitudes is easily violated. However, in reality the gap extents for more than several thousand km and L is rather a decaying length of the gap if one for example assumes a exponential relaxation of the gap. A more complex model, would remove this inconsistency, but the oscillatory behavior due to the repulsive azimuthal force should persist. We will address such a comprehensive non-linear model in the future. 4 Conclusion and Discussion In this paper we have presented results of N-Body simulations, which characterize the orbital evolution of the moonlet of the propeller Bl´eriot, being perturbed by the gravitational interactions with 15 outer and inner moons of Saturn. We found, that the 14:13 CER of Pandora and the 3:16:13 three-body resonances of Mimas and Pandora are the dominating perturbations of Bl´eriot’s orbit. Additionally, the chaotic interaction between Pandora and Prometheus has an eﬀect on Bl´eriot’s orbital evolution as well, resulting in changing libration frequencies over time. Our simulations yielded, that the gravitational interactions with the other moons cause similar libration frequencies of Bl´eriot as concluded from the Cassini ISS images, but the corresponding amplitudes are too small. Considering propeller-moonlet interactions with a new model, we have been able to ﬁnd a mechanism to amplify certain modes in form of a harmonic oscillator which is periodically driven by an external forcing. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: order

text: to

text: estimate

text: the

text: eigenlibration

text: period

text: for

text: the

text: moonlet,

text: we

text: use

text: values,

text: which

text: have

text: been

text: inspired

text: by

text: theoretical

text: and

text: observational

text: data

text: (Tiscareno

text: et

text: al.,

text: Spahn

text: and

text: Sremˇcevi´c,

text: We

text: can

text: estimate

text: the

text: length

text: of

text: the

text: gaps

text: from

text: the

text: analytical

text: model

text: derived

text: by

text: Sremˇcevi´c

text: et

text: al.

text: The

text: relative

text: mass

text: moved

text: out

text: of

text: the

text: gaps

text: along

text: the

text: azimuth

text: is

text: L

text: σ

text: σ

text: g

text: dy

text: σ

text: which

text: can

text: be

text: obtained

text: from

text: a

text: direct

text: numerical

text: integration.

text: Assuming

text: δσ/σ

text: for

text: the

text: rectangular

text: simpliﬁcation

text: of

text: the

text: gaps,

text: the

text: gap

text: length

text: is

text: L

text: and

text: thus,

text: for

text: Bl´eriot

text: L

text: where

text: the

text: scaling

text: length

text: aK

text: can

text: be

text: calculated

text: by

text: h

text: Ω

text: aK

text: s

text: ν

text: s

text: with

text: ν,

text: h

text: and

text: Ω

text: denoting

text: the

text: viscosity,

text: the

text: Hill

text: radius

text: and

text: the

text: Kepler

text: frequency.

text: The

text: value

text: of

text: ν

text: ﬁts

text: to

text: extrapolated

text: values

text: from

text: observations

text: of

text: Tiscareno

text: et

text: al.

text: and

text: simulations

text: of

text: Daisaka

text: et

text: al.

text: Further,

text: the

text: width

text: of

text: the

text: gap

text: As

text: a

text: result,

text: we

text: estimate

text: a

text: eigenlibration

text: period

text: of

text: that

text: ring-moonlet

text: harmonic

text: oscillator:

text: L

text: |δσ|

text: T

text: |δσ|

text: g

text: The

text: Hill

text: radius

text: h

text: of

text: the

text: propeller

text: moonlet

text: has

text: a

text: strong

text: inﬂuence

text: on

text: the

text: gap

text: length

text: L

text: m

text: and

text: thus

text: on

text: the

text: libration

text: period.

text: With

text: h

text: the

text: libration

text: period

text: is

text: T

text: If

text: the

text: propeller

text: moonlet

text: is

text: placed

text: in

text: the

text: Saturnian

text: ring

text: system

text: with

text: all

text: the

text: other

text: moons,

text: this

text: harmonic

text: oscillator

text: will

text: be

text: forced

text: with

text: a

text: zoo

text: of

text: frequencies

text: due

text: to

text: the

text: Saturnian

text: moons.

text: When

text: a

text: forcing

text: frequency

text: matches

text: the

text: eigenfrequency

text: close

text: enough,

text: a

text: large

text: gain

text: in

text: the

text: resulting

text: amplitude

text: is

text: possible.

text: For

text: example,

text: if

text: the

text: period

text: of

text: the

text: resonant

text: interaction

text: and

text: the

text: eigenperiod

text: of

text: the

text: oscillator

text: diﬀer

text: by

text: a

text: third

text: of

text: a

text: year,

text: the

text: amplitude

text: is

text: ampliﬁed

text: by

text: two

text: orders

text: of

text: magnitude.

text: The

text: closer

text: the

text: forcing

text: frequency

text: is

text: to

text: the

text: eigenfrequency,

text: the

text: larger

text: the

text: resulting

text: amplitudes.

text: Formally,

text: to

text: get

text: to

text: the

text: observed

text: amplitudes,

text: which

text: are

text: about

text: three

text: orders

text: of

text: magnitude

text: larger

text: than

text: the

text: values

text: from

text: Table

text: the

text: diﬀerence

text: of

text: the

text: periods

text: has

text: to

text: be

text: smaller

text: than

text: years.

text: The

text: longest

text: period

text: from

text: Table

text: falls

text: well

text: into

text: the

text: uncertainty

text: interval

text: of

text: the

text: estimated

text: libration

text: period

text: T,

text: so

text: that

text: our

text: model

text: can

text: indeed

text: induce

text: much

text: larger

text: libration

text: amplitudes.

text: One

text: should

text: note,

text: that

text: our

text: simple

text: model

text: is

text: generally

text: not

text: fully

text: consistent,

text: because

text: the

text: libration

text: amplitude

text: y

text: is

text: not

text: much

text: smaller

text: than

text: the

text: used

text: gap

text: length

text: L.

text: Thus,

text: the

text: condition

text: to

text: approximate

text: m

text: the

text: force

text: for

text: small

text: amplitudes

text: is

text: easily

text: violated.

text: However,

text: in

text: reality

text: the

text: gap

text: extents

text: for

text: more

text: than

text: several

text: thousand

text: km

text: and

text: L

text: is

text: rather

text: a

text: decaying

text: length

text: of

text: the

text: gap

text: if

text: one

text: for

text: example

text: assumes

text: a

text: exponential

text: relaxation

text: of

text: the

text: gap.

text: A

text: more

text: complex

text: model,

text: would

text: remove

text: this

text: inconsistency,

text: but

text: the

text: oscillatory

text: behavior

text: due

text: to

text: the

text: repulsive

text: azimuthal

text: force

text: should

text: persist.

text: We

text: will

text: address

text: such

text: a

text: comprehensive

text: non-linear

text: model

text: in

text: the

text: future.

text: Conclusion

text: and

text: Discussion

text: In

text: this

text: paper

text: we

text: have

text: presented

text: results

text: of

text: N-Body

text: simulations,

text: which

text: characterize

text: the

text: orbital

text: evolution

text: of

text: the

text: moonlet

text: of

text: the

text: propeller

text: Bl´eriot,

text: being

text: perturbed

text: by

text: the

text: gravitational

text: interactions

text: with

text: outer

text: and

text: inner

text: moons

text: of

text: Saturn.

text: We

text: found,

text: that

text: the

text: CER

text: of

text: Pandora

text: and

text: the

text: three-body

text: resonances

text: of

text: Mimas

text: and

text: Pandora

text: are

text: the

text: dominating

text: perturbations

text: of

text: Bl´eriot’s

text: orbit.

text: Additionally,

text: the

text: chaotic

text: interaction

text: between

text: Pandora

text: and

text: Prometheus

text: has

text: an

text: eﬀect

text: on

text: Bl´eriot’s

text: orbital

text: evolution

text: as

text: well,

text: resulting

text: in

text: changing

text: libration

text: frequencies

text: over

text: time.

text: Our

text: simulations

text: yielded,

text: that

text: the

text: gravitational

text: interactions

text: with

text: the

text: other

text: moons

text: cause

text: similar

text: libration

text: frequencies

text: of

text: Bl´eriot

text: as

text: concluded

text: from

text: the

text: Cassini

text: ISS

text: images,

text: but

text: the

text: corresponding

text: amplitudes

text: are

text: too

text: small.

text: Considering

text: propeller-moonlet

text: interactions

text: with

text: a

text: new

text: model,

text: we

text: have

text: been

text: able

text: to

text: ﬁnd

text: a

text: mechanism

text: to

text: amplify

text: certain

text: modes

text: in

text: form

text: of

text: a

text: harmonic

text: oscillator

text: which

text: is

text: periodically

text: driven

text: by

text: an

text: external

text: forcing.

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: END-TO-END ASR-FREE KEYWORD SEARCH FROM SPEECH Kartik Audhkhasi, Andrew Rosenberg, Abhinav Sethy, Bhuvana Ramabhadran, Brian Kingsbury IBM Watson, IBM T. J. Watson Research Center, Yorktown Heights, New York ABSTRACT Training a good ASR system is time-consuming and requires substantial amount of transcribed audio data. The main novelty of End-to-end (E2E) systems have achieved competitive results com- this paper is an end-to-end ASR-free KWS system motivated by the pared to conventional hybrid hidden Markov model (HMM)-deep recent success of E2E systems for ASR. Our fully-neural E2E KWS neural network based automatic speech recognition (ASR) systems. system lacks both an ASR system and a FST-index, which makes it Such E2E systems are attractive due to the lack of dependence on fast to train. We train our system in only 2 hours with minimal super- alignments between input acoustic and output grapheme or HMM vision and do not need fully transcribed training audio. Our system state sequence during training. This paper explores the design of an performs signiﬁcantly better than chance and respectably compared ASR-free end-to-end system for text query-based keyword search to a state-of-the-art hybrid HMM-DNN ASR-based system which (KWS) from speech trained with minimal supervision. Our E2E takes over 72 hours to train. KWS system consists of three sub-systems. The ﬁrst sub-system is a The next section gives an overview of related prior work. Sec- recurrent neural network (RNN)-based acoustic auto-encoder trained tion introduces our E2E ASR-free system by discussing its three to reconstruct the audio through a ﬁnite-dimensional representation. constituent sub-systems - an RNN-based acoustic encoder-decoder, The second sub-system is a character-level RNN language model us- a convolutional neural network (CNN)-RNN character language ing embeddings learned from a convolutional neural network. Since model (LM), and a feed-forward KWS network. Section discusses the acoustic and text query embeddings occupy different represen- the experimental setup, training of the E2E KWS system, and anal- tation spaces, they are input to a third feed-forward neural network y1s.1is. oPfrtihoerrWesuolrtks. Section gives directions for future work. that predicts whether the query occurs in the acoustic utterance or not. This E2E ASR-free KWS system performs respectably despite Most prior works relevant to this paper have focused on query- lacking a conventional ASR system and trains much faster. by-example (QbyE) retrieval of speech from a database. The user Index Terms— End-to-end systems, neural networks, keyword provides a speech utterance of the query to be searched, in contrast search, automatic speech recognition to a text query used in the KWS setup of this paper. Dynamic time warping (DTW) of acoustic features extracted from the speech query and speech utterances from the database is a classic technique in 1. INTRODUCTION such QbyE systems [9, 10]. The cost of DTW alignment serves as a matching score for retrieval. Deep neural networks (DNNs) have pushed the state-of-the-art Chen, Parada, and Sainath [11] present a system for QbyE where for automatic speech recognition (ASR) systems [1]. This has led to the audio database contains examples of certain key-phrases, such as signiﬁcant performance improvements on several well-known ASR “hello genie”. The last k state vectors from the ﬁnal hidden layer of benchmarks such as Switchboard [2, 3]. End-to-end (E2E) or fully- an RNN acoustic model give a ﬁxed-dimensional representation for neural architectures have become an alternative to the hybrid hidden both the speech query and each utterance in the speech database. The Markov model (HMM)-DNN architecture. These include the con- KWS detector then computes cosine similarity between the speech nectionist temporal classiﬁcation [4, 5] loss-based recurrent neural query and utterance representations. Levin et. al [12] compares network (RNN) and attention-based RNNs [6, 7]. several non-neural network-based techniques for computing ﬁxed- Automatic speech recognition is often not the end goal of real- dimensional representations of speech segments for QbyE, including world speech information processing systems. Instead, an impor- principal component analysis and Laplacian eigenmaps. Chung et. tant end goal is information retrieval, in particular keyword search al [13] also present a similar QbyE system using an acoustic RNN (KWS), that involves retrieving the speech utterances containing a encoder-decoder network. Kamper, Wang, and Livescu [14] use a user-speciﬁed text query from a large database. Conventional KWS Siamese convolutional neural network (CNN) for obtaining acous- from speech uses an ASR system as a front-end that converts the tic embeddings of the audio query and utterances from the database, speech database into a ﬁnite-state transducer (FST) index containing and train this network by minimizing a triplet hinge loss. all possible hypotheses word sequences with their associated con- Our paper is closely related to the recent work of Palaz, Syn- ﬁdence scores and time stamps [8]. The user-speciﬁc text query is naeve, and Collobert [15], where the authors propose a CNN-based then composed with this FST index to ﬁnd putative keyword loca- ASR system trained on a bag of words in the speech utterance. The tions and conﬁdence scores. CNN emits a time sequence of posterior distributions over the vo- The authors thank Dogan Can and Shrikanth Narayanan of the Signal cabulary, which is then aggregated over time to produce an estimate Analysis and Interpretation Lab, University of Southern California for use- of the bag of words. However, we note that their system is trained ful discussions. This paper uses the IARPA-Babel404b-v1.0a full language on a ﬁxed vocabulary of 1000 words compared with our proposed pack. Supported by the Intelligence Advanced Research Projects Activ- open-vocabulary system that does not use word identity. In addition, ity (IARPA) via Department of Defense U.S. Army Research Laboratory the training examples used in their approach use a stronger supervi- (DoD/ARL) contract number W911NF-12-C-0012. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes sion of word identity compared with a much weaker supervision in notwithstanding any copyright annotation thereon. our proposed model. The next section presents the architecture of ##LTLine##

text: END-TO-END

text: ASR-FREE

text: KEYWORD

text: SEARCH

text: FROM

text: SPEECH

text: Kartik

text: Audhkhasi,

text: Andrew

text: Rosenberg,

text: Abhinav

text: Sethy,

text: Bhuvana

text: Ramabhadran,

text: Brian

text: Kingsbury

text: IBM

label: author

text: Watson,

text: IBM

text: T.

text: J.

text: Watson

text: Research

text: Center,

text: Yorktown

text: Heights,

text: New

text: York

text: ABSTRACT

text: Training

text: a

text: good

text: ASR

text: system

text: is

text: time-consuming

text: and

text: requires

text: substantial

text: amount

text: of

text: transcribed

text: audio

text: data.

text: The

text: main

text: novelty

text: of

text: End-to-end

label: abstract

text: systems

text: have

text: achieved

text: competitive

text: results

text: com-

text: this

text: paper

text: is

text: an

text: end-to-end

text: ASR-free

text: KWS

text: system

text: motivated

text: by

text: the

text: pared

text: to

text: conventional

text: hybrid

text: hidden

text: Markov

text: model

text: (HMM)-deep

text: recent

text: success

text: of

text: systems

text: for

text: ASR.

text: Our

text: fully-neural

text: KWS

text: neural

text: network

text: based

text: automatic

text: speech

text: recognition

text: (ASR)

text: systems.

text: system

text: lacks

text: both

text: an

text: ASR

text: system

text: and

text: a

text: FST-index,

text: which

text: makes

text: it

text: Such

text: systems

text: are

text: attractive

text: due

text: to

text: the

text: lack

text: of

text: dependence

text: on

text: fast

text: to

text: train.

text: We

text: train

text: our

text: system

text: in

text: only

text: hours

text: with

text: minimal

text: super-

text: alignments

text: between

text: input

text: acoustic

text: and

text: output

text: grapheme

text: or

text: HMM

text: vision

text: and

text: do

text: not

text: need

text: fully

text: transcribed

text: training

text: audio.

text: Our

text: system

text: state

text: sequence

text: during

text: training.

text: This

text: paper

text: explores

text: the

text: design

text: of

text: an

text: performs

text: signiﬁcantly

text: better

text: than

text: chance

text: and

text: respectably

text: compared

text: ASR-free

text: end-to-end

text: system

text: for

text: text

text: query-based

text: keyword

text: search

text: to

text: a

text: state-of-the-art

text: hybrid

text: HMM-DNN

text: ASR-based

text: system

text: which

text: (KWS)

text: from

text: speech

text: trained

text: with

text: minimal

text: supervision.

text: Our

text: takes

text: over

text: hours

text: to

text: train.

text: KWS

text: system

text: consists

text: of

text: three

text: sub-systems.

text: The

text: ﬁrst

text: sub-system

text: is

text: a

text: The

text: next

text: section

text: gives

text: an

text: overview

text: of

text: related

text: prior

text: work.

text: Sec-

text: recurrent

text: neural

text: network

text: (RNN)-based

text: acoustic

text: auto-encoder

text: trained

text: tion

text: introduces

text: our

text: ASR-free

text: system

text: by

text: discussing

text: its

text: three

text: to

text: reconstruct

text: the

text: audio

text: through

text: a

text: ﬁnite-dimensional

text: representation.

text: constituent

text: sub-systems

text: an

text: RNN-based

text: acoustic

text: encoder-decoder,

text: The

text: second

text: sub-system

text: is

text: a

text: character-level

text: RNN

text: language

text: model

text: us-

text: a

text: convolutional

text: neural

text: network

text: (CNN)-RNN

text: character

text: language

text: ing

text: embeddings

text: learned

text: from

text: a

text: convolutional

text: neural

text: network.

text: Since

text: model

text: (LM),

text: and

text: a

text: feed-forward

text: KWS

text: network.

text: Section

text: discusses

text: the

text: acoustic

text: and

text: text

text: query

text: embeddings

text: occupy

text: different

text: represen-

text: the

text: experimental

text: setup,

text: training

text: of

text: the

text: KWS

text: system,

text: and

text: anal-

text: tation

text: spaces,

text: they

text: are

text: input

text: to

text: a

text: third

text: feed-forward

text: neural

text: network

text: oPfrtihoerrWesuolrtks.

text: Section

text: gives

text: directions

text: for

text: future

text: work.

text: that

text: predicts

text: whether

text: the

text: query

text: occurs

text: in

text: the

text: acoustic

text: utterance

text: or

text: not.

text: This

text: ASR-free

text: KWS

text: system

text: performs

text: respectably

text: despite

text: Most

text: prior

text: works

text: relevant

text: to

text: this

text: paper

text: have

text: focused

text: on

text: query-

text: lacking

text: a

text: conventional

text: ASR

text: system

text: and

text: trains

text: much

text: faster.

text: by-example

text: (QbyE)

text: retrieval

text: of

text: speech

text: from

text: a

text: database.

text: The

text: user

text: Index

text: Terms—

text: End-to-end

text: systems,

text: neural

text: networks,

text: keyword

text: provides

text: a

text: speech

text: utterance

text: of

text: the

text: query

text: to

text: be

text: searched,

text: in

text: contrast

text: search,

text: automatic

text: speech

text: recognition

text: to

text: a

text: text

text: query

text: used

text: in

text: the

text: KWS

text: setup

text: of

text: this

text: paper.

text: Dynamic

text: time

text: warping

text: (DTW)

text: of

text: acoustic

text: features

text: extracted

text: from

text: the

text: speech

text: query

text: and

text: speech

text: utterances

text: from

text: the

text: database

text: is

text: a

text: classic

text: technique

text: in

text: INTRODUCTION

text: such

text: QbyE

text: systems

text: The

text: cost

text: of

text: DTW

text: alignment

text: serves

text: as

text: a

text: matching

text: score

text: for

text: retrieval.

text: Deep

text: neural

text: networks

text: (DNNs)

text: have

text: pushed

text: the

text: state-of-the-art

text: Chen,

text: Parada,

text: and

text: Sainath

text: present

text: a

text: system

text: for

text: QbyE

text: where

text: for

text: automatic

text: speech

text: recognition

text: (ASR)

text: systems

text: This

text: has

text: led

text: to

text: the

text: audio

text: database

text: contains

text: examples

text: of

text: certain

text: key-phrases,

text: such

text: as

text: signiﬁcant

text: performance

text: improvements

text: on

text: several

text: well-known

text: ASR

text: “hello

text: genie”.

text: The

text: last

text: k

text: state

text: vectors

text: from

text: the

text: ﬁnal

text: hidden

text: layer

text: of

text: benchmarks

text: such

text: as

text: Switchboard

text: End-to-end

text: or

text: fully-

text: an

text: RNN

text: acoustic

text: model

text: give

text: a

text: ﬁxed-dimensional

text: representation

text: for

text: neural

text: architectures

text: have

text: become

text: an

text: alternative

text: to

text: the

text: hybrid

text: hidden

text: both

text: the

text: speech

text: query

text: and

text: each

text: utterance

text: in

text: the

text: speech

text: database.

text: The

text: Markov

text: model

text: (HMM)-DNN

text: architecture.

text: These

text: include

text: the

text: con-

text: KWS

text: detector

text: then

text: computes

text: cosine

text: similarity

text: between

text: the

text: speech

text: nectionist

text: temporal

text: classiﬁcation

text: loss-based

text: recurrent

text: neural

text: query

text: and

text: utterance

text: representations.

text: Levin

text: et.

text: al

text: compares

text: network

text: (RNN)

text: and

text: attention-based

text: RNNs

text: several

text: non-neural

text: network-based

text: techniques

text: for

text: computing

text: ﬁxed-

text: Automatic

text: speech

text: recognition

text: is

text: often

text: not

text: the

text: end

text: goal

text: of

text: real-

text: dimensional

text: representations

text: of

text: speech

text: segments

text: for

text: QbyE,

text: including

text: world

text: speech

text: information

text: processing

text: systems.

text: Instead,

text: an

text: impor-

text: principal

text: component

text: analysis

text: and

text: Laplacian

text: eigenmaps.

text: Chung

text: et.

text: tant

text: end

text: goal

text: is

text: information

text: retrieval,

text: in

text: particular

text: keyword

text: search

text: al

text: also

text: present

text: a

text: similar

text: QbyE

text: system

text: using

text: an

text: acoustic

text: RNN

text: (KWS),

text: that

text: involves

text: retrieving

text: the

text: speech

text: utterances

text: containing

text: a

text: encoder-decoder

text: network.

text: Kamper,

text: Wang,

text: and

text: Livescu

text: use

text: a

text: user-speciﬁed

text: text

text: query

text: from

text: a

text: large

text: database.

text: Conventional

text: KWS

text: Siamese

text: convolutional

text: neural

text: network

text: (CNN)

text: for

text: obtaining

text: acous-

text: from

text: speech

text: uses

text: an

text: ASR

text: system

text: as

text: a

text: front-end

text: that

text: converts

text: the

text: tic

text: embeddings

text: of

text: the

text: audio

text: query

text: and

text: utterances

text: from

text: the

text: database,

text: speech

text: database

text: into

text: a

text: ﬁnite-state

text: transducer

text: (FST)

text: index

text: containing

text: and

text: train

text: this

text: network

text: by

text: minimizing

text: a

text: triplet

text: hinge

text: loss.

text: all

text: possible

text: hypotheses

text: word

text: sequences

text: with

text: their

text: associated

text: con-

text: Our

text: paper

text: is

text: closely

text: related

text: to

text: the

text: recent

text: work

text: of

text: Palaz,

text: Syn-

text: ﬁdence

text: scores

text: and

text: time

text: stamps

text: The

text: user-speciﬁc

text: text

text: query

text: is

text: naeve,

text: and

text: Collobert

text: where

text: the

text: authors

text: propose

text: a

text: CNN-based

text: then

text: composed

text: with

text: this

text: FST

text: index

text: to

text: ﬁnd

text: putative

text: keyword

text: loca-

text: ASR

text: system

text: trained

text: on

text: a

text: bag

text: of

text: words

text: in

text: the

text: speech

text: utterance.

text: The

text: tions

text: and

text: conﬁdence

text: scores.

text: CNN

text: emits

text: a

text: time

text: sequence

text: of

text: posterior

text: distributions

text: over

text: the

text: vo-

text: The

text: authors

text: thank

text: Dogan

text: Can

text: and

text: Shrikanth

text: Narayanan

text: of

text: the

text: Signal

text: cabulary,

text: which

text: is

text: then

text: aggregated

text: over

text: time

text: to

text: produce

text: an

text: estimate

text: Analysis

text: and

text: Interpretation

text: Lab,

text: University

text: of

text: Southern

text: California

text: for

text: use-

text: of

text: the

text: bag

text: of

text: words.

text: However,

text: we

text: note

text: that

text: their

text: system

text: is

text: trained

text: ful

text: discussions.

text: This

text: paper

text: uses

text: the

text: full

text: language

text: on

text: a

text: ﬁxed

text: vocabulary

text: of

text: words

text: compared

text: with

text: our

text: proposed

text: pack.

text: Supported

text: by

text: the

text: Intelligence

text: Advanced

text: Research

text: Projects

text: Activ-

text: open-vocabulary

text: system

text: that

text: does

text: not

text: use

text: word

text: identity.

text: In

text: addition,

text: ity

text: (IARPA)

text: via

text: Department

text: of

text: Defense

text: U.S.

text: Army

text: Research

text: Laboratory

text: the

text: training

text: examples

text: used

text: in

text: their

text: approach

text: use

text: a

text: stronger

text: supervi-

text: (DoD/ARL)

text: contract

text: number

text: The

text: U.S.

text: Government

text: is

text: authorized

text: to

text: reproduce

text: and

text: distribute

text: reprints

text: for

text: Governmental

text: purposes

text: sion

text: of

text: word

text: identity

text: compared

text: with

text: a

text: much

text: weaker

text: supervision

text: in

text: notwithstanding

text: any

text: copyright

text: annotation

text: thereon.

text: our

text: proposed

text: model.

text: The

text: next

text: section

text: presents

text: the

text: architecture

text: of

text: ##LTLine##


text: ON THE PENROSE AND TAYLOR-SOCOLAR HEXAGONAL TILINGS JEONG-YUP LEE 1 , ROBERT V. MOODY 2 Abstract. We study the intimate relationship between the Penrose and the Taylor- Socolar tilings, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This uniﬁed approach produces both types of tilings together, clariﬁes their relationship, and oﬀers straight- forward proofs of their basic properties. Keywords: Penrose tiling, Taylor-Socolar tiling, double hexagon tiling, nested triangu- larizations, inverse sequences of hierarchical lattices 1. Introduction From the very beginning, aperiodic tilings have played a signiﬁcant role in unravelling the mysteries of aperiodic crystals. Knowing what is mathematically possible has often turned out to be a crucial element in conceiving what might be physically realizable. In this paper we discuss two remarkable aperiodic tilings of the plane that are built out of one of the most basic of all crystallographic structures: the standard periodic hexagonal lattice. Also from the very beginning, there arose the question of what might be the mini- mum number of diﬀerent prototiles necessary for a system of tiles and corresponding matching rules that permit, and only permit, aperiodic tilings. The very ﬁrst aperi- odic tilings involved thousands of prototiles. The famous aperiodic tilings of the plane like the rhombic Penrose and the Ammann-Beenker tilings are each based on just two prototiles, the allowable motions being translations and rotations. This of course im- mediately raises the question of whether aperiodic tilings based on just one prototile are possible. Taylor[7] and Taylor and Socolar[6] introduced a planar aperiodic tiling which can be built from a single hexagonal prototile allowing translations, rotations, and reﬂections. The tiling is based on the familiar hexagonal tiling of the plane, but if one distinguishes the prototile in its direct and reﬂected forms, then the matching rules allow only aperiodic tilings to appear. Their work revived interest in much earlier work of R. Penrose. In [4], he had introduced an aperiodic tiling, also based on marked hexagonal Date: 1 ##LTLine##

text: ON

text: THE

text: PENROSE

text: AND

text: TAYLOR-SOCOLAR

text: HEXAGONAL

text: TILINGS

text: JEONG-YUP

text: LEE

text: ROBERT

text: V.

text: MOODY

text: Abstract.

text: We

text: study

text: the

text: intimate

text: relationship

text: between

text: the

text: Penrose

text: and

text: the

text: Taylor-

text: Socolar

text: tilings,

text: within

text: both

text: the

text: context

text: of

text: double

text: hexagon

text: tiles

text: and

text: the

text: algebraic

text: context

text: of

text: hierarchical

text: inverse

text: sequences

text: of

text: triangular

text: lattices.

text: This

text: uniﬁed

text: approach

text: produces

text: both

text: types

text: of

text: tilings

text: together,

text: clariﬁes

text: their

text: relationship,

text: and

text: oﬀers

text: straight-

text: forward

text: proofs

text: of

text: their

text: basic

text: properties.

text: Keywords:

text: Penrose

text: tiling,

text: Taylor-Socolar

text: tiling,

text: double

text: hexagon

text: tiling,

text: nested

text: triangu-

text: larizations,

text: inverse

text: sequences

text: of

text: hierarchical

text: lattices

label: section

text: Introduction

text: From

text: the

text: very

text: beginning,

text: aperiodic

text: tilings

text: have

text: played

text: a

text: signiﬁcant

text: role

text: in

text: unravelling

text: the

text: mysteries

text: of

text: aperiodic

text: crystals.

text: Knowing

text: what

text: is

text: mathematically

text: possible

text: has

text: often

text: turned

text: out

text: to

text: be

text: a

text: crucial

text: element

text: in

text: conceiving

text: what

text: might

text: be

text: physically

text: realizable.

text: In

text: this

text: paper

text: we

text: discuss

text: two

text: remarkable

text: aperiodic

text: tilings

text: of

text: the

text: plane

text: that

text: are

text: built

text: out

text: of

text: one

text: of

text: the

text: most

text: basic

text: of

text: all

text: crystallographic

text: structures:

text: the

text: standard

text: periodic

text: hexagonal

text: lattice.

text: Also

text: from

text: the

text: very

text: beginning,

text: there

text: arose

text: the

text: question

text: of

text: what

text: might

text: be

text: the

text: mini-

text: mum

text: number

text: of

text: diﬀerent

text: prototiles

text: necessary

text: for

text: a

text: system

text: of

text: tiles

text: and

text: corresponding

text: matching

text: rules

text: that

text: permit,

text: and

text: only

text: permit,

text: aperiodic

text: tilings.

text: The

text: very

text: ﬁrst

text: aperi-

text: odic

text: tilings

text: involved

text: thousands

text: of

text: prototiles.

text: The

text: famous

text: aperiodic

text: tilings

text: of

text: the

text: plane

text: like

text: the

text: rhombic

text: Penrose

text: and

text: the

text: Ammann-Beenker

text: tilings

text: are

text: each

text: based

text: on

text: just

text: two

text: prototiles,

text: the

text: allowable

text: motions

text: being

text: translations

text: and

text: rotations.

text: This

text: of

text: course

text: im-

text: mediately

text: raises

text: the

text: question

text: of

text: whether

text: aperiodic

text: tilings

text: based

text: on

text: just

text: one

text: prototile

text: are

text: possible.

text: and

text: Taylor

text: and

text: introduced

text: a

text: planar

text: aperiodic

text: tiling

text: which

text: can

text: be

text: built

text: from

text: a

text: single

text: hexagonal

text: prototile

text: allowing

text: translations,

text: rotations,

text: and

text: reﬂections.

text: The

text: tiling

text: is

text: based

text: on

text: the

text: familiar

text: hexagonal

text: tiling

text: of

text: the

text: plane,

text: but

text: if

text: one

text: distinguishes

text: the

text: prototile

text: in

text: its

text: direct

text: and

text: reﬂected

text: forms,

text: then

text: the

text: matching

text: rules

text: allow

text: only

text: aperiodic

text: tilings

text: to

text: appear.

text: Their

text: work

text: revived

text: interest

text: in

text: much

text: earlier

text: work

text: of

text: R.

text: Penrose.

text: In

text: he

text: had

text: introduced

text: an

text: aperiodic

text: tiling,

text: also

text: based

text: on

text: marked

text: hexagonal

text: Date:

text: ##LTLine##


text: ON THE PENROSE AND TAYLOR-SOCOLAR HEXAGONAL TILINGS 19 v z u u’ v” v’ (cid:48) Figure 17. v, v are the ends of an edge e of a triangle T+ from the nesting determined T (cid:48) T by +(r). At its midpoint z we see the edge u, u of a triangle T from (q). The black T triangles all come from the triangulation of (q), the largest ones being of level 3. The edge T e is maximal, in the sense that it is not part of an edge of some larger triangle from +(r). (cid:48) Thus at its ends, the stripes of the large hexagons at v and v are in the directions of the (cid:48) other sides of T+. The inner hexagons along vv are centered at double hexagon centers and (cid:48) their stripes are all oriented in the same direction, namely perpendicular to the vv . At the (cid:48) left we have separated out the outer hexagons that overlay the small hexagons along vv . (cid:48) We see their matching arrows and how their stripes align to form the edge vv (in green). The colors (red/blue) of the short diameters of these large hexagons are determined by (or determine, whichever way one wants to put it) the color rule that we see in Fig. 10, though note that the stripe of the large hexagons is perpendicular to that of the small ones, so the right/left crossing rule is opposite! The fact that the stripe orientation changes at the end (cid:48) dictates that the edge vv is an interior edge of a larger triangle. The shift indicated by the orientation of the arrows matches this. 6. From nested triangulations to double hexagon tilings At this point it is rather clear that given any triangulation T (q) and any choice of one coset c + 3P leading to T +(r) ought to lead to a legal double hexagon tiling. Here are the details. We will assume that both triangulations are non-singular. This guarantees that there are no inﬁnite edges, we get proper triangulations, and they are nested. This nesting can be geometrically manifested by laterally shifting the edges as ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: THE

text: PENROSE

text: AND

text: TAYLOR-SOCOLAR

text: HEXAGONAL

text: TILINGS

text: v

font: default

text: z

text: u

text: u’

text: v”

text: v’

label: caption

text: Figure

text: v,

text: v

text: are

text: the

text: ends

text: of

text: an

text: edge

text: e

text: of

text: a

text: triangle

text: T+

text: from

text: the

text: nesting

text: determined

text: T

text: T

text: by

text: +(r).

text: At

text: its

text: midpoint

text: z

text: we

text: see

text: the

text: edge

text: u,

text: u

text: of

text: a

text: triangle

text: T

text: from

text: (q).

text: The

text: black

text: T

text: triangles

text: all

text: come

text: from

text: the

text: triangulation

text: of

text: (q),

text: the

text: largest

text: ones

text: being

text: of

text: level

text: The

text: edge

text: T

text: e

text: is

text: maximal,

text: in

text: the

text: sense

text: that

text: it

text: is

text: not

text: part

text: of

text: an

text: edge

text: of

text: some

text: larger

text: triangle

text: from

text: +(r).

text: Thus

text: at

text: its

text: ends,

text: the

text: stripes

text: of

text: the

text: large

text: hexagons

text: at

text: v

text: and

text: v

text: are

text: in

text: the

text: directions

text: of

text: the

text: other

text: sides

text: of

text: T+.

text: The

text: inner

text: hexagons

text: along

text: vv

text: are

text: centered

text: at

text: double

text: hexagon

text: centers

text: and

text: their

text: stripes

text: are

text: all

text: oriented

text: in

text: the

text: same

text: direction,

text: namely

text: perpendicular

text: to

text: the

text: vv

text: At

text: the

text: left

text: we

text: have

text: separated

text: out

text: the

text: outer

text: hexagons

text: that

text: overlay

text: the

text: small

text: hexagons

text: along

text: vv

text: We

text: see

text: their

text: matching

text: arrows

text: and

text: how

text: their

text: stripes

text: align

text: to

text: form

text: the

text: edge

text: vv

text: (in

text: green).

text: The

text: colors

text: (red/blue)

text: of

text: the

text: short

text: diameters

text: of

text: these

text: large

text: hexagons

text: are

text: determined

text: by

text: (or

text: determine,

text: whichever

text: way

text: one

text: wants

text: to

text: put

text: it)

text: the

text: color

text: rule

text: that

text: we

text: see

text: in

text: Fig.

text: though

text: note

text: that

text: the

text: stripe

text: of

text: the

text: large

text: hexagons

text: is

text: perpendicular

text: to

text: that

text: of

text: the

text: small

text: ones,

text: so

text: the

text: right/left

text: crossing

text: rule

text: is

text: opposite!

text: The

text: fact

text: that

text: the

text: stripe

text: orientation

text: changes

text: at

text: the

text: end

text: dictates

text: that

text: the

text: edge

text: vv

text: is

text: an

text: interior

text: edge

text: of

text: a

text: larger

text: triangle.

text: The

text: shift

text: indicated

text: by

text: the

text: orientation

text: of

text: the

text: arrows

text: matches

text: this.

text: From

text: nested

text: triangulations

text: to

text: double

text: hexagon

text: tilings

text: At

text: this

text: point

text: it

text: is

text: rather

text: clear

text: that

text: given

text: any

text: triangulation

text: T

text: (q)

text: and

text: any

text: choice

text: of

text: one

text: coset

text: c

text: leading

text: to

text: T

text: +(r)

text: ought

text: to

text: lead

text: to

text: a

text: legal

text: double

text: hexagon

text: tiling.

text: Here

text: are

text: the

text: details.

text: We

text: will

text: assume

text: that

text: both

text: triangulations

text: are

text: non-singular.

text: This

text: guarantees

text: that

text: there

text: are

text: no

text: inﬁnite

text: edges,

text: we

text: get

text: proper

text: triangulations,

text: and

text: they

text: are

text: nested.

text: This

text: nesting

text: can

text: be

text: geometrically

text: manifested

text: by

text: laterally

text: shifting

text: the

text: edges

text: as

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text: 20 JEONG-YUP LEE 1 , ROBERT V. MOODY 2 ≡ Figure 18. Starting from the pair (q, r), with r c mod 3P, hexagons centered at the 1 lattice points of Q are shown, with those centered on a coset c+3P indicated in yellow. The T nesting of the triangulation (q) arising from q is indicated. indicated by the nesting. The Voronoi cells of the lattices (nearest neighbor cells) are hexagons centered respectively on the points of T (q) and T +(r). We know that every hexagon from T (q) has a triangle edge passing through it and this edge will be shifted laterally in nesting. This is shown in Fig. 18. The hexagon is made into a well-arrowed hexagon by placing the pair of parallel arrows in the direction of the shift. The small hexagons now make a well-arrowed and well-matched hexagon tiling. Now we do the same thing with the triangulation T +(r), leading to the new hexag- onal tiling, again with arrows indicating edge shifting in the nesting, see Fig. 19. This is the second well-arrowed and well-matched hexagonal tiling. The outcome is that we have a tiling of double hexagon tiles which is non-singular and legal, see Fig. 2. ∈ T Figure 19. Continuing from Fig. 18, from r Q we obtain a nested triangulation +(r), 1 part of which is shown here. 7. Penrose tilings, Taylor-Socolar tilings, and beyond By deﬁnition a Penrose tiling is precisely a legal double hexagon tiling. Taylor- Socolar tilings (T-S tilings) are usually deﬁned by the T-S tiles shown in Fig. 20 and they are assembled as regular hexagonal tilings, but under the matching rules ##LTFigure## ##LTFigure## ##LTLine##

text: JEONG-YUP

text: LEE

text: ROBERT

text: V.

text: MOODY

text: Figure

text: Starting

text: from

text: the

text: pair

text: (q,

text: r),

text: with

text: r

text: c

text: mod

text: hexagons

text: centered

text: at

text: the

text: lattice

text: points

text: of

text: Q

text: are

text: shown,

text: with

text: those

text: centered

text: on

text: a

text: coset

text: indicated

text: in

text: yellow.

text: The

text: T

text: nesting

text: of

text: the

text: triangulation

text: (q)

text: arising

text: from

text: q

text: is

text: indicated.

text: indicated

text: by

text: the

text: nesting.

text: The

text: Voronoi

text: cells

text: of

text: the

text: lattices

text: (nearest

text: neighbor

text: cells)

text: are

text: hexagons

text: centered

text: respectively

text: on

text: the

text: points

text: of

text: T

text: (q)

text: and

text: T

text: +(r).

text: We

text: know

text: that

text: every

text: hexagon

text: from

text: T

text: (q)

text: has

text: a

text: triangle

text: edge

text: passing

text: through

text: it

text: and

text: this

text: edge

text: will

text: be

text: shifted

text: laterally

text: in

text: nesting.

text: This

text: is

text: shown

text: in

text: Fig.

text: The

text: hexagon

text: is

text: made

text: into

text: a

text: well-arrowed

text: hexagon

text: by

text: placing

text: the

text: pair

text: of

text: parallel

text: arrows

text: in

text: the

text: direction

text: of

text: the

text: shift.

text: The

text: small

text: hexagons

text: now

text: make

text: a

text: well-arrowed

text: and

text: well-matched

text: hexagon

text: tiling.

text: Now

text: we

text: do

text: the

text: same

text: thing

text: with

text: the

text: triangulation

text: T

text: +(r),

text: leading

text: to

text: the

text: new

text: hexag-

text: onal

text: tiling,

text: again

text: with

text: arrows

text: indicating

text: edge

text: shifting

text: in

text: the

text: nesting,

text: see

text: Fig.

text: This

text: is

text: the

text: second

text: well-arrowed

text: and

text: well-matched

text: hexagonal

text: tiling.

text: The

text: outcome

text: is

text: that

text: we

text: have

text: a

text: tiling

text: of

text: double

text: hexagon

text: tiles

text: which

text: is

text: non-singular

text: and

text: legal,

text: see

text: Fig.

text: T

text: Figure

text: Continuing

text: from

text: Fig.

text: from

text: r

text: Q

text: we

text: obtain

text: a

text: nested

text: triangulation

text: +(r),

text: part

text: of

text: which

text: is

text: shown

text: here.

text: Penrose

text: tilings,

text: Taylor-Socolar

text: tilings,

text: and

text: beyond

text: By

text: deﬁnition

text: a

text: Penrose

text: tiling

text: is

text: precisely

text: a

text: legal

text: double

text: hexagon

text: tiling.

text: Taylor-

text: Socolar

text: tilings

text: (T-S

text: tilings)

text: are

text: usually

text: deﬁned

text: by

text: the

text: T-S

text: tiles

text: shown

text: in

text: Fig.

text: and

text: they

text: are

text: assembled

text: as

text: regular

text: hexagonal

text: tilings,

text: but

text: under

text: the

text: matching

text: rules

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 8 JEONG-YUP LEE 1 , ROBERT V. MOODY 2 the decorations of the outer hexagons, it is possible to get an arrangement of triangles like the one shown in Fig. 6, which is visibly periodic. In all we shall see that triangles that are nested within appear on ever increasing scales, so there is a hierarchical structure. We shall call such an arrangement of triangles a nested triangulation. Figure 6. Periodic arrangements of triangles are possible if only decorations of the small hexagons are used. When we refer to the sizes of triangles in one of arrangements of triangles, we will always refer to the side lengths of the stretched out versions. Lengths are normalized so that the smallest triangles will be of side length equal to 1. We will see that with this normalization all lengths are powers of 2. In the sequel we will commonly use both versions of the triangles and nested triangles that emerge out of our discussion–the original ones that come from the decorated hexagons, and the stretched out ones that give us the arrangements of triangles. Once we have proved that all triangles are nested within, it is trivial to convert from one picture to the other. In the stretched out version, two triangles are said to make an opposite pair if they are of the same size and share a common edge, see Fig. 7. Notice that there is no speciﬁcation of how each of the two opposing triangles ﬁts into the overall arrangement of triangles. Proposition 4.1. Let D be a non-singular legal double hexagon tiling. Complete its small hexagons to a well-matched hexagonal tiling and let T be the resulting arrange- ment of expanded triangles formed from the decorations of the small hexagons. Then ##LTFigure##

text: JEONG-YUP

text: LEE

text: ROBERT

text: V.

text: MOODY

text: the

text: decorations

text: of

text: the

text: outer

text: hexagons,

text: it

text: is

text: possible

text: to

text: get

text: an

text: arrangement

text: of

text: triangles

text: like

text: the

text: one

text: shown

text: in

text: Fig.

text: which

text: is

text: visibly

text: periodic.

text: In

text: all

text: we

text: shall

text: see

text: that

text: triangles

text: that

text: are

text: nested

text: within

text: appear

text: on

text: ever

text: increasing

text: scales,

text: so

text: there

text: is

text: a

text: hierarchical

text: structure.

text: We

text: shall

text: call

text: such

text: an

text: arrangement

text: of

text: triangles

text: a

text: nested

text: triangulation.

text: Figure

text: Periodic

text: arrangements

text: of

text: triangles

text: are

text: possible

text: if

text: only

text: decorations

text: of

text: the

text: small

text: hexagons

text: are

text: used.

text: When

text: we

text: refer

text: to

text: the

text: sizes

text: of

text: triangles

text: in

text: one

text: of

text: arrangements

text: of

text: triangles,

text: we

text: will

text: always

text: refer

text: to

text: the

text: side

text: lengths

text: of

text: the

text: stretched

text: out

text: versions.

text: Lengths

text: are

text: normalized

text: so

text: that

text: the

text: smallest

text: triangles

text: will

text: be

text: of

text: side

text: length

text: equal

text: to

text: We

text: will

text: see

text: that

text: with

text: this

text: normalization

text: all

text: lengths

text: are

text: powers

text: of

text: In

text: the

text: sequel

text: we

text: will

text: commonly

text: use

text: both

text: versions

text: of

text: the

text: triangles

text: and

text: nested

text: triangles

text: that

text: emerge

text: out

text: of

text: our

text: discussion–the

text: original

text: ones

text: that

text: come

text: from

text: the

text: decorated

text: hexagons,

text: and

text: the

text: stretched

text: out

text: ones

text: that

text: give

text: us

text: the

text: arrangements

text: of

text: triangles.

text: Once

text: we

text: have

text: proved

text: that

text: all

text: triangles

text: are

text: nested

text: within,

text: it

text: is

text: trivial

text: to

text: convert

text: from

text: one

text: picture

text: to

text: the

text: other.

text: In

text: the

text: stretched

text: out

text: version,

text: two

text: triangles

text: are

text: said

text: to

text: make

text: an

text: opposite

text: pair

text: if

text: they

text: are

text: of

text: the

text: same

text: size

text: and

text: share

text: a

text: common

text: edge,

text: see

text: Fig.

text: Notice

text: that

text: there

text: is

text: no

text: speciﬁcation

text: of

text: how

text: each

text: of

text: the

text: two

text: opposing

text: triangles

text: ﬁts

text: into

text: the

text: overall

text: arrangement

text: of

text: triangles.

text: Proposition

text: Let

text: D

text: be

text: a

text: non-singular

text: legal

text: double

text: hexagon

text: tiling.

text: Complete

text: its

text: small

text: hexagons

text: to

text: a

text: well-matched

text: hexagonal

text: tiling

text: and

text: let

text: T

text: be

text: the

text: resulting

text: arrange-

text: ment

text: of

text: expanded

text: triangles

text: formed

text: from

text: the

text: decorations

text: of

text: the

text: small

text: hexagons.

text: Then

text: ##LTFigure##


text: 14 → ∞ → ∞ Now, we use the ansatz to compute the three terms in this equation in the regime t + , z + such that | − | (cid:48) − z vt = O(1). We will choose the direction η(z, t) = X (z vt). → − − ≈ We start with the left-hand side of (29). From x(z, t) X(z vt) and the approximation x(z, t + 1) x(z, t) − (cid:48) − vX (z vt), together with the special choice of η(z, t), we can rewrite the left hand side of (29) as (cid:90) (cid:0) (cid:1) (cid:24) (cid:24) (cid:24) (cid:48) − (cid:48) − (cid:48) v dz H X (z vt) ρ (X(z vt)) X (z, t) . (30) R (cid:24) Using the commutativity of the operator , this is equal to (cid:90) (cid:0) (cid:1) (cid:24) (cid:24) (cid:24) (cid:48) − (cid:48) − v dz H ρ (X(z vt)) X (z vt) 2 . (31) R − → Note that we can shift the argument in the integrals z vt z, and this term becomes independent of time. (cid:48) − Now, we consider the ﬁrst functional derivative on the right hand side of (29), when η(z, t) = X (z vt). It − → should be clear that we can immediately make the change of variables in the integrals z vt z which simpliﬁes the formulas. By the calculations in Appendix , we ﬁnd (cid:90) (cid:26) (cid:0) (cid:17) (cid:0) (cid:1)(cid:27) (cid:16) (cid:24) (cid:16) (cid:16) (cid:24) W (cid:24) (cid:16) (cid:24) (cid:24) δ (cid:48) (cid:48) (cid:48) − (cid:48) (cid:48) s [X (z)] = dz H X(z) [ρ (X(z)) X (z)] H c λ (ρ (X(z))) [ρ (X(z)) X (z)] . (32) δX R (cid:110) (cid:0) (cid:1)(cid:111) In order to simplify the above, we remark the following (cid:24) (cid:16) (cid:24) (cid:24) (cid:24) (cid:16) (cid:24) d 1 − − 1 (cid:48) H(R (X(z))) H(X(z) ρ (X(z))) + H(ρ (X(z))) (cid:48) H c(z) L (ρ (X(z))) dz R (1) (cid:24) (cid:24) (cid:24) L (1)(cid:24) (cid:24) (cid:24) (cid:24) (cid:24) (cid:24) (cid:48) − (cid:48) − (cid:48) (cid:48) (cid:48) (cid:48) = H(ρ (X(z)) X (z)) H(X (z) ρ (X(z))) H(X(z) ρ (X(z)) X (z)) + H(ρ (X(z)) X (z)) (cid:16) (cid:16) (cid:24) (cid:16) (cid:24) (cid:24) − (cid:48) (cid:48) H(c(z) λ (ρ (X(z))) [ρ (X(z)) X (z)]). Noticing that the ﬁrst two terms on the right-hand side cancel out, and using the duality rule (4) for the third term, we get the integrand in (32). In other words, the integr(cid:90)and in (32) equals ddzPs(z, X) = ddzWs(z, X(z)) and W δ (cid:48) d δXs [X (z)] = R dz dz Ws(X(z)) − = Ws(xBP) Ws(∆∞) = ∆E. (33) We now show that the functional derivative of the interaction part in (29) does not contribute when η(z, t) is (cid:48) replaced by X (z). By directly applying the deﬁnition of the functional derivative, we ﬁnd May 29, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Now,

text: we

text: use

text: the

text: ansatz

text: to

text: compute

text: the

text: three

text: terms

text: in

text: this

text: equation

text: in

text: the

text: regime

text: t

text: z

text: such

text: that

text: z

text: vt

text: We

text: will

text: choose

text: the

text: direction

text: η(z,

text: t)

text: X

text: (z

text: vt).

text: We

text: start

text: with

text: the

text: left-hand

text: side

text: of

text: From

text: x(z,

text: t)

text: X(z

text: vt)

text: and

text: the

text: approximation

text: x(z,

text: t

text: x(z,

text: t)

text: vX

text: (z

text: vt),

text: together

text: with

text: the

text: special

text: choice

text: of

text: η(z,

text: t),

text: we

text: can

text: rewrite

text: the

text: left

text: hand

text: side

text: of

text: as

text: v

text: dz

text: H

text: X

text: (z

text: vt)

text: ρ

text: (X(z

text: vt))

text: X

text: (z,

text: t)

text: R

text: Using

text: the

text: commutativity

text: of

text: the

text: operator

text: this

text: is

text: equal

text: to

text: v

text: dz

text: H

text: ρ

text: (X(z

text: vt))

text: X

text: (z

text: vt)

text: R

text: Note

text: that

text: we

text: can

text: shift

text: the

text: argument

text: in

text: the

text: integrals

text: z

text: vt

text: z,

text: and

text: this

text: term

text: becomes

text: independent

text: of

text: time.

text: Now,

text: we

text: consider

text: the

text: ﬁrst

text: functional

text: derivative

text: on

text: the

text: right

text: hand

text: side

text: of

text: when

text: η(z,

text: t)

text: X

text: (z

text: vt).

text: It

text: should

text: be

text: clear

text: that

text: we

text: can

text: immediately

text: make

text: the

text: change

text: of

text: variables

text: in

text: the

text: integrals

text: z

text: vt

text: z

text: which

text: simpliﬁes

text: the

text: formulas.

text: By

text: the

text: calculations

text: in

text: Appendix

text: we

text: ﬁnd

text: W

text: δ

text: s

text: [X

text: (z)]

text: dz

text: H

text: X(z)

text: [ρ

text: (X(z))

text: X

text: (z)]

text: H

text: c

text: λ

text: (ρ

text: (X(z)))

text: [ρ

text: (X(z))

text: X

text: (z)]

text: δX

text: R

text: In

text: order

text: to

text: simplify

text: the

text: above,

text: we

text: remark

text: the

text: following

text: d

text: H(R

text: (X(z)))

text: H(X(z)

text: ρ

text: (X(z)))

text: H(ρ

text: (X(z)))

text: H

text: c(z)

text: L

text: (ρ

text: (X(z)))

text: dz

text: R

text: L

text: H(ρ

text: (X(z))

text: X

text: (z))

text: H(X

text: (z)

text: ρ

text: (X(z)))

text: H(X(z)

text: ρ

text: (X(z))

text: X

text: (z))

text: H(ρ

text: (X(z))

text: X

text: (z))

text: H(c(z)

text: λ

text: (ρ

text: (X(z)))

text: [ρ

text: (X(z))

text: X

text: (z)]).

text: Noticing

text: that

text: the

text: ﬁrst

text: two

text: terms

text: on

text: the

text: right-hand

text: side

text: cancel

text: out,

text: and

text: using

text: the

text: duality

text: rule

text: for

text: the

text: third

text: term,

text: we

text: get

text: the

text: integrand

text: in

text: In

text: other

text: words,

text: the

text: in

text: equals

text: ddzPs(z,

text: X)

text: ddzWs(z,

text: X(z))

text: and

text: W

text: δ

text: d

text: δXs

text: [X

text: (z)]

text: R

text: dz

text: dz

text: Ws(X(z))

text: Ws(xBP)

text: Ws(∆∞)

text: ∆E.

text: We

text: now

text: show

text: that

text: the

text: functional

text: derivative

text: of

text: the

text: interaction

text: part

text: in

text: does

text: not

text: contribute

text: when

text: η(z,

text: t)

text: is

text: replaced

text: by

text: X

text: (z).

text: By

text: directly

text: applying

text: the

text: deﬁnition

text: of

text: the

text: functional

text: derivative,

text: we

text: ﬁnd

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 R → We also note the following identities valid for scalar maps f, g : [0, 1] (such as λ, ρ, L, R and their derivatives)   (cid:16)  − f (X(z)) = f(X(z))∆0 + (1 f(X(z)))∆∞, (cid:24) (39) − − − g (X(z)) = (1 g(1 X(z))∆0 + g(1 X(z))∆∞. Let us compute the denominator of (15). Using (3), (38), and (39) we have (cid:24) (cid:24) (cid:24) (cid:24) (cid:24) (cid:48) (cid:48) { − (cid:48) − (cid:48) − } { (cid:48) − (cid:48) } ρ (X(z)) X (z) 2 = (1 ρ (1 X(z)))∆0 + ρ (1 X(z))∆∞ (cid:24) X (z)∆0 X (z)∆∞ 2 { − (cid:48) − (cid:48) − } { (cid:48) − (cid:48) } = (1 ρ (1 X(z)))∆ + ρ (1 X(z))∆∞ X (z)2∆∞ X (z)2∆ 0 0 − (cid:48) − (cid:48) − − (cid:48) − (cid:48) = (1 ρ (1 X(z)))X (z)2∆ (1 ρ (1 X(z)))X (z)2∆ 0 0 (cid:48) − (cid:48) − (cid:48) − (cid:48) } + ρ (1 X(z))X (z)2∆∞ ρ (1 X(z))X (z)2∆ 0 (cid:48) − (cid:48) − (cid:48) − (cid:48) = ρ (1 X(z))X (z)2∆∞ ρ (1 X(z))X (z)2∆ , 0 (cid:0) (cid:1) and since H(∆ ) = 1, H(∆∞) = 0, and the entropy functional is linear, we obtain the denominator of (15) as 0 (cid:24) (cid:24) (cid:24) (cid:48) (cid:48) − (cid:48) − (cid:48) H ρ (X(z)) X (z) 2 = ρ (1 X(z))X (z)2. − For the numerator of (15), we have ∆E = W (x ) W (0), where the single system potential on the BEC is BEC BP BEC obtained from (6) using again (3) and (39). The exercise yields 1 − − − − − (cid:15) − − W (x) = (cid:48) (1 R(1 x)) xρ(1 x) (cid:48) L(1 ρ(1 x)). BEC R (1) L (1) Putting together these results, the velocity (15) becomes (cid:82) − − W (x ) W (0) v = BEC(cid:48) B−P BEC (cid:48) . (40) BEC R dz ρ (1 X(z))X (z)2 (Note that, with our normalizations, W (0) = 0 for all (cid:15).) The erasure proﬁle X(z) has to be computed from BEC the one-dimensional integral equation (cid:90) (cid:90) (cid:0) (cid:1) − (cid:48) 1 − 1 − − X(z) v X (z) = (cid:15) du λ 1 ds ρ(1 X(z u + s)) . (41) BEC 0 0 May 29, 2020 DRAFT ##LTLine## ##LTLine## ##LTLine##

text: R

text: We

text: also

text: note

text: the

text: following

text: identities

text: valid

text: for

text: scalar

text: maps

text: f,

text: g

text: (such

text: as

text: λ,

text: ρ,

text: L,

text: R

text: and

text: their

text: derivatives)

text: f

text: (X(z))

text: f(X(z)))∆∞,

text: g

text: (X(z))

text: X(z))∆∞.

text: Let

text: us

text: compute

text: the

text: denominator

text: of

text: Using

text: and

text: we

text: have

text: ρ

text: (X(z))

text: X

text: (z)

text: ρ

text: ρ

text: X(z))∆∞

text: X

text: X

text: (z)∆∞

text: ρ

text: X(z)))∆

text: ρ

text: X(z))∆∞

text: X

text: X

text: ρ

text: X(z)))X

text: ρ

text: X(z)))X

text: ρ

text: X(z))X

text: ρ

text: X(z))X

text: ρ

text: X(z))X

text: ρ

text: X(z))X

text: and

text: since

text: H(∆

text: H(∆∞)

text: and

text: the

text: entropy

text: functional

text: is

text: linear,

text: we

text: obtain

text: the

text: denominator

text: of

text: as

text: H

text: ρ

text: (X(z))

text: X

text: (z)

text: ρ

text: X(z))X

text: For

text: the

text: numerator

text: of

text: we

text: have

text: ∆E

text: W

text: (x

text: W

text: where

text: the

text: single

text: system

text: potential

text: on

text: the

text: BEC

text: is

text: BEC

text: BP

text: BEC

text: obtained

text: from

text: using

text: again

text: and

text: The

text: exercise

text: yields

text: W

text: (x)

text: x))

text: x)

text: x)).

text: BEC

text: R

text: L

text: Putting

text: together

text: these

text: results,

text: the

text: velocity

text: becomes

text: W

text: (x

text: W

text: v

text: B−P

text: BEC

text: BEC

text: R

text: dz

text: ρ

text: X(z))X

text: (Note

text: that,

text: with

text: our

text: normalizations,

text: W

text: for

text: all

text: The

text: erasure

text: proﬁle

text: X(z)

text: has

text: to

text: be

text: computed

text: from

text: BEC

text: the

text: one-dimensional

text: integral

text: equation

text: X(z)

text: v

text: X

text: (z)

text: du

text: λ

text: ds

text: X(z

text: u

text: s))

text: BEC

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##