text: Navier-Stokes Equations in Bounded Domain 19 y y y y y y (a) t=0.5 (b) t=1 (c) t=2 (d) t=3 (e) t=4 (f) t=5 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 x x x x x x y y y y y y (g) t=6 (h) t=8 (i) t=9 (j) t=10 (k) t=11 (l) t=12 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 x x x x x x Figure 10. The initial transients of bell lid cavity (2.5) at Re = 1000 and n = 256. Plotted ± ± ± ± ± ± ± ± ± ± ± ± iso-vorticity contours are 200, 100, 50, 20, 10, 7, 5, 4, 3, 2, 1 and 0.5. y y y y y y (a) t=1 (b) t=2 (c) t=4 (d) t=6 (e) t=8 (f) t=10 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 x x x x x x y y y y y y (g) t=12 (h) t=16 (i) t=20 (j) t=30 (k) t=40 (l) t=50 1 1 1 1 1 1 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 x x x x x x Figure 11. Navier-Stokes solutions of lid-driven cavity with ramp lid (2.8), Re = 1000, n = 512, shown at the contours of the preceding plot. At t = 50, the solutions almost reach the steady-state. The experimental data of Guermond et al. indicate that the initial ≈ ≈ ﬂow settled to a quasi-stationary state at t 12, and the start-up phase ﬁnalised at t 18. Even in the test, the lid shear near the start is uncertain. Nevertheless, these simulation snap-shots are broadly in agreement with the observation. We can identify how the main vortex is developed from the lid, and the way it grows and migrates into the cavity from the upper right corner. In addition, the appearance of the wall viscous layers is well-explained. 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text: 10 F. Lam y ζ y ψ=c 1 1 0.975 0.975 0.95 0.95 0.925 0.925 0.925 0.95 0.975 1 0.925 0.95 0.975 1 x x Figure 3. Numerical solution of ramp proﬁle (2.8) near the upper right corner at t=2. − × − ± ± ± Run at ν = 10 5, ∆t = 5 10 4, grid n = 1536. The ζ-contours are 500, 150, 100, ± ± ± ± ± ± ± ± − 75, 50, 35, 20, 15, 10, 5 and 1. The ψ-ﬁeld obtained from ∆ 1ζ is highly regular at the current mesh which is however considered as inadequately-resolved for the vorticity ﬁeld. It is reasonable to expect that the pure stream function formulation (3.6) would produce smooth, yet possibly spurious shears. By the method of separation o(cid:88)f variables and Fourier series, the solution ψ¯ can be expressed as ∞ ψ¯(x) = A sin(mπx) sinh(mπy), m m=1 (cid:90) by virtue of the no-slip on the side and lower walls. As usual, the Fourier coeﬃcients are given by 2 1 A = u (x) sin(mπx) dx. m sinh(mπ) Lid 0 − Because ∂ ψ¯ ∂ ψ¯ = 0, the solution ψ¯ does not aﬀect the vorticity. As we intend xx yy to iterate on the vorticity numerically, we need to solve the Poisson system (3.7) ∗ (cid:88) (cid:88) for ψ . It follows that, at every iteration, the solution for ψ is given by ∞ ∞ ψ(x) = B sin(qπx) sin(rπy), qr q=1(cid:90)r=(cid:90)1 where 4 1 1 B = ζ(x, y) sin(qπx) sin(rπy) dxdy. qr (qπ)2 + (rπ)2 0 0 This is an alternative to the full data (1.6); the selective use of homogeneous Dirich- let data ψ = 0 may be more eﬃcient in practical implementation of elliptic solvers. B † Our ideas may be generalised to arbitrary boundary data f(x) of C2 domains. † There is a deep-rooted misconception that considers the Poisson boundary-value problem (1.7) as over-speciﬁed because two sets of boundary conditions have to be satisﬁed, namely Dirich- let and Neumann data (i.e., ψ(x) = const, and ∂ψ/∂(cid:126)n(x) = const, where symbol (cid:126)n denotes the local outward normal). A host of numerical schemes then have been proposed to address the over- Article published on arXiv ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: F.

text: Lam

text: y

font: default

text: ζ

text: y

text: ψ=c

text: x

text: x

text: Figure

text: Numerical

label: caption

text: solution

text: of

text: ramp

text: proﬁle

text: near

text: the

text: upper

text: right

text: corner

text: at

text: Run

text: at

text: ν

text: ∆t

text: grid

text: n

text: The

text: ζ-contours

text: are

text: and

text: The

text: ψ-ﬁeld

text: obtained

text: from

text: 1ζ

text: is

text: highly

text: regular

text: at

text: the

text: current

text: mesh

text: which

text: is

text: however

text: considered

text: as

text: inadequately-resolved

text: for

text: the

text: vorticity

text: ﬁeld.

text: It

text: is

text: reasonable

text: to

text: expect

text: that

text: the

text: pure

text: stream

text: function

text: formulation

text: would

text: produce

text: smooth,

text: yet

text: possibly

text: spurious

text: shears.

text: By

text: the

text: method

text: of

text: separation

text: variables

text: and

text: Fourier

text: series,

text: the

text: solution

text: ψ¯

text: can

text: be

text: expressed

text: as

text: ψ¯(x)

text: A

text: sin(mπx)

text: sinh(mπy),

text: m

text: by

text: virtue

text: of

text: the

text: no-slip

text: on

text: the

text: side

text: and

text: lower

text: walls.

text: As

text: usual,

text: the

text: Fourier

text: coeﬃcients

text: are

text: given

text: by

text: A

text: u

text: (x)

text: sin(mπx)

text: dx.

text: m

text: sinh(mπ)

text: Lid

text: Because

text: ψ¯

text: ψ¯

text: the

text: solution

text: ψ¯

text: does

text: not

text: aﬀect

text: the

text: vorticity.

text: As

text: we

text: intend

text: xx

text: yy

text: to

text: iterate

text: on

text: the

text: vorticity

text: numerically,

text: we

text: need

text: to

text: solve

text: the

text: Poisson

text: system

text: for

text: ψ

text: It

text: follows

text: that,

text: at

text: every

text: iteration,

text: the

text: solution

text: for

text: ψ

text: is

text: given

text: by

text: ψ(x)

text: B

text: sin(qπx)

text: sin(rπy),

text: qr

text: where

text: B

text: ζ(x,

text: y)

text: sin(qπx)

text: sin(rπy)

text: dxdy.

text: qr

text: This

text: is

text: an

text: alternative

text: to

text: the

text: full

text: data

text: the

text: selective

text: use

text: of

text: homogeneous

text: Dirich-

text: let

text: data

text: ψ

text: may

text: be

text: more

text: eﬃcient

text: in

text: practical

text: implementation

text: of

text: elliptic

text: solvers.

text: B

text: Our

text: ideas

text: may

text: be

text: generalised

text: to

text: arbitrary

text: boundary

text: data

text: f(x)

text: of

text: domains.

text: There

text: is

text: a

text: deep-rooted

text: misconception

text: that

text: considers

text: the

text: Poisson

text: boundary-value

text: problem

text: as

text: over-speciﬁed

text: because

text: two

text: sets

text: of

text: boundary

text: conditions

text: have

text: to

text: be

text: satisﬁed,

text: namely

text: Dirich-

text: let

text: and

text: Neumann

text: data

text: (i.e.,

text: ψ(x)

text: const,

text: and

text: const,

text: where

text: symbol

text: denotes

text: the

text: local

text: outward

text: normal).

text: A

text: host

text: of

text: numerical

text: schemes

text: then

text: have

text: been

text: proposed

text: to

text: address

text: the

text: over-

text: Article

text: published

text: on

text: arXiv

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text: 4 However, if the electrical properties (cid:90)depend on position, this is not the case, but, rather, the right side of Eq. (2.1) would be ∇ 1 dω ∗ ∇ ∗ ∇ − · T + [E (ω) ( ε (ω))E (ω) + H (ω) ( µ (ω))H (ω)] . (2.8) i ij j i ij j 2T 2π For a recent review concerning electromagnetic stress tensors see Ref. [35]. For example, consider a dielectric body (µ = 1) immersed in a static classically imposed electric ﬁeld. Because there is no time dependence and no free charge, we have ∇ 1 ∇ · T = tr EE( )ε, (2.9) 2 ∇ where the trace is over the tensor indices, and the notation ( ) is a reminder that the free vector index is on the gradient operator. Suppose the body, which need not b(cid:73)e homogeneous, is immersed in a homogeneous medium of permittivity (cid:15). The force on the body is the momentum ﬂux into the body, F = − dS · T, (2.10) S since the local momentum conservation law(cid:90) holds there, wher(cid:90)e S is a surface that entirely surrounds the body. By the divergence theorem ∇ 1 ∇ F = − (dr) · T = − (dr) tr EE( )ε, (2.11) 2 V V where the spatial integral is over the interior of the body (because the permittivity is constant outside the body). This is a generalization of the familiar formula for the force on a dielectric, Eq. (11.44) of Ref. [34], to which it reduces for the isotropic case. We can immediately generalize this to the Casimir force by replacing in Eq. (2.8) (cid:48) ∗ (cid:48) 1 (cid:104)E(ω)E(ω ) (cid:105) = 2πδ(ω − ω ) Γ(ω), (2.12) i (cid:90) (cid:90) in terms of the Green’s dyadic Γ, so that the dispersion force on the dielectric body is 1 dω ∇ F = − (dr) tr Γ(r, r; ω)( )ε(r, ω). (2.13) Cas 2i 2π Here we have identiﬁed 2πδ(0) with the averaging time T. In particular, if the body has a homogeneous dielectric constant ε (cid:54)= (cid:15), then ∇ − ε = −ˆs(ε (cid:15))δ(s − s (r⊥)), (2.14) 0 where the surface of the body is given by s = s (r⊥), in terms of a coordinate s (outwardly) normal to the surface. 0 The other coordinates are denoted by r⊥. (Fo(cid:73)r the (cid:90)case of a planar body in the x-y plane, s = z.) Thus the Casimir force on the body is given by an integral over the surface of the body, 1 dω − F = dS tr(ε (cid:15))(r, ω)Γ(r, r; ω). (2.15) Cas 2i 2π S Again, this is an obvious(cid:90)generalization of known formulas.2 The general for(cid:90)m for the nonconservation of the vacuum expectation value of the electromagnetic stress tensor in a medium is ∇ 1 dω ∇ 1 dω · (cid:104)T(r)(cid:105) = tr Γ(r, r; ω)( )ε(r, ω), or ∂ (cid:104)T (cid:105)(r) = Γ (r, r; ω)∂ ε (r, ω). (2.16) j ji jk i kj 2i 2π 2i 2π This is, of course, quite analogous to the nonconservation equation satisﬁed by the stress tensor for a scalar ﬁeld in a background potential [28]. 2 For example, for the case of a dielectric ball, this formula leads immediately, upon use of the orthogonality relations for the vector spherical harmonics given in Ref. [34], p. 534, to the expression (5.19) for the total outward stress given in Ref. [1]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: However,

text: if

text: the

text: electrical

text: properties

text: on

text: position,

text: this

text: is

text: not

text: the

text: case,

text: but,

text: rather,

text: the

text: right

text: side

text: of

text: Eq.

text: would

text: be

text: dω

text: T

text: [E

text: (ω)

text: ε

text: (ω))E

text: (ω)

text: H

text: (ω)

text: µ

text: (ω))H

text: (ω)]

text: i

text: ij

text: j

text: i

text: ij

text: j

text: 2π

text: For

text: a

text: recent

text: review

text: concerning

text: electromagnetic

text: stress

text: tensors

text: see

text: Ref.

text: For

text: example,

text: consider

text: a

text: dielectric

text: body

text: (µ

text: immersed

text: in

text: a

text: static

text: classically

text: imposed

text: electric

text: ﬁeld.

text: Because

text: there

text: is

text: no

text: time

text: dependence

text: and

text: no

text: free

text: charge,

text: we

text: have

text: T

text: tr

text: EE(

text: )ε,

text: where

text: the

text: trace

text: is

text: over

text: the

text: tensor

text: indices,

text: and

text: the

text: notation

text: is

text: a

text: reminder

text: that

text: the

text: free

text: vector

text: index

text: is

text: on

text: the

text: gradient

text: operator.

text: Suppose

text: the

text: body,

text: which

text: need

text: not

text: homogeneous,

text: is

text: immersed

text: in

text: a

text: homogeneous

text: medium

text: of

text: permittivity

text: The

text: force

text: on

text: the

text: body

text: is

text: the

text: momentum

text: ﬂux

text: into

text: the

text: body,

text: F

text: dS

text: T,

text: S

text: since

text: the

text: local

text: momentum

text: conservation

text: holds

text: there,

text: S

text: is

text: a

text: surface

text: that

text: entirely

text: surrounds

text: the

text: body.

text: By

text: the

text: divergence

text: theorem

text: F

text: (dr)

text: T

text: (dr)

text: tr

text: EE(

text: )ε,

text: V

text: V

text: where

text: the

text: spatial

text: integral

text: is

text: over

text: the

text: interior

text: of

text: the

text: body

text: (because

text: the

text: permittivity

text: is

text: constant

text: outside

text: the

text: body).

text: This

text: is

text: a

text: generalization

text: of

text: the

text: familiar

text: formula

text: for

text: the

text: force

text: on

text: a

text: dielectric,

text: Eq.

text: of

text: Ref.

text: to

text: which

text: it

text: reduces

text: for

text: the

text: isotropic

text: case.

text: We

text: can

text: immediately

text: generalize

text: this

text: to

text: the

text: Casimir

text: force

text: by

text: replacing

text: in

text: Eq.

text: 2πδ(ω

text: ω

text: Γ(ω),

text: i

text: in

text: terms

text: of

text: the

text: Green’s

text: dyadic

text: Γ,

text: so

text: that

text: the

text: dispersion

text: force

text: on

text: the

text: dielectric

text: body

text: is

text: dω

text: F

text: (dr)

text: tr

text: Γ(r,

text: r;

text: ω)(

text: )ε(r,

text: ω).

text: Cas

text: 2π

text: Here

text: we

text: have

text: identiﬁed

text: with

text: the

text: averaging

text: time

text: T.

text: In

text: particular,

text: if

text: the

text: body

text: has

text: a

text: homogeneous

text: dielectric

text: constant

text: ε

text: then

text: ε

text: −ˆs(ε

text: s

text: (r⊥)),

text: where

text: the

text: surface

text: of

text: the

text: body

text: is

text: given

text: by

text: s

text: s

text: (r⊥),

text: in

text: terms

text: of

text: a

text: coordinate

text: s

text: (outwardly)

text: normal

text: to

text: the

text: surface.

text: The

text: other

text: coordinates

text: are

text: denoted

text: by

text: r⊥.

text: the

text: of

text: a

text: planar

text: body

text: in

text: the

text: x-y

text: plane,

text: s

text: z.)

text: Thus

text: the

text: Casimir

text: force

text: on

text: the

text: body

text: is

text: given

text: by

text: an

text: integral

text: over

text: the

text: surface

text: of

text: the

text: body,

text: dω

text: F

text: dS

text: tr(ε

text: ω)Γ(r,

text: r;

text: ω).

text: Cas

text: 2π

text: S

text: Again,

text: this

text: is

text: an

text: of

text: known

text: The

text: general

text: for

text: the

text: nonconservation

text: of

text: the

text: vacuum

text: expectation

text: value

text: of

text: the

text: electromagnetic

text: stress

text: tensor

text: in

text: a

text: medium

text: is

text: dω

text: dω

text: tr

text: Γ(r,

text: r;

text: ω)(

text: )ε(r,

text: ω),

text: or

text: Γ

text: (r,

text: r;

text: ω)∂

text: ε

text: (r,

text: ω).

text: j

text: ji

text: jk

text: i

text: kj

text: 2π

text: 2π

text: This

text: is,

text: of

text: course,

text: quite

text: analogous

text: to

text: the

text: nonconservation

text: equation

text: satisﬁed

text: by

text: the

text: stress

text: tensor

text: for

text: a

text: scalar

text: ﬁeld

text: in

text: a

text: background

text: potential

text: For

text: example,

text: for

text: the

text: case

text: of

text: a

text: dielectric

text: ball,

text: this

text: formula

text: leads

text: immediately,

text: upon

text: use

text: of

text: the

text: orthogonality

text: relations

text: for

text: the

text: vector

text: spherical

text: harmonics

text: given

text: in

text: Ref.

text: p.

text: to

text: the

text: expression

text: for

text: the

text: total

text: outward

text: stress

text: given

text: in

text: Ref.

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text: 7 which is independent of z, the term involving the reﬂection coeﬃcient having cancelled out. This is universally recognized as an irrelevant bulk term, since it has no contribution from the wall, and would be present if vacuum ﬁlled all space, so is to be omitted. It is the assertion of Ref. [24] that the same omission is to be done for the contribution to the stress tensor coming from the part of the Green’s function in the z > 0 region that is not proportional to the reﬂection coeﬃcient: In particular, they advocate omitting the stress tensor contribution arising from the term in the Green’s function (4.4) 1 F(z )G(z ), even though it is spatially varying, because this term would be there in the absence of the edge at α > < z = 0. This hypothesis may be suspect, but we will follow it for the moment. V. EXACTLY SOLVABLE EXAMPLES Now we examine two cases where both the TE and TM modes may be explicitly given. In the ﬁrst example, the permittivity has a singularity at a ﬁnite value of z, which is the natural boundary of the problem, and in the second the permittivity has an exponential behavior. A. A ﬁrst example Let us consider a planar medium described by λ a > z > 0 : µ˜ = 1, ε˜(z) = , (5.1) (a − z)2 which has a singularity at z = a. (This is a slightly generalized version of the medium considered in Ref. [24], where the potential was continuous, so (cid:15) ≡ ε˜(0) = λ/a2 = 1.) Because of that singularity, the right side of the wall has a ﬁnite depth, 0 < z < a; the region z > a is completely disconnected from the region containing the wall. This potential has the virtue of allowing explicit solutions: ± ± FE,H = (a − z) 1/2I (k(a − z)), GE,H = (a − z) 1/2K (k(a − z)), (5.2) ν ν (cid:114) where 1 1 ν = λζ2 + , = 1, λ. (5.3) 4 αE,H Here F is chosen to be ﬁnite as z → a. Indeed, Iν(0) = 0,(cid:16)Kν(+∞) =(cid:17) 0, consistent with the criteria stated in the previous section. Then the reﬂection coeﬃcients in the medium are (cid:16) (cid:17) (cid:48) ka K (ka) + κa ± 1 K (ka) R˜E,H = − 1,(cid:15) ν 2(1,(cid:15)) ν . (5.4) (cid:48) ka I (ka) + κa ± 1 I (ka) 1,(cid:15) ν 2(1,(cid:15)) ν The scattering part o(cid:26)f th(cid:20)e zz component of the re(cid:21)duced stress tensor (the part proportional to the reﬂection coeﬃcien(cid:27)ts) is 1 λζ2 − 1/4 (cid:48) (cid:48) ts,E,H(z) = R˜E,H − k2(a − z) + I2(k(a − z)) + k2(a − z)I 2(k(a − z)) ± kI (k(a − z))I (k(a − z)) . zz 2 a − z ν ν ν ν (5.5) As we wish to examine the(cid:32)stress just insi(cid:33)de the wall, we c(cid:114)an use the uniform(cid:32) asymptotic expans(cid:33)ion (UAE) for the (cid:88) (cid:88) Bessel functions, because it captures the short-distance behavior [37]. That expansion is, as ν → ∞: ∞ − ∞ Iν(νZ) ∼ √21πν (1 +eνηZ(Z2))1/4 1 + (cid:32) ukν(cid:88)(kt) , K(cid:33)ν(νZ) ∼ 2πν (1e(cid:114)+νZη(2Z)1)/4 1 + (−1)k(cid:32)ukν(kt) (cid:88), (5.(cid:33)6a) k=1 k=1 ∞ ∞ (cid:48) 1 (1 + Z2)1/4 v (t) (cid:48) π − (1 + Z2)1/4 v (t) I (νZ) ∼ √ eνη(Z) 1 + k , K (νZ) ∼ − e νη(Z) 1 + (−1)k k , ν 2πν Z νk ν 2ν Z νk k=1 k=1 (5.6b) − where u and v are polynomials in t = (1 + Z2) 1/2. The ﬁrst of these are k k 1 1 u (t) = (3t − 5t3), v (t) = (−9t + 7t3). (5.7) 1 1 24 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: is

text: independent

text: of

text: z,

text: the

text: term

text: involving

text: the

text: reﬂection

text: coeﬃcient

text: having

text: cancelled

text: out.

text: This

text: is

text: universally

text: recognized

text: as

text: an

text: irrelevant

text: bulk

text: term,

text: since

text: it

text: has

text: no

text: contribution

text: from

text: the

text: wall,

text: and

text: would

text: be

text: present

text: if

text: vacuum

text: ﬁlled

text: all

text: space,

text: so

text: is

text: to

text: be

text: omitted.

text: It

text: is

text: the

text: assertion

text: of

text: Ref.

text: that

text: the

text: same

text: omission

text: is

text: to

text: be

text: done

text: for

text: the

text: contribution

text: to

text: the

text: stress

text: tensor

text: coming

text: from

text: the

text: part

text: of

text: the

text: Green’s

text: function

text: in

text: the

text: z

text: region

text: that

text: is

text: not

text: proportional

text: to

text: the

text: reﬂection

text: coeﬃcient:

text: In

text: particular,

text: they

text: advocate

text: omitting

text: the

text: stress

text: tensor

text: contribution

text: arising

text: from

text: the

text: term

text: in

text: the

text: Green’s

text: function

text: F(z

text: )G(z

text: even

text: though

text: it

text: is

text: spatially

text: varying,

text: because

text: this

text: term

text: would

text: be

text: there

text: in

text: the

text: absence

text: of

text: the

text: edge

text: at

text: α

text: z

text: This

text: hypothesis

text: may

text: be

text: suspect,

text: but

text: we

text: will

text: follow

text: it

text: for

text: the

text: moment.

text: V.

text: EXACTLY

text: SOLVABLE

text: EXAMPLES

text: Now

text: we

text: examine

text: two

text: cases

text: where

text: both

text: the

text: TE

text: and

text: TM

text: modes

text: may

text: be

text: explicitly

text: given.

text: In

text: the

text: ﬁrst

text: example,

text: the

text: permittivity

text: has

text: a

text: singularity

text: at

text: a

text: ﬁnite

text: value

text: of

text: z,

text: which

text: is

text: the

text: natural

text: boundary

text: of

text: the

text: problem,

text: and

text: in

text: the

text: second

text: the

text: permittivity

text: has

text: an

text: exponential

text: behavior.

text: A.

text: A

text: ﬁrst

text: example

text: Let

text: us

text: consider

text: a

text: planar

text: medium

text: described

text: by

text: λ

text: a

text: z

text: µ˜

text: ε˜(z)

text: (a

text: which

text: has

text: a

text: singularity

text: at

text: z

text: a.

text: (This

text: is

text: a

text: slightly

text: generalized

text: version

text: of

text: the

text: medium

text: considered

text: in

text: Ref.

text: where

text: the

text: potential

text: was

text: continuous,

text: so

text: Because

text: of

text: that

text: singularity,

text: the

text: right

text: side

text: of

text: the

text: wall

text: has

text: a

text: ﬁnite

text: depth,

text: z

text: a;

text: the

text: region

text: z

text: a

text: is

text: completely

text: disconnected

text: from

text: the

text: region

text: containing

text: the

text: wall.

text: This

text: potential

text: has

text: the

text: virtue

text: of

text: allowing

text: explicit

text: solutions:

text: FE,H

text: (a

text: z)

text: (k(a

text: z)),

text: GE,H

text: (a

text: z)

text: (k(a

text: z)),

text: ν

text: ν

text: where

text: ν

text: λζ2

text: λ.

text: αE,H

text: Here

text: F

text: is

text: chosen

text: to

text: be

text: ﬁnite

text: as

text: z

text: a.

text: Indeed,

text: consistent

text: with

text: the

text: criteria

text: stated

text: in

text: the

text: previous

text: section.

text: Then

text: the

text: reﬂection

text: coeﬃcients

text: in

text: the

text: medium

text: are

text: ka

text: K

text: (ka)

text: κa

text: K

text: (ka)

text: R˜E,H

text: ν

text: ν

text: ka

text: I

text: (ka)

text: κa

text: I

text: (ka)

text: ν

text: ν

text: The

text: scattering

text: part

text: zz

text: component

text: of

text: the

text: stress

text: tensor

text: (the

text: part

text: proportional

text: to

text: the

text: reﬂection

text: is

text: λζ2

text: ts,E,H(z)

text: R˜E,H

text: z)

text: z))

text: z)I

text: z))

text: kI

text: (k(a

text: z))I

text: (k(a

text: z))

text: zz

text: a

text: z

text: ν

text: ν

text: ν

text: ν

text: As

text: we

text: wish

text: to

text: examine

text: just

text: the

text: wall,

text: we

text: use

text: the

text: asymptotic

text: (UAE)

text: for

text: the

text: Bessel

text: functions,

text: because

text: it

text: captures

text: the

text: short-distance

text: behavior

text: That

text: expansion

text: is,

text: as

text: ν

text: Iν(νZ)

text: √21πν

text: 2πν

text: v

text: (t)

text: π

text: v

text: (t)

text: I

text: (νZ)

text: eνη(Z)

text: k

text: K

text: (νZ)

text: e

text: νη(Z)

text: k

text: ν

text: 2πν

text: Z

text: νk

text: ν

text: 2ν

text: Z

text: νk

text: where

text: u

text: and

text: v

text: are

text: polynomials

text: in

text: t

text: The

text: ﬁrst

text: of

text: these

text: are

text: k

text: k

text: u

text: (t)

text: v

text: (t)

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text: 10 For the moment we examine only the leading term in the limit of this expression as z → 0, which is κ˜ tb,E,H → − , (5.25) zz 2 the obvious generalization of Eq. (4.9). When thi(cid:90)s is integrated over all frequencies and wavenumbers, and regulated by point-splitting as in Ref. [28], we obtain (see Appendix ) ∞ 1 sin κ˜δ 1 Tb,E,H ∼ − √ dκ˜ κ˜3 = √ , δ → 0, (5.26) zz 4π2 (cid:15) κ˜δ 2π2 (cid:15)δ4 0 exactly the leading bulk divergence seen for each scalar mode in Ref. [28], apart from the expected index of refraction factor. Later we shall encounter the subleading divergences dependent on the potential; beyond them, in the exact tb,E,H there are ﬁnite terms that presumably have physical signiﬁcance. B. Exponential permittivity (cid:26) Let us give another exactly solvable model. Consider the permittivity function 1, z < 0, ε(z) = (5.27) eαz, z > 0. (cid:27) (cid:26) For the two modes, the two fundamental solutions to Eq. (4.1) are for z > 0 [38] FE(z) (cid:27) K ((cid:26)2ζeαz/2/α), = ν (5.28a) GE(z) I (2ζeαz/2/α), ν FH(z) K (2ζeαz/2/α), = eαz/2 ν˜ (5.28b) GH(z) I (2ζeαz/2/α), (cid:114)ν˜ where 2k 4k2 ν = , ν˜ = 1 + . (5.29) α α2 Again, the second solution is unique, according to the criteria enunciated in Sec. , because I (0) = 0. In each case, ν the eﬀective Wronskian (4.5) is the same, α αE,H = . (5.30) 2 (cid:112) Using the UAE, the leading bulk stress tensor component is (cid:114) κ˘ tb,E = − , κ˘ = k2 + ζ2eαz, (5.31a) zz 2 κ˘2 α2 tb,H = − , κˆ = k2 + ζ2eαz + . (5.31b) zz 2κˆ 4 The scattering part of the reduced stress tensor, near the wall, has(cid:16)the(cid:17)form seen before in Eq. (5.19) if we replace ζ2 by κ2 cos2 θ, and a by 2/α, where α is the slope of the potential at the edge: (cid:16) (cid:17) ts,E,H ∼ − α2 e−2κz  2ζκ22 2 , (5.32) zz 16κ 1 − ζ2 2 . 2κ2 From this follows the same result for the stress tensor as in Eq. (5.20). We will see in the following section that this behavior is universal, as long as the potential is continuous and has a linear slope at the edge. VI. UNIVERSAL EDGE BEHAVIOR A. First-order perturbation theory Griniasty and Leonhardt [24] asserted that the behavior of the zz component of the subtracted stress tensor seen in Eq. (5.20) is universal. That is, it holds whenever the potential is continuous, but has a discontinuous slope at ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: the

text: moment

text: we

text: examine

text: only

text: the

text: leading

text: term

text: in

text: the

text: limit

text: of

text: this

text: expression

text: as

text: z

text: which

text: is

text: κ˜

text: tb,E,H

text: zz

text: the

text: obvious

text: generalization

text: of

text: Eq.

text: When

text: is

text: integrated

text: over

text: all

text: frequencies

text: and

text: wavenumbers,

text: and

text: regulated

text: by

text: point-splitting

text: as

text: in

text: Ref.

text: we

text: obtain

text: (see

text: Appendix

text: sin

text: κ˜δ

text: Tb,E,H

text: dκ˜

text: κ˜3

text: δ

text: zz

text: κ˜δ

text: exactly

text: the

text: leading

text: bulk

text: divergence

text: seen

text: for

text: each

text: scalar

text: mode

text: in

text: Ref.

text: apart

text: from

text: the

text: expected

text: index

text: of

text: refraction

text: factor.

text: Later

text: we

text: shall

text: encounter

text: the

text: subleading

text: divergences

text: dependent

text: on

text: the

text: potential;

text: beyond

text: them,

text: in

text: the

text: exact

text: tb,E,H

text: there

text: are

text: ﬁnite

text: terms

text: that

text: presumably

text: have

text: physical

text: signiﬁcance.

text: B.

text: Exponential

text: permittivity

text: Let

text: us

text: give

text: another

text: exactly

text: solvable

text: model.

text: Consider

text: the

text: permittivity

text: function

text: z

text: ε(z)

text: eαz,

text: z

text: For

text: the

text: two

text: modes,

text: the

text: two

text: fundamental

text: solutions

text: to

text: Eq.

text: are

text: for

text: z

text: FE(z)

text: K

text: ν

text: GE(z)

text: I

text: ν

text: FH(z)

text: K

text: ν˜

text: GH(z)

text: I

text: where

text: ν

text: ν˜

text: α

text: α2

text: Again,

text: the

text: second

text: solution

text: is

text: unique,

text: according

text: to

text: the

text: criteria

text: enunciated

text: in

text: Sec.

text: because

text: I

text: In

text: each

text: case,

text: ν

text: the

text: eﬀective

text: Wronskian

text: is

text: the

text: same,

text: α

text: αE,H

text: Using

text: the

text: UAE,

text: the

text: leading

text: bulk

text: stress

text: tensor

text: component

text: is

text: κ˘

text: tb,E

text: κ˘

text: zz

text: κ˘2

text: α2

text: tb,H

text: κˆ

text: zz

text: 2κˆ

text: The

text: scattering

text: part

text: of

text: the

text: reduced

text: stress

text: tensor,

text: near

text: the

text: wall,

text: seen

text: before

text: in

text: Eq.

text: if

text: we

text: replace

text: ζ2

text: by

text: κ2

text: θ,

text: and

text: a

text: by

text: where

text: α

text: is

text: the

text: slope

text: of

text: the

text: potential

text: at

text: the

text: edge:

text: ts,E,H

text: α2

text: zz

text: ζ2

text: From

text: this

text: follows

text: the

text: same

text: result

text: for

text: the

text: stress

text: tensor

text: as

text: in

text: Eq.

text: We

text: will

text: see

text: in

text: the

text: following

text: section

text: that

text: this

text: behavior

text: is

text: universal,

text: as

text: long

text: as

text: the

text: potential

text: is

text: continuous

text: and

text: has

text: a

text: linear

text: slope

text: at

text: the

text: edge.

text: VI.

text: UNIVERSAL

text: EDGE

text: BEHAVIOR

text: A.

text: First-order

text: perturbation

text: theory

text: Griniasty

text: and

text: Leonhardt

text: asserted

text: that

text: the

text: behavior

text: of

text: the

text: zz

text: component

text: of

text: the

text: subtracted

text: stress

text: tensor

text: seen

text: in

text: Eq.

text: is

text: universal.

text: That

text: is,

text: it

text: holds

text: whenever

text: the

text: potential

text: is

text: continuous,

text: but

text: has

text: a

text: discontinuous

text: slope

text: at

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text: Proof: First, we note that if the following conditions are where in (39) we use the fact that the number of observed satisﬁed, the user must make the the same decision as that probing costs is lower bounded by KD(t) and Lemma 6, and under the optimal ofﬂine policy: 1) The estimated channel the last inequality follows when the conditions on L and D (cid:4) means θˆ are ranked the same as θ . 2) For each channel i, in (37)(38) are satisﬁed. i i the order of θˆ and the estimated thresholds uˆ and ˆl are the Now, we are ready to prove Theorem 2. Denote ∆ as i i i same as that of θ , u , and l . A sufﬁcient condition to have the maximum regret generated in a single frame. Since the i i i the ﬁrst condition hold is to have |θ − θˆ | < Γ , ∀i ∈ [K]. maximum net reward in each frame is bounded by b − p , i i 1 0 0 For the second condition, we note that if all channels are and the minimum possible reward is above −p − Kc , we 0 0 ranked correctly, and the estimation errors on cˆ , ˆb and pˆ have ∆ ≤ b + Kc . Therefore, 0 0 0 0 0 are sufﬁciently small, we can related |uˆi − ui| and |ˆli − li| (cid:88) (cid:88) with them according to Lemma 5. R(T) Speciﬁcally, for the general scenario where θ (cid:54)= u , l , i i i P according to Lemma 5, if |θi − θˆi| < (cid:15), we must have ≤ ∆ + [take sub-(cid:88)opti(cid:18)mal (cid:19)policy in frame t]∆ uˆi ∈ (ui−Ai(cid:15), ui+Ai(cid:15)). Thus, when (cid:15) is sufﬁciently small s.t. t∈T (t) t∈/T (t) (Ai+1)(cid:15) < |ui−θi|, the order of uˆi, θˆi will be the same as that (cid:0) T 6 (cid:1) of ui, θi. Similarly, we can show that whe(cid:110)n (cid:15) is sufﬁcient(cid:111)ly ≤ K(D(T) + 1)(b0 + Kc0) + t2 ∆ (41) small s.t. (Bi + 1)(cid:15) < |li − θi|, the order of li, θi and the t=1 order of ˆli, θˆi will be the same. Combining |tho−se|tw|o−ca|ses ≤ KL(b0 + Kc0) log T + π2 + DK + 1 (b0 + Kc0) ttohgeettrhaenrs,mwiettenrotweitlhlamt wakheenth|eθis−amθˆie| ≤senmsiinng anuAidi+tθ1rian, sBlmiii+sθs1iion, := C1 log T + C2 (cid:0) (cid:1) decision over channel i as in Theorem 1. where C := KL(b + Kc ), C := π2 + DK + 1 (b + For the scenario where θ = u (or l ), according to 1 0 0 2 0 i i i Kc ). Theorem 1, this implies that taking actions guess/sense (or 0 quit/sense) won’t make any difference in the expected reward. D. Proof of Theorem 3 Thus, we only need to make sure the order of θˆ and ˆl (or i i uˆ ) is unchanged in order to make sure the expected reward Denote x(t) as the status of channels observed in frame t. i is the same as that under the optimal policy in Theorem 1. It contains the status of channels sensed or guessed in frame Combining all conditions together, we note that if all t. Let f(x(t); θ) be the probability mass function (PMF) of estimation errors are bounded by min{Γ1, Γ2}, ∀i ∈ [K], the observation x(t) under θ in frame t. (cid:48) user is able to make the same decision as that under the optimal Consider another parameter setting θ = (cid:48) ofﬂine policy. Then, {θ1, θ2(cid:48), . . . , θk, . .(cid:48). , θK}. The on∗ly difference between θ and θ is θ (cid:54)= θ , where k ∈ {K + 1, . . . , K}. We assume (cid:48) k k (cid:48) P[u(cid:88)ser takes suboptimal policy in exploration frame t] 1wo>ulθdkﬁ>rsθt 1e.xTamhuisn,ewcihthantnheelnke(cid:48)w. Lpeatraum(cid:48)etbeer vtheectcoorrθre,spthoenduisnegr (cid:48) k K upper threshold that θ would be compared with. We assume P P (cid:48) (cid:48) k (cid:48) ≤ [|θi − θˆi| > Γ] + [|c0 − cˆ0| > Γ] θk > uk, i.e., the optimal ofﬂine policy with θ is to transmit i=1 over channel k without sensing (i.e., guess). Besides, we (cid:88)+ P[|b0 − ˆb0| > Γ] + P[|p0 − pˆ0| > Γ] assume (1+(cid:15))kl(θk, max{θ1, u(cid:48)k}) = kl(θk, θk(cid:48) ) where (cid:15) > 0. ≤ K P(cid:20)[|θ − θˆ | > Γ] +(cid:21)P[|c − cˆ | > Γ] Dtrueen.oTteh1enx, awseahnaivnedicator function, which equals one if x is i i 0 0 i=1 (cid:20) (cid:12) (cid:21) (cid:20) (cid:21) + P Tp(t) ≤ θ2K D(cid:12)(cid:12)(cid:12) (t) ff((xx((tt));;θθ(cid:48))) = 1Xk(t)∈x(t) XXk((tt))θθk(cid:48) ++ ((11 −− XXk((tt))))((11 −− θθk(cid:48) )) (cid:20) (cid:12) (cid:21) (cid:20) (cid:21) k k k k + P |b0 − ˆb0|>Γ(cid:12)(cid:12)Tp(t)> θK D(t) P Tp(t)> θK D(t) + 1Xk(t)∈/x(t) (42) (cid:12) 2 2 (cid:18) (cid:19) P θK P θK Let E , P be the expectation and probability measure + |p0 − pˆ0|>Γ Tp(t)> D(t) Tp(t)> D(t) θ θ 2 2 associated with θ. Then, (cid:18) (cid:19) 2(cid:18)KD(t)Γ2 (cid:19) (cid:89) ≤ 2K exp(−2D(t)Γ2) + 2 exp − ∆2 c P T f(x(t); θ) P + 2 exp(cid:18) −θKD(t)Γ2 +(cid:19)2 exp −θKD(t)Γ2 d θ = (cid:89) f(x(t); θ(cid:48))d θ(cid:48) ∆2 ∆2 t=1 b p + exp − θK2 (D2(Kt) − 1) (39) = t=T1 1(cid:16)Xk(t)∈x(t(cid:17)) XXkk((tt))θθkk(cid:48) ++ ((11 −− XXkk((tt))))((11 −− θθkk(cid:48) ))dPθ(cid:48) 5 P ≤ (40) = exp kˆl d (cid:48) (43) t2 Nˆk(T) θ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof:

text: First,

text: we

text: note

text: that

text: if

text: the

text: following

text: conditions

text: are

text: where

text: in

text: we

text: use

text: the

text: fact

text: that

text: the

text: number

text: of

text: observed

text: satisﬁed,

text: the

text: user

text: must

text: make

text: the

text: the

text: same

text: decision

text: as

text: that

text: probing

text: costs

text: is

text: lower

text: bounded

text: by

text: KD(t)

text: and

text: Lemma

text: and

text: under

text: the

text: optimal

text: ofﬂine

text: policy:

text: The

text: estimated

text: channel

text: the

text: last

text: inequality

text: follows

text: when

text: the

text: conditions

text: on

text: L

text: and

text: D

text: means

text: θˆ

text: are

text: ranked

text: the

text: same

text: as

text: θ

text: For

text: each

text: channel

text: i,

text: in

text: are

text: satisﬁed.

text: i

text: i

text: the

text: order

text: of

text: θˆ

text: and

text: the

text: estimated

text: thresholds

text: uˆ

text: and

text: ˆl

text: are

text: the

text: Now,

text: we

text: are

text: ready

text: to

text: prove

text: Theorem

text: Denote

text: as

text: i

text: i

text: i

text: same

text: as

text: that

text: of

text: θ

text: u

text: and

text: l

text: A

text: sufﬁcient

text: condition

text: to

text: have

text: the

text: maximum

text: regret

text: generated

text: in

text: a

text: single

text: frame.

text: Since

text: the

text: i

text: i

text: i

text: the

text: ﬁrst

text: condition

text: hold

text: is

text: to

text: have

text: |θ

text: θˆ

text: Γ

text: ∀i

text: [K].

text: maximum

text: net

text: reward

text: in

text: each

text: frame

text: is

text: bounded

text: by

text: b

text: p

text: i

text: i

text: For

text: the

text: second

text: condition,

text: we

text: note

text: that

text: if

text: all

text: channels

text: are

text: and

text: the

text: minimum

text: possible

text: reward

text: is

text: above

text: −p

text: Kc

text: we

text: ranked

text: correctly,

text: and

text: the

text: estimation

text: errors

text: on

text: cˆ

text: ˆb

text: and

text: pˆ

text: have

text: b

text: Kc

text: Therefore,

text: are

text: sufﬁciently

text: small,

text: we

text: can

text: related

text: |uˆi

text: ui|

text: and

text: |ˆli

text: li|

text: with

text: them

text: according

text: to

text: Lemma

text: R(T)

text: Speciﬁcally,

text: for

text: the

text: general

text: scenario

text: where

text: θ

text: u

text: l

text: i

text: i

text: i

text: P

text: according

text: to

text: Lemma

text: if

text: |θi

text: θˆi|

text: we

text: must

text: have

text: [take

text: in

text: frame

text: t]∆

text: uˆi

text: Thus,

text: when

text: is

text: sufﬁciently

text: small

text: s.t.

text: t∈T

text: (t)

text: t∈/T

text: (t)

text: |ui−θi|,

text: the

text: order

text: of

text: uˆi,

text: θˆi

text: will

text: be

text: the

text: same

text: as

text: that

text: T

text: of

text: ui,

text: θi.

text: Similarly,

text: we

text: can

text: show

text: that

text: is

text: K(D(T)

text: small

text: s.t.

text: (Bi

text: |li

text: θi|,

text: the

text: order

text: of

text: li,

text: θi

text: and

text: the

text: order

text: of

text: ˆli,

text: θˆi

text: will

text: be

text: the

text: same.

text: Combining

text: |tho−se|tw|o−ca|ses

text: log

text: T

text: π2

text: DK

text: ttohgeettrhaenrs,mwiettenrotweitlhlamt

text: wakheenth|eθis−amθˆie|

text: ≤senmsiinng

text: log

text: T

text: decision

text: over

text: channel

text: i

text: as

text: in

text: Theorem

text: where

text: C

text: KL(b

text: Kc

text: C

text: π2

text: DK

text: (b

text: For

text: the

text: scenario

text: where

text: θ

text: u

text: (or

text: l

text: according

text: to

text: i

text: i

text: i

text: Kc

text: Theorem

text: this

text: implies

text: that

text: taking

text: actions

text: guess/sense

text: (or

text: quit/sense)

text: won’t

text: make

text: any

text: difference

text: in

text: the

text: expected

text: reward.

text: D.

text: Proof

text: of

text: Theorem

text: Thus,

text: we

text: only

text: need

text: to

text: make

text: sure

text: the

text: order

text: of

text: θˆ

text: and

text: ˆl

text: (or

text: i

text: i

text: uˆ

text: is

text: unchanged

text: in

text: order

text: to

text: make

text: sure

text: the

text: expected

text: reward

text: Denote

text: x(t)

text: as

text: the

text: status

text: of

text: channels

text: observed

text: in

text: frame

text: t.

text: i

text: is

text: the

text: same

text: as

text: that

text: under

text: the

text: optimal

text: policy

text: in

text: Theorem

text: It

text: contains

text: the

text: status

text: of

text: channels

text: sensed

text: or

text: guessed

text: in

text: frame

text: Combining

text: all

text: conditions

text: together,

text: we

text: note

text: that

text: if

text: all

text: t.

text: Let

text: f(x(t);

text: θ)

text: be

text: the

text: probability

text: mass

text: function

text: (PMF)

text: of

text: estimation

text: errors

text: are

text: bounded

text: by

text: ∀i

text: [K],

text: the

text: observation

text: x(t)

text: under

text: θ

text: in

text: frame

text: t.

text: user

text: is

text: able

text: to

text: make

text: the

text: same

text: decision

text: as

text: that

text: under

text: the

text: optimal

text: Consider

text: another

text: parameter

text: setting

text: θ

text: ofﬂine

text: policy.

text: Then,

text: θk,

text: θK}.

text: The

text: on∗ly

text: difference

text: between

text: θ

text: and

text: θ

text: is

text: θ

text: θ

text: where

text: k

text: {K

text: K}.

text: We

text: assume

text: k

text: k

text: takes

text: suboptimal

text: policy

text: in

text: exploration

text: frame

text: t]

text: vtheectcoorrθre,spthoenduisnegr

text: k

text: K

text: upper

text: threshold

text: that

text: θ

text: would

text: be

text: compared

text: with.

text: We

text: assume

text: P

text: P

text: k

text: [|θi

text: θˆi|

text: Γ]

text: Γ]

text: θk

text: uk,

text: i.e.,

text: the

text: optimal

text: ofﬂine

text: policy

text: with

text: θ

text: is

text: to

text: transmit

text: over

text: channel

text: k

text: without

text: sensing

text: (i.e.,

text: guess).

text: Besides,

text: we

text: Γ]

text: Γ]

text: assume

text: kl(θk,

text: where

text: K

text: θˆ

text: Γ]

text: cˆ

text: Γ]

text: awseahnaivnedicator

text: function,

text: which

text: equals

text: one

text: if

text: x

text: is

text: i

text: i

text: P

text: Tp(t)

text: (t)

text: k

text: k

text: k

text: k

text: P

text: θK

text: D(t)

text: P

text: Tp(t)>

text: θK

text: D(t)

text: P

text: θK

text: P

text: θK

text: Let

text: E

text: P

text: be

text: the

text: expectation

text: and

text: probability

text: measure

text: Tp(t)>

text: D(t)

text: Tp(t)>

text: D(t)

text: θ

text: θ

text: associated

text: with

text: θ.

text: Then,

text: exp

text: ∆2

text: c

text: P

text: T

text: f(x(t);

text: θ)

text: P

text: exp

text: d

text: θ

text: f(x(t);

text: ∆2

text: ∆2

text: b

text: p

text: exp

text: P

text: exp

text: kˆl

text: d

text: Nˆk(T)

text: θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##