text: 4 TABLE I SHRINKAGE FUNCTIONS Function Partial Derivative La Cost of Shrinkage (Cs) || || (cid:60) (cid:61) ≈ f(a) = (cid:80)a 1 ∂∂fa(a∗[[ii]]) = sgn(a) = sgn( [a]) + jsgn( [a]) sgn[aopt]sgn[aHopt] 2Mad + 4Mmult + 2Mdiv f(a) = Mm=1 log(1 + |am|/(cid:15)) ∂∂fa(a∗[[ii]]) = sgn((cid:60)(cid:0)[1a+])(cid:15)+||jas|g|n1((cid:61)[a]) ≈ 1s+gn(cid:15)[|aaoopptt]| 1s+gn(cid:15)[|aaHoopptt]| 4Mad + 7Mmult + 3Mdiv (cid:60) (cid:1) | | ≤ f(a) = ||a||0 ∂f∂(aa∗mm[i[i]]) = βjsgsng(n(cid:61)([a[mam])])+− β2am if am 1/β ≈ β2sgn[aopt]sgn[aHopt] 3Mad + 6Mmult + 2Mdiv (cid:80) 0 elsewhere ≈ − − | | − Mm=1(1 e β am ) β3sgn[aopt]aHopt − β3aoptsgn[aHopt] +β4aoptaHopt where λ and τ are the regularization terms. In order to derive an adaptive algorithm to minimize the cost function in (4) and perform system identiﬁcation, we employ an alternating optimization strategy. We compute the instantaneous gradient of (4) with respect to p[i] and w[i] and devise LMS-type algorithms: ∂C(p[i], w[i]) p[i + 1] = p[i] − ηλ ∗ ∂p [i] (cid:124)(cid:123)(cid:122)(cid:125) (5) ∗ ∂f(p[i]) = p[i] + ηe(w[i], p[i])W [i]x [i] − ηλ ∗ , ∂p [i] α ∂C(p[i], w[i]) w[i + 1] = w[i] − µ ∗ ∂w [i] (cid:124)(cid:123)(cid:122)(cid:125) (6) ∗ ∂f(w[i]) = w[i] + µe(w[i], p[i]) P [i]x[i] − µτ ∗ , ∂w [i] γ where e(w[i], p[i]) = d[i] − wH[i]P [i]x[i] is the error signal and µ and η are the step sizes of the LMS recursions, which are used in an alternating way. In Table I, different shrinkage functions are shown with their partial derivatives and other features. A key requirement of the proposed scheme is the initialization which results in the adjustment of p[i] to shrink the coefﬁcients corresponding to zero elements of the system and w[i] to estimate the non-zero coefﬁcients. Speciﬁcally, p[i] is initialized as an all-one vector (p[0] = 1 or P [0] = I) and w[i] is initialized as an all-zero vector (w[0] = 0 ). When p[i] is ﬁxed, the scheme is equivalent to a standard shrinkage algorithm. The two-step approach outperforms the single-step method since P[i] strives to perform the role of the Oracle algorithm (P ) by decreasing the values of its entries in the positions of the zero coefﬁcients. This or helps the recursion that adapts w[i] to perform the estimation of the non-zero coefﬁcients. This process is then alternated over the iterations, resulting in better performance. When P is employed, w[i] has the information about the actual positions of the zero or coefﬁcients. B. Computational Complexity ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I

text: SHRINKAGE

text: FUNCTIONS

text: Function

label: table

text: Partial

text: Derivative

text: La

text: Cost

text: of

text: Shrinkage

text: (Cs)

text: f(a)

text: ∂∂fa(a∗[[ii]])

text: sgn(a)

text: sgn(

text: [a])

text: jsgn(

text: [a])

text: sgn[aopt]sgn[aHopt]

text: f(a)

text: ∂∂fa(a∗[[ii]])

text: f(a)

text: ∂f∂(aa∗mm[i[i]])

text: if

text: am

text: elsewhere

text: e

text: β

text: am

text: where

text: λ

text: and

text: τ

text: are

text: the

text: regularization

text: terms.

text: In

text: order

text: to

text: derive

text: an

text: adaptive

text: algorithm

text: to

text: minimize

text: the

text: cost

text: function

text: in

text: and

text: perform

text: system

text: identiﬁcation,

text: we

text: employ

text: an

text: alternating

text: optimization

text: strategy.

text: We

text: compute

text: the

text: instantaneous

text: gradient

text: of

text: with

text: respect

text: to

text: p[i]

text: and

text: w[i]

text: and

text: devise

text: LMS-type

text: algorithms:

text: ∂C(p[i],

text: w[i])

text: p[i

text: p[i]

text: ηλ

text: ∂p

text: [i]

text: ∂f(p[i])

text: p[i]

text: ηe(w[i],

text: p[i])W

text: [i]x

text: [i]

text: ηλ

text: ∂p

text: [i]

text: α

text: ∂C(p[i],

text: w[i])

text: w[i

text: w[i]

text: µ

text: ∂w

text: [i]

text: ∂f(w[i])

text: w[i]

text: µe(w[i],

text: p[i])

text: P

text: [i]x[i]

text: µτ

text: ∂w

text: [i]

text: γ

text: where

text: e(w[i],

text: p[i])

text: d[i]

text: wH[i]P

text: [i]x[i]

text: is

text: the

text: error

text: signal

text: and

text: µ

text: and

text: η

text: are

text: the

text: step

text: sizes

text: of

text: the

text: LMS

text: recursions,

text: which

text: are

text: used

text: in

text: an

text: alternating

text: way.

text: In

text: Table

text: I,

text: different

text: shrinkage

text: functions

text: are

text: shown

text: with

text: their

text: partial

text: derivatives

text: and

text: other

text: features.

text: A

text: key

text: requirement

text: of

text: the

text: proposed

text: scheme

text: is

text: the

text: initialization

text: which

text: results

text: in

text: the

text: adjustment

text: of

text: p[i]

text: to

text: shrink

text: the

text: coefﬁcients

text: corresponding

text: to

text: zero

text: elements

text: of

text: the

text: system

text: and

text: w[i]

text: to

text: estimate

text: the

text: non-zero

text: coefﬁcients.

text: Speciﬁcally,

text: p[i]

text: is

text: initialized

text: as

text: an

text: all-one

text: vector

text: or

text: P

text: I)

text: and

text: w[i]

text: is

text: initialized

text: as

text: an

text: all-zero

text: vector

text: When

text: p[i]

text: is

text: ﬁxed,

text: the

text: scheme

text: is

text: equivalent

text: to

text: a

text: standard

text: shrinkage

text: algorithm.

text: The

text: two-step

text: approach

text: outperforms

text: the

text: single-step

text: method

text: since

text: P[i]

text: strives

text: to

text: perform

text: the

text: role

text: of

text: the

text: Oracle

text: algorithm

text: (P

text: by

text: decreasing

text: the

text: values

text: of

text: its

text: entries

text: in

text: the

text: positions

text: of

text: the

text: zero

text: coefﬁcients.

text: This

text: or

text: helps

text: the

text: recursion

text: that

text: adapts

text: w[i]

text: to

text: perform

text: the

text: estimation

text: of

text: the

text: non-zero

text: coefﬁcients.

text: This

text: process

text: is

text: then

text: alternated

text: over

text: the

text: iterations,

text: resulting

text: in

text: better

text: performance.

text: When

text: P

text: is

text: employed,

text: w[i]

text: has

text: the

text: information

text: about

text: the

text: actual

text: positions

text: of

text: the

text: zero

text: or

text: coefﬁcients.

text: B.

text: Computational

text: Complexity

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: 6 MSE = J + tr[R (K (cid:12) K )] min x w p (11) + tr[R (R (cid:12) K )] + tr[R (R (cid:12) K )], x wo p x or w where (cid:12) is the Hadamard product, R = E[x[i]xH[i]], K = E[e [i]eH[i]], K = E[e [i]eH[i]], R = E[w wH], and x w w w p p p wo o o R = E[p pH]. Using (5) and (6) into K and K , we obtain or o o p w K [i + 1] = (I − µR )K [i](I − µR ) w px w px (12) + µ2R J(i) (w ) + γ2L , px MSE o w K [i + 1] = (I − ηR )K [i](I − ηR ) p wx p wx (13) + η2R J(i) (p ) + α2L , wx MSE o p (cid:44) (cid:44) where J(i) (w ) E[|e(w , p[i])|2] and J(i) (p ) E[|e(w[i], p )|2] appear in (12) and (13). The other quantities are MSE o o MSE o o (cid:48) (cid:48) (cid:48) (cid:48) R = E[W [i]x[i]xH[i]W H[i]], L = E[f [w[i]]f H[w[i]], R = E[P [i]x[i]xH[i]P H[i]], L = E[f [p[i]]f H[p[i]] and wx w px p (cid:48) f (·) is the partial derivative with respect to the variable of the argument. In Table I, we use the variable a that plays the role of (cid:48) (cid:48) p[i] or w[i]. We obtained approximations for L = E[f [a[i]]f H[a[i]], where a is a generic function, to compute the matrices L a p and L for a given shrinkage function as shown in the 3rd column of Table I. w We can express R and R as R = R (cid:12) R [i]] and R = R (cid:12) R [i], where R = E[w[i]wH[i]] and R = wx px wx x w px x p w p E[p[i]pH[i]]. To simplify the analysis, we assume that the samples of the signal x[i] are uncorrelated, i.e., R = σ2I with σ2 being x x x the variance. Using the diagonal matrices R = Λ = σ2I, R = Λ [i] = σ2I (cid:12) R [i] and R = Λ [i] = σ2I (cid:12) R [i], x x x px px x p wx wx x w we can write K [i + 1] = (I − µΛ [i])K [i](I − µΛ [i]) w px w px (14) + µ2J(i) (w )Λ [i] + γ2L [i] MSE o px w K [i + 1] = (I − ηΛ [i])K [i](I − ηΛ [i]) p wx p wx (15) + η2J(i) (p )Λ [i] + α2L [i] MSE o wx p Due to the structure of the above equations, the approximations and the quantities involved, we can decouple them into Kn[i + 1] = (1 − µλn [i])Kn[i](1 − µλn [i]) w px w px (16) + µ2J(i) (w )λn [i] + γ2Ln [i] MSE o px w

label: paragraph

text: MSE

label: equation

text: J

text: tr[R

text: (K

text: K

text: min

text: x

text: w

text: p

text: tr[R

text: (R

text: K

text: tr[R

text: (R

text: K

text: x

text: wo

text: p

text: x

text: or

text: w

text: where

text: is

text: the

text: Hadamard

text: product,

text: R

text: E[x[i]xH[i]],

text: K

text: E[e

text: [i]eH[i]],

text: K

text: E[e

text: [i]eH[i]],

text: R

text: E[w

text: wH],

text: and

text: x

text: w

text: w

text: w

text: p

text: p

text: p

text: wo

text: o

text: o

text: R

text: E[p

text: pH].

text: Using

text: and

text: into

text: K

text: and

text: K

text: we

text: obtain

text: or

text: o

text: o

text: p

text: w

text: K

text: [i

text: (I

text: µR

text: )K

text: [i](I

text: µR

text: w

text: px

text: w

text: px

text: J(i)

text: (w

text: px

text: MSE

text: o

text: w

text: K

text: [i

text: (I

text: ηR

text: )K

text: [i](I

text: ηR

text: p

text: wx

text: p

text: wx

text: J(i)

text: (p

text: wx

text: MSE

text: o

text: p

text: where

text: J(i)

text: (w

text: E[|e(w

text: and

text: J(i)

text: (p

text: E[|e(w[i],

text: p

text: appear

text: in

text: and

text: The

text: other

text: quantities

text: are

text: MSE

text: o

text: o

text: MSE

text: o

text: o

text: R

text: E[W

text: [i]x[i]xH[i]W

text: H[i]],

text: L

text: E[f

text: [w[i]]f

text: H[w[i]],

text: R

text: E[P

text: [i]x[i]xH[i]P

text: H[i]],

text: L

text: E[f

text: [p[i]]f

text: H[p[i]]

text: and

text: wx

text: w

text: px

text: p

text: f

text: is

text: the

text: partial

text: derivative

text: with

text: respect

text: to

text: the

text: variable

text: of

text: the

text: argument.

text: In

text: Table

text: I,

text: we

text: use

text: the

text: variable

text: a

text: that

text: plays

text: the

text: role

text: of

text: p[i]

text: or

text: w[i].

text: We

text: obtained

text: approximations

text: for

text: L

text: E[f

text: [a[i]]f

text: H[a[i]],

text: where

text: a

text: is

text: a

text: generic

text: function,

text: to

text: compute

text: the

text: matrices

text: L

text: a

text: p

text: and

text: L

text: for

text: a

text: given

text: shrinkage

text: function

text: as

text: shown

text: in

text: the

text: column

text: of

text: Table

text: I.

text: w

text: We

text: can

text: express

text: R

text: and

text: R

text: as

text: R

text: R

text: R

text: [i]]

text: and

text: R

text: R

text: R

text: [i],

text: where

text: R

text: E[w[i]wH[i]]

text: and

text: R

text: wx

text: px

text: wx

text: x

text: w

text: px

text: x

text: p

text: w

text: p

text: E[p[i]pH[i]].

text: To

text: simplify

text: the

text: analysis,

text: we

text: assume

text: that

text: the

text: samples

text: of

text: the

text: signal

text: x[i]

text: are

text: uncorrelated,

text: i.e.,

text: R

text: with

text: σ2

text: being

text: x

text: x

text: x

text: the

text: variance.

text: Using

text: the

text: diagonal

text: matrices

text: R

text: Λ

text: R

text: Λ

text: [i]

text: R

text: [i]

text: and

text: R

text: Λ

text: [i]

text: R

text: [i],

text: x

text: x

text: x

text: px

text: px

text: x

text: p

text: wx

text: wx

text: x

text: w

text: we

text: can

text: write

text: K

text: [i

text: (I

text: µΛ

text: [i])K

text: [i](I

text: µΛ

text: [i])

text: w

text: px

text: w

text: px

text: (w

text: )Λ

text: [i]

text: [i]

text: MSE

text: o

text: px

text: w

text: K

text: [i

text: (I

text: ηΛ

text: [i])K

text: [i](I

text: ηΛ

text: [i])

text: p

text: wx

text: p

text: wx

text: (p

text: )Λ

text: [i]

text: [i]

text: MSE

text: o

text: wx

text: p

text: Due

text: to

text: the

text: structure

text: of

text: the

text: above

text: equations,

text: the

text: approximations

text: and

text: the

text: quantities

text: involved,

text: we

text: can

text: decouple

text: them

text: into

text: Kn[i

text: µλn

text: µλn

text: [i])

text: w

text: px

text: w

text: px

text: (w

text: )λn

text: [i]

text: [i]

text: MSE

text: o

text: px

text: w


text: MAPLE.2 A 3-dimensional plot of the function Qual (δ, ω) is shown in Figure 3 (left). In packing Figure 3 (middle and right) we highlight speciﬁc slices through the 3-dimensional plot to better explain the behavior. Figure 3 (middle) shows Qual (δ, ω) for three diﬀerent values of ω. We packing can observe that the FCC lattice (δ = 2) is indeed optimal for small values of allowed overlap ω. When ω = 0.1, the BCC (δ = 0.5) and the FCC lattice achieve approximately the same density, namely density = 1.03. Interestingly, for larger values of ω the BCC lattice attains the maximal density and surpasses the FCC lattice. Also observe that both lattices always achieve a better relaxed packing quality than the integer lattice (δ = 1). Looking at the density of the FCC and the BCC lattice depending on ω in Figure 3 (right), we can note that there is indeed only one switch ≈ of optimality (at ω 0.1). Our analysis indicates that the F CC and the BCC lattice are always locally optimal conﬁg- urations, and no other lattice from the family yields a better packing, independent of the allowed overlap. The natural next step would be to prove our observation. This problem can in theory be tackled with the same approach that we used in Section in the 2D case by relating the derivative of Qual (δ, ω) to the partial derivatives of vol_overlap using the implicit function theorem. For packing small values of ω, we were able to verify the claim, that is, prove monotonicity of the function in all branches with a substantial amount of symbolic computations. However, as soon as the expression for vol_overlap involves intersections of 2 and 3 spherical caps, the derivatives seem to become too complicated to be handled analytically. 5 Discussion This work has analyzed the problem of densest sphere packings while allowing some overlap among the spheres. We see our contributions as a ﬁrst step towards an interesting and important research direction, given the numerous applications of spheres with overlap in the natural sciences. For example, our analysis of the distance-based overlap measure showing that the FCC lattice is optimal independent of the amount of overlap, and hence independent of the scaling of the spheres, lays the theoretical foundations for [12], i.e., for analyzing the spatial organization of chromosomes in the cell nucleus as a sphere arrangement. A major restriction of our approach is our focus on a one-dimensional sub-lattice, the diagonally distorted lattices. Can we hope for an analysis of more general lattice families? This question should probably ﬁrst be considered in 2D, given the extremely involved proof of optimality already for the classical packing problem in 3D. References [1] R. P. Bambah: “On lattice coverings by spheres”. Proceedings of the National Institute of Sciences of India 20 (1954) 25–52. [2] J. Conway, N. Sloane: Sphere Packings, Lattices and Groups. Springer, 3rd edn., 1999. [3] T. Cremer, M. Cremer: “Chromosome territories”. Cold Spring Harbor Perspectives in Biology (2010). [4] S. H. DeVries, D. A. Baylor: “Mosaic arrangement of ganglion cell receptive ﬁelds in rabbit retina”. Journal of Neurophysiology 78 (1997) 2048–2060. 2http://www.maplesoft.com 11 ##LTLine##

text: A

text: plot

text: of

text: the

text: function

text: Qual

text: (δ,

text: ω)

text: is

text: shown

text: in

text: Figure

text: (left).

text: In

text: packing

text: Figure

text: (middle

text: and

text: right)

text: we

text: highlight

text: speciﬁc

text: slices

text: through

text: the

text: plot

text: to

text: better

text: explain

text: the

text: behavior.

text: Figure

text: (middle)

text: shows

text: Qual

text: (δ,

text: ω)

text: for

text: three

text: diﬀerent

text: values

text: of

text: ω.

text: We

text: packing

text: can

text: observe

text: that

text: the

text: FCC

text: lattice

text: (δ

text: is

text: indeed

text: optimal

text: for

text: small

text: values

text: of

text: allowed

text: overlap

text: ω.

text: When

text: ω

text: the

text: BCC

text: (δ

text: and

text: the

text: FCC

text: lattice

text: achieve

text: approximately

text: the

text: same

text: density,

text: namely

text: density

text: Interestingly,

text: for

text: larger

text: values

text: of

text: ω

text: the

text: BCC

text: lattice

text: attains

text: the

text: maximal

text: density

text: and

text: surpasses

text: the

text: FCC

text: lattice.

text: Also

text: observe

text: that

text: both

text: lattices

text: always

text: achieve

text: a

text: better

text: relaxed

text: packing

text: quality

text: than

text: the

text: integer

text: lattice

text: (δ

text: Looking

text: at

text: the

text: density

text: of

text: the

text: FCC

text: and

text: the

text: BCC

text: lattice

text: depending

text: on

text: ω

text: in

text: Figure

text: (right),

text: we

text: can

text: note

text: that

text: there

text: is

text: indeed

text: only

text: one

text: switch

text: of

text: optimality

text: (at

text: ω

text: Our

text: analysis

text: indicates

text: that

text: the

text: F

text: CC

text: and

text: the

text: BCC

text: lattice

text: are

text: always

text: locally

text: optimal

text: conﬁg-

text: urations,

text: and

text: no

text: other

text: lattice

text: from

text: the

text: family

text: yields

text: a

text: better

text: packing,

text: independent

text: of

text: the

text: allowed

text: overlap.

text: The

text: natural

text: next

text: step

text: would

text: be

text: to

text: prove

text: our

text: observation.

text: This

text: problem

text: can

text: in

text: theory

text: be

text: tackled

text: with

text: the

text: same

text: approach

text: that

text: we

text: used

text: in

text: Section

text: in

text: the

text: case

text: by

text: relating

text: the

text: derivative

text: of

text: Qual

text: (δ,

text: ω)

text: to

text: the

text: partial

text: derivatives

text: of

text: using

text: the

text: implicit

text: function

text: theorem.

text: For

text: packing

text: small

text: values

text: of

text: ω,

text: we

text: were

text: able

text: to

text: verify

text: the

text: claim,

text: that

text: is,

text: prove

text: monotonicity

text: of

text: the

text: function

text: in

text: all

text: branches

text: with

text: a

text: substantial

text: amount

text: of

text: symbolic

text: computations.

text: However,

text: as

text: soon

text: as

text: the

text: expression

text: for

text: involves

text: intersections

text: of

text: and

text: spherical

text: caps,

text: the

text: derivatives

text: seem

text: to

text: become

text: too

text: complicated

text: to

text: be

text: handled

text: analytically.

text: Discussion

text: This

text: work

text: has

text: analyzed

text: the

text: problem

text: of

text: densest

text: sphere

text: packings

text: while

text: allowing

text: some

text: overlap

text: among

text: the

text: spheres.

text: We

text: see

text: our

text: contributions

text: as

text: a

text: ﬁrst

text: step

text: towards

text: an

text: interesting

text: and

text: important

text: research

text: direction,

text: given

text: the

text: numerous

text: applications

text: of

text: spheres

text: with

text: overlap

text: in

text: the

text: natural

text: sciences.

text: For

text: example,

text: our

text: analysis

text: of

text: the

text: distance-based

text: overlap

text: measure

text: showing

text: that

text: the

text: FCC

text: lattice

text: is

text: optimal

text: independent

text: of

text: the

text: amount

text: of

text: overlap,

text: and

text: hence

text: independent

text: of

text: the

text: scaling

text: of

text: the

text: spheres,

text: lays

text: the

text: theoretical

text: foundations

text: for

text: i.e.,

text: for

text: analyzing

text: the

text: spatial

text: organization

text: of

text: chromosomes

text: in

text: the

text: cell

text: nucleus

text: as

text: a

text: sphere

text: arrangement.

text: A

text: major

text: restriction

text: of

text: our

text: approach

text: is

text: our

text: focus

text: on

text: a

text: one-dimensional

text: sub-lattice,

text: the

text: diagonally

text: distorted

text: lattices.

text: Can

text: we

text: hope

text: for

text: an

text: analysis

text: of

text: more

text: general

text: lattice

text: families?

text: This

text: question

text: should

text: probably

text: ﬁrst

text: be

text: considered

text: in

text: given

text: the

text: extremely

text: involved

text: proof

text: of

text: optimality

text: already

text: for

text: the

text: classical

text: packing

text: problem

text: in

text: References

text: R.

text: P.

text: Bambah:

text: “On

text: lattice

text: coverings

text: by

text: spheres”.

text: Proceedings

text: of

text: the

text: National

text: Institute

text: of

text: Sciences

text: of

text: India

text: J.

text: Conway,

text: N.

text: Sloane:

text: Sphere

text: Packings,

text: Lattices

text: and

text: Groups.

text: Springer,

text: edn.,

text: T.

text: Cremer,

text: M.

text: Cremer:

text: “Chromosome

text: territories”.

text: Cold

text: Spring

text: Harbor

text: Perspectives

text: in

text: Biology

text: S.

text: H.

text: DeVries,

text: D.

text: A.

text: Baylor:

text: “Mosaic

text: arrangement

text: of

text: ganglion

text: cell

text: receptive

text: ﬁelds

text: in

text: rabbit

text: retina”.

text: Journal

text: of

text: Neurophysiology

text: ##LTLine##


text: Figure 2: The Voronoi cell V for two diﬀerent values of δ < 1 (left and middle) and δ > 1 (right). On the left, the bisectors of type 2 are hit ﬁrst, whereas in the middle bisectors of type 1 are hit ﬁrst. Note that all lattice points neighboring the origin lie on a common circle around the origin. 4 Volume-based overlap In this section we analyze Qual for the volume-based overlap measure in dimension 2 and packing 3. As we have already pointed out in Lemma 2, we cannot expect the same behavior as for the distance-based overlap measure discussed in the previous section. However, the lemma only states that the FCC lattice, which is optimal for ω = 0 is worse than the BCC lattice for some value of ω. This does not rule out the possibility of other lattices being optimal. This section will perform a deeper investigation of the optimal lattice conﬁgurations, starting with the two-dimensional case. 4.1 Dimension 2 First of all, note that in dimension 2, the lattice for δ is a scaled version of the lattice for 1. ≤ δ Because of this symmetry, it suﬃces to study all lattices with 0 < δ 1. For analyzing the volume-based overlap measure, we ﬁrst derive a formula for the volume of ∩ V B . This requires the investigation of the Voronoi cell V in some detail. V is bounded by r ± ± six bisectors: four of them with the lattice points e(δ), e(δ), and two with the lattice points ± 1 2 (e(δ) + e(δ)). See Figure 2 for an illustration. We call the bisectors of type 1 and type 2, 1 2 respectively. Their distances to the origin are given by r and r , respectively, with √ 1 2 δ2√+ 1 √δ (cid:113) r := , r := . 1 2 2 2 2 Note that r > r if and only if δ < 1. 1 2 3 There are six boundary vertices of V and they all have the same distance to the origin, namely δ2√+ 1 r := , 3 2 2 ≥ { } which agrees with the covering radius computed in [6] (see also (11)). As expected, r max r , r , 3 1 2 with equality if and only if δ = 1. ∩ ≤ { } With this data we can derive a formula for the volume of V B : If r min r , r , B is r ≥ 1 2 r completely contained in V and the volume equals the volume of B . If r r , B contains all r 3 r boundary vertices of V and thus all of V (of volume δ), by convexity. In the last case where { } min r , r < r < r , the part of B that is not in V is the union of up to six circular segments. 1 2 3 r Their area is given by r2 − A = (Θ sin Θ), 2 7 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

label: caption

text: Voronoi

text: cell

text: V

text: for

text: two

text: diﬀerent

text: values

text: of

text: δ

text: (left

text: and

text: middle)

text: and

text: δ

text: (right).

text: On

text: the

text: left,

text: the

text: bisectors

text: of

text: type

text: are

text: hit

text: ﬁrst,

text: whereas

text: in

text: the

text: middle

text: bisectors

text: of

text: type

text: are

text: hit

text: ﬁrst.

text: Note

text: that

text: all

text: lattice

text: points

text: neighboring

text: the

text: origin

text: lie

text: on

text: a

text: common

text: circle

text: around

text: the

text: origin.

label: section

text: Volume-based

text: overlap

text: In

text: this

text: section

text: we

text: analyze

text: Qual

text: for

text: the

text: volume-based

text: overlap

text: measure

text: in

text: dimension

text: and

text: packing

text: As

text: we

text: have

text: already

text: pointed

text: out

text: in

text: Lemma

text: we

text: cannot

text: expect

text: the

text: same

text: behavior

text: as

text: for

text: the

text: distance-based

text: overlap

text: measure

text: discussed

text: in

text: the

text: previous

text: section.

text: However,

text: the

text: lemma

text: only

text: states

text: that

text: the

text: FCC

text: lattice,

text: which

text: is

text: optimal

text: for

text: ω

text: is

text: worse

text: than

text: the

text: BCC

text: lattice

text: for

text: some

text: value

text: of

text: ω.

text: This

text: does

text: not

text: rule

text: out

text: the

text: possibility

text: of

text: other

text: lattices

text: being

text: optimal.

text: This

text: section

text: will

text: perform

text: a

text: deeper

text: investigation

text: of

text: the

text: optimal

text: lattice

text: conﬁgurations,

text: starting

text: with

text: the

text: two-dimensional

text: case.

text: Dimension

text: First

text: of

text: all,

text: note

text: that

text: in

text: dimension

text: the

text: lattice

text: for

text: δ

text: is

text: a

text: scaled

text: version

text: of

text: the

text: lattice

text: for

text: δ

text: Because

text: of

text: this

text: symmetry,

text: it

text: suﬃces

text: to

text: study

text: all

text: lattices

text: with

text: δ

text: For

text: analyzing

text: the

text: volume-based

text: overlap

text: measure,

text: we

text: ﬁrst

text: derive

text: a

text: formula

text: for

text: the

text: volume

text: of

text: V

text: B

text: This

text: requires

text: the

text: investigation

text: of

text: the

text: Voronoi

text: cell

text: V

text: in

text: some

text: detail.

text: V

text: is

text: bounded

text: by

text: r

text: six

text: bisectors:

text: four

text: of

text: them

text: with

text: the

text: lattice

text: points

text: e(δ),

text: e(δ),

text: and

text: two

text: with

text: the

text: lattice

text: points

text: (e(δ)

text: e(δ)).

text: See

text: Figure

text: for

text: an

text: illustration.

text: We

text: call

text: the

text: bisectors

text: of

text: type

text: and

text: type

text: respectively.

text: Their

text: distances

text: to

text: the

text: origin

text: are

text: given

text: by

text: r

text: and

text: r

text: respectively,

text: with

text: δ2√+

text: √δ

text: r

text: r

text: Note

text: that

text: r

text: r

text: if

text: and

text: only

text: if

text: δ

text: There

text: are

text: six

text: boundary

text: vertices

text: of

text: V

text: and

text: they

text: all

text: have

text: the

text: same

text: distance

text: to

text: the

text: origin,

text: namely

text: δ2√+

text: r

text: which

text: agrees

text: with

text: the

text: covering

text: radius

text: computed

text: in

text: (see

text: also

text: As

text: expected,

text: r

text: max

text: r

text: r

text: with

text: equality

text: if

text: and

text: only

text: if

text: δ

text: With

text: this

text: data

text: we

text: can

text: derive

text: a

text: formula

text: for

text: the

text: volume

text: of

text: V

text: B

text: If

text: r

text: min

text: r

text: r

text: B

text: is

text: r

text: r

text: completely

text: contained

text: in

text: V

text: and

text: the

text: volume

text: equals

text: the

text: volume

text: of

text: B

text: If

text: r

text: r

text: B

text: contains

text: all

text: r

text: r

text: boundary

text: vertices

text: of

text: V

text: and

text: thus

text: all

text: of

text: V

text: (of

text: volume

text: δ),

text: by

text: convexity.

text: In

text: the

text: last

text: case

text: where

text: min

text: r

text: r

text: r

text: r

text: the

text: part

text: of

text: B

text: that

text: is

text: not

text: in

text: V

text: is

text: the

text: union

text: of

text: up

text: to

text: six

text: circular

text: segments.

text: r

text: Their

text: area

text: is

text: given

text: by

text: A

text: (Θ

text: sin

text: Θ),

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: which we want to maximize w.r.t. δ. This requires computing the derivative of r(δ, ω) w.r.t. δ. We do this by using the implicit function theorem ∂r − ∂F (δ, ω, r) (δ, ω) = ∂δ . ∂δ ∂F (δ, ω, r) ∂r  After some calculations we ﬁnd  √ (cid:16) (cid:17)  π 2r2−δ2 r ≤ r ≤ r , 0 < δ < √1 , 2δ arccos √√δ (cid:16) (cid:17) 2 1 3 ∂ density(δ, r(δ, ω))  π√(δ2−1)r 82r2−δ2−1 r ≤ r ≤ r , √1 < δ < 1, = 8δ√2 δ2+1(cid:18)arccos δ(cid:113)2+1 √ √(cid:16) (cid:17)(cid:19) 1 2 3 ∂δ − − − 8r2 − ≤ ≤ √8r2 δ2 1(δ2 1)+2δ 2r2 δ2 δ2+1 max(r , r ) r r , 0 < δ < 1. 1 2 3 4r δ2+1 2arccos δ2+1+arccos √δ 8r2 r 2 One can easily check that the ﬁrst derivative is non-negative for any δ, except if r equals the packing radius r corresponding to ω = 0, and we know the optimal packing for this case. Similarly, 2 the second derivative is non-positive except if r equals the packing radius r . The third derivative is 1 ≥ zero either if δ = √1 or if r equals the covering radius r corresponding to ω vol_overlap(δ, r ), 3 3 3 in which case the hexagonal lattice is optimal as we argued above. Moreover, for r < r , the 3 derivative is increasing for δ < √1 and decreasing for δ > √1 . This concludes the proof. 3 3 4.2 Dimension 3 In three dimensions, the symmetry between δ and 1 is lost, and we need to analyze both ≤ δ ∞ branches. We ﬁrst discuss the case 0 < δ 1: Imagine that r increases from 0 to . Initially, the ∩ volume of the intersection V B equals the volume of the ball. When increasing the ball radius r r, there are three possibilities: (i) We hit a bisector plane. From now on we have to subtract a spherical cap from the volume. There are a total of 14 bisector planes of three diﬀerent types. Their distance to the origin and number of occurrences are: (cid:114) (cid:114) √ δ2 + 2 2δ2 + 1 δ 3 r := (6 planes), r := (6 planes), r := (2 planes). 1 2 3 12 6 2 (ii) We hit a boundary edge of V , where two bisector planes are meeting. From now on, we have to add to the volume the intersection of the two spherical caps involved (because they are counted twice). There are a total of 36 trisector edges of two diﬀerent types. Their distance to the origin, number of occurrences, and types of bisector planes between the 3 involved (cid:112) spheres are: δ2√+ 2 (δ2 + 2√)(2δ2 + 1) r := (18 edges of type 1-1-2), r := (18 edges of type 1-2-3). 4 5 3 2 2 3 However, note that the volume of the cap intersection depends on the type of the bisector plane between the two spheres that are not centered at the origin. We get 5 diﬀerent subtypes, four of them appearing 6 times, and one appearing 12 times in the polytope. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: we

text: want

text: to

text: maximize

text: w.r.t.

text: δ.

text: This

text: requires

text: computing

text: the

text: derivative

text: of

text: r(δ,

text: ω)

text: w.r.t.

text: δ.

text: We

text: do

text: this

text: by

text: using

text: the

text: implicit

text: function

text: theorem

text: ∂r

text: ∂F

text: (δ,

text: ω,

text: r)

text: (δ,

text: ω)

text: ∂δ

text: ∂δ

text: ∂F

text: (δ,

text: ω,

text: r)

text: ∂r

text: After

text: some

text: calculations

text: we

text: ﬁnd

text: π

text: r

text: r

text: r

text: δ

text: √1

text: 2δ

text: arccos

text: √√δ

text: density(δ,

text: r(δ,

text: ω))

text: r

text: r

text: r

text: √1

text: δ

text: 8δ√2

text: ∂δ

text: δ2

text: δ2

text: max(r

text: r

text: r

text: r

text: δ

text: √δ

text: r

text: One

text: can

text: easily

text: check

text: that

text: the

text: ﬁrst

text: derivative

text: is

text: non-negative

text: for

text: any

text: δ,

text: except

text: if

text: r

text: equals

text: the

text: packing

text: radius

text: r

text: corresponding

text: to

text: ω

text: and

text: we

text: know

text: the

text: optimal

text: packing

text: for

text: this

text: case.

text: Similarly,

text: the

text: second

text: derivative

text: is

text: non-positive

text: except

text: if

text: r

text: equals

text: the

text: packing

text: radius

text: r

text: The

text: third

text: derivative

text: is

text: zero

text: either

text: if

text: δ

text: √1

text: or

text: if

text: r

text: equals

text: the

text: covering

text: radius

text: r

text: corresponding

text: to

text: ω

text: r

text: in

text: which

text: case

text: the

text: hexagonal

text: lattice

text: is

text: optimal

text: as

text: we

text: argued

text: above.

text: Moreover,

text: for

text: r

text: r

text: the

text: derivative

text: is

text: increasing

text: for

text: δ

text: √1

text: and

text: decreasing

text: for

text: δ

text: √1

text: This

text: concludes

text: the

text: proof.

text: Dimension

text: In

text: three

text: dimensions,

text: the

text: symmetry

text: between

text: δ

text: and

text: is

text: lost,

text: and

text: we

text: need

text: to

text: analyze

text: both

text: δ

text: branches.

text: We

text: ﬁrst

text: discuss

text: the

text: case

text: δ

text: Imagine

text: that

text: r

text: increases

text: from

text: to

text: Initially,

text: the

text: volume

text: of

text: the

text: intersection

text: V

text: B

text: equals

text: the

text: volume

text: of

text: the

text: ball.

text: When

text: increasing

text: the

text: ball

text: radius

text: r

text: r,

text: there

text: are

text: three

text: possibilities:

text: (i)

text: We

text: hit

text: a

text: bisector

text: plane.

text: From

text: now

text: on

text: we

text: have

text: to

text: subtract

text: a

text: spherical

text: cap

text: from

text: the

text: volume.

text: There

text: are

text: a

text: total

text: of

text: bisector

text: planes

text: of

text: three

text: diﬀerent

text: types.

text: Their

text: distance

text: to

text: the

text: origin

text: and

text: number

text: of

text: occurrences

text: are:

text: δ2

text: δ

text: r

text: planes),

text: r

text: planes),

text: r

text: planes).

text: (ii)

text: We

text: hit

text: a

text: boundary

text: edge

text: of

text: V

text: where

text: two

text: bisector

text: planes

text: are

text: meeting.

text: From

text: now

text: on,

text: we

text: have

text: to

text: add

text: to

text: the

text: volume

text: the

text: intersection

text: of

text: the

text: two

text: spherical

text: caps

text: involved

text: (because

text: they

text: are

text: counted

text: twice).

text: There

text: are

text: a

text: total

text: of

text: trisector

text: edges

text: of

text: two

text: diﬀerent

text: types.

text: Their

text: distance

text: to

text: the

text: origin,

text: number

text: of

text: occurrences,

text: and

text: types

text: of

text: bisector

text: planes

text: between

text: the

text: involved

text: spheres

text: are:

text: δ2√+

text: r

text: edges

text: of

text: type

text: r

text: edges

text: of

text: type

text: However,

text: note

text: that

text: the

text: volume

text: of

text: the

text: cap

text: intersection

text: depends

text: on

text: the

text: type

text: of

text: the

text: bisector

text: plane

text: between

text: the

text: two

text: spheres

text: that

text: are

text: not

text: centered

text: at

text: the

text: origin.

text: We

text: get

text: diﬀerent

text: subtypes,

text: four

text: of

text: them

text: appearing

text: times,

text: and

text: one

text: appearing

text: times

text: in

text: the

text: polytope.

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text: ##LTLine##

text: ##LTLine##


text: Hybrid Quantum-Classical Master Equations 3 Table 2. Reduced classical and quantum states. Conditional classical state. Missing conditional quantum state. (cid:80) Reduced Q Reduced C Conditional C Conditional Q (cid:54) ∃ ρˆ= ρˆ(x) ρ(x)=Tr ρˆ(x) ρˆ =ρˆ(x)/ρ(x) x x Having outlined the abstract features, let’s see an occasional list of very diﬀerent hybrid systems with their respective hybrid densities. In molecular physics ρˆ refers to quantized electrons and (r, p) stands for classical nuclear positions and momenta yielding ρˆ(r, p) [3]. In quantum optics ρˆ refers to quantized electrons and a, a(cid:63) stand for the complex amplitudes of classical e.m. ﬁeld modes yielding ρˆ(a, a(cid:63)) [7]. In nanophysics ρˆ refers to the quantum dot and n stands for the charge count yielding ρˆ(n). Last but not least, in quantum measurement ρˆ refers to the measured quantum system and x stands for the measurement outcome yielding ρˆ(x) [12], as discussed below. 3. Measurement What happens to the quantum state ρˆ of a quantum system, under measurement { } ˆ of the complete set of orthogonal projectors P ? We argue that hybrid formalism x and interpretation are exact alternatives to the standard ones in textbooks. Textbook formalism says that the pre-measurement state ρˆ jumps randomly to the post- measurement conditional quantum state ρˆ : x −→ 1 ˆ ˆ ρˆ ρˆ = P ρˆP (8) x x x ρ(x) ˆ ˆ with probability ρ(x) = Tr(P ρˆP ). x x In hybrid formalism, we say that ρˆ jumps deterministically into the post- measurement hybrid state: −→ ˆ ˆ ρˆ ρˆ(x) = P ρˆP . (9) x x The randomness of the outcome x is now expressed through the statistical interpretation (7), as well as (4-6), of the hybrid state. (cid:80) In complete generality, hybrid formalism is convenient for general (unsharp) ˆ ˆ quantum measurements deﬁned by Kraus operators M instead of projectors P , x x † ˆ ˆ ˆ satisfying completeness M M = I but no orthogonality or hermiticity. In hybrid x x x formalism, measurement is fully represented by the jump of the pre-measurement quantum state ρˆ into the post-measurement hybrid state: −→ † ˆ ˆ ρˆ ρˆ(x) = M ρˆM . (10) x x The statistical interpretation of ρˆ(x) reproduces the common rules (8) of measurement. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hybrid

label: title

text: Quantum-Classical

text: Master

text: Equations

text: Table

text: Reduced

text: classical

text: and

text: quantum

text: states.

text: Conditional

text: classical

text: state.

text: Missing

text: conditional

text: quantum

text: state.

text: Reduced

text: Q

text: Reduced

text: C

text: Conditional

text: C

text: Conditional

text: Q

font: default

text: ρˆ=

text: ρˆ(x)

text: ρ(x)=Tr

text: ρˆ(x)

text: ρˆ

text: =ρˆ(x)/ρ(x)

text: x

text: x

text: Having

text: outlined

text: the

text: abstract

text: features,

text: let’s

text: see

text: an

text: occasional

text: list

text: of

text: very

text: diﬀerent

text: hybrid

text: systems

text: with

text: their

text: respective

text: hybrid

text: densities.

text: In

text: molecular

text: physics

text: ρˆ

text: refers

text: to

text: quantized

text: electrons

text: and

text: (r,

text: p)

text: stands

text: for

text: classical

text: nuclear

text: positions

text: and

text: momenta

text: yielding

text: ρˆ(r,

text: p)

text: In

text: quantum

text: optics

text: ρˆ

text: refers

text: to

text: quantized

text: electrons

text: and

text: a,

text: stand

text: for

text: the

text: complex

text: amplitudes

text: of

text: classical

text: e.m.

text: ﬁeld

text: modes

text: yielding

text: ρˆ(a,

text: In

text: nanophysics

text: ρˆ

text: refers

text: to

text: the

text: quantum

text: dot

text: and

text: n

text: stands

text: for

text: the

text: charge

text: count

text: yielding

text: ρˆ(n).

text: Last

text: but

text: not

text: least,

text: in

text: quantum

text: measurement

text: ρˆ

text: refers

text: to

text: the

text: measured

text: quantum

text: system

text: and

text: x

text: stands

text: for

text: the

text: measurement

text: outcome

text: yielding

text: ρˆ(x)

text: as

text: discussed

text: below.

text: Measurement

text: What

text: happens

text: to

text: the

text: quantum

text: state

text: ρˆ

text: of

text: a

text: quantum

text: system,

text: under

text: measurement

text: ˆ

text: of

text: the

text: complete

text: set

text: of

text: orthogonal

text: projectors

text: P

text: We

text: argue

text: that

text: hybrid

text: formalism

text: x

text: and

text: interpretation

text: are

text: exact

text: alternatives

text: to

text: the

text: standard

text: ones

text: in

text: textbooks.

text: Textbook

text: formalism

text: says

text: that

text: the

text: pre-measurement

text: state

text: ρˆ

text: jumps

text: randomly

text: to

text: the

text: post-

text: measurement

text: conditional

text: quantum

text: state

text: ρˆ

text: x

text: ˆ

text: ˆ

text: ρˆ

text: ρˆ

text: P

text: ρˆP

text: x

text: x

text: x

text: ρ(x)

text: ˆ

text: ˆ

text: with

text: probability

text: ρ(x)

text: Tr(P

text: ρˆP

text: x

text: x

text: In

text: hybrid

text: formalism,

text: we

text: say

text: that

text: ρˆ

text: jumps

text: deterministically

text: into

text: the

text: post-

text: measurement

text: hybrid

text: state:

text: ˆ

text: ˆ

text: ρˆ

text: ρˆ(x)

text: P

text: ρˆP

text: x

text: x

text: The

text: randomness

text: of

text: the

text: outcome

text: x

text: is

text: now

text: expressed

text: through

text: the

text: statistical

text: interpretation

text: as

text: well

text: as

text: of

text: the

text: hybrid

text: state.

text: In

text: complete

text: generality,

text: hybrid

text: formalism

text: is

text: convenient

text: for

text: general

text: (unsharp)

text: ˆ

text: ˆ

text: quantum

text: measurements

text: deﬁned

text: by

text: Kraus

text: operators

text: M

text: instead

text: of

text: projectors

text: P

text: x

text: x

text: ˆ

text: ˆ

text: ˆ

text: satisfying

text: completeness

text: M

text: M

text: I

text: but

text: no

text: orthogonality

text: or

text: hermiticity.

text: In

text: hybrid

text: x

text: x

text: x

text: formalism,

text: measurement

text: is

text: fully

text: represented

text: by

text: the

text: jump

text: of

text: the

text: pre-measurement

text: quantum

text: state

text: ρˆ

text: into

text: the

text: post-measurement

text: hybrid

text: state:

text: ˆ

text: ˆ

text: ρˆ

text: ρˆ(x)

text: M

text: ρˆM

text: x

text: x

text: The

text: statistical

text: interpretation

text: of

text: ρˆ(x)

text: reproduces

text: the

text: common

text: rules

text: of

text: measurement.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Hybrid Quantum-Classical Master Equations 7 hybrid ME of monitoring is the following non-autonomous Fokker-Planck equation for the reduced classical density: dρ(X) − (cid:104) (cid:105) = ∂ qˆ ρ(X)+D∂2 ρ(X) , (27) dt X X X (cid:104) (cid:105) where qˆ = Tr(qˆρˆ ) is the conditional expectation value of qˆ. This equation X X expresses the diﬀusive noise superposed on the measured signal X, at diﬀusion coeﬃcient D. The hybrid ME (26) is equivalent with the standard theory of quantum monitoring [19] which prescribes two coupled Ito stochastic diﬀerential equations for the conditional state ρˆ and the measured signal X, respectively, instead of the hybrid ME for ρˆ(X). X The proof of equivalence of the two Ito equations with the hybrid ME is straightforward, will be shown elsewhere. 7. Summary, outlook We gave a short introduction into the concept of hybrid systems. We emphasized that quantum measurement has a natural hybrid formalism. A novel general structure of hybrid ME has been proposed, to unify the Pauli and Lindblad structures. An application to quantum continuous measurement (monitoring) has been shown. Ad hoc hybrid theories are often falling short. Certain ad hoc MEs violate the positivity of the hybrid density. This failure is abandoned by our class of hybrid MEs. Certain ad hoc hybrid theories don’t pass the ”free will test” [24]. The measurement- related theories, where the classical variable is the measured outcome, do pass it. Nonetheless, the consistency of hybrid theories is being under discussion, there can be a number of further consistency tests [25]. The present work is a deliberate outline of certain important features of hybrid systems, with an emphasis on quantum measurement. Some novel results, including the general structure of hybrid ME and the hybrid ME of quantum monitoring, will be detailed and further clariﬁed in forthcoming works [26] Acknowledgments Support by the Hungarian Scientiﬁc Research Fund under Grant No. 75129 and support by the EU COST Action MP1006 are acknowledged. The author is grateful to Thomas Konrad and Sandeep Goyal for many useful discussions. References [1] Imre K, O¨zizmir E, Rosenbaum M and Zwiefel P F 1967 J. Math. Phys. 5 1097 [2] Sherry T N and Sudarshan E C G 1978 Phys. Rev. D 18 4580 [3] Aleksandrov I V 1981 Z. Naturf. A 36 902 [4] Boucher W and Traschen J, 1988 Phys. Rev. D 37 3522 [5] Di´osi L 1995 Quantum dynamics with two Planck constants and the semiclassical limit Preprint quant-ph/9503023 ##LTLine##

text: Hybrid

text: Quantum-Classical

text: Master

text: Equations

text: hybrid

text: ME

text: of

text: monitoring

text: is

text: the

text: following

text: non-autonomous

text: Fokker-Planck

text: equation

text: for

text: the

text: reduced

text: classical

text: density:

text: dρ(X)

text: qˆ

text: ρ(X)

text: dt

text: X

text: X

text: X

text: where

text: qˆ

text: Tr(qˆρˆ

text: is

text: the

text: conditional

text: expectation

text: value

text: of

text: qˆ.

text: This

text: equation

text: X

text: X

text: expresses

text: the

text: diﬀusive

text: noise

text: superposed

text: on

text: the

text: measured

text: signal

text: X,

text: at

text: diﬀusion

text: coeﬃcient

text: D.

text: The

text: hybrid

text: ME

text: is

text: equivalent

text: with

text: the

text: standard

text: theory

text: of

text: quantum

text: monitoring

text: which

text: prescribes

text: two

text: coupled

text: Ito

text: stochastic

text: diﬀerential

text: equations

text: for

text: the

text: conditional

text: state

text: ρˆ

text: and

text: the

text: measured

text: signal

text: X,

text: respectively,

text: instead

text: of

text: the

text: hybrid

text: ME

text: for

text: ρˆ(X).

text: X

text: The

text: proof

text: of

text: equivalence

text: of

text: the

text: two

text: Ito

text: equations

text: with

text: the

text: hybrid

text: ME

text: is

text: straightforward,

text: will

text: be

text: shown

text: elsewhere.

text: Summary,

text: outlook

text: We

text: gave

text: a

text: short

text: introduction

text: into

text: the

text: concept

text: of

text: hybrid

text: systems.

text: We

text: emphasized

text: that

text: quantum

text: measurement

text: has

text: a

text: natural

text: hybrid

text: formalism.

text: A

text: novel

text: general

text: structure

text: of

text: hybrid

text: ME

text: has

text: been

text: proposed,

text: to

text: unify

text: the

text: Pauli

text: and

text: Lindblad

text: structures.

text: An

text: application

text: to

text: quantum

text: continuous

text: measurement

text: (monitoring)

text: has

text: been

text: shown.

text: Ad

text: hoc

text: hybrid

text: theories

text: are

text: often

text: falling

text: short.

text: Certain

text: ad

text: hoc

text: MEs

text: violate

text: the

text: positivity

text: of

text: the

text: hybrid

text: density.

text: This

text: failure

text: is

text: abandoned

text: by

text: our

text: class

text: of

text: hybrid

text: MEs.

text: Certain

text: ad

text: hoc

text: hybrid

text: theories

text: don’t

text: pass

text: the

text: ”free

text: will

text: test”

text: The

text: measurement-

text: related

text: theories,

text: where

text: the

text: classical

text: variable

text: is

text: the

text: measured

text: outcome,

text: do

text: pass

text: it.

text: Nonetheless,

text: the

text: consistency

text: of

text: hybrid

text: theories

text: is

text: being

text: under

text: discussion,

text: there

text: can

text: be

text: a

text: number

text: of

text: further

text: consistency

text: tests

text: The

text: present

text: work

text: is

text: a

text: deliberate

text: outline

text: of

text: certain

text: important

text: features

text: of

text: hybrid

text: systems,

text: with

text: an

text: emphasis

text: on

text: quantum

text: measurement.

text: Some

text: novel

text: results,

text: including

text: the

text: general

text: structure

text: of

text: hybrid

text: ME

text: and

text: the

text: hybrid

text: ME

text: of

text: quantum

text: monitoring,

text: will

text: be

text: detailed

text: and

text: further

text: clariﬁed

text: in

text: forthcoming

text: works

text: Acknowledgments

text: Support

text: by

text: the

text: Hungarian

text: Scientiﬁc

text: Research

text: Fund

text: under

text: Grant

text: No.

text: and

text: support

text: by

text: the

text: EU

text: COST

text: Action

text: are

text: acknowledged.

text: The

text: author

text: is

text: grateful

text: to

text: Thomas

text: Konrad

text: and

text: Sandeep

text: Goyal

text: for

text: many

text: useful

text: discussions.

text: References

text: Imre

text: K,

text: O¨zizmir

text: E,

text: Rosenbaum

text: M

text: and

text: Zwiefel

text: P

text: F

text: J.

text: Math.

text: Phys.

text: Sherry

text: T

text: N

text: and

text: Sudarshan

text: E

text: C

text: G

text: Phys.

text: Rev.

text: D

text: Aleksandrov

text: I

text: V

text: Z.

text: Naturf.

text: A

text: Boucher

text: W

text: and

text: Traschen

text: J,

text: Phys.

text: Rev.

text: D

text: Di´osi

text: L

text: Quantum

text: dynamics

text: with

text: two

text: Planck

text: constants

text: and

text: the

text: semiclassical

text: limit

text: Preprint

text: ##LTLine##


text: On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues ∗ ∗ † ‡ E. Kokabifar, G.B. Loghmani, A.M. Nazari, and S.M. Karbassi May 26, 2020 Abstract × Consider an n n matrix polynomial P(λ). A spectral norm distance from P(λ) to × ∈ C the set of n n matrix polynomials that have a given scalar µ as a multiple eigen- value was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of P(λ). In this paper, we extend this result to the case of two given distinct complex numbers µ and µ . First, we compute a lower bound for the spectral norm distance 1 2 from P(λ) to the set of matrix polynomials that have µ , µ as two eigenvalues. Then 1 2 we construct an associated perturbation of P(λ), such that the perturbed matrix poly- nomial has two given scalars µ and µ in its spectrum. Finally, we derive an upper 1 2 bound for the distance by the constructed perturbation of P(λ). Numerical examples are provided to illustrate the validity of the method. Keywords: Matrix polynomial, Eigenvalue, Perturbation, Singular value. AMS Classiﬁcation: 5A18, 65F35. 1 Introduction and preliminaries × × Let A be an n n complex matrix and let L be the set of complex n n matri- ces with a multiple zero eigenvalue. In 1999, Malyshev [8] obtained a formula for the spectral norm distance from A to L which can be considered as a theoretical solution to ∈ C × Wilkinson’s problem, that is, the calculation of the distance from a matrix A n n that × has all its eigenvalues simple to the n n matrices with multiple eigenvalues. Wilkinson ∗ Department of Mathematics, Faculty of Science, Yazd University, Yazd, Iran (e.kokabifar@stu.yazd.ac.ir, loghmani@yazd.ac.ir). † Department of Mathematics, Faculty of Science, Arak University, Arak, Iran (a-nazari@araku.ac.ir). ‡ Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Iran (mehdikar- bassi@gmail.com). 1 ##LTLine##

text: On

text: the

text: distance

text: from

text: a

text: matrix

text: polynomial

text: to

text: matrix

text: polynomials

text: with

text: two

text: prescribed

text: eigenvalues

label: author

text: E.

text: Kokabifar,

text: G.B.

text: Loghmani,

text: A.M.

text: Nazari,

text: and

text: S.M.

text: Karbassi

text: May

text: Abstract

label: abstract

text: Consider

text: an

text: n

text: n

text: matrix

text: polynomial

text: P(λ).

text: A

text: spectral

text: norm

text: distance

text: from

text: P(λ)

text: to

text: C

text: the

text: set

text: of

text: n

text: n

text: matrix

text: polynomials

text: that

text: have

text: a

text: given

text: scalar

text: µ

text: as

text: a

text: multiple

text: eigen-

text: value

text: was

text: introduced

text: and

text: obtained

text: by

text: Papathanasiou

text: and

text: Psarrakos.

text: They

text: computed

text: lower

text: and

text: upper

text: bounds

text: for

text: this

text: distance,

text: constructing

text: an

text: associated

text: perturbation

text: of

text: P(λ).

text: In

text: this

text: paper,

text: we

text: extend

text: this

text: result

text: to

text: the

text: case

text: of

text: two

text: given

text: distinct

text: complex

text: numbers

text: µ

text: and

text: µ

text: First,

text: we

text: compute

text: a

text: lower

text: bound

text: for

text: the

text: spectral

text: norm

text: distance

text: from

text: P(λ)

text: to

text: the

text: set

text: of

text: matrix

text: polynomials

text: that

text: have

text: µ

text: µ

text: as

text: two

text: eigenvalues.

text: Then

text: we

text: construct

text: an

text: associated

text: perturbation

text: of

text: P(λ),

text: such

text: that

text: the

text: perturbed

text: matrix

text: poly-

text: nomial

text: has

text: two

text: given

text: scalars

text: µ

text: and

text: µ

text: in

text: its

text: spectrum.

text: Finally,

text: we

text: derive

text: an

text: upper

text: bound

text: for

text: the

text: distance

text: by

text: the

text: constructed

text: perturbation

text: of

text: P(λ).

text: Numerical

text: examples

text: are

text: provided

text: to

text: illustrate

text: the

text: validity

text: of

text: the

text: method.

text: Keywords:

text: Matrix

text: polynomial,

text: Eigenvalue,

text: Perturbation,

text: Singular

text: value.

text: AMS

text: Classiﬁcation:

text: Introduction

text: and

text: preliminaries

text: Let

text: A

text: be

text: an

text: n

text: n

text: complex

text: matrix

text: and

text: let

text: L

text: be

text: the

text: set

text: of

text: complex

text: n

text: n

text: matri-

text: ces

text: with

text: a

text: multiple

text: zero

text: eigenvalue.

text: In

text: Malyshev

text: obtained

text: a

text: formula

text: for

text: the

text: spectral

text: norm

text: distance

text: from

text: A

text: to

text: L

text: which

text: can

text: be

text: considered

text: as

text: a

text: theoretical

text: solution

text: to

text: C

text: Wilkinson’s

text: problem,

text: that

text: is,

text: the

text: calculation

text: of

text: the

text: distance

text: from

text: a

text: matrix

text: A

text: n

text: n

text: that

text: has

text: all

text: its

text: eigenvalues

text: simple

text: to

text: the

text: n

text: n

text: matrices

text: with

text: multiple

text: eigenvalues.

text: Wilkinson

text: Department

text: of

text: Mathematics,

text: Faculty

text: of

text: Science,

text: Yazd

text: University,

text: Yazd,

text: Iran

text: (e.kokabifar@stu.yazd.ac.ir,

text: loghmani@yazd.ac.ir).

text: Department

text: of

text: Mathematics,

text: Faculty

text: of

text: Science,

text: Arak

text: University,

text: Arak,

text: Iran

text: (a-nazari@araku.ac.ir).

text: Department

text: of

text: Mathematics,

text: Yazd

text: Branch,

text: Islamic

text: Azad

text: University,

text: Yazd,

text: Iran

text: (mehdikar-

text: bassi@gmail.com).

text: ##LTLine##


text: Theorem 3.2. Let P (λ) be the matrix polynomial as in (1) and let µ and µ be two 1 2 given distinct complex numbers. Then for any γ > 0, ≤ D β (P, µ , µ , γ) (P, µ , µ ), low 1 2 1 2 where β (P, µ , µ , γ) is introduced in (15). In addition, if γ∗ > 0, then the matrix poly- low 1 2 nomial Qγ∗(λ) in (13) has µ1 and µ2 as two its eigenvalues co∈rresBponding to v1(γ∗) and vˆ(γ∗)Das two its ei≤genvectors, respectively. Furthermore, Qγ∗(λ) ∂ (P, βup(P, µ1, µ2, γ∗), w) and (P, µ , µ ) β (P, µ , µ , γ∗), where β (P, µ , µ , γ∗) is introduced in (16). 1 2 up 1 2 up 1 2 It should be pointed out that the bounds obtained are not necessarily optimal, how- D ever, it is assured that (P, µ , µ ) belongs to [β (P, µ , µ , γ∗), β (P, µ , µ , γ∗)]. Any- 1 2 low 1 2 up 1 2 how, the following remark can be used to obtain some close bounds. Remark 3.3. It is important to note that, Theorem 3.2 holds for any γ > 0 that assures 0 the matrix Vˆ (γ ) = is full (column) rank. Therefore, it can be an obvious expectation to 0 ﬁnd a value of γ > 0 that obtains the closest upper and lower bounds. For doing this, we can deﬁne the following nonnegative function − f(γ) = β (P, µ , µ , γ) β (P, , µ , µ , γ), up 1 2 low 1 2 and try to minimize this function by implementation of unconstrained optimization meth- ods (for example, see [10]). On the other hand, best lower bound and ﬁnest upper bound can be obtained by maximizing and minimizing β (P, µ , µ , γ) and β (P, µ , µ , γ), re- low 1 2 up 1 2 spectively. It is clear that values of γ which yield the smallest upper bound and the biggest lower bound may be diﬀerent. Now suppose that the singular value s − (F [P (µ , µ ); γ]) attains its maximum 2n 1 1 2 D ≤ value at γ = 0, i.e., γ∗ = 0. Next we compute an upper bound for (P, µ , µ ) 1 2 β (P, µ , µ , γ∗), constructing associated perturbations of P (λ). up 1 2 ∈ C Let u , v n, (i = 1, 2) be a pair of left and right singular vectors of P (µ ) cor- i i i responding to σ = s (P (µ )), (i = 1, 2), respectively, such that v and v are linearly i n i 1 2 independent. We deﬁne the matrix(cid:20) polynom(cid:21)ial ∆0(λ) as (cid:2) (cid:3) (cid:2) (cid:3) − σ 0 † ∆ (λ) = ∆ = u u 1 v v , (17) 0 (cid:2) 0 (cid:3) 1 2 0 σ 1 2 (cid:2) (cid:3) 2 † where v v is the Moore-Penrose pseudoinverse of v v . Then, 1 2 1 2 − Q (λ) = P (λ) + ∆ (λ) = A λm + A − λm 1 + ... + A λ + (A + ∆ ) , (18) 0 0 m m 1 1 0 0 11

text: Theorem

text: Let

text: P

text: (λ)

text: be

text: the

text: matrix

text: polynomial

text: as

text: in

text: and

text: let

text: µ

text: and

text: µ

text: be

text: two

text: given

text: distinct

text: complex

text: numbers.

text: Then

text: for

text: any

text: γ

text: D

text: β

text: (P,

text: µ

text: µ

text: γ)

text: (P,

text: µ

text: µ

text: low

text: where

text: β

text: (P,

text: µ

text: µ

text: γ)

text: is

text: introduced

text: in

text: In

text: addition,

text: if

text: γ∗

text: then

text: the

text: matrix

text: poly-

text: low

text: nomial

text: Qγ∗(λ)

text: in

text: has

text: µ1

text: and

text: µ2

text: as

text: two

text: its

text: eigenvalues

text: co∈rresBponding

text: to

text: and

text: vˆ(γ∗)Das

text: two

text: its

text: ei≤genvectors,

text: respectively.

text: Furthermore,

text: Qγ∗(λ)

text: (P,

text: βup(P,

text: γ∗),

text: w)

text: and

text: (P,

text: µ

text: µ

text: β

text: (P,

text: µ

text: µ

text: γ∗),

text: where

text: β

text: (P,

text: µ

text: µ

text: γ∗)

text: is

text: introduced

text: in

text: up

text: up

text: It

text: should

text: be

text: pointed

text: out

text: that

text: the

text: bounds

text: obtained

text: are

text: not

text: necessarily

text: optimal,

text: how-

text: D

text: ever,

text: it

text: is

text: assured

text: that

text: (P,

text: µ

text: µ

text: belongs

text: to

text: [β

text: (P,

text: µ

text: µ

text: γ∗),

text: β

text: (P,

text: µ

text: µ

text: γ∗)].

text: Any-

text: low

text: up

text: how,

text: the

text: following

text: remark

text: can

text: be

text: used

text: to

text: obtain

text: some

text: close

text: bounds.

text: Remark

text: It

text: is

text: important

text: to

text: note

text: that,

text: Theorem

text: holds

text: for

text: any

text: γ

text: that

text: assures

text: the

text: matrix

text: Vˆ

text: (γ

text: is

text: full

text: (column)

text: rank.

text: Therefore,

text: it

text: can

text: be

text: an

text: obvious

text: expectation

text: to

text: ﬁnd

text: a

text: value

text: of

text: γ

text: that

text: obtains

text: the

text: closest

text: upper

text: and

text: lower

text: bounds.

text: For

text: doing

text: this,

text: we

text: can

text: deﬁne

text: the

text: following

text: nonnegative

text: function

text: f(γ)

text: β

text: (P,

text: µ

text: µ

text: γ)

text: β

text: (P,

text: µ

text: µ

text: γ),

text: up

text: low

text: and

text: try

text: to

text: minimize

text: this

text: function

text: by

text: implementation

text: of

text: unconstrained

text: optimization

text: meth-

text: ods

text: (for

text: example,

text: see

text: On

text: the

text: other

text: hand,

text: best

text: lower

text: bound

text: and

text: ﬁnest

text: upper

text: bound

text: can

text: be

text: obtained

text: by

text: maximizing

text: and

text: minimizing

text: β

text: (P,

text: µ

text: µ

text: γ)

text: and

text: β

text: (P,

text: µ

text: µ

text: γ),

text: re-

text: low

text: up

text: spectively.

text: It

text: is

text: clear

text: that

text: values

text: of

text: γ

text: which

text: yield

text: the

text: smallest

text: upper

text: bound

text: and

text: the

text: biggest

text: lower

text: bound

text: may

text: be

text: diﬀerent.

text: Now

text: suppose

text: that

text: the

text: singular

text: value

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: attains

text: its

text: maximum

text: D

text: value

text: at

text: γ

text: i.e.,

text: γ∗

text: Next

text: we

text: compute

text: an

text: upper

text: bound

text: for

text: (P,

text: µ

text: µ

text: β

text: (P,

text: µ

text: µ

text: γ∗),

text: constructing

text: associated

text: perturbations

text: of

text: P

text: (λ).

text: up

text: C

text: Let

text: u

text: v

text: n,

text: (i

text: be

text: a

text: pair

text: of

text: left

text: and

text: right

text: singular

text: vectors

text: of

text: P

text: (µ

text: cor-

text: i

text: i

text: i

text: responding

text: to

text: σ

text: s

text: (P

text: (µ

text: (i

text: respectively,

text: such

text: that

text: v

text: and

text: v

text: are

text: linearly

text: i

text: n

text: i

text: independent.

text: We

text: deﬁne

text: the

text: as

text: σ

text: (λ)

text: u

text: u

text: v

text: v

text: σ

text: where

text: v

text: v

text: is

text: the

text: Moore-Penrose

text: pseudoinverse

text: of

text: v

text: v

text: Then,

text: Q

text: (λ)

text: P

text: (λ)

text: (λ)

text: A

text: λm

text: A

text: λm

text: A

text: λ

text: (A

text: m

text: m


text: [2] J.W. Demmel, On condition numbers and the distance to the nearest ill-posed problem, Numer. Math., 51 (1987), 251–289. [3] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [4] T. Kaczorek, Polynomial and Rational Matrices: Applications in Dynamical Systems Theory, Springer-Verlag, London, 2007. [5] P. Lancaster, Lambda-Matrices and Vibrating Systems, Dover Publications, 2002. [6] J.M. Gracia, Nearest matrix with two prescribed eigenvalues, Linear Algebra and its Appli- cations, 401 (2005), 277—294. [7] R.A. Lippert, Fixing two eigenvalues by a minimal perturbation, Linear Algebra Appl. 406 (2005) 177-–200. [8] A.N. Malyshev, A formula for the 2-norm distance from a matrix to the set of matrices with a multiple eigenvalue, Numer. Math. 83 (1999) 443-–454. [9] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math. Society, Providence, RI, Translations of Mathematical Monographs, Vol. 71, 1988. [10] J. Nocedal, S.J. Wright, Numerical Optimization, second edition, Springer Series in Operation Research and Financial Engineering, 2006. [11] N. Papathanasiou, P. Psarrakos, The distance from a matrix polynomial to matrix polyno- mials with a prescribed multiple eigenvalue, Linear Algebra and its Applications, 429 (2008), 1453—1477. [12] A. Ruhe, Properties of a matrix with a very ill-conditioned eigenproblem, Numer. Math., 15 (1970), 57–60. [13] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965. [14] J.H. Wilkinson, Note on matrices with a very ill-conditioned eigenproblem, Numer. Math., 19 (1972), 175–178. [15] J.H. Wilkinson, On neighbouring matrices with quadratic elementary divisors, Numer. Math., 44 (1984), 1–21. [16] J.H. Wilkinson, Sensitivity of eigenvalues, Util. Math., 25 (1984), 5–76. [17] J.H. Wilkinson, Sensitivity of eigenvalues II, Util. Math., 30 (1986), 243–286. 16

label: reference

text: J.W.

text: Demmel,

text: On

text: condition

text: numbers

text: and

text: the

text: distance

text: to

text: the

text: nearest

text: ill-posed

text: problem,

text: Numer.

text: Math.,

text: I.

text: Gohberg,

text: P.

text: Lancaster

text: and

text: L.

text: Rodman,

text: Matrix

text: Polynomials,

text: Academic

text: Press,

text: New

text: York,

text: T.

text: Kaczorek,

text: Polynomial

text: and

text: Rational

text: Matrices:

text: Applications

text: in

text: Dynamical

text: Systems

text: Theory,

text: Springer-Verlag,

text: London,

text: P.

text: Lancaster,

text: Lambda-Matrices

text: and

text: Vibrating

text: Systems,

text: Dover

text: Publications,

text: J.M.

text: Gracia,

text: Nearest

text: matrix

text: with

text: two

text: prescribed

text: eigenvalues,

text: Linear

text: Algebra

text: and

text: its

text: Appli-

text: cations,

text: R.A.

text: Lippert,

text: Fixing

text: two

text: eigenvalues

text: by

text: a

text: minimal

text: perturbation,

text: Linear

text: Algebra

text: Appl.

text: A.N.

text: Malyshev,

text: A

text: formula

text: for

text: the

text: distance

text: from

text: a

text: matrix

text: to

text: the

text: set

text: of

text: matrices

text: with

text: a

text: multiple

text: eigenvalue,

text: Numer.

text: Math.

text: A.S.

text: Markus,

text: Introduction

text: to

text: the

text: Spectral

text: Theory

text: of

text: Polynomial

text: Operator

text: Pencils,

text: Amer.

text: Math.

text: Society,

text: Providence,

text: RI,

text: Translations

text: of

text: Mathematical

text: Monographs,

text: Vol.

text: J.

text: Nocedal,

text: S.J.

text: Wright,

text: Numerical

text: Optimization,

text: second

text: edition,

text: Springer

text: Series

text: in

text: Operation

text: Research

text: and

text: Financial

text: Engineering,

text: N.

text: Papathanasiou,

text: P.

text: Psarrakos,

text: The

text: distance

text: from

text: a

text: matrix

text: polynomial

text: to

text: matrix

text: polyno-

text: mials

text: with

text: a

text: prescribed

text: multiple

text: eigenvalue,

text: Linear

text: Algebra

text: and

text: its

text: Applications,

text: A.

text: Ruhe,

text: Properties

text: of

text: a

text: matrix

text: with

text: a

text: very

text: ill-conditioned

text: eigenproblem,

text: Numer.

text: Math.,

text: J.H.

text: Wilkinson,

text: The

text: Algebraic

text: Eigenvalue

text: Problem,

text: Claredon

text: Press,

text: Oxford,

text: J.H.

text: Wilkinson,

text: Note

text: on

text: matrices

text: with

text: a

text: very

text: ill-conditioned

text: eigenproblem,

text: Numer.

text: Math.,

text: J.H.

text: Wilkinson,

text: On

text: neighbouring

text: matrices

text: with

text: quadratic

text: elementary

text: divisors,

text: Numer.

text: Math.,

text: J.H.

text: Wilkinson,

text: Sensitivity

text: of

text: eigenvalues,

text: Util.

text: Math.,

text: J.H.

text: Wilkinson,

text: Sensitivity

text: of

text: eigenvalues

text: II,

text: Util.

text: Math.,


text: 2 Properties of s − (F [P (µ , µ ); γ]) and its corresponding 2n 1 1 2 singular vectors In this section we study some properties of s − (F [P (µ , µ ); γ]) and its correspond- 2n 1 1 2 ing singular vectors. These properties are needed in the next section in order to obtain D bounds for (P, µ , µ ) and construct a perturbation of P (λ). In this section some def- 1 2 initions and lemmas of [4-6] are reconstructed for the case of two distinct eigenvalues. Proving of some lemmas is mostly similar to the proof of related lemmas in its references. Therefore, for convince, this proofs can be omitted. ∈ C (cid:54) (cid:54) Lemma 2.1. For µ , µ and for all γ = 0, we have either s − (F [P (µ , µ ); γ]) = 0 1 2 ≡ 2n 1 1 2 or s − (F [P (µ , µ ); γ]) 0 . 2n 1 1 2 (cid:3) Proof. Similar to Lemma 3.4 of [7] can be veriﬁed easily. Lemma 2.2. If µ and µ are two eigenvalues of the matrix polynomial Q(λ) = P (λ) + 1 (cid:54) 2 ∆(λ), then for any γ = 0 ≤ (cid:107) (cid:107) s − (F [P (µ , µ ); γ]) F [∆(µ , µ ); γ]) . 2n 1 1 2 1 2 (cid:54) Proof. Let µ and µ be two eigenvalues of Q(λ) = P (λ) + ∆(λ), then for any γ = 0 1 2 s − (F [Q(µ , µ ); γ]) = s (F [Q(µ , µ ); γ]) = 0, 2n 1 1 2 2n 1 2 applying the Weyl inequalities for singular values (for example, see Corollary 5.1 of [1]) for the above relation yields | − | ≤ (cid:107) (cid:107) s − (F [Q(µ , µ ); γ]) s − (F [P (µ , µ ); γ]) F [∆(µ , µ ); γ] , 2n 1 1 2 2n 1 1 2 1 2 combining two recent relation concludes ≤ (cid:107) (cid:107) (cid:3) s − (F [P (µ , µ ); γ]) F [∆(µ , µ ); γ] . 2n 1 1 2 1 2 D The two above lemmas will be used to obtain a lower bound for (P, µ , µ ). In re- 1 2 mainder of this section, some properties of singular vectors of s − (F [P (µ , µ ); γ]) are 2n 1 D 1 2 studied which will be (cid:20)necessar(cid:21)y f(cid:20)or compu(cid:21)tation an upper bound for (P, µ , µ ) and a 1 2 perturbation of P (λ) in next section. u (γ) v (γ) ∈ C ∈ C Deﬁnition 2.3. Let 1 , 1 2n(u (γ), v (γ) n, k = 1, 2) be a pair u (γ) v (γ) k k 2 2 × of left and right singular vectors of s − (F [P (µ , µ ); γ]) respectively. Deﬁne the n 2 2n 1 1 2 matrices U(γ) = [u (γ) u (γ)], and V (γ) = [v (γ) v (γ)]. 1 2 1 2 − − Now set uˆ(γ) = u (γ) θu (γ), vˆ(γ) = v (γ) θv (γ) where θ = −γ and deﬁne 2 1 2 1 µ1 µ2 Uˆ(γ) = [u (γ) uˆ(γ)], and Vˆ (γ) = [v (γ) vˆ(γ)]. 1 1 4 ##LTLine##

text: Properties

text: of

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: and

text: its

text: corresponding

text: singular

text: vectors

text: In

text: this

text: section

text: we

text: study

text: some

text: properties

text: of

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: and

text: its

text: correspond-

text: ing

text: singular

text: vectors.

text: These

text: properties

text: are

text: needed

text: in

text: the

text: next

text: section

text: in

text: order

text: to

text: obtain

text: D

text: bounds

text: for

text: (P,

text: µ

text: µ

text: and

text: construct

text: a

text: perturbation

text: of

text: P

text: (λ).

text: In

text: this

text: section

text: some

text: def-

text: initions

text: and

text: lemmas

text: of

text: are

text: reconstructed

text: for

text: the

text: case

text: of

text: two

text: distinct

text: eigenvalues.

text: Proving

text: of

text: some

text: lemmas

text: is

text: mostly

text: similar

text: to

text: the

text: proof

text: of

text: related

text: lemmas

text: in

text: its

text: references.

text: Therefore,

text: for

text: convince,

text: this

text: proofs

text: can

text: be

text: omitted.

text: C

text: Lemma

text: For

text: µ

text: µ

text: and

text: for

text: all

text: γ

text: we

text: have

text: either

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: or

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: Proof.

text: Similar

text: to

text: Lemma

text: of

text: can

text: be

text: veriﬁed

text: easily.

text: Lemma

text: If

text: µ

text: and

text: µ

text: are

text: two

text: eigenvalues

text: of

text: the

text: matrix

text: polynomial

text: Q(λ)

text: P

text: (λ)

text: ∆(λ),

text: then

text: for

text: any

text: γ

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: F

text: [∆(µ

text: µ

text: γ])

text: Proof.

text: Let

text: µ

text: and

text: µ

text: be

text: two

text: eigenvalues

text: of

text: Q(λ)

text: P

text: (λ)

text: ∆(λ),

text: then

text: for

text: any

text: γ

text: s

text: (F

text: [Q(µ

text: µ

text: γ])

text: s

text: (F

text: [Q(µ

text: µ

text: γ])

text: applying

text: the

text: Weyl

text: inequalities

text: for

text: singular

text: values

text: (for

text: example,

text: see

text: Corollary

text: of

text: for

text: the

text: above

text: relation

text: yields

text: s

text: (F

text: [Q(µ

text: µ

text: γ])

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: F

text: [∆(µ

text: µ

text: γ]

text: combining

text: two

text: recent

text: relation

text: concludes

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: F

text: [∆(µ

text: µ

text: γ]

text: D

text: The

text: two

text: above

text: lemmas

text: will

text: be

text: used

text: to

text: obtain

text: a

text: lower

text: bound

text: for

text: (P,

text: µ

text: µ

text: In

text: re-

text: mainder

text: of

text: this

text: section,

text: some

text: properties

text: of

text: singular

text: vectors

text: of

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: are

text: D

text: studied

text: which

text: will

text: be

text: an

text: upper

text: bound

text: for

text: (P,

text: µ

text: µ

text: and

text: a

text: perturbation

text: of

text: P

text: (λ)

text: in

text: next

text: section.

text: u

text: (γ)

text: v

text: (γ)

text: C

text: C

text: Deﬁnition

text: Let

text: (γ),

text: v

text: (γ)

text: n,

text: k

text: be

text: a

text: pair

text: u

text: (γ)

text: v

text: (γ)

text: k

text: k

text: of

text: left

text: and

text: right

text: singular

text: vectors

text: of

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: respectively.

text: Deﬁne

text: the

text: n

text: matrices

text: U(γ)

text: [u

text: (γ)

text: u

text: (γ)],

text: and

text: V

text: (γ)

text: [v

text: (γ)

text: v

text: (γ)].

text: Now

text: set

text: uˆ(γ)

text: u

text: (γ)

text: θu

text: (γ),

text: vˆ(γ)

text: v

text: (γ)

text: θv

text: (γ)

text: where

text: θ

text: −γ

text: and

text: deﬁne

text: µ1

text: µ2

text: Uˆ(γ)

text: [u

text: (γ)

text: uˆ(γ)],

text: and

text: Vˆ

text: (γ)

text: [v

text: (γ)

text: vˆ(γ)].

text: ##LTLine##


text: (cid:13)(cid:16) (cid:17)(cid:13) (cid:13)(cid:16) (cid:17)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Proof. At ﬁrst (cid:88)we have (cid:13) (cid:13) (cid:88) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) − (cid:13) (cid:13) − (cid:13) n (cid:13) µj µj (cid:13) n (cid:13) µj µj (cid:13) (cid:107) (cid:107) ≤ (cid:107) (cid:107) 1 2 ≤ 1 2 | | ∆[µ , µ ] ∆ − εw − = ε w[µ , µ ] , 1 2 j j 1 2 µ µ µ µ 1 2 1 2 j=1 j=1 (cid:13) (cid:13) (cid:13) (cid:13) and (cid:13)(cid:88) (cid:13) (cid:88) (cid:12)(cid:12) (cid:12)(cid:12) (cid:88) (cid:12)(cid:12) (cid:12)(cid:12) (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) n n n (cid:107) (cid:107) ≤ (cid:107) (cid:107) ≤ | | ∆(µ ) = ∆ µj ∆ µj εw µj =εw( µ ); i = 1, 2. i j i (cid:20) (cid:21) j i j i i j=1 j=1 j=1 x ∈ C ∈ C (cid:54) We can assume a unit vector 2n(x, y n) such that for any γ = 0, (cid:13)(cid:20)y (cid:21)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) (cid:13)(cid:20) ∆(µ ) 0 (cid:21) (cid:20)2 (cid:21)(cid:13) F [∆(µ , µ ); γ] 2 = (cid:13) 1 (cid:13) 1 2 (cid:13) γ∆[µ , µ ] ∆(µ ) (cid:13) (cid:13) 1 2 2 (cid:13) (cid:13)(cid:20) ∆(µ ) 0 x 2 (cid:21)(cid:13) = (cid:13) 1 (cid:13) (cid:13) γ∆[µ , µ ] ∆(µ ) y (cid:13) (cid:13) 1 2 2 (cid:13) | | (cid:107) (cid:107) ≤ (cid:13)(cid:20) εw( µ ) x (cid:21)(cid:13) 2 (cid:13)| | | | (cid:107)1(cid:107) | (cid:13)| (cid:107) (cid:107) (cid:13) γ ε w[µ , µ ] x + εw( µ(cid:13) ) y (cid:13) 1 2 2(cid:13) | | ≤ w( µ ) 0 2 ε2 | | | 1 | | | γ w[µ , µ ] w( µ ) 1 2 2 (cid:107) | | (cid:107) = ε2 F [ w(µ , µ ) ; γ]) . 1 2 (cid:3) Lemma 2.2 completes this proof . D Considering Deﬁnition 1.2 and Lemma 3.1, a lower bound for (P, µ , µ ) can be 1 2 obtained by minimizing the both sides of (9) as follows D ≥ s − (F [P (µ , µ ); γ]) (P, µ , µ ) 2(cid:107)n 1| 1 | 2 (cid:107) . (10) 1 2 F [ w(µ , µ ) ; γ]) 1 2 (cid:16) (cid:17) Let us now construct a perturbation of P (λ). First assume that γ∗ > 0 and P [µ , µ ] 1 2 is a nonsingular matrix. Therefore, Lemma 2.7 implies that rank Vˆ (γ∗) = 2. In this case, a matrix polynomial ∆γ∗(λ) is constructed such that µ1 and µ2 are the eigenvalues of the perturbation matrix polynomial Qγ∗(λ) = P (λ) + ∆γ∗(λ). 8 ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: At

text: ﬁrst

text: have

text: n

text: µj

text: µj

text: n

text: µj

text: µj

text: ∆[µ

text: µ

text: εw

text: ε

text: w[µ

text: µ

text: j

text: j

text: µ

text: µ

text: µ

text: µ

text: and

text: n

text: n

text: n

text: ∆(µ

text: µj

text: µj

text: εw

text: µj

text: =εw(

text: µ

text: i

text: i

text: j

text: i

text: j

text: i

text: j

text: i

text: i

text: x

text: C

text: C

text: We

text: can

text: assume

text: a

text: unit

text: vector

text: y

text: n)

text: such

text: that

text: for

text: any

text: γ

text: ∆(µ

text: F

text: [∆(µ

text: µ

text: γ]

text: γ∆[µ

text: µ

text: ∆(µ

text: ∆(µ

text: x

text: γ∆[µ

text: µ

text: ∆(µ

text: y

text: εw(

text: µ

text: x

text: γ

text: ε

text: w[µ

text: µ

text: x

text: εw(

text: y

text: w(

text: µ

text: ε2

text: γ

text: w[µ

text: µ

text: w(

text: µ

text: ε2

text: F

text: w(µ

text: µ

text: γ])

text: Lemma

text: completes

text: this

text: proof

text: D

text: Considering

text: Deﬁnition

text: and

text: Lemma

text: a

text: lower

text: bound

text: for

text: (P,

text: µ

text: µ

text: can

text: be

text: obtained

text: by

text: minimizing

text: the

text: both

text: sides

text: of

text: as

text: follows

text: D

text: s

text: (F

text: [P

text: (µ

text: µ

text: γ])

text: (P,

text: µ

text: µ

text: F

text: w(µ

text: µ

text: γ])

text: Let

text: us

text: now

text: construct

text: a

text: perturbation

text: of

text: P

text: (λ).

text: First

text: assume

text: that

text: γ∗

text: and

text: P

text: [µ

text: µ

text: is

text: a

text: nonsingular

text: matrix.

text: Therefore,

text: Lemma

text: implies

text: that

text: rank

text: Vˆ

text: (γ∗)

text: In

text: this

text: case,

text: a

text: matrix

text: polynomial

text: ∆γ∗(λ)

text: is

text: constructed

text: such

text: that

text: µ1

text: and

text: µ2

text: are

text: the

text: eigenvalues

text: of

text: the

text: perturbation

text: matrix

text: polynomial

text: Qγ∗(λ)

text: P

text: (λ)

text: ∆γ∗(λ).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:32) (cid:33) (cid:16) (cid:17) where − − 1 e ν + s e να D = − − − − − (cid:48) − , t e λα e λ e µ(1 Rµ )2 α2 + s 2 − E = e µ/21 , (4.10) with (cid:20)1 − (cid:16)− (cid:48) (cid:17) (cid:21) s = e (λ+µ+ν)/2( 2 + Rµ ) (4.11) 2 1 − − (cid:48) 2 − − 1/2 t = e (λ+µ+ν) 2e(λ+µ)/2 2eν/2 + eν/2Rµ e λα2 . (4.12) 4 As α only enters the action throug(cid:16)h the mass term in Eq.(cid:17)(2.18), and moreover non-derivatively, its equation of motion is purely algebraic. It reads, for any model of dRGT massive gravity: α β eµ + 4β eµ/2 + 6β = 0 . (4.13) 1 2 3 Although α = 0 is always a solution, there may exist other solutions for the non-minimal mod- els with β or/and β non-zero, depending on the signs and relative values of the parameters. 2 3 − However, in the minimal model — β = 2, β = β = 0 — Eq. (4.13) readily gives α = 0. In 1 2 3 other words, all spherically symmetric and stationary solutions of the minimal model are actually static. To proceed further with the remaining variables λ, µ, ν, let us write down the Einstein equations of motion in vacuum: G = m2T mass , (4.14) µν µν where the energy-momentum tensor derived from the mass term in Eq. (2.25) reads [35] − (cid:104) (cid:105) T mass = 3g + γ γ g . (4.15) µν µν µν µν (cid:18) (cid:19) The (T, T ) and (R, R) components of Eq. (4.14) read respectively: (cid:48) − λ 1 − eν λ + (eλ 1) = m2T mass , (4.16) R R2 TT (cid:48) ν 1 − + (1 eλ) = m2T mass , (4.17) R R2 RR (cid:16) (cid:17) where −1 − − − (cid:48) TTmTass = 2eν 12(λ+µ) 2 4 eλ/2 + 6 e12(λ+µ) + R µ , (4.18) − − − − T mass = eλ( 3 + 2 e µ/2 + e ν/2) . (4.19) RR To supplement these equations, one can use the (θ, θ) = (φ, φ) equation of motion. Alternatively, when Eqs. (4.16)-(4.17) are satisﬁed, it is equivalent to the Bianchi identity ≡ ∇ f µT mass = 0 , (4.20) g µR – 11 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: e

text: ν

text: s

text: e

text: να

text: D

text: t

text: e

text: λα

text: e

text: λ

text: e

text: Rµ

text: α2

text: s

text: E

text: e

text: with

text: s

text: e

text: Rµ

text: t

text: e

text: (λ+µ+ν)

text: e

text: λα2

text: As

text: α

text: only

text: enters

text: the

text: action

text: the

text: mass

text: term

text: in

text: and

text: moreover

text: non-derivatively,

text: its

text: equation

text: of

text: motion

text: is

text: purely

text: algebraic.

text: It

text: reads,

text: for

text: any

text: model

text: of

text: dRGT

text: massive

text: gravity:

text: α

text: β

text: eµ

text: 4β

text: 6β

text: Although

text: α

text: is

text: always

text: a

text: solution,

text: there

text: may

text: exist

text: other

text: solutions

text: for

text: the

text: non-minimal

text: mod-

text: els

text: with

text: β

text: or/and

text: β

text: non-zero,

text: depending

text: on

text: the

text: signs

text: and

text: relative

text: values

text: of

text: the

text: parameters.

text: However,

text: in

text: the

text: minimal

text: model

text: β

text: β

text: β

text: Eq.

text: readily

text: gives

text: α

text: In

text: other

text: words,

text: all

text: spherically

text: symmetric

text: and

text: stationary

text: solutions

text: of

text: the

text: minimal

text: model

text: are

text: actually

text: static.

text: To

text: proceed

text: further

text: with

text: the

text: remaining

text: variables

text: λ,

text: µ,

text: ν,

text: let

text: us

text: write

text: down

text: the

text: Einstein

text: equations

text: of

text: motion

text: in

text: vacuum:

text: G

text: mass

text: µν

text: µν

text: where

text: the

text: energy-momentum

text: tensor

text: derived

text: from

text: the

text: mass

text: term

text: in

text: Eq.

text: reads

text: T

text: mass

text: γ

text: γ

text: g

text: µν

text: µν

text: µν

text: µν

text: The

text: (T,

text: T

text: and

text: (R,

text: R)

text: components

text: of

text: Eq.

text: read

text: respectively:

text: λ

text: eν

text: λ

text: (eλ

text: mass

text: R

text: TT

text: ν

text: eλ)

text: mass

text: R

text: RR

text: where

text: −1

text: TTmTass

text: R

text: µ

text: T

text: mass

text: eλ(

text: e

text: e

text: RR

text: To

text: supplement

text: these

text: equations,

text: one

text: can

text: use

text: the

text: (θ,

text: θ)

text: (φ,

text: φ)

text: equation

text: of

text: motion.

text: Alternatively,

text: when

text: Eqs.

text: are

text: satisﬁed,

text: it

text: is

text: equivalent

text: to

text: the

text: Bianchi

text: identity

text: f

text: µT

text: mass

text: g

text: µR

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: decoupling limit. Yet, the absence of non-linearities at this level does not necessarily mean that they are ineﬀective to yield a Vainshtein mechanism. It solely implies that the decoupling limit Lagrangian does not suﬃce to decipher its existence or not in this model. Additionally, the analysis of Ref. [28] is restricted to spherically symmetric conﬁgurations. While this is a natural starting point and a very common assumption in the literature, we will see that the peculiarities of the minimal model might render it misleading as to the fate of the Vainshtein mechanism in real-world conditions, in which spherical symmetry is broken, even mildly, like in the solar system. For these reasons, we investigate the interactions of the minimal model arising at energies higher than Λ , in static spherically symmetric conﬁgurations and beyond. Working within the 3 Stu¨ckelberg formalism, and considering only the scalar graviton, we prove the remarkable fact that in static spherically symmetric conﬁgurations, all interactions arising below the Planck mass vanish identically, i.e. beyond the free, quadratic, action, the theory completely loses track of the graviton mass, barring energies larger than the Planck scale. This tantalizing fact points to- wards an absence of Vainshtein mechanism in this set up, but does not prove it. For this reason, we resort to the exact equations of motion in the metric formalism. We then show completely generally that in all vacuum stationary and spherically symmetric conﬁgurations, there exists an obstruction that precludes any recovery of General Relativity. While this could be seen as a proof that the minimal model is ruled out, we argue that our analysis of the energy scales of interactions in time-dependent or non-spherically symmetric conﬁgurations indicates that it would be premature to reach this conclusion without further study. The outline of the paper is as follows. In section , we introduce the theory of dRGT massive gravity, review the construction of its decoupling limit, and present the minimal model. In section , we investigate, within the Stu¨ckelberg formalism, energy scales of interactions in the minimal model beyond the decoupling limit. We study exact vacuum solutions of the theory in section and conclude in section . 2 dRGT massive gravity and its minimal model In this paper, we consider ghost-free dRGT massive gravity with Minkowski reference metric (cid:16) (cid:16) (cid:17)(cid:17) in four spacetime dimensions. Its Lagrangian is usually formulated as [9] √ L − M2 L K L K L K = g Pl R + m2 (2)( ) + α (3)( ) + α (4)( ) (2.1) MG 3 4 2 where R is the Ricci scalar of the spacetime metric g , m is the mass of the graviton, and µν L K (cid:104)K(cid:105) − (cid:104)K (cid:105) (2)( ) = 2 2 , (2.2) L K (cid:104)K(cid:105) − (cid:104)K(cid:105)(cid:104)K (cid:105) (cid:104)K (cid:105) (3)( ) = 3 3 2 + 2 3 , (2.3) L K (cid:104)K(cid:105) − (cid:104)K (cid:105)(cid:104)K(cid:105) (cid:104)K (cid:105)(cid:104)K(cid:105) (cid:104)K (cid:105) − (cid:104)K (cid:105) (4)( ) = 4 6 2 2 + 8 3 + 3 2 2 6 4 , (2.4) – 2 – ##LTLine## ##LTLine##

text: decoupling

text: limit.

text: Yet,

text: the

text: absence

text: of

text: non-linearities

text: at

text: this

text: level

text: does

text: not

text: necessarily

text: mean

text: that

text: they

text: are

text: ineﬀective

text: to

text: yield

text: a

text: Vainshtein

text: mechanism.

text: It

text: solely

text: implies

text: that

text: the

text: decoupling

text: limit

text: Lagrangian

text: does

text: not

text: suﬃce

text: to

text: decipher

text: its

text: existence

text: or

text: not

text: in

text: this

text: model.

text: Additionally,

text: the

text: analysis

text: of

text: Ref.

text: is

text: restricted

text: to

text: spherically

text: symmetric

text: conﬁgurations.

text: While

text: this

text: is

text: a

text: natural

text: starting

text: point

text: and

text: a

text: very

text: common

text: assumption

text: in

text: the

text: literature,

text: we

text: will

text: see

text: that

text: the

text: peculiarities

text: of

text: the

text: minimal

text: model

text: might

text: render

text: it

text: misleading

text: as

text: to

text: the

text: fate

text: of

text: the

text: Vainshtein

text: mechanism

text: in

text: real-world

text: conditions,

text: in

text: which

text: spherical

text: symmetry

text: is

text: broken,

text: even

text: mildly,

text: like

text: in

text: the

text: solar

text: system.

text: For

text: these

text: reasons,

text: we

text: investigate

text: the

text: interactions

text: of

text: the

text: minimal

text: model

text: arising

text: at

text: energies

text: higher

text: than

text: Λ

text: in

text: static

text: spherically

text: symmetric

text: conﬁgurations

text: and

text: beyond.

text: Working

text: within

text: the

text: Stu¨ckelberg

text: formalism,

text: and

text: considering

text: only

text: the

text: scalar

text: graviton,

text: we

text: prove

text: the

text: remarkable

text: fact

text: that

text: in

text: static

text: spherically

text: symmetric

text: conﬁgurations,

text: all

text: interactions

text: arising

text: below

text: the

text: Planck

text: mass

text: vanish

text: identically,

text: i.e.

text: beyond

text: the

text: free,

text: quadratic,

text: action,

text: the

text: theory

text: completely

text: loses

text: track

text: of

text: the

text: graviton

text: mass,

text: barring

text: energies

text: larger

text: than

text: the

text: Planck

text: scale.

text: This

text: tantalizing

text: fact

text: points

text: to-

text: wards

text: an

text: absence

text: of

text: Vainshtein

text: mechanism

text: in

text: this

text: set

text: up,

text: but

text: does

text: not

text: prove

text: it.

text: For

text: this

text: reason,

text: we

text: resort

text: to

text: the

text: exact

text: equations

text: of

text: motion

text: in

text: the

text: metric

text: formalism.

text: We

text: then

text: show

text: completely

text: generally

text: that

text: in

text: all

text: vacuum

text: stationary

text: and

text: spherically

text: symmetric

text: conﬁgurations,

text: there

text: exists

text: an

text: obstruction

text: that

text: precludes

text: any

text: recovery

text: of

text: General

text: Relativity.

text: While

text: this

text: could

text: be

text: seen

text: as

text: a

text: proof

text: that

text: the

text: minimal

text: model

text: is

text: ruled

text: out,

text: we

text: argue

text: that

text: our

text: analysis

text: of

text: the

text: energy

text: scales

text: of

text: interactions

text: in

text: time-dependent

text: or

text: non-spherically

text: symmetric

text: conﬁgurations

text: indicates

text: that

text: it

text: would

text: be

text: premature

text: to

text: reach

text: this

text: conclusion

text: without

text: further

text: study.

text: The

text: outline

text: of

text: the

text: paper

text: is

text: as

text: follows.

text: In

text: section

text: we

text: introduce

text: the

text: theory

text: of

text: dRGT

text: massive

text: gravity,

text: review

text: the

text: construction

text: of

text: its

text: decoupling

text: limit,

text: and

text: present

text: the

text: minimal

text: model.

text: In

text: section

text: we

text: investigate,

text: within

text: the

text: Stu¨ckelberg

text: formalism,

text: energy

text: scales

text: of

text: interactions

text: in

text: the

text: minimal

text: model

text: beyond

text: the

text: decoupling

text: limit.

text: We

text: study

text: exact

text: vacuum

text: solutions

text: of

text: the

text: theory

text: in

text: section

text: and

text: conclude

text: in

text: section

text: dRGT

text: massive

text: gravity

text: and

text: its

text: minimal

text: model

text: In

text: this

text: paper,

text: we

text: consider

text: ghost-free

text: dRGT

text: massive

text: gravity

text: with

text: Minkowski

text: reference

text: metric

text: in

text: four

text: spacetime

text: dimensions.

text: Its

text: Lagrangian

text: is

text: usually

text: formulated

text: as

text: L

text: L

text: K

text: L

text: K

text: L

text: K

text: g

text: Pl

text: R

text: α

text: α

text: MG

text: where

text: R

text: is

text: the

text: Ricci

text: scalar

text: of

text: the

text: spacetime

text: metric

text: g

text: m

text: is

text: the

text: mass

text: of

text: the

text: graviton,

text: and

text: µν

text: L

text: K

text: L

text: K

text: L

text: K

text: ##LTLine##

text: ##LTLine##