text: ≥ and where n + n + n 3 since we are considering interactions. Since m < M for realistic π A h Pl parameters, Λ is a decreasing function of α. For a generic mass term, there are interactions α whose energy scales lie below Λ . However, the dRGT Lagrangian (2.1) is built such that the 3 corresponding terms, in (∂2π)nπ and in (∂A)(∂2π)nπ, vanish identically. The lowest interaction scale is thus in general Λ = (M m2)1/3. By considering the so called decoupling limit (DL), 3 Pl such that → → ∞ m 0 , M , Λ ﬁxed , (2.12) Pl 3 one can therefore concentrate on the leading interactions of the theory. In this limit, the non- linearities in h and A disappear, and one can show that the full Lagrangian (2.1) boils down µν µ (cid:32) (cid:33) to (ignoring the free ﬁeld A in this limit) [8, 9] µ L −1 E 1 X(2) X(3) = hµν ˆαβh + hµν 2X(1) + (1 + 3α ) µν + (α + 4α ) µν , (2.13) DL 4 µν αβ 2 µν 3 Λ3 3 4 Λ6 3 3 E where ˆ is the Lichnerowicz operator, coming from the expansion of the Einstein-Hilbert action (cid:88) at quadratic order, and n − n! L − X(n) = ( 1)m − (Πm) (n m)(Π) , (2.14) µν 2(n m)! µν m=0 ≡ ∇ ∇ L ≡ L (cid:104) (cid:105) L where Π π, (0) 1, (1)(Π) = Π and (2,3,4) are deﬁned in Eqs. (2.2)-(2.4). µν µ ν The Lagrangian (2.13) kinetically mixes h and π. One can partially diagonalize it by use µν (3) of the transformation (the mixing in X can be eliminated as well but at the cost of a non-local µν ﬁeld redeﬁnition [8]) − 1 + 3α ∇ ∇ h = h¯ + π η 3 π π , (2.15) µν µν µν Λ3 µ ν 3 yielding (cid:88) L L −1 E 1 X(3) 5 (n) DL = 4h¯µν ˆµανβh¯αβ + 2(α3 + 4α4)h¯µν Λµ63ν + n=2 cn Λ33(Gn−al2) , (2.16) (cid:18) (cid:19) where −3 − 3 − 1 1 c = δ (1 + 3α )δ (1 + 3α )2 + (α + 4α ) δ n n,2 3 n,3 3 3 4 n,4 4 4 4 2 − 5 (1 + 3α )(α + 4α )δ (2.17) 3 3 4 n,5 8 L ≡ ∇ L − and where the Galileon Lagrangians are such that (n) ( π)2 (n 2)(Π). Gal – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: where

text: n

text: n

text: n

text: since

text: we

text: are

text: considering

text: interactions.

text: Since

text: m

text: M

text: for

text: realistic

text: π

text: A

text: h

text: Pl

text: parameters,

text: Λ

text: is

text: a

text: decreasing

text: function

text: of

text: α.

text: For

text: a

text: generic

text: mass

text: term,

text: there

text: are

text: interactions

text: α

text: whose

text: energy

text: scales

text: lie

text: below

text: Λ

text: However,

text: the

text: dRGT

text: Lagrangian

text: is

text: built

text: such

text: that

text: the

text: corresponding

text: terms,

text: in

text: and

text: in

text: vanish

text: identically.

text: The

text: lowest

text: interaction

text: scale

text: is

text: thus

text: in

text: general

text: Λ

text: (M

text: By

text: considering

text: the

text: so

text: called

text: decoupling

text: limit

text: (DL),

text: Pl

text: such

text: that

text: m

text: M

text: Λ

text: ﬁxed

text: Pl

text: one

text: can

text: therefore

text: concentrate

text: on

text: the

text: leading

text: interactions

text: of

text: the

text: theory.

text: In

text: this

text: limit,

text: the

text: non-

text: linearities

text: in

text: h

text: and

text: A

text: disappear,

text: and

text: one

text: can

text: show

text: that

text: the

text: full

text: Lagrangian

text: boils

text: down

text: µν

text: µ

text: to

text: (ignoring

text: the

text: free

text: ﬁeld

text: A

text: in

text: this

text: limit)

text: µ

text: L

text: −1

text: E

text: hµν

text: ˆαβh

text: hµν

text: 3α

text: µν

text: (α

text: 4α

text: µν

text: DL

text: µν

text: αβ

text: µν

text: Λ3

text: Λ6

text: E

text: where

text: ˆ

text: is

text: the

text: Lichnerowicz

text: operator,

text: coming

text: from

text: the

text: expansion

text: of

text: the

text: Einstein-Hilbert

text: action

text: at

text: quadratic

text: order,

text: and

text: n

text: n!

text: L

text: X(n)

text: (Πm)

text: (n

text: m)(Π)

text: µν

text: m)!

text: µν

text: L

text: L

text: L

text: where

text: Π

text: π,

text: Π

text: and

text: are

text: deﬁned

text: in

text: Eqs.

text: µν

text: µ

text: ν

text: The

text: Lagrangian

text: kinetically

text: mixes

text: h

text: and

text: π.

text: One

text: can

text: partially

text: diagonalize

text: it

text: by

text: use

text: µν

text: of

text: the

text: transformation

text: (the

text: mixing

text: in

text: X

text: can

text: be

text: eliminated

text: as

text: well

text: but

text: at

text: the

text: cost

text: of

text: a

text: non-local

text: µν

text: ﬁeld

text: redeﬁnition

text: 3α

text: h

text: h¯

text: π

text: η

text: π

text: π

text: µν

text: µν

text: µν

text: Λ3

text: µ

text: ν

text: yielding

text: L

text: L

text: −1

text: E

text: (n)

text: DL

text: 4h¯µν

text: ˆµανβh¯αβ

text: 4α4)h¯µν

text: Λµ63ν

text: cn

text: where

text: −3

text: c

text: δ

text: 3α

text: )δ

text: 3α

text: (α

text: 4α

text: δ

text: n

text: 3α

text: )(α

text: 4α

text: )δ

text: L

text: L

text: and

text: where

text: the

text: Galileon

text: Lagrangians

text: are

text: such

text: that

text: (n)

text: (n

text: Gal

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∼ interactions in (3.7) are either suppressed by the Planck mass, or of the type M Λ3Xn/Mn , ≥ Pl 3 Pl where n 3 and Xn stands for n products of either λ¯, ν¯ or π. The energy scales suppressing − − the latter terms are of the type Λ (2.11) with α = 1 2/(n 2) and are thus above the Planck α mass. As the interactions coming from the Einstein-Hilbert action are suppressed by the Planck mass (see Eq. (3.3)), we reach the conclusion that the lowest energy scale of interactions is M . Pl In other words, not only do the interactions at the energy Λ vanish, but also all the interactions 3 involving h¯ and π below the Planck mass! Had we used the formulation (2.1), this would µν L K have been obscured: each term (n)( ) contains non-trivial interactions below M , conspiring − Pl to cancel for the speciﬁc parameters of the minimal model α = 1/3, α = 1/12. As we have 3 4 seen, this becomes transparent with the formulation (2.18), relying on the fact that γ is linear in π/Λ3 (this is a consequence of considering a SSS conﬁguration) and that the minimal model 3 (2.24) is linear in γ itself. Beyond static spherically symmetric conﬁgurations. One can wonder to which extent the above conclusion relies on considering a SSS conﬁguration. To understand this, we ﬁrst consider a spherically symmetric conﬁguration o(cid:32)f the fo(cid:33)rm (3.3), but now with an additional time-dependence of ν¯, λ¯ and π. One then obtains − A 0 (g 1η˜)µ = (cid:32) (cid:0) ν (cid:1) 0 B (cid:0) (cid:1) (cid:33) (cid:0) (cid:1) (cid:0) (cid:1) where − − − (cid:48) − − (cid:48) − (cid:48)(cid:48) g 1 (1 + π¨˜)2 π˜˙ 2 g 1π˜˙ 2 π˜ + π¨˜ A = −tt (cid:48) (cid:18) − (cid:48)(cid:48) (cid:19) (cid:18)− tt −(cid:19) (cid:48)(cid:48) − (cid:48) , g 1π˜˙ 2 π˜ + π¨˜ g 1 (1 π˜ )2 π˜˙ 2 rr rr − (cid:48) π 1 − π˜ 2 B = 1 + 1 1 , (3.8) 2 M r (cid:16) Pl (cid:17) ≡ (cid:48) we recall that π = Λ3π˜, and ˙ d/dt. When π˜˙ = 0, a matrix square-root of A is 3 − − − − (cid:48)(cid:48) Diag ( g ) 1/2(1 + π¨˜), (g ) 1/2(1 π˜ ) , (3.9) tt rr then γµ is linear in π˜ = π/Λ3 and the ab(cid:32)ove arguments (cid:33)apply2. Similarly, when ν¯ = λ¯, one can ν 3 ﬁnd an “obvious” square-root of A: (cid:48) − 1 + π¨˜ π˜˙ g 1/2 − (cid:48) − (cid:48)(cid:48) , rr π˜˙ 1 π˜ γµ is still linear in π/Λ3 and the minimal energy of interactions is M . However, this does ν 3 × Pl not hold in full generality. Using that for any 2 2 matrix A, a square root R of A (the one (cid:16) (cid:112) (cid:17) (cid:16) (cid:112) (cid:17) connected to the identity in perturbation theory) reads − (cid:104) (cid:105) 1/2 R = A + 2 det(A) A + det(A)1 , (3.10) 2 2Note in particular that the latter apply when π is of the form π = π (r) + c t2, where c is a constant, as 0 considered for instance in Refs. [40, 41]. – 7 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: interactions

text: in

text: are

text: either

text: suppressed

text: by

text: the

text: Planck

text: mass,

text: or

text: of

text: the

text: type

text: M

text: Pl

text: Pl

text: where

text: n

text: and

text: Xn

text: stands

text: for

text: n

text: products

text: of

text: either

text: λ¯,

text: ν¯

text: or

text: π.

text: The

text: energy

text: scales

text: suppressing

text: the

text: latter

text: terms

text: are

text: of

text: the

text: type

text: Λ

text: with

text: α

text: and

text: are

text: thus

text: above

text: the

text: Planck

text: α

text: mass.

text: As

text: the

text: interactions

text: coming

text: from

text: the

text: Einstein-Hilbert

text: action

text: are

text: suppressed

text: by

text: the

text: Planck

text: mass

text: (see

text: Eq.

text: we

text: reach

text: the

text: conclusion

text: that

text: the

text: lowest

text: energy

text: scale

text: of

text: interactions

text: is

text: M

text: Pl

text: In

text: other

text: words,

text: not

text: only

text: do

text: the

text: interactions

text: at

text: the

text: energy

text: Λ

text: vanish,

text: but

text: also

text: all

text: the

text: interactions

text: involving

text: h¯

text: and

text: π

text: below

text: the

text: Planck

text: mass!

text: Had

text: we

text: used

text: the

text: formulation

text: this

text: would

text: µν

text: L

text: K

text: have

text: been

text: obscured:

text: each

text: term

text: (n)(

text: contains

text: non-trivial

text: interactions

text: below

text: M

text: conspiring

text: Pl

text: to

text: cancel

text: for

text: the

text: speciﬁc

text: parameters

text: of

text: the

text: minimal

text: model

text: α

text: α

text: As

text: we

text: have

text: seen,

text: this

text: becomes

text: transparent

text: with

text: the

text: formulation

text: relying

text: on

text: the

text: fact

text: that

text: γ

text: is

text: linear

text: in

text: π/Λ3

text: (this

text: is

text: a

text: consequence

text: of

text: considering

text: a

text: SSS

text: conﬁguration)

text: and

text: that

text: the

text: minimal

text: model

text: is

text: linear

text: in

text: γ

text: itself.

text: Beyond

text: static

text: spherically

text: symmetric

text: conﬁgurations.

text: One

text: can

text: wonder

text: to

text: which

text: extent

text: the

text: above

text: conclusion

text: relies

text: on

text: considering

text: a

text: SSS

text: conﬁguration.

text: To

text: understand

text: this,

text: we

text: ﬁrst

text: consider

text: a

text: spherically

text: symmetric

text: conﬁguration

text: the

text: but

text: now

text: with

text: an

text: additional

text: time-dependence

text: of

text: ν¯,

text: λ¯

text: and

text: π.

text: One

text: then

text: obtains

text: A

text: (g

text: 1η˜)µ

text: ν

text: B

text: where

text: g

text: π¨˜)2

text: π˜˙

text: g

text: 1π˜˙

text: π˜

text: π¨˜

text: A

text: −tt

text: tt

text: g

text: 1π˜˙

text: π˜

text: π¨˜

text: g

text: π˜

text: π˜˙

text: rr

text: rr

text: π

text: π˜

text: B

text: M

text: r

text: Pl

text: we

text: recall

text: that

text: π

text: Λ3π˜,

text: and

text: d/dt.

text: When

text: π˜˙

text: a

text: matrix

text: square-root

text: of

text: A

text: is

text: Diag

text: g

text: π¨˜),

text: (g

text: π˜

text: tt

text: rr

text: then

text: γµ

text: is

text: linear

text: in

text: π˜

text: π/Λ3

text: and

text: the

text: arguments

text: Similarly,

text: when

text: ν¯

text: λ¯,

text: one

text: can

text: ν

text: ﬁnd

text: an

text: “obvious”

text: square-root

text: of

text: A:

text: π¨˜

text: π˜˙

text: g

text: rr

text: π˜˙

text: π˜

text: γµ

text: is

text: still

text: linear

text: in

text: π/Λ3

text: and

text: the

text: minimal

text: energy

text: of

text: interactions

text: is

text: M

text: However,

text: this

text: does

text: ν

text: Pl

text: not

text: hold

text: in

text: full

text: generality.

text: Using

text: that

text: for

text: any

text: matrix

text: A,

text: a

text: square

text: root

text: R

text: of

text: A

text: (the

text: one

text: connected

text: to

text: the

text: identity

text: in

text: perturbation

text: theory)

text: reads

text: R

text: A

text: det(A)

text: A

text: in

text: particular

text: that

text: the

text: latter

text: apply

text: when

text: π

text: is

text: of

text: the

text: form

text: π

text: π

text: (r)

text: c

text: where

text: c

text: is

text: a

text: constant,

text: as

text: considered

text: for

text: instance

text: in

text: Refs.

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: (cid:18) (cid:19) (cid:18) (cid:19) and − (cid:48) π 1 − π˜ − π˜ 2 ,θ d = 1 + 1 Cot(θ) . (3.19) M r r2 Pl (cid:48) − The discussion then proceeds completely similarly as above: when rπ˜ π˜ = 0, a matrix (cid:32) (cid:18) (cid:19) (cid:18) ,θ(cid:19)(cid:33) ,θ square-root of C is − (cid:48) − − (cid:48)(cid:48) π 1/2 − π˜ − π˜ Diag (g ) 1/2(1 π˜ ), 1 + 1 ,θθ , (3.20) rr M r r2 Pl γµ is linear in π˜ = π/Λ3 and the minimal energy of interactions is M . Additionally, when ν (cid:18) 3 (cid:19) (cid:32) P(cid:33)l λ = 0, an “obvious” square-root of C reads: − − (cid:48)(cid:48) − − (cid:48) − π 1/2 1 π˜ r 1(rπ˜ π˜ ) 1 + − − (cid:48) − − (cid:48) ,−θ ,θ M r 3(rπ˜ π˜ ) 1 π˜ π˜,θθ Pl ,θ ,θ r r2 and the same conclusion holds. In the general case however, one can use Eq. (3.11) to ﬁnd the (cid:104) (cid:105) trace of the √relevant square root of C, and hence an explicit expression for γ . Expanding the − (cid:104) (cid:105) term M(cid:16) 2 m2 g(cid:17) γ in Eq. (2.25), one then ﬁnds, in addition to the terms suppressed by M and Pl Pl Λ − − , an inﬁnite number of interactions involving negative powers of Λ3, all proportional 1 2/(n 2) 3 to λ¯2 rπ˜,(cid:48)θ − π˜,θ 2. They include fo(cid:16)r example(cid:17)int(cid:0)eractions of the typ(cid:1)e λ¯2 rπ,(cid:48)θ − π,θ 2 π(cid:48)(cid:48) + π(cid:48)/r + π,θθ/r2 n b sin(θ) , (3.21) n r2Λ3M Λ3n 3 Pl 3 ≥ with non-zero b for all n 0, whose energy scales are again given by Eq. (3.14). n As a summary, although there is no interaction at the energy Λ in the minimal model, 3 and the lowest energy scale of interactions is M in static spherically symmetric conﬁgurations, Pl interactions arising at energy scales arbitrarily close to Λ are present in generic spherically sym- 3 metric but time-dependent conﬁgurations, or generic static but non-spherically symmetric ones. In the non-minimal models of massive gravity, the existence of a ﬁnite number of interactions arising at a well deﬁned lowest energy scale enables us to hope to capture the main physics of interactions by concentrating on this scale. This is the essence of the decoupling limit. Our analysis shows that no such hope is possible in the minimal model, either because there is no lowest energy scale of interaction — this is the generic case — or, in the presence of such a scale in SSS conﬁgurations, namely the Planck mass, because of the existence of an inﬁnite number of interactions arising at this scale, including for instance the ones of General Relativity. 4 Exact equations of motion and obstruction to a Vainshtein mechanism In the previous section, we have shown that in static spherically symmetric conﬁgurations, interactions below the Planck mass between h¯ and π vanish identically. This remarkable fact µν points towards the absence of a Vainshtein mechanism in this set up, but does not prove it. In this – 9 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: π

text: π˜

text: π˜

text: ,θ

text: d

text: Cot(θ)

text: M

text: r

text: Pl

text: The

text: discussion

text: then

text: proceeds

text: completely

text: similarly

text: as

text: above:

text: when

text: rπ˜

text: π˜

text: a

text: matrix

text: ,θ

text: square-root

text: of

text: C

text: is

text: π

text: π˜

text: π˜

text: Diag

text: (g

text: π˜

text: ,θθ

text: rr

text: M

text: r

text: Pl

text: γµ

text: is

text: linear

text: in

text: π˜

text: π/Λ3

text: and

text: the

text: minimal

text: energy

text: of

text: interactions

text: is

text: M

text: Additionally,

text: when

text: ν

text: λ

text: an

text: “obvious”

text: square-root

text: of

text: C

text: reads:

text: π

text: π˜

text: r

text: π˜

text: ,−θ

text: ,θ

text: M

text: r

text: π˜

text: π˜

text: π˜,θθ

text: Pl

text: ,θ

text: ,θ

text: r

text: and

text: the

text: same

text: conclusion

text: holds.

text: In

text: the

text: general

text: case

text: however,

text: one

text: can

text: use

text: Eq.

text: to

text: ﬁnd

text: the

text: trace

text: of

text: the

text: √relevant

text: square

text: root

text: of

text: C,

text: and

text: hence

text: an

text: explicit

text: expression

text: for

text: γ

text: Expanding

text: the

text: term

text: γ

text: in

text: Eq.

text: one

text: then

text: ﬁnds,

text: in

text: addition

text: to

text: the

text: terms

text: suppressed

text: by

text: M

text: and

text: Pl

text: Pl

text: Λ

text: an

text: inﬁnite

text: number

text: of

text: interactions

text: involving

text: negative

text: powers

text: of

text: all

text: proportional

text: to

text: λ¯2

text: π˜,θ

text: They

text: include

text: of

text: the

text: λ¯2

text: π,θ

text: n

text: b

text: sin(θ)

text: n

text: Pl

text: with

text: non-zero

text: b

text: for

text: all

text: n

text: whose

text: energy

text: scales

text: are

text: again

text: given

text: by

text: Eq.

text: n

text: As

text: a

text: summary,

text: although

text: there

text: is

text: no

text: interaction

text: at

text: the

text: energy

text: Λ

text: in

text: the

text: minimal

text: model,

text: and

text: the

text: lowest

text: energy

text: scale

text: of

text: interactions

text: is

text: M

text: in

text: static

text: spherically

text: symmetric

text: conﬁgurations,

text: Pl

text: interactions

text: arising

text: at

text: energy

text: scales

text: arbitrarily

text: close

text: to

text: Λ

text: are

text: present

text: in

text: generic

text: spherically

text: sym-

text: metric

text: but

text: time-dependent

text: conﬁgurations,

text: or

text: generic

text: static

text: but

text: non-spherically

text: symmetric

text: ones.

text: In

text: the

text: non-minimal

text: models

text: of

text: massive

text: gravity,

text: the

text: existence

text: of

text: a

text: ﬁnite

text: number

text: of

text: interactions

text: arising

text: at

text: a

text: well

text: deﬁned

text: lowest

text: energy

text: scale

text: enables

text: us

text: to

text: hope

text: to

text: capture

text: the

text: main

text: physics

text: of

text: interactions

text: by

text: concentrating

text: on

text: this

text: scale.

text: This

text: is

text: the

text: essence

text: of

text: the

text: decoupling

text: limit.

text: Our

text: analysis

text: shows

text: that

text: no

text: such

text: hope

text: is

text: possible

text: in

text: the

text: minimal

text: model,

text: either

text: because

text: there

text: is

text: no

text: lowest

text: energy

text: scale

text: of

text: interaction

text: this

text: is

text: the

text: generic

text: case

text: or,

text: in

text: the

text: presence

text: of

text: such

text: a

text: scale

text: in

text: SSS

text: conﬁgurations,

text: namely

text: the

text: Planck

text: mass,

text: because

text: of

text: the

text: existence

text: of

text: an

text: inﬁnite

text: number

text: of

text: interactions

text: arising

text: at

text: this

text: scale,

text: including

text: for

text: instance

text: the

text: ones

text: of

text: General

text: Relativity.

text: Exact

text: equations

text: of

text: motion

text: and

text: obstruction

text: to

text: a

text: Vainshtein

text: mechanism

text: In

text: the

text: previous

text: section,

text: we

text: have

text: shown

text: that

text: in

text: static

text: spherically

text: symmetric

text: conﬁgurations,

text: interactions

text: below

text: the

text: Planck

text: mass

text: between

text: h¯

text: and

text: π

text: vanish

text: identically.

text: This

text: remarkable

text: fact

text: µν

text: points

text: towards

text: the

text: absence

text: of

text: a

text: Vainshtein

text: mechanism

text: in

text: this

text: set

text: up,

text: but

text: does

text: not

text: prove

text: it.

text: In

text: this

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CHAPTER 2. FUNDAMENTALS 7 Deﬁnition 2.11. From M we can construct the dual matroid M∗. This has ground dual set equal to the ground set E of M, and the rank of any subset is found using the function r∗(X) = |X| + r(E∗ − X) − r(M). A basis of M∗ is is called a cobasis of M. Note that if B is a basis of M, then E −B is a cobasis of M. Similarly, the rank function, circuits and independent sets of M∗ are called the corank function, cocircuits and coindependent sets of M. Lemma 2.12. ([6, Proposition 2.1.7]) Let M be a matroid. Relax a circuit- relax hyperplane H of M to yield the matroid M(cid:48). Then (M(cid:48))∗ is identical to the matroid yielded from M∗ by relaxing the circuit-hyperplane E − H of M∗. Lemma 2.13. ([6, Proposition 3.3.5]) Let H be a cicuit-hyperplane of a matroid relax2 M, and let M(cid:48) be the matriod obtained from M by relaxing H. i. When e ∈ E(M) − H, M/e = M(cid:48)/e, and, unless e is a coloop of M, M(cid:48)\e is obtained from M\e by relaxing the circuit-hyperplane H of M\e. ii. Dually, when f ∈ H, M\ f = M(cid:48)\ f and, unless f is a loop of M, M(cid:48)/ f is obtained from M/ f by relaxing the circuit-hyperplane X − f of M/ f . 2.5 Transversals transversal Deﬁnition 2.14. Let S be any set. Take a family of subsets A = (A1,...,Ak) of S. A transversal or system of distinct representatives of A is a subset {s1,...,sm} of S such that si ∈ Ai for all i ∈ {1,...,m} and s1,...,sm are distinct. partial Deﬁnition 2.15. Let S be any set. X ⊆ S is a partial transversal of a family of subsets A = (A1,...,Aj) if X is a transversal of (A1,...,Ak) for some A1,...,Ak ⊆ S. transversals Lemma 2.16. Let A = (A1,...,Am) be a family of subsets of a set S. When A is a partition of S, the collection of partial transversals of A is the collection of independent sets of a matroid on S. This matroid is denoted by M A . [ ] If a matroid M is isomorphic to M[A ] for some family of subsets A , we say that M is a transversal matroid and that A is a presentation of M. Every transversal ##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: CHAPTER

text: FUNDAMENTALS

text: Deﬁnition

text: From

text: M

text: we

text: can

text: construct

text: the

text: dual

text: matroid

text: M∗.

text: This

text: has

text: ground

text: dual

text: set

text: equal

text: to

text: the

text: ground

text: set

text: E

text: of

text: M,

text: and

text: the

text: rank

text: of

text: any

text: subset

text: is

text: found

text: using

text: the

text: function

text: r∗(X)

text: |X|

text: r(E∗

text: X)

text: r(M).

text: A

text: basis

text: of

text: M∗

text: is

text: is

text: called

text: a

text: cobasis

text: of

text: M.

text: Note

text: that

text: if

text: B

text: is

text: a

text: basis

text: of

text: M,

text: then

text: E

text: −B

text: is

text: a

text: cobasis

text: of

text: M.

text: Similarly,

text: the

text: rank

text: function,

text: circuits

text: and

text: independent

text: sets

text: of

text: M∗

text: are

text: called

text: the

text: corank

text: function,

text: cocircuits

text: and

text: coindependent

text: sets

text: of

text: M.

text: Lemma

text: Proposition

text: Let

text: M

text: be

text: a

text: matroid.

text: Relax

text: a

text: circuit-

text: relax

text: hyperplane

text: H

text: of

text: M

text: to

text: yield

text: the

text: matroid

text: Then

text: is

text: identical

text: to

text: the

text: matroid

text: yielded

text: from

text: M∗

text: by

text: relaxing

text: the

text: circuit-hyperplane

text: E

text: H

text: of

text: M∗.

text: Lemma

text: Proposition

text: Let

text: H

text: be

text: a

text: cicuit-hyperplane

text: of

text: a

text: matroid

text: M,

text: and

text: let

text: be

text: the

text: matriod

text: obtained

text: from

text: M

text: by

text: relaxing

text: H.

text: i.

text: When

text: e

text: E(M)

text: H,

text: M/e

text: and,

text: unless

text: e

text: is

text: a

text: coloop

text: of

text: M,

text: is

text: obtained

text: from

text: M\e

text: by

text: relaxing

text: the

text: circuit-hyperplane

text: H

text: of

text: M\e.

text: ii.

text: Dually,

text: when

text: f

text: H,

text: M\

text: f

text: f

text: and,

text: unless

text: f

text: is

text: a

text: loop

text: of

text: M,

text: f

text: is

text: obtained

text: from

text: M/

text: f

text: by

text: relaxing

text: the

text: circuit-hyperplane

text: X

text: f

text: of

text: M/

text: f

text: Transversals

text: transversal

text: Deﬁnition

text: Let

text: S

text: be

text: any

text: set.

text: Take

text: a

text: family

text: of

text: subsets

text: A

text: of

text: S.

text: A

text: transversal

text: or

text: system

text: of

text: distinct

text: representatives

text: of

text: A

text: is

text: a

text: subset

text: of

text: S

text: such

text: that

text: si

text: Ai

text: for

text: all

text: i

text: and

text: are

text: distinct.

text: partial

text: Deﬁnition

text: Let

text: S

text: be

text: any

text: set.

text: X

text: S

text: is

text: a

text: partial

text: transversal

text: of

text: a

text: family

text: of

text: subsets

text: A

text: if

text: X

text: is

text: a

text: transversal

text: of

text: for

text: some

text: S.

text: transversals

text: Lemma

text: Let

text: A

text: be

text: a

text: family

text: of

text: subsets

text: of

text: a

text: set

text: S.

text: When

text: A

text: is

text: a

text: partition

text: of

text: S,

text: the

text: collection

text: of

text: partial

text: transversals

text: of

text: A

text: is

text: the

text: collection

text: of

text: independent

text: sets

text: of

text: a

text: matroid

text: on

text: S.

text: This

text: matroid

text: is

text: denoted

text: by

text: M

text: A

text: If

text: a

text: matroid

text: M

text: is

text: isomorphic

text: to

text: M[A

text: for

text: some

text: family

text: of

text: subsets

text: A

text: we

text: say

text: that

text: M

text: is

text: a

text: transversal

text: matroid

text: and

text: that

text: A

text: is

text: a

text: presentation

text: of

text: M.

text: Every

text: transversal

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CHAPTER 3. KINSER INEQUALITIES 19 n ∑ r(Xi ∩ E) + r((X1 ∪ X2) ∩ E) + r((X1 ∪ X3 ∪ Xn) ∩ E) i=3 n n + ∑ r((X2 ∪ Xi−1 ∪ Xi) ∩ E) + ∑ r(cid:48)(Xi ∩ E(cid:48)) + r(cid:48)((X1 ∪ X2) ∩ E(cid:48)) i=4 i=3 n + r(cid:48)((X1 ∪ X3 ∪ Xn) ∩ E(cid:48)) + ∑ r(cid:48)((X2 ∪ Xi−1 ∪ Xi) ∩ E(cid:48)) i=4 n ≤ r((X1 ∪ X3) ∩ E) + r((X1 ∪ Xn) ∩ E) + ∑ r((X2 ∪ Xi) ∩ E) i=3 n + ∑ r((Xi−1 ∪ Xi) ∩ E) + r(cid:48)((X1 ∪ X3) ∩ E(cid:48)) + r(cid:48)((X1 ∪ Xn) ∩ E(cid:48)) i=4 n n + ∑ r(cid:48)((X2 ∪ Xi) ∩ E(cid:48)) + ∑ r(cid:48)((Xi−1 ∪ Xi) ∩ E(cid:48)) i=3 i=4 (3.13.1) As M ∈ Kn, we have that n ∑ r(Xi ∩ E) + r((X1 ∪ X2) ∩ E) i=3 n + r((X1 ∪ X3 ∪ Xn) ∩ E) + ∑ r((X2 ∪ Xi−1 ∪ Xi) ∩ E) i=4 ≤ r((X1 ∪ X3) ∩ E) + r((X1 ∪ Xn) ∩ E) n n + ∑ r((X2 ∪ Xi) ∩ E) + ∑ r((Xi−1 ∪ Xi) ∩ E) i=3 i=4 As M(cid:48) ∈ Kn, we have that r(cid:48)(Xi ∩ E(cid:48)) + r(cid:48)((X1 ∪ X2) ∩ E(cid:48)) n + r(cid:48)((X1 ∪ X3 ∪ Xn) ∩ E(cid:48)) + ∑ r(cid:48)((X2 ∪ Xi−1 ∪ Xi) ∩ E(cid:48)) i=4 ≤ r(cid:48)((X1 ∪ X3) ∩ E(cid:48)) + r(cid:48)((X1 ∪ Xn) ∩ E(cid:48)) n n + ∑ r(cid:48)((X2 ∪ Xi) ∩ E(cid:48)) + ∑ r(cid:48)((Xi−1 ∪ Xi) ∩ E(cid:48)) i=3 i=4 The values of the terms on the left of inequality (3.0.21.1) are thus bounded by the terms on the right-hand side, and so the inequality holds. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHAPTER

text: KINSER

text: INEQUALITIES

text: n

label: equation

font: TVVISR+StandardSymL

text: r(Xi

text: E)

text: E)

text: Xn)

text: E)

text: n

text: n

text: Xi)

text: E)

text: n

text: Xn)

text: Xi)

text: n

text: E)

text: Xn)

text: E)

text: Xi)

text: E)

text: n

text: Xi)

text: E)

text: Xn)

text: n

text: n

text: Xi)

text: Xi)

text: As

text: M

text: Kn,

text: we

text: have

text: that

text: n

text: r(Xi

text: E)

text: E)

text: n

text: Xn)

text: E)

text: Xi)

text: E)

text: E)

text: Xn)

text: E)

text: n

text: n

text: Xi)

text: E)

text: Xi)

text: E)

text: As

text: Kn,

text: we

text: have

text: that

text: n

text: Xn)

text: Xi)

text: Xn)

text: n

text: n

text: Xi)

text: Xi)

text: The

text: values

text: of

text: the

text: terms

text: on

text: the

text: left

text: of

text: inequality

text: are

text: thus

text: bounded

text: by

text: the

text: terms

text: on

text: the

text: right-hand

text: side,

text: and

text: so

text: the

text: inequality

text: holds.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CHAPTER 4. KINSER HIERARCHY 25 K 4 Kin(4)− K 5 Kin(5)− K n Kin(6)− K∞ Representable Figure 4.2: Kinser classes (2) Next we show that the ﬁrst Kinser class, K , is dual-closed. 4 Lemma 4.4. Let M be an excluded minor for the class K . Let X ,...,X be a 4 1 4 bad family in M for inequality 4. Then X1,...,X4 is a partition of E(M). Proof. Suppose there is an element of E(M) which is not contained in some Xi. We could then delete this element and X1,...,Xn would still form a bad fam- ily, contradicting M being minor-minimal with respect to not being in K . Thus 4 X1,...,Xn must cover the entire ground set. Now assume the sets in the bad family are not disjoint, so there exists an x which is in Xi ∩ Xj for some i, j. Contract x, and for each set Xk, let Xk(cid:48) = Xk − x. Let L be some union of the Xi, so that r(L) appears somewhere in the inequality. Say that L is stable if rM/x(L − x) = rM(L). If none of the ﬁve terms on the right-hand side of the inequality are stable, then the sum on the right-hand side of the inequality decreases by exactly ﬁve when we contract x and remove x from each term. Since the left-hand side can decrease by at most ﬁve, this means that X1/x,...,Xn/x is a bad family in M/x, which is impossible as M is an excluded minor for K4, and so M/x ∈ K4. Therefore there is a stable set on the right-hand side. As x ∈ Xi ∩ Xj, the rank of any term that includes Xi or Xj decreases when we ##LTFigure##

text: CHAPTER

text: KINSER

text: HIERARCHY

text: K

font: default

text: K

text: K

text: n

text: K∞

text: Representable

text: Figure

text: Kinser

label: caption

text: classes

text: Next

text: we

text: show

text: that

text: the

text: ﬁrst

text: Kinser

text: class,

text: K

text: is

text: dual-closed.

text: Lemma

text: Let

text: M

text: be

text: an

text: excluded

text: minor

text: for

text: the

text: class

text: K

text: Let

text: X

text: ,...,X

text: be

text: a

text: bad

text: family

text: in

text: M

text: for

text: inequality

text: Then

text: is

text: a

text: partition

text: of

text: E(M).

text: Proof.

text: Suppose

text: there

text: is

text: an

text: element

text: of

text: E(M)

text: which

text: is

text: not

text: contained

text: in

text: some

text: Xi.

text: We

text: could

text: then

text: delete

text: this

text: element

text: and

text: would

text: still

text: form

text: a

text: bad

text: fam-

text: ily,

text: contradicting

text: M

text: being

text: minor-minimal

text: with

text: respect

text: to

text: not

text: being

text: in

text: K

text: Thus

text: must

text: cover

text: the

text: entire

text: ground

text: set.

text: Now

text: assume

text: the

text: sets

text: in

text: the

text: bad

text: family

text: are

text: not

text: disjoint,

text: so

text: there

text: exists

text: an

text: x

text: which

text: is

text: in

text: Xi

text: Xj

text: for

text: some

text: i,

text: j.

text: Contract

text: x,

text: and

text: for

text: each

text: set

text: Xk,

text: let

text: Xk

text: x.

text: Let

text: L

text: be

text: some

text: union

text: of

text: the

text: Xi,

text: so

text: that

text: r(L)

text: appears

text: somewhere

text: in

text: the

text: inequality.

text: Say

text: that

text: L

text: is

text: stable

text: if

text: rM/x(L

text: x)

text: rM(L).

text: If

text: of

text: the

text: ﬁve

text: terms

text: on

text: the

text: right-hand

text: side

text: of

text: the

text: inequality

text: are

text: stable,

text: then

text: the

text: sum

text: on

text: the

text: right-hand

text: side

text: of

text: the

text: inequality

text: decreases

text: by

text: exactly

text: ﬁve

text: when

text: we

text: contract

text: x

text: and

text: remove

text: x

text: from

text: each

text: term.

text: Since

text: the

text: left-hand

text: side

text: can

text: decrease

text: by

text: at

text: most

text: ﬁve,

text: this

text: means

text: that

text: is

text: a

text: bad

text: family

text: in

text: M/x,

text: which

text: is

text: impossible

text: as

text: M

text: is

text: an

text: excluded

text: minor

text: for

text: and

text: so

text: M/x

text: Therefore

text: there

text: is

text: a

text: stable

text: set

text: on

text: the

text: right-hand

text: side.

text: As

text: x

text: Xi

text: Xj,

text: the

text: rank

text: of

text: any

text: term

text: that

text: includes

text: Xi

text: or

text: Xj

text: decreases

text: when

text: we

text: ##LTFigure##


text: CHAPTER 5. K5 (cid:54)= K5∗ 35 X X 1 1 X X X X X X 5 2 3 5 2 3 X X 4 4 Subcase 1 Subcase 2 X X 1 1 X X X X X X 5 2 3 5 2 3 X X 4 4 Subcase 3 Subcase 4 In each subcase examined below, we are considering the circuit-hyperplanes to lie as indicated by the diagram at the start of each subcase. In all of these, there are multiple options for which elements are in what sets, as described below. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHAPTER

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: Subcase

text: Subcase

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: X

text: Subcase

text: Subcase

text: In

text: each

text: subcase

text: examined

text: below,

text: we

text: are

text: considering

text: the

text: circuit-hyperplanes

text: to

text: lie

text: as

text: indicated

text: by

text: the

text: diagram

text: at

text: the

text: start

text: of

text: each

text: subcase.

text: In

text: all

text: of

text: these,

text: there

text: are

text: multiple

text: options

text: for

text: which

text: elements

text: are

text: in

text: what

text: sets,

text: as

text: described

text: below.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CHAPTER 5. K5 (cid:54)= K5∗ 38 corank of X2 ∪X3 ∪X4 = V1 ∪V2 ∪Vi ∪Vj ∪Z must be unchanged as its complement Vk − Z is independent. Also, r∗(X3 ∪ X4) + 2 ≥ r∗(X3) + r∗(X4) − r∗(X3 ∩ X4) + 2 ≥ r∗(X3) + r∗(X4) − r∗(Z ∩ X4) + 2 ≥ r∗(X3) + 2 ≥ r∗(X3) Thus increasing the size of Z can only cause the left-hand side to decrease by more than the right-hand side, and so we cannot have a bad family for any choice of Z. Subcase 1.1b: X1 = Vj ∪Vk ∪ {a} where a ∈ Vi X V 2 = i X3 = V2 ∪ Z where Z ⊆ E(M) X4 = V1 ∪Vj X5 = V1 ∪Vk The only terms whose value changes from before is |X1 ∪ X5| which increases by one. So we must have that r∗(X3) + r∗(X1 ∪ X3 ∪ X4) + r∗(X2 ∪ X3 ∪ X4) > r∗(X1 ∪ X3) + r∗(X2 ∪ X3) + r∗(X3 ∪ X4) + 3 The same argument as before shows that we do not have a bad family, for any choice of Z. Subcase 1c: X1 = Vj ∪Vk ∪ {a,b} where a,b ∈ Vi X V 2 = i X3 = V2 ∪ Z where Z ⊆ E(M) X4 = V1 ∪Vj X5 = V1 ∪Vk Here, |X1 ∪ X5| is one higher than in the previous subcase, while r(X1 ∪ X5) is one lower, and the inequality is thus identical to subcase 1b. ##LTLine##

text: CHAPTER

text: corank

text: of

text: ∪Vi

text: ∪Vj

text: ∪Z

text: must

text: be

text: unchanged

text: as

text: its

text: complement

text: Vk

text: Z

text: is

text: independent.

text: Also,

text: r∗(Z

text: Thus

text: increasing

text: the

text: size

text: of

text: Z

text: can

text: only

text: cause

text: the

text: left-hand

text: side

text: to

text: decrease

text: by

text: more

text: than

text: the

text: right-hand

text: side,

text: and

text: so

text: we

text: cannot

text: have

text: a

text: bad

text: family

text: for

text: any

text: choice

text: of

text: Z.

text: Subcase

label: list

text: Vj

text: ∪Vk

text: {a}

text: where

text: a

text: Vi

text: X

text: V

text: i

text: Z

text: where

text: Z

text: E(M)

text: ∪Vj

text: ∪Vk

text: The

text: only

text: terms

text: whose

text: value

text: changes

text: from

text: before

text: is

text: which

text: increases

text: by

text: one.

text: So

text: we

text: must

text: have

text: that

text: The

text: same

text: argument

text: as

text: before

text: shows

text: that

text: we

text: do

text: not

text: have

text: a

text: bad

text: family,

text: for

text: any

text: choice

text: of

text: Z.

text: Subcase

text: Vj

text: ∪Vk

text: {a,b}

text: where

text: a,b

text: Vi

text: X

text: V

text: i

text: Z

text: where

text: Z

text: E(M)

text: ∪Vj

text: ∪Vk

text: Here,

text: is

text: one

text: higher

text: than

text: in

text: the

text: previous

text: subcase,

text: while

text: is

text: one

text: lower,

text: and

text: the

text: inequality

text: is

text: thus

text: identical

text: to

text: subcase

text: ##LTLine##


text: CHAPTER 5. K5 (cid:54)= K5∗ 58 Note that X2 ∪ X3 ∪ X4 = X2 ∪ X4 ∪ X5 = V1 ∪Vi ∪Vj ∪Vk = V2. LHS = r∗(X3) + r∗(X4) + r∗(X5) + r∗(X1 ∪ X2) + r∗(X1 ∪ X3 ∪ X5) + r∗(X2 ∪ X3 ∪ X4) + r∗(X2 ∪ X4 ∪ X5) = r∗(X1 ∪ X2) + r∗(X1 ∪ X3 ∪ X5) + |X3| + |X4| + |X5| + |X2 ∪ X3 ∪ X4| + |X2 ∪ X4 ∪ X5| + r(X3) + r(X4) + r(X5) + r(X2 ∪ X3 ∪ X4) + r(X2 ∪ X4 ∪ X5) − 5r(M) = r∗(X1 ∪ X2) + r∗(X1 ∪ X3 ∪ X5) − 5r(M) + 9 + 6 + 4 + 12 + 12 + 5 + 5 + 5 + 2 + 2 = r∗(X1 ∪ X2) + r∗(X1 ∪ X3 ∪ X5) − 5r(M) + 62 RHS = r∗(X1 ∪ X3) + r∗(X1 ∪ X5) + r∗(X2 ∪ X3) + r∗(X2 ∪ X4) + r∗(X2 ∪ X5) + r∗(X3 ∪ X4) + r∗(X4 ∪ X5) = r∗(X1 ∪ X3) + r∗(X1 ∪ X5) + |X2 ∪ X3| + |X2 ∪ X4| + |X2 ∪ X5| + |X3 ∪ X4| + |X4 ∪ X5| + r(X2 ∪ X3) + r(X2 ∪ X4) + r(X2 ∪ X5) + r(X3 ∪ X4) + r(X4 ∪ X5) − 5r(M) = r∗(X1 ∪ X3) + r∗(X1 ∪ X5) − 5r(M) + 12 + 9 + 9 + 12 + 9 + 2 + 4 + 4 + 2 + 4 = r∗(X1 ∪ X3) + r∗(X1 ∪ X5) − 5r(M) + 67 We thus must have that r∗(X1 ∪ X2) + r∗(X1 ∪ X3 ∪ X5) > r∗(X1 ∪ X3) + r∗(X1 ∪ X5) + 5 in order for this to be a bad family. ∅ Suppose Z = . We have that 7 + 9 > 9 + 6 + 5 which is untrue. Now suppose ∅ Z (cid:54)= . We have shown in Subcase 4.1a that r∗(X1 ∪ X3) cannot change, and that r∗(X1 ∪ X2) can increase by at most two. The sets X1,...,X3 are the same in the current subcase and hence the same facts apply. Note that X1 ∪ X3 ∪ X5 is equal to the entire ground set, and thus changing Z will have no effect on this term, as in Subcase 4.1a. Finally, note that X1 ∪ X5 = {a} ∪V2 ∪Vj ∪ Z and X1 ∪ X5 = ({b,c} ∪ Vi ∪ Vk) − Z. As {b,c} ∪ Vi ∪ Vk is a dependent set of rank 5, we can remove at most three elements from it without affecting the rank. Thus we can ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHAPTER

text: Note

text: that

text: ∪Vi

text: ∪Vj

text: ∪Vk

text: LHS

text: RHS

text: We

text: thus

text: must

text: have

text: that

text: in

text: order

text: for

text: this

text: to

text: be

text: a

text: bad

text: family.

text: Suppose

text: Z

text: We

text: have

text: that

text: which

text: is

text: untrue.

text: Now

text: suppose

text: Z

text: We

text: have

text: shown

text: in

text: Subcase

text: that

text: cannot

text: change,

text: and

text: that

text: can

text: increase

text: by

text: at

text: most

text: two.

text: The

text: sets

text: are

text: the

text: same

text: in

text: the

text: current

text: subcase

text: and

text: hence

text: the

text: same

text: facts

text: apply.

text: Note

text: that

text: is

text: equal

text: to

text: the

text: entire

text: ground

text: set,

text: and

text: thus

text: changing

text: Z

text: will

text: have

text: no

text: effect

text: on

text: this

text: term,

text: as

text: in

text: Subcase

text: Finally,

text: note

text: that

text: {a}

text: ∪Vj

text: Z

text: and

text: ({b,c}

text: Vi

text: Vk)

text: Z.

text: As

text: {b,c}

text: Vi

text: Vk

text: is

text: a

text: dependent

text: set

text: of

text: rank

text: we

text: can

text: remove

text: at

text: most

text: three

text: elements

text: from

text: it

text: without

text: affecting

text: the

text: rank.

text: Thus

text: we

text: can

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: CHAPTER 5. K5 (cid:54)= K5∗ 59 increase |X1 ∪ X5| by three without affecting r∗(X1 ∪ X5), but, after that, X1 ∪ X5 is coindependent, causing r∗(X1 ∪ X5) to be spanning for all such Z. Thus the right-hand side of the inequality can increase by at most three when we make Z non-empty. The left-hand side can increase by at most two, and so the inequality still cannot be satisﬁed. Subcase 4.2b: X1 = V2 ∪ Z where Z ⊆ E(M) X2 = Vi ∪ {a} where a ∈ V1 X3 = V3 ∪V4 ∪V5 X4 = Vk ∪V1 X5 = Vj ∪V1 Compared to Subcase 4.2a, X contains two less elements of V while X contains 2 1 5 two more. Consider the left-hand side of the inequality. Any terms not involving X or X are 2 5 unchanged from Subcase 4.2a. |X5| increases by two in comparision to Subcase 4.2a, while r(X5) remains the same, causing r∗(X5) to increase by two. As V1 ∈ X4, r∗(X2 ∪ X3 ∪ X4) and r∗(X2 ∪ X4 ∪ X5) remain the same. Note that X1 ∪ X3 ∪ X5 is equal to the entire ground set and thus must be spanning, as in Subcase 4.2a. Now note that X1 ∪ X2 = {a} ∪ V2 ∪ Vi ∪ Z and X1 ∪ X2 = ({b,c} ∪ Vi ∪ Vk) − Z. As {b,c}∪Vi ∪Vk is a dependent set of rank 5, we can remove at most three elements from it without affecting the rank. Thus we can increase |X1 ∪X2| by three without affecting r∗(X1 ∪X2), but, after that, X1 ∪ X2 is coindependent, causing r∗(X1 ∪X2) to be spanning for all such Z.. The left-hand side can thus increase by at most ﬁve in comparision to Subcase 4.2a. Take the right-hand side of the inequality. r∗(X2 ∪ X5) is unchanged. As V1 ⊂ X4, r∗(X2 ∪ X4) and r∗(X4 ∪ X5) are also unchanged. |X2 ∪ X3| decreases by two, but r(X2 ∪ X3) increases by two, meaning that r∗(X2 ∪ X3) is unchanged. Finally, take X1 ∪ X5 = V1 ∪V2 ∪Vj ∪ Z, where X1 ∪ X5 = (Vi ∪Vk) − Z. Suppose that, to begin ∅ with, Z = . In Subcase 4.2a, r∗(X1 ∪ X5) = 6. As X1 ∪ X5 is still spanning in the current subcase, r∗(X1 ∪ X5) increases by two due to the increase in |X1 ∪ X2|. Now suppose Z is non-empty. As Vi ∪Vk has rank 5, we can increase |X1 ∪ X5| by one without decreasing r(X1 ∪ X5). For any Z ⊆ Vi ∪Vk with a cardinality greater than or equal to one, X1 ∪ X5 is coindependent. This means X1 ∪ X5 is spanning ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHAPTER

text: increase

text: by

text: three

text: without

text: affecting

text: but,

text: after

text: that,

text: is

text: coindependent,

text: causing

text: to

text: be

text: spanning

text: for

text: all

text: such

text: Z.

text: Thus

text: the

text: right-hand

text: side

text: of

text: the

text: inequality

text: can

text: increase

text: by

text: at

text: most

text: three

text: when

text: we

text: make

text: Z

text: non-empty.

text: The

text: left-hand

text: side

text: can

text: increase

text: by

text: at

text: most

text: two,

text: and

text: so

text: the

text: inequality

text: still

text: cannot

text: be

text: satisﬁed.

text: Subcase

text: Z

text: where

text: Z

text: E(M)

text: Vi

text: {a}

text: where

text: a

text: Vk

text: Vj

text: Compared

text: to

text: Subcase

text: X

text: contains

text: two

text: less

text: elements

text: of

text: V

text: while

text: X

text: contains

text: two

text: more.

text: Consider

text: the

text: left-hand

text: side

text: of

text: the

text: inequality.

text: Any

text: terms

text: not

text: involving

text: X

text: or

text: X

text: are

text: unchanged

text: from

text: Subcase

text: increases

text: by

text: two

text: in

text: comparision

text: to

text: Subcase

text: while

text: remains

text: the

text: same,

text: causing

text: to

text: increase

text: by

text: two.

text: As

text: and

text: remain

text: the

text: same.

text: Note

text: that

text: is

text: equal

text: to

text: the

text: entire

text: ground

text: set

text: and

text: thus

text: must

text: be

text: spanning,

text: as

text: in

text: Subcase

text: Now

text: note

text: that

text: {a}

text: Vi

text: Z

text: and

text: ({b,c}

text: Vi

text: Vk)

text: Z.

text: As

text: {b,c}∪Vi

text: ∪Vk

text: is

text: a

text: dependent

text: set

text: of

text: rank

text: we

text: can

text: remove

text: at

text: most

text: three

text: elements

text: from

text: it

text: without

text: affecting

text: the

text: rank.

text: Thus

text: we

text: can

text: increase

text: by

text: three

text: without

text: affecting

text: but,

text: after

text: that,

text: is

text: coindependent,

text: causing

text: to

text: be

text: spanning

text: for

text: all

text: such

text: Z..

text: The

text: left-hand

text: side

text: can

text: thus

text: increase

text: by

text: at

text: most

text: ﬁve

text: in

text: comparision

text: to

text: Subcase

text: Take

text: the

text: right-hand

text: side

text: of

text: the

text: inequality.

text: is

text: unchanged.

text: As

text: and

text: are

text: also

text: unchanged.

text: decreases

text: by

text: two,

text: but

text: increases

text: by

text: two,

text: meaning

text: that

text: is

text: unchanged.

text: Finally,

text: take

text: ∪Vj

text: Z,

text: where

text: (Vi

text: ∪Vk)

text: Z.

text: Suppose

text: that,

text: to

text: begin

text: with,

text: Z

text: In

text: Subcase

text: As

text: is

text: still

text: spanning

text: in

text: the

text: current

text: subcase,

text: increases

text: by

text: two

text: due

text: to

text: the

text: increase

text: in

text: Now

text: suppose

text: Z

text: is

text: non-empty.

text: As

text: Vi

text: ∪Vk

text: has

text: rank

text: we

text: can

text: increase

text: by

text: one

text: without

text: decreasing

text: For

text: any

text: Z

text: Vi

text: ∪Vk

text: with

text: a

text: cardinality

text: greater

text: than

text: or

text: equal

text: to

text: one,

text: is

text: coindependent.

text: This

text: means

text: is

text: spanning

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Chapter 8 Conjectures Finally, we give some conjectures on the hierachy of the Kinser classes. Conjecture 8.1. Let n > 5. Kn (cid:54)= Kn∗. As shown in the previous chapter, verifying that a matroid satisﬁes a Kinser in- equality is very difﬁcult. Given the amount of difﬁculty involved in proving that the ﬁfth Kinser class is not dual closed, proving this result in general would in- volve an even greater amount of work. Based on that case, however, we give a strengthening of the above conjecture. Conjecture 8.2. Kin(n)− ∈ Kn∗ − Kn One further question about the structure of the hierarchy is how each dual class sits within the previous Kinser class. There are two possibilities here, and we conjecture that the following is true. conj Conjecture 8.3. Let n > 4. Kn∗+1 ⊆ Kn. The following conjecture is true if Conjecture 8.3 is as well. Conjecture 8.4. K∞∗ = K∞ To see that this follows from Conjecture 8.3, assume M ∈ K∞, but M (cid:54)∈ K∞∗. Then there exists an integer n such that M (cid:54)∈ K ∗. However, this contradicts Conjecture n 8.3, which gives that M ∈ Kn+1 ⊆ Kn∗. Assuming these conjectures to hold true, we have a ﬁnal diagram of the hierarchy. 74 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Chapter

text: Conjectures

text: Finally,

text: we

text: give

text: some

text: conjectures

text: on

text: the

text: hierachy

text: of

text: the

text: Kinser

text: classes.

text: Conjecture

text: Let

text: n

text: Kn

text: Kn∗.

text: As

text: shown

text: in

text: the

text: previous

text: chapter,

text: verifying

text: that

text: a

text: matroid

text: satisﬁes

text: a

text: Kinser

text: in-

text: equality

text: is

text: very

text: difﬁcult.

text: Given

text: the

text: amount

text: of

text: difﬁculty

text: involved

text: in

text: proving

text: that

text: the

text: ﬁfth

text: Kinser

text: class

text: is

text: not

text: dual

text: closed,

text: proving

text: this

text: result

text: in

text: general

text: would

text: in-

text: volve

text: an

text: even

text: greater

text: amount

text: of

text: work.

text: Based

text: on

text: that

text: case,

text: however,

text: we

text: give

text: a

text: strengthening

text: of

text: the

text: above

text: conjecture.

text: Conjecture

text: Kin(n)−

text: Kn∗

text: Kn

text: One

text: further

text: question

text: about

text: the

text: structure

text: of

text: the

text: hierarchy

text: is

text: how

text: each

text: dual

text: class

text: sits

text: within

text: the

text: previous

text: Kinser

text: class.

text: There

text: are

text: two

text: possibilities

text: here,

text: and

text: we

text: conjecture

text: that

text: the

text: following

text: is

text: true.

text: conj

text: Conjecture

text: Let

text: n

text: Kn.

text: The

text: following

text: conjecture

text: is

text: true

text: if

text: Conjecture

text: is

text: as

text: well.

text: Conjecture

text: K∞∗

text: K∞

text: To

text: see

text: that

text: this

text: follows

text: from

text: Conjecture

text: assume

text: M

text: K∞,

text: but

text: M

text: K∞∗.

text: Then

text: there

text: exists

text: an

text: integer

text: n

text: such

text: that

text: M

text: K

text: However,

text: this

text: contradicts

text: Conjecture

text: n

text: which

text: gives

text: that

text: M

text: Kn∗.

text: Assuming

text: these

text: conjectures

text: to

text: hold

text: true,

text: we

text: have

text: a

text: ﬁnal

text: diagram

text: of

text: the

text: hierarchy.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: We evaluate the performance of the methods for SUC on the dataset gathered by (Tur et al., 2012). It was compiled from utterances by users of a spoken dialog system. There are 16,000 training utterances, 2000 utterances for validation and 2000 utterances for testing. Each utterance is labelled with one of 25 domains. The hyper-parameters of the models are tuned on the validation set. The learning rate parameter of { } gradient descent is found by grid search with 0.1, 0.01, 0.001 . The number of layers is between 1 and 3. The number of hidden units is kept constant through layers and is found by sampling a random number from 300 to 800 units. We found that it was helpful to regularize the networks using dropout (Hinton et al., 2012). We sample the dropout rate randomly between 0% dropout and 20%. { } The λ of the zero-shot embedding method is found through grid-search with 0.1, 0.01, 0.001 . The models are trained on a cluster of computers with double quad-core Intel(R) Xeon(R) CPUs with 2.33GHz and 8Gb of RAM. Training either the ZDE method on the QCL data requires 4 hours of computation time. 8.2 Results First, we want to see what is learned by the embedding method described in Section . A ﬁrst step is to look at the nearest neighbor of words in the embedding space. Table 1 shows the nearest neighbours of speciﬁc words in the embedding space. We observe that the neighbors of the words al share the semantic domain of the word. This conﬁrms that the network learns some semantics of the language. Restaurant Hotel Flight Events Transportation steakhouse suites airline festivals distributing diner hyatt airfaire upcoming dfw seafood resorts plane ﬁreworks petroleum tavern ramada baggage happening hospitality Table 1: Nearest neighbours in the embedding space. Each column displays the 5 nearest neigh- bours of the word at the top. We can see that the embedding captures the semantics of the words. We can better visualize the embedding space using a network with a special architecture. Following (Hinton and Salakhutdinov, 2006), we train deep networks where the last hidden layer contains only 2 dimensions. The depth allows the network to progressively reduce the dimensionality of the data. This approach enables us to visualize exactly what the network has learned. Figure 1 shows the embedding a deep network with 3 layers (with size 200-10-2) trained on the QCL task. We observe that the embedding distinguishes between sentences related to movies and hotels. In Figure 3, we compare the embedding spaces of a DNN trained on the QCL (left) and a DNN trained using ZDE (right) both with hidden layers of sizes 200-10-2. The comparison suggests that minimizing the conditional entropy of the zero-shot classiﬁer successfully improves the clustering. Method Restaurant Hotel Flight Events Transportation ZSL with Bag-of-words 0.616 0.641 0.683 0.559 0.5 | ZSL with p(Y X) (LR) 0.779 0.821 0.457 0.677 0.472 | ZSL with p(Y X) (DNN) 0.838 0.862 0.46 0.631 0.503 ZSL with DNN Embedding 0.858 0.935 0.870 0.727 0.667 ZSL with ZDE Embedding 0.863 0.940 0.906 0.841 0.826 Representative URL heuristic (DNN) 0.798 0.892 0.769 0.707 0.577 Table 2: Comparison of several zero-shot semantic learning methods for 5 semantic classes. Our proposed zero-shot learning system with DNN embeddings outperforms other approaches. Second, we want to conﬁrm that good classiﬁcation results can be achieved using zero-shot semantic learning. To do this, we evaluate the classiﬁcation results of our method on the SUC task. Our results are given in Table 2. The performance is measured using the AUC (Area under the curve of the precision-recall curve) for which higher is better. We compare our ZDE method against various means of obtaining the semantic features H. We compare with using the bag-of-words | representation (denoted ZSL with Bag-of-words) as semantic features. ZSL with p(Y X) (LR) and | ZSL with p(Y X) (DNN) are models trained from the QCL to predict the website associated with 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: We

text: evaluate

text: the

text: performance

text: of

text: the

text: methods

text: for

text: SUC

text: on

text: the

text: dataset

text: gathered

text: by

text: (Tur

text: et

text: al.,

text: It

text: was

text: compiled

text: from

text: utterances

text: by

text: users

text: of

text: a

text: spoken

text: dialog

text: system.

text: There

text: are

text: training

text: utterances,

text: utterances

text: for

text: validation

text: and

text: utterances

text: for

text: testing.

text: Each

text: utterance

text: is

text: labelled

text: with

text: one

text: of

text: domains.

text: The

text: hyper-parameters

text: of

text: the

text: models

text: are

text: tuned

text: on

text: the

text: validation

text: set.

text: The

text: learning

text: rate

text: parameter

text: of

text: gradient

text: descent

text: is

text: found

text: by

text: grid

text: search

text: with

text: The

text: number

text: of

text: layers

text: is

text: between

text: and

text: The

text: number

text: of

text: hidden

text: units

text: is

text: kept

text: constant

text: through

text: layers

text: and

text: is

text: found

text: by

text: sampling

text: a

text: random

text: number

text: from

text: to

text: units.

text: We

text: found

text: that

text: it

text: was

text: helpful

text: to

text: regularize

text: the

text: networks

text: using

text: dropout

text: (Hinton

text: et

text: al.,

text: We

text: sample

text: the

text: dropout

text: rate

text: randomly

text: between

text: dropout

text: and

text: The

text: λ

text: of

text: the

text: zero-shot

text: embedding

text: method

text: is

text: found

text: through

text: grid-search

text: with

text: The

text: models

text: are

text: trained

text: on

text: a

text: cluster

text: of

text: computers

text: with

text: double

text: quad-core

text: Intel(R)

text: Xeon(R)

text: CPUs

text: with

text: and

text: of

text: RAM.

text: Training

text: either

text: the

text: ZDE

text: method

text: on

text: the

text: QCL

text: data

text: requires

text: hours

text: of

text: computation

text: time.

text: Results

text: First,

text: we

text: want

text: to

text: see

text: what

text: is

text: learned

text: by

text: the

text: embedding

text: method

text: described

text: in

text: Section

text: A

text: ﬁrst

text: step

text: is

text: to

text: look

text: at

text: the

text: nearest

text: neighbor

text: of

text: words

text: in

text: the

text: embedding

text: space.

text: Table

text: shows

text: the

text: nearest

text: neighbours

text: of

text: speciﬁc

text: words

text: in

text: the

text: embedding

text: space.

text: We

text: observe

text: that

text: the

text: neighbors

text: of

text: the

text: words

text: al

text: share

text: the

text: semantic

text: domain

text: of

text: the

text: word.

text: This

text: conﬁrms

text: that

text: the

text: network

text: learns

text: some

text: semantics

text: of

text: the

text: language.

text: Restaurant

text: Hotel

text: Flight

text: Events

text: Transportation

text: steakhouse

text: suites

text: airline

text: festivals

text: distributing

text: diner

text: hyatt

text: airfaire

text: upcoming

text: dfw

text: seafood

text: resorts

text: plane

text: ﬁreworks

text: petroleum

text: tavern

text: ramada

text: baggage

text: happening

text: hospitality

text: Table

text: Nearest

text: neighbours

text: in

text: the

text: embedding

text: space.

text: Each

text: column

text: displays

text: the

text: nearest

text: neigh-

text: bours

text: of

text: the

text: word

text: at

text: the

text: top.

text: We

text: can

text: see

text: that

text: the

text: embedding

text: captures

text: the

text: semantics

text: of

text: the

text: words.

text: We

text: can

text: better

text: visualize

text: the

text: embedding

text: space

text: using

text: a

text: network

text: with

text: a

text: special

text: architecture.

text: Following

text: (Hinton

text: and

text: Salakhutdinov,

text: we

text: train

text: deep

text: networks

text: where

text: the

text: last

text: hidden

text: layer

text: contains

text: only

text: dimensions.

text: The

text: depth

text: allows

text: the

text: network

text: to

text: progressively

text: reduce

text: the

text: dimensionality

text: of

text: the

text: data.

text: This

text: approach

text: enables

text: us

text: to

text: visualize

text: exactly

text: what

text: the

text: network

text: has

text: learned.

text: Figure

text: shows

text: the

text: embedding

text: a

text: deep

text: network

text: with

text: layers

text: (with

text: size

text: trained

text: on

text: the

text: QCL

text: task.

text: We

text: observe

text: that

text: the

text: embedding

text: distinguishes

text: between

text: sentences

text: related

text: to

text: movies

text: and

text: hotels.

text: In

text: Figure

text: we

text: compare

text: the

text: embedding

text: spaces

text: of

text: a

text: DNN

text: trained

text: on

text: the

text: QCL

text: (left)

text: and

text: a

text: DNN

text: trained

text: using

text: ZDE

text: (right)

text: both

text: with

text: hidden

text: layers

text: of

text: sizes

text: The

text: comparison

text: suggests

text: that

text: minimizing

text: the

text: conditional

text: entropy

text: of

text: the

text: zero-shot

text: classiﬁer

text: successfully

text: improves

text: the

text: clustering.

text: Method

text: Restaurant

text: Hotel

text: Flight

text: Events

text: Transportation

text: ZSL

text: with

text: Bag-of-words

text: ZSL

text: with

text: p(Y

text: X)

text: (LR)

text: ZSL

text: with

text: p(Y

text: X)

text: (DNN)

text: ZSL

text: with

text: DNN

text: Embedding

text: ZSL

text: with

text: ZDE

text: Embedding

text: Representative

text: URL

text: heuristic

text: (DNN)

text: Table

text: Comparison

text: of

text: several

text: zero-shot

text: semantic

text: learning

text: methods

text: for

text: semantic

text: classes.

text: Our

text: proposed

text: zero-shot

text: learning

text: system

text: with

text: DNN

text: embeddings

text: outperforms

text: other

text: approaches.

text: Second,

text: we

text: want

text: to

text: conﬁrm

text: that

text: good

text: classiﬁcation

text: results

text: can

text: be

text: achieved

text: using

text: zero-shot

text: semantic

text: learning.

text: To

text: do

text: this,

text: we

text: evaluate

text: the

text: classiﬁcation

text: results

text: of

text: our

text: method

text: on

text: the

text: SUC

text: task.

text: Our

text: results

text: are

text: given

text: in

text: Table

text: The

text: performance

text: is

text: measured

text: using

text: the

text: AUC

text: (Area

text: under

text: the

text: curve

text: of

text: the

text: precision-recall

text: curve)

text: for

text: which

text: higher

text: is

text: better.

text: We

text: compare

text: our

text: ZDE

text: method

text: against

text: various

text: means

text: of

text: obtaining

text: the

text: semantic

text: features

text: H.

text: We

text: compare

text: with

text: using

text: the

text: bag-of-words

text: representation

text: (denoted

text: ZSL

text: with

text: Bag-of-words)

text: as

text: semantic

text: features.

text: ZSL

text: with

text: p(Y

text: X)

text: (LR)

text: and

text: ZSL

text: with

text: p(Y

text: X)

text: (DNN)

text: are

text: models

text: trained

text: from

text: the

text: QCL

text: to

text: predict

text: the

text: website

text: associated

text: with

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:28) 0 < (cid:15) 1. (cid:99) (cid:98) (cid:99) (cid:98) Y → X Y X ∪ ∪ Let f : be a resolution of singularities so that is smooth and Θ E is a simple X − X − 0 normal crossing divisor where = f∗ 1( ), Θ = f∗ 1(Θ), and E is the exceptional locus of f. 0 0 Such a resolution is guaranteed by Hironaka [4]. (cid:28) − X X Fix 0 < (cid:15) 1 and put d = 1 (cid:15). Let c be the log canonical threshold so that ( , c(Θ+d )) X (cid:48) X (cid:48) 0 is log canonical but ( , c (Θ + d )) is not log canonical for any c > c. 0 (cid:48) Seeking a contradiction, suppose that c < 1 and put cd = d . This means that we may write the following equation: (cid:98) (cid:99) (cid:48)X ∗ (cid:48)X KY + cΘ + d = f (KX + cΘ + d ) + D 0 0 − − where D is exceptional. Write D as D = B A ∆ where A, B, ∆ are eﬀective, A, B are integral (cid:98) (cid:99) and nonzero, ∆ = 0, and A, B have no irreducible components in common. Now the equation becom(cid:98)es: (cid:99) (cid:48)X ∗ (cid:48)X − − KY + cΘ + d = f (KX + cΘ + d ) + B A ∆ 0 0 Which we can rewrite as: (cid:98) (cid:99) − ∗ (cid:48)X − − (cid:48)X f (KX + cΘ + d ) = B A (KY + cΘ + d + ∆) 0 0 ∗ Adding f (KX + Θ + KX ) to each side: 0 (cid:98) (cid:99) ∗ − − (cid:48) X ∗ X − − (cid:48)X f ((1 c)Θ + (1 d ) ) = (f (KX + Θ + ) + B A) (KY + cΘ + d + ∆) 0 0 0 Now we would like to apply Theorem 3 to the above situation. The conclusion is: O ∗ X − Rpπ∗Rqf∗ Y(f (KX + Θ + ) + B A) = 0 ((cid:63)) 0 ≥ for q 0 and p > 0. O − ⊂ O Since B is exceptional and A is non-zero, f∗ Y(B A) X is a non-trivial ideal sheaf. I ⊂ X I Call this ideal and let Z be the subscheme determined by . Consider the short exact sequence: → I → O → O → 0 X 0 Z O Tensoring by X (Θ) we get: → O ⊗ I → O → O → 0 X (Θ) X (Θ) (Θ) 0 Z Applying π∗, we arrive at the long exact sequence: → O ⊗ I → O → O −→ O ⊗ I → 0 π∗( X (Θ) ) π∗( X (Θ)) π∗( (Θ)) ρ R1π∗( X (Θ) ) . . . Z O X ⊗ I O ⊗ I By ((cid:63)), we have R1π∗( X (KX + Θ + ) ) = 0 and so R1π∗( X (Θ) ) = 0. Thus, we have O → O 0 that ρ : π∗( X (Θ)) π∗( (Θ)) is surjective. Z O X O Let us now consider the sheaf π∗ X (Θ). From Theorem 2 Part (4), we have that Hi( , X (Θ)) = 0 0 0 for all i > 0. Thus, by the cohomology and base change formula (see for example [3, III.12.11]) O O π∗ X (Θ) is locally free of rank 1. Moreover, all sections of π∗( X (Θ)) vanish on Z (since ⊂ Z Θ). Thus, ρ is in fact the zero map since it is merely the restriction to Z. We will reach a O contradiction if we can show that π∗( (Θ)) is not the zero sheaf. Z 4

text: Y

text: X

text: Y

text: X

text: Let

text: f

text: be

text: a

text: resolution

text: of

text: singularities

text: so

text: that

text: is

text: smooth

text: and

text: Θ

text: E

text: is

text: a

text: simple

text: X

text: X

text: normal

text: crossing

text: divisor

text: where

text: f∗

text: Θ

text: f∗

text: and

text: E

text: is

text: the

text: exceptional

text: locus

text: of

text: f.

text: Such

text: a

text: resolution

text: is

text: guaranteed

text: by

text: Hironaka

text: X

text: X

text: Fix

text: and

text: put

text: d

text: Let

text: c

text: be

text: the

text: log

text: canonical

text: threshold

text: so

text: that

text: c(Θ+d

text: X

text: X

text: is

text: log

text: canonical

text: but

text: c

text: (Θ

text: d

text: is

text: not

text: log

text: canonical

text: for

text: any

text: c

text: c.

text: Seeking

text: a

text: contradiction,

text: suppose

text: that

text: c

text: and

text: put

text: cd

text: d

text: This

text: means

text: that

text: we

text: may

text: write

text: the

text: following

text: equation:

text: KY

text: cΘ

text: d

text: f

text: (KX

text: cΘ

text: d

text: D

text: where

text: D

text: is

text: exceptional.

text: Write

text: D

text: as

text: D

text: B

text: A

text: where

text: A,

text: B,

text: are

text: eﬀective,

text: A,

text: B

text: are

text: integral

text: and

text: nonzero,

text: and

text: A,

text: B

text: have

text: no

text: irreducible

text: components

text: in

text: common.

text: Now

text: the

text: equation

text: KY

text: cΘ

text: d

text: f

text: (KX

text: cΘ

text: d

text: B

text: A

text: Which

text: we

text: can

text: rewrite

text: as:

text: f

text: (KX

text: cΘ

text: d

text: B

text: A

text: (KY

text: cΘ

text: d

text: Adding

text: f

text: (KX

text: Θ

text: KX

text: to

text: each

text: side:

text: X

text: X

text: f

text: c)Θ

text: d

text: (f

text: (KX

text: Θ

text: B

text: A)

text: (KY

text: cΘ

text: d

text: Now

text: we

text: would

text: like

text: to

text: apply

text: Theorem

text: to

text: the

text: above

text: situation.

text: The

text: conclusion

text: is:

text: O

text: X

text: Rpπ∗Rqf∗

text: Y(f

text: (KX

text: Θ

text: B

text: A)

text: for

text: q

text: and

text: p

text: O

text: O

text: Since

text: B

text: is

text: exceptional

text: and

text: A

text: is

text: non-zero,

text: f∗

text: Y(B

text: A)

text: X

text: is

text: a

text: non-trivial

text: ideal

text: sheaf.

text: I

text: X

text: I

text: Call

text: this

text: ideal

text: and

text: let

text: Z

text: be

text: the

text: subscheme

text: determined

text: by

text: Consider

text: the

text: short

text: exact

text: sequence:

text: I

text: O

text: O

text: X

text: Z

text: O

text: Tensoring

text: by

text: X

text: (Θ)

text: we

text: get:

text: O

text: I

text: O

text: O

text: X

text: (Θ)

text: X

text: (Θ)

text: (Θ)

text: Z

text: Applying

text: π∗,

text: we

text: arrive

text: at

text: the

text: long

text: exact

text: sequence:

text: O

text: I

text: O

text: O

text: O

text: I

text: π∗(

text: X

text: (Θ)

text: π∗(

text: X

text: (Θ))

text: π∗(

text: (Θ))

text: ρ

text: X

text: (Θ)

text: Z

text: O

text: X

text: I

text: O

text: I

text: By

text: we

text: have

text: X

text: (KX

text: Θ

text: and

text: so

text: X

text: (Θ)

text: Thus,

text: we

text: have

text: O

text: O

text: that

text: ρ

text: π∗(

text: X

text: (Θ))

text: π∗(

text: (Θ))

text: is

text: surjective.

text: Z

text: O

text: X

text: O

text: Let

text: us

text: now

text: consider

text: the

text: sheaf

text: π∗

text: X

text: (Θ).

text: From

text: Theorem

text: Part

text: we

text: have

text: that

text: Hi(

text: X

text: (Θ))

text: for

text: all

text: i

text: Thus,

text: by

text: the

text: cohomology

text: and

text: base

text: change

text: formula

text: (see

text: for

text: example

text: O

text: O

text: π∗

text: X

text: (Θ)

text: is

text: locally

text: free

text: of

text: rank

text: Moreover,

text: all

text: sections

text: of

text: π∗(

text: X

text: (Θ))

text: vanish

text: on

text: Z

text: (since

text: Z

text: Θ).

text: Thus,

text: ρ

text: is

text: in

text: fact

text: the

text: zero

text: map

text: since

text: it

text: is

text: merely

text: the

text: restriction

text: to

text: Z.

text: We

text: will

text: reach

text: a

text: O

text: contradiction

text: if

text: we

text: can

text: show

text: that

text: π∗(

text: (Θ))

text: is

text: not

text: the

text: zero

text: sheaf.

text: Z


text: ON CENTRAL EXTENSIONS OF SIMPLE DIFFERENTIAL ALGEBRAIC GROUPS 5 ⊂ U ∈ Z { } Let X n be an irreducible ∆-k-algebraic set. For all s ≥ , denote k X ≤ the image of { } { } → { } 0 s k y , . . . , y ≤ under the restriction map k y , . . . , y k X . The function 1 n s 1 n N (cid:51) (cid:55)→ { } s tr.deg Frac k X ≤ k s coincides, for all sufﬁciently large s, with the values of a polynomial, which we denote ω (s) X (see [15, Chapter 0.3] and [13, Chapter II.12] for details). If X is inﬁnite, we set τ(X) := deg ω , − X the degree of ω . If X is a point, there is a convention that τ(X) = 1. The number τ(X) is X called the ∆-type of X. For reducible X, the ∆-type is deﬁned to be the maximum among the − ≤ ≤ ∆-types of its irreducible components. One has 1 τ(X) m. Moreover, τ(X) is a birational ∈ invariant of X. In particular, it is deﬁned for all G LDAG . If X is inﬁnite and Zariski closed, k τ(X) = m. The ∆-type has the following properties ([15, Chapter IV.4], [7, Corollaries 2.2 and 2.4]): → (cid:48) (1) If µ : X X is an injective (resp., dominant) morphism of ∆-k-algebraic sets, then ≤ (cid:48) ≥ (cid:48) τ(X) τ(X ) (resp., τ(X) τ(X )). (cid:48) → (cid:16) (cid:48)(cid:48) { (cid:48) (cid:48)(cid:48) } (2) For every exact sequence G (cid:44) G G in LDAG , τ(G) = max τ(G ), τ(G ) . k (cid:48) → (cid:48)(cid:48) By an exact sequence above, we mean that G is the kernel of the epimorphism G G . ∈ U ⊂ For every G LDAG ( ), there exists a unique minimal normal closed subgroup G G k 0 (over k) such that τ(G/G ) < τ(G) ([7, Section 2.2.1]). It is called a strong identity component 0 of G. If G = G , G is called strongly connected. Strongly connected LDAGs are connected and, 0 → ⊂ moreover, (G ) = G . For every morphism of LDAGs f : G H, one has f(G ) H . 0 0 ∈ 0 (cid:48)0 → 0(cid:16) (cid:48)(cid:48) An inﬁnite G LDAG is called almost simple if, for all exact sequences G (cid:44) G G in (cid:48) (cid:48)(cid:48) k LDAG , τ(G ) < τ(G )). Almost simple LDAGs are strongly connected. Every exact sequence (cid:48) → k (cid:16) (cid:48)(cid:48) (cid:48)(cid:48) (cid:54) { } (cid:48) ⊂ G (cid:44) G G with almost simple G and G = e is central, that is, G Z(G) ([7, Corollary 2.14]). ∈ U ⊂ For a G LDAG ( ) there exists a unique maximal normal closed subgroup [G, G] G (over k k) such that G/[G, G] is commutative. We call G perfect if [G, G] = G. Note that perfectness of U ∈ G does not imply perfectness of the abstract group G( ). If is G LDAG is non-commutative ◦ k almost simple, then it is perfect [7, Proposition 3.4] and Z(G) is unipotent [7, Proposition 3.5]. ◦ Moreover, G/Z(G) is a simple LDAG (over k) [7, Corollary 2.15]. ∈ U For a G LDAG ( ), the radical Rad G of G is a maximal connected solvable subgroup k of G. As for algebraic groups, one shows that Rad G is unique. Moreover, it is deﬁned over k, { } which follows from [15, Corollary 2, p. 77]. The LDAG G is called semisimple if Rad G = e (equivalently, G does not contain inﬁnite commutative subgroups). It follows that semisimplicity is preserved under base ﬁeld extensions. × → Recall the group cohomology [2]. Let G, A be groups, where A is abelian, and let α : G A ∈ N { } A be an action of G on A. For an n , set G0 = e and { → } Cn(G, A) := all functions Gn A . These abelian groups are cochains in the complex with the differentials → ∈ N dn : Cn(G, A) Cn+1(G, A), n ,

text: ON

text: CENTRAL

text: EXTENSIONS

text: OF

text: SIMPLE

text: DIFFERENTIAL

text: ALGEBRAIC

text: GROUPS

text: U

text: Z

text: Let

text: X

text: n

text: be

text: an

text: irreducible

text: ∆-k-algebraic

text: set.

text: For

text: all

text: s

text: denote

text: k

text: X

text: the

text: image

text: of

text: s

text: k

text: y

text: y

text: under

text: the

text: restriction

text: map

text: k

text: y

text: y

text: k

text: X

text: The

text: function

text: n

text: s

text: n

text: N

text: s

text: tr.deg

text: Frac

text: k

text: X

text: k

text: s

text: coincides,

text: for

text: all

text: sufﬁciently

text: large

text: s,

text: with

text: the

text: values

text: of

text: a

text: polynomial,

text: which

text: we

text: denote

text: ω

text: (s)

text: X

text: (see

text: Chapter

text: and

text: Chapter

text: for

text: details).

text: If

text: X

text: is

text: inﬁnite,

text: we

text: set

text: τ(X)

text: deg

text: ω

text: X

text: the

text: degree

text: of

text: ω

text: If

text: X

text: is

text: a

text: point,

text: there

text: is

text: a

text: convention

text: that

text: τ(X)

text: The

text: number

text: τ(X)

text: is

text: X

text: called

text: the

text: ∆-type

text: of

text: X.

text: For

text: reducible

text: X,

text: the

text: ∆-type

text: is

text: deﬁned

text: to

text: be

text: the

text: maximum

text: among

text: the

text: ∆-types

text: of

text: its

text: irreducible

text: components.

text: One

text: has

text: τ(X)

text: m.

text: Moreover,

text: τ(X)

text: is

text: a

text: birational

text: invariant

text: of

text: X.

text: In

text: particular,

text: it

text: is

text: deﬁned

text: for

text: all

text: G

text: LDAG

text: If

text: X

text: is

text: inﬁnite

text: and

text: Zariski

text: closed,

text: k

text: τ(X)

text: m.

text: The

text: ∆-type

text: has

text: the

text: following

text: properties

text: Chapter

text: Corollaries

text: and

text: If

text: µ

text: X

text: X

text: is

text: an

text: injective

text: (resp.,

text: dominant)

text: morphism

text: of

text: ∆-k-algebraic

text: sets,

text: then

text: τ(X)

text: τ(X

text: (resp.,

text: τ(X)

text: τ(X

text: For

text: every

text: exact

text: sequence

text: G

text: G

text: G

text: in

text: LDAG

text: τ(G)

text: max

text: τ(G

text: τ(G

text: k

text: By

text: an

text: exact

text: sequence

text: above,

text: we

text: mean

text: that

text: G

text: is

text: the

text: kernel

text: of

text: the

text: epimorphism

text: G

text: G

text: U

text: For

text: every

text: G

text: LDAG

text: there

text: exists

text: a

text: unique

text: minimal

text: normal

text: closed

text: subgroup

text: G

text: G

text: k

text: (over

text: k)

text: such

text: that

text: τ(G/G

text: τ(G)

text: Section

text: It

text: is

text: called

text: a

text: strong

text: identity

text: component

text: of

text: G.

text: If

text: G

text: G

text: G

text: is

text: called

text: strongly

text: connected.

text: Strongly

text: connected

text: LDAGs

text: are

text: connected

text: and,

text: moreover,

text: (G

text: G

text: For

text: every

text: morphism

text: of

text: LDAGs

text: f

text: G

text: H,

text: one

text: has

text: f(G

text: H

text: An

text: inﬁnite

text: G

text: LDAG

text: is

text: called

text: almost

text: simple

text: if,

text: for

text: all

text: exact

text: sequences

text: G

text: G

text: G

text: in

text: k

text: LDAG

text: τ(G

text: τ(G

text: Almost

text: simple

text: LDAGs

text: are

text: strongly

text: connected.

text: Every

text: exact

text: sequence

text: k

text: G

text: G

text: G

text: with

text: almost

text: simple

text: G

text: and

text: G

text: e

text: is

text: central,

text: that

text: is,

text: G

text: Z(G)

text: Corollary

text: U

text: For

text: a

text: G

text: LDAG

text: there

text: exists

text: a

text: unique

text: maximal

text: normal

text: closed

text: subgroup

text: [G,

text: G]

text: G

text: (over

text: k

text: k)

text: such

text: that

text: G/[G,

text: G]

text: is

text: commutative.

text: We

text: call

text: G

text: perfect

text: if

text: [G,

text: G]

text: G.

text: Note

text: that

text: perfectness

text: of

text: U

text: G

text: does

text: not

text: imply

text: perfectness

text: of

text: the

text: abstract

text: group

text: G(

text: If

text: is

text: G

text: LDAG

text: is

text: non-commutative

text: k

text: almost

text: simple,

text: then

text: it

text: is

text: perfect

text: Proposition

text: and

text: Z(G)

text: is

text: unipotent

text: Proposition

text: Moreover,

text: G/Z(G)

text: is

text: a

text: simple

text: LDAG

text: (over

text: k)

text: Corollary

text: U

text: For

text: a

text: G

text: LDAG

text: the

text: radical

text: Rad

text: G

text: of

text: G

text: is

text: a

text: maximal

text: connected

text: solvable

text: subgroup

text: k

text: of

text: G.

text: As

text: for

text: algebraic

text: groups,

text: one

text: shows

text: that

text: Rad

text: G

text: is

text: unique.

text: Moreover,

text: it

text: is

text: deﬁned

text: over

text: k,

text: which

text: follows

text: from

text: Corollary

text: p.

text: The

text: LDAG

text: G

text: is

text: called

text: semisimple

text: if

text: Rad

text: G

text: e

text: (equivalently,

text: G

text: does

text: not

text: contain

text: inﬁnite

text: commutative

text: subgroups).

text: It

text: follows

text: that

text: semisimplicity

text: is

text: preserved

text: under

text: base

text: ﬁeld

text: extensions.

text: Recall

text: the

text: group

text: cohomology

text: Let

text: G,

text: A

text: be

text: groups,

text: where

text: A

text: is

text: abelian,

text: and

text: let

text: α

text: G

text: A

text: N

text: A

text: be

text: an

text: action

text: of

text: G

text: on

text: A.

text: For

text: an

text: n

text: set

text: e

text: and

text: Cn(G,

text: A)

text: all

text: functions

text: Gn

text: A

text: These

text: abelian

text: groups

text: are

text: cochains

text: in

text: the

text: complex

text: with

text: the

text: differentials

text: N

text: dn

text: Cn(G,

text: A)

text: A),

text: n


text: 2 havior in the Kondo regime was reached after a quantum The bath or lead Hamilt(cid:88)onian is: quench.52 Here we present two ways of obtaining spectral func- tions from real time CT-QMC in the strong correla- H = (cid:88) H , (3) B B(cid:96) tion limit. These methods were recently used to access (cid:96) nonequilibrium spectral properties;53 here we present H = ε a†a , (4) the details of the methodology, focusing for clarity on B(cid:96) ∈ k k k k (cid:96) the equilibrium aspects and applications. We empha- size, however, that the methods are equally applicable H is written here as a sum of N bath terms H to nonequilibrium situations. The ﬁrst method obtains B B B(cid:96) which may be characterized by diﬀerent dispersions ε real frequency spectral functions from two-time correla- k and diﬀerent thermodynamic parameters (such as tem- tion functions which are then Fourier transformed. The perature and chemical potential) entering through the second is based on an auxiliary current formulation, and initial conditions. In a molecular electronics scenario is more eﬃcient when one is interested in obtaining spec- with left and right leads, N = 2 and the index (cid:96) takes tral steady state or equilibrium properties. This second B the corresponding values L and R. H has inﬁnite de- method is compatible with the reduced dynamics tech- Bl † grees of freedom described by lead operators a( ) and a nique of Ref. 54 and with any numerical solver applicable k lead dispersion ε but is noninteracting. to transport. It simulates a direct measurement of the k The third term is the hybridization Hamiltonian V , spectral function by coupling auxiliary probes to the sys- which describes population transfer between the dot and tem, and is a generalization of the concept introduced leads. Here we as(cid:88)sume it to have the bilinear form for the wide band limit by Refs. 55 and 56, which was implemented in this limit within real-time path integral MoInntSeeCc.arIlIo.w57e deﬁne the class of models to which our V = (cid:88) V(cid:88)(cid:96), (cid:16) (cid:17) (5) (cid:96) method is applicable. In Sec. III we derive a general- ized version of the auxiliary current method in the spirit d † ∗ † V = t a d + t d a (6) of previous work, and in Sec. IV we introduce a new, (cid:96) ik k i ik i k ∈ fully general scheme compatible with CTQMC. In Sec. V i=1 k (cid:96) we provide an introduction to the numerically exact real (cid:80) characterized by a hopping t from dot to lead. time bold-line CTQMC method; as an example, we work ik The coupling densities Γ (ω), deﬁned as Γ (ω) = out the case of the Anderson impurity model, starting ∗ − (cid:96)ij (cid:96)ij 2π ∈ t t δ (ω ε ), fully deﬁne the properties of from strong-coupling expansion. In Sec. VI we present re- k (cid:96) ik jk k bath and hybridizations for the purpose of this paper. sults obtained from an implementation of the scheme for (cid:80) We will assume Γ = δ Γ to be diagonal in dot or- the Anderson impurity model within bold-line CTQMC, (cid:96)ij ij (cid:96) bital space but allow it to diﬀer for each lead, such and we explore the limitations of analytical continuation. ≡ | | − that Γ (ω) 2π ∈ t 2 δ (ω ε ). General coupling Finally, in Sec. VII, we summarize our ﬁndings and dis- (cid:96) k (cid:96) k k density matrices will be discussed brieﬂy in Sec. IV. cuss our conclusions. Information about thermodynamic parameters, e.g. II. MODEL the lead chemical potential or the lead temperature, is encapsulated in the initial conditions of the system. For We consider a general quantum impurity model de- example, fermionic leads at an inverse temperature β scribed by and a chemical potential µ are described by an initia(cid:96)l (cid:96) ≡ Fermi Dirac distribution f (ω) 1 . (cid:96) 1+eβ(cid:96)(ω−µ(cid:96)) III. AUXILIARY CURRENT METHOD H = H + H + V. (1) D B In their seminal 1992 paper, Meir and Wingreen H , the dot Hamiltonian, can be any (generally inter- showed that if initial correlations can be neglected, the D acting) Hamiltonian of the form steady-state current I out of lead (cid:96) can be written as:58 (cid:96) (cid:88) (cid:88) ˆ (cid:8) (cid:2) (cid:3)(cid:9) HD = d εid†idi + d Uikjld†id†jdkdl + · · · , (2) I(cid:96) = 2ieh dω (7) × − i=1 ijkl Tr Γ (ω) f (ω) [Gr (ω) Ga (ω)] + G< (ω) . (cid:96) (cid:96) † with d( ) denoting dot operators. d, the total number i of degrees of freedom on the dot, is assumed to be small. Here, e and h are the electron charge and Planck’s (cid:15) describes on-site energy levels, the interaction Ukl rep- constant (both are set to one from here onward); f is i ij (cid:96) resents the strength of four-operator (two-body) terms, the initial occupation of lead (cid:96); and Gr,Ga and G< are and the ellipsis all potential higher order interactions. the dot’s retarded, advanced and lesser Green’s functions, ##LTLine## ##LTLine##

text: havior

text: in

text: the

text: Kondo

text: regime

text: was

text: reached

text: after

text: a

text: quantum

text: The

text: bath

text: or

text: lead

text: is:

text: Here

text: we

text: present

text: two

text: ways

text: of

text: obtaining

text: spectral

text: func-

text: tions

text: from

text: real

text: time

text: CT-QMC

text: in

text: the

text: strong

text: correla-

text: H

text: H

text: B

text: tion

text: limit.

text: These

text: methods

text: were

text: recently

text: used

text: to

text: access

text: nonequilibrium

text: spectral

text: here

text: we

text: present

text: H

text: ε

text: a†a

text: the

text: details

text: of

text: the

text: methodology,

text: focusing

text: for

text: clarity

text: on

text: k

text: k

text: k

text: k

text: the

text: equilibrium

text: aspects

text: and

text: applications.

text: We

text: empha-

text: size,

text: however,

text: that

text: the

text: methods

text: are

text: equally

text: applicable

text: H

text: is

text: written

text: here

text: as

text: a

text: sum

text: of

text: N

text: bath

text: terms

text: H

text: to

text: nonequilibrium

text: situations.

text: The

text: ﬁrst

text: method

text: obtains

text: B

text: B

text: which

text: may

text: be

text: characterized

text: by

text: diﬀerent

text: dispersions

text: ε

text: real

text: frequency

text: spectral

text: functions

text: from

text: two-time

text: correla-

text: k

text: and

text: diﬀerent

text: thermodynamic

text: parameters

text: (such

text: as

text: tem-

text: tion

text: functions

text: which

text: are

text: then

text: Fourier

text: transformed.

text: The

text: perature

text: and

text: chemical

text: potential)

text: entering

text: through

text: the

text: second

text: is

text: based

text: on

text: an

text: auxiliary

text: current

text: formulation,

text: and

text: initial

text: conditions.

text: In

text: a

text: molecular

text: electronics

text: scenario

text: is

text: more

text: eﬃcient

text: when

text: one

text: is

text: interested

text: in

text: obtaining

text: spec-

text: with

text: left

text: and

text: right

text: leads,

text: N

text: and

text: the

text: index

text: takes

text: tral

text: steady

text: state

text: or

text: equilibrium

text: properties.

text: This

text: second

text: B

text: the

text: corresponding

text: values

text: L

text: and

text: R.

text: H

text: has

text: inﬁnite

text: de-

text: method

text: is

text: compatible

text: with

text: the

text: reduced

text: dynamics

text: tech-

text: Bl

text: grees

text: of

text: freedom

text: described

text: by

text: lead

text: operators

text: a(

text: and

text: a

text: nique

text: of

text: Ref.

text: and

text: with

text: any

text: numerical

text: solver

text: applicable

text: k

text: lead

text: dispersion

text: ε

text: but

text: is

text: noninteracting.

text: to

text: transport.

text: It

text: simulates

text: a

text: direct

text: measurement

text: of

text: the

text: k

text: The

text: third

text: term

text: is

text: the

text: hybridization

text: Hamiltonian

text: V

text: spectral

text: function

text: by

text: coupling

text: auxiliary

text: probes

text: to

text: the

text: sys-

text: which

text: describes

text: population

text: transfer

text: between

text: the

text: dot

text: and

text: tem,

text: and

text: is

text: a

text: generalization

text: of

text: the

text: concept

text: introduced

text: leads.

text: Here

text: we

text: it

text: to

text: have

text: the

text: bilinear

text: form

text: for

text: the

text: wide

text: band

text: limit

text: by

text: Refs.

text: and

text: which

text: was

text: implemented

text: in

text: this

text: limit

text: within

text: real-time

text: path

text: integral

text: deﬁne

text: the

text: class

text: of

text: models

text: to

text: which

text: our

text: V

text: method

text: is

text: applicable.

text: In

text: Sec.

text: III

text: we

text: derive

text: a

text: general-

text: ized

text: version

text: of

text: the

text: auxiliary

text: current

text: method

text: in

text: the

text: spirit

text: d

text: V

text: t

text: a

text: d

text: t

text: d

text: a

text: of

text: previous

text: work,

text: and

text: in

text: Sec.

text: IV

text: we

text: introduce

text: a

text: new,

text: ik

text: k

text: i

text: ik

text: i

text: k

text: fully

text: general

text: scheme

text: compatible

text: with

text: CTQMC.

text: In

text: Sec.

text: V

text: k

text: we

text: provide

text: an

text: introduction

text: to

text: the

text: numerically

text: exact

text: real

text: characterized

text: by

text: a

text: hopping

text: t

text: from

text: dot

text: to

text: lead.

text: time

text: bold-line

text: CTQMC

text: method;

text: as

text: an

text: example,

text: we

text: work

text: ik

text: The

text: coupling

text: densities

text: Γ

text: (ω),

text: deﬁned

text: as

text: Γ

text: (ω)

text: out

text: the

text: case

text: of

text: the

text: Anderson

text: impurity

text: model,

text: starting

text: 2π

text: t

text: t

text: δ

text: (ω

text: ε

text: fully

text: deﬁne

text: the

text: properties

text: of

text: from

text: strong-coupling

text: expansion.

text: In

text: Sec.

text: VI

text: we

text: present

text: re-

text: k

text: ik

text: jk

text: k

text: bath

text: and

text: hybridizations

text: for

text: the

text: purpose

text: of

text: this

text: paper.

text: sults

text: obtained

text: from

text: an

text: implementation

text: of

text: the

text: scheme

text: for

text: We

text: will

text: assume

text: Γ

text: δ

text: Γ

text: to

text: be

text: diagonal

text: in

text: dot

text: or-

text: the

text: Anderson

text: impurity

text: model

text: within

text: bold-line

text: CTQMC,

text: ij

text: bital

text: space

text: but

text: allow

text: it

text: to

text: diﬀer

text: for

text: each

text: lead,

text: such

text: and

text: we

text: explore

text: the

text: limitations

text: of

text: analytical

text: continuation.

text: that

text: Γ

text: (ω)

text: 2π

text: t

text: δ

text: (ω

text: ε

text: General

text: coupling

text: Finally,

text: in

text: Sec.

text: VII,

text: we

text: summarize

text: our

text: ﬁndings

text: and

text: dis-

text: k

text: k

text: k

text: density

text: matrices

text: will

text: be

text: discussed

text: brieﬂy

text: in

text: Sec.

text: IV.

text: cuss

text: our

text: conclusions.

text: Information

text: about

text: thermodynamic

text: parameters,

text: e.g.

text: II.

text: MODEL

text: the

text: lead

text: chemical

text: potential

text: or

text: the

text: lead

text: temperature,

text: is

text: encapsulated

text: in

text: the

text: initial

text: conditions

text: of

text: the

text: system.

text: For

text: We

text: consider

text: a

text: general

text: quantum

text: impurity

text: model

text: de-

text: example,

text: fermionic

text: leads

text: at

text: an

text: inverse

text: temperature

text: β

text: scribed

text: by

text: and

text: a

text: chemical

text: potential

text: µ

text: are

text: described

text: by

text: an

text: Fermi

text: Dirac

text: distribution

text: f

text: (ω)

text: III.

text: AUXILIARY

text: CURRENT

text: METHOD

text: H

text: H

text: H

text: V.

text: D

text: B

text: In

text: their

text: seminal

text: paper,

text: Meir

text: and

text: Wingreen

text: H

text: the

text: dot

text: Hamiltonian,

text: can

text: be

text: any

text: (generally

text: inter-

text: showed

text: that

text: if

text: initial

text: correlations

text: can

text: be

text: neglected,

text: the

text: D

text: acting)

text: Hamiltonian

text: of

text: the

text: form

text: steady-state

text: current

text: I

text: out

text: of

text: lead

text: can

text: be

text: written

text: ˆ

text: HD

text: d

text: εid†idi

text: d

text: Uikjld†id†jdkdl

text: dω

text: ijkl

text: Tr

text: Γ

text: (ω)

text: f

text: (ω)

text: [Gr

text: (ω)

text: Ga

text: (ω)]

text: G<

text: (ω)

text: with

text: d(

text: denoting

text: dot

text: operators.

text: d,

text: the

text: total

text: number

text: i

text: of

text: degrees

text: of

text: freedom

text: on

text: the

text: dot,

text: is

text: assumed

text: to

text: be

text: small.

text: Here,

text: e

text: and

text: h

text: are

text: the

text: electron

text: charge

text: and

text: Planck’s

text: describes

text: on-site

text: energy

text: levels,

text: the

text: interaction

text: Ukl

text: rep-

text: constant

text: (both

text: are

text: set

text: to

text: one

text: from

text: here

text: onward);

text: f

text: is

text: i

text: ij

text: resents

text: the

text: strength

text: of

text: four-operator

text: (two-body)

text: terms,

text: the

text: initial

text: occupation

text: of

text: lead

text: and

text: Gr,Ga

text: and

text: G<

text: are

text: and

text: the

text: ellipsis

text: all

text: potential

text: higher

text: order

text: interactions.

text: the

text: dot’s

text: retarded,

text: advanced

text: and

text: lesser

text: Green’s

text: functions,

text: ##LTLine##

text: ##LTLine##