text: (cid:48) We will concentrate on showing that [a, [b, c]] can be written as a sum of u[g, [b, c]]u . Consider · · · ∈ { } a = a1 ak, where each of ai x1, x2 . We a(cid:88)re done if we use the Leibniz Rule: k · · · · · · · · · [a, [b, c]] = [a a , [b, c]] = a a − [a , [b, c]]a a . 1 k 1 i 1 i i+1 k i=1 (cid:48) (cid:48) ∈ { } Next, we will show that [g, [b, c]] can be written as a sum of u[g, [g , d]]v, with g, g x , x 1 2 ∈ − and u, d, v A. We apply the Jacobi identity to get [g, [b, c]] = [b, [g, c]] [c, [g, b]]. Looking at the ﬁrst term [b, [g, c]], we can apply the Leibniz rule as before to show that it can be written as a sum (cid:48) of u[g, [g , d]]v. Since the second term is the same up to order as the ﬁrst term, we are done. (cid:48) Finally, we consider terms of the form [g, [g , d]], showing that they can be written as a sum of · · · ∈ { } the desired basis terms of u[g, y]v and uyvyw. Let d = d d , with each d x , x . We apply 1 j i 1 2 the Leibniz rule onc(cid:88)e again, this time to(cid:88)d. Thus, j j (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) [g, [g , d]] = [g, u [g , d ]v ] = (u [g, [g , d ]]v + u [g , d ][g, v ] + [g, u ][g , d ]v ) i i i i i i i i i i i i i=1 i=1 ∈ (cid:48) (cid:48) for some u , v A. The term u [g, [g , d ]]v is of the form of u[g, y]v already, as [g , d ] = y or 0. To i i i i i i ∈ show that [g, v ] (and simultaneously [g, u ]) is in the form of uyw (or is equal to 0) with u, w A, i i · · · ∈ { } we apply the Leibniz rule to vi(cid:88)= vi,1 vi,(cid:96) with vi,j x1, x2 . (cid:88) (cid:96) (cid:96) · · · · · · u y[g, v ] = u y v v − [g, v ]v v = u yw yv i i i i,1 i,j 1 i,j i,j+1 i,(cid:96) i j j j=1 j=1 (cid:54) vi,j=g ∈ (cid:3) with w , v A, which completes the proof. j j Then, we recall a theorem by [EKM09] that is the M analogue of Theorem 3.4. 4 Theorem 3.11 A presentation of A /M is given by the generators x , x the following relations: 2 4 1 2 [x , z ] = [x , z ] = [x , z ] = [x , z ] = 0, yz = yz = y3 = 0, z2 = z z = z2 = 0. 1 2 1 1 2 1 2 2 1 2 1 1 2 2 Armed with Lemma 3.10 and Theorem 3.11, we can ﬁnd a basis of M /M = N . 3 4 3 Proof Our aim is to rewrite the terms E and F in a normal form using rewriting rules from M ’s 4 ∈ ∈ { } basis, where E := u[x, y]v and F := uyvyw for u, v, w A and x x , x . 1 2 Using methods similar to those in Theorem 3.2, we ﬁnd that x and x in monomials like E 1 2 commute, and so E = xi xjz or xi xjz . 1 2 1 1 2 2 Next, we will rewrite F . We ﬁrst note that if there is more than one y present in any monomial, (cid:48) then all the y s commute with everything within that term, so F may be rewritten as uvwy2. Like

label: paragraph

text: We

text: will

text: concentrate

text: on

text: showing

text: that

text: [a,

text: [b,

text: c]]

text: can

text: be

text: written

text: as

text: a

text: sum

text: of

text: u[g,

text: [b,

text: c]]u

text: Consider

text: a

text: ak,

text: where

text: each

text: of

text: ai

text: We

text: done

text: if

text: we

text: use

text: the

text: Leibniz

text: Rule:

text: k

text: [a,

text: [b,

text: c]]

text: [a

text: a

text: [b,

text: c]]

text: a

text: a

text: [a

text: [b,

text: c]]a

text: a

text: k

text: i

text: i

text: k

text: Next,

text: we

text: will

text: show

text: that

text: [g,

text: [b,

text: c]]

text: can

text: be

text: written

text: as

text: a

text: sum

text: of

text: u[g,

text: [g

text: d]]v,

text: with

text: g,

text: g

text: x

text: x

text: and

text: u,

text: d,

text: v

text: A.

text: We

text: apply

text: the

text: Jacobi

text: identity

text: to

text: get

text: [g,

text: [b,

text: c]]

text: [b,

text: [g,

text: c]]

text: [c,

text: [g,

text: b]].

text: Looking

text: at

text: the

text: ﬁrst

text: term

text: [b,

text: [g,

text: c]],

text: we

text: can

text: apply

text: the

text: Leibniz

text: rule

text: as

text: before

text: to

text: show

text: that

text: it

text: can

text: be

text: written

text: as

text: a

text: sum

text: of

text: u[g,

text: [g

text: d]]v.

text: Since

text: the

text: second

text: term

text: is

text: the

text: same

text: up

text: to

text: order

text: as

text: the

text: ﬁrst

text: term,

text: we

text: are

text: done.

text: Finally,

text: we

text: consider

text: terms

text: of

text: the

text: form

text: [g,

text: [g

text: d]],

text: showing

text: that

text: they

text: can

text: be

text: written

text: as

text: a

text: sum

text: of

text: the

text: desired

text: basis

text: terms

text: of

text: u[g,

text: y]v

text: and

text: uyvyw.

text: Let

text: d

text: d

text: d

text: with

text: each

text: d

text: x

text: x

text: We

text: apply

text: j

text: i

text: the

text: Leibniz

text: rule

text: again,

text: this

text: time

text: Thus,

text: j

text: j

text: [g,

text: [g

text: d]]

text: [g,

text: u

text: [g

text: d

text: ]v

text: (u

text: [g,

text: [g

text: d

text: ]]v

text: u

text: [g

text: d

text: ][g,

text: v

text: [g,

text: u

text: ][g

text: d

text: ]v

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: for

text: some

text: u

text: v

text: A.

text: The

text: term

text: u

text: [g,

text: [g

text: d

text: ]]v

text: is

text: of

text: the

text: form

text: of

text: u[g,

text: y]v

text: already,

text: as

text: [g

text: d

text: y

text: or

text: To

text: i

text: i

text: i

text: i

text: i

text: i

text: show

text: that

text: [g,

text: v

text: (and

text: simultaneously

text: [g,

text: u

text: is

text: in

text: the

text: form

text: of

text: uyw

text: (or

text: is

text: equal

text: to

text: with

text: u,

text: w

text: A,

text: i

text: i

text: we

text: apply

text: the

text: Leibniz

text: rule

text: to

text: with

text: vi,j

text: u

text: y[g,

text: v

text: u

text: y

text: v

text: v

text: [g,

text: v

text: ]v

text: v

text: u

text: yw

text: yv

text: i

text: i

text: i

text: i,j

text: i,j

text: i

text: j

text: j

text: vi,j=g

text: with

text: w

text: v

text: A,

text: which

text: completes

text: the

text: proof.

text: j

text: j

text: Then,

text: we

text: recall

text: a

text: theorem

text: by

text: that

text: is

text: the

text: M

text: analogue

text: of

text: Theorem

text: Theorem

text: A

text: presentation

text: of

text: A

text: /M

text: is

text: given

text: by

text: the

text: generators

text: x

text: x

text: the

text: following

text: relations:

text: [x

text: z

text: [x

text: z

text: [x

text: z

text: [x

text: z

text: yz

text: yz

text: z

text: z

text: Armed

text: with

text: Lemma

text: and

text: Theorem

text: we

text: can

text: ﬁnd

text: a

text: basis

text: of

text: M

text: /M

text: N

text: Proof

text: Our

text: aim

text: is

text: to

text: rewrite

text: the

text: terms

text: E

text: and

text: F

text: in

text: a

text: normal

text: form

text: using

text: rewriting

text: rules

text: from

text: M

text: ’s

text: basis,

text: where

text: E

text: u[x,

text: y]v

text: and

text: F

text: uyvyw

text: for

text: u,

text: v,

text: w

text: A

text: and

text: x

text: x

text: x

text: Using

text: methods

text: similar

text: to

text: those

text: in

text: Theorem

text: we

text: ﬁnd

text: that

text: x

text: and

text: x

text: in

text: monomials

text: like

text: E

text: commute,

text: and

text: so

text: E

text: xi

text: xjz

text: or

text: xi

text: xjz

text: Next,

text: we

text: will

text: rewrite

text: F

text: We

text: ﬁrst

text: note

text: that

text: if

text: there

text: is

text: more

text: than

text: one

text: y

text: present

text: in

text: any

text: monomial,

text: then

text: all

text: the

text: y

text: s

text: commute

text: with

text: everything

text: within

text: that

text: term,

text: so

text: F

text: may

text: be

text: rewritten

text: as

text: Like


text: 4 Conclusion In this project, we programmed MAGMA [BCP97] to compute data about the dimensions and ranks of these lower central series ideal quotients for various algebras. Using this data, we formu- lated and proved conjectures concerning these quotients N (A). Just like how knowing suﬃciently i the divisors of an integer, we have proven a partial result about the substructure of an inﬁnite and complex family of algebras in Section . And, in Section we characterized the bigraded structure Z of N (A) and N (A) for algebras with two generators over . In addition, we have gathered over 2 3 250 bigraded tables and nearly 100 totally graded tables, which can aid further exploration of these algebraic structures and applications. 5 Future Work Z There is still much that may be explored in this topic. Over , we could describe the bi- graded structure of N (A ) by utilizing a recently published paper by [dCK13] that outlined 4 2 a basis of A/M . In addition, we could try to produce code and explore individual grading 5 of more than just 2 variables. In general, we would like to be able to describe N (A), where ∼ i Z(cid:104) · · · (cid:105) · · · A = x , , x /(xm1, , xmk). Potential further work is to perform individual grading on the 1 k 1 k B (A) deﬁned in the introduction. i There are several conjectures we were not able to prove by the time of submission: 1. By comparing Tables 5 and 6 in Section 5, where the only diﬀerence is that they were cal- Z F culated over versus , we seem to be able to recover Table 6’s data from the others’. We p Z Z Z mod out the free parts by p , leaving a copy of . If there was torsion in the Table over p m Z Z F , then there would be a copy of over . All our tables support this. (p,m) p 2. We have a conjecture about generators for the free part of N (A), that they are xi , xjv, where ∈ 4 1 2 v A has one of the following bidegrees: (1, 3), (2, 2), (3, 1), (2, 3), (3, 2), (3, 3). 3. Though we have a complete description of N (A), with A = A /(xm, xn), we have found 2 2 1 2 proofs of the same fashion that allow us to conjecture the number of generating terms there (cid:18) (cid:19) (cid:88) are in the basis N (A ): 2 k (cid:98) (cid:99) k 2 k . 2i i=1 By using a complex ﬁltration, a closed form of this expression can be found: Re((1 + i)k). 6 Methods and Tables In order to calculate free and torsion subgroups of N (A), we use preexisting code that calculated i Q N (A) over for one relation. This required us to modify the code to allow for multiple relations, i ##LTLine##

label: section

text: Conclusion

text: In

text: this

text: project,

text: we

text: programmed

text: MAGMA

text: to

text: compute

text: data

text: about

text: the

text: dimensions

text: and

text: ranks

text: of

text: these

text: lower

text: central

text: series

text: ideal

text: quotients

text: for

text: various

text: algebras.

text: Using

text: this

text: data,

text: we

text: formu-

text: lated

text: and

text: proved

text: conjectures

text: concerning

text: these

text: quotients

text: N

text: (A).

text: Just

text: like

text: how

text: knowing

text: suﬃciently

text: i

text: the

text: divisors

text: of

text: an

text: integer,

text: we

text: have

text: proven

text: a

text: partial

text: result

text: about

text: the

text: substructure

text: of

text: an

text: inﬁnite

text: and

text: complex

text: family

text: of

text: algebras

text: in

text: Section

text: And,

text: in

text: Section

text: we

text: characterized

text: the

text: bigraded

text: structure

text: Z

text: of

text: N

text: (A)

text: and

text: N

text: (A)

text: for

text: algebras

text: with

text: two

text: generators

text: over

text: In

text: addition,

text: we

text: have

text: gathered

text: over

text: bigraded

text: tables

text: and

text: nearly

text: totally

text: graded

text: tables,

text: which

text: can

text: aid

text: further

text: exploration

text: of

text: these

text: algebraic

text: structures

text: and

text: applications.

text: Future

text: Work

text: Z

text: There

text: is

text: still

text: much

text: that

text: may

text: be

text: explored

text: in

text: this

text: topic.

text: Over

text: we

text: could

text: describe

text: the

text: bi-

text: graded

text: structure

text: of

text: N

text: (A

text: by

text: utilizing

text: a

text: recently

text: published

text: paper

text: by

text: that

text: outlined

text: a

text: basis

text: of

text: A/M

text: In

text: addition,

text: we

text: could

text: try

text: to

text: produce

text: code

text: and

text: explore

text: individual

text: grading

text: of

text: more

text: than

text: just

text: variables.

text: In

text: general,

text: we

text: would

text: like

text: to

text: be

text: able

text: to

text: describe

text: N

text: (A),

text: where

text: i

text: A

text: x

text: x

text: xmk).

text: Potential

text: further

text: work

text: is

text: to

text: perform

text: individual

text: grading

text: on

text: the

text: k

text: k

text: B

text: (A)

text: deﬁned

text: in

text: the

text: introduction.

text: i

text: There

text: are

text: several

text: conjectures

text: we

text: were

text: not

text: able

text: to

text: prove

text: by

text: the

text: time

text: of

text: submission:

text: By

text: comparing

text: Tables

text: and

text: in

text: Section

text: where

text: the

text: only

text: diﬀerence

text: is

text: that

text: they

text: were

text: cal-

text: Z

text: F

text: culated

text: over

text: versus

text: we

text: seem

text: to

text: be

text: able

text: to

text: recover

text: Table

text: data

text: from

text: the

text: others’.

text: We

text: p

text: Z

text: Z

text: Z

text: mod

text: out

text: the

text: free

text: parts

text: by

text: p

text: leaving

text: a

text: copy

text: of

text: If

text: there

text: was

text: torsion

text: in

text: the

text: Table

text: over

text: p

text: m

text: Z

text: Z

text: F

text: then

text: there

text: would

text: be

text: a

text: copy

text: of

text: over

text: All

text: our

text: tables

text: support

text: this.

text: (p,m)

text: p

text: We

text: have

text: a

text: conjecture

text: about

text: generators

text: for

text: the

text: free

text: part

text: of

text: N

text: (A),

text: that

text: they

text: are

text: xi

text: xjv,

text: where

text: v

text: A

text: has

text: one

text: of

text: the

text: following

text: bidegrees:

text: Though

text: we

text: have

text: a

text: complete

text: description

text: of

text: N

text: (A),

text: with

text: A

text: A

text: /(xm,

text: xn),

text: we

text: have

text: found

text: proofs

text: of

text: the

text: same

text: fashion

text: that

text: allow

text: us

text: to

text: conjecture

text: the

text: number

text: of

text: generating

text: terms

text: there

text: are

text: in

text: the

text: basis

text: N

text: (A

text: k

text: k

text: k

text: By

text: using

text: a

text: complex

text: ﬁltration,

text: a

text: closed

text: form

text: of

text: this

text: expression

text: can

text: be

text: found:

text: i)k).

text: Methods

text: and

text: Tables

text: In

text: order

text: to

text: calculate

text: free

text: and

text: torsion

text: subgroups

text: of

text: N

text: (A),

text: we

text: use

text: preexisting

text: code

text: that

text: calculated

text: i

text: Q

text: N

text: (A)

text: over

text: for

text: one

text: relation.

text: This

text: required

text: us

text: to

text: modify

text: the

text: code

text: to

text: allow

text: for

text: multiple

text: relations,

text: i

text: ##LTLine##


text: · · · · · · − (m, n) 0 1 2 n 1 n · · · · · · · · · · · · · · · · · · 0 0 .. Z Z · · · · · · Z Z 1 . n .. Z Z · · · · · · Z Z s 2 . n ... ... ... ... ... ... ... ... ... ... ... ... ... ... − .. Z Z · · · · · · Z Z m 1 . n .. Z Z · · · · · · Z Z m . m m m (m,n) Table 1: Bigraded Description of N (A) 2 and · · · · · · − (m, n) 0 1 2 n 1 n n + 1 · · · · · · · · · · · · · · · · · · · · · 0 0 .. Z · · · · · · Z Z Z 1 . 0 f(n) 2 ... Z Z3 · · · · · · Z3 Z2 ⊕ Z Z ⊕ Z n n f(n) ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... m − 1 ... Z Z3 · · · · · · Z3 Z2 ⊕ Z Z ⊕ Z n n f(n) m ... Z Z2 ⊕ Z · · · · · · Z2 ⊕ Z Z ⊕ Z Z ⊕ Z m m m n f(n) (m,n) .. Z Z ⊕ Z · · · · · · Z ⊕ Z Z ⊕ Z Z m + 1 . f(m) m f(m) m f(m) f(m) (m,n) (m,n) Table 2: Bigraded Description of N (A) 3 We give an explicit basis of the non-torsion part and also compute the torsion in terms of m and n. 2 Divisibility of Total Dimensions in characteristic p The main tool of this section, Proposition 2.7, states that any ﬁnite dimensional module over ⊗ W (k), a sub algebra of the Weyl algebra with divided power structure, has dimension divisible j nj ⊗ by all pnj. We show in Theorem 2.9 that Ni(A) can be equipped with an action of jWnj(k), and as a corollary, one obtains (corollary 2.12) that dim(Ni(A)) is divisible by all pnj. 2.1 Weyl algebra with divided powers We ﬁrst recall the deﬁnition 2.1 of the algebra W (k) and then give in lemma 2.2 a system of ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: (m,

text: n)

text: n

text: n

text: Z

text: Z

text: Z

text: Z

text: n

text: Z

text: Z

text: Z

text: Z

text: s

text: n

text: Z

text: Z

text: Z

text: Z

text: m

text: n

text: Z

text: Z

text: Z

text: Z

text: m

text: m

text: m

text: m

text: (m,n)

text: Table

text: Bigraded

text: Description

text: of

text: N

text: (A)

text: and

text: (m,

text: n)

text: n

text: n

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text: (m, n) 0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 0 0 0 0 · 0 0 0 0 0 0 1 1 1 (2 5) (2) 0 0 0 0 · · · 0 0 0 0 0 2 1 3 3 (22 3 4 5) (22) 0 0 · 0 · · 0 · 0 0 0 3 1 3 5 (3) 4 (3 4) (2 43 32) (2 4) 0 · · · 0 · 0 0 0 4 (2) (22 32) (34 23) (34 23) (22 32) (2) 0 0 0 0 0 0 0 0 5 (3) (32) (32) (3) 0 0 0 0 0 0 0 6 0 0 0 0 0 0 7 0 0 0 0 0 8 0 0 0 0 9 0 0 0 10 0 0 11 0 Z(cid:104) (cid:105) Table 16: N : x, y /(x3, y4), Time: 912.87 sec, Memory: 789.53MB 4 (m, n) 0 1 2 3 4 5 6 7 8 9 10 11 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 · 0 (5) 0 0 0 0 2 0 0 0 2 5 4 (2 ·5) · 0 (24)· · · 0 (22) 0 0 3 0 0 2 6 · 9 (3) · 5 (22 · 32 · 42) 0 (26 · 3 · 4 5) 0 (23) 0 4 0 1 4 (2) · 6 (22 ·33) 6 (24 ·36) 2 (25 ·32 42) 0 (24 · 3 4) 0 (22) 5 0 0 0 (32 2) 0 (35 22) 0 (37 23) 0 (35 23) 0 (22 32) 6 0 0 0 (3) 0 (32) 0 (33) 0 (32) 7 0 0 0 0 0 8 0 0 0 0 9 0 0 0 10 0 0 11 0 Z(cid:104) (cid:105) Table 17: N : x, y /(x3, y4), Time: 912.87 sec, Memory: 789.53MB 5 7 Acknowledgements We would like to thank P. Etingof for suggesting and advising the project and conjectures, and T. Khovanova for a careful reading of a draft of this paper and suggestions to improve the presentation. Y.F. acknowledges support from Marie Curie Grant IOF-hqsmcf-274032 and from the Max Planck Institute for Mathematics. This research was partly supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation. I. 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text: (cid:3) which completes the induction step. ≥ Thus, W (k) is generated by x and D for all i 0. pi · · · Deﬁnition 2.3 Denote by W (k) the subalgebra generated by x and D , , D − . By Lemma n p pn 1 2.2, it is generated by x and all D with j < pn. j For example, W (k) is generated by x and D with relations [D, x] = 1 and Dp = 0. 1 We will need the following lemma in the proof of proposition 2.7. ∈ Lemma 2.4 For j < pn, all D W (k) are nilpotent. Moreover xpn is central in this algebra. j n (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Proof To show that all D are nilpotent, we ﬁrst show that all D are nilpotent. j pi Since D − D (=1) mpi D , an induction with Lucas’s theorem mpi (=3) m = m shows (m 1)pi pi pi mpi pi 1 that m!D = Dm. In particular, for m = p, we have that mpi pi Dp = p!D = 0. (6) pi pi+1 ≤ (cid:32) (cid:33) For arbitrary 0 j < pn, we have by(cid:89)the proof of L(cid:89)emma 2.2 that p Dp (=4) D = (D )p. j jsps jsps s s It remains to show that one of the terms in this product vanishes. Choosing any term in the product (cid:32) (cid:54) (cid:33) and noting that since all j < p, j ! = 0, one has s s Dp (=5) Dpjss p = (Dpjss)p = (Dpps)js (=6) 0js = 0. (7) jsps j ! j !p j !p j !p s s s s Thus, we have shown that all D are nilpotent. j It is clear that xpn commutes with x. We now show that it commutes with D as well. According j (cid:18) (cid:19) (cid:18) (cid:19) to lemma 2.2 it suﬃces to show it for D , with pi < pn. To this end, note that pi − pn + (cid:96) − − (cid:96) − [D , xpn]x(cid:96) = (cid:0)D (x(cid:1)pnx(cid:0)(cid:96)) (cid:1) xpn(D x(cid:96)) = xpn+(cid:96) pi xpn+(cid:96) pi. pi pi pi pi pi − Now, we show that 0 (cid:18)= ( pn+(cid:96)(cid:19) (cid:18)(cid:96) (cid:19)). Bu(cid:18)t, (cid:19)by Luc(cid:18)as’s(cid:19)Theorem(cid:18) w(cid:19)e have that pi pi (cid:89) (cid:89) − − pn + (cid:96) − (cid:96) 1 N 1 (cid:96) − N 1 (cid:96) (cid:3) s s = = 0. pi pi 0 pi pi s s s s ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: all

text: D

text: are

text: nilpotent.

text: j

text: It

text: is

text: clear

text: that

text: xpn

text: commutes

text: with

text: x.

text: We

text: now

text: show

text: that

text: it

text: commutes

text: with

text: D

text: as

text: well.

text: According

text: j

text: to

text: lemma

text: it

text: suﬃces

text: to

text: show

text: it

text: for

text: D

text: with

text: pi

text: pn.

text: To

text: this

text: end,

text: note

text: that

text: pi

text: pn

text: [D

text: xpn(D

text: pi

text: pi.

text: pi

text: pi

text: pi

text: pi

text: pi

text: Now,

text: we

text: show

text: that

text: have

text: that

text: pi

text: pi

text: pn

text: N

text: N

text: s

text: s

text: pi

text: pi

text: pi

text: pi

text: s

text: s

text: s

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 − − as Z 1 diverge as T 1/4 (in 3d - for 2d AFM ﬂuctua- tions, see Ref. 16), h is seen to nearly vanish at the QCP where θ(x) is the unit step function and we have taken − → (more precisely, h/h = 1 b at the QCP, which is a ξ 0, since we restrict ourselves to the critical region 0 very small quantity). In one-loop approximation of the We use this result(cid:90)in Eq. (8) to obtain self energy the most important eﬀect of Zeeman splitting is from the intermediate quasiparticle line (straddled by ∞ ≈ a ﬂuctuation propagator). The eﬀect on the AFM spin ImΣcr(ω) λ2 dνF(ν, ω)[Φ(ν; T, 0) + Φ(ν; T, h)] σ E −∞ ﬂuctuation propagator is small (h does not enter the Lan- ≈ dau damping term in lowest order). aT3/4[K(0, ω/T) + K(u/T, ω/T)], (12) Let us then look at the self energy expression (due to coupling to magnetic(cid:88)ene(cid:90)rgy ﬂuc(cid:90)tuations, as in Ref. 16): where u = h2/(cid:15) and we deﬁned a function K(z, y) F describing the scaling behavior as the magnetic ﬁeld and ∞ ≈ the energy ω is varied as follows: ImΣ (k, ω) λ2 (dq) dνF(ν, ω) σ E −∞ (cid:48) σ × ImχE (cid:48)(q, ν)ImG (cid:48)(k + q, ν + ω) (8) K(z, y) = I(cid:90)(z, y)/I(0, 0) (13) σσ σ − I(z, y) = I (z, y) + [I (0, y) I (z, y)]/z (14) where F(ν, ω) = f(ν+ω)+b(ν) and f(ω) = 1/(eω/T +1) 3/∞2 5/2 5/2 − and b(ω) = 1/(eω/T 1). The energy ﬂuctuation spec- I (z, y) = dxF(x, y)xµ (15) µ trum is given by (see Ref. 16): z | | where the thermal fa∝ctor−F(x, y) is deﬁned below Eq. ∝ ω( ω /Z2)3/2 (8). We also used λ Z 3 and we approximated Z by ImχE(q, ω) − | | (9) → ∞ E (q2 + ξ 2 + ω /Z2)2 Z(ω = 0; ξ , T, H) = const.(T/T )1/4 . 0 The wavevector q and the inverse correlation length ξ−1 The function K(h2/(cid:15)F T, 0) drops monotonically with increasing h, in a step-like fashion. The step width are in units of the Fermi wavenumber k and the ﬂuc- F scales with temperature T. This means that the spin tuation energy ω as well as all other energies (T, h) in ﬂip term shows threshold behavior as a function of h at ≈ units of the Fermi energy (cid:15)F , of the heavy quasiparticle h ((cid:15) T)1/2 . As mentioned above, the dependence F band of YRS, approximately 10 K. The decisive eﬀect of h on magnetic ﬁeld does not follow a simple func- of the Zeeman splitting is on the result of the angular ∗ tional form, so that the T line has to be determined (cid:90) integration over q in Eq. (8). numerically. In the following we will use the above result for the self (cid:90) dΩ energy to determine the contribution of qp spin-ﬂip scat- q ImG (cid:48)(k + q, ω) 4π σ tering to several of the quantities for which a threshold behavior as a function of H has been observed. In par- ≈ dΩ − ∗ − ∗ q Zδ(ω (cid:15) (cid:48) v q cos θ) ticular we will give a detailed comparison of the magne- 4π kσ F toresistance and Hall coeﬃcient data with our theoretical Z ∗ − | − ∗ | = ∗ θ(v q ω (cid:15) (cid:48) ) (10) result. v q F kσ IV. MAGNETORESISTIVITY F ∗ where v = Zv and v are the quasiparticle and bare F F F Fermi velocities. We need ImΣ (k, 0) at the Fermi en- The magnetoresistivity ρ(T, H) is determined by the ergy, i.e. (cid:15)∗ = 0 and (cid:15)∗ (cid:48) = h(σσ(cid:48)−σ)/2. Also, ω (cid:28) v∗ q, quasiparticle scattering rate due to impurities 1/τimp kσ kσ F as may be seen from the structure of Im χE(q, ω), so that and∗that due to scattering oﬀ the critical ﬂuctuations ω may be dropped in Eq. (10). We now see that the non- 1/τ = Z(ω)ImΣ(ω). At low enough temperature, im- (cid:48) spinﬂip term σ = σ gives rise to half of the contribution purity scattering dominates. We obtain ρ(T, H) from the we had previously (at H = 0). The spin-ﬂip term, how- Kubo formula for the ∗co(cid:28)nductivity by expanding in the ever, has the additional constraint on the q-integration small quantity τimp/τ 1 as follows (f is the Fermi ∗ | | v q > h . The q-integral of the spin ﬂip term in Eq. (8) function): (cid:34) (cid:90) (cid:35) F (cid:90) (cid:88) may be approximated by Φ(ω; ξ, T, h) deﬁned as ∗ ∗ Φ(ω; ξ, T(cid:90), h) = q2dq Imχ (q, ω) 1 θ(v∗ q − h) ρ(T, H) − ρ(0, H)) = (cid:34)(cid:88) (cid:90)dω ∂∂ωf N0 (ω)vτF2∗((ωω))(τimp)(cid:35)2 E v q F σ F − 2 ≈ − (cid:20) (qdq)ω5/2 − (cid:21) × (cid:90) ∂f ∗ ∗ Z 3 | | θ(q2 h2/Z2v2 ) (cid:88) dω N (ω)v 2(ω)τ (q2 + ω /Z2)2 F ∂ω 0 F imp | | | | − − − | | σ ≈ ω ω 3/2 θ( ω h2) θ(h2 Z2 ω ) ∂f | | + (11) = a dω ImΣ (k, ω) (16) Z3 ω /Z2 h2/Z2 ∂ω σ σ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: as

text: Z

text: diverge

text: as

text: T

text: (in

text: for

text: AFM

text: ﬂuctua-

text: tions,

text: see

text: Ref.

text: h

text: is

text: seen

text: to

text: nearly

text: vanish

text: at

text: the

text: QCP

text: where

text: θ(x)

text: is

text: the

text: unit

text: step

text: function

text: and

text: we

text: have

text: taken

text: (more

text: precisely,

text: h/h

text: b

text: at

text: the

text: QCP,

text: which

text: is

text: a

text: ξ

text: since

text: we

text: restrict

text: ourselves

text: to

text: the

text: critical

text: region

text: very

text: small

text: quantity).

text: In

text: one-loop

text: approximation

text: of

text: the

text: We

text: use

text: this

label: equation

text: Eq.

text: to

text: obtain

text: self

text: energy

text: the

text: most

text: important

text: eﬀect

text: of

text: Zeeman

text: splitting

text: is

text: from

text: the

text: intermediate

text: quasiparticle

text: line

text: (straddled

text: by

text: a

text: ﬂuctuation

text: propagator).

text: The

text: eﬀect

text: on

text: the

text: AFM

text: spin

text: ImΣcr(ω)

text: λ2

text: dνF(ν,

text: ω)[Φ(ν;

text: T,

text: Φ(ν;

text: T,

text: h)]

text: σ

text: E

text: ﬂuctuation

text: propagator

text: is

text: small

text: (h

text: does

text: not

text: enter

text: the

text: Lan-

text: dau

text: damping

text: term

text: in

text: lowest

text: order).

text: ω/T)

text: K(u/T,

text: ω/T)],

text: Let

text: us

text: then

text: look

text: at

text: the

text: self

text: energy

text: expression

text: (due

text: to

text: coupling

text: to

text: as

text: in

text: Ref.

text: where

text: u

text: and

text: we

text: deﬁned

text: a

text: function

text: K(z,

text: y)

text: F

text: describing

text: the

text: scaling

text: behavior

text: as

text: the

text: magnetic

text: ﬁeld

text: and

text: the

text: energy

text: ω

text: is

text: varied

text: as

text: follows:

text: ImΣ

text: (k,

text: ω)

text: λ2

text: (dq)

text: dνF(ν,

text: ω)

text: σ

text: E

text: σ

text: ImχE

text: ν)ImG

text: q,

text: ν

text: ω)

text: K(z,

text: y)

text: σσ

text: σ

text: I(z,

text: y)

text: I

text: (z,

text: y)

text: [I

text: y)

text: I

text: (z,

text: y)]/z

text: where

text: F(ν,

text: ω)

text: f(ν+ω)+b(ν)

text: and

text: f(ω)

text: and

text: b(ω)

text: The

text: energy

text: ﬂuctuation

text: spec-

text: I

text: (z,

text: y)

text: dxF(x,

text: y)xµ

text: µ

text: trum

text: is

text: given

text: by

text: (see

text: Ref.

text: z

text: where

text: the

text: thermal

text: fa∝ctor−F(x,

text: y)

text: is

text: deﬁned

text: below

text: Eq.

text: ω(

text: ω

text: We

text: also

text: used

text: λ

text: Z

text: and

text: we

text: approximated

text: Z

text: by

text: ImχE(q,

text: ω)

text: E

text: ξ

text: ω

text: Z(ω

text: ξ

text: T,

text: H)

text: const.(T/T

text: The

text: wavevector

text: q

text: and

text: the

text: inverse

text: correlation

text: length

text: ξ−1

text: The

text: function

text: T,

text: drops

text: monotonically

text: with

text: increasing

text: h,

text: in

text: a

text: step-like

text: fashion.

text: The

text: step

text: width

text: are

text: in

text: units

text: of

text: the

text: Fermi

text: wavenumber

text: k

text: and

text: the

text: ﬂuc-

text: F

text: scales

text: with

text: temperature

text: T.

text: This

text: means

text: that

text: the

text: spin

text: tuation

text: energy

text: ω

text: as

text: well

text: as

text: all

text: other

text: energies

text: (T,

text: h)

text: in

text: ﬂip

text: term

text: shows

text: threshold

text: behavior

text: as

text: a

text: function

text: of

text: h

text: at

text: units

text: of

text: the

text: Fermi

text: energy

text: of

text: the

text: heavy

text: quasiparticle

text: h

text: As

text: mentioned

text: above,

text: the

text: dependence

text: F

text: band

text: of

text: YRS,

text: approximately

text: K.

text: The

text: decisive

text: eﬀect

text: of

text: h

text: on

text: magnetic

text: ﬁeld

text: does

text: not

text: follow

text: a

text: simple

text: func-

text: of

text: the

text: Zeeman

text: splitting

text: is

text: on

text: the

text: result

text: of

text: the

text: angular

text: tional

text: form,

text: so

text: that

text: the

text: T

text: line

text: has

text: to

text: be

text: determined

text: integration

text: over

text: q

text: in

text: Eq.

text: numerically.

text: In

text: the

text: following

text: we

text: will

text: use

text: the

text: above

text: result

text: for

text: the

text: self

text: dΩ

text: energy

text: to

text: determine

text: the

text: contribution

text: of

text: qp

text: spin-ﬂip

text: scat-

text: q

text: ImG

text: q,

text: ω)

text: 4π

text: σ

text: tering

text: to

text: several

text: of

text: the

text: quantities

text: for

text: which

text: a

text: threshold

text: behavior

text: as

text: a

text: function

text: of

text: H

text: has

text: been

text: observed.

text: In

text: par-

text: dΩ

text: q

text: Zδ(ω

text: v

text: q

text: cos

text: θ)

text: ticular

text: we

text: will

text: give

text: a

text: detailed

text: comparison

text: of

text: the

text: magne-

text: 4π

text: kσ

text: F

text: toresistance

text: and

text: Hall

text: coeﬃcient

text: data

text: with

text: our

text: theoretical

text: Z

text: θ(v

text: q

text: ω

text: result.

text: v

text: q

text: F

text: kσ

text: IV.

text: MAGNETORESISTIVITY

text: F

text: where

text: v

text: Zv

text: and

text: v

text: are

text: the

text: quasiparticle

text: and

text: bare

text: F

text: F

text: F

text: Fermi

text: velocities.

text: We

text: need

text: ImΣ

text: (k,

text: at

text: the

text: Fermi

text: en-

text: The

text: magnetoresistivity

text: ρ(T,

text: H)

text: is

text: determined

text: by

text: the

text: ergy,

text: i.e.

text: and

text: Also,

text: ω

text: v∗

text: q,

text: quasiparticle

text: scattering

text: rate

text: due

text: to

text: impurities

text: kσ

text: kσ

text: F

text: as

text: may

text: be

text: seen

text: from

text: the

text: structure

text: of

text: Im

text: χE(q,

text: ω),

text: so

text: that

text: and∗that

text: due

text: to

text: scattering

text: oﬀ

text: the

text: critical

text: ﬂuctuations

text: ω

text: may

text: be

text: dropped

text: in

text: Eq.

text: We

text: now

text: see

text: that

text: the

text: non-

text: Z(ω)ImΣ(ω).

text: At

text: low

text: enough

text: temperature,

text: im-

text: spinﬂip

text: term

text: σ

text: σ

text: gives

text: rise

text: to

text: half

text: of

text: the

text: contribution

text: purity

text: scattering

text: dominates.

text: We

text: obtain

text: ρ(T,

text: H)

text: from

text: the

text: we

text: had

text: previously

text: (at

text: H

text: The

text: spin-ﬂip

text: term,

text: how-

text: Kubo

text: formula

text: for

text: the

text: by

text: expanding

text: in

text: the

text: ever,

text: has

text: the

text: additional

text: constraint

text: on

text: the

text: q-integration

text: small

text: quantity

text: τimp/τ

text: as

text: follows

text: (f

text: is

text: the

text: Fermi

text: v

text: q

text: h

text: The

text: q-integral

text: of

text: the

text: spin

text: ﬂip

text: term

text: in

text: Eq.

text: function):

text: F

text: may

text: be

text: approximated

text: by

text: Φ(ω;

text: ξ,

text: T,

text: h)

text: deﬁned

text: as

text: Φ(ω;

text: ξ,

text: h)

text: Imχ

text: (q,

text: ω)

text: θ(v∗

text: q

text: h)

text: ρ(T,

text: H)

text: H))

text: ∂∂ωf

text: E

text: v

text: q

text: F

text: σ

text: F

text: ∂f

text: Z

text: dω

text: N

text: (ω)v

text: ω

text: F

text: ∂ω

text: F

text: imp

text: σ

text: ω

text: ω

text: θ(

text: ω

text: ω

text: ∂f

text: a

text: dω

text: ImΣ

text: (k,

text: ω)

text: ω

text: ∂ω

text: σ

text: σ

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: 11 Q is a triple indicating that the query consid- makespan T is small enough (T ≤ − ) that they (t , R, T) T K A ers identical annealings of the embedded Hamiltonian have not been discriminated by the procedure described R for a single annealing time . To determine the el- in the subsection on timespan discrimination. Finally, H t emTents in Q we ﬁrst make somAe assumptions, namely, S is the set of valid solutions that can be discriminated i) suﬃcient statistics: for each query, R is suﬃciently (T ∈ (T − K, T]). Depending on what the timespan T of large to sample appropriately the ensembles deﬁned in the problem Hamiltonian and the optimal makespan H Eqs. (B7)–(B9); ii) generalized adiabaticity: t is opti- T are, the quantum annealeTr samples the following con- A mized (over the window of available annealing times) for ﬁguration space (reporting samples per query): R the best annealing performance in ﬁnding a ground state of (i.e., the annealing procedure is such that the to- talHexTpected annealing time required to evolve to a T −→ V S ∅ → (B7) ground state is as small as potAssRib(cid:63)le compared to the time T ∈ −T < −→ , V = ∅ → E0 >∈ ∆E, (T K, T] = E (0, ∆E], required to evolve to an excited state, with the same 0 (B8) probability). Both of these conditions can be achieved in T ≤ − −→ E V S (cid:54) ∅ → (B9) principle by measuring the appropriate optimal param- T K , , = E0 = 0. eters and through extensive test runs over R(cid:63)(q) t(cid:63) (q) random ensemblesAof instances. However, we note that Condition (B8) is the desired case where the ground verifying these assumptions experimentally is currently state of with energy corresponds to a valid sched- beyond the operational limits of the D-Wave Two device HT E0 ule with the optimal makespan we seek. The ground since the optimal for generalized adiabaticity is ex- t states corresponding to conditions (B7) and (B9) are A pected to be smaller than the minimum programmable instead, respectively, invalid schedules and valid sched- value [3]. Furthermore, we deemed the considerable ma- ules whose makespan could correspond to a global mini- chine time required for such a large-scale study too oner- mum (to be determined by subsequent calls). The above- ous in the context of this initial foray. Fortunately, the described assumption ii) is essential to justify aborting ﬁrst limitation is expected to be lifted with the next gen- the search when case (B8) is encountered. If and eration of chip, at which point nothing would prevent R tA are appropriately chosen, the ground state will be prefer- the proper determination of a family-speciﬁc choice of entially found instead of all other solutions such that one and . Given a speciﬁed annealing time, the num- R(cid:63) t(cid:63) can stop annealing reasonably soon (i.e., after a number ber of annAeals is determined by specifying , the target r of reads on the order of ) in the absence of the appear- probability of success for queries or conﬁden0ce level, and R(cid:63) ance of a zero-energy solution. We can then declare this measuring , the rate of occurrence of the ground state r minimum-energy conﬁguration, found within , to per repetitioqn for the following query: (0, ∆E] be the ground state and the associated makespan and schedule to be the optimal solution of the optimization − problem. On the other hand, we note that if , R(cid:63) = log[1 − r0]. (B5) a minimum of two calls are required to solve thKe p=rob0- log[1 rq] lem to optimality, one to show that no valid solution is found for T − and one to show that a zero-energy T = 1 The rate r depends on the family, T, and the conﬁguration is found for T . While for cases (B8) other parametqers. The minimum observed during pre- T = and (B9) the appearance of an energy less than or equal characterization should be used to guarantee the ground to heuristically determines the trigger that stops the ∆E state is found with at least the speciﬁed . Formally, r annealing of , for case (B7), we need to have a pre- 0 H the estimated time to solution of a problem is then given T scription, based on pre-characterization, on how long to by (cid:18) (cid:19) anneal in order to be conﬁdent that T . While op- (cid:88) T < timizing these times is a research program on its own − that requires extensive testing, we expect that the char- T = ∈Q tA lloogg[[11 − rr0q]] . (B6) awcitllerbiestoicnttihmeesfaomr eacohrideevrinogf mcoangdniittiuonde(Bfo8r)thwihseunnTkn=owTn q (cid:81) runtime. The total probability of success of solving the problem 3. Search strategy in time will consequently be . For the results T r presented in this paper, we used q q 500 000 and The ﬁnal important component of the computational R(cid:63) = t(cid:63) = s. A strategy consists in determining the sequence of times- min(t ) = 20 µ WeAcan deﬁne three diﬀerent sets of qubit conﬁgura- pans of the calls (i.e., the ensemble Q). Here we pro- tions that can be returned when the annealer is queried pose to select the queries based on an optimized binary with . E is the set of conﬁgurations whose energy is search that makes informed dichotomic decisions based q larger than as deﬁned in Section III of the paper. on the pre-characterization of the distribution of optimal ∆E These conﬁgurations represent invalid schedules. V is the makespans of the parametrized ensembles as described set of solutions with zero energy, i.e., the solutions whose in the previous sections. More speciﬁcally, the search is ##LTLine## ##LTLine##

text: Q

text: is

text: a

text: triple

text: indicating

text: that

text: the

text: query

text: consid-

text: makespan

text: T

text: is

text: small

text: enough

text: (T

text: that

text: they

text: (t

text: R,

text: T)

text: T

text: K

text: A

text: ers

text: identical

text: annealings

text: of

text: the

text: embedded

text: Hamiltonian

text: have

text: not

text: been

text: discriminated

text: by

text: the

text: procedure

text: described

text: R

text: for

text: a

text: single

text: annealing

text: time

text: To

text: determine

text: the

text: el-

text: in

text: the

text: subsection

text: on

text: timespan

text: discrimination.

text: Finally,

text: H

text: t

text: emTents

text: in

text: Q

text: we

text: ﬁrst

text: make

text: somAe

text: assumptions,

text: namely,

text: S

text: is

text: the

text: set

text: of

text: valid

text: solutions

text: that

text: can

text: be

text: discriminated

text: i)

text: suﬃcient

text: statistics:

text: for

text: each

text: query,

text: R

text: is

text: suﬃciently

text: (T

text: (T

text: K,

text: T]).

text: Depending

text: on

text: what

text: the

text: timespan

text: T

text: of

text: large

text: to

text: sample

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text: ensembles

text: deﬁned

text: in

text: the

text: problem

text: Hamiltonian

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text: the

text: optimal

text: makespan

text: H

text: Eqs.

text: ii)

text: generalized

text: adiabaticity:

text: t

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text: T

text: are,

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text: R

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text: (q)

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text: instances.

text: However,

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text: We

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text: The

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text: energy

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text: expect

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text: T

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text: awcitllerbiestoicnttihmeesfaomr

text: eacohrideevrinogf

text: q

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text: The

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text: sequence

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text: times-

text: min(t

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text: sets

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text: qubit

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text: pans

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text: the

text: ensemble

text: Q).

text: Here

text: we

text: pro-

text: tions

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text: can

text: be

text: returned

text: when

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text: annealer

text: is

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text: queries

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text: on

text: an

text: optimized

text: binary

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text: E

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text: the

text: set

text: of

text: conﬁgurations

text: whose

text: energy

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text: that

text: makes

text: informed

text: dichotomic

text: decisions

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text: q

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text: than

text: as

text: deﬁned

text: in

text: Section

text: III

text: of

text: the

text: paper.

text: on

text: the

text: pre-characterization

text: of

text: the

text: distribution

text: of

text: optimal

text: ∆E

text: These

text: conﬁgurations

text: represent

text: invalid

text: schedules.

text: V

text: is

text: the

text: makespans

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text: the

text: parametrized

text: ensembles

text: as

text: described

text: set

text: of

text: solutions

text: with

text: zero

text: energy,

text: i.e.,

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text: solutions

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text: in

text: the

text: previous

text: sections.

text: More

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text: ##LTLine##

text: ##LTLine##