text: 12 LEONID SLAVIN AND VASILY VASYUNIN (cid:104) (cid:105) (cid:104) (cid:105) ∈ (cid:107) (cid:107) Lemma 3.5. We have ϕ∗ = 0, ϕ2∗ = 1, and ϕ∗ BMOd(Q) with ϕ∗ = 1. Q Q BMOd(Q) ∈ Proof. It is clear that to verify that ϕ∗ BMOd(Q), and to compute the norm, it suﬃces to (cid:88) (cid:0) (cid:1)(cid:0) (cid:1) check the average oscillations of ϕ∗ only over the cubes Q . We have k ∞ (cid:104) (cid:105) − − − − − − ϕ∗ = 2n(k 1/2) 2 nj 2 n(j+1) j(2n 1) 1 Q k j=k − − = 2 n/2(2n 1)k (cid:88) (cid:0) (cid:1) and ∞ (cid:104) (cid:105) − − − − − ϕ2∗ = 2n(k 2)(2n 1) 2 nj j(2n 1) 1 2 Q k j=k − − = 1 + 2 n(2n 1)2k2. (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) − Setting k = 0, we obtain ϕ∗ = 0 and ϕ2∗ = 1. Furthermore, for any k, ϕ2∗ (cid:104) (cid:105) Q ∈ Q (cid:107) (cid:107) Qk (cid:3) ϕ∗ 2 = 1, which means that ϕ∗ BMOd(Q) and ϕ∗ = 1, as claimed. Q BMOd(Q) k ∈ Proof of Theorem 3.4. Take any ε (0, εd(n)) and let us write B for Bn, B for B − , and − ∈ 0 ∈ ε (cid:104) (cid:105) (cid:104) 2 (cid:105)n,ε δ for δ(2 n, ε). Fix a point x Ω and pick any ϕ BMOd(Q) such that ( ϕ , ϕ2 ) = x. ε Q Q By (3.4), ≥ (cid:104) (cid:105) B(x) eϕ . Q ≥ Taking the supremum over all such ϕ, we conclude that B(x) B(x). ∈ R To show the converse, we make use of the function ϕ∗ given by (3.5). Take any a and consider the corresponding point on Γ , P = (a, a2 + ε2). Let ϕ = εϕ∗ + a. Then (cid:104) (cid:105) (cid:104) (cid:105) (cid:107) (cid:107) ε (cid:88)a a ( ϕ , ϕ2 ) = P and ϕ = ε. Furthermore, a Q a Q a a BMOd(Q) ∞ (cid:104) (cid:105) − − − − − − eϕa = ea ε2n/2(1 2 n)(cid:88) (cid:16)2 njeε2 n/2(2n 1)(cid:17)j Q j=0 ∞ − − − − − − j = e ε2n/2(1 2 n) 2 neε2 n/2(2n 1) . j=0 This sum converges if and only if ε < 2n−/2 n log 2 = εd(n), in which case 2n 1 0 (cid:104) (cid:105) − − − 1 − eϕa Q = ea ε2n/2(1 2 n)1 − 2−neε2−n/2(2n−1) = ea K(2 n, ε) = B(Pa). ≥ Therefore, B(P ) B(P ). a a Thus, B = B on Γ . Similarly, by considering the constant function ψ = u corresponding ∈ ε u to a point (u, u2) Γ , we conclude that B = B on Γ . Now, as shown in [11], for any 0 0 − number c, the function B is linear along the line (cid:96) with the equation x = 2cx + δ2 c2; c 2 1 such lines are tangent to Γ and they foliate Ω . On the other hand, it is easy to show that δ ε B is locally concave in Ω (this is done is [11] for n = 1). ∈ ε Take any point x Ω and let (cid:96) be the unique tangent to Γ passing through x. Then (cid:96) ε δ intersects Γ and Γ at some points U and V, respectively, such that the segment [U, V ] lies 0 ε − entirely in Ω . We can write x as a convex combination of U and V : x = (1 γ)U + γV for ∈ ε some γ [0, 1]. Then, ≥ − − B(x) (1 γ) B(U) + γ B(V ) = (1 γ) B(U) + γ B(V ) = B(x). ≥ Thus B B everywhere in Ω . ε ##LTLine## ##LTLine##

text: LEONID

text: SLAVIN

text: AND

text: VASILY

text: VASYUNIN

text: Lemma

text: We

text: have

text: ϕ∗

text: ϕ2∗

text: and

text: ϕ∗

text: BMOd(Q)

text: with

text: ϕ∗

text: Q

text: Q

text: BMOd(Q)

text: Proof.

text: It

text: is

text: clear

text: that

text: to

text: verify

text: that

text: ϕ∗

text: BMOd(Q),

text: and

text: to

text: compute

text: the

text: norm,

text: it

text: suﬃces

text: to

text: check

text: the

text: average

text: oscillations

text: of

text: ϕ∗

text: only

text: over

text: the

text: cubes

text: Q

text: We

text: have

text: k

text: ϕ∗

text: nj

text: Q

text: k

text: j=k

text: and

text: ϕ2∗

text: nj

text: Q

text: k

text: j=k

text: Setting

text: k

text: we

text: obtain

text: ϕ∗

text: and

text: ϕ2∗

text: Furthermore,

text: for

text: any

text: k,

text: ϕ2∗

text: Q

text: Q

text: Qk

text: ϕ∗

text: which

text: means

text: that

text: ϕ∗

text: BMOd(Q)

text: and

text: ϕ∗

text: as

text: claimed.

text: Q

text: BMOd(Q)

text: k

text: Proof

text: of

text: Theorem

text: Take

text: any

text: ε

text: εd(n))

text: and

text: let

text: us

text: write

text: B

text: for

text: Bn,

text: B

text: for

text: B

text: and

text: ε

text: δ

text: for

text: n,

text: ε).

text: Fix

text: a

text: point

text: x

text: Ω

text: and

text: pick

text: any

text: ϕ

text: BMOd(Q)

text: such

text: that

text: ϕ

text: ϕ2

text: x.

text: ε

text: Q

text: Q

text: By

text: B(x)

text: eϕ

text: Q

text: Taking

text: the

text: supremum

text: over

text: all

text: such

text: ϕ,

text: we

text: conclude

text: that

text: B(x)

text: B(x).

text: R

text: To

text: show

text: the

text: converse,

text: we

text: make

text: use

text: of

text: the

text: function

text: ϕ∗

text: given

text: by

text: Take

text: any

text: a

text: and

text: consider

text: the

text: corresponding

text: point

text: on

text: Γ

text: P

text: (a,

text: Let

text: ϕ

text: εϕ∗

text: a.

text: Then

text: ε

text: a

text: ϕ

text: ϕ2

text: P

text: and

text: ϕ

text: ε.

text: Furthermore,

text: a

text: Q

text: a

text: Q

text: a

text: a

text: BMOd(Q)

text: eϕa

text: ea

text: Q

text: j

text: e

text: n)

text: This

text: sum

text: converges

text: if

text: and

text: only

text: if

text: ε

text: n

text: log

text: εd(n),

text: in

text: which

text: case

text: eϕa

text: Q

text: ea

text: ea

text: n,

text: ε)

text: B(Pa).

text: Therefore,

text: B(P

text: B(P

text: a

text: a

text: Thus,

text: B

text: B

text: on

text: Γ

text: Similarly,

text: by

text: considering

text: the

text: constant

text: function

text: ψ

text: u

text: corresponding

text: ε

text: u

text: to

text: a

text: point

text: (u,

text: Γ

text: we

text: conclude

text: that

text: B

text: B

text: on

text: Γ

text: Now,

text: as

text: shown

text: in

text: for

text: any

text: number

text: c,

text: the

text: function

text: B

text: is

text: linear

text: along

text: the

text: line

text: with

text: the

text: equation

text: x

text: δ2

text: c

text: such

text: lines

text: are

text: tangent

text: to

text: Γ

text: and

text: they

text: foliate

text: Ω

text: On

text: the

text: other

text: hand,

text: it

text: is

text: easy

text: to

text: show

text: that

text: δ

text: ε

text: B

text: is

text: locally

text: concave

text: in

text: Ω

text: (this

text: is

text: done

text: is

text: for

text: n

text: ε

text: Take

text: any

text: point

text: x

text: Ω

text: and

text: let

text: be

text: the

text: unique

text: tangent

text: to

text: Γ

text: passing

text: through

text: x.

text: Then

text: ε

text: δ

text: intersects

text: Γ

text: and

text: Γ

text: at

text: some

text: points

text: U

text: and

text: V,

text: respectively,

text: such

text: that

text: the

text: segment

text: [U,

text: V

text: lies

text: ε

text: entirely

text: in

text: Ω

text: We

text: can

text: write

text: x

text: as

text: a

text: convex

text: combination

text: of

text: U

text: and

text: V

text: x

text: γ)U

text: γV

text: for

text: ε

text: some

text: γ

text: Then,

text: B(x)

text: γ)

text: B(U)

text: γ

text: B(V

text: γ)

text: B(U)

text: γ

text: B(V

text: B(x).

text: Thus

text: B

text: B

text: everywhere

text: in

text: Ω

text: ε

text: ##LTLine##

text: ##LTLine##


text: α INEQUALITIES FOR BMO ON -TREES 15 Theorem 4.2. For any ε > 0, bn = b − . ε 2 n,ε Proof. The proof is similar to that of Theorem 3.4. Write b for bn and b for b − . Take ∈ ∈ (cid:104) (cid:105) (cid:104) (cid:105) ε 2 n,ε any x Ω and any ϕ BMOd(Q) such that ( ϕ , ϕ2 ) = x. As shown in the proof of ε ε Q Q Theorem 1.8 (equation (4.2) with X = Q and µ the Lebesgue measure), ≤ (cid:104)| |(cid:105) b(x) ϕ . Q ≤ Taking the inﬁmum over all such ϕ, we conclude that b b in Ω . ε To prove the converse, we again let Q = (0, 1)n and use the function ϕ∗ from (3.5). For | | ≥ − (cid:107) (cid:107) all a such that a 2 n/2ε, let ϕ = εϕ∗ + a. Recall that ϕ = ε and that (cid:104) (cid:105) (cid:104) (cid:105) a a BMOd(Q) ≥ − ( ϕ , ϕ2 ) = (a, a2 + ε2). Furthermore, ϕ is almost everywhere positive for a 2 n/2ε a Q a Q ≤ − − a and almost everywhere negative for a 2 n/2ε, thus ≤ (cid:104)| |(cid:105) |(cid:104) (cid:105) | | | b(a, a2 + ε2) ϕ = ϕ = a = b(a, a2 + ε2). a Q a Q ∩ Hence, b = b on Γ Ω . On the other hand, by considering the constant function ψ = u ε 1 ∈ u corresponding to a point (u, u2) Γ we conclude that b = b on Γ . ∈ 0 0 Take any x Ω and let (cid:96) be any line that passes through x, a point U on Γ , and a 1 x ⊂ 0 point V on Γ , and such that the segment [U, V ] Ω (there are inﬁnitely many such lines ε − ∈ 1 for each x). Write x = (1 γ)U + γV for some γ [0, 1]. Note that b is linear along [U, V ], while b is a locally convex function directly from its deﬁnition. Therefore, ≤ − − (4.4) b(x) (1 γ) b(U) + γ b(V ) = (1 γ) b(U) + γ b(V ) = b(x). Thus, b = b in Ω . 1∈ Now, take any x Ω and write x a convex combination of the three “corner” points of Ω , − − −0 − − 0 (0, 0), ( 2 n/2ε, (1 + 2 n)ε2), and (2 n/2ε, (1 + 2 n)ε2). Since at these three points b = b, b ≤ is linear in Ω , and b is locally convex, arguing as in (4.4) proves that b b in Ω . Putting 0 0 (cid:3) everything together, we conclude that b = b in Ω . ε Proof of Theorem 1.7. Arguing as in the proof of Theorem 1.3, we obtain (1.5) and (1.6) from Theorem 1.8. To prove sharpness of (1.5), we do not present an explicit function as ∈ D we did in Theorem 1.3. Instead, take any ε > 0 and any Q . Theorem 4.2 says that { } bn(0, ε2) = 2n/2 ε. Therefore, there exists a sequence of functions ϕ from BMOd(Q) such ε (cid:104) (cid:105) 2n+1 (cid:107) (cid:107) (cid:104) (cid:105) k that ϕ = 0, ϕ = ϕ2 1/2 = ε, and k Q k BMOd(Q) k Q (cid:104)| − (cid:104) (cid:105) |(cid:105) (cid:104)| |(cid:105) 2n/2 lim ϕ ϕ = lim ϕ = ε. k→∞ k k Q Q k→∞ k Q 2n + 1 R (cid:107) (cid:107) Extending each ϕk periodically to all of n we get ϕk BMOd(Rn) = ε, which completes th(cid:3)e proof. 5. α-martingales and continuous BMO Any metric measure space has a canonical BMO deﬁned on balls; we call this the continuous T T BMO. If the space is equipped with an α-tree , one also has BMO( ). It is natural to ask whether our theory for BMO on trees can be used to obtain sharp inequalities for the continuous BMO. If the tree in question is ﬁxed from the outset, the answer is no; however, if one allows each BMO function to generate its own tree and, thus, a martingale, then the answer is yes, provided all such martingales are regular in a particular uniform sense. ##LTLine## ##LTLine##

text: α

label: paragraph

text: INEQUALITIES

text: FOR

text: BMO

text: ON

text: -TREES

text: Theorem

text: For

text: any

text: ε

text: bn

text: b

text: ε

text: n,ε

text: Proof.

text: The

text: proof

text: is

text: similar

text: to

text: that

text: of

text: Theorem

text: Write

text: b

text: for

text: bn

text: and

text: b

text: for

text: b

text: Take

text: ε

text: n,ε

text: any

text: x

text: Ω

text: and

text: any

text: ϕ

text: BMOd(Q)

text: such

text: that

text: ϕ

text: ϕ2

text: x.

text: As

text: shown

text: in

text: the

text: proof

text: of

text: ε

text: ε

text: Q

text: Q

text: Theorem

text: (equation

text: with

text: X

text: Q

text: and

text: µ

text: the

text: Lebesgue

text: measure),

text: b(x)

text: ϕ

text: Q

text: Taking

text: the

text: inﬁmum

text: over

text: all

text: such

text: ϕ,

text: we

text: conclude

text: that

text: b

text: b

text: in

text: Ω

text: ε

text: To

text: prove

text: the

text: converse,

text: we

text: again

text: let

text: Q

text: and

text: use

text: the

text: function

text: ϕ∗

text: from

text: For

text: all

text: a

text: such

text: that

text: a

text: let

text: ϕ

text: εϕ∗

text: a.

text: Recall

text: that

text: ϕ

text: ε

text: and

text: that

text: a

text: a

text: BMOd(Q)

text: ϕ

text: ϕ2

text: (a,

text: Furthermore,

text: ϕ

text: is

text: almost

text: everywhere

text: positive

text: for

text: a

text: a

text: Q

text: a

text: Q

text: a

text: and

text: almost

text: everywhere

text: negative

text: for

text: a

text: thus

text: b(a,

text: ϕ

text: ϕ

text: a

text: b(a,

text: a

text: Q

text: a

text: Q

text: Hence,

text: b

text: b

text: on

text: Γ

text: Ω

text: On

text: the

text: other

text: hand,

text: by

text: considering

text: the

text: constant

text: function

text: ψ

text: u

text: ε

text: u

text: corresponding

text: to

text: a

text: point

text: (u,

text: Γ

text: we

text: conclude

text: that

text: b

text: b

text: on

text: Γ

text: Take

text: any

text: x

text: Ω

text: and

text: let

text: be

text: any

text: line

text: that

text: passes

text: through

text: x,

text: a

text: point

text: U

text: on

text: Γ

text: and

text: a

text: x

text: point

text: V

text: on

text: Γ

text: and

text: such

text: that

text: the

text: segment

text: [U,

text: V

text: Ω

text: (there

text: are

text: inﬁnitely

text: many

text: such

text: lines

text: ε

text: for

text: each

text: x).

text: Write

text: x

text: γ)U

text: γV

text: for

text: some

text: γ

text: Note

text: that

text: b

text: is

text: linear

text: along

text: [U,

text: V

text: while

text: b

text: is

text: a

text: locally

text: convex

text: function

text: directly

text: from

text: its

text: deﬁnition.

text: Therefore,

text: b(x)

text: γ)

text: b(U)

text: γ

text: b(V

text: γ)

text: b(U)

text: γ

text: b(V

text: b(x).

text: Thus,

text: b

text: b

text: in

text: Ω

text: 1∈

text: Now,

text: take

text: any

text: x

text: Ω

text: and

text: write

text: x

text: a

text: convex

text: combination

text: of

text: the

text: three

text: “corner”

text: points

text: of

text: Ω

text: −0

text: and

text: Since

text: at

text: these

text: three

text: points

text: b

text: b,

text: b

text: is

text: linear

text: in

text: Ω

text: and

text: b

text: is

text: locally

text: convex,

text: arguing

text: as

text: in

text: proves

text: that

text: b

text: b

text: in

text: Ω

text: Putting

text: everything

text: together,

text: we

text: conclude

text: that

text: b

text: b

text: in

text: Ω

text: ε

text: Proof

text: of

text: Theorem

text: Arguing

text: as

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: obtain

text: and

text: from

text: Theorem

text: To

text: prove

text: sharpness

text: of

text: we

text: do

text: not

text: present

text: an

text: explicit

text: function

text: as

text: D

text: we

text: did

text: in

text: Theorem

text: Instead,

text: take

text: any

text: ε

text: and

text: any

text: Q

text: Theorem

text: says

text: that

text: ε.

text: Therefore,

text: there

text: exists

text: a

text: sequence

text: of

text: functions

text: ϕ

text: from

text: BMOd(Q)

text: such

text: ε

text: k

text: that

text: ϕ

text: ϕ

text: ϕ2

text: ε,

text: and

text: k

text: Q

text: k

text: BMOd(Q)

text: k

text: Q

text: lim

text: ϕ

text: ϕ

text: lim

text: ϕ

text: ε.

text: k→∞

text: k

text: k

text: Q

text: Q

text: k→∞

text: k

text: Q

text: R

text: Extending

text: each

text: ϕk

text: periodically

text: to

text: all

text: of

text: n

text: we

text: get

text: ϕk

text: BMOd(Rn)

text: ε,

text: which

text: completes

text: proof.

text: α-martingales

text: and

text: continuous

text: BMO

text: Any

text: metric

text: measure

text: space

text: has

text: a

text: canonical

text: BMO

text: deﬁned

text: on

text: balls;

text: we

text: call

text: this

text: the

text: continuous

text: T

text: T

text: BMO.

text: If

text: the

text: space

text: is

text: equipped

text: with

text: an

text: α-tree

text: one

text: also

text: has

text: BMO(

text: It

text: is

text: natural

text: to

text: ask

text: whether

text: our

text: theory

text: for

text: BMO

text: on

text: trees

text: can

text: be

text: used

text: to

text: obtain

text: sharp

text: inequalities

text: for

text: the

text: continuous

text: BMO.

text: If

text: the

text: tree

text: in

text: question

text: is

text: ﬁxed

text: from

text: the

text: outset,

text: the

text: answer

text: is

text: no;

text: however,

text: if

text: one

text: allows

text: each

text: BMO

text: function

text: to

text: generate

text: its

text: own

text: tree

text: and,

text: thus,

text: a

text: martingale,

text: then

text: the

text: answer

text: is

text: yes,

text: provided

text: all

text: such

text: martingales

text: are

text: regular

text: in

text: a

text: particular

text: uniform

text: sense.

text: ##LTLine##

text: ##LTLine##


text: img23-eps-converted-to.pdf img24-eps-converted-to.pdf Figure 9. Azimuth angle resolution (left) and zenith angle resolution (right) as a function of primary,s zenith angle in the rectangular layout for the 10-fold condition. img25-eps-converted-to.pdf img26-eps-converted-to.pdf Figure 10. Azimuth angle resolution (left) and zenith angle resolution (right) as a function of primary,s zenith angle for the pentagon layout. are recorded. Moreover at higher energies (around the knee of the cosmic ray spectrum), the array is more eﬃcient. In the rectangular layout, four diﬀerent trigger conditions and four array sizes were considered. Study on the angular resolution showed that it obviously improves as the array size and number of triggered detectors increase. But limitation in the surface area of the array and, the signiﬁcant decrease in recording events for fully triggering of the array (20-fold – 12 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Azimuth

label: caption

text: angle

text: resolution

text: (left)

text: and

text: zenith

text: angle

text: resolution

text: (right)

text: as

text: a

text: function

text: of

text: primary,s

text: zenith

text: angle

text: in

text: the

text: rectangular

text: layout

text: for

text: the

text: condition.

text: Figure

text: Azimuth

text: angle

text: resolution

text: (left)

text: and

text: zenith

text: angle

text: resolution

text: (right)

text: as

text: a

text: function

text: of

text: primary,s

text: zenith

text: angle

text: for

text: the

text: pentagon

text: layout.

text: are

text: recorded.

text: Moreover

text: at

text: higher

text: energies

text: (around

text: the

text: knee

text: of

text: the

text: cosmic

text: ray

text: spectrum),

text: the

text: array

text: is

text: more

text: eﬃcient.

text: In

text: the

text: rectangular

text: layout,

text: four

text: diﬀerent

text: trigger

text: conditions

text: and

text: four

text: array

text: sizes

text: were

text: considered.

text: Study

text: on

text: the

text: angular

text: resolution

text: showed

text: that

text: it

text: obviously

text: improves

text: as

text: the

text: array

text: size

text: and

text: number

text: of

text: triggered

text: detectors

text: increase.

text: But

text: limitation

text: in

text: the

text: surface

text: area

text: of

text: the

text: array

text: and,

text: the

text: signiﬁcant

text: decrease

text: in

text: recording

text: events

text: for

text: fully

text: triggering

text: of

text: the

text: array

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: img1-eps-converted-to.pdf Figure 1. Cluster layout of the Alborz-I array where 20 SDs will be deployed in a square area. Distance between neighboring detectors in the central pentagon and also in the surrounding triangles around the central pentagon is 5 m. 2 The Alborz-I experiment The Alborz Observatory Array (AOA) has been designed to measure the ﬂux, arrival direction and energy of the primary cosmic ray around the knee. The AOA has been planned to have both Scintillation Detectors (SDs) and Water Cherenkov Detectors (WCDs). The Alborz-I as the ﬁrst phase of the AOA is realized at Sharif University of Technology ◦ (cid:48) ◦ (cid:48) (SUT), Tehran (35 43 N 51 20 E, 1200 m a.s.l corresponding to an average atmospheric depth − of 890 gcm 2), and consists of 20 scintillation detectors (each one 0.25 m2) covering an area of about 1600 m2. In the next phase, 10 WCDs will be added to it. The used detectors are optimized by a series of experiments and simulations [4, 5]. Each scintillation detector of the Alborz-I has a pyramidal galvanized iron light enclo- × × sure with 1 mm wall thickness and diﬀuse white coated inner surface, housing a 50 50 2 cm3 NE102A scintillator. Each detector is embedded in a station with a 5 mm cement ceiling. Scintillation light emitted due to passing a particle through the scintillator is read out from tip of the pyramidal light enclosure by a 5 cm photomultiplier tube (PMT, 9813B) at the distance of 20 cm of the scintillator sheet for timing measurements. Two diﬀerent array conﬁgurations, one cluster and other rectangular grid, are proposed. As shown in ﬁgure 1, in the cluster layout, 5 out of 20 detectors arranged on an internal pentagon with side length 5 m and the rest set on 5 surrounding triangles with the same side length 5 m which are placed on the corners of an external pentagon with side length about 18 m. In selection chain of the array properties, such as array layout, trigger condition and energy range of interest, several parameters including the trigger probability function, the number of recorded events and the angular resolution of the array are studied for diﬀerent – 2 – ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Cluster

text: layout

text: of

text: the

text: Alborz-I

text: array

text: where

text: SDs

text: will

text: be

text: deployed

text: in

text: a

text: square

text: area.

text: Distance

text: between

text: neighboring

text: detectors

text: in

text: the

text: central

text: pentagon

text: and

text: also

text: in

text: the

text: surrounding

text: triangles

text: around

text: the

text: central

text: pentagon

text: is

text: m.

label: section

text: The

text: Alborz-I

text: experiment

text: The

text: Alborz

text: Observatory

text: Array

text: (AOA)

text: has

text: been

text: designed

text: to

text: measure

text: the

text: ﬂux,

text: arrival

text: direction

text: and

text: energy

text: of

text: the

text: primary

text: cosmic

text: ray

text: around

text: the

text: knee.

text: The

text: AOA

text: has

text: been

text: planned

text: to

text: have

text: both

text: Scintillation

text: Detectors

text: (SDs)

text: and

text: Water

text: Cherenkov

text: Detectors

text: (WCDs).

text: The

text: Alborz-I

text: as

text: the

text: ﬁrst

text: phase

text: of

text: the

text: AOA

text: is

text: realized

text: at

text: Sharif

text: University

text: of

text: Technology

text: (SUT),

text: Tehran

text: N

text: E,

text: m

text: a.s.l

text: corresponding

text: to

text: an

text: average

text: atmospheric

text: depth

text: of

text: gcm

text: and

text: consists

text: of

text: scintillation

text: detectors

text: (each

text: one

text: covering

text: an

text: area

text: of

text: about

text: In

text: the

text: next

text: phase,

text: WCDs

text: will

text: be

text: added

text: to

text: it.

text: The

text: used

text: detectors

text: are

text: optimized

text: by

text: a

text: series

text: of

text: experiments

text: and

text: simulations

text: Each

text: scintillation

text: detector

text: of

text: the

text: Alborz-I

text: has

text: a

text: pyramidal

text: galvanized

text: iron

text: light

text: enclo-

text: sure

text: with

text: mm

text: wall

text: thickness

text: and

text: diﬀuse

text: white

text: coated

text: inner

text: surface,

text: housing

text: a

text: scintillator.

text: Each

text: detector

text: is

text: embedded

text: in

text: a

text: station

text: with

text: a

text: mm

text: cement

text: ceiling.

text: Scintillation

text: light

text: emitted

text: due

text: to

text: passing

text: a

text: particle

text: through

text: the

text: scintillator

text: is

text: read

text: out

text: from

text: tip

text: of

text: the

text: pyramidal

text: light

text: enclosure

text: by

text: a

text: cm

text: photomultiplier

text: tube

text: (PMT,

text: at

text: the

text: distance

text: of

text: cm

text: of

text: the

text: scintillator

text: sheet

text: for

text: timing

text: measurements.

text: Two

text: diﬀerent

text: array

text: conﬁgurations,

text: one

text: cluster

text: and

text: other

text: rectangular

text: grid,

text: are

text: proposed.

text: As

text: shown

text: in

text: ﬁgure

text: in

text: the

text: cluster

text: layout,

text: out

text: of

text: detectors

text: arranged

text: on

text: an

text: internal

text: pentagon

text: with

text: side

text: length

text: m

text: and

text: the

text: rest

text: set

text: on

text: surrounding

text: triangles

text: with

text: the

text: same

text: side

text: length

text: m

text: which

text: are

text: placed

text: on

text: the

text: corners

text: of

text: an

text: external

text: pentagon

text: with

text: side

text: length

text: about

text: m.

text: In

text: selection

text: chain

text: of

text: the

text: array

text: properties,

text: such

text: as

text: array

text: layout,

text: trigger

text: condition

text: and

text: energy

text: range

text: of

text: interest,

text: several

text: parameters

text: including

text: the

text: trigger

text: probability

text: function,

text: the

text: number

text: of

text: recorded

text: events

text: and

text: the

text: angular

text: resolution

text: of

text: the

text: array

text: are

text: studied

text: for

text: diﬀerent

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: dependencies, the trigger probability function, P(r, E), has been presented as a fraction of showers in the energies of interest (E) which fulﬁll the trigger condition in diﬀerent bins of shower core distance from the array center (r), as N (r, E) P (r, E) = trigger . (4.1) N (r, E) incident Where N (r, E) and N (r, E) are number of showers with energy E which trigger incident fulﬁll the trigger condition at distance r and total number of incident showers with given parameters r and E, respectively. The trigger probability function is an increasing function of energy, as the shower size gets bigger, and a decreasing function of distance from the shower core, as the density of particles gets smaller [10]. The trigger probability function changes between 0 and 1, which 0 happens for large shower core distances and 1 happens for the shower core distances close to the array center in high energy showers. In order to optimize the array size in a rectangular conﬁguration, four diﬀerent array sizes with detector spacings of 1.5 m, 3.5 m, 7 m and 14 m are studied. It should be noted that the surface area of the site at the SUT campus lets the maximum detector spacing of 7 m. However, in the last phase site of the array will be extended. The trigger conditions are provided by randomly at least 4-fold (4 out of 20), 6-fold (6 out of 20), 10-fold (10 out of 20) and ﬁnally fully triggering detectors. Due to the small surface of detectors, the fully triggering condition can provide more satisfactory statistical sample of charged particles in the shower disc for future studies on the energy resolutions and mass composition. Although for the fully triggering condition, the statistic of recorded events decreases signiﬁcantly. The trigger conditions at least 4-fold and 6-fold achieve larger statistics of recorded events. Though, as discussed in the next section for the at least 10-fold condition, the angular resolution accuracy improves signiﬁcantly in all zenith angles rather than at least 4-fold and 6-fold conditions. Thus, in the rectangular layout, we focus the analysis on the at least 10-fold condition. The trigger probability function from simulations for diﬀerent given detector spacing in diﬀerent bins of energy for the rectangular layout is shown in ﬁgures 2 and 3. As can be seen, the maximum shower core distance (r ) for detection in each energy is approximately max independent of the detector spacing while it increases with energy for a given detector spacing. Also the minimum energy of a shower that can be detected by the array obviously de- pends on the detector spacing so that for the smaller distances between neighboring detectors in the rectangular grid, eﬃciency on lower energies increases signiﬁcantly. In other words, the smaller array sizes allow to detect lower energies as the shower disc size becomes smaller. Therefore to focus on the knee, array size should be large enough. Beside the rectangular grid, a cluster layout is also studied. Through several imple- mented trigger conditions, those conditions with full triggered central cluster and two sur- rounding triangles and or three surrounding triangles and or four surrounding triangles, (i.e., 5 out of 5 in the central cluster and 2 adjacent triangles, 5 out of 5 in the central cluster and 3 adjacent triangles, and ﬁnally 5 out of 5 in the central cluster and 4 adjacent triangles) show – 4 – ##LTLine##

text: dependencies,

text: the

text: trigger

text: probability

text: function,

text: P(r,

text: E),

text: has

text: been

text: presented

text: as

text: a

text: fraction

text: of

text: showers

text: in

text: the

text: energies

text: of

text: interest

text: (E)

text: which

text: fulﬁll

text: the

text: trigger

text: condition

text: in

text: diﬀerent

text: bins

text: of

text: shower

text: core

text: distance

text: from

text: the

text: array

text: center

text: (r),

text: as

text: N

label: equation

text: (r,

text: E)

text: P

text: (r,

text: E)

text: trigger

text: N

text: (r,

text: E)

text: incident

text: Where

text: N

text: (r,

text: E)

text: and

text: N

text: (r,

text: E)

text: are

text: number

text: of

text: showers

text: with

text: energy

text: E

text: which

text: trigger

text: incident

text: fulﬁll

text: the

text: trigger

text: condition

text: at

text: distance

text: r

text: and

text: total

text: number

text: of

text: incident

text: showers

text: with

text: given

text: parameters

text: r

text: and

text: E,

text: respectively.

text: The

text: trigger

text: probability

text: function

text: is

text: an

text: increasing

text: function

text: of

text: energy,

text: as

text: the

text: shower

text: size

text: gets

text: bigger,

text: and

text: a

text: decreasing

text: function

text: of

text: distance

text: from

text: the

text: shower

text: core,

text: as

text: the

text: density

text: of

text: particles

text: gets

text: smaller

text: The

text: trigger

text: probability

text: function

text: changes

text: between

text: and

text: which

text: happens

text: for

text: large

text: shower

text: core

text: distances

text: and

text: happens

text: for

text: the

text: shower

text: core

text: distances

text: close

text: to

text: the

text: array

text: center

text: in

text: high

text: energy

text: showers.

text: In

text: order

text: to

text: optimize

text: the

text: array

text: size

text: in

text: a

text: rectangular

text: conﬁguration,

text: four

text: diﬀerent

text: array

text: sizes

text: with

text: detector

text: spacings

text: of

text: m,

text: m,

text: m

text: and

text: m

text: are

text: studied.

text: It

text: should

text: be

text: noted

text: that

text: the

text: surface

text: area

text: of

text: the

text: site

text: at

text: the

text: SUT

text: campus

text: lets

text: the

text: maximum

text: detector

text: spacing

text: of

text: m.

text: However,

text: in

text: the

text: last

text: phase

text: site

text: of

text: the

text: array

text: will

text: be

text: extended.

text: The

text: trigger

text: conditions

text: are

text: provided

text: by

text: randomly

text: at

text: least

text: out

text: of

text: out

text: of

text: out

text: of

text: and

text: ﬁnally

text: fully

text: triggering

text: detectors.

text: Due

text: to

text: the

text: small

text: surface

text: of

text: detectors,

text: the

text: fully

text: triggering

text: condition

text: can

text: provide

text: more

text: satisfactory

text: statistical

text: sample

text: of

text: charged

text: particles

text: in

text: the

text: shower

text: disc

text: for

text: future

text: studies

text: on

text: the

text: energy

text: resolutions

text: and

text: mass

text: composition.

text: Although

text: for

text: the

text: fully

text: triggering

text: condition,

text: the

text: statistic

text: of

text: recorded

text: events

text: decreases

text: signiﬁcantly.

text: The

text: trigger

text: conditions

text: at

text: least

text: and

text: achieve

text: larger

text: statistics

text: of

text: recorded

text: events.

text: Though,

text: as

text: discussed

text: in

text: the

text: next

text: section

text: for

text: the

text: at

text: least

text: condition,

text: the

text: angular

text: resolution

text: accuracy

text: improves

text: signiﬁcantly

text: in

text: all

text: zenith

text: angles

text: rather

text: than

text: at

text: least

text: and

text: conditions.

text: Thus,

text: in

text: the

text: rectangular

text: layout,

text: we

text: focus

text: the

text: analysis

text: on

text: the

text: at

text: least

text: condition.

text: The

text: trigger

text: probability

text: function

text: from

text: simulations

text: for

text: diﬀerent

text: given

text: detector

text: spacing

text: in

text: diﬀerent

text: bins

text: of

text: energy

text: for

text: the

text: rectangular

text: layout

text: is

text: shown

text: in

text: ﬁgures

text: and

text: As

text: can

text: be

text: seen,

text: the

text: maximum

text: shower

text: core

text: distance

text: (r

text: for

text: detection

text: in

text: each

text: energy

text: is

text: approximately

text: max

text: independent

text: of

text: the

text: detector

text: spacing

text: while

text: it

text: increases

text: with

text: energy

text: for

text: a

text: given

text: detector

text: spacing.

text: Also

text: the

text: minimum

text: energy

text: of

text: a

text: shower

text: that

text: can

text: be

text: detected

text: by

text: the

text: array

text: obviously

text: de-

text: pends

text: on

text: the

text: detector

text: spacing

text: so

text: that

text: for

text: the

text: smaller

text: distances

text: between

text: neighboring

text: detectors

text: in

text: the

text: rectangular

text: grid,

text: eﬃciency

text: on

text: lower

text: energies

text: increases

text: signiﬁcantly.

text: In

text: other

text: words,

text: the

text: smaller

text: array

text: sizes

text: allow

text: to

text: detect

text: lower

text: energies

text: as

text: the

text: shower

text: disc

text: size

text: becomes

text: smaller.

text: Therefore

text: to

text: focus

text: on

text: the

text: knee,

text: array

text: size

text: should

text: be

text: large

text: enough.

text: Beside

text: the

text: rectangular

text: grid,

text: a

text: cluster

text: layout

text: is

text: also

text: studied.

text: Through

text: several

text: imple-

text: mented

text: trigger

text: conditions,

text: those

text: conditions

text: with

text: full

text: triggered

text: central

text: cluster

text: and

text: two

text: sur-

text: rounding

text: triangles

text: and

text: or

text: three

text: surrounding

text: triangles

text: and

text: or

text: four

text: surrounding

text: triangles,

text: (i.e.,

text: out

text: of

text: in

text: the

text: central

text: cluster

text: and

text: adjacent

text: triangles,

text: out

text: of

text: in

text: the

text: central

text: cluster

text: and

text: adjacent

text: triangles,

text: and

text: ﬁnally

text: out

text: of

text: in

text: the

text: central

text: cluster

text: and

text: adjacent

text: triangles)

text: show

text: ##LTLine##


text: (cid:52) diﬀerent bins of energy (N (E, E + E)) that fulﬁll the implemented trigger condition for diﬀerent array layouts can be calculated by a convolution of the trigger probability function P(r, E) and diﬀerential ﬂux of primary cosmic rays Φ(E) (per area, time, solid angle and (cid:90) (cid:90) (cid:90) (cid:90) energy) (eq.(4.2)), (cid:52) (cid:52) E+ E N(E, E + E) = P (r, E)Φ(E) dS dE dΩ dt (4.2) S E where the intervals are selected according to the default initial conditions in the simu- lations. S is the surface of a square grid which considered to study showers arriving inside and outside of the array while triggered it. Φ(E) (eqations (4.3a), (4.3b) and (4.3c)) is a primary all-particle spectrum in the knee region determined with the Tibet air shower array which is located at an almost ideal atmospheric depth for this energy range and is highly · instrumented [10]. In the energy range below the knee, for E<5.62 1014 eV , · − E − ± − − − − Φ(E) = 1.5 10 20( · ) 2.60 0.04 [m 2s 1sr 1eV 1], (4.3a) 5.62 1014 · and above the knee, for E > 7.08 1015 eV , · − E − ± − − − − Φ(E) = 1.2 10 23( · ) 3.00 0.05 [m 2s 1sr 1eV 1], (4.3b) 7.08 1015 · and in the knee, at E = 1.78 1015 eV , · − − − − − Φ(E) = 6.7 10 22 [m 2s 1sr 1eV 1]. (4.3c) Figure 5 shows the energy distribution of triggered events per day that fulﬁll the 10-fold condition for diﬀerent array sizes in the rectangular layout. The results show an energy de- pendence of maximum triggered events on the detector spacing so that it changes from 1014 × eV in the small array sizes (detector spacing of 1.5 m and 3.5 m) to 3 1014 eV in the large enough array sizes (detector spacing of 7 m and 14 m). The energy distribution of triggered events per day for diﬀerent trigger conditions in × the cluster layout is shown in ﬁgure 6. As can be seen, the maximum happens in 3 1014 eV so that the array size is the same as the rectangular grid with detector spacing 7 m. It should be noted that the errors are due to systematic uncertainties of the cosmic ray spectrum (eqations (4.3a), (4.3b) and (4.3c)) which is used in the Monte Carlo simulations to estimate the number of triggered events. 5 Angular resolution The angular resolution of the Alborz-I is discussed by considering the second shower set including 12000 proton and alpha showers (with the same previous composition) distributed × × ◦ ◦ over a continuous range from 2 1014 eV to 4 1014 eV with θ between 0 and 60 in steps ◦ of 5 . This energy range is selected according to receiving maximum number of events in – 7 – ##LTLine## ##LTLine##

text: diﬀerent

text: bins

text: of

text: energy

text: (N

text: (E,

text: E

text: E))

text: that

text: fulﬁll

text: the

text: implemented

text: trigger

text: condition

text: for

text: diﬀerent

text: array

text: layouts

text: can

text: be

text: calculated

text: by

text: a

text: convolution

text: of

text: the

text: trigger

text: probability

text: function

text: P(r,

text: E)

text: and

text: diﬀerential

text: ﬂux

text: of

text: primary

text: cosmic

text: rays

text: Φ(E)

text: (per

text: area,

text: time,

text: solid

text: angle

text: and

text: energy)

text: E+

text: E

text: N(E,

text: E

text: E)

text: P

text: (r,

text: E)Φ(E)

text: dS

text: dE

text: dΩ

text: dt

text: S

text: E

text: where

text: the

text: intervals

text: are

text: selected

text: according

text: to

text: the

text: default

text: initial

text: conditions

text: in

text: the

text: simu-

text: lations.

text: S

text: is

text: the

text: surface

text: of

text: a

text: square

text: grid

text: which

text: considered

text: to

text: study

text: showers

text: arriving

text: inside

text: and

text: outside

text: of

text: the

text: array

text: while

text: triggered

text: it.

text: Φ(E)

text: (eqations

text: and

text: is

text: a

text: primary

text: all-particle

text: spectrum

text: in

text: the

text: knee

text: region

text: determined

text: with

text: the

text: Tibet

text: air

text: shower

text: array

text: which

text: is

text: located

text: at

text: an

text: almost

text: ideal

text: atmospheric

text: depth

text: for

text: this

text: energy

text: range

text: and

text: is

text: highly

text: instrumented

text: In

text: the

text: energy

text: range

text: below

text: the

text: knee,

text: for

text: eV

text: E

text: Φ(E)

text: [m

text: and

text: above

text: the

text: knee,

text: for

text: E

text: eV

text: E

text: Φ(E)

text: [m

text: and

text: in

text: the

text: knee,

text: at

text: E

text: eV

text: Φ(E)

text: [m

text: Figure

text: shows

text: the

text: energy

text: distribution

text: of

text: triggered

text: events

text: per

text: day

text: that

text: fulﬁll

text: the

text: condition

text: for

text: diﬀerent

text: array

text: sizes

text: in

text: the

text: rectangular

text: layout.

text: The

text: results

text: show

text: an

text: energy

text: de-

text: pendence

text: of

text: maximum

text: triggered

text: events

text: on

text: the

text: detector

text: spacing

text: so

text: that

text: it

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text: array

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text: (detector

text: spacing

text: of

text: m

text: and

text: m).

text: The

text: energy

text: distribution

text: of

text: triggered

text: events

text: per

text: day

text: for

text: diﬀerent

text: trigger

text: conditions

text: in

text: the

text: cluster

text: layout

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text: shown

text: in

text: ﬁgure

text: As

text: can

text: be

text: seen,

text: the

text: maximum

text: happens

text: in

text: eV

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text: that

text: the

text: array

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text: is

text: the

text: same

text: as

text: the

text: rectangular

text: grid

text: with

text: detector

text: spacing

text: m.

text: It

text: should

text: be

text: noted

text: that

text: the

text: errors

text: are

text: due

text: to

text: systematic

text: uncertainties

text: of

text: the

text: cosmic

text: ray

text: spectrum

text: (eqations

text: and

text: which

text: is

text: used

text: in

text: the

text: Monte

text: Carlo

text: simulations

text: to

text: estimate

text: the

text: number

text: of

text: triggered

text: events.

text: Angular

text: resolution

text: The

text: angular

text: resolution

text: of

text: the

text: Alborz-I

text: is

text: discussed

text: by

text: considering

text: the

text: second

text: shower

text: set

text: including

text: proton

text: and

text: alpha

text: showers

text: (with

text: the

text: same

text: previous

text: composition)

text: distributed

text: over

text: a

text: continuous

text: range

text: from

text: eV

text: to

text: eV

text: with

text: θ

text: between

text: and

text: in

text: steps

text: of

text: This

text: energy

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text: selected

text: according

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text: number

text: of

text: events

text: in

text: ##LTLine##

text: ##LTLine##


text: LIFTS OF NON-COMPACT CONVEX SETS AND CONE FACTORIZATIONS CHU WANG AND LIHONG ZHI Abstract. In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we deﬁne a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an aﬃne slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly deﬁned slack operator. Key words: lift; convex set; recession cone; polyhedron; cone factorization; non- negative rank; positive semideﬁnite rank. 1. Introduction Given a linear programming problem, how to reformulate it to a standard form with fewer constraints is an important problem. In [8], Yannakakis proved that the nonnegative rank of a slack matrix of a polytope P is the minimum k such that P is the linear image of an aﬃne slice of the nonnegative quadrant. In [1, 4], Yannakakis’s result was generalized to decide whether a convex body C (a compact convex set containing the origin in its interior) is the linear image of an aﬃne slice of a given convex cone K (K-lift) via cone factorizations of slack operators. Although it was claimed that results in [4] hold for all convex sets, we notice that it is more complicated to identify whether a non-compact convex set C containing no lines has a K-lift since C could be generated by not only extreme points but also extreme directions. Moreover, if a convex set contains lines, then it has no extreme points or extreme directions. Furthermore, if the convex set does not contain the origin in its interior, linear functions corresponding to its polar can not characterize the convex set completely (see Example 3.1). These facts motivate us to study how to extend the deﬁnitions of K-lift and slack operator to a general convex set and show the relationship between lifts of convex sets and cone factorizations of slack operators when the convex set is not a convex body. R Our contribution: Let C be a closed convex set in n and K a closed convex R cone in m. We consider how to generalize the factorization theorem in [4] to a broader class of convex sets. Our main results are as follows. • When C contains the origin, since C = Coo, C can be described by all vectors in Co. When C does not contain the origin, we show that a convex set C can be characterized completely by linear functions deﬁned by Co = Date: May 27, 2020. 1 ##LTLine##

text: LIFTS

text: OF

text: NON-COMPACT

text: CONVEX

text: SETS

text: AND

text: CONE

text: FACTORIZATIONS

text: CHU

text: WANG

text: AND

text: LIHONG

text: ZHI

text: Abstract.

text: In

label: abstract

text: this

text: paper

text: we

text: generalize

text: the

text: factorization

text: theorem

text: of

text: Gouveia,

text: Parrilo

text: and

text: Thomas

text: to

text: a

text: broader

text: class

text: of

text: convex

text: sets.

text: Given

text: a

text: general

text: convex

text: set,

text: we

text: deﬁne

text: a

text: slack

text: operator

text: associated

text: to

text: the

text: set

text: and

text: its

text: polar

text: according

text: to

text: whether

text: the

text: convex

text: set

text: is

text: full

text: dimensional,

text: whether

text: it

text: is

text: a

text: translated

text: cone

text: and

text: whether

text: it

text: contains

text: lines.

text: We

text: strengthen

text: the

text: condition

text: of

text: a

text: cone

text: lift

text: by

text: requiring

text: not

text: only

text: the

text: convex

text: set

text: is

text: the

text: image

text: of

text: an

text: aﬃne

text: slice

text: of

text: a

text: given

text: closed

text: convex

text: cone,

text: but

text: also

text: its

text: recession

text: cone

text: is

text: the

text: image

text: of

text: the

text: linear

text: slice

text: of

text: the

text: closed

text: convex

text: cone.

text: We

text: show

text: that

text: the

text: generalized

text: lift

text: of

text: a

text: convex

text: set

text: can

text: also

text: be

text: characterized

text: by

text: the

text: cone

text: factorization

text: of

text: a

text: properly

text: deﬁned

text: slack

text: operator.

text: Key

text: words:

text: lift;

text: convex

text: set;

text: recession

text: cone;

text: polyhedron;

text: cone

text: factorization;

text: non-

text: negative

text: rank;

text: positive

text: semideﬁnite

text: rank.

text: Introduction

text: Given

text: a

text: linear

text: programming

text: problem,

text: how

text: to

text: reformulate

text: it

text: to

text: a

text: standard

text: form

text: with

text: fewer

text: constraints

text: is

text: an

text: important

text: problem.

text: In

text: Yannakakis

text: proved

text: that

text: the

text: nonnegative

text: rank

text: of

text: a

text: slack

text: matrix

text: of

text: a

text: polytope

text: P

text: is

text: the

text: minimum

text: k

text: such

text: that

text: P

text: is

text: the

text: linear

text: image

text: of

text: an

text: aﬃne

text: slice

text: of

text: the

text: nonnegative

text: quadrant.

text: In

text: Yannakakis’s

text: result

text: was

text: generalized

text: to

text: decide

text: whether

text: a

text: convex

text: body

text: C

text: (a

text: compact

text: convex

text: set

text: containing

text: the

text: origin

text: in

text: its

text: interior)

text: is

text: the

text: linear

text: image

text: of

text: an

text: aﬃne

text: slice

text: of

text: a

text: given

text: convex

text: cone

text: K

text: (K-lift)

text: via

text: cone

text: factorizations

text: of

text: slack

text: operators.

text: Although

text: it

text: was

text: claimed

text: that

text: results

text: in

text: hold

text: for

text: all

text: convex

text: sets,

text: we

text: notice

text: that

text: it

text: is

text: more

text: complicated

text: to

text: identify

text: whether

text: a

text: non-compact

text: convex

text: set

text: C

text: containing

text: no

text: lines

text: has

text: a

text: K-lift

text: since

text: C

text: could

text: be

text: generated

text: by

text: not

text: only

text: extreme

text: points

text: but

text: also

text: extreme

text: directions.

text: Moreover,

text: if

text: a

text: convex

text: set

text: contains

text: lines,

text: then

text: it

text: has

text: no

text: extreme

text: points

text: or

text: extreme

text: directions.

text: Furthermore,

text: if

text: the

text: convex

text: set

text: does

text: not

text: contain

text: the

text: origin

text: in

text: its

text: interior,

text: linear

text: functions

text: corresponding

text: to

text: its

text: polar

text: can

text: not

text: characterize

text: the

text: convex

text: set

text: completely

text: (see

text: Example

text: These

text: facts

text: motivate

text: us

text: to

text: study

text: how

text: to

text: extend

text: the

text: deﬁnitions

text: of

text: K-lift

text: and

text: slack

text: operator

text: to

text: a

text: general

text: convex

text: set

text: and

text: show

text: the

text: relationship

text: between

text: lifts

text: of

text: convex

text: sets

text: and

text: cone

text: factorizations

text: of

text: slack

text: operators

text: when

text: the

text: convex

text: set

text: is

text: not

text: a

text: convex

text: body.

text: R

text: Our

text: contribution:

text: Let

text: C

text: be

text: a

text: closed

text: convex

text: set

text: in

text: n

text: and

text: K

text: a

text: closed

text: convex

text: R

text: cone

text: in

text: m.

text: We

text: consider

text: how

text: to

text: generalize

text: the

text: factorization

text: theorem

text: in

text: to

text: a

text: broader

text: class

text: of

text: convex

text: sets.

text: Our

text: main

text: results

text: are

text: as

text: follows.

text: When

text: C

text: contains

text: the

text: origin,

text: since

text: C

text: Coo,

text: C

text: can

text: be

text: described

text: by

text: all

text: vectors

text: in

text: Co.

text: When

text: C

text: does

text: not

text: contain

text: the

text: origin,

text: we

text: show

text: that

text: a

text: convex

text: set

text: C

text: can

text: be

text: characterized

text: completely

text: by

text: linear

text: functions

text: deﬁned

text: by

text: Co

text: Date:

text: May

text: ##LTLine##


text: 12 CHU WANG AND LIHONG ZHI Figure 3. Example 3.14 Example 3.14. Let C be a 3-dimensional polyhedron in R3 deﬁned by thefollowing inequaClit=y: (x1, x2, x3) ∈ R3 :  −−10011 −−2√√2√333√√√3333333 00000   xxx123  ≤  111111  . − 0 1 3 0 3 − 0 0 1 { } The six vertices of C are (cos(iπ/3), sin(iπ/3), 0), i = 0, . . . , 5 and 0+C = { } cone ( (0, 0, 1) ). According to Deﬁnition 4.1, its slack matrix is    0 0 1 2 2 1 0     1 0 0 1 2 2 0     2 1 0 0 1 2 0    S :=  2 2 1 0 0 1 0  . 1 2 2 1 0 0 0 0 1 2 2 1 0 0 0 0 0 0 0 0 1 × It has been shown in [4, Example 2] that the ﬁrst 6 6 submatrix S of S has R H R a 5+-factorization(cid:18). Howeve(cid:19)r, we(cid:18)claim that t(cid:19)he(cid:18)matrix S do(cid:19)es not have a 5+- factorization. If it does, we can assume it has the following nonnegative decompo- sition: S 0 A A B B H = 11 12 11 12 0 1 A A B B 21 22 21 22 Since A B + A B = 0, we have A B = 0. We claim that A = 0. 21 11 22 21 22 21 22 Otherwise, B will be zero and S = A B . This contradicts to the fact that R21 H 11 11 S has no 4 -factorization. From A B + A B = 1 and A = 0, we have H + 21 12 22 22 22 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHU

text: WANG

text: AND

text: LIHONG

text: ZHI

text: Figure

text: Example

text: Example

text: Let

text: C

text: be

text: a

text: polyhedron

text: in

text: deﬁned

text: by

text: thefollowing

text: inequaClit=y:

text: (x1,

text: The

text: six

text: vertices

text: of

text: C

text: are

text: i

text: and

text: cone

text: According

text: to

text: Deﬁnition

text: its

text: slack

text: matrix

text: is

text: S

text: It

text: has

text: been

text: shown

text: in

text: Example

text: that

text: the

text: ﬁrst

text: submatrix

text: S

text: of

text: S

text: has

text: R

text: H

text: R

text: a

text: that

text: S

text: not

text: have

text: a

text: factorization.

text: If

text: it

text: does,

text: we

text: can

text: assume

text: it

text: has

text: the

text: following

text: nonnegative

text: decompo-

text: sition:

text: S

text: A

text: A

text: B

text: B

text: H

text: A

text: A

text: B

text: B

text: Since

text: A

text: B

text: A

text: B

text: we

text: have

text: A

text: B

text: We

text: claim

text: that

text: A

text: Otherwise,

text: B

text: will

text: be

text: zero

text: and

text: S

text: A

text: B

text: This

text: contradicts

text: to

text: the

text: fact

text: that

text: H

text: S

text: has

text: no

text: -factorization.

text: From

text: A

text: B

text: A

text: B

text: and

text: A

text: we

text: have

text: H

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Gianluca Giovannetti,1,1 Luca de’ Medici,1 Markus Aichhorn,1 and Massimo Capone1,1 We show that the insulating character of the iron-selenide La O Fe Se can be explained in 2 3 2 2 terms of Mott localization in sharp contrast with the metallic behavior of FeSe and other parent parent compounds of iron superconductors. We demonstrate that the key ingredient that makes La O Fe Se a Mott insulator, rather than a correlated metal dominated by the Hund’s coupling is 2 3 2 2 the enhanced crystal-ﬁeld splitting, accompanied by a smaller orbital-resolved kinetic energy. The strong deviation from orbital degeneracy introduced by the crystal-ﬁeld splitting also pushes this materials close to an orbital-selective Mott transition. We predict that either doping or uniaxial external pressure can drive the material into an orbital-selective Mott state, where only one or few orbitals are metallized while the others remain insulating. PACS numbers: 71.30.+h, 71.10.Fd, 71.27.+a I. INTRODUCTION hole per iron site, isostructural materials where iron is re- placed with manganese are characterized by the d5 half- ﬁlled conﬁguration and they are indeed antiferromagnetic The link between high-temperature superconductiv- Mott insulators.12–14. However, the evidence for strong- ity and strong electron-electron correlations has been correlation physics in the FeSC does can not rely on the forged and strengthened by decades of investigation in presence of actual Mott states directly connected with the copper-based superconductors (cuprates). In this the superconducting compounds. Indeed no Mott insu- light, the debate about the strength and the role of elec- lator exists in the 122 family (BaFe As or isoelectronic tron correlations in iron-based superconductors (FeSC) 2 2 compounds doped either with holes and electrons), the maintains a crucial importance. The overall phenomenol- 1111 family which originates doping LaFeAsO, the 11 ogy of these materials does not provide a self-evident selenides FeSe and FeTe, and LiFeAS. While this experi- answer. In these materials superconductivity appears mental fact is indeed completely compatible with the sce- doping a metallic spin-density-wave parent compound, nario based on the Hund’s coupling, it may cast doubts rather than the Mott insulator of the cuprates, and, on the whole relevance of correlations. while the metallic state is highly incoherent, the stan- dard ﬁngerprints of strong correlations, like the Hubbard A close relative of FeSC with insulating behavior is bands, are not universally observed. La O Fe Se . This system is based on a square lattice of 2 3 2 2 Furthermore non-perturbative studies of the eﬀect of Fe ions with nominal valence 2+ as in all of iron pnictides the interactions, mainly based on Dynamical Mean-Field and chalcogenides. However, the resistivity as a function Theory (DMFT) and related methods, have highlighted of temperature shows clearly an insulating behavior with a novel behavior, in which the Hund’s coupling J to play an activation energy gap of about 0.19 eV.15 On the ba- h a major role in determining the degree of correlations1,2. sis of this experimental evidence combined with Density- The electrons in the d-orbitals (the parent compounds Functional Theory (DFT) supplemented by mean-ﬁeld have a nominal ﬁlling of six electrons for each iron atom) treatment of the Hubbard U (DFT+U) in the ordered are strongly correlated, as measured by the bad-metallic magnetic phase it has been argued that La2O3Fe2Se2 behavior with small coherence scales shown in many ex- is in a Mott state with low-temperature antiferromag- periments, but the Hubbard repulsion U is substantially netic ordering.15 This evidence raises the natural ques- smaller than the critical value for the Mott transition3,4. tion about the reason why the Hund’s metallic behav- This regime is often labeled as a “Hund’s metal” and ior of undoped FeSC is replaced with a Mott state in displays characteristic properties5,6 including a remark- this compound. A simple bandwidth reduction is a very able tendency to “orbital selectivity”, i.e., to a neat dif- unlikely answer, since one of the deﬁning properties of ferentiation in the degree of correlation of the diﬀerent the Hund’s metal regime is that the critical value for a orbitals, leading to the simultaneous presence of weakly Mott transition is pushed to very large values, signiﬁ- and strongly correlated electrons4,7–10. In Ref.10 it has cantly far from the experimental estimates.16 This calls been shown that the Hund’s coupling decouples the or- for the identiﬁcation of speciﬁc aspects of the bandstruc- bitals quenching the inter-orbital ﬂuctuations and that ture which make the present material insulating. this leads to a picture of ﬁve doped single-band Mott in- In this work we investigate the diﬀerences of the elec- sulators. Consequently the degree of correlation is con- tronic structure of La O Fe Se with respect to parent 2 3 2 2 trolled by the distance of each individual orbital from the compounds of the FeSC using the reference case of FeSe. half-ﬁlled d5 conﬁguration. The relevance of the d5 Mott We identify in a strong lifting of the orbital degeneracy phase in the phase diagram of models for iron supercon- the main diﬀerence of the material under consideration ductors has been observed also in 11. with respect to most parent compounds of iron-based su- While no iron-based material can be doped with one perconductors. This leads to a marked tendency towards

text: Gianluca

label: author

text: Luca

text: de’

text: Markus

text: and

text: Massimo

text: We

text: show

text: that

text: the

text: insulating

text: character

text: of

text: the

text: iron-selenide

text: La

text: O

text: Fe

text: Se

text: can

text: be

text: explained

text: in

text: terms

text: of

text: Mott

text: localization

text: in

text: sharp

text: contrast

text: with

text: the

text: metallic

text: behavior

text: of

text: FeSe

text: and

text: other

text: parent

text: parent

text: compounds

text: of

text: iron

text: superconductors.

text: We

text: demonstrate

text: that

text: the

text: key

text: ingredient

text: that

text: makes

text: La

text: O

text: Fe

text: Se

text: a

text: Mott

text: insulator,

text: rather

text: than

text: a

text: correlated

text: metal

text: dominated

text: by

text: the

text: Hund’s

text: coupling

text: is

text: the

text: enhanced

text: crystal-ﬁeld

text: splitting,

text: accompanied

text: by

text: a

text: smaller

text: orbital-resolved

text: kinetic

text: energy.

text: The

text: strong

text: deviation

text: from

text: orbital

text: degeneracy

text: introduced

text: by

text: the

text: crystal-ﬁeld

text: splitting

text: also

text: pushes

text: this

text: materials

text: close

text: to

text: an

text: orbital-selective

text: Mott

text: transition.

text: We

text: predict

text: that

text: either

text: doping

text: or

text: uniaxial

text: external

text: pressure

text: can

text: drive

text: the

text: material

text: into

text: an

text: orbital-selective

text: Mott

text: state,

text: where

text: only

text: one

text: or

text: few

text: orbitals

text: are

text: metallized

text: while

text: the

text: others

text: remain

text: insulating.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: hole

text: per

text: iron

text: site,

text: isostructural

text: materials

text: where

text: iron

text: is

text: re-

text: placed

text: with

text: manganese

text: are

text: characterized

text: by

text: the

text: half-

text: ﬁlled

text: conﬁguration

text: and

text: they

text: are

text: indeed

text: antiferromagnetic

text: The

text: link

text: between

text: high-temperature

text: superconductiv-

text: Mott

text: However,

text: the

text: evidence

text: for

text: strong-

text: ity

text: and

text: strong

text: electron-electron

text: correlations

text: has

text: been

text: correlation

text: physics

text: in

text: the

text: FeSC

text: does

text: can

text: not

text: rely

text: on

text: the

text: forged

text: and

text: strengthened

text: by

text: decades

text: of

text: investigation

text: in

text: presence

text: of

text: actual

text: Mott

text: states

text: directly

text: connected

text: with

text: the

text: copper-based

text: superconductors

text: (cuprates).

text: In

text: this

text: the

text: superconducting

text: compounds.

text: Indeed

text: no

text: Mott

text: insu-

text: light,

text: the

text: debate

text: about

text: the

text: strength

text: and

text: the

text: role

text: of

text: elec-

text: lator

text: exists

text: in

text: the

text: family

text: (BaFe

text: As

text: or

text: isoelectronic

text: tron

text: correlations

text: in

text: iron-based

text: superconductors

text: (FeSC)

text: compounds

text: doped

text: either

text: with

text: holes

text: and

text: electrons),

text: the

text: maintains

text: a

text: crucial

text: importance.

text: The

text: overall

text: phenomenol-

text: family

text: which

text: originates

text: doping

text: LaFeAsO,

text: the

text: ogy

text: of

text: these

text: materials

text: does

text: not

text: provide

text: a

text: self-evident

text: selenides

text: FeSe

text: and

text: FeTe,

text: and

text: LiFeAS.

text: While

text: this

text: experi-

text: answer.

text: In

text: these

text: materials

text: superconductivity

text: appears

text: mental

text: fact

text: is

text: indeed

text: completely

text: compatible

text: with

text: the

text: sce-

text: doping

text: a

text: metallic

text: spin-density-wave

text: parent

text: compound,

text: nario

text: based

text: on

text: the

text: Hund’s

text: coupling,

text: it

text: may

text: cast

text: doubts

text: rather

text: than

text: the

text: Mott

text: insulator

text: of

text: the

text: cuprates,

text: and,

text: on

text: the

text: whole

text: relevance

text: of

text: correlations.

text: while

text: the

text: metallic

text: state

text: is

text: highly

text: incoherent,

text: the

text: stan-

text: dard

text: ﬁngerprints

text: of

text: strong

text: correlations,

text: like

text: the

text: Hubbard

text: A

text: close

text: relative

text: of

text: FeSC

text: with

text: insulating

text: behavior

text: is

text: bands,

text: are

text: not

text: universally

text: observed.

text: La

text: O

text: Fe

text: Se

text: This

text: system

text: is

text: based

text: on

text: a

text: square

text: lattice

text: of

text: Furthermore

text: non-perturbative

text: studies

text: of

text: the

text: eﬀect

text: of

text: Fe

text: ions

text: with

text: nominal

text: valence

text: as

text: in

text: all

text: of

text: iron

text: pnictides

text: the

text: interactions,

text: mainly

text: based

text: on

text: Dynamical

text: Mean-Field

text: and

text: chalcogenides.

text: However,

text: the

text: resistivity

text: as

text: a

text: function

text: Theory

text: (DMFT)

text: and

text: related

text: methods,

text: have

text: highlighted

text: of

text: temperature

text: shows

text: clearly

text: an

text: insulating

text: behavior

text: with

text: a

text: novel

text: behavior,

text: in

text: which

text: the

text: Hund’s

text: coupling

text: J

text: to

text: play

text: an

text: activation

text: energy

text: gap

text: of

text: about

text: On

text: the

text: ba-

text: h

text: a

text: major

text: role

text: in

text: determining

text: the

text: degree

text: of

text: sis

text: of

text: this

text: experimental

text: evidence

text: combined

text: with

text: Density-

text: The

text: electrons

text: in

text: the

text: d-orbitals

text: (the

text: parent

text: compounds

text: Functional

text: Theory

text: (DFT)

text: supplemented

text: by

text: mean-ﬁeld

text: have

text: a

text: nominal

text: ﬁlling

text: of

text: six

text: electrons

text: for

text: each

text: iron

text: atom)

text: treatment

text: of

text: the

text: Hubbard

text: U

text: (DFT+U)

text: in

text: the

text: ordered

text: are

text: strongly

text: correlated,

text: as

text: measured

text: by

text: the

text: bad-metallic

text: magnetic

text: phase

text: it

text: has

text: been

text: argued

text: that

text: behavior

text: with

text: small

text: coherence

text: scales

text: shown

text: in

text: many

text: ex-

text: is

text: in

text: a

text: Mott

text: state

text: with

text: low-temperature

text: antiferromag-

text: periments,

text: but

text: the

text: Hubbard

text: repulsion

text: U

text: is

text: substantially

text: netic

text: This

text: evidence

text: raises

text: the

text: natural

text: ques-

text: smaller

text: than

text: the

text: critical

text: value

text: for

text: the

text: Mott

text: tion

text: about

text: the

text: reason

text: why

text: the

text: Hund’s

text: metallic

text: behav-

text: This

text: regime

text: is

text: often

text: labeled

text: as

text: a

text: “Hund’s

text: metal”

text: and

text: ior

text: of

text: undoped

text: FeSC

text: is

text: replaced

text: with

text: a

text: Mott

text: state

text: in

text: displays

text: characteristic

text: including

text: a

text: remark-

text: this

text: compound.

text: A

text: simple

text: bandwidth

text: reduction

text: is

text: a

text: very

text: able

text: tendency

text: to

text: “orbital

text: selectivity”,

text: i.e.,

text: to

text: a

text: neat

text: dif-

text: unlikely

text: answer,

text: since

text: one

text: of

text: the

text: deﬁning

text: properties

text: of

text: ferentiation

text: in

text: the

text: degree

text: of

text: correlation

text: of

text: the

text: diﬀerent

text: the

text: Hund’s

text: metal

text: regime

text: is

text: that

text: the

text: critical

text: value

text: for

text: a

text: orbitals,

text: leading

text: to

text: the

text: simultaneous

text: presence

text: of

text: weakly

text: Mott

text: transition

text: is

text: pushed

text: to

text: very

text: large

text: values,

text: signiﬁ-

text: and

text: strongly

text: correlated

text: In

text: it

text: has

text: cantly

text: far

text: from

text: the

text: experimental

text: This

text: calls

text: been

text: shown

text: that

text: the

text: Hund’s

text: coupling

text: decouples

text: the

text: or-

text: for

text: the

text: identiﬁcation

text: of

text: speciﬁc

text: aspects

text: of

text: the

text: bandstruc-

text: bitals

text: quenching

text: the

text: inter-orbital

text: ﬂuctuations

text: and

text: that

text: ture

text: which

text: make

text: the

text: present

text: material

text: insulating.

text: this

text: leads

text: to

text: a

text: picture

text: of

text: ﬁve

text: doped

text: single-band

text: Mott

text: in-

text: In

text: this

text: work

text: we

text: investigate

text: the

text: diﬀerences

text: of

text: the

text: elec-

text: sulators.

text: Consequently

text: the

text: degree

text: of

text: correlation

text: is

text: con-

text: tronic

text: structure

text: of

text: La

text: O

text: Fe

text: Se

text: with

text: respect

text: to

text: parent

text: trolled

text: by

text: the

text: distance

text: of

text: each

text: individual

text: orbital

text: from

text: the

text: compounds

text: of

text: the

text: FeSC

text: using

text: the

text: reference

text: case

text: of

text: FeSe.

text: half-ﬁlled

text: conﬁguration.

text: The

text: relevance

text: of

text: the

text: Mott

text: We

text: identify

text: in

text: a

text: strong

text: lifting

text: of

text: the

text: orbital

text: degeneracy

text: phase

text: in

text: the

text: phase

text: diagram

text: of

text: models

text: for

text: iron

text: supercon-

text: the

text: main

text: diﬀerence

text: of

text: the

text: material

text: under

text: consideration

text: ductors

text: has

text: been

text: observed

text: also

text: in

text: with

text: respect

text: to

text: most

text: parent

text: compounds

text: of

text: iron-based

text: su-

text: While

text: no

text: iron-based

text: material

text: can

text: be

text: doped

text: with

text: one

text: perconductors.

text: This

text: leads

text: to

text: a

text: marked

text: tendency

text: towards


text: First, th(cid:80)e term in front of the state sum contributes a−2r(c+−c−). Next we get a−12 j rot((cid:15)j) from the MO(cid:80)Y state sum. This term is computed in the following lemma: − Lemma 3. If (cid:15) sums to γ then rot((cid:15) ) = r(s s−), where s and j j + + s− are the numbers of positive and negative Seifert circles of the diagram. Proof. We say two elementary ﬂows α, β intersect if α contains edges in more than one region complementary to β. Suppose our state (cid:15) contains a (cid:48) pair of intersecting ﬂows, α, β. We may create a new state (cid:15) by replacing (cid:80) two intersecting components by their resolution as in Seifert’s algorithm. This new state has the same number of components and the same rotation numbers. Hence we may compute the sum rot((cid:15) ) on a state (cid:15) without j j intersecting ﬂows. In such states all elementary ﬂows are parallel to the Seifert-circles of the underlying knot diagram. Around each Seifert circle there are r components of elementary ﬂows with rotation number equal − to the sign of the Seifert circle. Therefore the sum equals r(s s−). + − Finally we need to estimate maxdeg E (a, 1, q). According to a rot,P(cid:15) q Lemma 2 this is the maximum of rot over the vertices of the order polytope P . This is the sum of the ﬂows with positive rotation number. For states (cid:15) where each ﬂow has a single component this is maximal and equal to rs . ≤ − +− Adding everything up we ﬁnd maxdeg P (D; a, q) r( c + c− − a r 2 + s s−) proving part a) of theorem 1. + As claimed in the introduction this agrees with Morton’s bound when r = 1. To properly compare the formulas we note that we do not divide by the unknot; our colored HOMFLY is unreduced. Also Morton’s variable v is our a−12 and his variable z is our q 12 − q−12 . 4.2 Bound on the q-degree As with the a-bound our strategy is to estimate max maxdeg (i, (cid:15)) i,(cid:15) q by studying the states (i, (cid:15)) carefully. (cid:80) −(c+−c−)r(r−1) First the framing term in front of the state sum gives q 2 . − c∈C σ(c)ic Next the choice i of resolving the crossings yields q 2 . According r(s+−s−) to Lemma 3 the rotation factor gives q 2 . Next comes the weight − q w((cid:15)) and ﬁnally the Ehrhart polynomial. For the latter we use Lemma 2 to get a maximal contribution of q0 since each order polytope contains the origin. Now assume that there exists a state (cid:15) attaining the minimal possible weight w((cid:15)). Combining all the contributions and the upper bound for Ehrhart this means that every crossing c contributes at most 1 − − − − ((r i )i + (r i )r σ(c)(i + r(r 1))) 2 c c c c ∈ { } This is maximal for i 0, 1 for negative c and for positive c only when c− − i = 0. We get 1((r2 σ(c)r(r 1)). In conclusion our upper bound is: c 2 − − ≤ s s− + c + c−(2r 1) maxdeg P (L; a, q) r + + q r 2 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: First,

text: term

text: in

text: front

text: of

text: the

text: state

text: sum

text: contributes

text: Next

text: we

text: get

text: j

text: from

text: the

text: state

text: sum.

text: This

text: term

text: is

text: computed

text: in

text: the

text: following

text: lemma:

text: Lemma

text: If

text: sums

text: to

text: γ

text: then

text: r(s

text: s−),

text: where

text: s

text: and

text: j

text: j

text: s−

text: are

text: the

text: numbers

text: of

text: positive

text: and

text: negative

text: Seifert

text: circles

text: of

text: the

text: diagram.

text: Proof.

text: We

text: say

text: two

text: elementary

text: ﬂows

text: α,

text: β

text: intersect

text: if

text: α

text: contains

text: edges

text: in

text: more

text: than

text: one

text: region

text: complementary

text: to

text: β.

text: Suppose

text: our

text: state

text: contains

text: a

text: pair

text: of

text: intersecting

text: ﬂows,

text: α,

text: β.

text: We

text: may

text: create

text: a

text: new

text: state

text: by

text: replacing

text: two

text: intersecting

text: components

text: by

text: their

text: resolution

text: as

text: in

text: Seifert’s

text: algorithm.

text: This

text: new

text: state

text: has

text: the

text: same

text: number

text: of

text: components

text: and

text: the

text: same

text: rotation

text: numbers.

text: Hence

text: we

text: may

text: compute

text: the

text: sum

text: on

text: a

text: state

text: without

text: j

text: j

text: intersecting

text: ﬂows.

text: In

text: such

text: states

text: all

text: elementary

text: ﬂows

text: are

text: parallel

text: to

text: the

text: Seifert-circles

text: of

text: the

text: underlying

text: knot

text: diagram.

text: Around

text: each

text: Seifert

text: circle

text: there

text: are

text: r

text: components

text: of

text: elementary

text: ﬂows

text: with

text: rotation

text: number

text: equal

text: to

text: the

text: sign

text: of

text: the

text: Seifert

text: circle.

text: Therefore

text: the

text: sum

text: equals

text: r(s

text: s−).

text: Finally

text: we

text: need

text: to

text: estimate

text: maxdeg

text: E

text: (a,

text: q).

text: According

text: to

text: a

text: q

text: Lemma

text: this

text: is

text: the

text: maximum

text: of

text: rot

text: over

text: the

text: vertices

text: of

text: the

text: order

text: polytope

text: P

text: This

text: is

text: the

text: sum

text: of

text: the

text: ﬂows

text: with

text: positive

text: rotation

text: number.

text: For

text: states

text: where

text: each

text: ﬂow

text: has

text: a

text: single

text: component

text: this

text: is

text: maximal

text: and

text: equal

text: to

text: rs

text: Adding

text: everything

text: up

text: we

text: ﬁnd

text: maxdeg

text: P

text: (D;

text: a,

text: q)

text: r(

text: c

text: c−

text: a

text: r

text: s

text: s−)

text: proving

text: part

text: a)

text: of

text: theorem

text: As

text: claimed

text: in

text: the

text: introduction

text: this

text: agrees

text: with

text: Morton’s

text: bound

text: when

text: r

text: To

text: properly

text: compare

text: the

text: formulas

text: we

text: note

text: that

text: we

text: do

text: not

text: divide

text: by

text: the

text: unknot;

text: our

text: colored

text: HOMFLY

text: is

text: unreduced.

text: Also

text: Morton’s

text: variable

text: v

text: is

text: our

text: and

text: his

text: variable

text: z

text: is

text: our

text: q

text: Bound

text: on

text: the

text: q-degree

text: As

text: with

text: the

text: a-bound

text: our

text: strategy

text: is

text: to

text: estimate

text: max

text: maxdeg

text: (i,

text: q

text: by

text: studying

text: the

text: states

text: (i,

text: carefully.

text: First

text: the

text: framing

text: term

text: in

text: front

text: of

text: the

text: state

text: sum

text: gives

text: q

text: c∈C

text: σ(c)ic

text: Next

text: the

text: choice

text: i

text: of

text: resolving

text: the

text: crossings

text: yields

text: q

text: According

text: r(s+−s−)

text: to

text: Lemma

text: the

text: rotation

text: factor

text: gives

text: q

text: Next

text: comes

text: the

text: weight

text: q

text: and

text: ﬁnally

text: the

text: Ehrhart

text: polynomial.

text: For

text: the

text: latter

text: we

text: use

text: Lemma

text: to

text: get

text: a

text: maximal

text: contribution

text: of

text: since

text: each

text: order

text: polytope

text: contains

text: the

text: origin.

text: Now

text: assume

text: that

text: there

text: exists

text: a

text: state

text: attaining

text: the

text: minimal

text: possible

text: weight

text: Combining

text: all

text: the

text: contributions

text: and

text: the

text: upper

text: bound

text: for

text: Ehrhart

text: this

text: means

text: that

text: every

text: crossing

text: c

text: contributes

text: at

text: most

text: ((r

text: i

text: )i

text: (r

text: i

text: )r

text: σ(c)(i

text: r(r

text: c

text: c

text: c

text: c

text: This

text: is

text: maximal

text: for

text: i

text: for

text: negative

text: c

text: and

text: for

text: positive

text: c

text: only

text: when

text: c−

text: i

text: We

text: get

text: σ(c)r(r

text: In

text: conclusion

text: our

text: upper

text: bound

text: is:

text: c

text: s

text: s−

text: c

text: maxdeg

text: P

text: (L;

text: a,

text: q)

text: r

text: q

text: r

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: [GL11] S. Garoufalidis and T. Le, Nahm sums, stability and the colored Jones polynomial, arXiv:1112.3905, 2011. [GNS15] S. Gukov, S. Nawata, and M. Stosic, Colored homlfy homology of knots and links, In preparation, 2015. [GvdV13] S. Garoufalidis and R. van der Veen, A generating series for Murakami-Ohtsuki-Yamada graph evaluations, arXiv:1312.2117, 2013. [GvdV14] , Quadratic integer programming and the slope conjec- ture, arXiv:1405.5088, 2014. [Hat82] A. Hatcher, On the boundary curves of incompressible surfaces, Paciﬁc J. Math. 99 (1982), 373–377. [KM13] T. Kalman and H. Murakami, Root polytopes, parking functions and the HOMFLY polynomial, 1305.4925.pdf, 2013. [Man04] V. Manturov, Knot theory, CRC Press, 2004. [Mor86] H. Morton, Seifert circles and knot polynomials, Math. Proc. Camb. Phil. Soc. 99 (1986), 107–109. [MOY98] H. Murakami, T. Ohtsuki, and S. Yamada, HOMFLY polyno- mial via an invariant of trivalent graphs, Enseign. Math. 44 (1998), 325–360. [Sta72] R. Stanley, Ordered structures and partitions, Memoirs of the American Mathe- matical Society, vol. 119, AMS, 1972. [Sta78] J. Stallings, Constructions of ﬁbred knots and links, Algebraic and geometric topology, Proc. Sympos. Pure Math. 32 (1978), 55–60. [vdV13] R. van der Veen, A slope conjecture for links, arXiv:1306.3275, 2013. [Zhu13] S. Zhu, colored HOMFLY polynomials via skein theory, J. High Energy Physics 229 (2013). 18 ##LTLine##

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text: and

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text: preparation,

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text: and

text: R.

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text: der

text: Veen,

text: A

text: generating

text: series

text: for

text: Murakami-Ohtsuki-Yamada

text: graph

text: evaluations,

text: Quadratic

text: integer

text: programming

text: and

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text: slope

text: conjec-

text: ture,

text: A.

text: Hatcher,

text: On

text: the

text: boundary

text: curves

text: of

text: incompressible

text: surfaces,

text: Paciﬁc

text: J.

text: Math.

text: T.

text: Kalman

text: and

text: H.

text: Murakami,

text: Root

text: polytopes,

text: parking

text: functions

text: and

text: the

text: HOMFLY

text: polynomial,

text: V.

text: Manturov,

text: Knot

text: theory,

text: CRC

text: Press,

text: H.

text: Morton,

text: Seifert

text: circles

text: and

text: knot

text: polynomials,

text: Math.

text: Proc.

text: Camb.

text: Phil.

text: Soc.

text: H.

text: Murakami,

text: T.

text: Ohtsuki,

text: and

text: S.

text: Yamada,

text: HOMFLY

text: polyno-

text: mial

text: via

text: an

text: invariant

text: of

text: trivalent

text: graphs,

text: Enseign.

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text: R.

text: Stanley,

text: Ordered

text: structures

text: and

text: partitions,

text: Memoirs

text: of

text: the

text: American

text: Mathe-

text: matical

text: Society,

text: vol.

text: AMS,

text: J.

text: Stallings,

text: Constructions

text: of

text: ﬁbred

text: knots

text: and

text: links,

text: Algebraic

text: and

text: geometric

text: topology,

text: Proc.

text: Sympos.

text: Pure

text: Math.

text: R.

text: van

text: der

text: Veen,

text: A

text: slope

text: conjecture

text: for

text: links,

text: S.

text: Zhu,

text: colored

text: HOMFLY

text: polynomials

text: via

text: skein

text: theory,

text: J.

text: High

text: Energy

text: Physics

text: ##LTLine##


text: Our treatment relies on the standard theory of generating functions for lattice point ennumeration and Stanley reciprocity, see for example [BR07]. The present development of q-Ehrhart polynomials is heavily inspired by the preprint by Chapoton [Cha13]. The special case where the polytope is generic and the linear form positive so that no polymomials in N appear is due to Chapoton. ⊂ R (cid:88) For a polytope Q m and a linear form λ we consider the weighted sum W (Q, q) = qλ(x) λ ∈Z ∩ x m NQ We would like to study how W chang(cid:40)es as Q gets scaled as NQ. For example when Q = [0, 1] then(cid:88) W (NQ, q) = N qλx = 11−−qqNλλ if λ (cid:54)= 0 λ N if λ = 0 x=0 Theorem 3. (q-Ehrhart theorem): Let Q be a lattice polytope and a linear form λ. ∈ Q ± a. There exists a two variable Laurent polynomial E (a, b, q) (q)[a 1, b] ∈ N λ,Q such that for every N : E (qN, N, q) = W (NQ, q) λ,Q λ b. Denote by (NQ)o the interior lattice points of the polytope then we have the following form of Ehrhart reciprocity: − − − E (qN, N, q 1) = ( 1)dim(Q)W ((NQ)o, q) λ,Q λ (cid:80) Proof. By triangulating and the inclusion-exclusion principle we may re- duce to the special case where Q is a (lattice) simplex [BR07]. For such a ∞ simplex Q we consider the generating series G (q, z) = W (Q, q)zN. (cid:81) λ,Q N=0 λ By Theorem 3.5 on the integer point transform [BR07] we know that G(q, z) is a rational function in q, z whose denominator can be factored as − (cid:80) ∈ (1 qλ(w)z). Here the product runs over the set of vertices V (Q) w V (Q) of the simplex Q. If the denominator had only one such factor then we − − could expand (1 qrz) s = (N + s) . . . (N + 1)qNzN to obtain part N 1) of the theorem. We can reduce to this case by expanding G (q, z) Q λ,Q into partial fractions with respect to z with coeﬃcients in (q). For the reciprocity statement of part 2) we apply Stanley’s reciprocity (cid:88) theorem, Theorem 4.3 in [BR07], to the generating function ∞ Wλ,Qo(q)zN = Gλ,Qo(q, z) N=0 to obtain (cid:88) − (cid:88) − − ( 1)1+dim(Q)Gλ,Qo(q, z) = Gλ,Q(q 1, z 1) ∞ −∞ − − − − − = E (q N, N, q 1)z N = E (qN, N, q 1)zN λ,Q λ,Q − N=0 N= 1 5 ##LTLine##

text: Our

text: treatment

text: relies

text: on

text: the

text: standard

text: theory

text: of

text: generating

text: functions

text: for

text: lattice

text: point

text: ennumeration

text: and

text: Stanley

text: reciprocity,

text: see

text: for

text: example

text: The

text: present

text: development

text: of

text: q-Ehrhart

text: polynomials

text: is

text: heavily

text: inspired

text: by

text: the

text: preprint

text: by

text: Chapoton

text: The

text: special

text: case

text: where

text: the

text: polytope

text: is

text: generic

text: and

text: the

text: linear

text: form

text: positive

text: so

text: that

text: no

text: polymomials

text: in

text: N

text: appear

text: is

text: due

text: to

text: Chapoton.

text: R

text: For

text: a

text: polytope

text: Q

text: m

text: and

text: a

text: linear

text: form

text: λ

text: we

text: consider

text: the

text: weighted

text: sum

text: W

text: (Q,

text: q)

text: qλ(x)

text: λ

text: ∈Z

text: x

text: m

text: NQ

text: We

text: would

text: like

text: to

text: study

text: how

text: W

text: as

text: Q

text: gets

text: scaled

text: as

text: NQ.

text: For

text: example

text: when

text: Q

text: W

text: (NQ,

text: q)

text: N

text: qλx

text: if

text: λ

text: λ

text: N

text: if

text: λ

text: Theorem

text: (q-Ehrhart

text: theorem):

text: Let

text: Q

text: be

text: a

text: lattice

text: polytope

text: and

text: a

text: linear

text: form

text: λ.

text: Q

text: a.

text: There

text: exists

text: a

text: two

text: variable

text: Laurent

text: polynomial

text: E

text: (a,

text: b,

text: q)

text: (q)[a

text: b]

text: N

text: λ,Q

text: such

text: that

text: for

text: every

text: N

text: E

text: (qN,

text: N,

text: q)

text: W

text: (NQ,

text: q)

text: λ,Q

text: λ

text: b.

text: Denote

text: by

text: (NQ)o

text: the

text: interior

text: lattice

text: points

text: of

text: the

text: polytope

text: then

text: we

text: have

text: the

text: following

text: form

text: of

text: Ehrhart

text: reciprocity:

text: E

text: (qN,

text: N,

text: q

text: ((NQ)o,

text: q)

text: λ,Q

text: λ

text: Proof.

text: By

text: triangulating

text: and

text: the

text: inclusion-exclusion

text: principle

text: we

text: may

text: re-

text: duce

text: to

text: the

text: special

text: case

text: where

text: Q

text: is

text: a

text: (lattice)

text: simplex

text: For

text: such

text: a

text: simplex

text: Q

text: we

text: consider

text: the

text: generating

text: series

text: G

text: (q,

text: z)

text: W

text: (Q,

text: q)zN.

text: λ,Q

text: λ

text: By

text: Theorem

text: on

text: the

text: integer

text: point

text: transform

text: we

text: know

text: that

text: G(q,

text: z)

text: is

text: a

text: rational

text: function

text: in

text: q,

text: z

text: whose

text: denominator

text: can

text: be

text: factored

text: as

text: qλ(w)z).

text: Here

text: the

text: product

text: runs

text: over

text: the

text: set

text: of

text: vertices

text: V

text: (Q)

text: w

text: V

text: (Q)

text: of

text: the

text: simplex

text: Q.

text: If

text: the

text: denominator

text: had

text: only

text: one

text: such

text: factor

text: then

text: we

text: could

text: expand

text: qrz)

text: s

text: (N

text: s)

text: (N

text: to

text: obtain

text: part

text: N

text: of

text: the

text: theorem.

text: We

text: can

text: reduce

text: to

text: this

text: case

text: by

text: expanding

text: G

text: (q,

text: z)

text: Q

text: λ,Q

text: into

text: partial

text: fractions

text: with

text: respect

text: to

text: z

text: with

text: coeﬃcients

text: in

text: (q).

text: For

text: the

text: reciprocity

text: statement

text: of

text: part

text: we

text: apply

text: Stanley’s

text: reciprocity

text: theorem,

text: Theorem

text: in

text: to

text: the

text: generating

text: function

text: Wλ,Qo(q)zN

text: Gλ,Qo(q,

text: z)

text: to

text: obtain

text: z)

text: Gλ,Q(q

text: z

text: E

text: (q

text: N,

text: N,

text: q

text: N

text: E

text: (qN,

text: N,

text: q

text: λ,Q

text: λ,Q

text: N=

text: ##LTLine##


text: number of each ﬂow. As shown there are only three possible sequences of elementary ﬂows that add up to the ﬂow of D 1, namely ((cid:15) , (cid:15)−), ((cid:15) , (cid:15)−) 1 + + and ((cid:15)0). These will be the three terms in the sum for (cid:32)[D11]. Since(cid:33)the w(cid:16)eight and the sum of rotation numbers a(cid:17)re 0 we have [D ](a, q) = 11 − a − a12 − a−12 2 Erot,P((cid:15)+,(cid:15)−) + Erot,P((cid:15)−,(cid:15)+) Erot,P((cid:15)0) (q , 1, q) = q 12 − q−12 Here Erot,P((cid:15)0)(−a, b, q)−= b+1 and Erot,P((cid:15)+,−(cid:15)−)(a,−b, q) = Erot,P((cid:15)(cid:48)+,(cid:15)(cid:48)−)(a, b, q) = Etioront,P. (W(cid:15)−e,(cid:15)+h)a(vae E1, b, q 1) = E(aro,t−,P(1(cid:15),(cid:48)−q,)(cid:15)(cid:48)+=)(a a1−,1b−, q1 1) was computed in sec- (cid:16) ro(cid:104)t,P((cid:15)+,(cid:15)(cid:105)−) −q (cid:104) (cid:105) (q12 −q−(cid:17)12 )2 Likewise since (cid:15) , (cid:15)− = (cid:15)−, (cid:15) = 1 we have [D ](a, q) = + + 2 01 −1 1 a − q 2 Erot,P((cid:15)+,(cid:15)−) + q 2 Erot,P((cid:15)−,(cid:15)+) (q , 1, q) = (a12 − a−21 )(a12 q 12 − a−12 q−12 ) = (q 12 − q−21 )2 (cid:104) (cid:48) (cid:48) (cid:105) (cid:18) and ﬁnally (cid:15) , (cid:15)− = 0 so [D ](a, q) = (cid:19) + 00 − a − q 1Erot,P((cid:15)+,(cid:15)−) + Erot,P((cid:15)(cid:48)+,(cid:15)(cid:48)−) + Erot,P((cid:15)(cid:48)−,(cid:15)(cid:48)+) + qErot,P((cid:15)−,(cid:15)+) (q , 1, q) (a12 − a−12 )(a12 q 12 − a−12 q−12 )(q 12 + q−12 ) = (q 12 − q−12 )2 Adding up we get the following value for the Hopf link: P1(H; a, q) = −((qa1212 −−qa−−1212))2 (a−32 − a−12 (q − 1 + q−1)) in perfect agreement with the 2-braid formula from the introduction for c = 2, r = 1. 4 The maximal degree of the colored HOM- FLY polynomial In this section we take a closer look at the terms in the state sum to prove our bounds on the maximal degree in both a and q. Since the col- ored HOMLFY polynomial not exactly a polynomial but rather a rational function of q we should perhaps clarify the meaning of maxdeg . For any q rational function we can consider its Laurent series at inﬁnity. It has a ﬁnite highest degree term and its exponent is what we call maxdeg . q 4.1 Bound on the a-degree Fix a state i, (cid:15) in the state sum. We will compute its maxdeg (i, (cid:15)). a Since several states can cancel max maxdeg (i, (cid:15)) yields an upper bound i,(cid:15) a for maxdeg P (D; a, q). a r 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: number

text: of

text: each

text: ﬂow.

text: As

text: shown

text: there

text: are

text: only

text: three

text: possible

text: sequences

text: of

text: elementary

text: ﬂows

text: that

text: add

text: up

text: to

text: the

text: ﬂow

text: of

text: D

text: namely

text: and

text: These

text: will

text: be

text: the

text: three

text: terms

text: in

text: the

text: sum

text: for

text: and

text: the

text: sum

text: of

text: rotation

text: numbers

text: we

text: have

text: [D

text: ](a,

text: q)

text: a

text: (q

text: q)

text: q

text: Here

text: b,

text: q)−=

text: and

text: q)

text: b,

text: q)

text: Etioront,P.

text: b,

text: q

text: was

text: computed

text: in

text: sec-

text: −q

text: Likewise

text: since

text: we

text: have

text: [D

text: ](a,

text: q)

text: −1

text: a

text: q

text: q

text: (q

text: q)

text: q

text: (q

text: and

text: ﬁnally

text: so

text: [D

text: ](a,

text: q)

text: a

text: q

text: (q

text: q)

text: q

text: )(q

text: (q

text: Adding

text: up

text: we

text: get

text: the

text: following

text: value

text: for

text: the

text: Hopf

text: link:

text: a,

text: q)

text: (q

text: in

text: perfect

text: agreement

text: with

text: the

text: formula

text: from

text: the

text: introduction

text: for

text: c

text: r

text: The

text: maximal

text: degree

text: of

text: the

text: colored

text: HOM-

text: FLY

text: polynomial

text: In

text: this

text: section

text: we

text: take

text: a

text: closer

text: look

text: at

text: the

text: terms

text: in

text: the

text: state

text: sum

text: to

text: prove

text: our

text: bounds

text: on

text: the

text: maximal

text: degree

text: in

text: both

text: a

text: and

text: q.

text: Since

text: the

text: col-

text: ored

text: HOMLFY

text: polynomial

text: not

text: exactly

text: a

text: polynomial

text: but

text: rather

text: a

text: rational

text: function

text: of

text: q

text: we

text: should

text: perhaps

text: clarify

text: the

text: meaning

text: of

text: maxdeg

text: For

text: any

text: q

text: rational

text: function

text: we

text: can

text: consider

text: its

text: Laurent

text: series

text: at

text: inﬁnity.

text: It

text: has

text: a

text: ﬁnite

text: highest

text: degree

text: term

text: and

text: its

text: exponent

text: is

text: what

text: we

text: call

text: maxdeg

text: q

text: Bound

text: on

text: the

text: a-degree

text: Fix

text: a

text: state

text: i,

text: in

text: the

text: state

text: sum.

text: We

text: will

text: compute

text: its

text: maxdeg

text: (i,

text: a

text: Since

text: several

text: states

text: can

text: cancel

text: max

text: maxdeg

text: (i,

text: yields

text: an

text: upper

text: bound

text: a

text: for

text: maxdeg

text: P

text: (D;

text: a,

text: q).

text: a

text: r

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