text: GLOBAL MINIMIZATION OF TIKHONOV FUNCTIONALS 7 (cid:0) (cid:1) holds and ﬁnally (cid:107) − (cid:107) − (cid:107) (cid:107) − (cid:107) (cid:107) † (cid:54) (cid:107) (cid:107) F (xδ ) yδ α ω 2 + 2(1 c ω )αD (xδ , x ) (δ + α ω )2 α fp α (cid:3) implies both error estimates in (2.9). † One easily veriﬁes that the right hand side in the estimate for D (xδ , x ) in (2.9) (cid:107)fp(cid:107) α is minimized if the regularization parameter is chosen as α = δ/ ω . This a-priori parameter choice rule would thus give optimal convergence rates estimates in terms of the Bregman distance. Since the source element ω itself will usually not be at (cid:107) (cid:107) (cid:54) hand, we might use the estimate ω (cid:37) from Assumption 2.5 (iii) and choose α = δ/(cid:37) instead. This is indeed feasible, if the scaling parameter s = 3 and the d-TIGRA algorithm will approximate α = δ/(cid:37) from above (cf. Proposition 4.2). (cid:62) ∗ In what follows, we will consider values α α , where ∗ δ (2.10) α := − (s 2)(cid:37) and for such α (2.9) yields (cid:107) − (cid:107) (cid:54) (cid:107) (cid:107) (cid:54) (2.11) F (xδ ) yδ 2α ω + δ s(cid:37)α α which will turn out to be of great use. Thus, the scaling parameter s > 2 in Assumption 2.5 (iii) can be regarded as a trade oﬀ between the smallness condition ∗ (cid:37) < 1 and the choice of the regularization parameter α . sc We conclude this section with two important consequences of Assumption 2.5. It is worth noting, that the bounds obtained below are not necessarily optimal. Similar yet more elaborate estimates were already used in [26]. For our purposes, however, it is the existence of uniform bounds in α which is primarily important. (cid:16) (cid:17) Lemma 2.8. If Assumption 2.5 holds, then (cid:107) (cid:107) (cid:54) p 1/p (cid:107) − (cid:107) (2.12) xδ A := ∗ F (0) yδ 2/p α 2α (cid:107) (cid:48) (cid:107) (cid:54) (cid:107) (cid:48) (cid:107) (2.13) F (xδ ) K := LA + F (0) , α (cid:62) ∗ for all α α . Proof. From the minimizing property of xδ we obtain α (cid:54) (cid:54) (cid:54) 1(cid:107) − (cid:107) αf (xδ ) Φ (xδ ) Φ (0) F (0) yδ 2. p α α α α 2 (cid:62) ∗ Thus, for α α , (cid:107) (cid:107) (cid:54) p (cid:107) − (cid:107) (cid:54) p (cid:107) − (cid:107) xδ p F (0) yδ 2 ∗ F (0) yδ 2. α 2α 2α Now, (2.12) readily follows by taking the p-th root. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: GLOBAL

text: MINIMIZATION

text: OF

text: TIKHONOV

text: FUNCTIONALS

text: holds

text: and

text: ﬁnally

text: F

text: (xδ

text: yδ

text: α

text: ω

text: c

text: ω

text: )αD

text: (xδ

text: x

text: (δ

text: α

text: ω

text: α

text: fp

text: α

text: implies

text: both

text: error

text: estimates

text: in

text: One

text: easily

text: veriﬁes

text: that

text: the

text: right

text: hand

text: side

text: in

text: the

text: estimate

text: for

text: D

text: (xδ

text: x

text: in

text: α

text: is

text: minimized

text: if

text: the

text: regularization

text: parameter

text: is

text: chosen

text: as

text: α

text: δ/

text: ω

text: This

text: a-priori

text: parameter

text: choice

text: rule

text: would

text: thus

text: give

text: optimal

text: convergence

text: rates

text: estimates

text: in

text: terms

text: of

text: the

text: Bregman

text: distance.

text: Since

text: the

text: source

text: element

text: ω

text: itself

text: will

text: usually

text: not

text: be

text: at

text: hand,

text: we

text: might

text: use

text: the

text: estimate

text: ω

text: from

text: Assumption

text: (iii)

text: and

text: choose

text: α

text: instead.

text: This

text: is

text: indeed

text: feasible,

text: if

text: the

text: scaling

text: parameter

text: s

text: and

text: the

text: d-TIGRA

text: algorithm

text: will

text: approximate

text: α

text: from

text: above

text: (cf.

text: Proposition

text: In

text: what

text: follows,

text: we

text: will

text: consider

text: values

text: α

text: α

text: where

text: δ

text: α

text: (s

text: and

text: for

text: such

text: α

text: yields

text: F

text: (xδ

text: yδ

text: 2α

text: ω

text: δ

text: α

text: which

text: will

text: turn

text: out

text: to

text: be

text: of

text: great

text: use.

text: Thus,

text: the

text: scaling

text: parameter

text: s

text: in

text: Assumption

text: (iii)

text: can

text: be

text: regarded

text: as

text: a

text: trade

text: oﬀ

text: between

text: the

text: smallness

text: condition

text: and

text: the

text: choice

text: of

text: the

text: regularization

text: parameter

text: α

text: sc

text: We

text: conclude

text: this

text: section

text: with

text: two

text: important

text: consequences

text: of

text: Assumption

text: It

text: is

text: worth

text: noting,

text: that

text: the

text: bounds

text: obtained

text: below

text: are

text: not

text: necessarily

text: optimal.

text: Similar

text: yet

text: more

text: elaborate

text: estimates

text: were

text: already

text: used

text: in

text: For

text: our

text: purposes,

text: however,

text: it

text: is

text: the

text: existence

text: of

text: uniform

text: bounds

text: in

text: α

text: which

text: is

text: primarily

text: important.

text: Lemma

text: If

text: Assumption

text: holds,

text: then

text: p

text: xδ

text: A

text: F

text: yδ

text: α

text: 2α

text: F

text: (xδ

text: K

text: LA

text: F

text: α

text: for

text: all

text: α

text: α

text: Proof.

text: From

text: the

text: minimizing

text: property

text: of

text: xδ

text: we

text: obtain

text: α

text: αf

text: (xδ

text: Φ

text: (xδ

text: Φ

text: F

text: yδ

text: p

text: α

text: α

text: α

text: α

text: Thus,

text: for

text: α

text: α

text: p

text: p

text: xδ

text: p

text: F

text: yδ

text: F

text: yδ

text: α

text: 2α

text: 2α

text: Now,

text: readily

text: follows

text: by

text: taking

text: the

text: p-th

text: root.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 WEI WANG, STEPHAN W. ANZENGRUBER, RONNY RAMLAU, AND BO HAN (cid:107) (cid:48) (cid:107) For the estimate on F (xδ ) we observe that by Assumption 2.5 (i) α (cid:107) (cid:48) (cid:107) (cid:54) (cid:107) (cid:48) − (cid:48) (cid:107) (cid:107) (cid:48) (cid:107) (cid:54) (cid:107) (cid:107) (cid:107) (cid:48) (cid:107) F (xδ ) F (xδ ) F (0) + F (0) L xδ + F (0) . α α α (cid:3) Thus (2.13) follows using (2.12). Under our standing assumptions, the distance of minimizers xδ and xδ cor- α | α¯− | responding to nearby values α and α¯ turns out to be of the order of α α¯ . This result has essential implications regarding the outer update step where we decrease the regularization parameter α by a factor q¯ < 1 (cf. Lemma 3.10 and Proposition 3.11). The rather technical proof is postponed to Section 12. (cid:62) ∗ Proposition 2.9. Let Assumption 2.5 hold and let α¯ α . Then both xδ and α F (xδ ) are continuous from the right with respect to α¯, in the sense that α (cid:107) − (cid:107) (cid:107) − (cid:107) lim xδ xδ = 0, lim F (xδ ) F (xδ ) = 0. → α α¯ → α α¯ α +α¯ α +α¯ (cid:54) (cid:54) In particular, for α¯ α α¯ with q¯0 2sc(cid:37) (2.14) q¯ := < 1, 0 1 + sc(cid:37) we have (cid:107) − (cid:107) (cid:54) − (2.15) xδ xδ σ(α α¯), α α¯ − − where σ = cA(12−s(cid:37)sKc(cid:37))α∗ and cA := p21(3A)p 2 with A from Lemma 2.8. Remark 2.10. Note, that Proposition 2.9 implies uniqueness of the minimizer xδ (cid:62) ∗ α for α α . Indeed, if we let α¯ = α with arbitrarily chosen minimizers xδ , xδ , (cid:107) − (cid:107) α α¯ then (2.15) yields xδ xδ = 0. α¯ α 3. The iterated dual gradient descent method In this section we analyze a dual gradient descent method for the computation of a minimizer of the Tikhonov functional with sparsity constraints using the discrep- ancy principle as a global stopping rule. Several technical proofs are collected in Section 12 for improved readability. The algorithm can be summarized as follows: There are several requirements we need to verify in order to prove that the above algorithm is well-deﬁned and globally convergent. First of all, for each j the dual gradient descent algorithm in lines 7-(cid:107)1∇1 starting f(cid:107)rom xj,0 should converge to a minimizer xδ of Φ (x) and, secondly, Φ (x ) should tend to zero as → ∞ αj αj αj j,k k , so that each inner iteration terminates after a ﬁnite number of steps. Additionally we need to ensure that the discrepancy principle can be used as a stopping rule for the outer iteration. In the following, we would like to investigate the local directional convexity of the Tikhonov functional Φ (x) near a minimizer xδ , i.e. the convexity of the functions α α ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: WEI

text: WANG,

text: STEPHAN

text: W.

text: ANZENGRUBER,

text: RONNY

text: RAMLAU,

text: AND

text: BO

text: HAN

text: For

text: the

text: estimate

text: on

text: F

text: (xδ

text: we

text: observe

text: that

text: by

text: Assumption

text: (i)

text: α

text: F

text: (xδ

text: F

text: (xδ

text: F

text: F

text: L

text: xδ

text: F

text: α

text: α

text: α

text: Thus

text: follows

text: using

text: Under

text: our

text: standing

text: assumptions,

text: the

text: distance

text: of

text: minimizers

text: xδ

text: and

text: xδ

text: cor-

text: α

text: α¯−

text: responding

text: to

text: nearby

text: values

text: α

text: and

text: α¯

text: turns

text: out

text: to

text: be

text: of

text: the

text: order

text: of

text: α

text: α¯

text: This

text: result

text: has

text: essential

text: implications

text: regarding

text: the

text: outer

text: update

text: step

text: where

text: we

text: decrease

text: the

text: regularization

text: parameter

text: α

text: by

text: a

text: factor

text: q¯

text: (cf.

text: Lemma

text: and

text: Proposition

text: The

text: rather

text: technical

text: proof

text: is

text: postponed

text: to

text: Section

text: Proposition

text: Let

text: Assumption

text: hold

text: and

text: let

text: α¯

text: α

text: Then

text: both

text: xδ

text: and

text: α

text: F

text: (xδ

text: are

text: continuous

text: from

text: the

text: right

text: with

text: respect

text: to

text: α¯,

text: in

text: the

text: sense

text: that

text: α

text: lim

text: xδ

text: xδ

text: lim

text: F

text: (xδ

text: F

text: (xδ

text: α

text: α¯

text: α

text: α¯

text: α

text: +α¯

text: α

text: +α¯

text: In

text: particular,

text: for

text: α¯

text: α

text: α¯

text: with

text: q¯

text: we

text: have

text: xδ

text: xδ

text: σ(α

text: α¯),

text: α

text: α¯

text: where

text: σ

text: and

text: cA

text: with

text: A

text: from

text: Lemma

text: Remark

text: Note,

text: that

text: Proposition

text: implies

text: uniqueness

text: of

text: the

text: minimizer

text: xδ

text: α

text: for

text: α

text: α

text: Indeed,

text: if

text: we

text: let

text: α¯

text: α

text: with

text: arbitrarily

text: chosen

text: minimizers

text: xδ

text: xδ

text: α

text: α¯

text: then

text: yields

text: xδ

text: xδ

text: α¯

text: α

text: The

text: iterated

text: dual

text: gradient

text: descent

text: method

text: In

text: this

text: section

text: we

text: analyze

text: a

text: dual

text: gradient

text: descent

text: method

text: for

text: the

text: computation

text: of

text: a

text: minimizer

text: of

text: the

text: Tikhonov

text: functional

text: with

text: sparsity

text: constraints

text: using

text: the

text: discrep-

text: ancy

text: principle

text: as

text: a

text: global

text: stopping

text: rule.

text: Several

text: technical

text: proofs

text: are

text: collected

text: in

text: Section

text: for

text: improved

text: readability.

text: The

text: algorithm

text: can

text: be

text: summarized

text: as

text: follows:

text: There

text: are

text: several

text: requirements

text: we

text: need

text: to

text: verify

text: in

text: order

text: to

text: prove

text: that

text: the

text: above

text: algorithm

text: is

text: well-deﬁned

text: and

text: globally

text: convergent.

text: First

text: of

text: all,

text: for

text: each

text: j

text: the

text: dual

text: gradient

text: descent

text: algorithm

text: in

text: lines

text: starting

text: should

text: converge

text: to

text: a

text: minimizer

text: xδ

text: of

text: Φ

text: (x)

text: and,

text: secondly,

text: Φ

text: (x

text: should

text: tend

text: to

text: zero

text: as

text: αj

text: αj

text: αj

text: j,k

text: k

text: so

text: that

text: each

text: inner

text: iteration

text: terminates

text: after

text: a

text: ﬁnite

text: number

text: of

text: steps.

text: Additionally

text: we

text: need

text: to

text: ensure

text: that

text: the

text: discrepancy

text: principle

text: can

text: be

text: used

text: as

text: a

text: stopping

text: rule

text: for

text: the

text: outer

text: iteration.

text: In

text: the

text: following,

text: we

text: would

text: like

text: to

text: investigate

text: the

text: local

text: directional

text: convexity

text: of

text: the

text: Tikhonov

text: functional

text: Φ

text: (x)

text: near

text: a

text: minimizer

text: xδ

text: i.e.

text: the

text: convexity

text: of

text: the

text: functions

text: α

text: α

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 WEI WANG, STEPHAN W. ANZENGRUBER, RONNY RAMLAU, AND BO HAN | | | | (cid:54) (cid:26) (cid:114) (cid:27) holds for t , t r , where R is as deﬁned in (3.2) and 1 2 α Φα 2√ γα γα (3.3) r := min , . α 1 + 2 csK 10cL − In particual, ϕ (t) in (3.1) is thus convex on [ r , r ]. α,h α α Of course, Theorem 3.2 does not ensure that Φ (x) is itself convex in any neigh- α bourhood of xδ . We will say that Φ (x) is directionally convex on B (xδ ), tacitly α α rα α assuming that only lines through xδ are considered. Here B (x) denotes the ball ∈ α r of radius r centered at x X, i.e. { ∈ (cid:107) − (cid:107) (cid:54) } B (xδ ) = x X : x xδ r . rα α α α → ∞ → ∞ Note that r does not depend on the direction h and that r as α . α α As a ﬁrst consequence of Theorem 3.2, we obtain an estimate on the directional derivative of Φ (x) in direction xδ . α α ∈ Proposition 3.3. Let Assumption 2.5 hold and let x B (xδ ) with r from (cid:62) ∗ rα α α Theorem 3.2 for some α α , then (cid:104)∇ − (cid:105) (cid:62) (3.4) Φ (x), x xδ γαD (xδ , x) α α fp α − holds with γ = 1 sc(cid:37). 2 Proof. Using the minimizing property of xδ and Theorem 3.2, we obtain α −(cid:104)∇ − (cid:105) − Φ (x), x xδ = R (xδ , x) Φ (xδ ) + Φ (x) α α Φα α α α α (cid:62) (cid:62) R (xδ , x) γαD (xδ , x), Φα α fp α (cid:3) and the proof is complete. Remark 3.4. Note that, due to the strict convexity of f (x), Proposition 3.3 in p particular implies that there are no critical points other than xδ inside B (xδ ). → ∞ α rα α The size r of this region increases as α and due to Proposition 2.9 there α (cid:54) ∈ (cid:48) ∗ exists α such that xδ = 0, but 0 B (xδ ). Consequently, F (0) cannot be the α ∇ rα α zero mapping as otherwise Φ (0) = 0 would hold and thus x = 0 would be a α critical point of Φ (x). α Our convergence analysis below relies primarily on the Bregman distance and for the strictly convex functionals f (x) it is well known that p → D (xδ , x ) 0 fp αj j,k (cid:107) − (cid:107) → also yields xδ x 0. But at the same time it will be important to ensure αj j,k that the iterates x remain inside the region of directional convexity B (xδ ) j,k rαj αj ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WEI

text: WANG,

text: STEPHAN

text: W.

text: ANZENGRUBER,

text: RONNY

text: RAMLAU,

text: AND

text: BO

text: HAN

label: equation

text: holds

text: for

text: t

text: t

text: r

text: where

text: R

text: is

text: as

text: deﬁned

text: in

text: and

text: α

text: Φα

text: 2√

text: γα

text: γα

text: r

text: min

text: α

text: csK

text: In

text: particual,

text: ϕ

text: (t)

text: in

text: is

text: thus

text: convex

text: on

text: r

text: r

text: α,h

text: α

text: α

text: Of

text: course,

text: Theorem

text: does

text: not

text: ensure

text: that

text: Φ

text: (x)

text: is

text: itself

text: convex

text: in

text: any

text: neigh-

text: α

text: bourhood

text: of

text: xδ

text: We

text: will

text: say

text: that

text: Φ

text: (x)

text: is

text: directionally

text: convex

text: on

text: B

text: (xδ

text: tacitly

text: α

text: α

text: rα

text: α

text: assuming

text: that

text: only

text: lines

text: through

text: xδ

text: are

text: considered.

text: Here

text: B

text: (x)

text: denotes

text: the

text: ball

text: α

text: r

text: of

text: radius

text: r

text: centered

text: at

text: x

text: X,

text: i.e.

text: B

text: (xδ

text: x

text: X

text: x

text: xδ

text: r

text: rα

text: α

text: α

text: α

text: Note

text: that

text: r

text: does

text: not

text: depend

text: on

text: the

text: direction

text: h

text: and

text: that

text: r

text: as

text: α

text: α

text: α

text: As

text: a

text: ﬁrst

text: consequence

text: of

text: Theorem

text: we

text: obtain

text: an

text: estimate

text: on

text: the

text: directional

text: derivative

text: of

text: Φ

text: (x)

text: in

text: direction

text: xδ

text: α

text: α

text: Proposition

text: Let

text: Assumption

text: hold

text: and

text: let

text: x

text: B

text: (xδ

text: with

text: r

text: from

text: rα

text: α

text: α

text: Theorem

text: for

text: some

text: α

text: α

text: then

text: Φ

text: (x),

text: x

text: xδ

text: γαD

text: (xδ

text: x)

text: α

text: α

text: fp

text: α

text: holds

text: with

text: γ

text: Proof.

text: Using

text: the

text: minimizing

text: property

text: of

text: xδ

text: and

text: Theorem

text: we

text: obtain

text: α

text: Φ

text: (x),

text: x

text: xδ

text: R

text: (xδ

text: x)

text: Φ

text: (xδ

text: Φ

text: (x)

text: α

text: α

text: Φα

text: α

text: α

text: α

text: α

text: R

text: (xδ

text: x)

text: γαD

text: (xδ

text: x),

text: Φα

text: α

text: fp

text: α

text: and

text: the

text: proof

text: is

text: complete.

text: Remark

text: Note

text: that,

text: due

text: to

text: the

text: strict

text: convexity

text: of

text: f

text: (x),

text: Proposition

text: in

text: p

text: particular

text: implies

text: that

text: there

text: are

text: no

text: critical

text: points

text: other

text: than

text: xδ

text: inside

text: B

text: (xδ

text: α

text: rα

text: α

text: The

text: size

text: r

text: of

text: this

text: region

text: increases

text: as

text: α

text: and

text: due

text: to

text: Proposition

text: there

text: α

text: exists

text: α

text: such

text: that

text: xδ

text: but

text: B

text: (xδ

text: Consequently,

text: F

text: cannot

text: be

text: the

text: α

text: rα

text: α

text: zero

text: mapping

text: as

text: otherwise

text: Φ

text: would

text: hold

text: and

text: thus

text: x

text: would

text: be

text: a

text: α

text: critical

text: point

text: of

text: Φ

text: (x).

text: α

text: Our

text: convergence

text: analysis

text: below

text: relies

text: primarily

text: on

text: the

text: Bregman

text: distance

text: and

text: for

text: the

text: strictly

text: convex

text: functionals

text: f

text: (x)

text: it

text: is

text: well

text: known

text: that

text: p

text: D

text: (xδ

text: x

text: fp

text: αj

text: j,k

text: also

text: yields

text: xδ

text: x

text: But

text: at

text: the

text: same

text: time

text: it

text: will

text: be

text: important

text: to

text: ensure

text: αj

text: j,k

text: that

text: the

text: iterates

text: x

text: remain

text: inside

text: the

text: region

text: of

text: directional

text: convexity

text: B

text: (xδ

text: j,k

text: rαj

text: αj

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 ∗ → (cid:48) FIG. 1: Leading order amplitude for DVπ0(η)P, γ + P M + P . Crossed diagrams are not shown in the ﬁgure. enters the exclusive p(cid:101)rocess scattering amplitudes always multiplied by the transverse momentum transfer ∆⊥) while ⊥ the TMD h is related to the imaginary part of the same GTMD. 1 In a similar way, H , involves a double spin ﬂip and its contribution therefore also vanishes in the forward limit. T This GPD emerges as the diﬀerence between the canonical transversity non-ﬂip amplitudes (ATY ), and the planar transversity non-ﬂip amplitudes (ATX). Since in the forward direction there is no distinction between canonical transversity and planar transversity, clearly then, this GPD requires non-forward scattering to be non-zero. The target polar(cid:101)ized transversely at an azimuthal angle diﬀerent from zero or π/2 will allow this double correlation to be probed. Finally,(cid:101)E is the most elusive of the chiral odd GPDs since because(cid:101)of Parity and Time reversal constraints it T vanishes for the skewness parameter ξ = 0, and it is therefore zero in the forward limit; its ﬁrst moment in x is also zero [6]. E can therefore be consi(cid:101)dered an entirely oﬀ-forward product. E (x, ξ, t) describes a transversely polarized T T quark (along the x axis) in a longitudinally polarized proton. It is T-even and directly connected to the ﬁrst moment ⊥ of the TMD h . Our interest in E stems from the fact that it manifests a similar spin structure to the chiral-ev(cid:101)en 1L T twist three GPD, G , that was shown to enter the sum rule component for partonic Orbital Angular Momentum 2 (OAM) in Ref.[14, 15]. Correlated, although indirect information on OAM therefore can be obtained from both E ⊥ T and h measurements. 1L In Ref.[4] we evaluated the contributions of the various chiral odd GPDs to the observables in exclusive Deeply Virtual π0 electro-Production (DVπ0P) recently measured at Jeﬀerson Lab [16, 17]. In particular, there exist mea- surements for the various unpolarized scattering components, for the beam spin asymmetry A , and for the target LU spin asymmetries for longitudinal polarization, A (unpolarized beam) and A (polarized beam). Our analysis UL LL uses a ﬂexible parametrization which is based on the reggeized diquark model, that is a spectator model with variable mass of the spectator, M , which reproduces the Regge behavior in the low x, large M limit [18, 19]. The presence X X of scalar and axial-vector diquarks allows us to model the u and d quark distributions separately, i.e. distinguishing between the diﬀerent isospin projections, ud and uu, as it follows from the use of SU(4) symmetry in the nucleon. The model’s parameters were determined from quantitative ﬁts of the chiral even GPDs using a compilation of: i) ﬂavor separated Dirac and Pauli nucleon form factor data [20], and axial [21] and pseudo-scalar [22] form factor data; ii) DVCS data [23]; and iii) reproducing the forward limit of GPDs for both the unpolarized and polarized parton distribution functions from DIS data. For the latter we evolved the model from its initial low scale to the scale of the data using leading order Perturbative QCD (PQCD) GPD evolution equations (see e.g. Refs.[24, 25]). For the various asymmetries, in particular, we predicted that the spin modulations which are ideal for measuring chiral odd GPDs using a longitudinally polarized target are Asin2φ, and the constant in φ term, A . The latter UL LL are, in fact, proportional to transverse polarization terms only, and therefore they involve only chiral odd GPDs. The other modulations, Asinφ, and Acosφ, contain contributions from both longitudinal and transverse photons, their UL LL description will always contain a mixture of chiral odd and chiral even GPDs, and thus, they are not most appropriate for a clean extraction of the chiral odd sector. Using our model, we found that the dominant contribution to these ##LTFigure## ##LTFigure##

label: caption

text: FIG.

text: Leading

text: order

text: amplitude

text: for

text: γ

text: P

text: M

text: P

text: Crossed

text: diagrams

text: are

text: not

text: shown

text: in

text: the

text: ﬁgure.

text: enters

text: the

text: exclusive

text: scattering

text: amplitudes

text: always

text: multiplied

text: by

text: the

text: transverse

text: momentum

text: transfer

text: while

text: the

text: TMD

text: h

text: is

text: related

text: to

text: the

text: imaginary

text: part

text: of

text: the

text: same

text: GTMD.

text: In

text: a

text: similar

text: way,

text: H

text: involves

text: a

text: double

text: spin

text: ﬂip

text: and

text: its

text: contribution

text: therefore

text: also

text: vanishes

text: in

text: the

text: forward

text: limit.

text: T

text: This

text: GPD

text: emerges

text: as

text: the

text: diﬀerence

text: between

text: the

text: canonical

text: transversity

text: non-ﬂip

text: amplitudes

text: (ATY

text: and

text: the

text: planar

text: transversity

text: non-ﬂip

text: amplitudes

text: (ATX).

text: Since

text: in

text: the

text: forward

text: direction

text: there

text: is

text: no

text: distinction

text: between

text: canonical

text: transversity

text: and

text: planar

text: transversity,

text: clearly

text: then,

text: this

text: GPD

text: requires

text: non-forward

text: scattering

text: to

text: be

text: non-zero.

text: The

text: target

text: transversely

text: at

text: an

text: azimuthal

text: angle

text: diﬀerent

text: from

text: zero

text: or

text: will

text: allow

text: this

text: double

text: correlation

text: to

text: be

text: probed.

text: is

text: the

text: most

text: elusive

text: of

text: the

text: chiral

text: odd

text: GPDs

text: since

text: Parity

text: and

text: Time

text: reversal

text: constraints

text: it

text: T

text: vanishes

text: for

text: the

text: skewness

text: parameter

text: ξ

text: and

text: it

text: is

text: therefore

text: zero

text: in

text: the

text: forward

text: limit;

text: its

text: ﬁrst

text: moment

text: in

text: x

text: is

text: also

text: zero

text: E

text: can

text: therefore

text: be

text: an

text: entirely

text: oﬀ-forward

text: product.

text: E

text: (x,

text: ξ,

text: t)

text: describes

text: a

text: transversely

text: polarized

text: T

text: T

text: quark

text: (along

text: the

text: x

text: axis)

text: in

text: a

text: longitudinally

text: polarized

text: proton.

text: It

text: is

text: T-even

text: and

text: directly

text: connected

text: to

text: the

text: ﬁrst

text: moment

text: of

text: the

text: TMD

text: h

text: Our

text: interest

text: in

text: E

text: stems

text: from

text: the

text: fact

text: that

text: it

text: manifests

text: a

text: similar

text: spin

text: structure

text: to

text: the

text: T

text: twist

text: three

text: GPD,

text: G

text: that

text: was

text: shown

text: to

text: enter

text: the

text: sum

text: rule

text: component

text: for

text: partonic

text: Orbital

text: Angular

text: Momentum

text: (OAM)

text: in

text: Correlated,

text: although

text: indirect

text: information

text: on

text: OAM

text: therefore

text: can

text: be

text: obtained

text: from

text: both

text: E

text: T

text: and

text: h

text: measurements.

text: In

text: we

text: evaluated

text: the

text: contributions

text: of

text: the

text: various

text: chiral

text: odd

text: GPDs

text: to

text: the

text: observables

text: in

text: exclusive

text: Deeply

text: Virtual

text: π0

text: electro-Production

text: recently

text: measured

text: at

text: Jeﬀerson

text: Lab

text: In

text: particular,

text: there

text: exist

text: mea-

text: surements

text: for

text: the

text: various

text: unpolarized

text: scattering

text: components,

text: for

text: the

text: beam

text: spin

text: asymmetry

text: A

text: and

text: for

text: the

text: target

text: LU

text: spin

text: asymmetries

text: for

text: longitudinal

text: polarization,

text: A

text: (unpolarized

text: beam)

text: and

text: A

text: (polarized

text: beam).

text: Our

text: analysis

text: UL

text: LL

text: uses

text: a

text: ﬂexible

text: parametrization

text: which

text: is

text: based

text: on

text: the

text: reggeized

text: diquark

text: model,

text: that

text: is

text: a

text: spectator

text: model

text: with

text: variable

text: mass

text: of

text: the

text: spectator,

text: M

text: which

text: reproduces

text: the

text: Regge

text: behavior

text: in

text: the

text: low

text: x,

text: large

text: M

text: limit

text: The

text: presence

text: X

text: X

text: of

text: scalar

text: and

text: axial-vector

text: diquarks

text: allows

text: us

text: to

text: model

text: the

text: u

text: and

text: d

text: quark

text: distributions

text: separately,

text: i.e.

text: distinguishing

text: between

text: the

text: diﬀerent

text: isospin

text: projections,

text: ud

text: and

text: uu,

text: as

text: it

text: follows

text: from

text: the

text: use

text: of

text: symmetry

text: in

text: the

text: nucleon.

text: The

text: model’s

text: parameters

text: were

text: determined

text: from

text: quantitative

text: ﬁts

text: of

text: the

text: chiral

text: even

text: GPDs

text: using

text: a

text: compilation

text: of:

text: i)

text: ﬂavor

text: separated

text: Dirac

text: and

text: Pauli

text: nucleon

text: form

text: factor

text: data

text: and

text: axial

text: and

text: pseudo-scalar

text: form

text: factor

text: data;

text: ii)

text: DVCS

text: data

text: and

text: iii)

text: reproducing

text: the

text: forward

text: limit

text: of

text: GPDs

text: for

text: both

text: the

text: unpolarized

text: and

text: polarized

text: parton

text: distribution

text: functions

text: from

text: DIS

text: data.

text: For

text: the

text: latter

text: we

text: evolved

text: the

text: model

text: from

text: its

text: initial

text: low

text: scale

text: to

text: the

text: scale

text: of

text: the

text: data

text: using

text: leading

text: order

text: Perturbative

text: QCD

text: (PQCD)

text: GPD

text: evolution

text: equations

text: (see

text: e.g.

text: For

text: the

text: various

text: asymmetries,

text: in

text: particular,

text: we

text: predicted

text: that

text: the

text: spin

text: modulations

text: which

text: are

text: ideal

text: for

text: measuring

text: chiral

text: odd

text: GPDs

text: using

text: a

text: longitudinally

text: polarized

text: target

text: are

text: and

text: the

text: constant

text: in

text: φ

text: term,

text: A

text: The

text: latter

text: UL

text: LL

text: are,

text: in

text: fact,

text: proportional

text: to

text: transverse

text: polarization

text: terms

text: only,

text: and

text: therefore

text: they

text: involve

text: only

text: chiral

text: odd

text: GPDs.

text: The

text: other

text: modulations,

text: Asinφ,

text: and

text: Acosφ,

text: contain

text: contributions

text: from

text: both

text: longitudinal

text: and

text: transverse

text: photons,

text: their

text: UL

text: LL

text: description

text: will

text: always

text: contain

text: a

text: mixture

text: of

text: chiral

text: odd

text: and

text: chiral

text: even

text: GPDs,

text: and

text: thus,

text: they

text: are

text: not

text: most

text: appropriate

text: for

text: a

text: clean

text: extraction

text: of

text: the

text: chiral

text: odd

text: sector.

text: Using

text: our

text: model,

text: we

text: found

text: that

text: the

text: dominant

text: contribution

text: to

text: these

text: ##LTFigure##

text: ##LTFigure##


text: 12 (cid:118) (cid:118) (cid:117) and (cid:117) (cid:117) (cid:117) (cid:117) (cid:116) (cid:16) (cid:17) (cid:116) (cid:16) (cid:17) (cid:104) (cid:105) (cid:104) (cid:105) k⊥2 k˜⊥2 a = , a˜ = (cid:104) (cid:105) 2 − (cid:104) (cid:105) 2 X2 + k⊥2 2Mζ2 (X ζ)2 + k˜⊥2 2Mζ2 Q2 Q2 (cid:104) (cid:105) ≈ (cid:104) (cid:105) (cid:104) − (cid:105) ≈ (cid:104) (cid:105) − | | where k⊥2 1/2 0.3 GeV, and k˜⊥2 1/2 = (k⊥ + (1 X)∆)2 1/2 k⊥2 1/2 + (1 X) ∆ . The chiral even GPDs used to evaluate Eqs.(34) and (35) were taken from the parametrization developed in Ref.[19]. In the forward limit one obtains, (cid:101) − −M(1 x) H(0) = E(0) (36a) T m + Mx − M(2 x) + m E(cid:101)(0) = E(0) (36b) T m + Mx (cid:101) E(0) = 0 (36c) T H(0) + H(0) H(0) = (36d) T 2 and for S = 1, (cid:101) H(1) = 0 (37a) (cid:101)T E(1) = 2aE(1) (37b) T (cid:101) E1 = 0 (37c) T − 2x H(1) + H(1) H(1) = (37d) T 1 + x2 2 Note that once the chiral odd GPDs are calculated using the equations above, at a given initial scale of the model, (cid:46) Q2 1 GeV2, their evolution to the scale of the data must proceed according to the equations developed for the chiral o odd sector in [43, 44]. The parametric form for the chiral even GPDs was given in Ref.[? ]. As both new DVCS and meson electroproduction data become available, it will be possible to perform a global ﬁt using simultaneously all sets of data. At the present stage our approach guarantees a better control over the various kinematical dependences. E. Flavor Dependence The u and d quark chiral odd distributions can be readily obtained from Eqs.(34) and (35) by using the SU(4) symmetry of the prot(cid:115)on wave f(cid:34)unction, (cid:35) √ | ↑(cid:105) 2 √a | ↑ (cid:105) √1 | ↑ (cid:105) − 1 | ↓ (cid:105) − 1 | ↑ (cid:105) 2 | ↓ (cid:105) p = S u S0 + u T0 u T1 d T0 + d T1 (38) 1 + a2 2 0 3 2 0 3 0 3 1 3 1 S ≡ ≡ where S0 SS3 is the scalar diquark with isospin 0 and spin component 0, T0,1 TS3 is the axial vector (triplet) 0 I3 0,1 I3 diquark with indicated isospin and spin components, and the parameter a = 1 for SU(4) symmetry and can diﬀer S from 1 to allow for s(cid:115)ymmetry(cid:34)breaking [41, 42]. Se(cid:18)parating out the spin dep(cid:19)e(cid:32)nd(cid:114)ence leaves, (cid:114) (cid:33)(cid:35) | ↑(cid:105) 2 √a | (cid:105) | ↑(cid:105) √1 | (cid:105) − √1 | (cid:105) 2 | ↓(cid:105) − 1 | ↑(cid:105) p = S uS 0, + d T u T 1, 0, . (39) 1 + a2 2 0 3 1 6 0 3 3 S ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: and

text: k˜⊥2

text: a

text: a˜

text: (X

text: k˜⊥2

text: where

text: GeV,

text: and

text: k˜⊥2

text: (k⊥

text: X)

text: The

text: chiral

text: even

text: GPDs

text: used

text: to

text: evaluate

text: and

text: were

text: taken

text: from

text: the

text: parametrization

text: developed

text: in

text: In

text: the

text: forward

text: limit

text: one

text: obtains,

text: x)

text: T

text: m

text: Mx

text: x)

text: m

text: T

text: m

text: Mx

text: T

text: T

text: and

text: for

text: S

text: T

text: T

text: T

text: Note

text: that

text: once

text: the

text: chiral

text: odd

text: GPDs

text: are

text: calculated

text: using

text: the

text: equations

text: above,

text: at

text: a

text: given

text: initial

text: scale

text: of

text: the

text: model,

text: their

text: evolution

text: to

text: the

text: scale

text: of

text: the

text: data

text: must

text: proceed

text: according

text: to

text: the

text: equations

text: developed

text: for

text: the

text: chiral

text: o

text: odd

text: sector

text: in

text: The

text: parametric

text: form

text: for

text: the

text: chiral

text: even

text: GPDs

text: was

text: given

text: in

text: Ref.[?

text: As

text: both

text: new

text: DVCS

text: and

text: meson

text: electroproduction

text: data

text: become

text: available,

text: it

text: will

text: be

text: possible

text: to

text: perform

text: a

text: global

text: ﬁt

text: using

text: simultaneously

text: all

text: sets

text: of

text: data.

text: At

text: the

text: present

text: stage

text: our

text: approach

text: guarantees

text: a

text: better

text: control

text: over

text: the

text: various

text: kinematical

text: dependences.

text: E.

text: Flavor

text: Dependence

text: The

text: u

text: and

text: d

text: quark

text: chiral

text: odd

text: distributions

text: can

text: be

text: readily

text: obtained

text: from

text: and

text: by

text: using

text: the

text: symmetry

text: of

text: the

text: wave

text: √a

text: √1

text: p

text: S

text: u

text: u

text: u

text: d

text: d

text: S

text: where

text: is

text: the

text: scalar

text: diquark

text: with

text: isospin

text: and

text: spin

text: component

text: is

text: the

text: axial

text: vector

text: (triplet)

text: diquark

text: with

text: indicated

text: isospin

text: and

text: spin

text: components,

text: and

text: the

text: parameter

text: a

text: for

text: symmetry

text: and

text: can

text: diﬀer

text: S

text: from

text: to

text: allow

text: for

text: out

text: the

text: spin

text: leaves,

text: √a

text: √1

text: √1

text: p

text: S

text: uS

text: d

text: T

text: u

text: T

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 (cid:18) (cid:19) ≡ (cid:32) + M2 (cid:33) P P , , 0) + P ∆ ≡ (cid:32)ξ (2P+), t + ∆2T , ∆ ) (cid:33) (3a) + T 2ξP P ≡ (cid:32)(1 + ξ)P+, M2 + ∆2T /4, ∆ /2 (cid:33) + T (1 + ξ)P P(cid:48) ≡ (1 − ξ)P+, M2−+ ∆2T+/4, −∆T /2 , (1 ξ)P where in the DGLAP region (here we consid(cid:16)er x > ξ) the coordinates of(cid:17)the oﬀ-shell struck parton are, (cid:16) (cid:17) ≡ + − k (x + ξ)P , k , k + ∆ /2 , T T (cid:48) ≡ − + (cid:48)− − k (x ξ)P , k , k ∆ /2 (4a) T T Other useful variables can be written as, ≈ − (cid:48) − ≈ − ≈ sˆ = (k + q)2 Q2(x ξ)/2ξ, uˆ = (k q)2 Q2(x + ξ)/2ξ, q (Pq)/P+ = Q2/(4ξ)(1 + ξ)P+. ≡ The loop diagram in Fig.1 integrated over the struck quark’s momentum is performed using the variables: d4k − ≡ − dk+dk d2k⊥ P+dXdk d2k⊥. B. Helicity Amplitudes Structure To describe spin dependent observables we next introduce the helicity amplitudes (for a detailed description of the helicity amplitudes formalism in deeply virtual scattering processes see also Ref.[27]). For pseudoscalar meson production one has [2, 18], (cid:88) (cid:48) (cid:48) ⊗ fΛΛ (ξ, t, Q2) = gλλ (x, ξ, t, Q2) A (cid:48) (cid:48) (x, ξ, t), (5) Λγ0 (cid:48) Λγ0 Λ λ ,Λλ λ,λ where the helicities of the virtual photon and the initial proton are, Λ , Λ, and the helicities of the produced pion (cid:48) γ and ﬁnal proton are 0, and Λ , respectively. Notice that both longitudinal and transverse polarizations of the virtual ∗ ∗ photon γ can, in principle, contribute. While γ was shown to be the leading contribution in Ref.[1], in [2, 18] L(T) L ∗ → a possible scenario beyond collinear factorization was presented for γ p Mp which explains the large transverse T photon polarization contributions observed in the experimental data in terms of chiral odd GPDs. In Eq.(5) we (cid:48) ∗ → describe the factorization into a “hard part”, gλλ for the partonic subprocess γ + q π0 + q, which appears now Λγ0 T (cid:82) at twist three, and the quark-proton helicity amplitudes, A (cid:48) (cid:48) that contain the chiral odd GPDs. Λ ,λ ;Λ,λ The amplitudes A (cid:48) (cid:48) implicitly involve an integration over the unobserved quark’s transerve momentum, k , and Λ λ ,Λλ ≈ − ⊗ → T are functions of x = Q2/2Mν 2ξ/(1 ξ2), t and Q2. The convolution integral in Eq.(5) is given by −1 dx. Bj 1 The connection with the correlator i(cid:90)s carried out by considering, − dz − (cid:104) (cid:48) (cid:48) | O | (cid:105)| A (cid:48) (cid:48) = eixP+z p , Λ (cid:48) (z) p, Λ , (6) Λ λ ,Λλ 2π λ λ z+=0,zT=0 where, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P

text: P

text: P

text: t

text: T

text: P

text: ξ)P+,

text: T

text: ξ)P

text: ξ)P+,

text: −∆T

text: ξ)P

text: where

text: in

text: the

text: DGLAP

text: region

text: (here

text: we

text: x

text: ξ)

text: the

text: coordinates

text: oﬀ-shell

text: struck

text: parton

text: are,

text: k

text: (x

text: ξ)P

text: k

text: k

text: T

text: T

text: k

text: (x

text: ξ)P

text: k

text: k

text: T

text: T

text: Other

text: useful

text: variables

text: can

text: be

text: written

text: as,

text: sˆ

text: (k

text: uˆ

text: (k

text: q

text: (Pq)/P+

text: ξ)P+.

text: The

text: loop

text: diagram

text: in

text: integrated

text: over

text: the

text: struck

text: quark’s

text: momentum

text: is

text: performed

text: using

text: the

text: variables:

text: dk+dk

text: P+dXdk

text: B.

text: Helicity

text: Amplitudes

text: Structure

text: To

text: describe

text: spin

text: dependent

text: observables

text: we

text: next

text: introduce

text: the

text: helicity

text: amplitudes

text: (for

text: a

text: detailed

text: description

text: of

text: the

text: helicity

text: amplitudes

text: formalism

text: in

text: deeply

text: virtual

text: scattering

text: processes

text: see

text: also

text: For

text: pseudoscalar

text: meson

text: production

text: one

text: has

text: fΛΛ

text: (ξ,

text: t,

text: gλλ

text: (x,

text: ξ,

text: t,

text: A

text: (x,

text: ξ,

text: t),

text: Λγ0

text: Λγ0

text: Λ

text: λ

text: ,Λλ

text: λ,λ

text: where

text: the

text: helicities

text: of

text: the

text: virtual

text: photon

text: and

text: the

text: initial

text: proton

text: are,

text: Λ

text: Λ,

text: and

text: the

text: helicities

text: of

text: the

text: produced

text: pion

text: γ

text: and

text: ﬁnal

text: proton

text: are

text: and

text: Λ

text: respectively.

text: Notice

text: that

text: both

text: longitudinal

text: and

text: transverse

text: polarizations

text: of

text: the

text: virtual

text: photon

text: γ

text: can,

text: in

text: principle,

text: contribute.

text: While

text: γ

text: was

text: shown

text: to

text: be

text: the

text: leading

text: contribution

text: in

text: in

text: L(T)

text: L

text: a

text: possible

text: scenario

text: beyond

text: collinear

text: factorization

text: was

text: presented

text: for

text: γ

text: p

text: Mp

text: which

text: explains

text: the

text: large

text: transverse

text: T

text: photon

text: polarization

text: contributions

text: observed

text: in

text: the

text: experimental

text: data

text: in

text: terms

text: of

text: chiral

text: odd

text: GPDs.

text: In

text: we

text: describe

text: the

text: factorization

text: into

text: a

text: “hard

text: part”,

text: gλλ

text: for

text: the

text: partonic

text: subprocess

text: γ

text: q

text: π0

text: q,

text: which

text: appears

text: now

text: Λγ0

text: T

text: at

text: twist

text: three,

text: and

text: the

text: quark-proton

text: helicity

text: amplitudes,

text: A

text: that

text: contain

text: the

text: chiral

text: odd

text: GPDs.

text: Λ

text: ,λ

text: ;Λ,λ

text: The

text: amplitudes

text: A

text: implicitly

text: involve

text: an

text: integration

text: over

text: the

text: unobserved

text: quark’s

text: transerve

text: momentum,

text: k

text: and

text: Λ

text: λ

text: ,Λλ

text: T

text: are

text: functions

text: of

text: x

text: t

text: and

text: The

text: convolution

text: integral

text: in

text: is

text: given

text: by

text: −1

text: dx.

text: Bj

text: The

text: connection

text: with

text: the

text: correlator

text: carried

text: out

text: by

text: considering,

text: dz

text: O

text: A

text: eixP+z

text: p

text: Λ

text: (z)

text: p,

text: Λ

text: Λ

text: λ

text: ,Λλ

text: 2π

text: λ

text: λ

text: where,

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: 10 structure of partonic OAM, and we conﬁrmed from an alternative perspective, that OAM is twist three, and that it is uniquely observable through DVCS type measurements of G (see also [33]). In particular, we pointed out that 2 based on Parity constraints, the twist two chiral even GTMD labeled F in [7], describing an unpolarized quark in a 14 longitudinally polarized proton, could not contribute to OAM. We also pointed out that another possible(cid:101)candidate, ⊥ the chiral odd TMD, h [34], does not display the necessary spin structure of partonic OAM. To conﬁrm this 1T ⊥ picture, it would be important to obtain correlated, although indirect information on OAM from both E and h T 1L measurements. D. Chiral Odd GPDs in Spectator Model Eqs.(19) provide the helicity amplitudes that enter directly the observables for pseudoscalar meson electroproduc- tion. However, in practical calculations we ﬁnd an outstanding problem: diﬀerently from the chiral even sector, even using available models for the chiral odd GPDs both the normalization to the form factors and the forward limit for these quantities are largely undetermined from independent measurements (with the exclusion, perhaps, of transver- sity, h ). This makes it diﬃcult to estimate the magnitude of the chiral odd GPDs. In the spectator model, that 1 we describe below, one can overcome this diﬃculty by exploiting Parity relations among the helicity amplitudes that allow us to connect the chiral odd GPDs to the chiral even ones (analogous relations were found to hold for TMDs in Ref.[42]). Although this is a model dependent procedure, we consider it a necessary step at an initial stage of both theoretical and experimental investigations. To extract the chiral odd GPDs from DVπ0(η)P data we propose a parametrization based on the reggeized diquark model developed in Refs.[18, 19, 35, 36]. This is, in a nutshell, a spectator model with variable mass of the spectator, → ∞ → M , which allows us to reproduce the Regge behavior in the limit M (x 0). The model was evolved from X X its initial low scale to the scale of the data using leading order Perturbative QCD (PQCD) GPD evolution equations (see e.g. Refs.[24, 25]). In the chiral even sector we determined the model’s parameters from a ﬁt using a compilation of three diﬀerent data sets, namely, nucleon form factor data (ﬂavor separated Dirac and Pauli form factors [20], axial [21] and pseudo-scalar form factor [22]), DVCS data [23], and DIS data on Fp(n). For the latter we accurately 2 reproduced the valence quarks PDFs obtained from global ﬁts [37]. In Ref.[18] our approach lead us to succesfully reproduce data on various observables in DVCS besides the ones used in the ﬁt, namely the charge [38, 39], and transverse [38, 39] single spin asymmetries. In the chiral odd sector the GPDs are largely unconstrained. This is mostly due to the fact that, diﬀerently from the GPDs in the chiral even sector, no experimental measurements from t dependent form factors exist, which would provide normalizations for the GPDs. In the analysis of π0 and η electroproduction data it is, however, important to be able to gauge the size of the various chiral odd GPDs contributions. We therefore developed an alternative procedure by exploiting Parity relations within the reggeized diquark model. The extension of this model to the chiral odd sector is explained in detail in Ref.[4] where we obtained predictions for both the unpolarized and longitudinally polarized observables in π0 electroproduction, namely the beam spin asymmetry, A , and the polarized target asyemmetries, A and A . LU UL LL The Parity relations between chiral even and chiral odd GPDs which are valid, in general, within a class of models including any type of spectator model with diquark spin S = 0, 1 and angular momentum L = 0 5 yield, respectively, for S = 0, A(0) −− = A(0) (32a) ++, ++,++ − ∗ A(0) − = A(0) − (32b) ++,+ ++, + (cid:20) (cid:21) − ∗(cid:112) A(0−) = A(−0) (cid:101) (32c) + ,++ +,++ − t t − X˜ − ξ A(0−) − = 0 1 ξ2 (cid:48) E − E , (32d) + , + 4M m + MX 1 ξ2 (cid:54) 5 More complicated intermediate state conﬁgurations with L = 0 have been considered recently in Ref.[40]. The Parity relations would be diﬀerent in this case. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: structure

text: of

text: partonic

text: OAM,

text: and

text: we

text: conﬁrmed

text: from

text: an

text: alternative

text: perspective,

text: that

text: OAM

text: is

text: twist

text: three,

text: and

text: that

text: it

text: is

text: uniquely

text: observable

text: through

text: DVCS

text: type

text: measurements

text: of

text: G

text: (see

text: also

text: In

text: particular,

text: we

text: pointed

text: out

text: that

text: based

text: on

text: Parity

text: constraints,

text: the

text: twist

text: two

text: chiral

text: even

text: GTMD

text: labeled

text: F

text: in

text: describing

text: an

text: unpolarized

text: quark

text: in

text: a

text: longitudinally

text: polarized

text: proton,

text: could

text: not

text: contribute

text: to

text: OAM.

text: We

text: also

text: pointed

text: out

text: that

text: another

text: the

text: chiral

text: odd

text: TMD,

text: h

text: does

text: not

text: display

text: the

text: necessary

text: spin

text: structure

text: of

text: partonic

text: OAM.

text: To

text: conﬁrm

text: this

text: picture,

text: it

text: would

text: be

text: important

text: to

text: obtain

text: correlated,

text: although

text: indirect

text: information

text: on

text: OAM

text: from

text: both

text: E

text: and

text: h

text: T

text: measurements.

text: D.

text: Chiral

text: Odd

text: GPDs

text: in

text: Spectator

text: Model

text: provide

text: the

text: helicity

text: amplitudes

text: that

text: enter

text: directly

text: the

text: observables

text: for

text: pseudoscalar

text: meson

text: electroproduc-

text: tion.

text: However,

text: in

text: practical

text: calculations

text: we

text: ﬁnd

text: an

text: outstanding

text: problem:

text: diﬀerently

text: from

text: the

text: chiral

text: even

text: sector,

text: even

text: using

text: available

text: models

text: for

text: the

text: chiral

text: odd

text: GPDs

text: both

text: the

text: normalization

text: to

text: the

text: form

text: factors

text: and

text: the

text: forward

text: limit

text: for

text: these

text: quantities

text: are

text: largely

text: undetermined

text: from

text: independent

text: measurements

text: (with

text: the

text: exclusion,

text: perhaps,

text: of

text: transver-

text: sity,

text: h

text: This

text: makes

text: it

text: diﬃcult

text: to

text: estimate

text: the

text: magnitude

text: of

text: the

text: chiral

text: odd

text: GPDs.

text: In

text: the

text: spectator

text: model,

text: that

text: we

text: describe

text: below,

text: one

text: can

text: overcome

text: this

text: diﬃculty

text: by

text: exploiting

text: Parity

text: relations

text: among

text: the

text: helicity

text: amplitudes

text: that

text: allow

text: us

text: to

text: connect

text: the

text: chiral

text: odd

text: GPDs

text: to

text: the

text: chiral

text: even

text: ones

text: (analogous

text: relations

text: were

text: found

text: to

text: hold

text: for

text: TMDs

text: in

text: Although

text: this

text: is

text: a

text: model

text: dependent

text: procedure,

text: we

text: consider

text: it

text: a

text: necessary

text: step

text: at

text: an

text: initial

text: stage

text: of

text: both

text: theoretical

text: and

text: experimental

text: investigations.

text: To

text: extract

text: the

text: chiral

text: odd

text: GPDs

text: from

text: data

text: we

text: propose

text: a

text: parametrization

text: based

text: on

text: the

text: reggeized

text: diquark

text: model

text: developed

text: in

text: This

text: is,

text: in

text: a

text: nutshell,

text: a

text: spectator

text: model

text: with

text: variable

text: mass

text: of

text: the

text: spectator,

text: M

text: which

text: allows

text: us

text: to

text: reproduce

text: the

text: Regge

text: behavior

text: in

text: the

text: limit

text: M

text: (x

text: The

text: model

text: was

text: evolved

text: from

text: X

text: X

text: its

text: initial

text: low

text: scale

text: to

text: the

text: scale

text: of

text: the

text: data

text: using

text: leading

text: order

text: Perturbative

text: QCD

text: (PQCD)

text: GPD

text: evolution

text: equations

text: (see

text: e.g.

text: In

text: the

text: chiral

text: even

text: sector

text: we

text: determined

text: the

text: model’s

text: parameters

text: from

text: a

text: ﬁt

text: using

text: a

text: compilation

text: of

text: three

text: diﬀerent

text: data

text: sets,

text: namely,

text: nucleon

text: form

text: factor

text: data

text: (ﬂavor

text: separated

text: Dirac

text: and

text: Pauli

text: form

text: factors

text: axial

text: and

text: pseudo-scalar

text: form

text: factor

text: DVCS

text: data

text: and

text: DIS

text: data

text: on

text: Fp(n).

text: For

text: the

text: latter

text: we

text: accurately

text: reproduced

text: the

text: valence

text: quarks

text: PDFs

text: obtained

text: from

text: global

text: ﬁts

text: In

text: our

text: approach

text: lead

text: us

text: to

text: succesfully

text: reproduce

text: data

text: on

text: various

text: observables

text: in

text: DVCS

text: besides

text: the

text: ones

text: used

text: in

text: the

text: ﬁt,

text: namely

text: the

text: charge

text: and

text: transverse

text: single

text: spin

text: asymmetries.

text: In

text: the

text: chiral

text: odd

text: sector

text: the

text: GPDs

text: are

text: largely

text: unconstrained.

text: This

text: is

text: mostly

text: due

text: to

text: the

text: fact

text: that,

text: diﬀerently

text: from

text: the

text: GPDs

text: in

text: the

text: chiral

text: even

text: sector,

text: no

text: experimental

text: measurements

text: from

text: t

text: dependent

text: form

text: factors

text: exist,

text: which

text: would

text: provide

text: normalizations

text: for

text: the

text: GPDs.

text: In

text: the

text: analysis

text: of

text: π0

text: and

text: η

text: electroproduction

text: data

text: it

text: is,

text: however,

text: important

text: to

text: be

text: able

text: to

text: gauge

text: the

text: size

text: of

text: the

text: various

text: chiral

text: odd

text: GPDs

text: contributions.

text: We

text: therefore

text: developed

text: an

text: alternative

text: procedure

text: by

text: exploiting

text: Parity

text: relations

text: within

text: the

text: reggeized

text: diquark

text: model.

text: The

text: extension

text: of

text: this

text: model

text: to

text: the

text: chiral

text: odd

text: sector

text: is

text: explained

text: in

text: detail

text: in

text: where

text: we

text: obtained

text: predictions

text: for

text: both

text: the

text: unpolarized

text: and

text: longitudinally

text: polarized

text: observables

text: in

text: π0

text: electroproduction,

text: namely

text: the

text: beam

text: spin

text: asymmetry,

text: A

text: and

text: the

text: polarized

text: target

text: asyemmetries,

text: A

text: and

text: A

text: LU

text: UL

text: LL

text: The

text: Parity

text: relations

text: between

text: chiral

text: even

text: and

text: chiral

text: odd

text: GPDs

text: which

text: are

text: valid,

text: in

text: general,

text: within

text: a

text: class

text: of

text: models

text: including

text: any

text: type

text: of

text: spectator

text: model

text: with

text: diquark

text: spin

text: S

text: and

text: angular

text: momentum

text: L

text: yield,

text: respectively,

text: for

text: S

text: t

text: t

text: X˜

text: ξ

text: ξ2

text: E

text: E

text: m

text: MX

text: ξ2

text: More

text: complicated

text: intermediate

text: state

text: conﬁgurations

text: with

text: L

text: have

text: been

text: considered

text: recently

text: in

text: The

text: Parity

text: relations

text: would

text: be

text: diﬀerent

text: in

text: this

text: case.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [31] C.A. Haselwandter, D.D. Vvedensky, Int. J. Mod. Phys. B 22, 3721 (2008) [32] F.D.A. Aara˜o, Phys. Rev. E 81, 041605 (2010) [33] J.M. Kim, J.M. Kosterlitz, Phys. Rev. Lett. 62, 2289 (1989) [34] Y. Kim, D.K. Park, J.M. Kim, J. Phys. A: Math. Gen. 27, L533 (1994) [35] F. Reif, Statistical and thermal physics, McGraw-Hill, New York, 1965 [36] D.A. Kessler, H. Levine, L.M.Sander, Phys. Rev. Lett. 69, 100 (1991) [37] C.M. Arizmendi, J.M. Sanchez, J. Phys.: Condens. Matter 5, A103 (1993) [38] S. Das Sarma, S.V. Ghaisas, J.M. Kim, Phys. Rev. E 49, 122 (1994) [39] S. Das Sarma, C.J. Lanczycki, R. Kotlyar, S.V. Ghaisas, Phys. Rev. E 53, 359 (1996) [40] S. Das Sarma, in: Z. Zhang and M.G. Lagally (Ed.), Morphological Or- ganization in Epitaxial Growth and Removal, World Scientiﬁc, Singapore, 1998. [41] M.R. Baklanov, K. Maex, Phil. Trans. R. Soc. A 364, 201 (2006) 14

text: C.A.

text: Haselwandter,

text: D.D.

text: Vvedensky,

text: Int.

text: J.

text: Mod.

text: Phys.

text: B

text: F.D.A.

text: Aara˜o,

text: Phys.

text: Rev.

text: E

text: J.M.

text: Kim,

text: J.M.

text: Kosterlitz,

text: Phys.

text: Rev.

text: Lett.

text: Y.

text: Kim,

text: D.K.

text: Park,

text: J.M.

text: Kim,

text: J.

text: Phys.

text: A:

text: Math.

text: Gen.

text: F.

text: Reif,

text: Statistical

text: and

text: thermal

text: physics,

text: McGraw-Hill,

text: New

text: York,

text: D.A.

text: Kessler,

text: H.

text: Levine,

text: L.M.Sander,

text: Phys.

text: Rev.

text: Lett.

text: C.M.

text: Arizmendi,

text: J.M.

text: Sanchez,

text: J.

text: Phys.:

text: Condens.

text: Matter

text: S.

text: Das

text: Sarma,

text: S.V.

text: Ghaisas,

text: J.M.

text: Kim,

text: Phys.

text: Rev.

text: E

text: S.

text: Das

text: Sarma,

text: C.J.

text: Lanczycki,

text: R.

text: Kotlyar,

text: S.V.

text: Ghaisas,

text: Phys.

text: Rev.

text: E

text: S.

text: Das

text: Sarma,

text: in:

text: Z.

text: Zhang

text: and

text: M.G.

text: Lagally

text: (Ed.),

text: Morphological

text: Or-

text: ganization

text: in

text: Epitaxial

text: Growth

text: and

text: Removal,

text: World

text: Scientiﬁc,

text: Singapore,

text: M.R.

text: Baklanov,

text: K.

text: Maex,

text: Phil.

text: Trans.

text: R.

text: Soc.

text: A


text: R λ ξ 4.7 × 102 3.60 -3.54 1.4 × 103 4.27 -3.96 4.1 × 103 5.61 -6.12 1.2 × 104 6.69 -6.29 3.6 × 104 7.99 -9.12 1.0 × 105 9.81 -14.37 Table 1: The values of λ and ξ used in the deﬁnition of g(θ; λ, ξ). 116600 250 w (θ), R = 3.6×104 ww ((θθ)),, RR == 33..66××110044 w (θ), R = 1.0×105 114400 ww ((θθ)),, RR == 11..00××110055 w˜1(θ) ww ((θθ)),, RR == 33..11××110055 ˜w2 200 w˜2(θ) 112200 ww˜˜11((θθ)) ˜wand1 150 wand˜wwand˜w11 1188000000 w, RR SR 100 MSMS 6600 M RR R 4400 50 2200 0 00 0 5000 10000 15000 20000 25000 30000 35000 00 22000000 44000000 66000000 88000000 1100000000 1122000000 θ [ML] θθ [[MMLL]] Figure 2: Approximation of w simultaneously by w˜ and w˜ (a) and by w˜ 1 2 1 (b) for diﬀerent values of R. 1144 ww ((θθ)),, RR == 99..11××110055 ww ((θθ)),, RR == 44..11××110033 ww ((θθ)),, RR == 22..77××110066 ww ((θθ)),, RR == 11..22××110044 1122 ww ((θθ)),, RR == 77..99××110066 220000 ww ((θθ)),, RR == 33..66××110044 ww˜˜11((θθ)) ww˜˜22((θθ)) ˜w˜w11 1100 ˜w˜w22 dd dd 115500 nn nn aa 88 aa ww ww SRSR 66 SRSR 110000 MM MM RR 44 RR 5500 22 00 00 00 22000000 44000000 66000000 88000000 1100000000 00 55000000 1100000000 1155000000 2200000000 2255000000 θθ [[MMLL]] θθ [[MMLL]] Figure 3: Approximation of w by w˜ for higher values of R (left) and by w˜ 1 2 for smaller R (right). (right). These observations are clear evidences that the scaling behavior of the epitaxial model under consideration is described in asymptotic limit by w(R, θ) ∼ θ1/2, (12) that is, for θ → ∞ it coincides with the model of random deposition. Further- 9 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

label: table

text: λ

text: ξ

text: Table

text: The

text: values

text: of

text: λ

text: and

text: ξ

text: used

text: in

text: the

text: deﬁnition

text: of

text: g(θ;

text: λ,

text: ξ).

font: default

text: w

text: (θ),

text: R

text: ww

text: ((θθ)),,

text: RR

text: w

text: (θ),

text: R

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: w,

text: RR

text: SR

text: MSMS

text: M

text: RR

text: R

text: θ

text: [ML]

text: θθ

text: [[MMLL]]

text: Figure

text: Approximation

text: of

text: w

text: simultaneously

text: by

text: w˜

text: and

text: w˜

text: (a)

text: and

text: by

text: w˜

text: (b)

text: for

text: diﬀerent

text: values

text: of

text: R.

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: ww

text: ((θθ)),,

text: RR

text: dd

text: dd

text: nn

text: nn

text: aa

text: aa

text: ww

text: ww

text: SRSR

text: SRSR

text: MM

text: MM

text: RR

text: RR

text: θθ

text: [[MMLL]]

text: θθ

text: [[MMLL]]

text: Figure

text: Approximation

text: of

text: w

text: by

text: w˜

text: for

text: higher

text: values

text: of

text: R

text: (left)

text: and

text: by

text: w˜

text: for

text: smaller

text: R

text: (right).

text: (right).

text: These

text: observations

text: are

text: clear

text: evidences

text: that

text: the

text: scaling

text: behavior

text: of

text: the

text: epitaxial

text: model

text: under

text: consideration

text: is

text: described

text: in

text: asymptotic

text: limit

text: by

text: w(R,

text: θ)

text: that

text: is,

text: for

text: θ

text: it

text: coincides

text: with

text: the

text: model

text: of

text: random

text: deposition.

text: Further-

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text: 2 S. S. KAZEMIPOOR AND M. ZAKERI ∗ ≡ ≡ For 2 < p < 2 and the weight functions f h 1, the authors Ambrosetti-Brezis-Cerami [4]have investigated Eq. (E λ ) in [4]. They ( ) found that there exists λ > 0 such that Eq. (E λ ) admits at least two ∈0 ( ) positive solutions for λ (0, λ ), has a positive solution for λ = λ and 0 0 no positive solution exists for λ > λ . For more general cases, we refer 0 the reader to Afrouzi-Ala-Kazemipoor [1], Ambrosetti-Azorezo-Peral [3], de Figueiredo-Gossez-Ubilla [13], EL Hamidi [15] and Wu [25], etc. Recently, in [25] the author has considered a semilinear scalar ellip- tic equation involving concave-convex nonlinearities and sign-changing weight function, and showed multiplicity results with respect to the parameter λ via the extraction of Palais-Smale sequences in the Ne- hari manifold, where for the deﬁnition of Nehari manifolds we refer the reader to Nehari [18] or Willem [23]. For the semilinear elliptic systems with concave-convex nonlinearities,  the authorsAdriouch- EL Hamidi [5] considered the following prob- lems: − | | − | |  ∆u = λu + α u α 2u v β in Ω, − | | −α+β | | | | − ∆v = µ v q 2v + β u α v β 2v in Ω, α+β u = v = 0 on ∂Ω. They proved this system has at least two positive solutions when the R pair of the parameters (λ, µ) belongs to a certain subset of 2. For more similar problems, we refer the reader to Ahammou [2], Alves-de Morais Filho-Souto [6], Bozhkov-Mitidieri [8], Cle´ment-de Figueiredo- Mitidieri [11], de Figueiredo-Felmer [12], El Hamidi [16], Squassina [19] and ve´lin [22], etc. By the above results we know that the existence and multiplicity of positive solutions of semilinear elliptic problems depend on the nonlin- earity term . Let S be the best Sobolev constant for the embedding of H1(Ω) in Lα+β(Ω). Then we have the following result. 0 In this paper, we give a very simple variational proof which is similar to proof of Brown-Wu (see [9]) to prove the existence of at least two positive solutions of system (E ) which is similar to the Wu system λ [26]. (cid:90) (cid:90) × We shall throughout use the function space W = W 1,2(Ω) W 1,2(Ω) 0 0 with norm || || |∇ | |∇ | (u, v) = ( u 2dx + v 2dx)1 2 × Ω Ω and the standard Lp(Ω) Lp(Ω) spaces whose norms we denote by || || (u, v) . p 2. Fibering Maps and The Nehari Manifold ##LTLine## ##LTLine## ##LTLine##

text: S.

text: S.

text: KAZEMIPOOR

text: AND

text: M.

text: ZAKERI

text: For

text: p

text: and

text: the

text: weight

text: functions

text: f

text: h

text: the

text: authors

text: Ambrosetti-Brezis-Cerami

text: investigated

text: Eq.

text: (E

text: λ

text: in

text: They

text: found

text: that

text: there

text: exists

text: λ

text: such

text: that

text: Eq.

text: (E

text: λ

text: admits

text: at

text: least

text: two

text: ∈0

text: positive

text: solutions

text: for

text: λ

text: λ

text: has

text: a

text: positive

text: solution

text: for

text: λ

text: λ

text: and

text: no

text: positive

text: solution

text: exists

text: for

text: λ

text: λ

text: For

text: more

text: general

text: cases,

text: we

text: refer

text: the

text: reader

text: to

text: Afrouzi-Ala-Kazemipoor

text: Ambrosetti-Azorezo-Peral

text: de

text: Figueiredo-Gossez-Ubilla

text: EL

text: Hamidi

text: and

text: Wu

text: etc.

text: Recently,

text: in

text: the

text: author

text: has

text: considered

text: a

text: semilinear

text: scalar

text: ellip-

text: tic

text: equation

text: involving

text: concave-convex

text: nonlinearities

text: and

text: sign-changing

text: weight

text: function,

text: and

text: showed

text: multiplicity

text: results

text: with

text: respect

text: to

text: the

text: parameter

text: λ

text: via

text: the

text: extraction

text: of

text: Palais-Smale

text: sequences

text: in

text: the

text: Ne-

text: hari

text: manifold,

text: where

text: for

text: the

text: deﬁnition

text: of

text: Nehari

text: manifolds

text: we

text: refer

text: the

text: reader

text: to

text: Nehari

text: or

text: Willem

text: For

text: the

text: semilinear

text: elliptic

text: systems

text: with

text: concave-convex

text: nonlinearities,

text: the

text: authorsAdriouch-

text: EL

text: Hamidi

text: considered

text: the

text: following

text: prob-

text: lems:

text: ∆u

text: λu

text: α

text: u

text: α

text: v

text: β

text: in

text: Ω,

text: −α+β

text: ∆v

text: µ

text: v

text: q

text: β

text: u

text: α

text: v

text: β

text: in

text: Ω,

text: α+β

text: u

text: v

text: on

text: ∂Ω.

text: They

text: proved

text: this

text: system

text: has

text: at

text: least

text: two

text: positive

text: solutions

text: when

text: the

text: R

text: pair

text: of

text: the

text: parameters

text: (λ,

text: µ)

text: belongs

text: to

text: a

text: certain

text: subset

text: of

text: For

text: more

text: similar

text: problems,

text: we

text: refer

text: the

text: reader

text: to

text: Ahammou

text: Alves-de

text: Morais

text: Filho-Souto

text: Bozhkov-Mitidieri

text: Cle´ment-de

text: Figueiredo-

text: Mitidieri

text: de

text: Figueiredo-Felmer

text: El

text: Hamidi

text: Squassina

text: and

text: ve´lin

text: etc.

text: By

text: the

text: above

text: results

text: we

text: know

text: that

text: the

text: existence

text: and

text: multiplicity

text: of

text: positive

text: solutions

text: of

text: semilinear

text: elliptic

text: problems

text: depend

text: on

text: the

text: nonlin-

text: earity

text: term

text: Let

text: S

text: be

text: the

text: best

text: Sobolev

text: constant

text: for

text: the

text: embedding

text: of

text: in

text: Lα+β(Ω).

text: Then

text: we

text: have

text: the

text: following

text: result.

text: In

text: this

text: paper,

text: we

text: give

text: a

text: very

text: simple

text: variational

text: proof

text: which

text: is

text: similar

text: to

text: proof

text: of

text: Brown-Wu

text: (see

text: to

text: prove

text: the

text: existence

text: of

text: at

text: least

text: two

text: positive

text: solutions

text: of

text: system

text: (E

text: which

text: is

text: similar

text: to

text: the

text: Wu

text: system

text: λ

text: We

text: shall

text: throughout

text: use

text: the

text: function

text: space

text: W

text: W

text: W

text: with

text: norm

text: (u,

text: v)

text: u

text: v

text: Ω

text: Ω

text: and

text: the

text: standard

text: Lp(Ω)

text: Lp(Ω)

text: spaces

text: whose

text: norms

text: we

text: denote

text: by

text: (u,

text: v)

text: p

text: Fibering

text: Maps

text: and

text: The

text: Nehari

text: Manifold

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ELEMENTARY APPROACH TO. . . 12 [5] D. Bezdek and K. Bezdek. Shortest billiard trajectories. Geometriae Dedicata, 141:197–206, 2009. [6] K. Bezdek and R. Connelly. Covering curves by translates of a convex set. The American Mathe- matical Monthly, 96(9):789–806, 1989. [7] G. Kuperberg. From the Mahler conjecture to Gauss linking integrals. Geometric and Functional Analysis, 18(3):870–892, 2008. [8] K. Mahler. Ein U¨bertragungsprinzip fu¨r konvexe K¨orper. Cˇasopis pro P˘estov´an´ı Matematiky a Fysiky, 68:93–102, 1939. [9] S. Tabachnikov. Geometry and billiards, volume 30. Amer Mathematical Society, 2005. [10] T. Tao. Open question: the Mahler conjecture on convex bodies. 2007. terrytao.wordpress.com. Email address: akopjan@gmail.com Email address: alexey m39@mail.ru Email address: r n karasev@mail.ru URL: http://www.rkarasev.ru/en/ Email address: independsharik@yandex.ru ♠ Institute of Science and Technology Austria (IST Austria), Am Campus 1, 3400 Klosterneuburg, Austria ♣♦♥ Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700 ♠♣♦ Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994 ##LTLine## ##LTLine## ##LTLine##

text: ELEMENTARY

text: APPROACH

text: TO.

label: reference

text: D.

text: Bezdek

text: and

text: K.

text: Bezdek.

text: Shortest

text: billiard

text: trajectories.

text: Geometriae

text: Dedicata,

text: K.

text: Bezdek

text: and

text: R.

text: Connelly.

text: Covering

text: curves

text: by

text: translates

text: of

text: a

text: convex

text: set.

text: The

text: American

text: Mathe-

text: matical

text: Monthly,

text: G.

text: Kuperberg.

text: From

text: the

text: Mahler

text: conjecture

text: to

text: Gauss

text: linking

text: integrals.

text: Geometric

text: and

text: Functional

text: Analysis,

text: K.

text: Mahler.

text: Ein

text: U¨bertragungsprinzip

text: fu¨r

text: konvexe

text: K¨orper.

text: Cˇasopis

text: pro

text: P˘estov´an´ı

text: Matematiky

text: a

text: Fysiky,

text: S.

text: Tabachnikov.

text: Geometry

text: and

text: billiards,

text: volume

text: Amer

text: Mathematical

text: Society,

text: T.

text: Tao.

text: Open

text: question:

text: the

text: Mahler

text: conjecture

text: on

text: convex

text: bodies.

text: terrytao.wordpress.com.

text: Email

text: address:

text: akopjan@gmail.com

text: Email

text: address:

text: alexey

text: Email

text: address:

text: r

text: n

text: karasev@mail.ru

text: URL:

text: Email

text: address:

text: independsharik@yandex.ru

text: Institute

text: of

text: Science

text: and

text: Technology

text: Austria

text: (IST

text: Austria),

text: Am

text: Campus

text: Klosterneuburg,

text: Austria

text: Moscow

text: Institute

text: of

text: Physics

text: and

text: Technology,

text: Institutskiy

text: per.

text: Dolgoprudny,

text: Russia

text: Institute

text: for

text: Information

text: Transmission

text: Problems

text: RAS,

text: Bolshoy

text: Karetny

text: per.

text: Moscow,

text: Russia

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ELEMENTARY APPROACH TO. . . 7 z y o x Fig. 4. Explanation of the proof of Theorem 3.2. Remark 3.4. Let K be strictly convex. If the length of P were 4 then in the above (cid:107) − (cid:107) (cid:107) − (cid:107) argument the equality z x + z y = 2 will hold at most once on either half of P . So a translate of K covers K and P has at most 2 bounces. Actually, one bounce is impossible, so a 2-bouncing trajectory is the only case of equality, and this trajectory − ∈ must be the segment [q, q] for some q ∂K. If K is not strictly convex then other minimal trajectories also exist. Remark 3.5. If K is a square in the plane, which is not smooth and not strictly convex, then there are plenty of minimal trajectories. Here a minimal trajectory is understood as something providing the minimum to the right hand side of the deﬁning equation in Theorem 2.1. Any segment connecting the two opposite sides of K is such, and some of the quadrangles with vertices on the four sides of K are also such. As another simple exercise, we establish the following result: Theorem 3.6. Let K be 2-periodic with respect to T and let T be centrally symmetric. ◦ Then K + λT is also 2-periodic with respect to T for any λ. Proof. Consider one of the shortest closed billiard trajectories in K bouncing at q and q . ◦ {− 1} ∈ 2◦ From Theorem 3.2 we also know that ξ (T ) = 4 and we can ﬁnd a pair q, q ∂T T ◦ that gives a shortest closed billiard trajectory in T with length 4 and such that q is − { } proportional to q q . The minimality assumption for q , q implies that the normals − 2 1 1 2◦ − p and p to K at q and q are the same as the normals to T at q and q respectively. 1 {2 − } ◦ Then the pair of points q λq, q +λq is in the boundary of K +λT and the normals ◦ 1 2 − { − } to K + λT at these points are again p and p. Now it follows that q λq, q + λq is ◦ ◦ 1 2 a closed billiard trajectory in K + λT of length ξ (K) + λξ (T ). From (3.3) it follows T T (cid:3) that this trajectory is minimal. 4. Attempt toward the non-symmetric Mahler’s conjecture · ◦ ≥ In [3] Mahler’s conjecture vol K vol K 4n for centrally symmetric convex n- n! dimensional K was reduced, assuming the Viterbo conjecture on symplectic capacities, to proving that ≥ ξ ◦(K) 4, K which is true, see Theorem 3.2 in the previous section. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ELEMENTARY

text: APPROACH

text: TO.

text: z

text: y

text: o

text: x

text: Fig.

text: Explanation

text: of

text: the

text: proof

text: of

text: Theorem

text: Remark

text: Let

text: K

text: be

text: strictly

text: convex.

text: If

text: the

text: length

text: of

text: P

text: were

text: then

text: in

text: the

text: above

text: argument

text: the

text: equality

text: z

text: x

text: z

text: y

text: will

text: hold

text: at

text: most

text: once

text: on

text: either

text: half

text: of

text: P

text: So

text: a

text: translate

text: of

text: K

text: covers

text: K

text: and

text: P

text: has

text: at

text: most

text: bounces.

text: Actually,

text: one

text: bounce

text: is

text: impossible,

text: so

text: a

text: trajectory

text: is

text: the

text: only

text: case

text: of

text: equality,

text: and

text: this

text: trajectory

text: must

text: be

text: the

text: segment

text: [q,

text: q]

text: for

text: some

text: q

text: ∂K.

text: If

text: K

text: is

text: not

text: strictly

text: convex

text: then

text: other

text: minimal

text: trajectories

text: also

text: exist.

text: Remark

text: If

text: K

text: is

text: a

text: square

text: in

text: the

text: plane,

text: which

text: is

text: not

text: smooth

text: and

text: not

text: strictly

text: convex,

text: then

text: there

text: are

text: plenty

text: of

text: minimal

text: trajectories.

text: Here

text: a

text: minimal

text: trajectory

text: is

text: understood

text: as

text: something

text: providing

text: the

text: minimum

text: to

text: the

text: right

text: hand

text: side

text: of

text: the

text: deﬁning

text: equation

text: in

text: Theorem

text: Any

text: segment

text: connecting

text: the

text: two

text: opposite

text: sides

text: of

text: K

text: is

text: such,

text: and

text: some

text: of

text: the

text: quadrangles

text: with

text: vertices

text: on

text: the

text: four

text: sides

text: of

text: K

text: are

text: also

text: such.

text: As

text: another

text: simple

text: exercise,

text: we

text: establish

text: the

text: following

text: result:

text: Theorem

text: Let

text: K

text: be

text: with

text: respect

text: to

text: T

text: and

text: let

text: T

text: be

text: centrally

text: symmetric.

text: Then

text: K

text: λT

text: is

text: also

text: with

text: respect

text: to

text: T

text: for

text: any

text: λ.

text: Proof.

text: Consider

text: one

text: of

text: the

text: shortest

text: closed

text: billiard

text: trajectories

text: in

text: K

text: bouncing

text: at

text: q

text: and

text: q

text: 2◦

text: From

text: Theorem

text: we

text: also

text: know

text: that

text: ξ

text: (T

text: and

text: we

text: can

text: ﬁnd

text: a

text: pair

text: q,

text: q

text: ∂T

text: T

text: that

text: gives

text: a

text: shortest

text: closed

text: billiard

text: trajectory

text: in

text: T

text: with

text: length

text: and

text: such

text: that

text: q

text: is

text: proportional

text: to

text: q

text: q

text: The

text: minimality

text: assumption

text: for

text: q

text: q

text: implies

text: that

text: the

text: normals

text: 2◦

text: p

text: and

text: p

text: to

text: K

text: at

text: q

text: and

text: q

text: are

text: the

text: same

text: as

text: the

text: normals

text: to

text: T

text: at

text: q

text: and

text: q

text: respectively.

text: Then

text: the

text: pair

text: of

text: points

text: q

text: λq,

text: q

text: +λq

text: is

text: in

text: the

text: boundary

text: of

text: K

text: +λT

text: and

text: the

text: normals

text: to

text: K

text: λT

text: at

text: these

text: points

text: are

text: again

text: p

text: and

text: p.

text: Now

text: it

text: follows

text: that

text: q

text: λq,

text: q

text: λq

text: is

text: a

text: closed

text: billiard

text: trajectory

text: in

text: K

text: λT

text: of

text: length

text: ξ

text: (K)

text: λξ

text: (T

text: From

text: it

text: follows

text: T

text: T

text: that

text: this

text: trajectory

text: is

text: minimal.

text: Attempt

text: toward

text: the

text: non-symmetric

text: Mahler’s

text: conjecture

text: In

text: Mahler’s

text: conjecture

text: vol

text: K

text: vol

text: K

text: for

text: centrally

text: symmetric

text: convex

text: n-

text: n!

text: dimensional

text: K

text: was

text: reduced,

text: assuming

text: the

text: Viterbo

text: conjecture

text: on

text: symplectic

text: capacities,

text: to

text: proving

text: that

text: ξ

text: ◦(K)

text: K

text: which

text: is

text: true,

text: see

text: Theorem

text: in

text: the

text: previous

text: section.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: ELEMENTARY APPROACH TO. . . 8 Dropping the assumption of the central symmetry, the corresponding version of Mahler’s conjecture becomes (see [2]): · ◦ ≥ (n + 1)n+1 vol K vol K (n!)2 ⊂ R for convex bodies K n containing the origin in the interior. Again, assuming Viterbo’s conjecture, in order to deduce from it the non-symmetric Mahler conjecture, one would have to prove: (cid:18) (cid:19) ≥ (n + 1)n+1 1/n (4.1) ξ ◦(K) , K n! the right hand side being asymptotically e by Stirling’s formula. In fact, already for n = 2 it is easy to check by hand, or look at Theorem 4.1 below, that the sharp estimate is ≥ ξ ◦(K) 3, K(cid:18) (cid:19) while (4.1) gives the number 33 1/2 , 2 which is greater than 3. For higher dimensions, there also remains a gap between the actual lower bound for the billiard trajectory length and the bound needed to establish the non-symmetric Mahler conjecture, assuming the Viterbo conjecture. Namely, we are going to prove: ⊂ R Theorem 4.1. If K n is a convex body containing the origin in its interior then ≥ ξ ◦(K) 2 + 2/n, K and the bound is sharp. This theorem shows that the non-symmetric Mahler conjecture is out of reach of the billiard approach of [3]. Proof. We invoke Theorem 2.1 and consider a closed polygonal line P not ﬁtting into a smaller homothet of K. By the same theorem we can assume that P has vertices q , . . . , q ≤ 1 m with m n + 1. qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq11111111111111111111111111111111111111111111111111111111111111111 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq11111111111111111111111111111111111111111111111111111111111111111 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv33333333333333333333333333333333333333333333333333333333333333333 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq11111111111111111111111111111111111111111111111111111111111111111 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv33333333333333333333333333333333333333333333333333333333333333333 o qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq22222222222222222222222222222222222222222222222222222222222222222 o qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq22222222222222222222222222222222222222222222222222222222222222222 o qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq22222222222222222222222222222222222222222222222222222222222222222 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv11111111111111111111111111111111111111111111111111111111111111111 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv22222222222222222222222222222222222222222222222222222222222222222 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq33333333333333333333333333333333333333333333333333333333333333333 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv11111111111111111111111111111111111111111111111111111111111111111 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv22222222222222222222222222222222222222222222222222222222222222222 qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq33333333333333333333333333333333333333333333333333333333333333333 vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv11111111111111111111111111111111111111111111111111111111111111111 Fig. 5. Measuring the length of a directed Fig. 6. Replacing K with L. segment. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ELEMENTARY

text: APPROACH

text: TO.

text: Dropping

text: the

text: assumption

text: of

text: the

text: central

text: symmetry,

text: the

text: corresponding

text: version

text: of

text: Mahler’s

text: conjecture

text: becomes

text: (see

text: (n

text: vol

text: K

text: vol

text: K

text: R

text: for

text: convex

text: bodies

text: K

text: n

text: containing

text: the

text: origin

text: in

text: the

text: interior.

text: Again,

text: assuming

text: Viterbo’s

text: conjecture,

text: in

text: order

text: to

text: deduce

text: from

text: it

text: the

text: non-symmetric

text: Mahler

text: conjecture,

text: one

text: would

text: have

text: to

text: prove:

text: (n

text: ξ

text: ◦(K)

text: K

text: n!

text: the

text: right

text: hand

text: side

text: being

text: asymptotically

text: e

text: by

text: Stirling’s

text: formula.

text: In

text: fact,

text: already

text: for

text: n

text: it

text: is

text: easy

text: to

text: check

text: by

text: hand,

text: or

text: look

text: at

text: Theorem

text: below,

text: that

text: the

text: sharp

text: estimate

text: is

text: ξ

text: ◦(K)

text: while

text: gives

text: the

text: number

text: which

text: is

text: greater

text: than

text: For

text: higher

text: dimensions,

text: there

text: also

text: remains

text: a

text: gap

text: between

text: the

text: actual

text: lower

text: bound

text: for

text: the

text: billiard

text: trajectory

text: length

text: and

text: the

text: bound

text: needed

text: to

text: establish

text: the

text: non-symmetric

text: Mahler

text: conjecture,

text: assuming

text: the

text: Viterbo

text: conjecture.

text: Namely,

text: we

text: are

text: going

text: to

text: prove:

text: R

text: Theorem

text: If

text: K

text: n

text: is

text: a

text: convex

text: body

text: containing

text: the

text: origin

text: in

text: its

text: interior

text: then

text: ξ

text: ◦(K)

text: K

text: and

text: the

text: bound

text: is

text: sharp.

text: This

text: theorem

text: shows

text: that

text: the

text: non-symmetric

text: Mahler

text: conjecture

text: is

text: out

text: of

text: reach

text: of

text: the

text: billiard

text: approach

text: of

text: Proof.

text: We

text: invoke

text: Theorem

text: and

text: consider

text: a

text: closed

text: polygonal

text: line

text: P

text: not

text: ﬁtting

text: into

text: a

text: smaller

text: homothet

text: of

text: K.

text: By

text: the

text: same

text: theorem

text: we

text: can

text: assume

text: that

text: P

text: has

text: vertices

text: q

text: q

text: m

text: with

text: m

text: n

text: o

text: o

text: o

text: Fig.

text: Measuring

text: the

text: length

text: of

text: a

text: directed

text: Fig.

text: Replacing

text: K

text: with

text: L.

text: segment.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ELEMENTARY APPROACH TO. . . 10 (cid:54) extend this segment to a half-line beyond q it must leave L at q . Assuming q = q i+1 i+1 i i+1 (otherwise we have less points and the dimension drops) we obtain, in particular, that the point q can only lie in the relative interior of its respective F . i i We see that q is the projection of q onto F parallel to v . If we apply these projections i ∈ i+1 i i cyclically starting from q F and ending at the same point then we obtain a map that i i takes F into its relative interior and that is linear on F . Such a map has a unique i i ﬁxed point. So it follows that having chosen L with a cyclic order on its facets we can (cid:80) reconstruct the considered polygonal line P uniquely. Another way to show the uniqueness is to observe that the condition (4.2) implies n t v = 0 and therefore determines the t uniquely up to a positive multiple. Hence i=0 i i i the polygonal line P is determined uniquely up to translation and a positive homothety, ∈ and the additional property q F ﬁxes it completely. i i Now we are going to consider everything in barycentric coordinates. Let (m , . . . , m ) 0 n (cid:88) be the barycentric coordinate of the origin in L. Then it is not hard to express the q in i terms of the v . We are going to index everything cyclically modulo n + 1 and we put i M = m m . k l ≤ ≤ 0 k<l n From the Schur concavity of the elementary symmetric functions it follows that M takes · · · ≤ its maximum value at m = = m = 1 and therefore M n . We have already 0 n n+1 2n+2 shown the uniqueness of the q after the choice of the order of the projections along the i v to facets. It remains to guess the exp(cid:80)ress(cid:80)ion for q and prove that it gives the solution. i i Our guess is − j 1 m m v (cid:54) j k j j=i k=i q = , i M − − where the inner summation goes cyclically from i to j 1, so it is allowed that j 1 < i. First, it is easy to observe that the sum of all coeﬃcients in the numerator equals M, ∈ because every monomial m m is used precisely once. Therefore we have q F . Then −k l i i (cid:80) (cid:80) (cid:80) (cid:80) (cid:80) (cid:80) we express the vector q q : i+1 i − − j 1 − j 1 − (cid:80)− (cid:54) mjmk(cid:80)vj (cid:54) mjmkvj (cid:54) mimjvi (cid:80)(cid:54) mjmivj j=i+1 k=i+1 j=i k=i j=i j=i q q = = . i+1 i (cid:80) M M − Since m v = 0, we obtain m m v = m2v . And from m = 1 we get j j (cid:54) j i j i i j j j=i m m v + m2v = m v . Finally, (cid:54) i j i i i i i j=i − m v m i i i q q = and t = . (cid:88)i+1 (cid:88)i M i M Now we can bound the sum of t from below: i m 1 ≥ 2n + 2 i t = = . i M M n i i This means that the length of P in the norm with unit ball L is at least 2 + 2/n, and with all m equal this bound is actually attained. i Since it is possible to approximate L by a smooth body, whose polar is also smooth, keeping the trajectory and its length the same, we conclude that the bound is sharp even (cid:3) in the class of smooth bodies K with smooth polars. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ELEMENTARY

text: APPROACH

text: TO.

text: extend

text: this

text: segment

text: to

text: a

text: half-line

text: beyond

text: q

text: it

text: must

text: leave

text: L

text: at

text: q

text: Assuming

text: q

text: q

text: i

text: (otherwise

text: we

text: have

text: less

text: points

text: and

text: the

text: dimension

text: drops)

text: we

text: obtain,

text: in

text: particular,

text: that

text: the

text: point

text: q

text: can

text: only

text: lie

text: in

text: the

text: relative

text: interior

text: of

text: its

text: respective

text: F

text: i

text: i

text: We

text: see

text: that

text: q

text: is

text: the

text: projection

text: of

text: q

text: onto

text: F

text: parallel

text: to

text: v

text: If

text: we

text: apply

text: these

text: projections

text: i

text: i

text: i

text: cyclically

text: starting

text: from

text: q

text: F

text: and

text: ending

text: at

text: the

text: same

text: point

text: then

text: we

text: obtain

text: a

text: map

text: that

text: i

text: i

text: takes

text: F

text: into

text: its

text: relative

text: interior

text: and

text: that

text: is

text: linear

text: on

text: F

text: Such

text: a

text: map

text: has

text: a

text: unique

text: i

text: i

text: ﬁxed

text: point.

text: So

text: it

text: follows

text: that

text: having

text: chosen

text: L

text: with

text: a

text: cyclic

text: order

text: on

text: its

text: facets

text: we

text: can

text: reconstruct

text: the

text: considered

text: polygonal

text: line

text: P

text: uniquely.

text: Another

text: way

text: to

text: show

text: the

text: uniqueness

text: is

text: to

text: observe

text: that

text: the

text: condition

text: implies

text: n

text: t

text: v

text: and

text: therefore

text: determines

text: the

text: t

text: uniquely

text: up

text: to

text: a

text: positive

text: multiple.

text: Hence

text: i

text: i

text: i

text: the

text: polygonal

text: line

text: P

text: is

text: determined

text: uniquely

text: up

text: to

text: translation

text: and

text: a

text: positive

text: homothety,

text: and

text: the

text: additional

text: property

text: q

text: F

text: ﬁxes

text: it

text: completely.

text: i

text: i

text: Now

text: we

text: are

text: going

text: to

text: consider

text: everything

text: in

text: barycentric

text: coordinates.

text: Let

text: (m

text: m

text: n

text: be

text: the

text: barycentric

text: coordinate

text: of

text: the

text: origin

text: in

text: L.

text: Then

text: it

text: is

text: not

text: hard

text: to

text: express

text: the

text: q

text: in

text: i

text: terms

text: of

text: the

text: v

text: We

text: are

text: going

text: to

text: index

text: everything

text: cyclically

text: modulo

text: n

text: and

text: we

text: put

text: i

text: M

text: m

text: m

text: k

text: l

text: k<l

text: n

text: From

text: the

text: Schur

text: concavity

text: of

text: the

text: elementary

text: symmetric

text: functions

text: it

text: follows

text: that

text: M

text: takes

text: its

text: maximum

text: value

text: at

text: m

text: m

text: and

text: therefore

text: M

text: n

text: We

text: have

text: already

text: n

text: shown

text: the

text: uniqueness

text: of

text: the

text: q

text: after

text: the

text: choice

text: of

text: the

text: order

text: of

text: the

text: projections

text: along

text: the

text: i

text: v

text: to

text: facets.

text: It

text: remains

text: to

text: guess

text: the

text: for

text: q

text: and

text: prove

text: that

text: it

text: gives

text: the

text: solution.

text: i

text: i

text: Our

text: guess

text: is

text: j

text: m

text: m

text: v

text: j

text: k

text: j

text: j=i

text: k=i

text: q

text: i

text: M

text: where

text: the

text: inner

text: summation

text: goes

text: cyclically

text: from

text: i

text: to

text: j

text: so

text: it

text: is

text: allowed

text: that

text: j

text: i.

text: First,

text: it

text: is

text: easy

text: to

text: observe

text: that

text: the

text: sum

text: of

text: all

text: coeﬃcients

text: in

text: the

text: numerator

text: equals

text: M,

text: because

text: every

text: monomial

text: m

text: m

text: is

text: used

text: precisely

text: once.

text: Therefore

text: we

text: have

text: q

text: F

text: Then

text: −k

text: l

text: i

text: i

text: we

text: express

text: the

text: vector

text: q

text: q

text: i

text: j

text: j

text: mjmkvj

text: mimjvi

text: mjmivj

text: j=i

text: k=i

text: j=i

text: j=i

text: q

text: q

text: i

text: M

text: M

text: Since

text: m

text: v

text: we

text: obtain

text: m

text: m

text: v

text: And

text: from

text: m

text: we

text: get

text: j

text: j

text: j

text: i

text: j

text: i

text: i

text: j

text: j

text: j=i

text: m

text: m

text: v

text: m

text: v

text: Finally,

text: i

text: j

text: i

text: i

text: i

text: i

text: i

text: j=i

text: m

text: v

text: m

text: i

text: i

text: i

text: q

text: q

text: and

text: t

text: M

text: i

text: M

text: Now

text: we

text: can

text: bound

text: the

text: sum

text: of

text: t

text: from

text: below:

text: i

text: m

text: i

text: t

text: i

text: M

text: M

text: n

text: i

text: i

text: This

text: means

text: that

text: the

text: length

text: of

text: P

text: in

text: the

text: norm

text: with

text: unit

text: ball

text: L

text: is

text: at

text: least

text: and

text: with

text: all

text: m

text: equal

text: this

text: bound

text: is

text: actually

text: attained.

text: i

text: Since

text: it

text: is

text: possible

text: to

text: approximate

text: L

text: by

text: a

text: smooth

text: body,

text: whose

text: polar

text: is

text: also

text: smooth,

text: keeping

text: the

text: trajectory

text: and

text: its

text: length

text: the

text: same,

text: we

text: conclude

text: that

text: the

text: bound

text: is

text: sharp

text: even

text: in

text: the

text: class

text: of

text: smooth

text: bodies

text: K

text: with

text: smooth

text: polars.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##