text: 54 SIMONETTA ABENDA AND PETR G. GRINEVICH ∈ ∈ vertex V , r [k] and l [0, N ] (see Figures 10 and 15). Finally, the vertexes V irjl r irjNr (2) (3) have two edges which we label e and e ; irjl irjl ∈ (3) The families of matrices Eˆ(r) and vectors cˆ(r), r [k], as in Theorem 5.1 associated to the RREF matrix A representing [A]. (cid:80) Finally, Eθ((cid:126)t) = (eθ1((cid:126)t), . . . , eθn((cid:126)t)), where θj((cid:126)t) = l≥1 κljtl and (cid:126)t = (t1 = x, t2 = y, t3 = t, t4, . . . ) ≺ · · (cid:31) are the KP times, and , denotes the usual scalar product. In the statements of Theorems and in the Deﬁnitions of this section we do not explicitly mention such data. Figure 16. Left: the linear relations for the v.v.w. Φ(m)((cid:126)t) at white and black vertexes. irjl Right: the normalized v.v.w Φˆ(m)((cid:126)t) takes the same value at all the edges at a given black irjl (cid:48) vertex V and diﬀerent values at the edges at a given trivalent white vertex V . irjl irjl ∈ ∈ By construction, for each ﬁxed r [k] and l [0, N ] we may assign the vector E(r)[j ] to the r l vertical edge ending at the white vertex V . We now introduce a simple rule to assign a vector irjl at each horizontal edge in such a way that, if we move along the graph in the opposite direction w.r.t. its orientation along all edges except the pivot edges, where we move up, we obtain the (3) r-th row of the RREF matrix A when we reach the edge e . When we pass a black vertex, we ir multiply the vector of its left horizontal edge by the weight of the right horizontal edge. When we pass a white vertex, we sum up the vectors of its left and vertical vertexes. N In section , we associate consistent systems of vectors to any orientation of any network associated to the given point [A]. Deﬁnition 6.2. Vertex vectors (v.v.) and vacuum vertex wavefunction (v.v.w.) on N (cid:48) N (cid:48) . Let the soliton data and the notations be ﬁxed as in Remark 6.1. To each edge of , we T T ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: vertex

text: V

text: r

text: [k]

text: and

text: l

text: N

text: (see

text: Figures

text: and

text: Finally,

text: the

text: vertexes

text: V

text: irjl

text: r

text: irjNr

text: have

text: two

text: edges

text: which

text: we

text: label

text: e

text: and

text: e

text: irjl

text: irjl

text: The

text: families

text: of

text: matrices

text: Eˆ(r)

text: and

text: vectors

text: cˆ(r),

text: r

text: [k],

text: as

text: in

text: Theorem

text: associated

text: to

text: the

text: RREF

text: matrix

text: A

text: representing

text: [A].

text: Finally,

text: where

text: κljtl

text: and

text: x,

text: y,

text: t,

text: are

text: the

text: KP

text: times,

text: and

text: denotes

text: the

text: usual

text: scalar

text: product.

text: In

text: the

text: statements

text: of

text: Theorems

text: and

text: in

text: the

text: Deﬁnitions

text: of

text: this

text: section

text: we

text: do

text: not

text: explicitly

text: mention

text: such

text: data.

text: Figure

text: Left:

text: the

text: linear

text: relations

text: for

text: the

text: v.v.w.

text: at

text: white

text: and

text: black

text: vertexes.

text: irjl

text: Right:

text: the

text: normalized

text: v.v.w

text: takes

text: the

text: same

text: value

text: at

text: all

text: the

text: edges

text: at

text: a

text: given

text: black

text: irjl

text: vertex

text: V

text: and

text: diﬀerent

text: values

text: at

text: the

text: edges

text: at

text: a

text: given

text: trivalent

text: white

text: vertex

text: V

text: irjl

text: irjl

text: By

text: construction,

text: for

text: each

text: ﬁxed

text: r

text: [k]

text: and

text: l

text: N

text: we

text: may

text: assign

text: the

text: vector

text: E(r)[j

text: to

text: the

text: r

text: l

text: vertical

text: edge

text: ending

text: at

text: the

text: white

text: vertex

text: V

text: We

text: now

text: introduce

text: a

text: simple

text: rule

text: to

text: assign

text: a

text: vector

text: irjl

text: at

text: each

text: horizontal

text: edge

text: in

text: such

text: a

text: way

text: that,

text: if

text: we

text: move

text: along

text: the

text: graph

text: in

text: the

text: opposite

text: direction

text: w.r.t.

text: its

text: orientation

text: along

text: all

text: edges

text: except

text: the

text: pivot

text: edges,

text: where

text: we

text: move

text: up,

text: we

text: obtain

text: the

text: r-th

text: row

text: of

text: the

text: RREF

text: matrix

text: A

text: when

text: we

text: reach

text: the

text: edge

text: e

text: When

text: we

text: pass

text: a

text: black

text: vertex,

text: we

text: ir

text: multiply

text: the

text: vector

text: of

text: its

text: left

text: horizontal

text: edge

text: by

text: the

text: weight

text: of

text: the

text: right

text: horizontal

text: edge.

text: When

text: we

text: pass

text: a

text: white

text: vertex,

text: we

text: sum

text: up

text: the

text: vectors

text: of

text: its

text: left

text: and

text: vertical

text: vertexes.

text: N

text: In

text: section

text: we

text: associate

text: consistent

text: systems

text: of

text: vectors

text: to

text: any

text: orientation

text: of

text: any

text: network

text: associated

text: to

text: the

text: given

text: point

text: [A].

text: Deﬁnition

text: Vertex

text: vectors

text: (v.v.)

text: and

text: vacuum

text: vertex

text: wavefunction

text: (v.v.w.)

text: on

text: N

text: N

text: Let

text: the

text: soliton

text: data

text: and

text: the

text: notations

text: be

text: ﬁxed

text: as

text: in

text: Remark

text: To

text: each

text: edge

text: of

text: we

text: T

text: T

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

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text: ##LTFigure##

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text: ##LTFigure##

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text: 56 SIMONETTA ABENDA AND PETR G. GRINEVICH i.e. the sum of the contributions from the edges to the left and below the white vertex V ; irjl ∈ (3) (7) To the horizontal edge e at the white source vertex V , r [k], we associate the v.v. ir ir and the v.v.w. ≺ (cid:31) (47) E(3) = E(1) + E(2) , Φ(3)((cid:126)t) = E(3), E ((cid:126)t) . ir irj1 irj1 ir ir0 θ 1 1 1 1 Figure 17. Same as in Fig. 17 at the bivalent white vertex V [left] and at the irjNr pivot vertex V [right]. ir ∈ ∈ We illustrate Deﬁnition 6.2 in Figures 16 and 17. For any r [k], l [N ], Deﬁnition 6.2 r implies: (1) the value of the v.v. (respectively of the v.v.w.) on the edge to the right of each white vertex V is the sum of the v.v. (respectively of the v.v.w.) on the edges to the left irjl and below V ; irjl (2) the value of the v.v. (respectively of the v.v.w.) on the edges to the right of and above (cid:48) each black vertex V is the proportional to the value of the v.v. (respectively of the irjl (cid:48) v.v.w.) on the edge to the left of V . irjl Comparing Theorem 5.1 and Deﬁnition 6.2, it is not diﬃcult to prove that at the vertex V ir (3) E = A[r]. ir Lemma 6.1. Let the soliton data and the notations be ﬁxed as in Remark 6.1. Let the v.v. and v.v.w. be as in Deﬁnition 6.2. Then, ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: i.e.

label: list

text: the

text: sum

text: of

text: the

text: contributions

text: from

text: the

text: edges

text: to

text: the

text: left

text: and

text: below

text: the

text: white

text: vertex

text: V

text: irjl

text: To

text: the

text: horizontal

text: edge

text: e

text: at

text: the

text: white

text: source

text: vertex

text: V

text: r

text: [k],

text: we

text: associate

text: the

text: v.v.

text: ir

text: ir

text: and

text: the

text: v.v.w.

label: equation

text: E

text: ir

text: ir

text: θ

font: default

text: Figure

text: Same

label: caption

text: as

text: in

text: Fig.

text: at

text: the

text: bivalent

text: white

text: vertex

text: V

text: [left]

text: and

text: at

text: the

text: irjNr

text: pivot

text: vertex

text: V

text: [right].

text: ir

text: We

text: illustrate

text: Deﬁnition

text: in

text: Figures

text: and

text: For

text: any

text: r

text: [k],

text: l

text: [N

text: Deﬁnition

text: r

text: implies:

text: the

text: value

text: of

text: the

text: v.v.

text: (respectively

text: of

text: the

text: v.v.w.)

text: on

text: the

text: edge

text: to

text: the

text: right

text: of

text: each

text: white

text: vertex

text: V

text: is

text: the

text: sum

text: of

text: the

text: v.v.

text: (respectively

text: of

text: the

text: v.v.w.)

text: on

text: the

text: edges

text: to

text: the

text: left

text: irjl

text: and

text: below

text: V

text: irjl

text: the

text: value

text: of

text: the

text: v.v.

text: (respectively

text: of

text: the

text: v.v.w.)

text: on

text: the

text: edges

text: to

text: the

text: right

text: of

text: and

text: above

text: each

text: black

text: vertex

text: V

text: is

text: the

text: proportional

text: to

text: the

text: value

text: of

text: the

text: v.v.

text: (respectively

text: of

text: the

text: irjl

text: v.v.w.)

text: on

text: the

text: edge

text: to

text: the

text: left

text: of

text: V

text: irjl

text: Comparing

text: Theorem

text: and

text: Deﬁnition

text: it

text: is

text: not

text: diﬃcult

text: to

text: prove

text: that

text: at

text: the

text: vertex

text: V

text: ir

text: E

text: A[r].

text: ir

text: Lemma

text: Let

text: the

text: soliton

text: data

text: and

text: the

text: notations

text: be

text: ﬁxed

text: as

text: in

text: Remark

text: Let

text: the

text: v.v.

text: and

text: v.v.w.

text: be

text: as

text: in

text: Deﬁnition

text: Then,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 57 ∈ ∈ (cid:48) (1) for any r [k] and l [N ], the vertex vector at the edge joining V , V is r irjl irjl (cid:88)Nr (48) E(3) = cˆr Eˆ(r)[j ]; irjl js s s=l ∈ (3) (2) for any r [k] the vertex vector at the edge e is the r–th row of the RREF matrix ir ∈ representing [A] GrTNN(k, n) (3) (49) E = A[r], irjl and the v.v.w. is the r–th generator of the dressing transformation associated to the K soliton data ( , [A]). (50) Φ(3)((cid:126)t) = f(r)((cid:126)t). ir The proof of the Lemma easily follows comparing (48), (49) and (50) with (31), (36), (37) and (39) in Theorem 5.1 and with (41)–(47) in Deﬁnition 6.2. As a consequence, both the vertex vectors and vertex wavefunction associated above to the bipartite Le–diagram may be recursively computed using Theorem 5.1 starting from the last row of the Le–diagram which corresponds to the index r = k and moving up to r = 1. For simplicity, we illustrate the algorithm for the vertex vectors only (see also Figures 16 and 17 for the vertex wavefunction). ∈ Algorithm for the vertex vectors: For any r [k]: (1) If N = 0, i.e. all the boxes of the r–th line of the Le–diagram contain zeros, assign to r ≡ the white vertex V the vector Eˆ(r)[i ] Eˆ(k+1)[i ] and proceed to (3); r r r (2) Otherwise: (a) Start from the leftmost white vertex of the line, V and assign to the edge joining irjNr (cid:48) V and V the vertex vector irjNr irjNr E(3) = Eˆ(r)[j ] irjNr Nr

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: for

text: any

text: r

text: [k]

text: and

text: l

text: [N

text: the

text: vertex

text: vector

text: at

text: the

text: edge

text: joining

text: V

text: V

text: is

text: r

text: irjl

text: irjl

text: cˆr

text: Eˆ(r)[j

text: irjl

text: js

text: s

text: s=l

text: for

text: any

text: r

text: [k]

text: the

text: vertex

text: vector

text: at

text: the

text: edge

text: e

text: is

text: the

text: r–th

text: row

text: of

text: the

text: RREF

text: matrix

text: ir

text: representing

text: [A]

text: GrTNN(k,

text: n)

text: E

text: A[r],

text: irjl

text: and

text: the

text: v.v.w.

text: is

text: the

text: r–th

text: generator

text: of

text: the

text: dressing

text: transformation

text: associated

text: to

text: the

text: K

text: soliton

text: data

text: [A]).

text: ir

text: The

text: proof

text: of

text: the

text: Lemma

text: easily

text: follows

text: comparing

text: and

text: with

text: and

text: in

text: Theorem

text: and

text: with

text: in

text: Deﬁnition

text: As

text: a

text: consequence,

text: both

text: the

text: vertex

text: vectors

text: and

text: vertex

text: wavefunction

text: associated

text: above

text: to

text: the

text: bipartite

text: Le–diagram

text: may

text: be

text: recursively

text: computed

text: using

text: Theorem

text: starting

text: from

text: the

text: last

text: row

text: of

text: the

text: Le–diagram

text: which

text: corresponds

text: to

text: the

text: index

text: r

text: k

text: and

text: moving

text: up

text: to

text: r

text: For

text: simplicity,

text: we

text: illustrate

text: the

text: algorithm

text: for

text: the

text: vertex

text: vectors

text: only

text: (see

text: also

text: Figures

text: and

text: for

text: the

text: vertex

text: wavefunction).

text: Algorithm

text: for

text: the

text: vertex

text: vectors:

text: For

text: any

text: r

text: [k]:

text: If

text: N

text: i.e.

text: all

text: the

text: boxes

text: of

text: the

text: r–th

text: line

text: of

text: the

text: Le–diagram

text: contain

text: zeros,

text: assign

text: to

text: r

text: the

text: white

text: vertex

text: V

text: the

text: vector

text: Eˆ(r)[i

text: and

text: proceed

text: to

text: r

text: r

text: r

text: Otherwise:

text: (a)

text: Start

text: from

text: the

text: leftmost

text: white

text: vertex

text: of

text: the

text: line,

text: V

text: and

text: assign

text: to

text: the

text: edge

text: joining

text: irjNr

text: V

text: and

text: V

text: the

text: vertex

text: vector

text: irjNr

text: irjNr

text: Eˆ(r)[j

text: irjNr

text: Nr


text: 76 SIMONETTA ABENDA AND PETR G. GRINEVICH In Figure 24 we illustrate these relations at trivalent vertexes assuming that the incoming edges do not intersect the gauge boundary rays. E3 w3 w2w E2 SS 12 :: EE 3 3 == ww33(( EE 2 1 −−E E 12 )) E2 w2 w1 E1 SS 12 :: EE 3 3 == −+ww33 EE11 EE 2 2 == +−ww22 EE11 1E1 S 3 : E 3 = w3( E 2 + E 1 ) E3 w3 S 3 : E 3 = +w3 E1 E 2 = +w2 E1 l S l S S 2 S 2 3 S 3 S 1 1 Figure 24. The linear system at black and white vertexes. − For instance, if l belongs to the sector S , at the white vertex E = w (E E ), where E 1 3 3 2 1 j ∈ denotes the vector associated to the edge e , j [3], while at the black vertex E = w E and j 3 3 1 − E = w E . 2 2 1 According to our deﬁnition all components of vectors associated either to a boundary source ∈ N or to an isolated boundary sink are zero for any edge e . In this section we ﬁrst show that the components of E are rational expression in the weights e with subtraction-free denominator. As a consequence, the j–th component of E may be null e even if there exist paths starting at e and ending at b . j Proposition 7.1. Rational representation for the components of vectors E Given e N O ∈ N ( , , l), let e such that there is a path starting at e and ending at the boundary sink b . j Then (E ) is a rational expression in term of the weights on the graph with subtraction-free e j denominator. Proof. The proof of this statement is a straightforward adaptation of the proof of Lemma 4.4 in [52]. Let us ﬁx an edge e in the graph and a boundary sink b such that there exits at least one j directed path from e to b . Following [52], we proof the statement by induction on the number j of internal edges. If e is directly connected to b , nothing has to be proven. Otherwise, let V be the internal j vertex connected to b , and, following [52], pick an internal edge e at V , and construct a new j V ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: In

text: Figure

text: we

text: illustrate

text: these

text: relations

text: at

text: trivalent

text: vertexes

text: assuming

text: that

text: the

text: incoming

text: edges

text: do

text: not

text: intersect

text: the

text: gauge

text: boundary

text: rays.

text: SS

text: EE

text: EE

text: −−E

text: E

text: SS

text: EE

text: EE

text: S

text: E

text: E

text: E

text: S

text: E

text: E

text: l

text: S

text: l

text: S

text: S

text: S

text: S

text: S

text: Figure

text: The

text: linear

text: system

text: at

text: black

text: and

text: white

text: vertexes.

text: For

text: instance,

text: if

text: l

text: belongs

text: to

text: the

text: sector

text: S

text: at

text: the

text: white

text: vertex

text: E

text: w

text: (E

text: E

text: where

text: E

text: j

text: denotes

text: the

text: vector

text: associated

text: to

text: the

text: edge

text: e

text: j

text: while

text: at

text: the

text: black

text: vertex

text: E

text: w

text: E

text: and

text: j

text: E

text: w

text: E

text: According

text: to

text: our

text: deﬁnition

text: all

text: components

text: of

text: vectors

text: associated

text: either

text: to

text: a

text: boundary

text: source

text: N

text: or

text: to

text: an

text: isolated

text: boundary

text: sink

text: are

text: zero

text: for

text: any

text: edge

text: e

text: In

text: this

text: section

text: we

text: ﬁrst

text: show

text: that

text: the

text: components

text: of

text: E

text: are

text: rational

text: expression

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text: subtraction-free

text: denominator.

text: As

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text: consequence,

text: the

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text: b

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text: Proposition

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text: Proof.

text: The

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text: e

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text: proof

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text: statement

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text: the

text: number

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text: If

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text: b

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text: Otherwise,

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text: the

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text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 77 N (cid:48) (cid:48)(cid:48) network ˜ by erasing e , and adding a boundary sink b and a boundary source b respectively, V connected to the initial and the ﬁnal vertex of e by unit edges. Let us denote by E˜(a, b) the V N b-th component of the vector associated to the edge a in ˜ . N N To obtain from ˜ a network equivalent to , it is suﬃcient to add an additional edge from (cid:48) (cid:48)(cid:48) b to b with the weight q, which was the weight of e . V We have 3 possible conﬁgurations: (cid:48)(cid:48) (cid:48) (1) The vertex V is white and it is the initial vertex of e (see Figure 25). In this case E˜(b , b ) V (cid:48)(cid:48) and E˜(b , b ) are positive while E˜(e, b ) and E(e, b ) have opposite sign. Moreover, for j j j (cid:48) (cid:48) any edge a E˜(a, b ) = pE˜(a, b ). We assume here that b and b are suﬃciently close to j j each other. Therefore − (cid:48) (cid:48)(cid:48) − (cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) · · · E(e, b ) = E˜(e, b ) + E˜(e, b )qE˜(b , b ) E˜(e, b )qE˜(b , b )qE˜(b , b ) + j j j j (cid:48) (cid:48)(cid:48) − − E˜(e, b )qE˜(b , b ) E˜(e, b ) = E˜(e, bj) + 1 + qE˜(b(cid:48)(cid:48), b(cid:48))j = 1 + qE˜(b(cid:48)(cid:48)j, b(cid:48)) e e V q V e p V p bj bj b’ q b" Figure 25. (2) The vertex V is white and it is the ﬁnal vertex of e (see Figure 26). In this case E˜(e, b ) = V j (cid:48)(cid:48) (cid:48) (cid:48) (cid:48)(cid:48) 0, E˜(b , b ) is positive and E˜(e, b ) has the same sign as E(e, b ), and E˜(b , b ) = p. j j (cid:48) (cid:48) (cid:48)(cid:48) − (cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) · · · qpE˜(e, b ) E(e, bj) = E˜(e, b )qE˜(b , bj) E˜(e, b )qE˜(b , b )qE˜(b , bj) + = 1 + qE˜(b(cid:48)(cid:48), b(cid:48)) e e V q V e p V p b’ bj bj b" q Figure 26. (cid:48)(cid:48) (cid:48) (3) The vertex V is black and it is the ﬁnal vertex of e (see Figure 27) In this case E˜(b , b ) = V 0. All paths connecting e to b form two groups: the paths reaching b either through f j j v ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: N

text: network

text: by

text: erasing

text: e

text: and

text: adding

text: a

text: boundary

text: sink

text: b

text: and

text: a

text: boundary

text: source

text: b

text: respectively,

text: V

text: connected

text: to

text: the

text: initial

text: and

text: the

text: ﬁnal

text: vertex

text: of

text: e

text: by

text: unit

text: edges.

text: Let

text: us

text: denote

text: by

text: E˜(a,

text: b)

text: the

text: V

text: N

text: b-th

text: component

text: of

text: the

text: vector

text: associated

text: to

text: the

text: edge

text: a

text: in

text: N

text: N

text: To

text: obtain

text: from

text: a

text: network

text: equivalent

text: to

text: it

text: is

text: suﬃcient

text: to

text: add

text: an

text: additional

text: edge

text: from

text: b

text: to

text: b

text: with

text: the

text: weight

text: q,

text: which

text: was

text: the

text: weight

text: of

text: e

text: V

text: We

text: have

text: possible

text: conﬁgurations:

text: The

text: vertex

text: V

text: is

text: white

text: and

text: it

text: is

text: the

text: initial

text: vertex

text: of

text: e

text: (see

text: Figure

text: In

text: this

text: case

text: E˜(b

text: b

text: V

text: and

text: E˜(b

text: b

text: are

text: positive

text: while

text: E˜(e,

text: b

text: and

text: E(e,

text: b

text: have

text: opposite

text: sign.

text: Moreover,

text: for

text: j

text: j

text: j

text: any

text: edge

text: a

text: E˜(a,

text: b

text: pE˜(a,

text: b

text: We

text: assume

text: here

text: that

text: b

text: and

text: b

text: are

text: suﬃciently

text: close

text: to

text: j

text: j

text: each

text: other.

text: Therefore

text: E(e,

text: b

text: E˜(e,

text: b

text: E˜(e,

text: b

text: )qE˜(b

text: b

text: E˜(e,

text: b

text: )qE˜(b

text: b

text: )qE˜(b

text: b

text: j

text: j

text: j

text: j

text: E˜(e,

text: b

text: )qE˜(b

text: b

text: E˜(e,

text: b

text: E˜(e,

text: bj)

text: e

text: e

text: V

text: q

text: V

text: e

text: p

text: V

text: p

text: bj

text: bj

text: b’

text: q

text: b"

text: Figure

text: The

text: vertex

text: V

text: is

text: white

text: and

text: it

text: is

text: the

text: ﬁnal

text: vertex

text: of

text: e

text: (see

text: Figure

text: In

text: this

text: case

text: E˜(e,

text: b

text: V

text: j

text: E˜(b

text: b

text: is

text: positive

text: and

text: E˜(e,

text: b

text: has

text: the

text: same

text: sign

text: as

text: E(e,

text: b

text: and

text: E˜(b

text: b

text: p.

text: j

text: j

text: qpE˜(e,

text: b

text: E(e,

text: bj)

text: E˜(e,

text: b

text: )qE˜(b

text: bj)

text: E˜(e,

text: b

text: )qE˜(b

text: b

text: )qE˜(b

text: bj)

text: e

text: e

text: V

text: q

text: V

text: e

text: p

text: V

text: p

text: b’

text: bj

text: bj

text: b"

text: q

text: Figure

text: The

text: vertex

text: V

text: is

text: black

text: and

text: it

text: is

text: the

text: ﬁnal

text: vertex

text: of

text: e

text: (see

text: Figure

text: In

text: this

text: case

text: E˜(b

text: b

text: V

text: All

text: paths

text: connecting

text: e

text: to

text: b

text: form

text: two

text: groups:

text: the

text: paths

text: reaching

text: b

text: either

text: through

text: f

text: j

text: j

text: v

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 85 F F E2 E1 E2 E1 E2 f E1 E2 f E1 w2 f w1 w2−1 f w−11 w2 w1 w2−1 w−11 F F F= F EF 1= = fwE12E2 j0 i0 EE 12 == ww12−−11EE21 EF 2= = fwE2−11E2 j0 i0 EF 1= = −wf1EE22 j0 i0 EEF= 12 −== Fww12−−11EE12 EF 2= = fwE2−11E1 j0 i0 Figure 33. The consistency of the linear system in both orientations at trivalent black edges. F F E2 E1 E2 E1 E2w2f Ew11 Ew22−f1 wE−111 Fw2 f w1 Fw2−1 f w−11 F= F E 1 =w1(E 2 +F) j0 i0 EE 21 == ww21−−11EE12 E 2 =w2−1(E 1 −F) j0 i0 E 1 =w1(E 2 +F) j0 i0 EEF= 12 −==Fww12−−11EE12 E 2 =w2−1(E 1 +F) j0 i0 Figure 34. The consistency of the linear system in both orientations at trivalent white edges. − since E = A[i ] E , Eˆ = E and ei0 r i0 ei0 i0 (cid:15)(e ) = (cid:15) (e ) + (cid:15) (e ) = 1, i0 1 i0 2 i0 − since, either e has a + region to the left and the pair (e, l) is negatively oriented or it has a i0 (cid:3) region to the left and the pair (e, l) is positively oriented (see Figure 35). l l i i 0 0 Figure 35. The rule of the sign at e . i0 N O Let us now describe the change of signs if we change the orientation in ( , , l) along a closed Q Q simple path . divides the interior of the disk into two regions: we mark the external region 0 0 Q − to with a + and the internal region with . 0 (cid:54)∈ Q If e , then the edge lies in one such region and we 0 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: F

text: F

text: f

text: f

text: f

text: f

text: F

text: F

text: F=

text: F

text: EF

text: EE

text: EF

text: EF

text: EEF=

text: EF

text: Figure

text: The

text: consistency

text: of

text: the

text: linear

text: system

text: in

text: both

text: orientations

text: at

text: trivalent

text: black

text: edges.

text: F

text: F

text: f

text: f

text: F=

text: F

text: E

text: +F)

text: EE

text: E

text: −F)

text: E

text: +F)

text: EEF=

text: E

text: +F)

text: Figure

text: The

text: consistency

text: of

text: the

text: linear

text: system

text: in

text: both

text: orientations

text: at

text: trivalent

text: white

text: edges.

text: since

text: E

text: A[i

text: E

text: Eˆ

text: E

text: and

text: r

text: (e

text: (e

text: since,

text: either

text: e

text: has

text: a

text: region

text: to

text: the

text: left

text: and

text: the

text: pair

text: (e,

text: l)

text: is

text: negatively

text: oriented

text: or

text: it

text: has

text: a

text: region

text: to

text: the

text: left

text: and

text: the

text: pair

text: (e,

text: l)

text: is

text: positively

text: oriented

text: (see

text: Figure

text: l

text: l

text: i

text: i

text: Figure

text: The

text: rule

text: of

text: the

text: sign

text: at

text: e

text: N

text: O

text: Let

text: us

text: now

text: describe

text: the

text: change

text: of

text: signs

text: if

text: we

text: change

text: the

text: orientation

text: in

text: l)

text: along

text: a

text: closed

text: Q

text: Q

text: simple

text: path

text: divides

text: the

text: interior

text: of

text: the

text: disk

text: into

text: two

text: regions:

text: we

text: mark

text: the

text: external

text: region

text: Q

text: to

text: with

text: a

text: and

text: the

text: internal

text: region

text: with

text: Q

text: If

text: e

text: then

text: the

text: edge

text: lies

text: in

text: one

text: such

text: region

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text: we

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 89 1 1 1 1 1 1 Figure 37. The unicolored edge contraction/uncontraction Since we work with trivalent graphs, there are two cases of interest to us: the insertion/removal of a Darboux vertex next to a boundary vertex (see Figure 38) and the ﬂip move (see Figure 39). ℓ ℓ ℓ ℓ ℓ 0 0 A[r] Figure 38. The insertion/removal of a Darboux vertex next to a boundary sink [left] and to a boundary source [right]. The insertion/removal of a Darboux vertex leaves invariant the system of vectors except for (D) (D) the edges e = [b , V ], e = V , V , more precisely l l l l l l (D) (1) The insertion of a univalent white Darboux vertex V at the vertex V next to the js js ∈ − (D) (D) boundary sink b , s [n k], produces a null vector on the edge e joining V to js js js V and keeps invariant the vectors of the other edges at V ; js js (D) (2) At a boundary source b , we combine the insertion of the Darboux vertex V , the ir ir (D) (D) change of orientation of the edges e = [b , V ] and e = [V , V ] and the translation ir ir ir ir ir ir ∈ (D) of the boundary ray from b to V , r [k]. It is not restrictive to assume that the ir ir (D) vertexes b , V and V are so near to each other that moving the r–th boundary ir ir ir ray does not eﬀect its intersections with the rest of the graph and that there is no (D) intersection of boundary rays with e . Then the eﬀect of the composition of these ir ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: Figure

text: The

text: unicolored

text: edge

text: contraction/uncontraction

text: Since

text: we

text: work

text: with

text: trivalent

text: graphs,

text: there

text: are

text: two

text: cases

text: of

text: interest

text: to

text: us:

text: the

text: insertion/removal

text: of

text: a

text: Darboux

text: vertex

text: next

text: to

text: a

text: boundary

text: vertex

text: (see

text: Figure

text: and

text: the

text: ﬂip

text: move

text: (see

text: Figure

text: ℓ

text: ℓ

text: ℓ

text: ℓ

text: ℓ

text: A[r]

text: Figure

text: The

text: insertion/removal

text: of

text: a

text: Darboux

text: vertex

text: next

text: to

text: a

text: boundary

text: sink

text: [left]

text: and

text: to

text: a

text: boundary

text: source

text: [right].

text: The

text: insertion/removal

text: of

text: a

text: Darboux

text: vertex

text: leaves

text: invariant

text: the

text: system

text: of

text: vectors

text: except

text: for

text: (D)

text: (D)

text: the

text: edges

text: e

text: [b

text: V

text: e

text: V

text: V

text: more

text: precisely

text: l

text: l

text: l

text: l

text: l

text: l

text: (D)

text: The

text: insertion

text: of

text: a

text: univalent

text: white

text: Darboux

text: vertex

text: V

text: at

text: the

text: vertex

text: V

text: next

text: to

text: the

text: js

text: js

text: (D)

text: (D)

text: boundary

text: sink

text: b

text: s

text: [n

text: k],

text: produces

text: a

text: vector

text: on

text: the

text: edge

text: e

text: joining

text: V

text: to

text: js

text: js

text: js

text: V

text: and

text: keeps

text: invariant

text: the

text: vectors

text: of

text: the

text: other

text: edges

text: at

text: V

text: js

text: js

text: (D)

text: At

text: a

text: boundary

text: source

text: b

text: we

text: combine

text: the

text: insertion

text: of

text: the

text: Darboux

text: vertex

text: V

text: the

text: ir

text: ir

text: (D)

text: (D)

text: change

text: of

text: orientation

text: of

text: the

text: edges

text: e

text: [b

text: V

text: and

text: e

text: [V

text: V

text: and

text: the

text: translation

text: ir

text: ir

text: ir

text: ir

text: ir

text: ir

text: (D)

text: of

text: the

text: boundary

text: ray

text: from

text: b

text: to

text: V

text: r

text: [k].

text: It

text: is

text: not

text: restrictive

text: to

text: assume

text: that

text: the

text: ir

text: ir

text: (D)

text: vertexes

text: b

text: V

text: and

text: V

text: are

text: so

text: near

text: to

text: each

text: other

text: that

text: moving

text: the

text: r–th

text: boundary

text: ir

text: ir

text: ir

text: ray

text: does

text: not

text: eﬀect

text: its

text: intersections

text: with

text: the

text: rest

text: of

text: the

text: graph

text: and

text: that

text: there

text: is

text: no

text: (D)

text: intersection

text: of

text: boundary

text: rays

text: with

text: e

text: Then

text: the

text: eﬀect

text: of

text: the

text: composition

text: of

text: these

text: ir

text: ##LTFigure##

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text: KP THEORY, TOTAL NON–NEGATIVITY AND M–CURVES 95 − D − D a degree g +n k vacuum divisor N(cid:48) and a degree g +n k dressed divisor N(cid:48) to vac, dr, N (cid:48) and characterize its properties w.r.t. changes of gauge ray direction and orientation in the graph; (3) The vacuum divisor and the vacuum wavefunction on Γ The vacuum divisor D D − on Γ is obtained from N(cid:48) eliminating the n k Darboux sink points. By vac,Γ vac, D M construction depends on the base in the matroid . The normalized vacuum vac,Γ wavefunction φˆ on Γ is obtained imposing that it coincides for all times at each marked point with the value of the normalized vacuum vertex wavefunction at the corresponding N (cid:48) ≤ edge in . By construction it is meromorphic of degree d g and its eﬀective pole D divisor is ; vac,Γ (4) The KP divisor and the KP wavefunction on Γ By construction the KP divisor D D K = ( , [A]) is contained in the union of the ovals of Γ and it is obtained from KP,Γ KP,Γ D N(cid:48) eliminating the n Darboux sink and source divisor points and adding the degree dr, k Sato divisor; M (5) The invariance of the KP divisor with respect to changes of base in . On CP N N each component Γ = 1 of Γ( ) corresponding to a trivalent white vertex V of , we l l have four real ordered marked points corresponding to the edges at V and to the dressed l N point divisor. A change of orientation of induces a change in the local coordinate used on Γ , that is it leaves invariant the order and the position of the four points on Γ . l l N O To obtain a consistent system of linear equations on the whole network ( , , l), such that ∈ at each edge ending at a boundary vertex b , s [n], the vector E is the s–th vector of the s bs N (cid:48) O canonical basis, we generalize the construction in Section and consider the network ( , , l) where: N (cid:48) N (1) is the network obtained from adding an edge with unit weight and a white vertex ∈ (D) V at each bivalent white vertex V associated to the boundary vertex b , s [n] (see s s s Figure 38); O N (cid:48) (2) is the orientation of such that ∈ (D) (D) (a) For any s [n], the edge joining V and V points towards V if b is a boundary s s s s sink, otherwise it points towards V ; s ∈ (b) The edge joining V to the bounndary source b points towards b for any r [k]; ir ir ir N O (c) All other edges are oriented as in ( , );

text: KP

text: THEORY,

text: TOTAL

text: NON–NEGATIVITY

text: AND

text: M–CURVES

text: D

text: D

text: a

text: degree

text: g

text: +n

text: k

text: vacuum

text: divisor

text: and

text: a

text: degree

text: g

text: +n

text: k

text: dressed

text: divisor

text: to

text: vac,

text: dr,

text: N

text: and

text: characterize

text: its

text: properties

text: w.r.t.

text: changes

text: of

text: gauge

text: ray

text: direction

text: and

text: orientation

text: in

text: the

text: graph;

text: The

text: vacuum

text: divisor

text: and

text: the

text: vacuum

text: wavefunction

text: on

text: Γ

text: The

text: vacuum

text: divisor

text: D

text: D

text: on

text: Γ

text: is

text: obtained

text: from

text: eliminating

text: the

text: n

text: k

text: Darboux

text: sink

text: points.

text: By

text: vac,Γ

text: vac,

text: D

text: M

text: construction

text: depends

text: on

text: the

text: base

text: in

text: the

text: matroid

text: The

text: normalized

text: vacuum

text: vac,Γ

text: wavefunction

text: φˆ

text: on

text: Γ

text: is

text: obtained

text: imposing

text: that

text: it

text: coincides

text: for

text: all

text: times

text: at

text: each

text: marked

text: point

text: with

text: the

text: value

text: of

text: the

text: normalized

text: vacuum

text: vertex

text: wavefunction

text: at

text: the

text: corresponding

text: N

text: edge

text: in

text: By

text: construction

text: it

text: is

text: meromorphic

text: of

text: degree

text: d

text: g

text: and

text: its

text: eﬀective

text: pole

text: D

text: divisor

text: is

text: vac,Γ

text: The

text: KP

text: divisor

text: and

text: the

text: KP

text: wavefunction

text: on

text: Γ

text: By

text: construction

text: the

text: KP

text: divisor

text: D

text: D

text: K

text: [A])

text: is

text: contained

text: in

text: the

text: union

text: of

text: the

text: ovals

text: of

text: Γ

text: and

text: it

text: is

text: obtained

text: from

text: KP,Γ

text: KP,Γ

text: D

text: eliminating

text: the

text: n

text: Darboux

text: sink

text: and

text: source

text: divisor

text: points

text: and

text: adding

text: the

text: degree

text: dr,

text: k

text: Sato

text: divisor;

text: M

text: The

text: invariance

text: of

text: the

text: KP

text: divisor

text: with

text: respect

text: to

text: changes

text: of

text: base

text: in

text: On

text: CP

text: N

text: N

text: each

text: component

text: Γ

text: of

text: Γ(

text: corresponding

text: to

text: a

text: trivalent

text: white

text: vertex

text: V

text: of

text: we

text: l

text: l

text: have

text: four

text: real

text: ordered

text: marked

text: points

text: corresponding

text: to

text: the

text: edges

text: at

text: V

text: and

text: to

text: the

text: dressed

text: l

text: N

text: point

text: divisor.

text: A

text: change

text: of

text: orientation

text: of

text: induces

text: a

text: change

text: in

text: the

text: local

text: coordinate

text: used

text: on

text: Γ

text: that

text: is

text: it

text: leaves

text: invariant

text: the

text: order

text: and

text: the

text: position

text: of

text: the

text: four

text: points

text: on

text: Γ

text: l

text: l

text: N

text: O

text: To

text: obtain

text: a

text: consistent

text: system

text: of

text: linear

text: equations

text: on

text: the

text: whole

text: network

text: l),

text: such

text: that

text: at

text: each

text: edge

text: ending

text: at

text: a

text: boundary

text: vertex

text: b

text: s

text: [n],

text: the

text: vector

text: E

text: is

text: the

text: s–th

text: vector

text: of

text: the

text: s

text: bs

text: N

text: O

text: canonical

text: basis,

text: we

text: generalize

text: the

text: construction

text: in

text: Section

text: and

text: consider

text: the

text: network

text: l)

text: where:

text: N

text: N

text: is

text: the

text: network

text: obtained

text: from

text: adding

text: an

text: edge

text: with

text: unit

text: weight

text: and

text: a

text: white

text: vertex

text: (D)

text: V

text: at

text: each

text: bivalent

text: white

text: vertex

text: V

text: associated

text: to

text: the

text: boundary

text: vertex

text: b

text: s

text: [n]

text: (see

text: s

text: s

text: s

text: Figure

text: O

text: N

text: is

text: the

text: orientation

text: of

text: such

text: that

text: (D)

text: (D)

text: (a)

text: For

text: any

text: s

text: [n],

text: the

text: edge

text: joining

text: V

text: and

text: V

text: points

text: towards

text: V

text: if

text: b

text: is

text: a

text: boundary

text: s

text: s

text: s

text: s

text: sink,

text: otherwise

text: it

text: points

text: towards

text: V

text: s

text: (b)

text: The

text: edge

text: joining

text: V

text: to

text: the

text: bounndary

text: source

text: b

text: points

text: towards

text: b

text: for

text: any

text: r

text: [k];

text: ir

text: ir

text: ir

text: N

text: O

text: (c)

text: All

text: other

text: edges

text: are

text: oriented

text: as

text: in


text: 100 SIMONETTA ABENDA AND PETR G. GRINEVICH (cid:16) (cid:17) − − − (86) Φe3,O,l((cid:126)t) = (−1)int(e3) w3 (cid:16)(−1)wind(e3,e1)Φe1,O,l((cid:126)t) + (−1)wind(e3,e2) Φe2,O,l((cid:126)t) (cid:17), Ψe3,O,l((cid:126)t) = ( 1)int(e3) w3 ( 1)wind(e3,e1)Ψe1,O,l((cid:126)t) + ( 1)wind(e3,e2) Ψe2,O,l((cid:126)t) ; (3) At any trivalent black vertex incident with the incoming edges e , e and the outcoming 2 3 edge e , 1 − Φ O ((cid:126)t) = ( 1)int(em)+wind(em,e1) w Φ O ((cid:126)t), m = 2, 3, em, ,l m e1, ,l − Ψ O ((cid:126)t) = ( 1)int(em)+wind(em,e1) w Ψ O ((cid:126)t), m = 2, 3. em, ,l m e1, ,l In view of the above we associate a vacuum network divisor and dressed network divisor to N (cid:48) O ( , , l) as follows. D Deﬁnition 9.2. The vacuum network divisor N(cid:48) and the dressed network divisor vac, D N (cid:48) O N(cid:48). At each trivalent white vertex V of ( , , l), let Φ O ((cid:126)t) be the vacuum v.w. on the dr, em, ,l ∈ edge e , m [3], deﬁned above. Then we assign the following vacuum network divisor number m γ to V : vac,V − (87) γvac,V = (−1)wind(e3,e1()Φe11),wOi,ln((cid:126)dt0()e3+,e1()−Φe11),wOi,ln((cid:126)dt0()e3,e2)Φe2,O,l((cid:126)t0), where wind(e , e ) is the winding number of the directed path [e , e ], i = 1, 2. 3 i 3 i D { ∈ − } N (cid:48) We call N(cid:48) = γ , l [g + n k] the vacuum network divisor on , where V , vac, vac,Vl l ∈ − l [g + n k], are the trivalent white vertices of the network. N (cid:48) O At each trivalent white vertex V of ( , , l), let Ψ O ((cid:126)t) be the dressed v.w. on the edge em, ,l ∈ e , m [3], deﬁned above. Then, to V we assign the dressed network divisor number m − (88) γdr,V = (−1)wind(e3,e1()Ψ1e1),wOi,ln((cid:126)dt0(e)3+,e1()−Ψ1e1),wOi,ln((cid:126)dt0(e)3,e2)Ψe2,O,l((cid:126)t0), with wind(e , e ) as above. If V is a trivalent vertex, associated to a Darboux source, the 3 i ∞ denominator in (88) is zero, and we set γ = . dr,V ##LTLine## ##LTLine##

text: SIMONETTA

text: ABENDA

text: AND

text: PETR

text: G.

text: GRINEVICH

text: At

text: any

text: trivalent

text: black

text: vertex

text: incident

text: with

text: the

text: incoming

text: edges

text: e

text: e

text: and

text: the

text: outcoming

text: edge

text: e

text: Φ

text: O

text: w

text: Φ

text: O

text: m

text: em,

text: ,l

text: m

text: ,l

text: Ψ

text: O

text: w

text: Ψ

text: O

text: m

text: em,

text: ,l

text: m

text: ,l

text: In

text: view

text: of

text: the

text: above

text: we

text: associate

text: a

text: vacuum

text: network

text: divisor

text: and

text: dressed

text: network

text: divisor

text: to

text: N

text: O

text: l)

text: as

text: follows.

text: D

text: Deﬁnition

text: The

text: vacuum

text: network

text: divisor

text: and

text: the

text: dressed

text: network

text: divisor

text: vac,

text: D

text: N

text: O

text: At

text: each

text: trivalent

text: white

text: vertex

text: V

text: of

text: l),

text: let

text: Φ

text: O

text: be

text: the

text: vacuum

text: v.w.

text: on

text: the

text: dr,

text: em,

text: ,l

text: edge

text: e

text: m

text: deﬁned

text: above.

text: Then

text: we

text: assign

text: the

text: following

text: vacuum

text: network

text: divisor

text: number

text: m

text: γ

text: to

text: V

text: vac,V

text: γvac,V

text: where

text: wind(e

text: e

text: is

text: the

text: winding

text: number

text: of

text: the

text: directed

text: path

text: [e

text: e

text: i

text: i

text: i

text: D

text: N

text: We

text: call

text: γ

text: l

text: [g

text: n

text: k]

text: the

text: vacuum

text: network

text: divisor

text: on

text: where

text: V

text: vac,

text: vac,Vl

text: l

text: l

text: [g

text: n

text: k],

text: are

text: the

text: trivalent

text: white

text: vertices

text: of

text: the

text: network.

text: N

text: O

text: At

text: each

text: trivalent

text: white

text: vertex

text: V

text: of

text: l),

text: let

text: Ψ

text: O

text: be

text: the

text: dressed

text: v.w.

text: on

text: the

text: edge

text: em,

text: ,l

text: e

text: m

text: deﬁned

text: above.

text: Then,

text: to

text: V

text: we

text: assign

text: the

text: dressed

text: network

text: divisor

text: number

text: m

text: γdr,V

text: with

text: wind(e

text: e

text: as

text: above.

text: If

text: V

text: is

text: a

text: trivalent

text: vertex,

text: associated

text: to

text: a

text: Darboux

text: source,

text: the

text: i

text: denominator

text: in

text: is

text: zero,

text: and

text: we

text: set

text: γ

text: dr,V

text: ##LTLine##

text: ##LTLine##


text: Figure 7: True value and estimated value for the parameters in the model, where data is generated for 20 locations and 5 years (refer to Equation (3.6)). The choices for φ were (0.001, 0.005, 0.01, 0.05, 0.1, 0.3, 0.5) while s the same for φ were (0.5, 0.75, 1, 1.5). We used 80% of the data as our training data and t the rest 20% were used for validation purposes in this regard. We found that the combination of φ = 0.01 and φ = 1 was the best one for the data at s t hand. To put it into the perspective of the actual spatial and temporal scale, we can say that these choices correspond to a signiﬁcant correlation in an approximate range of 300 kilometers and in a time span of 3 weeks, respectively. On the other hand, the LM test 21 ##LTFigure##

text: Figure

text: True

text: value

text: and

text: estimated

text: value

text: for

text: the

text: parameters

text: in

text: the

text: model,

text: where

text: data

text: is

text: generated

text: for

text: locations

text: and

text: years

text: (refer

text: to

text: Equation

text: The

text: choices

text: for

text: φ

text: were

text: while

text: s

text: the

text: same

text: for

text: φ

text: were

text: We

text: used

text: of

text: the

text: data

text: as

text: our

text: training

text: data

text: and

text: t

text: the

text: rest

text: were

text: used

text: for

text: validation

text: purposes

text: in

text: this

text: regard.

text: We

text: found

text: that

text: the

text: combination

text: of

text: φ

text: and

text: φ

text: was

text: the

text: best

text: one

text: for

text: the

text: data

text: at

text: s

text: t

text: hand.

text: To

text: put

text: it

text: into

text: the

text: perspective

text: of

text: the

text: actual

text: spatial

text: and

text: temporal

text: scale,

text: we

text: can

text: say

text: that

text: these

text: choices

text: correspond

text: to

text: a

text: signiﬁcant

text: correlation

text: in

text: an

text: approximate

text: range

text: of

text: kilometers

text: and

text: in

text: a

text: time

text: span

text: of

text: weeks,

text: respectively.

text: On

text: the

text: other

text: hand,

text: the

text: LM

text: test

text: ##LTFigure##


text: Figure 8: Estimates of the space-time interaction terms for diﬀerent regions capture the eﬀects of Taiwan PM data. We evaluate the residuals after ﬁtting the model 2.5 and plot them using standard statistical procedures. First, the top panel in Figure 10 shows that there is no particular pattern in the residuals, thereby establishing that the residuals can be assumed to be uncorrelated. Next, the plot of the residuals corresponding to diﬀerent months are presented in the bottom panel of the same ﬁgure. It shows that the variances are more or less evenly distributed across the months, thereby showing that the issue of heteroskedasticity has been taken care of. Furthermore, the left panel of Figure 11 shows that the histogram is approximately nor- 24 ##LTFigure##

text: Figure

text: Estimates

text: of

text: the

text: space-time

text: interaction

text: terms

text: for

text: diﬀerent

text: regions

text: capture

text: the

text: eﬀects

text: of

text: Taiwan

text: PM

text: data.

text: We

text: evaluate

text: the

text: residuals

text: after

text: ﬁtting

text: the

text: model

text: and

text: plot

text: them

text: using

text: standard

text: statistical

text: procedures.

text: First,

text: the

text: top

text: panel

text: in

text: Figure

text: shows

text: that

text: there

text: is

text: no

text: particular

text: pattern

text: in

text: the

text: residuals,

text: thereby

text: establishing

text: that

text: the

text: residuals

text: can

text: be

text: assumed

text: to

text: be

text: uncorrelated.

text: Next,

text: the

text: plot

text: of

text: the

text: residuals

text: corresponding

text: to

text: diﬀerent

text: months

text: are

text: presented

text: in

text: the

text: bottom

text: panel

text: of

text: the

text: same

text: ﬁgure.

text: It

text: shows

text: that

text: the

text: variances

text: are

text: more

text: or

text: less

text: evenly

text: distributed

text: across

text: the

text: months,

text: thereby

text: showing

text: that

text: the

text: issue

text: of

text: heteroskedasticity

text: has

text: been

text: taken

text: care

text: of.

text: Furthermore,

text: the

text: left

text: panel

text: of

text: Figure

text: shows

text: that

text: the

text: histogram

text: is

text: approximately

text: nor-

text: ##LTFigure##


text: particular, we consider a separable structure for the spatial and temporal dependence, and that condition can be relaxed to address a more general setup. Moreover, in the aforementioned data analysis example, it was found that maximum wind speed at every location plays an important role in the air pollution, while there is signiﬁcant space-time interaction eﬀect as well. Combined, they raise an important question - how much eﬀect does the wind ﬂux, calculated from the wind speed, wind direction and the coordinates of diﬀerent locations, have on the pollution? In order to address that, it will be necessary to know about the physical behavior of the wind, and that process can be incorporated in the model to develop more eﬃcient techniques. Another important aspect of this topic is to identify the eﬀect of air pollution on human life. Pollution is responsible for many respiratory and cardiovascular diseases, and it is immensely important to use appropriate modeling techniques for such problems. It has the potential to impact health sciences signiﬁcantly. 6 Appendix Proof of Theorem 3.1. Note that the set-up of our problem is similar to a generalized ∼ least squares (GLS) problem, where Y = Xθ + ε, such that ε N(0, σ2Ω). Following our previous notations, Ω = (Σ + D), where D is a diagonal matrix with diagonal elements v equal to some τ2. j ˆ Now, for proving the required result, we deﬁne three diﬀerent estimators of θ. Below, θ is ˆ the estimator we are considering in this study, θ denotes the usual GLS estimator, and G ˆ θ is a feasible GLS estimator. F (cid:48) − − − (cid:48) − − θˆ = (X Ωˆ 1/2W Ωˆ 1/2X) 1(X Ωˆ 1/2W Ωˆ 1/2Y ) (cid:48) − − (cid:48) − θˆ = (X Ω 1X) 1(X Ω 1Y ) G (cid:48) − − (cid:48) − θˆ = (X Ωˆ 1X) 1(X Ωˆ 1Y ) F ˆ In the above, W is the weight matrix as deﬁned in ?? and Ω is our estimate of the covari- ance matrix. For co√nvenience, as use √N = nT hereafter. Following Baltagi (2011, Chapter − − ˆ ˆ 9), we know that N(θ θ) and N(θ θ) have the same asymptotic distribution G F − (cid:48) − → ∞ (cid:48) − − − −→ N(0, σ2Q 1), where Q = lim(X Ω 1X/N) as N , if X (Ωˆ 1 Ω 1)X/N P 0 and (cid:48) − − − −→ X (Ωˆ 1 Ω 1)ε/N P 0. Further, a suﬃcient condition for this to hold is that Ωˆ is a consistent estimator for Ω and that X has a satisfactory limiting behavior. 28 ##LTLine## ##LTLine##

text: particular,

text: we

text: consider

text: a

text: separable

text: structure

text: for

text: the

text: spatial

text: and

text: temporal

text: dependence,

text: and

text: that

text: condition

text: can

text: be

text: relaxed

text: to

text: address

text: a

text: more

text: general

text: setup.

text: Moreover,

text: in

text: the

text: aforementioned

text: data

text: analysis

text: example,

text: it

text: was

text: found

text: that

text: maximum

text: wind

text: speed

text: at

text: every

text: location

text: plays

text: an

text: important

text: role

text: in

text: the

text: air

text: pollution,

text: while

text: there

text: is

text: signiﬁcant

text: space-time

text: interaction

text: eﬀect

text: as

text: well.

text: Combined,

text: they

text: raise

text: an

text: important

text: question

text: how

text: much

text: eﬀect

text: does

text: the

text: wind

text: ﬂux,

text: calculated

text: from

text: the

text: wind

text: speed,

text: wind

text: direction

text: and

text: the

text: coordinates

text: of

text: diﬀerent

text: locations,

text: have

text: on

text: the

text: pollution?

text: In

text: order

text: to

text: address

text: that,

text: it

text: will

text: be

text: necessary

text: to

text: know

text: about

text: the

text: physical

text: behavior

text: of

text: the

text: wind,

text: and

text: that

text: process

text: can

text: be

text: incorporated

text: in

text: the

text: model

text: to

text: develop

text: more

text: eﬃcient

text: techniques.

text: Another

text: important

text: aspect

text: of

text: this

text: topic

text: is

text: to

text: identify

text: the

text: eﬀect

text: of

text: air

text: pollution

text: on

text: human

text: life.

text: Pollution

text: is

text: responsible

text: for

text: many

text: respiratory

text: and

text: cardiovascular

text: diseases,

text: and

text: it

text: is

text: immensely

text: important

text: to

text: use

text: appropriate

text: modeling

text: techniques

text: for

text: such

text: problems.

text: It

text: has

text: the

text: potential

text: to

text: impact

text: health

text: sciences

text: signiﬁcantly.

text: Appendix

text: Proof

text: of

text: Theorem

text: Note

text: that

text: the

text: set-up

text: of

text: our

text: problem

text: is

text: similar

text: to

text: a

text: generalized

text: least

text: squares

text: (GLS)

text: problem,

text: where

text: Y

text: Xθ

text: ε,

text: such

text: that

text: ε

text: Following

text: our

text: previous

text: notations,

text: Ω

text: (Σ

text: D),

text: where

text: D

text: is

text: a

text: diagonal

text: matrix

text: with

text: diagonal

text: elements

text: v

text: equal

text: to

text: some

text: j

text: ˆ

text: Now,

text: for

text: proving

text: the

text: required

text: result,

text: we

text: deﬁne

text: three

text: diﬀerent

text: estimators

text: of

text: θ.

text: Below,

text: θ

text: is

text: ˆ

text: the

text: estimator

text: we

text: are

text: considering

text: in

text: this

text: study,

text: θ

text: denotes

text: the

text: usual

text: GLS

text: estimator,

text: and

text: G

text: ˆ

text: θ

text: is

text: a

text: feasible

text: GLS

text: estimator.

text: F

text: θˆ

text: (X

text: Ωˆ

text: Ωˆ

text: Ωˆ

text: Ωˆ

text: θˆ

text: (X

text: Ω

text: Ω

text: G

text: θˆ

text: (X

text: Ωˆ

text: Ωˆ

text: F

text: ˆ

text: In

text: the

text: above,

text: W

text: is

text: the

text: weight

text: matrix

text: as

text: deﬁned

text: in

text: and

text: Ω

text: is

text: our

text: estimate

text: of

text: the

text: covari-

text: ance

text: matrix.

text: For

text: co√nvenience,

text: as

text: use

text: √N

text: nT

text: hereafter.

text: Following

text: Baltagi

text: Chapter

text: ˆ

text: ˆ

text: we

text: know

text: that

text: N(θ

text: θ)

text: and

text: N(θ

text: θ)

text: have

text: the

text: same

text: asymptotic

text: distribution

text: G

text: F

text: where

text: Q

text: lim(X

text: Ω

text: as

text: N

text: if

text: X

text: (Ωˆ

text: Ω

text: P

text: and

text: X

text: (Ωˆ

text: Ω

text: P

text: Further,

text: a

text: suﬃcient

text: condition

text: for

text: this

text: to

text: hold

text: is

text: that

text: Ωˆ

text: is

text: a

text: consistent

text: estimator

text: for

text: Ω

text: and

text: that

text: X

text: has

text: a

text: satisfactory

text: limiting

text: behavior.

text: ##LTLine##

text: ##LTLine##


text: remarks and the scopes of future work. 2 Preliminary Analysis 2.1 Data The PM data we analyze are collected from 71 oﬃcial monitoring stations across Tai- 2.5 wan. However, for ﬁve of those stations, we do not have the necessary information on the covariates considered and so, we decide to exclude them from this study. The remaining 66 monitoring stations are irregularly located in space, spread over the whole Island with some concentrations around big cities or industrial areas. The minimum and maximum of the pairwise distances of these stations are 0.58 kilometers and 366.7 kilometers re- spectively while the arithmatic mean of the same is 140.7 kilometers. One major talking point of our study is that we consider the issue of space-time interaction as a property of a region, rather than that of every individual station. For that, we divide the data into several clusters, based on the latitude and longitude of the stations. More on this is discussed in the following sections. Figure 1 shows the exact locations of the 66 stations. The concentration of the stations on the west coast is understandable because it is the most populated area of the Island. Diﬀerent colors of the map indicate diﬀerent regions (clusters) in our study. Turn to the temporal scale of our study. The data are obtained on an hourly basis for ten years, starting from January 1, 2006 to December 31, 2015. However, the sampling frequencies vary from station to station and for some sites, there are missing values. More- over, as we are mainly interested in identifying space-time interaction for the air pollution data, lower temporal resolution is desired and simpler. Thus, in this analysis, we decide to aggregate the hourly data into weekly averages based on all available measurements within a week. It is worth mentioning that this is a common practice while dealing with monitoring data, as discussed in Smith et al. (2003). In order to maintain continuity in the time scale, we consider the whole set of 3652 days (from 2006 to 2015) together and divide it into 522 weeks. Hence, the total number of data points considered in this study × is 66 522 = 34452. Throughout the study, whenever needed, we would use one year’s × (2015) data, i.e. 52 66 = 3432 points (approximately the last 10% for each site), for validation purposes. A detailed discussion on this is presented in the following sections. 5

text: remarks

text: and

text: the

text: scopes

text: of

text: future

text: work.

label: section

text: Preliminary

text: Analysis

text: Data

text: The

text: PM

text: data

text: we

text: analyze

text: are

text: collected

text: from

text: oﬃcial

text: monitoring

text: stations

text: across

text: Tai-

text: wan.

text: However,

text: for

text: ﬁve

text: of

text: those

text: stations,

text: we

text: do

text: not

text: have

text: the

text: necessary

text: information

text: on

text: the

text: covariates

text: considered

text: and

text: so,

text: we

text: decide

text: to

text: exclude

text: them

text: from

text: this

text: study.

text: The

text: remaining

text: monitoring

text: stations

text: are

text: irregularly

text: located

text: in

text: space,

text: spread

text: over

text: the

text: whole

text: Island

text: with

text: some

text: concentrations

text: around

text: big

text: cities

text: or

text: industrial

text: areas.

text: The

text: minimum

text: and

text: maximum

text: of

text: the

text: pairwise

text: distances

text: of

text: these

text: stations

text: are

text: kilometers

text: and

text: kilometers

text: re-

text: spectively

text: while

text: the

text: arithmatic

text: mean

text: of

text: the

text: same

text: is

text: kilometers.

text: One

text: major

text: talking

text: point

text: of

text: our

text: study

text: is

text: that

text: we

text: consider

text: the

text: issue

text: of

text: space-time

text: interaction

text: as

text: a

text: property

text: of

text: a

text: region,

text: rather

text: than

text: that

text: of

text: every

text: individual

text: station.

text: For

text: that,

text: we

text: divide

text: the

text: data

text: into

text: several

text: clusters,

text: based

text: on

text: the

text: latitude

text: and

text: longitude

text: of

text: the

text: stations.

text: More

text: on

text: this

text: is

text: discussed

text: in

text: the

text: following

text: sections.

text: Figure

text: shows

text: the

text: exact

text: locations

text: of

text: the

text: stations.

text: The

text: concentration

text: of

text: the

text: stations

text: on

text: the

text: west

text: coast

text: is

text: understandable

text: because

text: it

text: is

text: the

text: most

text: populated

text: area

text: of

text: the

text: Island.

text: Diﬀerent

text: colors

text: of

text: the

text: map

text: indicate

text: diﬀerent

text: regions

text: (clusters)

text: in

text: our

text: study.

text: Turn

text: to

text: the

text: temporal

text: scale

text: of

text: our

text: study.

text: The

text: data

text: are

text: obtained

text: on

text: an

text: hourly

text: basis

text: for

text: ten

text: years,

text: starting

text: from

text: January

text: to

text: December

text: However,

text: the

text: sampling

text: frequencies

text: vary

text: from

text: station

text: to

text: station

text: and

text: for

text: some

text: sites,

text: there

text: are

text: missing

text: values.

text: More-

text: over,

text: as

text: we

text: are

text: mainly

text: interested

text: in

text: identifying

text: space-time

text: interaction

text: for

text: the

text: air

text: pollution

text: data,

text: lower

text: temporal

text: resolution

text: is

text: desired

text: and

text: simpler.

text: Thus,

text: in

text: this

text: analysis,

text: we

text: decide

text: to

text: aggregate

text: the

text: hourly

text: data

text: into

text: weekly

text: averages

text: based

text: on

text: all

text: available

text: measurements

text: within

text: a

text: week.

text: It

text: is

text: worth

text: mentioning

text: that

text: this

text: is

text: a

text: common

text: practice

text: while

text: dealing

text: with

text: monitoring

text: data,

text: as

text: discussed

text: in

text: Smith

text: et

text: al.

text: In

text: order

text: to

text: maintain

text: continuity

text: in

text: the

text: time

text: scale,

text: we

text: consider

text: the

text: whole

text: set

text: of

text: days

text: (from

text: to

text: together

text: and

text: divide

text: it

text: into

text: weeks.

text: Hence,

text: the

text: total

text: number

text: of

text: data

text: points

text: considered

text: in

text: this

text: study

text: is

text: Throughout

text: the

text: study,

text: whenever

text: needed,

text: we

text: would

text: use

text: one

text: year’s

text: data,

text: i.e.

text: points

text: (approximately

text: the

text: last

text: for

text: each

text: site),

text: for

text: validation

text: purposes.

text: A

text: detailed

text: discussion

text: on

text: this

text: is

text: presented

text: in

text: the

text: following

text: sections.


text: 12 Park & Ratra Table 8 Non-ﬂat ΛCDM model parameters constrained with Planck TT + lowP + lensing, JLA, NewBAO, H(z), and fσ8 data (mean and 68.3% conﬁdence limits). Parameter TT+lowP+lensing TT+lowP+lensing+JLA TT+lowP+lensing+NewBAO ± ± ± Ωbh2 0.02305 0.00020 0.02304 0.00020 0.02303 0.00020 ± ± ± Ωch2 0.1091 0.0010 0.1091 0.0011 0.1093 0.0010 ± ± ± 100θMC 1.04237 0.00042 1.04233 0.00041 1.04230 0.00042 ± ± ± τ 0.101 0.021 0.107 0.017 0.115 0.011 ± ± ± ln(1010As) 3.110 0.041 3.120 0.034 3.139 0.022 ± ± ± Ωk −0.0160 0.0087 −0.0133 0.0063 −0.0087 0.0017 ± ± ± H0 [km s−1 Mpc−1] 65.1 3.3 66.0 2.5 67.81 0.66 ± ± ± Ωm 0.316 0.033 0.306 0.023 0.2893 0.0057 ± ± ± σ8 0.799 0.021 0.805 0.017 0.8148 0.0097 Parameter TT+lowP+lensing+H(z) TT+lowP+lensing+JLA+NewBAO TT+lowP+lensing+JLA+NewBAO+H(z) ± ± ± Ωbh2 0.02304 0.00020 0.02302 0.00020 0.02303 0.00019 ± ± ± Ωch2 0.1094 0.0010 0.1094 0.0010 0.1094 0.0010 ± ± ± 100θMC 1.04231 0.00041 1.04228 0.00042 1.04229 0.00041 ± ± ± τ 0.119 0.015 0.115 0.011 0.115 0.011 ± ± ± ln(1010As) 3.145 0.029 3.138 0.022 3.139 0.022 ± ± ± Ωk −0.0075 0.0042 −0.0086 0.0017 −0.0083 0.0016 ± ± ± H0 [km s−1 Mpc−1] 68.4 1.9 67.82 0.66 67.93 0.63 ± ± ± Ωm 0.285 0.016 0.2893 0.0058 0.2885 0.0055 ± ± ± σ8 0.818 0.014 0.8148 0.0098 0.8156 0.0098 Parameter TT+lowP+lensing+fσ8 TT+lowP+lensing+NewBAO+fσ8 TT+lowP+lensing+JLA+NewBAO+H(z)+fσ8 ± ± ± Ωbh2 0.02305 0.00020 0.02303 0.00020 0.02305 0.00020 ± ± ± Ωch2 0.1090 0.0011 0.1091 0.0011 0.1092 0.0010 ± ± ± 100θMC 1.04229 0.00041 1.04229 0.00041 1.04226 0.00041 ± ± ± τ 0.117 0.019 0.112 0.011 0.113 0.011 ± ± ± ln(1010As) 3.141 0.037 3.132 0.022 3.134 0.022 ± ± ± Ωk −0.0076 0.0064 −0.0086 0.0017 −0.0082 0.0016 ± ± ± H0 [km s−1 Mpc−1] 68.7 3.0 67.93 0.67 68.07 0.63 ± ± ± Ωm 0.283 0.024 0.2877 0.0058 0.2868 0.0055 ± ± ± σ8 0.815 0.019 0.8111 0.0098 0.8124 0.0095 MCMC chains.7 We present the individual contribution of number of degrees of freedom of the Planck CMB data is not each data set used to constrain the model parameters. The to- available and the absolute value of χ2 is arbitrary, only the dif- tal χ2 is the sum of those from the high-(cid:96) CMB TT likelihood ference of χ2 of one model relative to the other is meaningful (χ2 ), the low-(cid:96) CMB power spectra (χ2 ), lensing for the Planck CMB data. In Table 9, for the non-ﬂat ΛCDM PlikTT lowTEB (χ2 ), JLA SNIa (χ2 ), NewBAO (χ2 ), H(z) (χ2 ), model, we list ∆χ2, the excess χ2 over the value of the tilted lensing JLA NewBAO H(z) sfaσn8cdeaptaar(aχm2fσe8te),rsan(dχ2the )c.onTthriebuntoionnstfarnodmartdhenfoorrmegalriozuantidonnuoif- ﬂdaatta-ΛseCtDs.MFomr tohdeenl ocno-nCstMraBineddatwa istehtst,htehesanmuemcboemrsboifnadteiognreoesf prior the Planck 2015 CMB data likelihoods render the number of of freedom are 735, 15, 31, 10 for JLA SNIa, NewBAO, H(z), Planck 2015 CMB degrees of freedom ambiguous. Since the fσ8 data sets, respectively, for a total of 791 degrees of free- dom. The reduced χ2’s for the individual non-CMB data sets (cid:46) 7 Note that our estimation of the best-ﬁt position is approximate and can are χ2/ν 1. There are 189 points in the TT + lowP Planck differ from the true best-ﬁt position. COSMOMC has a built-in routine 2015 data (binned angular power spectrum) and 197 when the that obtains the minimum χ2 by using Powell’s minimization method that CMB lensing observations are included. searches for the local minimum by differentiating the likelihood distribu- Let us ﬁrst focus on how the model ﬁts the individual data tion. Although this method is efﬁcient at ﬁnding the accurate location of the minimum χ2, it has the limitation that it sometimes stays in a local min- sets. Compared to the tilted ﬂat-ΛCDM model, the non-ﬂat imum. Here we list the minimum χ2 estimated directly from the MCMC ΛCDM model constrained with the Planck CMB data alone chains without relying on Powell’s minimization method. Therefore our min- (excluding and including CMB lensing data) does worse at imum χ2 values differ from those supplied by the Planck team. For the tilted ﬁtting the Planck high-(cid:96) C ’s while it ﬁts the low-(cid:96) ones a bit ﬂat-ΛCDM model constrained with TT + lowP data, the Planck team pro- better. Inclusion of the n(cid:96)on-CMB data with the CMB data vides χ2 estimated from Powell’s minimization method: χ2 = 763.37, PlikTT also results in the best-ﬁt non-ﬂat model providing a poorer χ2 = 10496.47, χ2 = 2.08, with total χ2 = 11261.9. The Planck team lowTEB prior ﬁt to the high-(cid:96) TT measurements, both with and without the also provides the MCMC chains for this model, from which the minimum χ2 lensing data, compared to the tilted ﬂat-ΛCDM case. Adding can be estimated to be χ2PlikTT = 766.43, χ2lowTEB = 10495.95, χ2prior = 2.66, JLA SNIa or NewBAO data to the Planck TT + lowP + lensing with total χ2 = 11265.1. The latter results are very similar to our values listed data improves the non-ﬂat model ﬁt to the Planck low-(cid:96) TEB in Table 9. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Park

text: Ratra

text: Table

text: Non-ﬂat

text: ΛCDM

text: model

text: parameters

text: constrained

text: with

text: Planck

text: TT

text: lowP

text: lensing,

text: JLA,

text: NewBAO,

text: H(z),

text: and

text: data

text: (mean

text: and

text: conﬁdence

text: limits).

text: Parameter

label: table

text: TT+lowP+lensing

text: TT+lowP+lensing+JLA

text: TT+lowP+lensing+NewBAO

text: τ

text: Ωk

text: [km

text: Ωm

text: σ8

text: Parameter

text: TT+lowP+lensing+H(z)

text: TT+lowP+lensing+JLA+NewBAO

text: TT+lowP+lensing+JLA+NewBAO+H(z)

text: τ

text: Ωk

text: [km

text: Ωm

text: σ8

text: Parameter

text: τ

text: Ωk

text: [km

text: Ωm

text: σ8

text: MCMC

text: We

text: present

text: the

text: individual

text: contribution

text: of

text: number

text: of

text: degrees

text: of

text: freedom

text: of

text: the

text: Planck

text: CMB

text: data

text: is

text: not

text: each

text: data

text: set

text: used

text: to

text: constrain

text: the

text: model

text: parameters.

text: The

text: to-

text: available

text: and

text: the

text: absolute

text: value

text: of

text: χ2

text: is

text: arbitrary,

text: only

text: the

text: dif-

text: tal

text: χ2

text: is

text: the

text: sum

text: of

text: those

text: from

text: the

text: CMB

text: TT

text: likelihood

text: ference

text: of

text: χ2

text: of

text: one

text: model

text: relative

text: to

text: the

text: other

text: is

text: meaningful

text: the

text: CMB

text: power

text: spectra

text: lensing

text: for

text: the

text: Planck

text: CMB

text: data.

text: In

text: Table

text: for

text: the

text: non-ﬂat

text: ΛCDM

text: PlikTT

text: lowTEB

text: JLA

text: SNIa

text: NewBAO

text: H(z)

text: model,

text: we

text: list

text: ∆χ2,

text: the

text: excess

text: χ2

text: over

text: the

text: value

text: of

text: the

text: tilted

text: lensing

text: JLA

text: NewBAO

text: H(z)

text: ﬂdaatta-ΛseCtDs.MFomr

text: tohdeenl

text: ocno-nCstMraBineddatwa

text: prior

text: the

text: Planck

text: CMB

text: data

text: likelihoods

text: render

text: the

text: number

text: of

text: of

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text: 14 Park & Ratra Flat TT+lowP Flat TT+lowP 0.96 Nonﬂat TT+lowP 0.96 Nonﬂat TT+lowP Nonﬂat TT+lowP+fσ8 Nonﬂat TT+lowP+NewBAO Nonﬂat TT+lowP+JLA Nonﬂat TT+lowP+NewBAO+fσ8 0.90 Nonﬂat TT+lowP+H(z) 0.90 Nonﬂat TT+lowP+NewBAO+JLA+H(z) Nonﬂat TT+lowP+NewBAO Nonﬂat TT+lowP+NewBAO+JLA+H(z)+fσ8 0.84 0.84 0.78 0.78 0.72 0.72 0.66 DES Y1 All 0.66 DES Y1 All 0.60 0.60 0.30 0.45 0.60 0.75 0.90 1.05 1.20 0.30 0.45 0.60 0.75 0.90 1.05 1.20 Ω Ω m m Flat TT+lowP+lensing Flat TT+lowP+lensing 0.96 Nonﬂat TT+lowP+lensing 0.96 Nonﬂat TT+lowP+lensing Nonﬂat TT+lowP+lensing+fσ8 Nonﬂat TT+lowP+lensing+NewBAO Nonﬂat TT+lowP+lensing+JLA Nonﬂat TT+lowP+lensing+NewBAO+fσ8 0.90 Nonﬂat TT+lowP+lensing+H(z) 0.90 Nonﬂat TT+lowP+lensing+NewBAO+JLA+H(z) Nonﬂat TT+lowP+lensing+NewBAO Nonﬂat TT+lowP+lensing+NewBAO+JLA+H(z)+fσ8 0.84 0.84 0.78 0.78 0.72 0.72 0.66 DES Y1 All 0.66 DES Y1 All 0.60 0.60 0.20 0.24 0.28 0.32 0.36 0.40 0.20 0.24 0.28 0.32 0.36 0.40 Ω Ω m m Figure 7. Same as Fig. 6 but for the non-ﬂat ΛCDM model. fully gathered compilation of observational data, the largest ﬂat ΛCDM model is more consistent with the recent such collection utilized to date. DES measurement (DES Collaboration 2017a). Our main results, in summary, are: • Using a consistent power spectrum for energy den- • Ωbh2, τ, Ωch2, and some of the other measured cos- sity inhomogeneities in the non-ﬂat model, we conﬁrm, mological parameter values are quite model dependent. with greater signiﬁcance, the Ooba et al. (2017a) re- For such parameters, caution is called for when com- sult that cosmological data does not demand spatially- paring a value measured in a cosmological model to a ﬂat hypersurfaces. These data (including CMB lensing value determined using another technique. measurements) favor a closed Universe at more than 5σ signiﬁcance, with spatial curvature contributing about a Overall, the tilted ﬂat-ΛCDM model has a somewhat lower percent to the current cosmological energy budget. χ2 than the non-ﬂat ΛCDM case and so is more favored. How- ever it is unclear how to translate this into a quantitative rela- • The best-ﬁt non-ﬂat ΛCDM model provides a better ﬁt tive probability. On the other hand, the non-ﬂat ΛCDM model to the low-(cid:96) C(cid:96)’s and better agrees with the σ8–Ωm DES has other advantages. It is possible that a more complete constraints, but does worse than the best-ﬁt tilted ﬂat- understanding of systematic differences between constraints ΛCDM model in ﬁtting the higher-(cid:96) C ’s. derived using the lower-(cid:96) and higher-(cid:96) CMB anisotropy data (cid:96) might have some bearing on these issues. • H0 measured in both models are almost identical, and consistent with most other measurements of H . How- 0 ever, as is well known, an estimate of the local expan- We acknowledge valuable discussions with J. Ooba. C.- sion rate (Anderson & Riess 2017) is 2.7σ larger. G.P. was supported by the Basic Science Research Pro- gram through the National Research Foundation of Ko- • σ8 measured in both models are identical and consistent rea (NRF) funded by the Ministry of Education (No. with the recent DES measurement (DES Collaboration 2017R1D1A1B03028384). B.R. was supported in part by 2017a). DOE grant DE-SC0011840. • The measured Ωm is more model dependent than the REFERENCES measured σ8 and the Ωm value measured using the non- σ σ 8 8 σ σ 8 8 ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Park

text: Ratra

text: Flat

text: TT+lowP

text: Flat

text: TT+lowP

text: Nonﬂat

text: TT+lowP

text: Nonﬂat

text: TT+lowP

text: Nonﬂat

text: Nonﬂat

text: TT+lowP+NewBAO

text: Nonﬂat

text: TT+lowP+JLA

text: Nonﬂat

text: Nonﬂat

text: TT+lowP+H(z)

text: Nonﬂat

text: TT+lowP+NewBAO+JLA+H(z)

text: Nonﬂat

text: TT+lowP+NewBAO

text: Nonﬂat

text: DES

text: All

text: DES

text: All

text: Ω

text: Ω

text: m

text: m

text: Flat

text: TT+lowP+lensing

text: Flat

text: TT+lowP+lensing

text: Nonﬂat

text: TT+lowP+lensing

text: Nonﬂat

text: TT+lowP+lensing

text: Nonﬂat

text: Nonﬂat

text: TT+lowP+lensing+NewBAO

text: Nonﬂat

text: TT+lowP+lensing+JLA

text: Nonﬂat

text: Nonﬂat

text: TT+lowP+lensing+H(z)

text: Nonﬂat

text: TT+lowP+lensing+NewBAO+JLA+H(z)

text: Nonﬂat

text: TT+lowP+lensing+NewBAO

text: Nonﬂat

text: DES

text: All

text: DES

text: All

text: Ω

text: Ω

text: m

text: m

text: Figure

text: Same

text: as

text: Fig.

text: but

text: for

text: the

text: non-ﬂat

text: ΛCDM

text: model.

text: fully

text: gathered

text: compilation

text: of

text: observational

text: data,

text: the

text: largest

text: ﬂat

text: ΛCDM

text: model

text: is

text: more

text: consistent

text: with

text: the

text: recent

text: such

text: collection

text: utilized

text: to

text: date.

text: DES

text: measurement

text: (DES

text: Collaboration

text: Our

text: main

text: results,

text: in

text: summary,

text: are:

text: Using

text: a

text: consistent

text: power

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text: energy

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text: τ,

text: and

text: some

text: of

text: the

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text: in

text: the

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text: model,

text: we

text: conﬁrm,

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text: are

text: quite

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text: with

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text: the

text: Ooba

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text: For

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text: caution

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text: sult

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text: does

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text: a

text: value

text: measured

text: in

text: a

text: cosmological

text: model

text: to

text: a

text: ﬂat

text: hypersurfaces.

text: These

text: data

text: (including

text: CMB

text: lensing

text: value

text: determined

text: using

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text: measurements)

text: favor

text: a

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text: than

text: 5σ

text: signiﬁcance,

text: with

text: spatial

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text: a

text: Overall,

text: the

text: tilted

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text: model

text: has

text: a

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text: lower

text: percent

text: to

text: the

text: current

text: cosmological

text: energy

text: budget.

text: χ2

text: than

text: the

text: non-ﬂat

text: ΛCDM

text: case

text: and

text: so

text: is

text: more

text: favored.

text: How-

text: ever

text: it

text: is

text: unclear

text: how

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text: this

text: into

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text: The

text: best-ﬁt

text: non-ﬂat

text: ΛCDM

text: model

text: provides

text: a

text: better

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text: tive

text: probability.

text: On

text: the

text: other

text: hand,

text: the

text: non-ﬂat

text: ΛCDM

text: model

text: to

text: the

text: and

text: better

text: agrees

text: with

text: the

text: σ8–Ωm

text: DES

text: has

text: other

text: advantages.

text: It

text: is

text: possible

text: that

text: a

text: more

text: complete

text: constraints,

text: but

text: does

text: worse

text: than

text: the

text: best-ﬁt

text: tilted

text: ﬂat-

text: understanding

text: of

text: systematic

text: differences

text: between

text: constraints

text: ΛCDM

text: model

text: in

text: ﬁtting

text: the

text: C

text: ’s.

text: derived

text: using

text: the

text: and

text: CMB

text: anisotropy

text: data

text: might

text: have

text: some

text: bearing

text: on

text: these

text: issues.

text: measured

text: in

text: both

text: models

text: are

text: almost

text: identical,

text: and

text: consistent

text: with

text: most

text: other

text: measurements

text: of

text: H

text: How-

text: ever,

text: as

text: is

text: well

text: known,

text: an

text: estimate

text: of

text: the

text: local

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text: We

text: acknowledge

text: valuable

text: discussions

text: with

text: J.

text: Ooba.

text: C.-

text: sion

text: rate

text: (Anderson

text: Riess

text: is

text: larger.

text: G.P.

text: was

text: supported

text: by

text: the

text: Basic

text: Science

text: Research

text: Pro-

text: gram

text: through

text: the

text: National

text: Research

text: Foundation

text: of

text: Ko-

text: σ8

text: measured

text: in

text: both

text: models

text: are

text: identical

text: and

text: consistent

text: rea

text: (NRF)

text: funded

text: by

text: the

text: Ministry

text: of

text: Education

text: (No.

text: with

text: the

text: recent

text: DES

text: measurement

text: (DES

text: Collaboration

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text: was

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text: DOE

text: grant

text: The

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text: Ωm

text: is

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text: model

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text: than

text: the

text: REFERENCES

text: measured

text: σ8

text: and

text: the

text: Ωm

text: value

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text: σ

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