text: SEMI-SIMPLICIAL SPACES 29 A and use that -equivalences satisfy the 2-out-of-3 property. A similar argument (using also Remark 2.8) shows that k• is homotopy cartesian. The square (cid:107) (cid:107) (cid:107)k•(cid:107) (cid:47)(cid:47) (cid:107) (cid:107) W• X• (cid:107) (cid:107) (cid:107) (cid:107) g• f• (cid:15)(cid:15) (cid:15)(cid:15) (cid:107) (cid:107) (cid:107)h•(cid:107) (cid:47)(cid:47) (cid:107) (cid:107) Z• Y• is homotopy cartesian. This follows by applying Theorem 2.12 to both h• and k•, using that (6.11) is homotopy cartesian for p = 0 and using Remark 2.8. Since (6.11) for p = 0 is homotopy cartesian, comparing homotopy ﬁbres gives a commutative square (cid:39) (cid:47)(cid:47) hoﬁb (g ) hoﬁb (f ) z 0 h0(z) 0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:107) (cid:107) (cid:39) (cid:47)(cid:47) (cid:107) (cid:107) hoﬁb ( g• ) hoﬁb ( f• ) ι(z) ι(z) → A in which the horizontal maps are weak equivalences. Since g• : W• Z• is -cartesian, and A by assumption Z• is basic, it follows that the left vertical map is an -equivalence. Therefore, A ∈ the right vertical map is also an -equivalence. This holds for any z Z , but the map h is 0 0 (cid:3) 0-connected, which ﬁnishes the proof. The next lemma provides an appropriate resolution of a simplicial set by a contractible simplicial space. Lemma 6.12. Let Y• be a 0-connected simplicial set. Then there is a simplicial space QY• with (cid:107) (cid:107) (cid:39) ∗ → QY• and a homotopy cartesian morphism f• : QY• Y•, such that f is 0-connected. 0 The same statement is true for semi-simplicial sets, with the same proof. ∈ ∈ → (cid:107) (cid:107) Proof. Fix a vertex y Y . For each simplex σ Y , we let χ : ∆p Y• denote its character- 0(cid:97) ⊂ p σ istic map. Furthermore, we view ∆p ∆p+1 as the last face, i.e. the face opposite to e . We p+1 let { | → (cid:107) (cid:107) | } QY := (σ, h) h : ∆p+1 Y• ; h = χ ; h(e ) = y , p ∆p σ p+1 ∈ σ Yp × (cid:107) (cid:107) → ◦ topologised as a subspace of Y Y• ∆p. Deﬁne d : QY QY − by d (σ, h) := (d σ, h di) p i p →p 1 i i (and the degeneracy maps in an analogous way) and f : QY Y by f (σ, h) := σ. Then → p p p p f• : QY• Y• is a map of simplicial spaces. This should be viewed as an analogue of the path ﬁbration, and we now verify that indeed it has − → − the characteristic properties of that construction. The maps d : f 1(σ) f −1 (σ) are homotopy i p p(cid:107) 1 (cid:107) (cid:39) ∗ equivalences, so that f• is homotopy cartesian. It remains to be shown that QY• . − (cid:107) (cid:107) (cid:107) (cid:107) First observe that the ﬁbre f 1(y) is the based loop space Ω Y• . Let P Y• denote the path 0(cid:107) (cid:107) y y space: the space of all paths in Y• with endp(cid:16)oint y. The map (cid:17) (cid:107) (cid:107) −→ (cid:107) (cid:107) g : QY• P Y• y (cid:55)−→ (cid:55)→ − (σ, h, t) s h((1 s)t, s) makes the diagram (cid:107) (cid:107) (cid:47)(cid:47) (cid:107) (cid:107) g (cid:47)(cid:47) (cid:107) (cid:107) Ω Y• QY• P Y• y y (cid:107) (cid:107) f• ev0 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) { } (cid:47)(cid:47) (cid:107) (cid:107) (cid:107) (cid:107) y Y• Y• commute, by inspection. Since f• is homotopy cartesian, the left-hand square is homotopy carte- sian. The outer rectangle is also hompotopy cartesian, as it is cartesian and ev is a ﬁbration. Thus 0 the map between vertical homotopy ﬁbres over y of the right-hand square is an equivalence: this ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLICIAL

text: SPACES

text: A

text: and

text: use

text: that

text: -equivalences

text: satisfy

text: the

text: property.

text: A

text: similar

text: argument

text: (using

text: also

text: Remark

text: shows

text: that

text: k•

text: is

text: homotopy

text: cartesian.

text: The

text: square

text: W•

text: X•

text: g•

text: f•

text: Z•

text: Y•

text: is

text: homotopy

text: cartesian.

text: This

text: follows

text: by

text: applying

text: Theorem

text: to

text: both

text: h•

text: and

text: k•,

text: using

text: that

text: is

text: homotopy

text: cartesian

text: for

text: p

text: and

text: using

text: Remark

text: Since

text: for

text: p

text: is

text: homotopy

text: cartesian,

text: comparing

text: homotopy

text: ﬁbres

text: gives

text: a

text: commutative

text: square

text: hoﬁb

text: (g

text: hoﬁb

text: (f

text: z

text: hoﬁb

text: g•

text: hoﬁb

text: f•

text: ι(z)

text: ι(z)

text: A

text: in

text: which

text: the

text: horizontal

text: maps

text: are

text: weak

text: equivalences.

text: Since

text: g•

text: W•

text: Z•

text: is

text: -cartesian,

text: and

text: A

text: by

text: assumption

text: Z•

text: is

text: basic,

text: it

text: follows

text: that

text: the

text: left

text: vertical

text: map

text: is

text: an

text: -equivalence.

text: Therefore,

text: A

text: the

text: right

text: vertical

text: map

text: is

text: also

text: an

text: -equivalence.

text: This

text: holds

text: for

text: any

text: z

text: Z

text: but

text: the

text: map

text: h

text: is

text: which

text: ﬁnishes

text: the

text: proof.

text: The

text: next

text: lemma

text: provides

text: an

text: appropriate

text: resolution

text: of

text: a

text: simplicial

text: set

text: by

text: a

text: contractible

text: simplicial

text: space.

text: Lemma

text: Let

text: Y•

text: be

text: a

text: simplicial

text: set.

text: Then

text: there

text: is

text: a

text: simplicial

text: space

text: QY•

text: with

text: QY•

text: and

text: a

text: homotopy

text: cartesian

text: morphism

text: f•

text: QY•

text: Y•,

text: such

text: that

text: f

text: is

text: The

text: same

text: statement

text: is

text: true

text: for

text: semi-simplicial

text: sets,

text: with

text: the

text: same

text: proof.

text: Proof.

text: Fix

text: a

text: vertex

text: y

text: Y

text: For

text: each

text: simplex

text: σ

text: Y

text: we

text: let

text: χ

text: ∆p

text: Y•

text: denote

text: its

text: character-

text: p

text: σ

text: istic

text: map.

text: Furthermore,

text: we

text: view

text: ∆p

text: as

text: the

text: last

text: face,

text: i.e.

text: the

text: face

text: opposite

text: to

text: e

text: We

text: let

text: QY

text: (σ,

text: h)

text: h

text: Y•

text: h

text: χ

text: h(e

text: y

text: p

text: ∆p

text: σ

text: σ

text: Yp

text: topologised

text: as

text: a

text: subspace

text: of

text: Y

text: Y•

text: ∆p.

text: Deﬁne

text: d

text: QY

text: QY

text: by

text: d

text: (σ,

text: h)

text: (d

text: σ,

text: h

text: di)

text: p

text: i

text: p

text: →p

text: i

text: i

text: (and

text: the

text: degeneracy

text: maps

text: in

text: an

text: analogous

text: way)

text: and

text: f

text: QY

text: Y

text: by

text: f

text: (σ,

text: h)

text: σ.

text: Then

text: p

text: p

text: p

text: p

text: f•

text: QY•

text: Y•

text: is

text: a

text: map

text: of

text: simplicial

text: spaces.

text: This

text: should

text: be

text: viewed

text: as

text: an

text: analogue

text: of

text: the

text: path

text: ﬁbration,

text: and

text: we

text: now

text: verify

text: that

text: indeed

text: it

text: has

text: the

text: characteristic

text: properties

text: of

text: that

text: construction.

text: The

text: maps

text: d

text: f

text: f

text: −1

text: (σ)

text: are

text: homotopy

text: i

text: p

text: equivalences,

text: so

text: that

text: f•

text: is

text: homotopy

text: cartesian.

text: It

text: remains

text: to

text: be

text: shown

text: that

text: QY•

text: First

text: observe

text: that

text: the

text: ﬁbre

text: f

text: is

text: the

text: based

text: loop

text: space

text: Ω

text: Y•

text: Let

text: P

text: Y•

text: denote

text: the

text: path

text: y

text: y

text: space:

text: the

text: space

text: of

text: all

text: paths

text: in

text: Y•

text: with

text: y.

text: The

text: map

text: g

text: QY•

text: P

text: Y•

text: y

text: (σ,

text: h,

text: t)

text: s

text: s)t,

text: s)

text: makes

text: the

text: diagram

text: g

text: Ω

text: Y•

text: QY•

text: P

text: Y•

text: y

text: y

text: f•

text: y

text: Y•

text: Y•

text: commute,

text: by

text: inspection.

text: Since

text: f•

text: is

text: homotopy

text: cartesian,

text: the

text: left-hand

text: square

text: is

text: homotopy

text: carte-

text: sian.

text: The

text: outer

text: rectangle

text: is

text: also

text: hompotopy

text: cartesian,

text: as

text: it

text: is

text: cartesian

text: and

text: ev

text: is

text: a

text: ﬁbration.

text: Thus

text: the

text: map

text: between

text: vertical

text: homotopy

text: ﬁbres

text: over

text: y

text: of

text: the

text: right-hand

text: square

text: is

text: an

text: equivalence:

text: this

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 JOHANNES EBERT AND OSCAR RANDAL-WILLIAMS (cid:107) (cid:107) (cid:107) (cid:107) → If (Y•, Y− , (cid:15)) is an augmented semi-simplicial space, then there is an induced map (cid:15)• : Y• 1 Y− . There is a standard technique for easily showing that such maps are homotopy equivalences, 1 which goes under the name of “having an extra degeneracy”. Lemma 1.12. Let (Y•, Y− , (cid:15)) be an augmented semi-simplicial space, and suppose there are maps → ≥ −1 h : Y Y for p 1 such that p+1 p p+1 d h = Id , p+1 p+1 Yp ≤ d h = h d for 0 i < p + 1, i p+1 p i (cid:15)0h0 = IdY−1 (cid:107) (cid:107) (cid:107) (cid:107) → then (cid:15)• : Y• Y− is a homotopy equivalence. 1 → ≥ − Dually, if there are maps g : Y Y for p 1 such that p+1 p p+1 d g = Id , 0 p+1 Yp ≤ d g = g d − for 0 < i p + 1, i p+1 p i 1 (cid:15)0g0 = IdY−1 then the same conclusion holds. In the ﬁrst case the conditions on the maps h are formally identical to the conditions relating p+1 face maps d to degeneracy maps s , except that h behaves like a hypothetical degeneracy map i i p+1 s , whereas in the deﬁnition of a simplicial object there are only degeneracy maps s , s , . . . , s : p+→1 0 1 p Y Y . For this reason such a collection of maps h is often called an extra degeneracy. p p+1 p+1 → (Similarly, g behaves like a hypothetical degeneracy map s− : Y Y .) p+1 1 p p+1 → ⊂ (cid:107) (cid:107) (cid:107) (cid:107) ◦ Proof. Let us ju×st co×nsider→the ﬁrst×case. W→e (cid:107)hav(cid:107)e h0 : Y−1 Y0 Y• and (cid:15)• h0 = IdY−1. The maps [0, 1] Y ∆p Y ∆p+1 Y• , deﬁned by p p+1 (cid:55)→ − − (s; x; t , . . . , t ) (h (x); (1 s)t , . . . , (1 s)t , s), 0 p p+1 0 p respect the equivalence relation used in the deﬁnition of the geometric realisation. Since taking products and taking quotients commutes in compactly generated spaces, this yields a homotopy × (cid:107) (cid:107) → (cid:107) (cid:107) − − ◦ (cid:107) (cid:107) (cid:3) H : [0, 1] Y• Y• , and one veriﬁes that H(0, ) = Id(cid:107) (cid:107) and that H(1, ) = h (cid:15)• . Y• 0 ∗ Any semi-simplicial space Y• is augmented over a point in a unique way. The data of an extra ∗ → degeneracy in this case gives in particular a point y : Y , and the homotopy in the proof (cid:107) (cid:107) { } ⊂ ⊂0 (cid:107) (cid:107) 0 gives a contraction of Y• to the point y Y Y• . This can be generalised to maps of 0 0 semi-simplicial spaces, as follows. → ∈ Lemma 1.13. Let f• : X• Y• be a map of semi-simplicial spaces and y Y . A semi-simplicial 0→ 0 nullhomotopy from f• to y is a collection of continuous maps h : X Y such that 0 p+1 p p+1 d h = f , p+1 p+1 p ≤ ≤ ≥ d h = h d for 0 i p and p 1, i p+1 p i ≡ d h y . 0 1 0 (cid:107) (cid:107) (cid:107) (cid:107) → Such a semi-simplicial nullhomotopy induces a homotopy from f• to the constant map X• { } ⊂ ⊂ (cid:107) (cid:107) y Y Y• . 0 0 × Proof. Use the same formula as in the proof of Lemma 1.12 to obtain a homotopy H : [0, 1] (cid:107) (cid:107) → (cid:107) (cid:107) − (cid:107) (cid:107) − (cid:3) X• Y• with H(0, ) = f• and H(1, ) the constant map with value y . 0

label: paragraph

text: JOHANNES

text: EBERT

text: AND

text: OSCAR

text: RANDAL-WILLIAMS

text: If

text: (Y•,

text: Y−

text: is

text: an

text: augmented

text: semi-simplicial

text: space,

text: then

text: there

text: is

text: an

text: induced

text: map

text: Y•

text: Y−

text: There

text: is

text: a

text: standard

text: technique

text: for

text: easily

text: showing

text: that

text: such

text: maps

text: are

text: homotopy

text: equivalences,

text: which

text: goes

text: under

text: the

text: name

text: of

text: “having

text: an

text: extra

text: degeneracy”.

text: Lemma

text: Let

text: (Y•,

text: Y−

text: be

text: an

text: augmented

text: semi-simplicial

text: space,

text: and

text: suppose

text: there

text: are

text: maps

text: −1

text: h

text: Y

text: Y

text: for

text: p

text: such

text: that

text: p

text: d

text: h

text: Id

text: Yp

text: d

text: h

text: h

text: d

text: for

text: i

text: p

text: i

text: p

text: i

text: then

text: Y•

text: Y−

text: is

text: a

text: homotopy

text: equivalence.

text: Dually,

text: if

text: there

text: are

text: maps

text: g

text: Y

text: Y

text: for

text: p

text: such

text: that

text: p

text: d

text: g

text: Id

text: Yp

text: d

text: g

text: g

text: d

text: for

text: i

text: p

text: i

text: p

text: i

text: then

text: the

text: same

text: conclusion

text: holds.

text: In

text: the

text: ﬁrst

text: case

text: the

text: conditions

text: on

text: the

text: maps

text: h

text: are

text: formally

text: identical

text: to

text: the

text: conditions

text: relating

text: face

text: maps

text: d

text: to

text: degeneracy

text: maps

text: s

text: except

text: that

text: h

text: behaves

text: like

text: a

text: hypothetical

text: degeneracy

text: map

text: i

text: i

text: s

text: whereas

text: in

text: the

text: deﬁnition

text: of

text: a

text: simplicial

text: object

text: there

text: are

text: only

text: degeneracy

text: maps

text: s

text: s

text: s

text: p

text: Y

text: Y

text: For

text: this

text: reason

text: such

text: a

text: collection

text: of

text: maps

text: h

text: is

text: often

text: called

text: an

text: extra

text: degeneracy.

text: p

text: (Similarly,

text: g

text: behaves

text: like

text: a

text: hypothetical

text: degeneracy

text: map

text: s−

text: Y

text: Y

text: p

text: Proof.

text: Let

text: us

text: ju×st

text: co×nsider→the

text: ﬁrst×case.

text: W→e

text: Y•

text: and

text: The

text: maps

text: Y

text: ∆p

text: Y

text: Y•

text: deﬁned

text: by

text: p

text: (s;

text: x;

text: t

text: t

text: (h

text: (x);

text: s)t

text: s)t

text: s),

text: p

text: p

text: respect

text: the

text: equivalence

text: relation

text: used

text: in

text: the

text: deﬁnition

text: of

text: the

text: geometric

text: realisation.

text: Since

text: taking

text: products

text: and

text: taking

text: quotients

text: commutes

text: in

text: compactly

text: generated

text: spaces,

text: this

text: yields

text: a

text: homotopy

text: H

text: Y•

text: Y•

text: and

text: one

text: veriﬁes

text: that

text: and

text: that

text: h

text: Y•

text: Any

text: semi-simplicial

text: space

text: Y•

text: is

text: augmented

text: over

text: a

text: point

text: in

text: a

text: unique

text: way.

text: The

text: data

text: of

text: an

text: extra

text: degeneracy

text: in

text: this

text: case

text: gives

text: in

text: particular

text: a

text: point

text: y

text: Y

text: and

text: the

text: homotopy

text: in

text: the

text: proof

text: ⊂0

text: gives

text: a

text: contraction

text: of

text: Y•

text: to

text: the

text: point

text: y

text: Y

text: Y•

text: This

text: can

text: be

text: generalised

text: to

text: maps

text: of

text: semi-simplicial

text: spaces,

text: as

text: follows.

text: Lemma

text: Let

text: f•

text: X•

text: Y•

text: be

text: a

text: map

text: of

text: semi-simplicial

text: spaces

text: and

text: y

text: Y

text: A

text: semi-simplicial

text: 0→

text: nullhomotopy

text: from

text: f•

text: to

text: y

text: is

text: a

text: collection

text: of

text: continuous

text: maps

text: h

text: X

text: Y

text: such

text: that

text: p

text: d

text: h

text: f

text: p

text: d

text: h

text: h

text: d

text: for

text: i

text: p

text: and

text: p

text: i

text: p

text: i

text: d

text: h

text: y

text: Such

text: a

text: semi-simplicial

text: nullhomotopy

text: induces

text: a

text: homotopy

text: from

text: f•

text: to

text: the

text: constant

text: map

text: X•

text: y

text: Y

text: Y•

text: Proof.

text: Use

text: the

text: same

text: formula

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: to

text: obtain

text: a

text: homotopy

text: H

text: X•

text: Y•

text: with

text: f•

text: and

text: the

text: constant

text: map

text: with

text: value

text: y


text: SEMI-SIMPLICIAL SPACES 9 Remark 2.3. Theorem 2.2 is false in general for the thin geometric realisation of simplicial spaces. This is the main reason why—even for simplicial spaces—it is often preferable to consider the fat geometric realisation. A concrete counterexample was given by Lawson in response to a question on MathOverﬂow [17]. Theorem 2.2 has the following useful generalisation. → → − Lemma 2.4. Let f• : X• Y• be a map of semi-simplicial spaces. If f : X Y is (k p)- (cid:107) (cid:107) p p p connected for all p, then f• is k-connected. (cid:107) (cid:107) (cid:107) (cid:107) → (cid:107) (cid:107) Proof. By Lemma 2.1 it is enough to show that f• (n) : X• (n) Y• (n) is k-connected for ∅ each n. The case n = 0 is trivial. For the induction step, we may as well suppose that X = Y = (cid:107) (cid:107) − i i for i > n and that f• (n 1) is k-connected. We factorise f• as (2.5) X• −j→• W• −g→• Z• −h→• Y• ∅ as follows. The semi-simplicial space W• has W = Y for i < n, W = X and W = for i > n. → i i◦ ◦ n n i The face maps W W − are the compositions f − d = d f , and the other face maps are n n 1 n 1 i i n the same as those for X•. The map j is the identity, and j = f for i < n. n i i Then factorise f as n −→ −→ f : X gn Z hn Y n n n n where h is a weak homotopy equivalence, and Z is obtained from X by attaching cells of n − n n ∅ dimension at least (k n + 1). For i < n let Z = Y , and for i > n let Z = . The map g is the → i i i i identity for i < n, and h : Z Y is the identity as well. This yields the factorisation (2.5). i i i (cid:107) (cid:107) The map h• is a levelwise weak equivalence, and so h• is a weak equivalence by Theorem 2.2. (cid:107) (cid:107) − (cid:107) (cid:107) − × × Moreover, W• (n 1) = Z• (n 1) and the pair (Z ∆n, Z ∂∆n) is obtained from the pair × × n n (cid:107) (cid:107) (W ∆n, W ∂∆n) by attaching cells of dimension at least (k + 1), so Z• (n) is obtained from (cid:107) n(cid:107) n (cid:107) (cid:107) (cid:107) (cid:107) → (cid:107) (cid:107) W• (n) by attaching cells of dimension at least (k + 1): in particular, g• : W• (n) Z• (n) (cid:107) (cid:107) − (cid:107) (cid:107) − → (cid:107) (cid:107) − is k-connected. By the inductive hypothesis, j• (n 1) : X• (n 1) W• (n 1) is k-connected and j is the identity. Using the notation introduced in the proof of Theorem 2.2, we get that →n → ∩ → ∩ UX UW is k-connected, while UX UW and UX UX UW UW are weak equivalences. 0 0 1 (cid:107)1 (cid:107) 0 1 0 1 (cid:3) From [32, Theorem 6.7.9], it follows that j• is k-connected. Using this we can now prove the analogue of Lemma 1.8 for semi-simplicial spaces, rather than semi-simplicial sets. (cid:107) (cid:107) → (cid:107) (cid:107) Lemma 2.6. For each semi-simplicial space X•, the map X• EX• is a weak homotopy equivalence. Proof. Consider the bi-semi-simplicial set Sing (X ), with Sing (EX•) = E(Sing X•) so giving a p q p p commutative square (cid:107) (cid:107) (cid:47)(cid:47) (cid:107) (cid:107) (cid:107) (cid:107) Sing•X• E(Sing•X•) Sing•(EX•) (cid:15)(cid:15) (cid:15)(cid:15) (cid:107) (cid:107) (cid:47)(cid:47) (cid:107) (cid:107) X• EX• . The vertical maps are weak equivalences by Lemma 1.11 and Theorem 2.2, and the top map is a (cid:3) weak equivalence by Lemma 1.8 and Theorem 2.2; hence the bottom map is a weak equivalence. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SEMI-SIMPLICIAL

text: SPACES

text: Remark

text: Theorem

text: is

text: false

text: in

text: general

text: for

text: the

text: thin

text: geometric

text: realisation

text: of

text: simplicial

text: spaces.

text: This

text: is

text: the

text: main

text: reason

text: why—even

text: for

text: simplicial

text: spaces—it

text: is

text: often

text: preferable

text: to

text: consider

text: the

text: fat

text: geometric

text: realisation.

text: A

text: concrete

text: counterexample

text: was

text: given

text: by

text: Lawson

text: in

text: response

text: to

text: a

text: question

text: on

text: MathOverﬂow

text: Theorem

text: has

text: the

text: following

text: useful

text: generalisation.

text: Lemma

text: Let

text: f•

text: X•

text: Y•

text: be

text: a

text: map

text: of

text: semi-simplicial

text: spaces.

text: If

text: f

text: X

text: Y

text: is

text: (k

text: p)-

text: p

text: p

text: p

text: connected

text: for

text: all

text: p,

text: then

text: f•

text: is

text: k-connected.

text: Proof.

text: By

text: Lemma

text: it

text: is

text: enough

text: to

text: show

text: that

text: f•

text: (n)

text: X•

text: (n)

text: Y•

text: (n)

text: is

text: k-connected

text: for

text: each

text: n.

text: The

text: case

text: n

text: is

text: trivial.

text: For

text: the

text: induction

text: step,

text: we

text: may

text: as

text: well

text: suppose

text: that

text: X

text: Y

text: i

text: i

text: for

text: i

text: n

text: and

text: that

text: f•

text: (n

text: is

text: k-connected.

text: We

text: factorise

text: f•

text: as

text: X•

text: −j→•

text: W•

text: −g→•

text: Z•

text: −h→•

text: Y•

text: as

text: follows.

text: The

text: semi-simplicial

text: space

text: W•

text: has

text: W

text: Y

text: for

text: i

text: n,

text: W

text: X

text: and

text: W

text: for

text: i

text: n.

text: i

text: i◦

text: n

text: n

text: i

text: The

text: face

text: maps

text: W

text: W

text: are

text: the

text: compositions

text: f

text: d

text: d

text: f

text: and

text: the

text: other

text: face

text: maps

text: are

text: n

text: n

text: n

text: i

text: i

text: n

text: the

text: same

text: as

text: those

text: for

text: X•.

text: The

text: map

text: j

text: is

text: the

text: identity,

text: and

text: j

text: f

text: for

text: i

text: n.

text: n

text: i

text: i

text: Then

text: factorise

text: f

text: as

text: n

text: f

text: X

text: gn

text: Z

text: hn

text: Y

text: n

text: n

text: n

text: n

text: where

text: h

text: is

text: a

text: weak

text: homotopy

text: equivalence,

text: and

text: Z

text: is

text: obtained

text: from

text: X

text: by

text: attaching

text: cells

text: of

text: n

text: n

text: n

text: dimension

text: at

text: least

text: (k

text: n

text: For

text: i

text: n

text: let

text: Z

text: Y

text: and

text: for

text: i

text: n

text: let

text: Z

text: The

text: map

text: g

text: is

text: the

text: i

text: i

text: i

text: i

text: identity

text: for

text: i

text: n,

text: and

text: h

text: Z

text: Y

text: is

text: the

text: identity

text: as

text: well.

text: This

text: yields

text: the

text: factorisation

text: i

text: i

text: i

text: The

text: map

text: h•

text: is

text: a

text: levelwise

text: weak

text: equivalence,

text: and

text: so

text: h•

text: is

text: a

text: weak

text: equivalence

text: by

text: Theorem

text: Moreover,

text: W•

text: (n

text: Z•

text: (n

text: and

text: the

text: pair

text: (Z

text: ∆n,

text: Z

text: ∂∆n)

text: is

text: obtained

text: from

text: the

text: pair

text: n

text: n

text: (W

text: ∆n,

text: W

text: ∂∆n)

text: by

text: attaching

text: cells

text: of

text: dimension

text: at

text: least

text: (k

text: so

text: Z•

text: (n)

text: is

text: obtained

text: from

text: n

text: W•

text: (n)

text: by

text: attaching

text: cells

text: of

text: dimension

text: at

text: least

text: (k

text: in

text: particular,

text: g•

text: W•

text: (n)

text: Z•

text: (n)

text: is

text: k-connected.

text: By

text: the

text: inductive

text: hypothesis,

text: j•

text: (n

text: X•

text: (n

text: W•

text: (n

text: is

text: k-connected

text: and

text: j

text: is

text: the

text: identity.

text: Using

text: the

text: notation

text: introduced

text: in

text: the

text: proof

text: of

text: Theorem

text: we

text: get

text: that

text: →n

text: UX

text: UW

text: is

text: k-connected,

text: while

text: UX

text: UW

text: and

text: UX

text: UX

text: UW

text: UW

text: are

text: weak

text: equivalences.

text: From

text: Theorem

text: it

text: follows

text: that

text: j•

text: is

text: k-connected.

text: Using

text: this

text: we

text: can

text: now

text: prove

text: the

text: analogue

text: of

text: Lemma

text: for

text: semi-simplicial

text: spaces,

text: rather

text: than

text: semi-simplicial

text: sets.

text: Lemma

text: For

text: each

text: semi-simplicial

text: space

text: X•,

text: the

text: map

text: X•

text: EX•

text: is

text: a

text: weak

text: homotopy

text: equivalence.

text: Proof.

text: Consider

text: the

text: bi-semi-simplicial

text: set

text: Sing

text: (X

text: with

text: Sing

text: (EX•)

text: E(Sing

text: X•)

text: so

text: giving

text: a

text: p

text: q

text: p

text: p

text: commutative

text: square

text: Sing•X•

text: E(Sing•X•)

text: Sing•(EX•)

text: X•

text: EX•

text: The

text: vertical

text: maps

text: are

text: weak

text: equivalences

text: by

text: Lemma

text: and

text: Theorem

text: and

text: the

text: top

text: map

text: is

text: a

text: weak

text: equivalence

text: by

text: Lemma

text: and

text: Theorem

text: hence

text: the

text: bottom

text: map

text: is

text: a

text: weak

text: equivalence.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 JOHANNES EBERT AND OSCAR RANDAL-WILLIAMS Deﬁnition 2.7. A commutative square X k1 (cid:47)(cid:47) Y 1 1 f g (cid:15)(cid:15) (cid:15)(cid:15) X k0 (cid:47)(cid:47) Y 0 0 ∈ → is called homotopy cartesian if for each basepoint x X , the map hoﬁb (f) hoﬁb (g), 0 x k0(x) induced by k and k , is a weak homotopy equivalence. 0 1 ∈ Remark 2.8. Equivalently, one can express this condition by saying that for all y Y , the induced → 1 map hoﬁb (k ) hoﬁb (k ) is a weak homotopy equivalence. y 1 g(y) 0 More symmetrically, one can express this condition by saying that the canonical map from X 1 to the homotopy ﬁbre product × { ∈ × × | } X h Y := (x , y , γ) X Y map([0, 1], Y ) γ(0) = k (x ), γ(1) = g(y ) 0 Y0 1 0 1 0 1 0 0 0 1 is a weak homotopy equivalence. Let us record the 2-out-of-3 properties enjoyed by homotopy cartesian squares. Given adjacent commutative squares X k1 (cid:47)(cid:47) Y l1 (cid:47)(cid:47) Z 1 1 1 f g h (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) X k0 (cid:47)(cid:47) Y l0 (cid:47)(cid:47) Z 0 0 0 then (i) if the left and right squares are homotopy cartesian, the outer square is homotopy cartesian; (ii) is the right and outer squares are homotopy cartesian, the left square is homotopy cartesian; (iii) if the left and outer squares are homotopy cartesian, and k is 0-connected, the right square 0 is homotopy cartesian. → Deﬁnition 2.9. A map f• : X• Y• of semi-simplicial spaces is called homotopy cartesian if for ≥ ≤ ≤ each p 1 and each 0 i p, the square X di (cid:47)(cid:47) X − p p 1 (2.10) fp fp−1 (cid:15)(cid:15) (cid:15)(cid:15) Y di (cid:47)(cid:47)(cid:47)(cid:47) Y − p p 1 is homotopy cartesian. For each p there are p + 1 conditions to be checked. The next lemma shows that the number of conditions to be checked can be drastically reduced. → Lemma 2.11. To prove that f• : X• Y• is homotopy cartesian, it is enough to verify that (2.10) is homotopy cartesian for those (p, i) with i = 0 and for (p, i) = (1, 1). Dually, it is enough to verify that (2.10) is homotopy cartesian for those (p, i) with i = p and for (p, i) = (1, 0). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JOHANNES

text: EBERT

text: AND

text: OSCAR

text: RANDAL-WILLIAMS

text: Deﬁnition

text: A

text: commutative

text: square

text: X

font: default

text: Y

text: f

text: g

text: X

text: Y

text: is

text: called

text: homotopy

text: cartesian

text: if

text: for

text: each

text: basepoint

text: x

text: X

text: the

text: map

text: hoﬁb

text: (f)

text: hoﬁb

text: (g),

text: x

text: induced

text: by

text: k

text: and

text: k

text: is

text: a

text: weak

text: homotopy

text: equivalence.

text: Remark

text: Equivalently,

text: one

text: can

text: express

text: this

text: condition

text: by

text: saying

text: that

text: for

text: all

text: y

text: Y

text: the

text: induced

text: map

text: hoﬁb

text: (k

text: hoﬁb

text: (k

text: is

text: a

text: weak

text: homotopy

text: equivalence.

text: y

text: g(y)

text: More

text: symmetrically,

text: one

text: can

text: express

text: this

text: condition

text: by

text: saying

text: that

text: the

text: canonical

text: map

text: from

text: X

text: to

text: the

text: homotopy

text: ﬁbre

text: product

text: X

text: h

text: Y

text: (x

text: y

text: γ)

text: X

text: Y

text: Y

text: k

text: (x

text: g(y

text: is

text: a

text: weak

text: homotopy

text: equivalence.

text: Let

text: us

text: record

text: the

text: properties

text: enjoyed

text: by

text: homotopy

text: cartesian

text: squares.

text: Given

text: adjacent

text: commutative

text: squares

text: X

text: Y

text: Z

text: f

text: g

text: h

text: X

text: Y

text: Z

text: then

text: (i)

text: if

text: the

text: left

text: and

text: right

text: squares

text: are

text: homotopy

text: cartesian,

text: the

text: outer

text: square

text: is

text: homotopy

text: cartesian;

text: (ii)

text: is

text: the

text: right

text: and

text: outer

text: squares

text: are

text: homotopy

text: cartesian,

text: the

text: left

text: square

text: is

text: homotopy

text: cartesian;

text: (iii)

text: if

text: the

text: left

text: and

text: outer

text: squares

text: are

text: homotopy

text: cartesian,

text: and

text: k

text: is

text: the

text: right

text: square

text: is

text: homotopy

text: cartesian.

text: Deﬁnition

text: A

text: map

text: f•

text: X•

text: Y•

text: of

text: semi-simplicial

text: spaces

text: is

text: called

text: homotopy

text: cartesian

text: if

text: for

text: each

text: p

text: and

text: each

text: i

text: p,

text: the

text: square

text: X

text: di

text: X

text: p

text: p

text: fp

text: Y

text: di

text: Y

text: p

text: p

text: is

text: homotopy

text: cartesian.

text: For

text: each

text: p

text: there

text: are

text: p

text: conditions

text: to

text: be

text: checked.

text: The

text: next

text: lemma

text: shows

text: that

text: the

text: number

text: of

text: conditions

text: to

text: be

text: checked

text: can

text: be

text: drastically

text: reduced.

text: Lemma

text: To

text: prove

text: that

text: f•

text: X•

text: Y•

text: is

text: homotopy

text: cartesian,

text: it

text: is

text: enough

text: to

text: verify

text: that

text: is

text: homotopy

text: cartesian

text: for

text: those

text: (p,

text: i)

text: with

text: i

text: and

text: for

text: (p,

text: i)

text: Dually,

text: it

text: is

text: enough

text: to

text: verify

text: that

text: is

text: homotopy

text: cartesian

text: for

text: those

text: (p,

text: i)

text: with

text: i

text: p

text: and

text: for

text: (p,

text: i)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: (cid:34) (cid:40) (cid:82) (cid:35) 17 (cid:82) | | ∞ − D = π c√1/2 F (ξ) +(cid:40)2(cid:82)|c|3/2 ξ√+ −dx F1∞/3(x) F2/3(−ξ) c > 0 , (A62) 1/3 + 16 3 (cid:82) π 3 ξ dx F1/3(x) F2/3(ξ) c < 0 ∞ − 5 √+ dx F (x) F (ξ) c > 0 Ie = 2 |c|3/2 F1/3 (ξ) + πξ 3 − ξ+1∞/3dx F1/3(2x/3) − F2/3(ξ) c < 0 . (A63) 6. Sixth case I f and perform integrations over τ and x using (A5). + We compute (cid:90) We then perform the change of variable (A24) to obtain ∞ I = (cid:12)(cid:12)(cid:90)√4 d(cid:20)x (cid:18)x4 (cid:19)(cid:21)(cid:12)(cid:12) (A64) (cid:90) (cid:104) (cid:16) (cid:17)(cid:105) f (cid:12)π 3c2 −∞ (cid:12) (cid:12) (cid:12) ∞ dτ exp ı (c + x2)τ + τ3 2 . I = √24 lim −i|π| ∞ dy e 32ξı ysc+y33 , −∞ 3 f π 3c2 (cid:15)→0 32 c −∞ y3 Performing the c(cid:90)hange of v(cid:90)ariable (A(cid:90)11) we get that ∞ ∞ ∞ √4 (cid:20) (cid:18) (cid:19)(cid:21) where sc = sign(c). I = 2 (cid:2) (cid:3) dx x4 dτ dτ− (A65) f π 3c2 −∞ −∞ + −∞ Here we recognize in the limit integral (A60), which value is given in (A62). Therefore the ﬁnal result is τ−2 exp 2ıτ−τ2 exp 2ıτ− (c + x2) + . + 3 Is is not possible to to exchange integration over x with integration over τ−. We work around this by inserting a (cid:40) (cid:82)1 positive real paramete(cid:90)r (cid:15) (cid:90) (cid:90) If = 2 |c|3/2 F1/(cid:82)3 (ξ) + (A67) ∞ − √+ dx F (x) F (ξ) c > 0 ∞ ∞ ∞ ξ − 1∞/3 2/3 − . √8 + If = π 3c2 (cid:15)l→im0 −(cid:20)∞ dx x4 (cid:18)−∞ dτ+ −∞ dτ(cid:19)−(cid:21)(A66) π 3 ξ dx F1/3(x) F2/3(ξ) c < 0 e[2ı(τ−+i(cid:15))τ+2]e 2ı(τ−+i(cid:15)) (c+x2)+(τ−+3i(cid:15))2 , [1] G. Voisin, S. Bonazzola, and F. Mottez, Physical Review D [1(22]01A7.).K. Harding and R. Preece, The Astrophysical Journal 319, 939 (1987). [2] J. Arons, in Neutron Stars and Pulsars, Astrophysics and Sp[1a3ce] IS.cCieanbcaellLeirboraarnydNJ.o.W35il7m,se,dMiteedmobryieWd.elBlaecSkoecrie(tSapArinstgreornomica Italiana 83, 230 (2012). Berlin Heidelberg, 2009) pp. 373–420. [14] J. Schwinger, Proceedings of the National Academy of Sciences 40, 132 (1954). [3] G. Voisin, S. Bonazzola, and F. Mottez, in SF2A-2016: P[1r5o]ceJe.diSncghswoinfgtehreaAndnnWua.-ly.mTeesatiin, gAnofnatlhseoFf rPehnycshicSso1c1ie0t,y6o3f(1978). Astronomy and Astrophysics (2016). [16] M. Le Bellac, Physique quantique (EDP Sciences, Les Ulis; Paris, 2003). [4] M. A. Ruderman and P. G. Sutherland, The Astrophysical J[o1u7r]nVal.1B9.6B, e5r1es(t1e9t7sk5)ii., E. M. Lifshitz, and L. P. Pitaevski, Quantum electrodynamics (Oxford, Royaume-Uni, 1982). [5] K. S. Cheng and J. L. Zhang, The Astrophysical Journal 463[1,82]7J1.(D19.9J6a)c.kson, Classical Electrodynamics, 3rd Edition (1998). [6] J. L. Zhang and Y. F. Yuan, The Astrophysical Journal 493,[1892]6J(.1S9c9h8w).inger, Physical Review 75, 1912 (1949). [7] T. Harko and K. S. Cheng, Monthly Notices of the Royal As[t2r0o]noFm. iWca.lJS.oOcilevteyr 3an3d5,N9a9ti(o2n0a0l2I)n. stitute of Standards and Technology (U.S.), eds., NIST handbook of mathematical functions [8] D. Vigano`, D. F. Torres, K. Hirotani, and M. E. Pessah, Month(lyCaNmobtirciedsgeofUtnhieveRrsoiytaylPArsetsrso:noNmISicTa,l CSoacmiebtryid4g4e7;,N1e1w64York, 2010). (2014). [21] G. Voisin, S. Bonazzola, and F. Mottez, in Gamma2016 (2016). [9] S. R. Kelner, A. Y. Prosekin, and F. A. Aharonian, The Ast[r2o2n]oIm. Sic.aGl Jraodusrhntaely1n4, 9I., M33. (R2y01zh5)ik,,aArX. iJve:ﬀ1r5e0y,1.0a4n9d94D.. Zwillinger, Table of Integrals, Series, and Products (2000). [10] A. A. Sokolov and I. M. Ternov, Synchrotron Radiation (Aka[d23em] Gie.-VNe.rlWaga,t1so9n6,8)A. treatise on the theory of Bessel functions, 2nd ed. (Cambridge UniversityPress, Cambridge, 1966). [11] H. G. Latal, The Astrophysical Journal 309, 372 (1986). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: D

text: π

text: F

text: (ξ)

text: ξ√+

text: −dx

text: c

text: π

text: ξ

text: dx

text: c

text: dx

text: F

text: (x)

text: F

text: (ξ)

text: c

text: Ie

text: (ξ)

text: πξ

text: c

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text: Sixth

text: case

text: I

text: f

text: and

text: perform

text: integrations

text: over

text: τ

text: and

text: x

text: using

text: We

text: compute

text: We

text: then

text: perform

text: the

text: change

text: of

text: variable

text: to

text: obtain

text: I

text: f

text: dτ

text: exp

text: ı

text: (c

text: τ3

text: I

text: lim

text: −i|π|

text: dy

text: e

text: 32ξı

text: f

text: π

text: c

text: Performing

text: the

text: of

text: we

text: get

text: that

text: √4

text: where

text: sc

text: sign(c).

text: I

text: dx

text: dτ

text: dτ−

text: f

text: π

text: Here

text: we

text: recognize

text: in

text: the

text: limit

text: integral

text: which

text: value

text: is

text: given

text: in

text: Therefore

text: the

text: ﬁnal

text: result

text: is

text: τ−2

text: exp

text: 2ıτ−τ2

text: exp

text: 2ıτ−

text: (c

text: Is

text: is

text: not

text: possible

text: to

text: to

text: exchange

text: integration

text: over

text: x

text: with

text: integration

text: over

text: τ−.

text: We

text: work

text: around

text: this

text: by

text: inserting

text: a

text: positive

text: real

text: If

text: (ξ)

text: dx

text: F

text: (x)

text: F

text: (ξ)

text: c

text: ξ

text: √8

text: If

text: π

text: dx

text: dτ+

text: π

text: ξ

text: dx

text: c

text: G.

text: Voisin,

text: S.

text: Bonazzola,

text: and

text: F.

text: Mottez,

text: Physical

text: Review

text: D

text: Harding

text: and

text: R.

text: Preece,

text: The

text: Astrophysical

text: Journal

text: J.

text: Arons,

text: in

text: Neutron

text: Stars

text: and

text: Pulsars,

text: Astrophysics

text: and

text: Italiana

text: Berlin

text: Heidelberg,

text: pp.

text: J.

text: Schwinger,

text: Proceedings

text: of

text: the

text: National

text: Academy

text: of

text: Sciences

text: G.

text: Voisin,

text: S.

text: Bonazzola,

text: and

text: F.

text: Mottez,

text: in

text: gAnofnatlhseoFf

text: Astronomy

text: and

text: Astrophysics

text: M.

text: Le

text: Bellac,

text: Physique

text: quantique

text: (EDP

text: Sciences,

text: Les

text: Ulis;

text: Paris,

text: M.

text: A.

text: Ruderman

text: and

text: P.

text: G.

text: Sutherland,

text: The

text: Astrophysical

text: E.

text: M.

text: Lifshitz,

text: and

text: L.

text: P.

text: Pitaevski,

text: Quantum

text: electrodynamics

text: (Oxford,

text: Royaume-Uni,

text: K.

text: S.

text: Cheng

text: and

text: J.

text: L.

text: Zhang,

text: The

text: Astrophysical

text: Journal

text: Classical

text: Electrodynamics,

text: Edition

text: J.

text: L.

text: Zhang

text: and

text: Y.

text: F.

text: Yuan,

text: The

text: Astrophysical

text: Journal

text: Physical

text: Review

text: T.

text: Harko

text: and

text: K.

text: S.

text: Cheng,

text: Monthly

text: Notices

text: of

text: the

text: Royal

text: iWca.lJS.oOcilevteyr

text: stitute

text: of

text: Standards

text: and

text: Technology

text: (U.S.),

text: eds.,

text: NIST

text: handbook

text: of

text: mathematical

text: functions

text: D.

text: Vigano`,

text: D.

text: F.

text: Torres,

text: K.

text: Hirotani,

text: and

text: M.

text: E.

text: Pessah,

text: G.

text: Voisin,

text: S.

text: Bonazzola,

text: and

text: F.

text: Mottez,

text: in

text: S.

text: R.

text: Kelner,

text: A.

text: Y.

text: Prosekin,

text: and

text: F.

text: A.

text: Aharonian,

text: The

text: Sic.aGl

text: Zwillinger,

text: Table

text: of

text: Integrals,

text: Series,

text: and

text: Products

text: A.

text: A.

text: Sokolov

text: and

text: I.

text: M.

text: Ternov,

text: Synchrotron

text: Radiation

text: treatise

text: on

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text: theory

text: of

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text: functions,

text: ed.

text: (Cambridge

text: UniversityPress,

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text: H.

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text: 5 Therefore there is a range of magnetic ﬁelds and elec- ∼ tron energies (remember that ω ω ) for which this crit − amplitude is small. For example, for a ”typical” pulsar Icos = 2π(ω/2Ω) 1/3Ai(x), (33) with B = 108 Teslas, γ = 107 and a dipolar magnetic I = −2π(ω/2Ω)−2/3Ai(cid:48)(x). (34) sin ﬁeld with curvature next to the pole of ρ = 104 m (see (35) e.g. [2]) one has For practical calculations, the Airy integrals can be λω /c (cid:39) − − crit 0.05γ2ρ 1B 1/2. (26) changed into modiﬁed Besse(cid:115)l functions K γ 7 4 8 ν For now, we can legitimately consider these corrections 3 K (ξ) = π | |Ai(x), (36) to be negligible. This amounts to consider that the parti- 1/3 √x cle is inﬁnitely conﬁned, λ = 0, as in the classical theory. We bring back the deconﬁnement corrections in the next − 3 (cid:48) K (ξ) = π Ai (x), (37) sections. 2/3 x We proceed to integrate the expressions in the current | | (15). Integration over φ simply yields a factor 2π since with ξ = 2 x 3/2 and assuming x > 0 . 3 within our approximation of inﬁnite conﬁnement there is We now calculate the intensities. We need to compute − no explicit dependence in φ. Integration over x of xe x2 the matrix elements (5) for both parallel and perpendic- yields a factor 1/2. To integrate over θ, we use the fact ular polarizations. We seek a result to the lowest ultra- that sin θ and cos θ are slowly varying compared to the relativistic order. For this, it is useful to see that owing (cid:29) ∼ exponential for θ 1/γ to develop them to ﬁrst order in to the θ factor in (28) I 1 I . Further, the polariza- sin γ cos ∼ θ. Moreover, we extent the boundaries(cid:28)to inﬁnity since tion vectors (18) are expanded to ﬁrst order in κ 1/γ the contributing part is centered on θ 1. We get the such that the squared matrix elements for respectively two following inte(cid:90)grals parallel and perpendicular polarizations are (cid:16) (cid:17) ∞ Icos = (cid:90)−+∞ ei2ωΩ θ(cid:16)(κ2+γ1˜ )+θ33 (cid:17)dθ , (27) M0(cid:107)02 = 2(cid:15) (cid:126)eω2 V Is2in, (38) 0 k ∞ Isin = −+∞ θei2ωΩ θ(κ2+γ1˜ )+θ33 dθ , (28) M0⊥02 = 2(cid:15)0(cid:126)eω2 kV κ2Ic2os. (39) (29) Inserting the above matrix elements in the expression of the intensity (8) and expressing I and I with mod- cos sin and iﬁed Bessel functions one ob(cid:18)tains (cid:19) (cid:107) 1 d2I 1 e2ω2 (cid:18) 1 2(cid:19) j = (0, I , I ) . (30) 00 = κ2 + K2 (ξ), (40) 00 2π cos sin dodω 2πΩ 12π3(cid:15) c γ2 2/3 0 ⊥ d2I 1 e2ω2 1 00 = (cid:12) (cid:12) κ2 κ2 + K2 (ξ), (41) We recognize in (27) an Airy integral and its derivative dodω 2πΩ(cid:12) 12π3(cid:15)0(cid:12)c γ2 1/3 (cid:12) (cid:12) in (28). We use in this paper the deﬁnitions of special functions of [20] where th(cid:90)e Airy function is given by 3/2 where ξ = ω κ2 + 1 . These expressions are (cid:16) (cid:17) 3Ω γ2 ∞ identical to expressions found in the classical theory (see 1 i xt+t3 e.g. [18]. Ai(x) = dt e 3 . (31) 2π −∞ IV. GENERAL CALCULATION OF After performing the change of variable (cid:16) (cid:17) SYNCHRO-CURVATURE INCLUDING QUANTUM CORRECTIONS − → ω 1/3 θ˜ θ = (cid:16) θ˜ (cid:17) (32) We now generalize the calculation of the previous sec- 2Ω tion to transitions between states of any initial perpen- (cid:48) dicular quantum number n to a ﬁnal number n including one identiﬁes x = (ω/2Ω)2/3 κ2 + γ˜12 and obtains quantum corrections up to second order in (cid:126)Eω. The need ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Therefore

text: there

text: is

text: a

text: range

text: of

text: magnetic

text: ﬁelds

text: and

text: elec-

text: tron

text: energies

text: (remember

text: that

text: ω

text: ω

text: for

text: which

text: this

text: crit

text: amplitude

text: is

text: small.

text: For

text: example,

text: for

text: a

text: ”typical”

text: pulsar

text: Icos

text: with

text: B

text: Teslas,

text: γ

text: and

text: a

text: dipolar

text: magnetic

text: I

text: sin

text: ﬁeld

text: with

text: curvature

text: next

text: to

text: the

text: pole

text: of

text: ρ

text: m

text: (see

text: e.g.

text: one

text: has

text: For

text: practical

text: calculations,

text: the

text: Airy

text: integrals

text: can

text: be

text: λω

text: /c

text: crit

text: changed

text: into

text: modiﬁed

text: functions

text: K

text: γ

text: ν

text: For

text: now,

text: we

text: can

text: legitimately

text: consider

text: these

text: corrections

text: K

text: (ξ)

text: π

text: |Ai(x),

text: to

text: be

text: negligible.

text: This

text: amounts

text: to

text: consider

text: that

text: the

text: parti-

text: √x

text: cle

text: is

text: inﬁnitely

text: conﬁned,

text: λ

text: as

text: in

text: the

text: classical

text: theory.

text: We

text: bring

text: back

text: the

text: deconﬁnement

text: corrections

text: in

text: the

text: next

text: K

text: (ξ)

text: π

text: Ai

text: (x),

text: sections.

text: x

text: We

text: proceed

text: to

text: integrate

text: the

text: expressions

text: in

text: the

text: current

text: Integration

text: over

text: φ

text: simply

text: yields

text: a

text: factor

text: 2π

text: since

text: with

text: ξ

text: x

text: and

text: assuming

text: x

text: within

text: our

text: approximation

text: of

text: inﬁnite

text: conﬁnement

text: there

text: is

text: We

text: now

text: calculate

text: the

text: intensities.

text: We

text: need

text: to

text: compute

text: no

text: explicit

text: dependence

text: in

text: φ.

text: Integration

text: over

text: x

text: of

text: xe

text: the

text: matrix

text: elements

text: for

text: both

text: parallel

text: and

text: perpendic-

text: yields

text: a

text: factor

text: To

text: integrate

text: over

text: θ,

text: we

text: use

text: the

text: fact

text: ular

text: polarizations.

text: We

text: seek

text: a

text: result

text: to

text: the

text: lowest

text: ultra-

text: that

text: sin

text: θ

text: and

text: cos

text: θ

text: are

text: slowly

text: varying

text: compared

text: to

text: the

text: relativistic

text: order.

text: For

text: this,

text: it

text: is

text: useful

text: to

text: see

text: that

text: owing

text: exponential

text: for

text: θ

text: to

text: develop

text: them

text: to

text: ﬁrst

text: order

text: in

text: to

text: the

text: θ

text: factor

text: in

text: I

text: I

text: Further,

text: the

text: polariza-

text: sin

text: γ

text: cos

text: θ.

text: Moreover,

text: we

text: extent

text: the

text: inﬁnity

text: since

text: tion

text: vectors

text: are

text: expanded

text: to

text: ﬁrst

text: order

text: in

text: κ

text: the

text: contributing

text: part

text: is

text: centered

text: on

text: θ

text: We

text: get

text: the

text: such

text: that

text: the

text: squared

text: matrix

text: elements

text: for

text: respectively

text: two

text: following

text: parallel

text: and

text: perpendicular

text: polarizations

text: are

text: Icos

text: V

text: k

text: Isin

text: θei2ωΩ

text: θ(κ2+γ1˜

text: dθ

text: kV

text: Inserting

text: the

text: above

text: matrix

text: elements

text: in

text: the

text: expression

text: of

text: the

text: intensity

text: and

text: expressing

text: I

text: and

text: I

text: with

text: mod-

text: cos

text: sin

text: and

text: iﬁed

text: Bessel

text: functions

text: one

text: j

text: I

text: I

text: κ2

text: (ξ),

text: 2π

text: cos

text: sin

text: dodω

text: 2πΩ

text: c

text: γ2

text: κ2

text: κ2

text: (ξ),

text: We

text: recognize

text: in

text: an

text: Airy

text: integral

text: and

text: its

text: derivative

text: dodω

text: γ2

text: in

text: We

text: use

text: in

text: this

text: paper

text: the

text: deﬁnitions

text: of

text: special

text: functions

text: of

text: where

text: Airy

text: function

text: is

text: given

text: by

text: where

text: ξ

text: ω

text: κ2

text: These

text: expressions

text: are

text: 3Ω

text: γ2

text: identical

text: to

text: expressions

text: found

text: in

text: the

text: classical

text: theory

text: (see

text: i

text: e.g.

text: Ai(x)

text: dt

text: e

text: 2π

text: IV.

text: GENERAL

text: CALCULATION

text: OF

text: After

text: performing

text: the

text: change

text: of

text: variable

text: SYNCHRO-CURVATURE

text: INCLUDING

text: QUANTUM

text: CORRECTIONS

text: ω

text: θ˜

text: θ

text: θ˜

text: We

text: now

text: generalize

text: the

text: calculation

text: of

text: the

text: previous

text: sec-

text: 2Ω

text: tion

text: to

text: transitions

text: between

text: states

text: of

text: any

text: initial

text: perpen-

text: dicular

text: quantum

text: number

text: n

text: to

text: a

text: ﬁnal

text: number

text: n

text: including

text: one

text: identiﬁes

text: x

text: κ2

text: γ˜12

text: and

text: obtains

text: quantum

text: corrections

text: up

text: to

text: second

text: order

text: in

text: The

text: need

text: ##LTLine##

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text: ##LTLine##


text: 7 ∼ ∼ We then integr(cid:90)ate over x with only integrals of the type κ θ 1/γ˜, (49) ∞ − p! dx e x2x2p+1 = , (55) and the critical pulsation 2 0 3 where p is a positive integer. ω˜crit = 2Ωγ˜3. (50) We are n(cid:90)ow left with integrals over θ of the type (cid:16)(cid:16) (cid:17) (cid:17) evaWnte hniogwh-pernoecregeydatcocuinratecgyrtaoteseocvoenrdφo. rTdoeroobntaeisnepthaerarteels- Bpq = −π dθ ei2ωΩ κ2+γ12 θ+θ33 cosp θ sinq θ (56) π the imaginary exponential in (44) as in (21) and notices that, similarly to (25), its argu(cid:18)ment (cid:19)is of order where p, q are positive integers. As in the previous ∼ section, the smallness of contributing values of θ 1/γ˜ λk ∼ √ γ (cid:126)ω 2B 1/2 allows to extend boundaries to inﬁnity. Moreover, to x n c , (51) leading ultra-relati(cid:90)vistic order one has γ˜ γ˜ E B (cid:16)(cid:16) (cid:17) (cid:17) ∞ radwihalerdeiswtaenucseedofthane fealcetcttrhoantitsh∼e a√venraagsedexnpolraminaeldizeidn ∀(p, q), Bpq = −∞ dθ ei2ωΩ κ2+γ12 θ+θ33 θq. (57) paper 1. Assuming (51) is small compared to one, we expand the−second factor of (21) to second order in the Using deﬁnition (31) one sees that Bpq is proportional argument ixλk (cos φ sin κ + sin φ cos κ sin θ). We are to the q-th derivative of the Airy function. Reminding left to integrate t(cid:90)erms of the form that the Airy function veriﬁes the relation [20] π (cid:48)(cid:48) A = dφ eipφ (a cos φ + b sin φ)q (52) Ai (x) = xAi(x) (58) pq − π ≥ (cid:48) where p and q are integers and q 0, a and b can have one is able to express every Bpq in terms of Ai and Ai , any value independent of φ. One can show that and from that in terms of Icos and Isin ((27),(28)). In  particular, we need(cid:18)the follo(cid:19)wing expressions  | | q < p Apq = 0 if or− | | . (53) B = − κ2 + 1 I (cid:18) , (cid:19) (59) q p odd p2 γ cos 2 (cid:0) (cid:1) 1 4ω˜ − 1 B = crit I κ2 + I , (60) For this reason, the only transitions yielding terms of p3 γ˜3 3ω cos γ sin order lower or equal to (cid:126)ω 2 once current (6) is inserted 2 (61) E (cid:48) (cid:48) in the squared matrix element (5) are for n = n ,n = (cid:0) −(cid:1) (cid:48) − n 1 and n = n 2. Moreover, one can see that the → next non-null term of the expansion of (23) is of order where the replacement 1/γ2 1/γ is assumed in I 2 cos (cid:126)Eω 4, pushing further the validity of our approximation. and Isin. (cid:0) (cid:1) In practice, we need Squarring (44), inserting it into the matrix element (5) (cid:80)  and using formula (8) we obtain all the relevant intensi- − ties to order (cid:126)ω 2. These intensities are proportional  ππ((aa2++ibb)2) pp == 0 1 qq == 12 to (ζζ(cid:48))2 and tEherefore the spin average 12 ζ,ζ(cid:48)=±1 is A = − . (54) immediate, giving pq π(a ib) p = 1 q = 1 π(a + ib)2 p = 2 q = 2 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: then

text: over

text: x

text: with

text: only

text: integrals

text: of

text: the

text: type

text: κ

text: θ

text: p!

text: dx

text: e

text: and

text: the

text: critical

text: pulsation

text: where

text: p

text: is

text: a

text: positive

text: integer.

text: ω˜crit

text: 2Ωγ˜3.

text: We

text: are

text: left

text: with

text: integrals

text: over

text: θ

text: of

text: the

text: type

text: evaWnte

text: rTdoeroobntaeisnepthaerarteels-

text: Bpq

text: −π

text: dθ

text: cosp

text: θ

text: sinq

text: θ

text: π

text: the

text: imaginary

text: exponential

text: in

text: as

text: in

text: and

text: notices

text: that,

text: similarly

text: to

text: its

text: of

text: order

text: where

text: p,

text: q

text: are

text: positive

text: integers.

text: As

text: in

text: the

text: previous

text: section,

text: the

text: smallness

text: of

text: contributing

text: values

text: of

text: θ

text: 1/γ˜

text: λk

text: γ

text: allows

text: to

text: extend

text: boundaries

text: to

text: inﬁnity.

text: Moreover,

text: to

text: x

text: n

text: c

text: leading

text: order

text: one

text: has

text: γ˜

text: γ˜

text: E

text: B

text: radwihalerdeiswtaenucseedofthane

text: fealcetcttrhoantitsh∼e

text: a√venraagsedexnpolraminaeldizeidn

text: ∀(p,

text: q),

text: Bpq

text: dθ

text: θq.

text: paper

text: Assuming

text: is

text: small

text: compared

text: to

text: one,

text: we

text: expand

text: the−second

text: factor

text: of

text: to

text: second

text: order

text: in

text: the

text: Using

text: deﬁnition

text: one

text: sees

text: that

text: Bpq

text: is

text: proportional

text: argument

text: ixλk

text: (cos

text: φ

text: sin

text: κ

text: sin

text: φ

text: cos

text: κ

text: sin

text: θ).

text: We

text: are

text: to

text: the

text: q-th

text: derivative

text: of

text: the

text: Airy

text: function.

text: Reminding

text: left

text: to

text: integrate

text: of

text: the

text: form

text: that

text: the

text: Airy

text: function

text: veriﬁes

text: the

text: relation

text: π

text: A

text: dφ

text: eipφ

text: (a

text: cos

text: φ

text: b

text: sin

text: φ)q

text: Ai

text: (x)

text: xAi(x)

text: pq

text: π

text: where

text: p

text: and

text: q

text: are

text: integers

text: and

text: q

text: a

text: and

text: b

text: can

text: have

text: one

text: is

text: able

text: to

text: express

text: every

text: Bpq

text: in

text: terms

text: of

text: Ai

text: and

text: Ai

text: any

text: value

text: independent

text: of

text: φ.

text: One

text: can

text: show

text: that

text: and

text: from

text: that

text: in

text: terms

text: of

text: Icos

text: and

text: Isin

text: In

text: particular,

text: we

text: expressions

text: q

text: p

text: Apq

text: if

text: or−

text: B

text: κ2

text: I

text: q

text: p

text: odd

text: γ

text: cos

text: 4ω˜

text: B

text: crit

text: I

text: κ2

text: I

text: For

text: this

text: reason,

text: the

text: only

text: transitions

text: yielding

text: terms

text: of

text: γ˜3

text: 3ω

text: cos

text: γ

text: sin

text: order

text: lower

text: or

text: equal

text: to

text: once

text: current

text: is

text: inserted

text: E

text: in

text: the

text: squared

text: matrix

text: element

text: are

text: for

text: n

text: n

text: ,n

text: n

text: and

text: n

text: n

text: Moreover,

text: one

text: can

text: see

text: that

text: the

text: next

text: non-null

text: term

text: of

text: the

text: expansion

text: of

text: is

text: of

text: order

text: where

text: the

text: replacement

text: is

text: assumed

text: in

text: I

text: cos

text: pushing

text: further

text: the

text: validity

text: of

text: our

text: approximation.

text: and

text: Isin.

text: In

text: practice,

text: we

text: need

text: Squarring

text: inserting

text: it

text: into

text: the

text: matrix

text: element

text: and

text: using

text: formula

text: we

text: obtain

text: all

text: the

text: relevant

text: intensi-

text: ties

text: to

text: order

text: These

text: intensities

text: are

text: proportional

text: pp

text: qq

text: to

text: and

text: tEherefore

text: the

text: spin

text: average

text: is

text: A

text: immediate,

text: giving

text: pq

text: π(a

text: ib)

text: p

text: q

text: π(a

text: p

text: q

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:19) (cid:18) (cid:19) (cid:32)(cid:18) (cid:19) (cid:33) (cid:32) (cid:18) (cid:19) (cid:33) (cid:18) (cid:19) (cid:18) (cid:19) Note that | | ≤ rt k ≤ r t n t r t n/t ≤ r t2 r n E (cid:112) kt (rt)t(cid:112) − e − (rn)t − en/t − . t n/t r 1 t r 1 r 1 r 1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) ≤ Put t := n . Since r = o( n ), one can give an estimate (rn)t n2t = e2tlnn = eo(n/r). Also, lnn lnn t2 o(n) | | n(cid:4) (cid:5) −r = −r and en/t = eo(n/r). Summing up, E(H) = −r + o(1) . r 1 r 1 r 1 Let us show that H has no panchromatic coloring. Suppose the contrary and consider a panchromatic × { } coloring. Let us call a color q a minor color for a line [k] i if it has at most k vertices. By pigeonhole × { } (cid:0) (cid:1) r ⊂ principle every line [k] i has a minor color. Again, by pigeonhole principle there is a set A [rt] of lines | | ≥ { } with the same minor color q such tha(cid:0)t A(cid:1) t. Next, for any ﬁxed β the proportion of such i ∈ that { | ∈ } − α α A ((β + i ) mod k, α) α A has no color q, is at least r 1 t. By the linearity of expectation there is a α{ } − ∈r { | ∈ } choice of i ∈ such that at least k r 1 t = n indices β B give q-free sets ((β + i ) mod k, α) α A . α α A r t α So there is an edge without color q, which gives a contradiction. Acknowledgements The author is supported by the Russian Science Foundation grant 16-11-10014. Also the author is grateful to Misha Basok, Roman Prosanov and Andrei Raigorodskii for very careful reading of the draft of the paper and to Fedya Petrov for some motivating remarks. References [1] Noga Alon. Choice numbers of graphs: a probabilistic approach. Combinatorics, Probability and Computing, 1(02):107–114, 1992. [2] Noga Alon, Joel H. Spencer, and Paul Erd˝os. The probabilistic method. John Wiley & Sons, 2016. [3] Paul Erd˝os, Arthur L. Rubin, and Herbert Taylor. Choosability in graphs. Congr. Numer, 26:125–157, 1979. [4] Paul Erd˝os. On a combinatorial problem, II. Acta Mathematica Hungarica, 15(3-4):445–447, 1964. [5] Paul Erd˝os and L´aszl´o Lov´asz. Problems and results on 3-chromatic hypergraphs and some related questions. Inﬁnite and ﬁnite sets, 10(2):609–627, 1975. [6] Heidi Gebauer. On the construction of 3-chromatic hypergraphs with few edges. Journal of Combina- torial Theory, Series A, 120(7):1483–1490, 2013. [7] Alexandr Kostochka. On a theorem of Erd˝os, Rubin, and Taylor on choosability of complete bipartite graphs. The Electronic Journal of Combinatorics, 9(9):1, 2002. [8] Alexandr Kostochka. Color-critical graphs and hypergraphs with few edges: a survey. In More sets, graphs and numbers, pages 175–197. Springer, 2006. [9] Alexandr V. Kostochka and Douglas R. Woodall. Density conditions for panchromatic colourings of hypergraphs. Combinatorica, 21(4):515–541, 2001. [10] Andr´as Pluh´ar. Greedy colorings of uniform hypergraphs. Random Structures & Algorithms, 35(2):216– 221, 2009. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: rt

text: k

text: r

text: t

text: n

text: t

text: r

text: t

text: n/t

text: r

text: r

text: n

text: E

text: kt

text: e

text: (rn)t

text: en/t

text: t

text: n/t

text: r

text: t

text: r

text: r

text: r

text: Put

text: t

text: n

text: Since

text: r

text: o(

text: n

text: one

text: can

text: give

text: an

text: estimate

text: (rn)t

text: eo(n/r).

text: Also,

text: lnn

text: lnn

text: o(n)

text: −r

text: −r

text: and

text: en/t

text: eo(n/r).

text: Summing

text: up,

text: E(H)

text: −r

text: r

text: r

text: r

text: Let

text: us

text: show

text: that

text: H

text: has

text: no

text: panchromatic

text: coloring.

text: Suppose

text: the

text: contrary

text: and

text: consider

text: a

text: panchromatic

text: coloring.

text: Let

text: us

text: call

text: a

text: color

text: q

text: a

text: minor

text: color

text: for

text: a

text: line

text: [k]

text: i

text: if

text: it

text: has

text: at

text: most

text: k

text: vertices.

text: By

text: pigeonhole

text: r

text: principle

text: every

text: line

text: [k]

text: i

text: has

text: a

text: minor

text: color.

text: Again,

text: by

text: pigeonhole

text: principle

text: there

text: is

text: a

text: set

text: A

text: [rt]

text: of

text: lines

text: with

text: the

text: same

text: minor

text: color

text: q

text: such

text: t.

text: Next,

text: for

text: any

text: ﬁxed

text: β

text: the

text: proportion

text: of

text: such

text: i

text: that

text: α

text: α

text: A

text: ((β

text: i

text: mod

text: k,

text: α)

text: α

text: A

text: has

text: no

text: color

text: q,

text: is

text: at

text: least

text: r

text: t.

text: By

text: the

text: linearity

text: of

text: expectation

text: there

text: is

text: a

text: α{

text: ∈r

text: choice

text: of

text: i

text: such

text: that

text: at

text: least

text: k

text: r

text: t

text: n

text: indices

text: β

text: B

text: give

text: q-free

text: sets

text: ((β

text: i

text: mod

text: k,

text: α)

text: α

text: A

text: α

text: α

text: A

text: r

text: t

text: α

text: So

text: there

text: is

text: an

text: edge

text: without

text: color

text: q,

text: which

text: gives

text: a

text: contradiction.

text: Acknowledgements

text: The

text: author

text: is

text: supported

text: by

text: the

text: Russian

text: Science

text: Foundation

text: grant

text: Also

text: the

text: author

text: is

text: grateful

text: to

text: Misha

text: Basok,

text: Roman

text: Prosanov

text: and

text: Andrei

text: Raigorodskii

text: for

text: very

text: careful

text: reading

text: of

text: the

text: draft

text: of

text: the

text: paper

text: and

text: to

text: Fedya

text: Petrov

text: for

text: some

text: motivating

text: remarks.

text: References

text: Noga

text: Alon.

text: Choice

text: numbers

text: of

text: graphs:

text: a

text: probabilistic

text: approach.

text: Combinatorics,

text: Probability

text: and

text: Computing,

text: Noga

text: Alon,

text: Joel

text: H.

text: Spencer,

text: and

text: Paul

text: Erd˝os.

text: The

text: probabilistic

text: method.

text: John

text: Wiley

text: Sons,

text: Paul

text: Erd˝os,

text: Arthur

text: L.

text: Rubin,

text: and

text: Herbert

text: Taylor.

text: Choosability

text: in

text: graphs.

text: Congr.

text: Numer,

text: Paul

text: Erd˝os.

text: On

text: a

text: combinatorial

text: problem,

text: II.

text: Acta

text: Mathematica

text: Hungarica,

text: Paul

text: Erd˝os

text: and

text: L´aszl´o

text: Lov´asz.

text: Problems

text: and

text: results

text: on

text: hypergraphs

text: and

text: some

text: related

text: questions.

text: Inﬁnite

text: and

text: ﬁnite

text: sets,

text: Heidi

text: Gebauer.

text: On

text: the

text: construction

text: of

text: hypergraphs

text: with

text: few

text: edges.

text: Journal

text: of

text: Combina-

text: torial

text: Theory,

text: Series

text: A,

text: Alexandr

text: Kostochka.

text: On

text: a

text: theorem

text: of

text: Erd˝os,

text: Rubin,

text: and

text: Taylor

text: on

text: choosability

text: of

text: complete

text: bipartite

text: graphs.

text: The

text: Electronic

text: Journal

text: of

text: Combinatorics,

text: Alexandr

text: Kostochka.

text: Color-critical

text: graphs

text: and

text: hypergraphs

text: with

text: few

text: edges:

text: a

text: survey.

text: In

text: More

text: sets,

text: graphs

text: and

text: numbers,

text: pages

text: Springer,

text: Alexandr

text: V.

text: Kostochka

text: and

text: Douglas

text: R.

text: Woodall.

text: Density

text: conditions

text: for

text: panchromatic

text: colourings

text: of

text: hypergraphs.

text: Combinatorica,

text: Andr´as

text: Pluh´ar.

text: Greedy

text: colorings

text: of

text: uniform

text: hypergraphs.

text: Random

text: Structures

text: Algorithms,

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: [11] Andrei M. Raigorodskii and Dmitrii A. Shabanov. The Erd˝os-Hajnal problem of hypergraph colouring, its generalizations, and related problems. Russian Mathematical Surveys, 66(5):933, 2011. [12] Anastasiya P. Rozovskaya and Dmitrii A. Shabanov. Improvement of the lower bound in the Kostochka problem of panchromatic coloring of a hypergraph. Mathematical Notes, 89(5):903–906, 2011. [13] Dmitrii A. Shabanov. The existence of panchromatic colourings for uniform hypergraphs. Sbornik: Mathematics, 201(4):607, 2010. [14] Dmitry A. Shabanov. On a generalization of Rubin’s theorem. Journal of Graph Theory, 67(3):226–234, 2011. [15] Alexander Sidorenko. What we know and what we do not know about Tur´an numbers. Graphs and Combinatorics, 11(2):179–199, 1995. 8

label: reference

text: Andrei

text: M.

text: Raigorodskii

text: and

text: Dmitrii

text: A.

text: Shabanov.

text: The

text: Erd˝os-Hajnal

text: problem

text: of

text: hypergraph

text: colouring,

text: its

text: generalizations,

text: and

text: related

text: problems.

text: Russian

text: Mathematical

text: Surveys,

text: Anastasiya

text: P.

text: Rozovskaya

text: and

text: Dmitrii

text: A.

text: Shabanov.

text: Improvement

text: of

text: the

text: lower

text: bound

text: in

text: the

text: Kostochka

text: problem

text: of

text: panchromatic

text: coloring

text: of

text: a

text: hypergraph.

text: Mathematical

text: Notes,

text: Dmitrii

text: A.

text: Shabanov.

text: The

text: existence

text: of

text: panchromatic

text: colourings

text: for

text: uniform

text: hypergraphs.

text: Sbornik:

text: Mathematics,

text: Dmitry

text: A.

text: Shabanov.

text: On

text: a

text: generalization

text: of

text: Rubin’s

text: theorem.

text: Journal

text: of

text: Graph

text: Theory,

text: Alexander

text: Sidorenko.

text: What

text: we

text: know

text: and

text: what

text: we

text: do

text: not

text: know

text: about

text: Tur´an

text: numbers.

text: Graphs

text: and

text: Combinatorics,


text: 8 concurrent with the noise-increasing processes. The mi- is related to the reported glassy properties of the cyto- croscopic interpretation of this result is unclear. One plasm system [11], which should also aﬀect the nucleoid. possibility is that this increased diﬀusivity is due to non- In this framework, active relocation might cause a ﬂu- thermal active ﬂuctuations [13]. However, nonthermal idization eﬀect, releasing local portions of cytoplasm and random forces are expected to dominate at long time nucleoid from “cages” where they are otherwise con- lags. The noise increase has been proven to be ATP- ﬁned with limited mobility. In this case, the diﬀerential dependent, but not associated to any speciﬁc process noise would be due to heterogeneity in the sub-diﬀusion such as the activity to DNA gyrase (Topoisomerase favor- process [35, 38]. Possibly more general sources of hetero- ing relaxation of positive supercoiling) or depolymeriza- geneity than glassyness could also lead to similar eﬀects. tion of MreB (cell wall biosynthesis), and is only weakly Such kind of disorder has been recently implicated for linked to RNA polymerase activity. Active relocations the motion of cytoplasmic particles [12]. A more pre- have been previously interpreted as relaxation of stress cise dissection of such hypothesis requires a theoretical (generated by process such as DNA replication and tran- approach that incorporates explicitly the more complex scription) due to release of internal tethering interactions physical ingredient of crowding and glassy behavior. (e.g., by bridging proteins such as H-NS and Fis or con- V. ACKNOWLEDGMENTS densins such as MukBEF) [1, 8, 9]. This kind of motion is not necessarily associated to increased noise, because the We are grateful to Pietro Cicuta, Kevin Dorfman, stress-release events might be well-separated temporally Alessandro Taloni for helpful discussions. This work was from the stress-generating ones. supported by the International Human Frontier Science Another possibile interpretation (complementary to Program Organization, grant RGY0070/2014. MT is the previous one) of the increased mobility in presence grateful for the ﬁnancial support of the EU-Horizon2020 of active movements might explain this behavior. This IRSES program DIONICOS (612707). ∗ [17] E. Lutz, Phys. Rev. E 64, 051106 (2001). † marco.cosentino-lagomarsino@upmc.fr [18] E. Barkai and J. Klafter, Physical review letters 81, 1134 (1998). [1] N. Kleckner, J. K. Fisher, M. Stouf, M. A. White, D. Bates, [1a9n]dAG. .TWaloitnzi,, CAu.rCrhOepchinkiMn,icaronbdioJl.2K2l,af1t2e7r,(P20h1y4s)ic.al review letters 104, 160602 (2010). [2] M. Cosentino Lagomarsino, O. Esp´eli, and I. Junier, FEBS L[2e0t]t N58. 9J., K29u9w6a(d2a0,1K5)..C. Cheveralls, B. Traxler, and P. A. Wiggins, Nucleic Acids Res. 41, 1 (2013). [3] V. Benza, B. Bassetti, K. Dorfman, V. Scolari, K. Bromek, P[2.1C]icTu.tSa,akaanude aMn.dCTo.seSnatiitnoo, SLoafgtoMmaatrtseinro,(2R0e1p6.).Prog. Phys. 75, 076602 (2012). [22] H. Vandebroek and C. Vanderzande, Phys. Rev. E 92, 060601 (2015). [4] A. Javer, Z. Long, E. Nugent, M. Grisi, K. Siriwatwetchakul,[2K3.] DA..DYo.rGfmroasnb,ePrg. CanicdutJa.,-Fa.nJdoaMnn. yC,oPshenytsi.nRoeLva.gEom92ar,s0in3o2,118 (2015). Nat. Commun. 4, 3003 (2013). [24] D. Osmanovic and Y. Rabin, Soft Matter 13, 963 (2017). [5] S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev.[2L5e]tCt..1L0o4v,er2d3o8,10O2. (B2´e0n1i0c)h.ou, M. Moreau, and R. Voituriez, Phys Rev E Stat Nonlin Soft Matter Phys 80, 031146 (2009). [6] S. C. Weber, J. A. Theriot, and A. J. Spakowitz, Phys. Rev.[2E6]8O2,. 0B1´e1n9ic1h3o(u2,0C10.)L. overdo, M. 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Deng and E. Barkai, Phys. Rev. E 79, 011112 (2009). [11] B. R. Parry, I. V. Surovtsev, M. T. Cabeen, C. S. O’Hern, E[.3R3]. DH.ufQreiasnne,,“ParnodceCss.eJsawcoitbhs-lWonagg-nraern,gCe eclolr1re5l6a,ti1o8n3s,”(20(1S4p)r.inger Berlin Heidelberg, 2003) Chap. Fractional Brownian Motion [12] T. J. Lampo, S. Stylianidou, M. P. Backlund, P. A. Wiggins, andanAd. FJ.raScptaioknoawliGtza,uBsisoiapnhyNsiociasle.Journal 112, 532 (2017). [13] S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Proc. Natl[.3A4]caMd.. MSc.i.MUeSeArsc1h0a9er,t7a3n38d (F2.0S12a)b.zikar, Statistics & Probability Letters 83, 2269 (2013). [14] T. J. Lampo, A. S. Kennard, and A. J. Spakowitz, Biophysi[c3a5l]joBu.rWnaaln1g1,0J,.3K3u8o(,2S0.16C).. Bae, and S. Granick, Nature materials 11, 481 (2012). [15] K. Polovnikov, M. Gherardi, M. Cosentino Lagomarsino, an[d36M] .I.TBamromns,hatreXini,v:I1.7K03a.n1t0e8r,41E.(K20e1p7te).n, M. Lindner, S. Berezin, Y. Shav-Tal, and Y. Garini, Nucleus 7, 27 (2016). [16] M. Gherardi and M. Cosentino Lagomarsino, Procedures for [m37o]dIe.l-Bguroidnesdtedina,taYa.nIasrlyaseils, Eof. cKherpotmeno,soSm. aMl aloi,ciYd.ySnhaamv-iTcsala,tE. Barkai, and Y. Garini, Phys. Rev. Lett. 103, 018102 (2009). short time scales, edited by O. Espeli, Methods in Molecular B[3I8o]loBgy.,WVoaln.gT, hSe. BMa.ctAernitahloNnuyc, lSeo.idC.- BMaeet,hoadnsdanSd. PGrroatnoiccokl,sProceedings of the National Academy of Sciences of the United (Springer Publishing Company, Incorporated, 2017). States of America 106, 15160 (2009). ∗ Nat. Commun. 4, 3003 (2013). † marco.cosentino-lagomarsino@upmc.fr [5] S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev. Lett. 104, 238102 (2010). [1] N. Kleckner, J. K. Fisher, M. Stouf, M. A. White, D. Bates, a[6n]dSG. C. W. Witezb, eCr,urJr. OAp. iTnhMeriicorto,biaonld22A,.1J2.7S(p2a0k1o4w).itz, Phys. Rev. E 82, 011913 (2010). [2] M. Cosentino Lagomarsino, O. Esp´eli, and I. Junier, FEBS L[e7t]t S5.8C9,. 2W9e9b6e(r2,0M15.)A. . Thompson, W. E. Moerner, A. J. Spakowitz, and J. A. Theriot, Biophys. J. 102, 2443 (2012). [3] V. Benza, B. Bassetti, K. Dorfman, V. Scolari, K. Bromek, P.[8C]icAu.taJ,avaern,dNM. .J.CKosuewnatidnao, LZa.gLoomnagr,sVin.oG, R. eBpe.nPzaro, gK..PDh.ysD. orfman, P. A. Wiggins, P. Cicuta, and M. C. Lagomarsino, Nat 75, 076602 (2012). Commun 5, 3854 (2014). [4] A. Javer, Z. Long, E. Nugent, M. Grisi, K. Siriwatwetchakul, K[9.] DJ.. KD.orFfmishaenr,,PA..CBicouutran,iqaunedl, MG.. WCoitsze,ntBin. oWLeaingeorm, aMrs.iPnor,entiss, and N. Kleckner, Cell 153, 882 (2013). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: concurrent

text: with

text: the

text: noise-increasing

text: processes.

text: The

text: mi-

text: is

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text: the

text: reported

text: glassy

text: properties

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text: the

text: cyto-

text: croscopic

text: interpretation

text: of

text: this

text: result

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text: One

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text: The

text: noise

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text: be

text: ATP-

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text: with

text: limited

text: mobility.

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text: or

text: con-

text: V.

text: ACKNOWLEDGMENTS

text: densins

text: such

text: as

text: MukBEF)

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text: necessarily

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text: noise,

text: because

text: the

text: We

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text: grateful

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text: Pietro

text: Cicuta,

text: Kevin

text: Dorfman,

text: stress-release

text: events

text: might

text: be

text: well-separated

text: temporally

text: Alessandro

text: Taloni

text: for

text: helpful

text: discussions.

text: This

text: work

text: was

text: from

text: the

text: stress-generating

text: ones.

text: supported

text: by

text: the

text: International

text: Human

text: Frontier

text: Science

text: Another

text: possibile

text: interpretation

text: (complementary

text: to

text: Program

text: Organization,

text: grant

text: MT

text: is

text: the

text: previous

text: one)

text: of

text: the

text: increased

text: mobility

text: in

text: presence

text: grateful

text: for

text: the

text: ﬁnancial

text: support

text: of

text: the

text: of

text: active

text: movements

text: might

text: explain

text: this

text: behavior.

text: This

text: IRSES

text: program

text: DIONICOS

text: E.

text: Lutz,

text: Phys.

text: Rev.

text: E

text: marco.cosentino-lagomarsino@upmc.fr

text: E.

text: Barkai

text: and

text: J.

text: Klafter,

text: Physical

text: review

text: letters

text: N.

text: Kleckner,

text: J.

text: K.

text: Fisher,

text: M.

text: Stouf,

text: M.

text: A.

text: White,

text: D.

text: Bates,

text: .TWaloitnzi,,

text: review

text: letters

text: M.

text: Cosentino

text: Lagomarsino,

text: O.

text: Esp´eli,

text: and

text: I.

text: Junier,

text: FEBS

text: Cheveralls,

text: B.

text: Traxler,

text: and

text: P.

text: A.

text: Wiggins,

text: Nucleic

text: Acids

text: Res.

text: V.

text: Benza,

text: B.

text: Bassetti,

text: K.

text: Dorfman,

text: V.

text: Scolari,

text: K.

text: Bromek,

text: aMn.dCTo.seSnatiitnoo,

text: Phys.

text: H.

text: Vandebroek

text: and

text: C.

text: Vanderzande,

text: Phys.

text: Rev.

text: E

text: A.

text: Javer,

text: Z.

text: Long,

text: E.

text: Nugent,

text: M.

text: Grisi,

text: K.

text: DA..DYo.rGfmroasnb,ePrg.

text: CanicdutJa.,-Fa.nJdoaMnn.

text: Nat.

text: Commun.

text: D.

text: Osmanovic

text: and

text: Y.

text: Rabin,

text: Soft

text: Matter

text: S.

text: C.

text: Weber,

text: A.

text: J.

text: Spakowitz,

text: and

text: J.

text: A.

text: Theriot,

text: Phys.

text: M.

text: Moreau,

text: and

text: R.

text: Voituriez,

text: Phys

text: Rev

text: E

text: Stat

text: Nonlin

text: Soft

text: Matter

text: Phys

text: S.

text: C.

text: Weber,

text: J.

text: A.

text: Theriot,

text: and

text: A.

text: J.

text: Spakowitz,

text: Phys.

text: overdo,

text: M.

text: Moreau,

text: and

text: R.

text: Voituriez,

text: Phys

text: Chem

text: Chem

text: Phys

text: S.

text: C.

text: Weber,

text: M.

text: A.

text: Thompson,

text: W.

text: E.

text: Moerner,

text: A.

text: J.

text: Mhe.riToat,mBmio,pahnyds.

text: OJ..

text: rnal

text: of

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text: Physics

text: A.

text: Javer,

text: N.

text: J.

text: Kuwada,

text: Z.

text: Long,

text: V.

text: G.

text: Benza,

text: K.

text: D.

text: PO..AB.´enWicihgoguin,sD,

text: P.

text: .GCreicbuetnak,ova,nPd.

text: MLe.vCit.z,LCag.oLmovaersrdinoo,,

text: Nanadt

text: R.

text: Voituriez,

text: Phys

text: Rev

text: Lett

text: Commun

text: C.

text: P.

text: Brangwynne,

text: G.

text: H.

text: Koenderink,

text: F.

text: C.

text: MacKintosh,

text: and

text: D.

text: A.

text: Weitz,

text: Trends

text: Cell

text: Biol

text: J.

text: K.

text: Fisher,

text: A.

text: Bourniquel,

text: G.

text: Witz,

text: B.

text: Weiner,

text: M.

text: E

text: Stat

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text: Soft

text: Matter

text: Phys

text: M.

text: C.

text: Joshi,

text: A.

text: Bourniquel,

text: J.

text: Fisher,

text: B.

text: T.

text: Ho,

text: D.

text: Magnan,

text: ]KAle.ckPnaerrm,

text: eagngidanDi,.

text: TBa.

text: tFersa,nPorsocch.,Naantdl.

text: W.

text: Deng

text: and

text: E.

text: Barkai,

text: Phys.

text: Rev.

text: E

text: B.

text: R.

text: Parry,

text: I.

text: V.

text: Surovtsev,

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text: Cabeen,

text: C.

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text: Berlin

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text: Brownian

text: Motion

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text: Lagomarsino,

text: Procedures

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text: aMl

text: aloi,ciYd.ySnhaamv-iTcsala,tE.

text: Barkai,

text: and

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text: Garini,

text: Phys.

text: Rev.

text: Lett.

text: short

text: time

text: scales,

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text: Methods

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text: hSe.

text: BMa.ctAernitahloNnuyc,

text: lSeo.idC.-

text: BMaeet,hoadnsdanSd.

text: PGrroatnoiccokl,sProceedings

text: of

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text: States

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text: Hybrid Isolation Forest - Application to Intrusion Detection 17 : The dependence of the AUC values to the other meta parameter settings, namely the number of iTrees and the sample size assigned to each iTree, is presented in Fig 13. (a) (b) 0.90 0.90 0.85 0.85 0.80 0.80 0.75 0.70 0.75 0.65 IF 0.70 IF HIF1 HIF1 0.60 HIF2 HIF2 23 24 25 26 27 28 29 23 24 25 26 27 28 29 t (# iTrees) ψ (# samples) Fig. 13. AUC values: (a) when the number of trees varies while the sample size remains constant equal to 128 samples; (b) when the sample size varies while the number of trees in the forest remains constant equal to 128 trees. For this experiment that is characterized with a dataset size of n = 1000 samples used to train the isolation forest, one can see that the HIF reaches good and relatively stable AUC values with fewer trees ((a) sub-figure) and lower sample size ((b) sub-figure) which is quite advantageous in terms of memory space and response time. 3.5 Setting up meta parameters α1 and α2 We consider here that ’normal’ data are available for training. [ ] : in this case we only need to setup . Figures 6 and 8 shows that a default value for can be setup in 2; 5 . If α1 α1 . . numerous unlabeled anomalies exist in the test data as well as ’normal’ test data, then an exploratory analysis of the distribution of the HIF1 score may be used to isolate anomaly modes from the ’normal’ distribution. By varying , the α1 separation of the distribution modes may vary and the seek for an ’optimal’ separation may be carried out. As an example, Fig. 14 shows, for α1 = .4 the distribution HIF1 scores for ’normal’ train data distribution (in blue) and the test data (in red) in which ’normal’ and ’red’ anomalies are mixed up. Clearly, the tail of the test data distribution is much longer towards the high scores than for the train data. This is obviously cues for anomaly identification. Hence, one can sample few data with high score and expertise them to decide whether they correspond to some kind of anomalies, and doing so, we may be able to provide some labeled anomaly data. This is the basis for designing an some active learning process. : in that case, labeled anomalies can be used in a supervised learning process. A simple grid search can be setup to optimize the detection of the anomalies as we did in our previous test to estimate the impact of the HIF additional Manuscript submitted to ACM CUA CUA ##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##

text: Hybrid

label: title

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text: Isolation

text: Forest

text: Application

text: to

text: Intrusion

text: Detection

text: The

text: dependence

text: of

text: the

text: AUC

text: values

text: to

text: the

text: other

text: meta

text: parameter

text: settings,

text: namely

text: the

text: number

text: of

text: iTrees

text: and

text: the

text: sample

text: size

text: assigned

text: to

text: each

text: iTree,

text: is

text: presented

text: in

text: Fig

text: (a)

text: (b)

text: IF

text: IF

text: t

text: iTrees)

text: ψ

text: samples)

text: Fig.

text: AUC

text: values:

text: (a)

text: when

text: the

text: number

text: of

text: trees

text: varies

text: while

text: the

text: sample

text: size

text: remains

text: constant

text: equal

text: to

text: samples;

text: (b)

text: when

text: the

text: sample

text: size

text: varies

text: while

text: the

text: number

text: of

text: trees

text: in

text: the

text: forest

text: remains

text: constant

text: equal

text: to

text: trees.

text: For

text: this

text: experiment

text: that

text: is

text: characterized

text: with

text: a

text: dataset

text: size

text: of

text: n

text: samples

text: used

text: to

text: train

text: the

text: isolation

text: forest,

text: one

text: can

text: see

text: that

text: the

text: HIF

text: reaches

text: good

text: and

text: relatively

text: stable

text: AUC

text: values

text: with

text: fewer

text: trees

text: ((a)

text: sub-figure)

text: and

text: lower

text: sample

text: size

text: ((b)

text: sub-figure)

text: which

text: is

text: quite

text: advantageous

text: in

text: terms

text: of

text: memory

text: space

text: and

text: response

text: time.

text: Setting

text: up

text: meta

text: parameters

text: α1

text: and

text: α2

text: We

text: consider

text: here

text: that

text: ’normal’

text: data

text: are

text: available

text: for

text: training.

text: in

text: this

text: case

text: we

text: only

text: need

text: to

text: setup

text: Figures

text: and

text: shows

text: that

text: a

text: default

text: value

text: for

text: can

text: be

text: setup

text: in

text: If

text: α1

text: α1

text: numerous

text: unlabeled

text: anomalies

text: exist

text: in

text: the

text: test

text: data

text: as

text: well

text: as

text: ’normal’

text: test

text: data,

text: then

text: an

text: exploratory

text: analysis

text: of

text: the

text: distribution

text: of

text: the

text: score

text: may

text: be

text: used

text: to

text: isolate

text: anomaly

text: modes

text: from

text: the

text: ’normal’

text: distribution.

text: By

text: varying

text: the

text: α1

text: separation

text: of

text: the

text: distribution

text: modes

text: may

text: vary

text: and

text: the

text: seek

text: for

text: an

text: ’optimal’

text: separation

text: may

text: be

text: carried

text: out.

text: As

text: an

text: example,

text: Fig.

text: shows,

text: for

text: α1

text: the

text: distribution

text: scores

text: for

text: ’normal’

text: train

text: data

text: distribution

text: (in

text: blue)

text: and

text: the

text: test

text: data

text: (in

text: red)

text: in

text: which

text: ’normal’

text: and

text: ’red’

text: anomalies

text: are

text: mixed

text: up.

text: Clearly,

text: the

text: tail

text: of

text: the

text: test

text: data

text: distribution

text: is

text: much

text: longer

text: towards

text: the

text: high

text: scores

text: than

text: for

text: the

text: train

text: data.

text: This

text: is

text: obviously

text: cues

text: for

text: anomaly

text: identification.

text: Hence,

text: one

text: can

text: sample

text: few

text: data

text: with

text: high

text: score

text: and

text: expertise

text: them

text: to

text: decide

text: whether

text: they

text: correspond

text: to

text: some

text: kind

text: of

text: anomalies,

text: and

text: doing

text: so,

text: we

text: may

text: be

text: able

text: to

text: provide

text: some

text: labeled

text: anomaly

text: data.

text: This

text: is

text: the

text: basis

text: for

text: designing

text: an

text: some

text: active

text: learning

text: process.

text: in

text: that

text: case,

text: labeled

text: anomalies

text: can

text: be

text: used

text: in

text: a

text: supervised

text: learning

text: process.

text: A

text: simple

text: grid

text: search

text: can

text: be

text: setup

text: to

text: optimize

text: the

text: detection

text: of

text: the

text: anomalies

text: as

text: we

text: did

text: in

text: our

text: previous

text: test

text: to

text: estimate

text: the

text: impact

text: of

text: the

text: HIF

text: additional

text: Manuscript

text: submitted

text: to

text: ACM

text: CUA

text: CUA

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