text: Alternative Description of Magnetic Monopoles in Quantum Mechanics Samuel Kov´aˇcik Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland E-mail: skovacik@stp.dias.ie Peter Preˇsnajder Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Mlynska´ dolina, Bratislava, 842 48, Slovakia E-mail: presnajder@fmph.uniba.sk Abstract. We present an alternative description of magnetic monopoles by lifting quantum mechanics from 3-dimensional space into a one with 2 complex dimensions. Magnetic monopoles are realized by generalization of the considered states. Usual algebraic relations and magnetic ﬁelds describing monopoles are reproduced, with the Dirac quantisation condition satisﬁed naturally. Ref. number: DIAS-STP-18-04 1. Introduction Magnetic monopoles are a vital part of many theoretical models, despite never being actually observed. They have a long stretching history – in classical electromagnetism they appeared as a generalization of Maxwell equations (by adding magnetic charge sources). The situation is more intricate in quantum mechanics, where electromagnetism is described by electromagnet potentials, which were deﬁned to make the theory free of magnetic monopoles by default (as ∼ div rot A ρ = 0). M However, there is a workaround – one can use potentials singular on a (half-)line, so called Dirac strings, [1]. This idea allowed Dirac to study magnetic monopoles in the context of quantum mechanics (QM) and to derive the famous Dirac quantisation condition, which states that the product of electric and magnetic charge, denoted µ, has to be quantized. Therefore, if at least one magnetic monopole exists in the Universe the electric charge has to be quantised – as is indeed observed. This description was later refurbished by Yang and Wu [2, 3], who instead of using one singular potential used two non-singular potentials deﬁned in diﬀerent, but overlapping, regions. This connected the theory of magnetic monopoles not only to the mathematical theory of sections, but also, in restrospective, to topology. Two potentials deﬁne two solutions which are related by a phase factor transformation. On the overlapping region, one can deﬁne a non-contractible path and by moving along the entire loop, the phase has to change by an integer multiple of 2π. The winding number integer corresponds to the quantised µ. Important studies of magnetic

text: Alternative

label: title

text: Description

text: of

text: Magnetic

text: Monopoles

text: in

text: Quantum

text: Mechanics

text: Samuel

label: author

text: Kov´aˇcik

text: Dublin

text: Institute

text: for

text: Advanced

text: Studies,

text: Burlington

text: Road,

text: Dublin

text: Ireland

text: E-mail:

text: skovacik@stp.dias.ie

label: footer

text: Peter

text: Preˇsnajder

text: Faculty

text: of

text: Mathematics,

text: Physics

text: and

text: Informatics,

text: Comenius

text: University

text: Bratislava,

text: Mlynska´

text: dolina,

text: Bratislava,

text: Slovakia

text: E-mail:

text: presnajder@fmph.uniba.sk

text: Abstract.

text: We

label: abstract

text: present

text: an

text: alternative

text: description

text: of

text: magnetic

text: monopoles

text: by

text: lifting

text: quantum

text: mechanics

text: from

text: space

text: into

text: a

text: one

text: with

text: complex

text: dimensions.

text: Magnetic

text: monopoles

text: are

text: realized

text: by

text: generalization

text: of

text: the

text: considered

text: states.

text: Usual

text: algebraic

text: relations

text: and

text: magnetic

text: ﬁelds

text: describing

text: monopoles

text: are

text: reproduced,

text: with

text: the

text: Dirac

text: quantisation

text: condition

text: satisﬁed

text: naturally.

text: Ref.

text: number:

text: Introduction

text: Magnetic

text: monopoles

text: are

text: a

text: vital

text: part

text: of

text: many

text: theoretical

text: models,

text: despite

text: never

text: being

text: actually

text: observed.

text: They

text: have

text: a

text: long

text: stretching

text: history

text: in

text: classical

text: electromagnetism

text: they

text: appeared

text: as

text: a

text: generalization

text: of

text: Maxwell

text: equations

text: (by

text: adding

text: magnetic

text: charge

text: sources).

text: The

text: situation

text: is

text: more

text: intricate

text: in

text: quantum

text: mechanics,

text: where

text: electromagnetism

text: is

text: described

text: by

text: electromagnet

text: potentials,

text: which

text: were

text: deﬁned

text: to

text: make

text: the

text: theory

text: free

text: of

text: magnetic

text: monopoles

text: by

text: default

text: (as

text: div

text: rot

text: A

text: ρ

text: M

text: However,

text: there

text: is

text: a

text: workaround

text: one

text: can

text: use

text: potentials

text: singular

text: on

text: a

text: (half-)line,

text: so

text: called

text: Dirac

text: strings,

text: This

text: idea

text: allowed

text: Dirac

text: to

text: study

text: magnetic

text: monopoles

text: in

text: the

text: context

text: of

text: quantum

text: mechanics

text: (QM)

text: and

text: to

text: derive

text: the

text: famous

text: Dirac

text: quantisation

text: condition,

text: which

text: states

text: that

text: the

text: product

text: of

text: electric

text: and

text: magnetic

text: charge,

text: denoted

text: µ,

text: has

text: to

text: be

text: quantized.

text: Therefore,

text: if

text: at

text: least

text: one

text: magnetic

text: monopole

text: exists

text: in

text: the

text: Universe

text: the

text: electric

text: charge

text: has

text: to

text: be

text: quantised

text: as

text: is

text: indeed

text: observed.

text: This

text: description

text: was

text: later

text: refurbished

text: by

text: Yang

text: and

text: Wu

text: who

text: instead

text: of

text: using

text: one

text: singular

text: potential

text: used

text: two

text: non-singular

text: potentials

text: deﬁned

text: in

text: diﬀerent,

text: but

text: overlapping,

text: regions.

text: This

text: connected

text: the

text: theory

text: of

text: magnetic

text: monopoles

text: not

text: only

text: to

text: the

text: mathematical

text: theory

text: of

text: sections,

text: but

text: also,

text: in

text: restrospective,

text: to

text: topology.

text: Two

text: potentials

text: deﬁne

text: two

text: solutions

text: which

text: are

text: related

text: by

text: a

text: phase

text: factor

text: transformation.

text: On

text: the

text: overlapping

text: region,

text: one

text: can

text: deﬁne

text: a

text: non-contractible

text: path

text: and

text: by

text: moving

text: along

text: the

text: entire

text: loop,

text: the

text: phase

text: has

text: to

text: change

text: by

text: an

text: integer

text: multiple

text: of

text: The

text: winding

text: number

text: integer

text: corresponds

text: to

text: the

text: quantised

text: µ.

text: Important

text: studies

text: of

text: magnetic


text: Given our choice of ξ , the only nontrivial component (in spherical coordinates) is κ δ + κ cos(θ) A = , (14) φ 2r sin(θ) which deﬁnes the following magnetic ﬁelds −κ x i B = (rot A) = . (15) i i 2 r3 (cid:0) (cid:1) As we can see, the resulting Coulomb-like ﬁeld depends only on κ, not δ. The potential A φ can be split into two parts, the ﬁrst one Aκ = κ cot(θ) describes the magnetic ﬁeld while the ∇ φ 2r other, Aδ = δ csc(θ) = δ φ can be gauged away. Both of them appear in the analysis of φ 2r φ 2 Yang, see [3]. It is obvious now that κ/2 can be identiﬁed with the magnetic charge µ. Dirac quantisation (cid:16) (cid:17) condition that µ has to be a half-integer is satisﬁed naturally as κ counts the diﬀerence in the ∗ powers of z and z , which is in our construction an integer. (cid:16) (cid:17) (cid:16) (cid:17) κ If we set δ = 0 we have ξ = z∗1z2∗ 4 and the resulting potential A is singular for both κ z1z2 φ (cid:16) (cid:17) ± κ κ θ = 0, π. On the other hand choosing δ = κ yields ξ = z1∗ 2 or ξ = z2∗ 2 , which leads to (cid:16) (cid:17) κ z κ z 1 2 a potential singular only on the north or the south pole. Therefore we conclude that z1∗ and z 1 (cid:0) (cid:1) z2∗ correspond to Dirac semi-strings with µ = κ and the general choice (13) is a combination z 2 2 of both. ∗ − It is also possible to deﬁne additional operator Vˆ = 1 z ∂ ∗ z ∂ which acts as 4 2r α zα α zα Vˆ = 1∂ on the considered states and measures their monopole charge. 4 r γ 4. Conclusion We have shown a slightly unusual construction of QM. The three-dimensional theory was reformulated in four-dimensional (or two-complex-dimensional) space by considering a restricted class of wave-functions of the form Φ(z¯σz). This construction is alluring as it oﬀers some new possibilities. As we have shown, the extra dimension, which in the context of Hopf ﬁbrations has a topology of a circle, allowed us to describe monopoles states in a regular way. Also, as this space is naturally equipped with Poisson structure, it can be (canonically) { } − → quantised to obtain noncommutative space: z , z¯ = iδ [z , z+] = δ and [x , x ] = α β αβ α β αβ i j 2iε x . Such spaces, whose close points cannot be distinguished below some fundamental scale, ijk k are a rather general aspect of theories of quantum gravity. After this quantisation, one can deﬁne QM the same way as was done in this paper to obtained so-called noncommutative QM, see [13, 14, 17, 18, 19]. By considering a similarly generalized class of states as in this paper (containing an unequal number of bosonic creation and annihilation operators), one can realize magnetic monopoles in this theory, see [15, 16]. The fact that lifting the theory one dimension up oﬀered new possibilities shall be of no surprise – when Kaluza and Klein studied general relativity in 5 dimensions, with the ﬁfth dimension tightly curled up, electromagnetism ﬁtted in in a very natural way. Also not so surprisingly, there are known gravitational solutions in 5 dimensions that look like magnetic monopoles in 4 dimensions, [9, 10]. Our choice of the form of ξ ensured that the factor which introduced monopoles was just k || || || || || || a phase, not aﬀecting the norm of the states Φ 2 = Φ(x)ξ 2 = Φ(x) 2. If we were more κ k κ careless and chose, for exa−mple, ξk = (z1z2) 2 , following the same steps we would lead to the same monopole ﬁeld B = κ xi . The fact that such choice changes the (r, θ, φ) dependence of i 2 r3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Given

label: paragraph

text: our

text: choice

text: of

text: ξ

text: the

text: only

text: nontrivial

text: component

text: (in

text: spherical

text: coordinates)

text: is

text: κ

text: δ

label: equation

text: κ

text: cos(θ)

text: A

text: φ

text: sin(θ)

text: which

text: deﬁnes

text: the

text: following

text: magnetic

text: ﬁelds

text: −κ

text: x

text: i

text: B

text: (rot

text: A)

text: i

text: i

text: As

text: we

text: can

text: see,

text: the

text: resulting

text: Coulomb-like

text: ﬁeld

text: depends

text: only

text: on

text: κ,

text: not

text: δ.

text: The

text: potential

text: A

text: φ

text: can

text: be

text: split

text: into

text: two

text: parts,

text: the

text: ﬁrst

text: one

text: Aκ

text: κ

text: cot(θ)

text: describes

text: the

text: magnetic

text: ﬁeld

text: while

text: the

text: φ

text: other,

text: Aδ

text: δ

text: csc(θ)

text: δ

text: φ

text: can

text: be

text: gauged

text: away.

text: Both

text: of

text: them

text: appear

text: in

text: the

text: analysis

text: of

text: φ

text: φ

text: Yang,

text: see

text: It

text: is

text: obvious

text: now

text: that

text: can

text: be

text: identiﬁed

text: with

text: the

text: magnetic

text: charge

text: µ.

text: Dirac

text: quantisation

text: condition

text: that

text: µ

text: has

text: to

text: be

text: a

text: half-integer

text: is

text: satisﬁed

text: naturally

text: as

text: κ

text: counts

text: the

text: diﬀerence

text: in

text: the

text: powers

text: of

text: z

text: and

text: z

text: which

text: is

text: in

text: our

text: construction

text: an

text: integer.

text: κ

text: If

text: we

text: set

text: δ

text: we

text: have

text: ξ

text: and

text: the

text: resulting

text: potential

text: A

text: is

text: singular

text: for

text: both

text: κ

text: φ

text: κ

text: κ

text: θ

text: π.

text: On

text: the

text: other

text: hand

text: choosing

text: δ

text: κ

text: yields

text: ξ

text: or

text: ξ

text: which

text: leads

text: to

text: κ

text: z

text: κ

text: z

text: a

text: potential

text: singular

text: only

text: on

text: the

text: north

text: or

text: the

text: south

text: pole.

text: Therefore

text: we

text: conclude

text: that

text: and

text: z

text: correspond

text: to

text: Dirac

text: semi-strings

text: with

text: µ

text: κ

text: and

text: the

text: general

text: choice

text: is

text: a

text: combination

text: z

text: of

text: both.

text: It

text: is

text: also

text: possible

text: to

text: deﬁne

text: additional

text: operator

text: Vˆ

text: z

text: z

text: which

text: acts

text: as

text: α

text: zα

text: α

text: zα

text: Vˆ

text: 1∂

text: on

text: the

text: considered

text: states

text: and

text: measures

text: their

text: monopole

text: charge.

text: r

text: γ

text: Conclusion

text: We

text: have

text: shown

text: a

text: slightly

text: unusual

text: construction

text: of

text: QM.

text: The

text: three-dimensional

text: theory

text: was

text: reformulated

text: in

text: four-dimensional

text: (or

text: two-complex-dimensional)

text: space

text: by

text: considering

text: a

text: restricted

text: class

text: of

text: wave-functions

text: of

text: the

text: form

text: Φ(z¯σz).

text: This

text: construction

text: is

text: alluring

text: as

text: it

text: oﬀers

text: some

text: new

text: possibilities.

text: As

text: we

text: have

text: shown,

text: the

text: extra

text: dimension,

text: which

text: in

text: the

text: context

text: of

text: Hopf

text: ﬁbrations

text: has

text: a

text: topology

text: of

text: a

text: circle,

text: allowed

text: us

text: to

text: describe

text: monopoles

text: states

text: in

text: a

text: regular

text: way.

text: Also,

text: as

text: this

text: space

text: is

text: naturally

text: equipped

text: with

text: Poisson

text: structure,

text: it

text: can

text: be

text: (canonically)

text: quantised

text: to

text: obtain

text: noncommutative

text: space:

text: z

text: z¯

text: iδ

text: [z

text: z+]

text: δ

text: and

text: [x

text: x

text: α

text: β

text: αβ

text: α

text: β

text: αβ

text: i

text: j

text: x

text: Such

text: spaces,

text: whose

text: close

text: points

text: cannot

text: be

text: distinguished

text: below

text: some

text: fundamental

text: scale,

text: ijk

text: k

text: are

text: a

text: rather

text: general

text: aspect

text: of

text: theories

text: of

text: quantum

text: gravity.

text: After

text: this

text: quantisation,

text: one

text: can

text: deﬁne

text: QM

text: the

text: same

text: way

text: as

text: was

text: done

text: in

text: this

text: paper

text: to

text: obtained

text: so-called

text: noncommutative

text: QM,

text: see

text: By

text: considering

text: a

text: similarly

text: generalized

text: class

text: of

text: states

text: as

text: in

text: this

text: paper

text: (containing

text: an

text: unequal

text: number

text: of

text: bosonic

text: creation

text: and

text: annihilation

text: operators),

text: one

text: can

text: realize

text: magnetic

text: monopoles

text: in

text: this

text: theory,

text: see

text: The

text: fact

text: that

text: lifting

text: the

text: theory

text: one

text: dimension

text: up

text: oﬀered

text: new

text: possibilities

text: shall

text: be

text: of

text: no

text: surprise

text: when

text: Kaluza

text: and

text: Klein

text: studied

text: general

text: relativity

text: in

text: dimensions,

text: with

text: the

text: ﬁfth

text: dimension

text: tightly

text: curled

text: up,

text: electromagnetism

text: ﬁtted

text: in

text: in

text: a

text: very

text: natural

text: way.

text: Also

text: not

text: so

text: surprisingly,

text: there

text: are

text: known

text: gravitational

text: solutions

text: in

text: dimensions

text: that

text: look

text: like

text: magnetic

text: monopoles

text: in

text: dimensions,

text: Our

text: choice

text: of

text: the

text: form

text: of

text: ξ

text: ensured

text: that

text: the

text: factor

text: which

text: introduced

text: monopoles

text: was

text: just

text: k

text: a

text: phase,

text: not

text: aﬀecting

text: the

text: norm

text: of

text: the

text: states

text: Φ

text: Φ(x)ξ

text: Φ(x)

text: If

text: we

text: were

text: more

text: κ

text: k

text: κ

text: careless

text: and

text: chose,

text: for

text: exa−mple,

text: ξk

text: following

text: the

text: same

text: steps

text: we

text: would

text: lead

text: to

text: the

text: same

text: monopole

text: ﬁeld

text: B

text: κ

text: xi

text: The

text: fact

text: that

text: such

text: choice

text: changes

text: the

text: (r,

text: θ,

text: φ)

text: dependence

text: of

text: i

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: We set M [Ψ] = 1 if there exists P , Q , P and Q such that the following i j1 j1 j2 j2 conditions are satisﬁed: (i) P = Merge(P , P ), (ii) Q = Merge(Q , Q ), i j1 j2 i j1 j2 (iii) M [A , B , P , Q ] = 1, and (iv) M [A , B , P , Q ] = 1. j1 i i j1 j1 j2 i i j2 j2 The graph has valid bipartitioning if there exists a Ψ such that M [Ψ] = 1. r Where r is the root node of the nice tree decomposition. The correctness of the algorithm is implied by the correctness of M [Ψ] values, which can i be proved using a bottom up induction on the nice tree decomposition. The ∗ time complexity of the algorithm is O (22tr). Thus we get the following: · Theorem 4. There is an 2O(tr) n time algorithm that solves the BWS-H | | problem for any arbitrary ﬁxed H ( V (H) = r), on graphs with tree-width at most t. 6 H-Free Bipartitioning Problem The techniques described in Section 1 can also be used to solve the H-Free Bipartitioning Problem . As we are looking for bipartitioning without H as an induced subgraph. Deﬁnition 1 and ((cid:63)) conditions at the introduced node are modiﬁed as below. Deﬁnition 5 (Induced Subgraph Legal Sequence in Γ with re- ∈ Ai spect to A). A sequence s = (w , w , w , . . . , w ) Γ is legal if the 1 (cid:48) 2 3 r Ai sequence s corresponds to subgraph H of H within A as follows. { | } { | } \{ ∪ Let FV (s) = (cid:96) w = fg , DC(s) = (cid:96) w = dc and V I(s) = [r] FV (s) } (cid:48) (cid:96) (cid:96) ∈ { ∪ DC(s) . Let H be the induced subgraph of H formed by u , (cid:96) V I(s) } (cid:48) { | ∈ ∪ } (cid:96) FV (s) . That is H = H[ u (cid:96) V I(s) FV (s) ]. | | (cid:96) ∈ \ If there exist FV (s) distinct vertices z A A corresponding to each index (cid:48) (cid:96) { | i∈ }∪{ | ∈ } in FV (s) such that H is isomorphic to G[ w (cid:96) V I(s) z (cid:96) FV (s) ], (cid:96) (cid:96) then s is legal. Otherwise, the sequence is illegal. ((cid:63)) conditions at the introduced node: ∃ (cid:54) ∈ { } ∈ (a) (cid:96) = (cid:96) , such that w = v, w A , u , u E(H) but { 1 } 2∈ (cid:96)1 (cid:96)2 i (cid:96)1 (cid:96)2 v, w / E(G). ∃ (cid:54) (cid:96)2 ∈ { } ∈ (b) (cid:96) = (cid:96) , such that w = v, w A , u , u / E(H) but { 1 } 2∈ (cid:96)1 (cid:96)2 i (cid:96)1 (cid:96)2 v, w E(G). ∃ (cid:54) (cid:96)2 { } ∈ (c) (cid:96) = (cid:96) , such that w = v, w = fg, u , u E(H). 1 2 (cid:96)1 ∈(cid:96)2 \ (cid:96)1 (cid:96)2 (d) Let s = (w , w , w , . . . , w ) Γ P . There exists (cid:96) such that 1 2 3 (cid:54) r Ai ∈ i ∪ { } 1 (cid:54) w = v and for all (cid:96) = (cid:96) w A dc . For all (cid:96) = (cid:96) (cid:96)1 ∈ { }2 ∈ 1 ⇐(cid:96)2⇒ { i } ∈ 1 2 w , w A , u , u E(H) w , w E(G). (cid:96)1 (cid:96)2 i (cid:96)1 (cid:96)2 (cid:96)1 (cid:96)2 Thus we get the following: · Theorem 5. There is an 2O(tr) n time algorithm that solves the H-Free | | Bipartitioning Problem for any arbitrary ﬁxed H ( V (H) = r), on graphs with tree-width at most t. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

label: list

text: set

text: M

text: [Ψ]

text: if

text: there

text: exists

text: P

text: Q

text: P

text: and

text: Q

text: such

text: that

text: the

text: following

text: i

text: conditions

text: are

text: satisﬁed:

text: (i)

text: P

text: Merge(P

text: P

text: (ii)

text: Q

text: Merge(Q

text: Q

text: i

text: i

text: (iii)

text: M

text: [A

text: B

text: P

text: Q

text: and

text: (iv)

text: M

text: [A

text: B

text: P

text: Q

text: i

text: i

text: i

text: i

text: The

text: graph

text: has

text: valid

text: bipartitioning

text: if

text: there

text: exists

text: a

text: Ψ

text: such

text: that

text: M

text: [Ψ]

text: r

text: Where

text: r

text: is

text: the

text: root

text: node

text: of

text: the

text: nice

text: tree

text: decomposition.

text: The

text: correctness

text: of

text: the

text: algorithm

text: is

text: implied

text: by

text: the

text: correctness

text: of

text: M

text: [Ψ]

text: values,

text: which

text: can

text: i

text: be

text: proved

text: using

text: a

text: bottom

text: up

text: induction

text: on

text: the

text: nice

text: tree

text: decomposition.

text: The

text: time

text: complexity

text: of

text: the

text: algorithm

text: is

text: O

text: Thus

text: we

text: get

text: the

text: following:

text: Theorem

text: There

text: is

text: an

text: n

text: time

text: algorithm

text: that

text: solves

text: the

text: BWS-H

text: problem

text: for

text: any

text: arbitrary

text: ﬁxed

text: H

text: V

text: (H)

text: r),

text: on

text: graphs

text: with

text: tree-width

text: at

text: most

text: t.

text: H-Free

text: Bipartitioning

text: Problem

text: The

text: techniques

text: described

text: in

text: Section

text: can

text: also

text: be

text: used

text: to

text: solve

text: the

text: H-Free

text: Bipartitioning

text: Problem

text: As

text: we

text: are

text: looking

text: for

text: bipartitioning

text: without

text: H

text: as

text: an

text: induced

text: subgraph.

text: Deﬁnition

text: and

text: conditions

text: at

text: the

text: introduced

text: node

text: are

text: modiﬁed

text: as

text: below.

text: Deﬁnition

text: (Induced

text: Subgraph

text: Legal

text: Sequence

text: in

text: Γ

text: with

text: re-

text: Ai

text: spect

text: to

text: A).

text: A

text: sequence

text: s

text: (w

text: w

text: w

text: w

text: Γ

text: is

text: legal

text: if

text: the

text: r

text: Ai

text: sequence

text: s

text: corresponds

text: to

text: subgraph

text: H

text: of

text: H

text: within

text: A

text: as

text: follows.

text: Let

text: FV

text: (s)

text: w

text: fg

text: DC(s)

text: w

text: dc

text: and

text: V

text: I(s)

text: [r]

text: FV

text: (s)

text: DC(s)

text: Let

text: H

text: be

text: the

text: induced

text: subgraph

text: of

text: H

text: formed

text: by

text: u

text: V

text: I(s)

text: FV

text: (s)

text: That

text: is

text: H

text: H[

text: u

text: V

text: I(s)

text: FV

text: (s)

text: If

text: there

text: exist

text: FV

text: (s)

text: distinct

text: vertices

text: z

text: A

text: A

text: corresponding

text: to

text: each

text: index

text: i∈

text: in

text: FV

text: (s)

text: such

text: that

text: H

text: is

text: isomorphic

text: to

text: G[

text: w

text: V

text: I(s)

text: z

text: FV

text: (s)

text: then

text: s

text: is

text: legal.

text: Otherwise,

text: the

text: sequence

text: is

text: illegal.

text: conditions

text: at

text: the

text: introduced

text: node:

text: (a)

text: such

text: that

text: w

text: v,

text: w

text: A

text: u

text: u

text: E(H)

text: but

text: 2∈

text: i

text: v,

text: w

text: E(G).

text: (b)

text: such

text: that

text: w

text: v,

text: w

text: A

text: u

text: u

text: E(H)

text: but

text: 2∈

text: i

text: v,

text: w

text: E(G).

text: (c)

text: such

text: that

text: w

text: v,

text: w

text: fg,

text: u

text: u

text: E(H).

text: (d)

text: Let

text: s

text: (w

text: w

text: w

text: w

text: Γ

text: P

text: There

text: exists

text: such

text: that

text: r

text: Ai

text: i

text: w

text: v

text: and

text: for

text: all

text: w

text: A

text: dc

text: For

text: all

text: i

text: w

text: w

text: A

text: u

text: u

text: E(H)

text: w

text: w

text: E(G).

text: i

text: Thus

text: we

text: get

text: the

text: following:

text: Theorem

text: There

text: is

text: an

text: n

text: time

text: algorithm

text: that

text: solves

text: the

text: H-Free

text: Bipartitioning

text: Problem

text: for

text: any

text: arbitrary

text: ﬁxed

text: H

text: V

text: (H)

text: r),

text: on

text: graphs

text: with

text: tree-width

text: at

text: most

text: t.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11. Diestel, R.: Graph Theory. Springer-Verlag Heidelberg (2005) 12. Cygan, M., Fomin, F.V., Kowalik, L(cid:32) ., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer (2015) 13. Robertson, N., Seymour, P.: Graph minors. X. Obstructions to tree- decomposition. Journal of Combinatorial Theory, Series B 52(2) (1991) 153– 190 14. Kloks, T., ed. In: Treewidth: Computations and Approximations. Lecture Notes in Computer Science, Springer (1994)

label: reference

text: Diestel,

text: R.:

text: Graph

text: Theory.

text: Springer-Verlag

text: Heidelberg

text: Cygan,

text: M.,

text: Fomin,

text: F.V.,

text: Kowalik,

text: Lokshtanov,

text: D.,

text: Marx,

text: D.,

text: Pilipczuk,

text: M.,

text: Pilipczuk,

text: M.,

text: Saurabh,

text: S.:

text: Parameterized

text: Algorithms.

text: Springer

text: Robertson,

text: N.,

text: Seymour,

text: P.:

text: Graph

text: minors.

text: X.

text: Obstructions

text: to

text: tree-

text: decomposition.

text: Journal

text: of

text: Combinatorial

text: Theory,

text: Series

text: B

text: Kloks,

text: T.,

text: ed.

text: In:

text: Treewidth:

text: Computations

text: and

text: Approximations.

text: Lecture

text: Notes

text: in

text: Computer

text: Science,

text: Springer


text: i v u u 1 2 v v 1 2 u u 3 4 u u 5 6 Fig. 2. Forming H at an introduce node. Fig. 1. An example graph H. Sequence s = (v, v2, v1, fg, fg, fg). a valid bipartitioning if there exists a 4-tuple Ψ such that M [Ψ] = 1, where r is r the root of the nice tree decomposition. The time complexity at each of the nodes in the tree decomposition is as ∗ ∗ follows: constant time at leaf nodes, O (2t+t2) time at insert nodes, O (22t+t2) ∗ time at forget nodes and O (2t+2t2) time at join nodes. This gives the following: Theorem 3. There is an O(2O(t2)n˙ ) time algorithm that solves the BWS-H Problem when H = C on graphs with tree-width at most t. 4 5 Bipartitioning without H Let X be a bag at node i of the nice tree decomposition. Let (A , B ) be i i i a partition of X . We can easily check if G[A ] or G[B ] has H as a subgraph. i i i ⊆ Otherwise, we need to see if there is a partition (A, B) of V (T ) such that A A, ⊆ i i B B and both G[A ] and G[B ] do not have H has a subgraph. If there is i i i (cid:48) such a partition (A, B), then G[A] and G[B] may have subgraph H , an induced subgraph of H which can lead to H at some ancestral node (introduce node or join node) of the nice tree decomposition (See Figures 2 and 3). We perform dynamic programming over the nice tree decomposition. At each node i we guess a partition (A , B ) of X and possible induced subgraphs of H i i i that are part of A and B respectively. We check if such a partition is possible. Below we explain the algorithm in detail. Let the vertices of the graph H are labeled as u , u , u , . . . , u . Let (A , B ) 1 2 3 r i i be a partition of vertices in the bag X . Let (A, B) be a partition of V (T ) such ⊇ ⊇ i i that A A and B B . We deﬁne Γ as follows: i i Ai { | ∈ { ∪ { }} S = (w , w , w , . . . , w ) w A fg, dc , Ai ∀ 1(cid:54) 2 3 r ⇒(cid:96) ∈i{ }} (cid:96) = (cid:96) , w = w = w fg, dc . { 1 2 (cid:96)1 (cid:96)2 ∈(cid:96)1 | (cid:54) I = s = (w , w , w , . . . , w ) S there exists (cid:96) = (cid:96) Ai 1 2 3 r Ai { } ∈1 2 } such that w = fg, w = dc and u , u E(H) \ (cid:96)1 (cid:96)2 (cid:96)1 (cid:96)2 Γ =S I Ai Ai Ai ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: i

font: default

text: v

text: u

text: u

text: v

text: v

text: u

text: u

text: u

text: u

text: Fig.

text: Forming

label: caption

text: H

text: at

text: an

text: introduce

text: node.

text: Fig.

text: An

text: example

text: graph

text: H.

text: Sequence

text: s

text: (v,

text: fg,

text: fg,

text: fg).

text: a

text: valid

text: bipartitioning

text: if

text: there

text: exists

text: a

text: Ψ

text: such

text: that

text: M

text: [Ψ]

text: where

text: r

text: is

text: r

text: the

text: root

text: of

text: the

text: nice

text: tree

text: decomposition.

text: The

text: time

text: complexity

text: at

text: each

text: of

text: the

text: nodes

text: in

text: the

text: tree

text: decomposition

text: is

text: as

text: follows:

text: constant

text: time

text: at

text: leaf

text: nodes,

text: O

text: time

text: at

text: insert

text: nodes,

text: O

text: time

text: at

text: forget

text: nodes

text: and

text: O

text: time

text: at

text: join

text: nodes.

text: This

text: gives

text: the

text: following:

text: Theorem

text: There

text: is

text: an

text: time

text: algorithm

text: that

text: solves

text: the

text: BWS-H

text: Problem

text: when

text: H

text: C

text: on

text: graphs

text: with

text: tree-width

text: at

text: most

text: t.

text: Bipartitioning

text: without

text: H

text: Let

text: X

text: be

text: a

text: bag

text: at

text: node

text: i

text: of

text: the

text: nice

text: tree

text: decomposition.

text: Let

text: (A

text: B

text: be

text: i

text: i

text: i

text: a

text: partition

text: of

text: X

text: We

text: can

text: easily

text: check

text: if

text: G[A

text: or

text: G[B

text: has

text: H

text: as

text: a

text: subgraph.

text: i

text: i

text: i

text: Otherwise,

text: we

text: need

text: to

text: see

text: if

text: there

text: is

text: a

text: partition

text: (A,

text: B)

text: of

text: V

text: (T

text: such

text: that

text: A

text: A,

text: i

text: i

text: B

text: B

text: and

text: both

text: G[A

text: and

text: G[B

text: do

text: not

text: have

text: H

text: has

text: a

text: subgraph.

text: If

text: there

text: is

text: i

text: i

text: i

text: such

text: a

text: partition

text: (A,

text: B),

text: then

text: G[A]

text: and

text: G[B]

text: may

text: have

text: subgraph

text: H

text: an

text: induced

text: subgraph

text: of

text: H

text: which

text: can

text: lead

text: to

text: H

text: at

text: some

text: ancestral

text: node

text: (introduce

text: node

text: or

text: join

text: node)

text: of

text: the

text: nice

text: tree

text: decomposition

text: (See

text: Figures

text: and

text: We

text: perform

text: dynamic

text: programming

text: over

text: the

text: nice

text: tree

text: decomposition.

text: At

text: each

text: node

text: i

text: we

text: guess

text: a

text: partition

text: (A

text: B

text: of

text: X

text: and

text: possible

text: induced

text: subgraphs

text: of

text: H

text: i

text: i

text: i

text: that

text: are

text: part

text: of

text: A

text: and

text: B

text: respectively.

text: We

text: check

text: if

text: such

text: a

text: partition

text: is

text: possible.

text: Below

text: we

text: explain

text: the

text: algorithm

text: in

text: detail.

text: Let

text: the

text: vertices

text: of

text: the

text: graph

text: H

text: are

text: labeled

text: as

text: u

text: u

text: u

text: u

text: Let

text: (A

text: B

text: r

text: i

text: i

text: be

text: a

text: partition

text: of

text: vertices

text: in

text: the

text: bag

text: X

text: Let

text: (A,

text: B)

text: be

text: a

text: partition

text: of

text: V

text: (T

text: such

text: i

text: i

text: that

text: A

text: A

text: and

text: B

text: B

text: We

text: deﬁne

text: Γ

text: as

text: follows:

text: i

text: i

text: Ai

text: S

text: (w

text: w

text: w

text: w

text: w

text: A

text: fg,

text: dc

text: Ai

text: r

text: ∈i{

text: w

text: w

text: w

text: fg,

text: dc

text: I

text: s

text: (w

text: w

text: w

text: w

text: S

text: there

text: exists

text: Ai

text: r

text: Ai

text: ∈1

text: such

text: that

text: w

text: fg,

text: w

text: dc

text: and

text: u

text: u

text: E(H)

text: Γ

text: =S

text: I

text: Ai

text: Ai

text: Ai

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A MIXED DISCRETE–CONTINUOUS FRAGMENTATION MODEL GRAHAM BAIRD AND ENDRE SU¨LI Abstract. Motivated by the occurrence of “shattering” mass-loss observed in purely con- tinuous fragmentation models, this work concerns the development and the mathematical analysis of a new class of hybrid discrete–continuous fragmentation models. Once estab- lished, the model, which takes the form of an integro-diﬀerential equation coupled with a system of ordinary diﬀerential equations, is subjected to a rigorous mathematical analysis, using the theory and methods of operator semigroups and their generators. Most notably, by applying the theory relating to the Kato–Voigt perturbation theorem, honest substochastic semigroups and operator matrices, the existence of a unique, diﬀerentiable solution to the model is established. This solution is also shown to preserve nonnegativity and conserve mass. 1. Introduction The mathematical modelling of fragmentation, and the reverse coagulation process have a long history, with the ﬁrst work dating back to [1]. Such models have found applications in areas as diverse as polymer science [2], population dynamics [3] and astrophysics [4]. The models of such processes typically classify the entities within the system according to some physical state variable, for example their volume, area or mass, the aim being then to de- termine the evolution of the system with respect to this variable as time progresses. Models are typically classiﬁed as either discrete or continuous, depending on the nature of the state variable of interest. Generally, there are great similarities between the two forms, with each continuous model having an analogous discrete version and vice versa. The selection of the form to use is largely a modelling choice, and depends on the scale of the phenomenon to be described. In this paper we shall be exclusively considering fragmentation processes, with no coagula- tion mechanism involved. With continuous models of such processes, diﬃculties can arise when the break-up rate for particles blows up as their size goes to zero and particles are allowed to get too small too quickly. The unbounded fragmentation rate can result in a runaway fragmentation process and a loss of mass unaccounted for in the model formula- tion. This loss of mass was observed by McGrady and Ziﬀ in [5], the process was termed ‘shattering’ and attributed to the creation of ‘dust’ particles with zero size but positive mass. In [6], Huang et al. suggest that such a runaway fragmentation process is unphysical, and that at some point particles become too small to break-up any further. Their proposed model includes a cut-oﬀ size x > 0, above which particles are able to fragment as usual. c However, once a particle’s size drops below x it ceases to be able to fragment, becoming c 2010 Mathematics Subject Classiﬁcation. 35F10, 45K05, 47D06, 47N20. Key words and phrases. Fragmentation models, mixed discrete–continuous fragmentation model, sub- stochastic semigroups, existence and uniqueness of solution. 1 ##LTLine##

text: A

text: MIXED

text: DISCRETE–CONTINUOUS

text: FRAGMENTATION

text: MODEL

text: GRAHAM

text: BAIRD

text: AND

text: ENDRE

text: SU¨LI

text: Abstract.

text: Motivated

text: by

text: the

text: occurrence

text: of

text: “shattering”

text: mass-loss

text: observed

text: in

text: purely

text: con-

text: tinuous

text: fragmentation

text: models,

text: this

text: work

text: concerns

text: the

text: development

text: and

text: the

text: mathematical

text: analysis

text: of

text: a

text: new

text: class

text: of

text: hybrid

text: discrete–continuous

text: fragmentation

text: models.

text: Once

text: estab-

text: lished,

text: the

text: model,

text: which

text: takes

text: the

text: form

text: of

text: an

text: integro-diﬀerential

text: equation

text: coupled

text: with

text: a

text: system

text: of

text: ordinary

text: diﬀerential

text: equations,

text: is

text: subjected

text: to

text: a

text: rigorous

text: mathematical

text: analysis,

text: using

text: the

text: theory

text: and

text: methods

text: of

text: operator

text: semigroups

text: and

text: their

text: generators.

text: Most

text: notably,

text: by

text: applying

text: the

text: theory

text: relating

text: to

text: the

text: Kato–Voigt

text: perturbation

text: theorem,

text: honest

text: substochastic

text: semigroups

text: and

text: operator

text: matrices,

text: the

text: existence

text: of

text: a

text: unique,

text: diﬀerentiable

text: solution

text: to

text: the

text: model

text: is

text: established.

text: This

text: solution

text: is

text: also

text: shown

text: to

text: preserve

text: nonnegativity

text: and

text: conserve

text: mass.

text: Introduction

text: The

text: mathematical

text: modelling

text: of

text: fragmentation,

text: and

text: the

text: reverse

text: coagulation

text: process

text: have

text: a

text: long

text: history,

text: with

text: the

text: ﬁrst

text: work

text: dating

text: back

text: to

text: Such

text: models

text: have

text: found

text: applications

text: in

text: areas

text: as

text: diverse

text: as

text: polymer

text: science

text: population

text: dynamics

text: and

text: astrophysics

text: The

text: models

text: of

text: such

text: processes

text: typically

text: classify

text: the

text: entities

text: within

text: the

text: system

text: according

text: to

text: some

text: physical

text: state

text: variable,

text: for

text: example

text: their

text: volume,

text: area

text: or

text: mass,

text: the

text: aim

text: being

text: then

text: to

text: de-

text: termine

text: the

text: evolution

text: of

text: the

text: system

text: with

text: respect

text: to

text: this

text: variable

text: as

text: time

text: progresses.

text: Models

text: are

text: typically

text: classiﬁed

text: as

text: either

text: discrete

text: or

text: continuous,

text: depending

text: on

text: the

text: nature

text: of

text: the

text: state

text: variable

text: of

text: interest.

text: Generally,

text: there

text: are

text: great

text: similarities

text: between

text: the

text: two

text: forms,

text: with

text: each

text: continuous

text: model

text: having

text: an

text: analogous

text: discrete

text: version

text: and

text: vice

text: versa.

text: The

text: selection

text: of

text: the

text: form

text: to

text: use

text: is

text: largely

text: a

text: modelling

text: choice,

text: and

text: depends

text: on

text: the

text: scale

text: of

text: the

text: phenomenon

text: to

text: be

text: described.

text: In

text: this

text: paper

text: we

text: shall

text: be

text: exclusively

text: considering

text: fragmentation

text: processes,

text: with

text: no

text: coagula-

text: tion

text: mechanism

text: involved.

text: With

text: continuous

text: models

text: of

text: such

text: processes,

text: diﬃculties

text: can

text: arise

text: when

text: the

text: break-up

text: rate

text: for

text: particles

text: blows

text: up

text: as

text: their

text: size

text: goes

text: to

text: zero

text: and

text: particles

text: are

text: allowed

text: to

text: get

text: too

text: small

text: too

text: quickly.

text: The

text: unbounded

text: fragmentation

text: rate

text: can

text: result

text: in

text: a

text: runaway

text: fragmentation

text: process

text: and

text: a

text: loss

text: of

text: mass

text: unaccounted

text: for

text: in

text: the

text: model

text: formula-

text: tion.

text: This

text: loss

text: of

text: mass

text: was

text: observed

text: by

text: McGrady

text: and

text: Ziﬀ

text: in

text: the

text: process

text: was

text: termed

text: ‘shattering’

text: and

text: attributed

text: to

text: the

text: creation

text: of

text: ‘dust’

text: particles

text: with

text: zero

text: size

text: but

text: positive

text: mass.

text: In

text: Huang

text: et

text: al.

text: suggest

text: that

text: such

text: a

text: runaway

text: fragmentation

text: process

text: is

text: unphysical,

text: and

text: that

text: at

text: some

text: point

text: particles

text: become

text: too

text: small

text: to

text: break-up

text: any

text: further.

text: Their

text: proposed

text: model

text: includes

text: a

text: cut-oﬀ

text: size

text: x

text: above

text: which

text: particles

text: are

text: able

text: to

text: fragment

text: as

text: usual.

text: c

text: However,

text: once

text: a

text: particle’s

text: size

text: drops

text: below

text: x

text: it

text: ceases

text: to

text: be

text: able

text: to

text: fragment,

text: becoming

text: c

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases.

text: Fragmentation

text: models,

text: mixed

text: discrete–continuous

text: fragmentation

text: model,

text: sub-

text: stochastic

text: semigroups,

text: existence

text: and

text: uniqueness

text: of

text: solution.

text: ##LTLine##


text: A MIXED DISCRETE–CONTINUOUS FRAGMENTATION MODEL 11 If we take(cid:90)the(cid:18)s(cid:90)econd term from abo(cid:19)ve, split the inner integral in two and then change the order of integration for the integral over the bounded domain, then we get ∞ R(cid:90) (cid:18)(cid:90) | (cid:19) (cid:90) (cid:18)(cid:90) (cid:19) a(y)b(x y)g(y) dy x dx N x ∞ (cid:90) R R (cid:18)(cid:90) | (cid:19) (cid:90) R (cid:18)(cid:90) | (cid:19) = a(y)b(x y)g(y) dy x dx + a(y)b(x y)g(y) dy x dx N x N R ∞ R y | R | = a(y)g(y) xb(x y) dx dy + a(y)b(x y)g(y) dy x dx. N (cid:90) N N R Substituting this back into (3.7) then yields ∞ − (cid:90) (cid:18) B(cid:90) (cid:19) ( a(x)g(x) + ( g) (x)) x dx N − (cid:90) R (cid:18)(cid:90)− x | (cid:19) = lim x yb(y x) dy a(x)g(x) dx →∞ R N N ∞ R (cid:90) (cid:18)(cid:90) | (cid:19) + lim a(y)b(x y)g(y) dy x dx →∞ R N R ∞ − R | = c(g) + lim a(y)b(x y)g(y) dy x dx. →∞ R N R − The nonnegativity of the additional term accompanying c(g) gives us (3.6), and with that (cid:3) the honesty of the semigroup (G (t)) ≥ . K t 0 Having obtained an X -valued solution to our abstract Cauchy problem equation (3.2), we C now examine whether this provides a scalar-valued solution to our original continuous regime equation (1.1). But ﬁrst we require the following result, which we have delayed until now as it concerns the extended operators introduced above. Lemma 3.6. Suppose that (A , D(A )) and (B , D(B )) satisfy the conditions of Theo- C C C C rem 2.3 along with conditions (3.3) and (3.4). Additionally let us deﬁne the operator T with ⊂ D(T) = LF X by C − − Tu = u L 1u, which is permitted since L is one-to-one [8, Theorem 6.18]. Then, the extension K of A +B C C that generates a substochastic semigroup on X is given by C (cid:110) (cid:111) Ku = Tu + Bu, with ∈ ∩ ∈ (cid:107) (cid:107) D(K) = u D(T) D(B) : Tu + Bu X , and lim (LB)nu = 0 . C →∞ n (cid:3) Proof. See [8, Theorem 6.20] Theorem 3.7. There exists a measurable scalar-valued representation u (x, t) of the semi- C group solution (G (t)c )(x), such that u (x, t) is absolutely continuous with respect to t, the K 0 C ∞ × ∞ partial derivative ∂ u (x, t) exists almost everywhere on (N, ) [0, ) and u (x, t) satisﬁes t C C equation (1.1) for almost all x > N and t > 0. Further, this representation is unique up to sets of measure zero.

text: A

text: MIXED

text: DISCRETE–CONTINUOUS

text: FRAGMENTATION

text: MODEL

text: If

text: we

text: term

text: from

text: split

text: the

text: inner

text: integral

text: in

text: two

text: and

text: then

text: change

text: the

text: order

text: of

text: integration

text: for

text: the

text: integral

text: over

text: the

text: bounded

text: domain,

text: then

text: we

text: get

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx

text: N

text: x

text: R

text: R

text: R

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx

text: N

text: x

text: N

text: R

text: R

text: y

text: R

text: a(y)g(y)

text: xb(x

text: y)

text: dx

text: dy

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx.

text: N

text: N

text: N

text: R

text: Substituting

text: this

text: back

text: into

text: then

text: yields

text: a(x)g(x)

text: g)

text: (x))

text: x

text: dx

text: N

text: R

text: x

text: lim

text: x

text: yb(y

text: x)

text: dy

text: a(x)g(x)

text: dx

text: R

text: N

text: N

text: R

text: lim

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx

text: R

text: N

text: R

text: R

text: c(g)

text: lim

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx.

text: R

text: N

text: R

text: The

text: nonnegativity

text: of

text: the

text: additional

text: term

text: accompanying

text: c(g)

text: gives

text: us

text: and

text: with

text: that

text: the

text: honesty

text: of

text: the

text: semigroup

text: (G

text: (t))

text: K

text: t

text: Having

text: obtained

text: an

text: X

text: -valued

text: solution

text: to

text: our

text: abstract

text: Cauchy

text: problem

text: equation

text: we

text: C

text: now

text: examine

text: whether

text: this

text: provides

text: a

text: scalar-valued

text: solution

text: to

text: our

text: original

text: continuous

text: regime

text: equation

text: But

text: ﬁrst

text: we

text: require

text: the

text: following

text: result,

text: which

text: we

text: have

text: delayed

text: until

text: now

text: as

text: it

text: concerns

text: the

text: extended

text: operators

text: introduced

text: above.

text: Lemma

text: Suppose

text: that

text: (A

text: D(A

text: and

text: (B

text: D(B

text: satisfy

text: the

text: conditions

text: of

text: Theo-

text: C

text: C

text: C

text: C

text: rem

text: along

text: with

text: conditions

text: and

text: Additionally

text: let

text: us

text: deﬁne

text: the

text: operator

text: T

text: with

text: D(T)

text: LF

text: X

text: by

text: C

text: Tu

text: u

text: L

text: which

text: is

text: permitted

text: since

text: L

text: is

text: one-to-one

text: Theorem

text: Then,

text: the

text: extension

text: K

text: of

text: A

text: +B

text: C

text: C

text: that

text: generates

text: a

text: substochastic

text: semigroup

text: on

text: X

text: is

text: given

text: by

text: C

text: Ku

text: Tu

text: Bu,

text: with

text: D(K)

text: u

text: D(T)

text: D(B)

text: Tu

text: Bu

text: X

text: and

text: lim

text: (LB)nu

text: C

text: n

text: Proof.

text: See

text: Theorem

text: Theorem

text: There

text: exists

text: a

text: measurable

text: scalar-valued

text: representation

text: u

text: (x,

text: t)

text: of

text: the

text: semi-

text: C

text: group

text: solution

text: (G

text: (t)c

text: )(x),

text: such

text: that

text: u

text: (x,

text: t)

text: is

text: absolutely

text: continuous

text: with

text: respect

text: to

text: t,

text: the

text: K

text: C

text: partial

text: derivative

text: u

text: (x,

text: t)

text: exists

text: almost

text: everywhere

text: on

text: (N,

text: and

text: u

text: (x,

text: t)

text: satisﬁes

text: t

text: C

text: C

text: equation

text: for

text: almost

text: all

text: x

text: N

text: and

text: t

text: Further,

text: this

text: representation

text: is

text: unique

text: up

text: to

text: sets

text: of

text: measure

text: zero.


text: 18 GRAHAM BAIRD AND ENDRE SU¨LI set of unique classical solutions to the equations (1.2). Having determined the existence of such solutions, in the upcoming section we establish some key properties displayed by them. 6. Solution Properties (Nonnegativity and Mass Conservation) The existence of the C -semigroup (T(t)) ≥ provides us with a unique strong solution, 0 t 0 u(t) = T(t)u , to the abstract Cauchy problem (5.1). For the solution to be physically 0 relevant given the problem setting, we would expect it to possess a number of properties. In particular, we hope that the solution preserves nonnegativity and we demonstrate conserva- (cid:0) (cid:1) tion of mass. First oﬀ we establish the nonnegativity of the strong solution. ∈ Lemma 6.1. Provided that the initial data are nonnegative, i.e. u = d0 D(A) = × 0 c0 + X D(K) , then the strong solution u(t) = T(t)u , emanating from u , remains nonneg- D+ + ∈ ≥ 0 0 ative, that is u(t) D(A) for all t 0. + ∈ Proof. Let u D(A) . Then the two components of the solution u(t) are given by (5.5). 0 + Since (G (t)) ≥ is a substochastic semigroup, it preserves nonnegativity of the initial da- K t 0 ∈ ∈ ≥ tum. Therefore, when c D(K) we have G (t)f D(K) for all t 0, hence the 0 + ∈ K 0 ≥ + continuous regime solution satisﬁes u (t) D(K) for all t 0. C + As an integral operator with a nonnegative kernel, the operator C is easily seen to be a pos- itive operator. The semigroup (T (t)) ≥ is a positive semigroup as was shown in Lemma 4.1. t 0 Together with the nonnegativity of the continuous regime solution, these ensure that the (cid:0) (cid:1) ∈ integral term is a nonnegative contribution to u (t) whenever c D(K) . As stated, D ∈ 0 ≥ + (T (t)) ≥ is a positive semigroup, and therefore u (t) X for all t 0 when the initial t 0 ∈ × D D+ system state satisﬁes u = d0 X D(K) . 0 c0 D+ + (cid:0) (cid:1) ∈ ∈ ∈ (cid:3) Together, having u (t) X and u (t) D(K) gives us u(t) D(A) as required. D D+ C + + Let u(t) = uD(t) be the solution to the abstract Cauchy problem (5.1) with nonnegative uC(t) initial data. Then the masses in the discrete and cont(cid:90)inuous regimes, at time t, are given by (cid:88) ∞ N M (t) = iu (t) and M (t) = (u (t)) (x) x dx, D Di C C N i=1 respectively. Summing these gives us the total mass in the system M(t) = M (t)+M (t) at D C time t. In each fragmentation event, mass is redistributed from larger particles to the smaller resulting particles, but the total mass involved should be conserved. This mass-conservation was built into our model in the form of the conditions (1.3) and (1.4). Therefore, although M (t) may increase and M (t) decrease as larger particles break into smaller pieces, we D C would expect the total mass, M(t), to remain constant. Lemma 6.2. The total mass within the system is conserved, that is M(t) remains constant ≥ for all t 0. Proof. Since (G (t)) ≥ is an honest semigroup, by Deﬁnition 2.5 we have K t 0

text: GRAHAM

text: BAIRD

text: AND

text: ENDRE

text: SU¨LI

text: set

text: of

text: unique

text: classical

text: solutions

text: to

text: the

text: equations

text: Having

text: determined

text: the

text: existence

text: of

text: such

text: solutions,

text: in

text: the

text: upcoming

text: section

text: we

text: establish

text: some

text: key

text: properties

text: displayed

text: by

text: them.

label: section

text: Solution

text: Properties

text: (Nonnegativity

text: and

text: Mass

text: Conservation)

text: The

text: existence

text: of

text: the

text: C

text: -semigroup

text: (T(t))

text: provides

text: us

text: with

text: a

text: unique

text: strong

text: solution,

text: t

text: u(t)

text: T(t)u

text: to

text: the

text: abstract

text: Cauchy

text: problem

text: For

text: the

text: solution

text: to

text: be

text: physically

text: relevant

text: given

text: the

text: problem

text: setting,

text: we

text: would

text: expect

text: it

text: to

text: possess

text: a

text: number

text: of

text: properties.

text: In

text: particular,

text: we

text: hope

text: that

text: the

text: solution

text: preserves

text: nonnegativity

text: and

text: we

text: demonstrate

text: conserva-

text: tion

text: of

text: mass.

text: First

text: oﬀ

text: we

text: establish

text: the

text: nonnegativity

text: of

text: the

text: strong

text: solution.

text: Lemma

text: Provided

text: that

text: the

text: initial

text: data

text: are

text: nonnegative,

text: i.e.

text: u

text: D(A)

text: X

text: D(K)

text: then

text: the

text: strong

text: solution

text: u(t)

text: T(t)u

text: emanating

text: from

text: u

text: remains

text: nonneg-

text: D+

text: ative,

text: that

text: is

text: u(t)

text: D(A)

text: for

text: all

text: t

text: Proof.

text: Let

text: u

text: D(A)

text: Then

text: the

text: two

text: components

text: of

text: the

text: solution

text: u(t)

text: are

text: given

text: by

text: Since

text: (G

text: (t))

text: is

text: a

text: substochastic

text: semigroup,

text: it

text: preserves

text: nonnegativity

text: of

text: the

text: initial

text: da-

text: K

text: t

text: tum.

text: Therefore,

text: when

text: c

text: D(K)

text: we

text: have

text: G

text: (t)f

text: D(K)

text: for

text: all

text: t

text: hence

text: the

text: K

text: continuous

text: regime

text: solution

text: satisﬁes

text: u

text: (t)

text: D(K)

text: for

text: all

text: t

text: C

text: As

text: an

text: integral

text: operator

text: with

text: a

text: nonnegative

text: kernel,

text: the

text: operator

text: C

text: is

text: easily

text: seen

text: to

text: be

text: a

text: pos-

text: itive

text: operator.

text: The

text: semigroup

text: (T

text: (t))

text: is

text: a

text: positive

text: semigroup

text: as

text: was

text: shown

text: in

text: Lemma

text: t

text: Together

text: with

text: the

text: nonnegativity

text: of

text: the

text: continuous

text: regime

text: solution,

text: these

text: ensure

text: that

text: the

text: integral

text: term

text: is

text: a

text: nonnegative

text: contribution

text: to

text: u

text: (t)

text: whenever

text: c

text: D(K)

text: As

text: stated,

text: D

text: (T

text: (t))

text: is

text: a

text: positive

text: semigroup,

text: and

text: therefore

text: u

text: (t)

text: X

text: for

text: all

text: t

text: when

text: the

text: initial

text: t

text: D

text: D+

text: system

text: state

text: satisﬁes

text: u

text: X

text: D(K)

text: D+

text: Together,

text: having

text: u

text: (t)

text: X

text: and

text: u

text: (t)

text: D(K)

text: gives

text: us

text: u(t)

text: D(A)

text: as

text: required.

text: D

text: D+

text: C

text: Let

text: u(t)

text: uD(t)

text: be

text: the

text: solution

text: to

text: the

text: abstract

text: Cauchy

text: problem

text: with

text: nonnegative

text: uC(t)

text: initial

text: data.

text: Then

text: the

text: masses

text: in

text: the

text: discrete

text: and

text: regimes,

text: at

text: time

text: t,

text: are

text: given

text: by

text: N

text: M

text: (t)

text: iu

text: (t)

text: and

text: M

text: (t)

text: (u

text: (t))

text: (x)

text: x

text: dx,

text: D

text: Di

text: C

text: C

text: N

text: respectively.

text: Summing

text: these

text: gives

text: us

text: the

text: total

text: mass

text: in

text: the

text: system

text: M(t)

text: M

text: (t)+M

text: (t)

text: at

text: D

text: C

text: time

text: t.

text: In

text: each

text: fragmentation

text: event,

text: mass

text: is

text: redistributed

text: from

text: larger

text: particles

text: to

text: the

text: smaller

text: resulting

text: particles,

text: but

text: the

text: total

text: mass

text: involved

text: should

text: be

text: conserved.

text: This

text: mass-conservation

text: was

text: built

text: into

text: our

text: model

text: in

text: the

text: form

text: of

text: the

text: conditions

text: and

text: Therefore,

text: although

text: M

text: (t)

text: may

text: increase

text: and

text: M

text: (t)

text: decrease

text: as

text: larger

text: particles

text: break

text: into

text: smaller

text: pieces,

text: we

text: D

text: C

text: would

text: expect

text: the

text: total

text: mass,

text: M(t),

text: to

text: remain

text: constant.

text: Lemma

text: The

text: total

text: mass

text: within

text: the

text: system

text: is

text: conserved,

text: that

text: is

text: M(t)

text: remains

text: constant

text: for

text: all

text: t

text: Proof.

text: Since

text: (G

text: (t))

text: is

text: an

text: honest

text: semigroup,

text: by

text: Deﬁnition

text: we

text: have

text: K

text: t


text: 8 GRAHAM BAIRD AND ENDRE SU¨LI This allows us to form the operator A + B with domain D(A ). Equation (1.1) is then C C C reformulated in the setting of X as the abstract Cauchy problem: C d ∈ (3.2) u (t) = K[u (t)], t > 0; u (0) = c D(K), C C C 0 dt where K is some extension of the operator A + B . The Kato–Voigt perturbation theorem C C (Theorem 2.3) will allow us to prove the existence of such an operator K, which generates a semigroup. Theorem 3.2. There exists an extension (K, D(K)) of (A + B , D(A )), which generates C C C a substochastic semigroup (G (t)) ≥ . K t 0 Proof. To establish this result we show that the three conditions set out in Theorem 2.3 are satisﬁed for our particular operators A and B . C C (1) It is clear that (A , D(A )) generates a substochastic semigroup C C − ∈ (G (t)) ≥ on X , where (G (t)f) (x) = exp( a(x)t)f(x), for f X . AC t 0 C AC ⊆ C (2) We have shown in Lemma 3.1 that D(A ) D(B ). The nonnegativity of a and b C ∈ C ∈ (cid:90) (cid:90) (cid:18) (cid:90) (cid:19) imply that B a positive operator, so that B f X for all f D(B ) . ∈ C C C+ C + (3) For all f D(A ) we have that C + ∞ ∞ ∞ (cid:90) (cid:90) (cid:18)−(cid:90) (cid:19)| (A f + B f) x dx = a(x)f(x) + a(y)b(x y)f(y) dy x dx C C N N x ∞ ∞ ∞ − (cid:90) (cid:90) (cid:18)(cid:90) | (cid:19) = a(x)f(x)x dx + a(y)b(x y)f(y) dy x dx N N x ∞ ∞ − (cid:90) (cid:18) (cid:90) (cid:19) y | = a(x)f(x)x dx + a(y)f(y) xb(x y) dx dy N N N ∞ − − x | − ≤ = x yb(y x) dy a(x)f(x) dx =: c(f) 0. N N (cid:82) We have introduced the notation c to represent the ﬁnal integral expression, and this func- tional will have signiﬁcance in the analysis which follows. The nonnegativity of c comes | ≤ x as a result of the earlier statement regarding yb(y x) dy x. The conditions of Theo- N rem 2.3 have been shown to hold in our case; hence there exists an extension (K, D(K)) of (cid:3) (A + B , D(A )), which generates a substochastic semigroup (G (t)) ≥ . C C C K t 0 This theorem proves only the existence of a generating extension K, and oﬀers no indication of how exactly K relates to A + B . The nature of the generator K is closely related to C C the concept of the honesty of the semigroup (Deﬁnition 2.5) and in turn the occurrence of ‘shattering’. This relationship is discussed in [17], where a range of possibilities for K are considered and it is shown that the cases in which shattering occurs coincide with those in which the semigroup generated by K is dishonest. In order to establish the honesty of the semigroup (G (t)) ≥ , we follow a similar approach to K t 0 that taken in [8, Section 6.3]. Let us denote by E the set of all measurable functions deﬁned ∞ on (N, ), which take values within the extended reals. By E we denote the subspace of E f ##LTLine##

text: GRAHAM

text: BAIRD

text: AND

text: ENDRE

text: SU¨LI

text: This

text: allows

text: us

text: to

text: form

text: the

text: operator

text: A

text: B

text: with

text: domain

text: D(A

text: Equation

text: is

text: then

text: C

text: C

text: C

text: reformulated

text: in

text: the

text: setting

text: of

text: X

text: as

text: the

text: abstract

text: Cauchy

text: problem:

text: C

text: d

text: u

text: (t)

text: K[u

text: (t)],

text: t

text: u

text: c

text: D(K),

text: C

text: C

text: C

text: dt

text: where

text: K

text: is

text: some

text: extension

text: of

text: the

text: operator

text: A

text: B

text: The

text: Kato–Voigt

text: perturbation

text: theorem

text: C

text: C

text: (Theorem

text: will

text: allow

text: us

text: to

text: prove

text: the

text: existence

text: of

text: such

text: an

text: operator

text: K,

text: which

text: generates

text: a

text: semigroup.

text: Theorem

text: There

text: exists

text: an

text: extension

text: (K,

text: D(K))

text: of

text: (A

text: B

text: D(A

text: which

text: generates

text: C

text: C

text: C

text: a

text: substochastic

text: semigroup

text: (G

text: (t))

text: K

text: t

text: Proof.

text: To

text: establish

text: this

text: result

text: we

text: show

text: that

text: the

text: three

text: conditions

text: set

text: out

text: in

text: Theorem

text: are

text: satisﬁed

text: for

text: our

text: particular

text: operators

text: A

text: and

text: B

text: C

text: C

text: It

text: is

text: clear

text: that

text: (A

text: D(A

text: generates

text: a

text: substochastic

text: semigroup

text: C

text: C

text: (G

text: (t))

text: on

text: X

text: where

text: (G

text: (t)f)

text: (x)

text: exp(

text: a(x)t)f(x),

text: for

text: f

text: X

text: AC

text: t

text: C

text: AC

text: C

text: We

text: have

text: shown

text: in

text: Lemma

text: that

text: D(A

text: D(B

text: The

text: nonnegativity

text: of

text: a

text: and

text: b

text: C

text: C

text: imply

text: that

text: B

text: a

text: positive

text: operator,

text: so

text: that

text: B

text: f

text: X

text: for

text: all

text: f

text: D(B

text: C

text: C

text: C+

text: C

text: For

text: all

text: f

text: D(A

text: we

text: have

text: that

text: C

text: (A

text: f

text: B

text: f)

text: x

text: dx

text: a(x)f(x)

text: a(y)b(x

text: y)f(y)

text: dy

text: x

text: dx

text: C

text: C

text: N

text: N

text: x

text: a(x)f(x)x

text: dx

text: a(y)b(x

text: y)f(y)

text: dy

text: x

text: dx

text: N

text: N

text: x

text: y

text: a(x)f(x)x

text: dx

text: a(y)f(y)

text: xb(x

text: y)

text: dx

text: dy

text: N

text: N

text: N

text: x

text: x

text: yb(y

text: x)

text: dy

text: a(x)f(x)

text: dx

text: c(f)

text: N

text: N

text: We

text: have

text: introduced

text: the

text: notation

text: c

text: to

text: represent

text: the

text: ﬁnal

text: integral

text: expression,

text: and

text: this

text: func-

text: tional

text: will

text: have

text: signiﬁcance

text: in

text: the

text: analysis

text: which

text: follows.

text: The

text: nonnegativity

text: of

text: c

text: comes

text: x

text: as

text: a

text: result

text: of

text: the

text: earlier

text: statement

text: regarding

text: yb(y

text: x)

text: dy

text: x.

text: The

text: conditions

text: of

text: Theo-

text: N

text: rem

text: have

text: been

text: shown

text: to

text: hold

text: in

text: our

text: case;

text: hence

text: there

text: exists

text: an

text: extension

text: (K,

text: D(K))

text: of

text: (A

text: B

text: D(A

text: which

text: generates

text: a

text: substochastic

text: semigroup

text: (G

text: (t))

text: C

text: C

text: C

text: K

text: t

text: This

text: theorem

text: proves

text: only

text: the

text: existence

text: of

text: a

text: generating

text: extension

text: K,

text: and

text: oﬀers

text: no

text: indication

text: of

text: how

text: exactly

text: K

text: relates

text: to

text: A

text: B

text: The

text: nature

text: of

text: the

text: generator

text: K

text: is

text: closely

text: related

text: to

text: C

text: C

text: the

text: concept

text: of

text: the

text: honesty

text: of

text: the

text: semigroup

text: (Deﬁnition

text: and

text: in

text: turn

text: the

text: occurrence

text: of

text: ‘shattering’.

text: This

text: relationship

text: is

text: discussed

text: in

text: where

text: a

text: range

text: of

text: possibilities

text: for

text: K

text: are

text: considered

text: and

text: it

text: is

text: shown

text: that

text: the

text: cases

text: in

text: which

text: shattering

text: occurs

text: coincide

text: with

text: those

text: in

text: which

text: the

text: semigroup

text: generated

text: by

text: K

text: is

text: dishonest.

text: In

text: order

text: to

text: establish

text: the

text: honesty

text: of

text: the

text: semigroup

text: (G

text: (t))

text: we

text: follow

text: a

text: similar

text: approach

text: to

text: K

text: t

text: that

text: taken

text: in

text: Section

text: Let

text: us

text: denote

text: by

text: E

text: the

text: set

text: of

text: all

text: measurable

text: functions

text: deﬁned

text: on

text: (N,

text: which

text: take

text: values

text: within

text: the

text: extended

text: reals.

text: By

text: E

text: we

text: denote

text: the

text: subspace

text: of

text: E

text: f

text: ##LTLine##


text: 10 GRAHAM BAIRD AND ENDRE SU¨LI → The ﬁnal items of notation which we must introduce are the mappings B : D(B) E , → + f,+ where D(B) = G and L : F X deﬁned by + C+ ∈ Bf := sup B f , f D(B) , ∈N C n + n ∈ Lf := sup R(1, A )f , f F , ∈N C n + n ≤ ≤ ∈ N where 0 f f for all n and sup ∈N f = f. n n+1 n n With the set notations and extension operators deﬁned, we can now detail the key generator characterisation result, which will enable us to establish the honesty of our semigroup. ∈ − ∈ (cid:90) (cid:90) Theorem 3.4. If for all f F such that f + BLf X and c(Lf) exists it is true that + C ∞ ∞ − ≥ − (3.5) Lfx dx + ( f + BLf) x dx c (Lf) , N N then K = A + B . C C (cid:3) Proof. See [8, Theorem 6.22]. Theorem 3.5. If the fragmentation rate, a(x), is such that ∞ ∈ ∞ lim sup a(x) < and a L∞ (N, ), → ,loc x N+ then the semigroup (G (t)) ≥ is honest. K t 0 − Proof. The proof of this result follows closely that of [8, Theorem 8.5]. Since A f = af, { ∈ − ∈ } C − as in [9, Corollary 3.1], we have that F = f E : (1 + a) 1f X and Lf = (1 + a) 1f, C whilst by Lebesgue’s monotone convergence theorem the operator B is given by the integral B expression . ∈ − ∈ For f F , let g = Lf = (1 + a) 1f X . Then we see that the condition (3.5) is satisﬁed +∈ − B ∈C+ if for all g X such that ag + g X and c(g) exists, we have: C+ (cid:90) C ∞ − B ≥ − (3.6) ( a(x)g(x) + ( g) (x)) x dx c (g) . N ∈ By our assumptions regarding the function a, we have ag L ((N, R], x dx) for any N < ∞ B ∈ − B ∈ 1 R < with g L ((N, R], x dx) also, since ag + g X . We may write the left-hand 1 C side of (3.(cid:90)6) as (cid:90) ∞ − (cid:26) (cid:90) B (cid:90) (cid:18)(cid:90) R − (cid:19)B (cid:27) ( a(x)g(x) + ( g) (x)) x dx = lim ( a(x)g(x) + ( g) (x)) x dx →∞ R N N ∞ − R R | (3.7) = lim a(x)g(x)x dx + a(y)b(x y)g(y) dy x dx . →∞ R N N x ##LTLine##

text: GRAHAM

text: BAIRD

text: AND

text: ENDRE

text: SU¨LI

text: The

text: ﬁnal

text: items

text: of

text: notation

text: which

text: we

text: must

text: introduce

text: are

text: the

text: mappings

text: B

text: D(B)

text: E

text: f,+

text: where

text: D(B)

text: G

text: and

text: L

text: F

text: X

text: deﬁned

text: by

text: C+

text: Bf

text: sup

text: B

text: f

text: f

text: D(B)

text: ∈N

text: C

text: n

text: n

text: Lf

text: sup

text: A

text: )f

text: f

text: F

text: ∈N

text: C

text: n

text: n

text: N

text: where

text: f

text: f

text: for

text: all

text: n

text: and

text: sup

text: ∈N

text: f

text: f.

text: n

text: n

text: n

text: With

text: the

text: set

text: notations

text: and

text: extension

text: operators

text: deﬁned,

text: we

text: can

text: now

text: detail

text: the

text: key

text: generator

text: characterisation

text: result,

text: which

text: will

text: enable

text: us

text: to

text: establish

text: the

text: honesty

text: of

text: our

text: semigroup.

text: Theorem

text: If

text: for

text: all

text: f

text: F

text: such

text: that

text: f

text: BLf

text: X

text: and

text: c(Lf)

text: exists

text: it

text: is

text: true

text: that

text: C

text: Lfx

text: dx

text: f

text: BLf)

text: x

text: dx

text: c

text: (Lf)

text: N

text: N

text: then

text: K

text: A

text: B

text: C

text: C

text: Proof.

text: See

text: Theorem

text: Theorem

text: If

text: the

text: fragmentation

text: rate,

text: a(x),

text: is

text: such

text: that

text: lim

text: sup

text: a(x)

text: and

text: a

text: L∞

text: (N,

text: ,loc

text: x

text: N+

text: then

text: the

text: semigroup

text: (G

text: (t))

text: is

text: honest.

text: K

text: t

text: Proof.

text: The

text: proof

text: of

text: this

text: result

text: follows

text: closely

text: that

text: of

text: Theorem

text: Since

text: A

text: f

text: af,

text: C

text: as

text: in

text: Corollary

text: we

text: have

text: that

text: F

text: f

text: E

text: a)

text: X

text: and

text: Lf

text: a)

text: C

text: whilst

text: by

text: Lebesgue’s

text: monotone

text: convergence

text: theorem

text: the

text: operator

text: B

text: is

text: given

text: by

text: the

text: integral

text: B

text: expression

text: For

text: f

text: F

text: let

text: g

text: Lf

text: a)

text: X

text: Then

text: we

text: see

text: that

text: the

text: condition

text: is

text: satisﬁed

text: B

text: ∈C+

text: if

text: for

text: all

text: g

text: X

text: such

text: that

text: ag

text: g

text: X

text: and

text: c(g)

text: exists,

text: we

text: have:

text: C+

text: C

text: B

text: a(x)g(x)

text: g)

text: (x))

text: x

text: dx

text: c

text: (g)

text: N

text: By

text: our

text: assumptions

text: regarding

text: the

text: function

text: a,

text: we

text: have

text: ag

text: L

text: ((N,

text: R],

text: x

text: dx)

text: for

text: any

text: N

text: B

text: B

text: R

text: with

text: g

text: L

text: ((N,

text: R],

text: x

text: dx)

text: also,

text: since

text: ag

text: g

text: X

text: We

text: may

text: write

text: the

text: left-hand

text: C

text: side

text: of

text: as

text: B

text: R

text: a(x)g(x)

text: g)

text: (x))

text: x

text: dx

text: lim

text: a(x)g(x)

text: g)

text: (x))

text: x

text: dx

text: R

text: N

text: N

text: R

text: R

text: lim

text: a(x)g(x)x

text: dx

text: a(y)b(x

text: y)g(y)

text: dy

text: x

text: dx

text: R

text: N

text: N

text: x

text: ##LTLine##


text: 16 68 N. Reyren, S. T. S, A. D. Caviglia, L. F. Kourkoutis, G. Hamm89erAl,.CB.aRlaitcshktyera,nCd. EW. .ASbcrhanheaidmesr,, PTh.yKs.oRppe,v.AB. S4.5R, u1e3t1s2c5hi(,1992). D. Jaccard, M. Gabay, D. Muller, J. M. Triscone, and J. Man9n0haVr.t,JS. cEiemnecrey3a1n7d, S11. 9K6iv(e2l0s0o7n),.Phys. Rev. B 46, 10812 (1992). 69 L. P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (92100P1.).Coleman, E. Miranda, and A. Tsvelik, Phys. Rev. B 49, 8955 (1994). 70 P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys.92REev..ALbertath. a9m2,s,09A7.0B01al(a2t0sk0y4,).D. J. Scalapino, and J. R. Schrieﬀer, Phys. Rev. B 52, 1271 (1995). 71 S. Fujimoto, J. Phys. Soc. Jpn. 76, 051008 (2007). 93 P. Coleman, A. Georges, and A. M. Tsvelik, J. Phys. Condens. Matter 9, 345 (1997). 72 M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009). 94 D. Belitz and T. R. Kirkpatrick, Phys. Rev. B 60, 3485 (1999). 73 Y. Tanaka, Y. Nazarov, and S. Kashiwaya, Phys. Rev. Lett. 9905, M16.7V00o3jta(2a0n0d3)E. . Dagotto, Phys. Rev. B 59, R713 (1999). 74 S. Kobayashi, Y. Tanaka, and M. Sato, ArXiv e-prints (2015)9,6aYrX. iFvu:1se5y1a0,.0H14. 1K1o[hcnoon,d-amnadt.Ksu.pMr-icyoank]e., J. Phys. Soc. Jpn. 72, 2914 (2003). 75 A. Umerski, Phys. Rev. B 55, 5266 (1997). 97 K. Shigeta, S. Onari, K. Yada, and Y. Tanaka, Phys. Rev. B 79, 174507 (2009). 76 A. Furusaki, M. Matsumoto, and M. Sigrist, Phys. Rev. B 649,805T4.5H14ot(t2a0,0J1.).Phys. Soc. Jpn. 78, 123710 (2009). 77 X.-L. Qi, T. L. Hughes, S. Raghu, and S.-C. Zhang, Phys. Re9v9. LKe.ttS.h1ig0e2t,a,18Y7.0T01an(a2k0a0,9K).. Kuroki, S. Onari, and H. Aizawa, Phys. Rev. B 83, 140509(R) (2011). 78 G.R.Volovik, The Universe in a Helium Droplet (Oxford Scien10c0e DPu. bSloicleantoiovn,sI,.2M00a3r)t.in, and D. Mozyrsky, Phys. Rev. B 79, 132502 (2009). 79 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). 101 Y. Fuseya and K. Miyake, J. Phys. Soc. Jpn. 80, 054705 (2011). 80 M. Matsumoto and H. Shiba, J. Phys. Soc. Jpn 64, 1703 (199150)2. H. Kusunose, Y. Fuseya, and K. Miyake, J. Phys. Soc. Jpn. 80, 054702 (2011). 81 Y. Tanaka, Y. Nazarov, A. Golubov, and S. Kashiwaya, Phys1.03RHev..KBu6su9n,o1s4e4,5Y1.9F(u2s0e0y4a),. and K. Miyake, J. Phys. Soc. Jpn. 80, 044711 (2011). 82 K. Yada, S. Onari, Y. Tanaka, and J. Inoue, Phys. Rev. B 8010,414K0.5F09uk(u2i00a9n)d. Y. Kato, Journal of the Physical Society of Japan 87, 014706 (2018). 83 Y. Tanaka, Y. Mizuno, T. Yokoyama, K. Yada, and M. Sato,10P5hYys..ARseavn.oL, eYtt..V1.0F5o,m09in7o0v0,2 (a2n0d10Y).. Tanaka, Phys. Rev. B 90, 094512 (2014). 84 K. Yada, M. Sato, Y. Tanaka, and T. Yokoyama, Phys. Rev.1B06. 8Y3.,V0.6F45o0m5in(o2v0,11Y).. Tanaka, Y. Asano, and M. Eschrig, Phys. Rev. B 91, 144514 (2015). 85 S. Wu and K. V. Samokhin, Phys. Rev. B 82, 184501 (2010).107 S. Hoshino, Phys. Rev. B 90, 115154 (2014). 86 S. Wu and K. V. Samokhin, Phys. Rev. B 81, 214506 (2010).108 S. Hoshino, K. Yada, and Y. Tanaka, Phys. Rev. B 93, 224511 (2016). 87 T. R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. 66, 1533 (1991). 88 D. Belitz and T. R. Kirkpatrick, Phys. Rev. B 46, 8393 (1992).

text: N.

text: Reyren,

text: S.

text: T.

text: S,

text: A.

text: D.

text: Caviglia,

text: L.

text: F.

text: Kourkoutis,

text: G.

text: EW.

text: .ASbcrhanheaidmesr,,

text: PTh.yKs.oRppe,v.AB.

text: D.

text: Jaccard,

text: M.

text: Gabay,

text: D.

text: Muller,

text: J.

text: M.

text: Triscone,

text: and

text: J.

text: Rev.

text: B

text: L.

text: P.

text: Gor’kov

text: and

text: E.

text: I.

text: Rashba,

text: Phys.

text: Rev.

text: Lett.

text: E.

text: Miranda,

text: and

text: A.

text: Tsvelik,

text: Phys.

text: Rev.

text: B

text: P.

text: A.

text: Frigeri,

text: D.

text: F.

text: Agterberg,

text: A.

text: Koga,

text: and

text: M.

text: Sigrist,

text: J.

text: Scalapino,

text: and

text: J.

text: R.

text: Schrieﬀer,

text: Phys.

text: Rev.

text: B

text: S.

text: Fujimoto,

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: P.

text: Coleman,

text: A.

text: Georges,

text: and

text: A.

text: M.

text: Tsvelik,

text: J.

text: Phys.

text: Condens.

text: Matter

text: M.

text: Sato

text: and

text: S.

text: Fujimoto,

text: Phys.

text: Rev.

text: B

text: D.

text: Belitz

text: and

text: T.

text: R.

text: Kirkpatrick,

text: Phys.

text: Rev.

text: B

text: Y.

text: Tanaka,

text: Y.

text: Nazarov,

text: and

text: S.

text: Kashiwaya,

text: Phys.

text: Rev.

text: Lett.

text: Dagotto,

text: Phys.

text: Rev.

text: B

text: S.

text: Kobayashi,

text: Y.

text: Tanaka,

text: and

text: M.

text: Sato,

text: ArXiv

text: e-prints

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: A.

text: Umerski,

text: Phys.

text: Rev.

text: B

text: K.

text: Shigeta,

text: S.

text: Onari,

text: K.

text: Yada,

text: and

text: Y.

text: Tanaka,

text: Phys.

text: Rev.

text: B

text: A.

text: Furusaki,

text: M.

text: Matsumoto,

text: and

text: M.

text: Sigrist,

text: Phys.

text: Rev.

text: B

text: Soc.

text: Jpn.

text: X.-L.

text: Qi,

text: T.

text: L.

text: Hughes,

text: S.

text: Raghu,

text: and

text: S.-C.

text: Zhang,

text: Phys.

text: Kuroki,

text: S.

text: Onari,

text: and

text: H.

text: Aizawa,

text: Phys.

text: Rev.

text: B

text: G.R.Volovik,

text: The

text: Universe

text: in

text: a

text: Helium

text: Droplet

text: (Oxford

text: DPu.

text: and

text: D.

text: Mozyrsky,

text: Phys.

text: Rev.

text: B

text: X.-L.

text: Qi

text: and

text: S.-C.

text: Zhang,

text: Rev.

text: Mod.

text: Phys.

text: Y.

text: Fuseya

text: and

text: K.

text: Miyake,

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: M.

text: Matsumoto

text: and

text: H.

text: Shiba,

text: J.

text: Phys.

text: Soc.

text: Jpn

text: H.

text: Kusunose,

text: Y.

text: Fuseya,

text: and

text: K.

text: Miyake,

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: Y.

text: Tanaka,

text: Y.

text: Nazarov,

text: A.

text: Golubov,

text: and

text: S.

text: Kashiwaya,

text: and

text: K.

text: Miyake,

text: J.

text: Phys.

text: Soc.

text: Jpn.

text: K.

text: Yada,

text: S.

text: Onari,

text: Y.

text: Tanaka,

text: and

text: J.

text: Inoue,

text: Phys.

text: Rev.

text: B

text: Y.

text: Kato,

text: Journal

text: of

text: the

text: Physical

text: Society

text: of

text: Japan

text: Y.

text: Tanaka,

text: Y.

text: Mizuno,

text: T.

text: Yokoyama,

text: K.

text: Yada,

text: and

text: M.

text: Tanaka,

text: Phys.

text: Rev.

text: B

text: K.

text: Yada,

text: M.

text: Sato,

text: Y.

text: Tanaka,

text: and

text: T.

text: Yokoyama,

text: Phys.

text: Tanaka,

text: Y.

text: Asano,

text: and

text: M.

text: Eschrig,

text: Phys.

text: Rev.

text: B

text: S.

text: Wu

text: and

text: K.

text: V.

text: Samokhin,

text: Phys.

text: Rev.

text: B

text: S.

text: Hoshino,

text: Phys.

text: Rev.

text: B

text: S.

text: Wu

text: and

text: K.

text: V.

text: Samokhin,

text: Phys.

text: Rev.

text: B

text: S.

text: Hoshino,

text: K.

text: Yada,

text: and

text: Y.

text: Tanaka,

text: Phys.

text: Rev.

text: B

text: T.

text: R.

text: Kirkpatrick

text: and

text: D.

text: Belitz,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: Belitz

text: and

text: T.

text: R.

text: Kirkpatrick,

text: Phys.

text: Rev.

text: B