text: ∅ Lemma 2.10. Assume D (S) = . Then for all ﬁnite index subgroup G < G, there f ∅ 0 exists a symmetric generating set S for which D (S ) = . 0 f 0 ∈ D ∗ Proof. We have to ﬁnd S such that for any x ˚ such that Reg(x, S ) is inﬁnite, we have ∗ 0 that Reg(x, S ) is inﬁnite. First, note that this is automatic if the G-orbit of x contains 0 no discontinuity point of S. Indeed, it then follows that for any symmetric generating set ∗ S of G , Reg(x, S ) is G .x which is inﬁnite. 0 0 ∈ 0 0 Given x Disc(S) we claim that we can ﬁnd a symmetric generating set S of G ∗ ∈ ∩ D x 0 such that Reg(y, S ) is inﬁnite for all y G.x ˚. This will conclude the proof since x ∈ one can take for S the union of all sets S for x Disc(S). Let Γ be the orbit graph 0 x of x, i. e. the graph whose vertices are the points in G.x and there is an edge labelled ∈ s S between y and z if z = sy. Since S is ﬁnite, this is a connected locally ﬁnite graph. ∩ D Let Γ be the subgraph of Γ whose vertices are the points in G.x ˚ and with an edge reg labelled s between y and z = sy if s is continuous at y. The connected components of ∗ ∈ D Γ are precisely the sets Reg(y, S ), for y G.x. Since Disc(S) and ˚ are ﬁnite, Γ reg ∅ reg has ﬁnitely many connected components, all of which are inﬁnite since D (S) = . f We now focus on some connected component Λ of Γ . We will prove that for each ∈ reg ∗ y Λ, one can ﬁnd a symmetric generating set S of G such that Reg(y, S ) is inﬁnite ∩ y 0 ∩ y and and contains (Λ G .y). Since Λ is the ﬁnite union of the sets (Λ G .y), the lemma 0 0 will follow. ⊂ ∈ Choose U G a ﬁnite set of representatives of right cosets G u of G in G with 1 U. ⊂ ∈ 0 0 Choose T Λ be a maximal tree. For each z Λ, denote by w the element of G labelled z by the path in T joining y to z. Then w (y) = z and w is continuous at y. Then consider ∈ ∈ z z ∈ (cid:48) ⊂ u U such that u w G ; in particular, u (z) = u w (y) G .y. Let Λ Λ be the z z z 0 z z z 0 \ (cid:48) set of vertices z such that u is continous at z. Since Disc(U) is ﬁnite, Λ Λ is ﬁnite, and z (cid:48) since Λ is locally ﬁnite and connected, Λ has ﬁnitely many connected components, and so ∩ (cid:48) (cid:48) ∩ (cid:48) does T Λ (here, we view Λ and T Λ as induced subgraphs of Λ and T respectively). ∈ (cid:48) ∈ Let ∆ be the set of points the form u (z) for z Λ . Since 1 U has no discontinuity ∩ ⊂ z point, Λ G .y ∆. Given a symmetric generating set S of G , we view ∆ as the 0 (cid:48) (cid:48)y 0 vertex set of a graph where we put an edge between y and y in ∆ if there is a generator ∈ (cid:48) (cid:48) (cid:48) 1 2 s S sending y to y and continuous on y . We claim that we can ﬁnd S so that y 1 2 1 ⊂ y ∗ ∆ is connected. This will conclude the proof since this implies that ∆ Reg(y, S ), so ∗ ∩ y Reg(y, S ) is inﬁnite and contains Λ G .y. y 0 First take for S a ﬁnite symmetric generating set of G containing all the elements −y 0 of the form u su 1 where s is the label of an edge in T joining z to z (this takes z2 z1 ∈ (cid:48) 1 2 ﬁnitely many values). Then for any two points z , z Λ joined by an edge in T , the (cid:48) (cid:48) 1 2 corresponding points z = u (z ) and z = u (z ) are connected by an edge in ∆. Since (cid:48) 1 z1 1 2 z2 2 Λ has ﬁnitely many connected components, so does ∆. ∩ (cid:48) Now choose points y , y , . . . , y in each connected component of T Λ with y = y, (cid:48) ∈ 0 1 ∈n { } 0 and let y = u (y ) ∆. For all i 1, . . . , n , let w be the label of the path in T joining i yi i ∈ (cid:48) i y to y . Then the element u w G sends y to y and is continuous at y. Thus adding i yi i 0 i the elements u w , . . . , u w to S makes ∆ connected which concludes the proof. y1 1 yn n y Let G < G be a ﬁnite index subgroup. Let J , . . . , J be the irreducible compo- 0 D 1 n(G0) nents of G in . 0 ∅ By the lemma, there is a ﬁnite symmetric generating set S of G such that D (S ) = . ≤ 0 0 f 0 Imanishi theorem then says that each i n(G ), the irreducible component J is a union D D 0 i D of connected component of . Let n( ) be the number of connected component of . D − The quantity n( ) n(G ) is therefore non-negative, and when we go to a ﬁnite index 0 subgroup, this quantity decreases. D − Thus taking a ﬁnite index subgroup minimizing the quantity n( ) n(G ) gives a 0 stable group. 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: Assume

text: D

text: (S)

text: Then

text: for

text: all

text: ﬁnite

text: index

text: subgroup

text: G

text: G,

text: there

text: f

text: exists

text: a

text: symmetric

text: generating

text: set

text: S

text: for

text: which

text: D

text: (S

text: f

text: D

text: Proof.

text: We

text: have

text: to

text: ﬁnd

text: S

text: such

text: that

text: for

text: any

text: x

text: such

text: that

text: Reg(x,

text: S

text: is

text: inﬁnite,

text: we

text: have

text: that

text: Reg(x,

text: S

text: is

text: inﬁnite.

text: First,

text: note

text: that

text: this

text: is

text: automatic

text: if

text: the

text: G-orbit

text: of

text: x

text: contains

text: no

text: discontinuity

text: point

text: of

text: S.

text: Indeed,

text: it

text: then

text: follows

text: that

text: for

text: any

text: symmetric

text: generating

text: set

text: S

text: of

text: G

text: Reg(x,

text: S

text: is

text: G

text: .x

text: which

text: is

text: inﬁnite.

text: Given

text: x

text: Disc(S)

text: we

text: claim

text: that

text: we

text: can

text: ﬁnd

text: a

text: symmetric

text: generating

text: set

text: S

text: of

text: G

text: D

text: x

text: such

text: that

text: Reg(y,

text: S

text: is

text: inﬁnite

text: for

text: all

text: y

text: G.x

text: This

text: will

text: conclude

text: the

text: proof

text: since

text: x

text: one

text: can

text: take

text: for

text: S

text: the

text: union

text: of

text: all

text: sets

text: S

text: for

text: x

text: Disc(S).

text: Let

text: Γ

text: be

text: the

text: orbit

text: graph

text: x

text: of

text: x,

text: i.

text: e.

text: the

text: graph

text: whose

text: vertices

text: are

text: the

text: points

text: in

text: G.x

text: and

text: there

text: is

text: an

text: edge

text: labelled

text: s

text: S

text: between

text: y

text: and

text: z

text: if

text: z

text: sy.

text: Since

text: S

text: is

text: ﬁnite,

text: this

text: is

text: a

text: connected

text: locally

text: ﬁnite

text: graph.

text: D

text: Let

text: Γ

text: be

text: the

text: subgraph

text: of

text: Γ

text: whose

text: vertices

text: are

text: the

text: points

text: in

text: G.x

text: and

text: with

text: an

text: edge

text: reg

text: labelled

text: s

text: between

text: y

text: and

text: z

text: sy

text: if

text: s

text: is

text: continuous

text: at

text: y.

text: The

text: connected

text: components

text: of

text: D

text: Γ

text: are

text: precisely

text: the

text: sets

text: Reg(y,

text: S

text: for

text: y

text: G.x.

text: Since

text: Disc(S)

text: and

text: are

text: ﬁnite,

text: Γ

text: reg

text: reg

text: has

text: ﬁnitely

text: many

text: connected

text: components,

text: all

text: of

text: which

text: are

text: inﬁnite

text: since

text: D

text: (S)

text: f

text: We

text: now

text: focus

text: on

text: some

text: connected

text: component

text: Λ

text: of

text: Γ

text: We

text: will

text: prove

text: that

text: for

text: each

text: reg

text: y

text: Λ,

text: one

text: can

text: ﬁnd

text: a

text: symmetric

text: generating

text: set

text: S

text: of

text: G

text: such

text: that

text: Reg(y,

text: S

text: is

text: inﬁnite

text: y

text: y

text: and

text: and

text: contains

text: (Λ

text: G

text: .y).

text: Since

text: Λ

text: is

text: the

text: ﬁnite

text: union

text: of

text: the

text: sets

text: (Λ

text: G

text: .y),

text: the

text: lemma

text: will

text: follow.

text: Choose

text: U

text: G

text: a

text: ﬁnite

text: set

text: of

text: representatives

text: of

text: right

text: cosets

text: G

text: u

text: of

text: G

text: in

text: G

text: with

text: U.

text: Choose

text: T

text: Λ

text: be

text: a

text: maximal

text: tree.

text: For

text: each

text: z

text: Λ,

text: denote

text: by

text: w

text: the

text: element

text: of

text: G

text: labelled

text: z

text: by

text: the

text: path

text: in

text: T

text: joining

text: y

text: to

text: z.

text: Then

text: w

text: (y)

text: z

text: and

text: w

text: is

text: continuous

text: at

text: y.

text: Then

text: consider

text: z

text: z

text: u

text: U

text: such

text: that

text: u

text: w

text: G

text: in

text: particular,

text: u

text: (z)

text: u

text: w

text: (y)

text: G

text: .y.

text: Let

text: Λ

text: Λ

text: be

text: the

text: z

text: z

text: z

text: z

text: z

text: z

text: set

text: of

text: vertices

text: z

text: such

text: that

text: u

text: is

text: continous

text: at

text: z.

text: Since

text: Disc(U)

text: is

text: ﬁnite,

text: Λ

text: Λ

text: is

text: ﬁnite,

text: and

text: z

text: since

text: Λ

text: is

text: locally

text: ﬁnite

text: and

text: connected,

text: Λ

text: has

text: ﬁnitely

text: many

text: connected

text: components,

text: and

text: so

text: does

text: T

text: Λ

text: (here,

text: we

text: view

text: Λ

text: and

text: T

text: Λ

text: as

text: induced

text: subgraphs

text: of

text: Λ

text: and

text: T

text: respectively).

text: Let

text: be

text: the

text: set

text: of

text: points

text: the

text: form

text: u

text: (z)

text: for

text: z

text: Λ

text: Since

text: U

text: has

text: no

text: discontinuity

text: z

text: point,

text: Λ

text: G

text: .y

text: Given

text: a

text: symmetric

text: generating

text: set

text: S

text: of

text: G

text: we

text: view

text: as

text: the

text: vertex

text: set

text: of

text: a

text: graph

text: where

text: we

text: put

text: an

text: edge

text: between

text: y

text: and

text: y

text: in

text: if

text: there

text: is

text: a

text: generator

text: s

text: S

text: sending

text: y

text: to

text: y

text: and

text: continuous

text: on

text: y

text: We

text: claim

text: that

text: we

text: can

text: ﬁnd

text: S

text: so

text: that

text: y

text: y

text: is

text: connected.

text: This

text: will

text: conclude

text: the

text: proof

text: since

text: this

text: implies

text: that

text: Reg(y,

text: S

text: so

text: y

text: Reg(y,

text: S

text: is

text: inﬁnite

text: and

text: contains

text: Λ

text: G

text: .y.

text: y

text: First

text: take

text: for

text: S

text: a

text: ﬁnite

text: symmetric

text: generating

text: set

text: of

text: G

text: containing

text: all

text: the

text: elements

text: −y

text: of

text: the

text: form

text: u

text: su

text: where

text: s

text: is

text: the

text: label

text: of

text: an

text: edge

text: in

text: T

text: joining

text: z

text: to

text: z

text: (this

text: takes

text: ﬁnitely

text: many

text: values).

text: Then

text: for

text: any

text: two

text: points

text: z

text: z

text: Λ

text: joined

text: by

text: an

text: edge

text: in

text: T

text: the

text: corresponding

text: points

text: z

text: u

text: (z

text: and

text: z

text: u

text: (z

text: are

text: connected

text: by

text: an

text: edge

text: in

text: Since

text: Λ

text: has

text: ﬁnitely

text: many

text: connected

text: components,

text: so

text: does

text: Now

text: choose

text: points

text: y

text: y

text: y

text: in

text: each

text: connected

text: component

text: of

text: T

text: Λ

text: with

text: y

text: y,

text: ∈n

text: and

text: let

text: y

text: u

text: (y

text: For

text: all

text: i

text: n

text: let

text: w

text: be

text: the

text: label

text: of

text: the

text: path

text: in

text: T

text: joining

text: i

text: yi

text: i

text: i

text: y

text: to

text: y

text: Then

text: the

text: element

text: u

text: w

text: G

text: sends

text: y

text: to

text: y

text: and

text: is

text: continuous

text: at

text: y.

text: Thus

text: adding

text: i

text: yi

text: i

text: i

text: the

text: elements

text: u

text: w

text: u

text: w

text: to

text: S

text: makes

text: connected

text: which

text: concludes

text: the

text: proof.

text: yn

text: n

text: y

text: Let

text: G

text: G

text: be

text: a

text: ﬁnite

text: index

text: subgroup.

text: Let

text: J

text: J

text: be

text: the

text: irreducible

text: compo-

text: D

text: nents

text: of

text: G

text: in

text: By

text: the

text: lemma,

text: there

text: is

text: a

text: ﬁnite

text: symmetric

text: generating

text: set

text: S

text: of

text: G

text: such

text: that

text: D

text: (S

text: f

text: Imanishi

text: theorem

text: then

text: says

text: that

text: each

text: i

text: n(G

text: the

text: irreducible

text: component

text: J

text: is

text: a

text: union

text: D

text: D

text: i

text: D

text: of

text: connected

text: component

text: of

text: Let

text: n(

text: be

text: the

text: number

text: of

text: connected

text: component

text: of

text: D

text: The

text: quantity

text: n(

text: n(G

text: is

text: therefore

text: non-negative,

text: and

text: when

text: we

text: go

text: to

text: a

text: ﬁnite

text: index

text: subgroup,

text: this

text: quantity

text: decreases.

text: D

text: Thus

text: taking

text: a

text: ﬁnite

text: index

text: subgroup

text: minimizing

text: the

text: quantity

text: n(

text: n(G

text: gives

text: a

text: stable

text: group.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 8 f=149GHz (a) e- jump NO MW 7 X(0)= X (0)-X (0) 2 1 6 X (0) X (0) X (0) 5 fix1ed orbits 3 2 E F 4 (b) with MW e- jump 3 dark 2 A sin(w ) E F 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 X1(t) X (t) X (t) B(T) 3 2 orbits move FIG. 2: Calculated magnetoresistance, Rxx, vs magnetic ﬁeld, B, for dark and radiation of frequency f = 149 GHz, in (c) M W averaged time a high-mobility 2DES with strong RSOI of Rashba parameter × − · α = 0.6 10 11eV m. For the dark curve we obtain, as ) X( A sin(w ) expected, a beating pattern proﬁle made up of a system of nodes and antinodes. The radiation curve exhibits similar X (0) beating pattern but modulated by the rise of the system of 2 peaks and valleys of RIRO. (T=1K). X (0) X (0)=X (0)-A sin(w ) 1 3 2 DISTA NCE placement model includes, in a perturbative way, electron promotion with the corresponding photons absorption to higher Landau states and subsequent electron advances and recoils via scattering to explain R peaks and val- xx leys. Yet, the radiation-driven electron orbit theory does not consider any radiation-assisted electron promotion. FIG. 1: Schematic ﬁgure describing elastic scattering between As a further evolution of the displacement model, our Landau states or orbits. a) Dark scenario. The scattering theory proceeds in an alternative approach starting from jump takes place between ﬁxed orbits. τ is the time it takes the exact solution of the time-dependent Schrodinger the electron to complete the jump and reach the ﬁnal orbit. equation for an electron under magnetic ﬁeld and radia- b) Under radiation the scattering jump is between oscillating tion. The obtained exact wave function represents a Lan- orbits according to A sin wt. c) Scattering under radiation dau state where the guiding center is harmonically driven for an averaged time scenario, i.e., steady state. The station- back and forth by radiation at the same frequency. Inter- ary advanced distance turns out to be smaller than the dark estingly, the Landau states guiding center follow a clas- one. This corresponds to a valley in the radiation-induced sical trajectory given by the solution of the driven clas- magnetoresistance oscillations. sical oscillator. According to this theory, the interaction of the driven Landau states with charged impurities ends up giving rise to shorter and longer average advanced kind of systems. We could also predict that with samples distance by the scattered electrons. These distance are with even higher mobilities and at high enough radiation reﬂected on irradiated R as valleys and peaks respec- xx intensity, 2DES systems with RSOI can give rise to ZRS. tively. An interesting trait of this model is that it encom- Thus, we start oﬀ based on the previous theory of the passes quantum and classical concepts unlike the inelastic radiation-driven electron orbit10–14. This theory stems and the displacement models that are fully quantum. At from the displacement model7 and shares with it the in- this point we can highlight a recent theoretical contribu- terplay between charged impurity scattering and radia- tion by Beltukov29 et al. that is fully classical. It should tion to be at the heart of RIRO. As a reminder, the dis- not be surprising to take into account and include totally YGREN E ) ( R XX ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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##

label: paragraph

font: default

text: (a)

text: e-

text: jump

text: NO

text: MW

text: X

text: X

text: X

text: X

text: orbits

text: E

text: F

text: (b)

text: with

text: MW

text: e-

text: jump

text: dark

text: A

text: sin(w

text: E

text: F

text: X

text: (t)

text: X

text: (t)

text: B(T)

text: orbits

text: move

text: FIG.

text: Calculated

text: magnetoresistance,

text: Rxx,

text: vs

text: magnetic

text: ﬁeld,

text: B,

text: for

text: dark

text: and

text: radiation

text: of

text: frequency

text: f

text: GHz,

text: in

text: (c)

text: M

text: W

text: averaged

text: time

text: a

text: high-mobility

text: with

text: strong

text: RSOI

text: of

text: Rashba

text: parameter

text: α

text: m.

text: For

text: the

text: dark

text: curve

text: we

text: obtain,

text: as

text: X(

text: A

text: sin(w

text: expected,

text: a

text: beating

text: pattern

text: proﬁle

text: made

text: up

text: of

text: a

text: system

text: of

text: nodes

text: and

text: antinodes.

text: The

text: radiation

text: curve

text: exhibits

text: similar

text: X

text: beating

text: pattern

text: but

text: modulated

text: by

text: the

text: rise

text: of

text: the

text: system

text: of

text: peaks

text: and

text: valleys

text: of

text: RIRO.

text: X

text: X

text: sin(w

text: DISTA

text: NCE

text: placement

text: model

text: includes,

text: in

text: a

text: perturbative

text: way,

text: electron

text: promotion

text: with

text: the

text: corresponding

text: photons

text: absorption

text: to

text: higher

text: Landau

text: states

text: and

text: subsequent

text: electron

text: advances

text: and

text: recoils

text: via

text: scattering

text: to

text: explain

text: R

text: peaks

text: and

text: val-

text: xx

text: leys.

text: Yet,

text: the

text: radiation-driven

text: electron

text: orbit

text: theory

text: does

text: not

text: consider

text: any

text: radiation-assisted

text: electron

text: promotion.

text: FIG.

text: Schematic

text: ﬁgure

text: describing

text: elastic

text: scattering

text: between

text: As

text: a

text: further

text: evolution

text: of

text: the

text: displacement

text: model,

text: our

text: Landau

text: states

text: or

text: orbits.

text: a)

text: Dark

text: scenario.

text: The

text: scattering

text: theory

text: proceeds

text: in

text: an

text: alternative

text: approach

text: starting

text: from

text: jump

text: takes

text: place

text: between

text: ﬁxed

text: orbits.

text: τ

text: is

text: the

text: time

text: it

text: takes

text: the

text: exact

text: solution

text: of

text: the

text: time-dependent

text: Schrodinger

text: the

text: electron

text: to

text: complete

text: the

text: jump

text: and

text: reach

text: the

text: ﬁnal

text: orbit.

text: equation

text: for

text: an

text: electron

text: under

text: magnetic

text: ﬁeld

text: and

text: radia-

text: b)

text: Under

text: radiation

text: the

text: scattering

text: jump

text: is

text: between

text: oscillating

text: tion.

text: The

text: obtained

text: exact

text: wave

text: function

text: represents

text: a

text: Lan-

text: orbits

text: according

text: to

text: A

text: sin

text: wt.

text: c)

text: Scattering

text: under

text: radiation

text: dau

text: state

text: where

text: the

text: guiding

text: center

text: is

text: harmonically

text: driven

text: for

text: an

text: averaged

text: time

text: scenario,

text: i.e.,

text: steady

text: state.

text: The

text: station-

text: back

text: and

text: forth

text: by

text: radiation

text: at

text: the

text: same

text: frequency.

text: Inter-

text: ary

text: advanced

text: distance

text: turns

text: out

text: to

text: be

text: smaller

text: than

text: the

text: dark

text: estingly,

text: the

text: Landau

text: states

text: guiding

text: center

text: follow

text: a

text: clas-

text: one.

text: This

text: corresponds

text: to

text: a

text: valley

text: in

text: the

text: radiation-induced

text: sical

text: trajectory

text: given

text: by

text: the

text: solution

text: of

text: the

text: driven

text: clas-

text: magnetoresistance

text: oscillations.

text: sical

text: oscillator.

text: According

text: to

text: this

text: theory,

text: the

text: interaction

text: of

text: the

text: driven

text: Landau

text: states

text: with

text: charged

text: impurities

text: ends

text: up

text: giving

text: rise

text: to

text: shorter

text: and

text: longer

text: average

text: advanced

text: kind

text: of

text: systems.

text: We

text: could

text: also

text: predict

text: that

text: with

text: samples

text: distance

text: by

text: the

text: scattered

text: electrons.

text: These

text: distance

text: are

text: with

text: even

text: higher

text: mobilities

text: and

text: at

text: high

text: enough

text: radiation

text: reﬂected

text: on

text: irradiated

text: R

text: as

text: valleys

text: and

text: peaks

text: respec-

text: xx

text: intensity,

text: systems

text: with

text: RSOI

text: can

text: give

text: rise

text: to

text: ZRS.

text: tively.

text: An

text: interesting

text: trait

text: of

text: this

text: model

text: is

text: that

text: it

text: encom-

text: Thus,

text: we

text: start

text: oﬀ

text: based

text: on

text: the

text: previous

text: theory

text: of

text: the

text: passes

text: quantum

text: and

text: classical

text: concepts

text: unlike

text: the

text: inelastic

text: radiation-driven

text: electron

text: This

text: theory

text: stems

text: and

text: the

text: displacement

text: models

text: that

text: are

text: fully

text: quantum.

text: At

text: from

text: the

text: displacement

text: and

text: shares

text: with

text: it

text: the

text: in-

text: this

text: point

text: we

text: can

text: highlight

text: a

text: recent

text: theoretical

text: contribu-

text: terplay

text: between

text: charged

text: impurity

text: scattering

text: and

text: radia-

text: tion

text: by

text: et

text: al.

text: that

text: is

text: fully

text: classical.

text: It

text: should

text: tion

text: to

text: be

text: at

text: the

text: heart

text: of

text: RIRO.

text: As

text: a

text: reminder,

text: the

text: dis-

text: not

text: be

text: surprising

text: to

text: take

text: into

text: account

text: and

text: include

text: totally

text: YGREN

text: E

text: R

text: XX

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text: 3 (cid:114) or partially classical approaches explaining RIRO consid- ering the very low magnetic ﬁelds where this eﬀect rise up. (cid:126) ± 1 (cid:126) − 8α2 E ± = w N ( w gµ B)2 (cid:113)+ N (3) We have added to the same total Hamiltonian of the N c 2 c B R2 radiation-driven electron orbit theory the Rashba inter- action, solving exactly the corresponding time-dependent (cid:126) where R is the magnetic length, R = . The cor- Schrodinger equation. In other words, we develop a semi- eB responding wave function for t(cid:18)he + branch(cid:19)is, classical model based on the exact solution of the elec- tronic wave function in the presence of a static B with (cid:112) important Rashba coupling interacting with radiation 1 cos θφ − achnadrgaepdeirmtuprubraittiioesn. tWreaetmcoennstidfeorr tehlaisstticypsecaotftesrciantgtefrroinmg ψN+ = Ly eikyy sin2θ2φNN 1 (4) as the most probable at the low temperatures normally used in experiments. Following the previous model and and for the - branch, (cid:18) (cid:19) applying a Boltzmann transport model we are able to (cid:112) obtain an expression for R with radiation and RSOI. − xx 1 sin θφ − In the simulations we obtain, ﬁrst without radiation, the ψ − = eikyy (cid:20) 2 N 1(cid:21) (5) well-known beating pattern with the system of nodes and N Ly cos θ2φN antinodes of the Rashba magnetoresistance30–39. Then, √ √ we switched on light obtaining Rxx that shows a strong where θ is given by θ = arctan 2 R2α−(cid:126)N and φ is the deformation of the previous beating pattern where the gµBB wc nodes and antinodes follow the peaks and valleys of normalized quantum harmonic oscillator wave function, RIRO. We study the dependence on power and frequency i.e., Landau state, N being the corresponding Landau including the terahertz regime. 2DES with RSOI share level index. According to these results the Rashba spin- similar Hamiltonian with 3D topological insulators, then orbit interaction mixes spin-down and spin-up states of we consider that the results that we present in this arti- adjacent Landau levels to give rise to two new energy cle could be of application in the ﬁeld of 3D topological branches of eigenstates of the Hamiltonian H0. insulators under the inﬂuence of radiation. To analyze magnetotransport in 2DES with RSOI we II. THEORETICAL MODEL calculate the longitudinal conductivity σxx following the Boltzmann t(cid:90)ransport theory40–42, wher(cid:18)e σ is g(cid:19)iven by: xx − We consider a 2DES in the x y plane with strong ∞ Rashba coupling subjected to a static and perpendic- −df(E) ular B and a DC electric ﬁeld parallel to the x di- σxx = e2 dEρi(E)[∆X(0)]2WI dE (6) rection. Using Landau gauge for the potential vector, 0 A = (0, Bx, 0), the hamiltonian of such a system, H 0 being E the energy, ρ (E) the density of initial states reads: H = (H + H ) where the diﬀerent compo- i 0(cid:20) B SO (cid:21) and f(E) the electron distribution function. ∆X(0) nents of H are: 0 is the shift of the guiding center coordinate for the eigenstates involved in the scattering event, (cid:20) p2 1 − (cid:21) 1 HB = 2mx∗ + 2m∗wc2 (x X0)2 σ0 + 2gµBσzB + −eEdcX0 + 12m∗ EBd22c σ0 (1) ∆X(0) = [X2(0) − X1(0)] (cid:39) 2Rc (7) −α − − HSO = (cid:126) [σypx σxeB (x X0)] (2) (cid:16) (cid:17) X (0) and X (0) being the guiding center coordinates 2 1 for ﬁnal and initial states respectively and R the cy- X is the center of the orbit for the electron spiral mo- c tion:0 X0 = − (cid:126)ekBy − meE∗wdc2 , Edc is the DC electric ﬁeld ctelortinrognrraatdeiubse.caWusIeiswtehecornesmidoetre tchhaatrgaetd vimerpyurloitwy stceamt-- c →p−arallel to the x direction, σ stands for the unit matrix, peratures (T) this is the most likely source of scattering 0 σ = (σ , σ , σ ) are the Pauli spin matrices, g the Zee- for electrons in high mobility 2DES. According to the x y z man factor, µ the Bhor magneton, α the Rashba spin- Fermi’s Golden Rule W is given by B I orbit coupling parameter and w the cyclotron frequency. c The Schrodinger equation corresponding to the Hamilto- 2π | | | | − nian H0 can be exactly solved and the resulting states are WI = (cid:126) NI < ψf± Vs ψi± > 2δ(Ef Ei) (8) labeled by the quantum number N. For N = 0 there is (cid:126) − only one level of energy given by E = ( w gµ B)/2. ≥ 0 c B For N 1 we obtain two branches of levels labelled by where N is the impurity density and E and E are I i f + and - and with energies: the energies of the initial and ﬁnal states respectively. V s ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: or

text: partially

text: classical

text: approaches

text: explaining

text: RIRO

text: consid-

text: ering

text: the

text: very

text: low

text: magnetic

text: ﬁelds

text: where

text: this

text: eﬀect

text: rise

text: up.

text: E

text: w

text: N

text: w

text: gµ

text: N

text: We

text: have

text: added

text: to

text: the

text: same

text: total

text: Hamiltonian

text: of

text: the

text: N

text: c

text: c

text: B

text: radiation-driven

text: electron

text: orbit

text: theory

text: the

text: Rashba

text: inter-

text: action,

text: solving

text: exactly

text: the

text: corresponding

text: time-dependent

text: where

text: R

text: is

text: the

text: magnetic

text: length,

text: R

text: The

text: cor-

text: Schrodinger

text: equation.

text: In

text: other

text: words,

text: we

text: develop

text: a

text: semi-

text: eB

text: responding

text: wave

text: function

text: for

text: classical

text: model

text: based

text: on

text: the

text: exact

text: solution

text: of

text: the

text: elec-

text: tronic

text: wave

text: function

text: in

text: the

text: presence

text: of

text: a

text: static

text: B

text: with

text: important

text: Rashba

text: coupling

text: interacting

text: with

text: radiation

text: cos

text: θφ

text: achnadrgaepdeirmtuprubraittiioesn.

text: tWreaetmcoennstidfeorr

text: ψN+

text: Ly

text: eikyy

text: as

text: the

text: most

text: probable

text: at

text: the

text: low

text: temperatures

text: normally

text: used

text: in

text: experiments.

text: Following

text: the

text: previous

text: model

text: and

text: and

text: for

text: the

text: branch,

text: applying

text: a

text: Boltzmann

text: transport

text: model

text: we

text: are

text: able

text: to

text: obtain

text: an

text: expression

text: for

text: R

text: with

text: radiation

text: and

text: RSOI.

text: xx

text: sin

text: θφ

text: In

text: the

text: simulations

text: we

text: obtain,

text: ﬁrst

text: without

text: radiation,

text: the

text: ψ

text: eikyy

text: N

text: well-known

text: beating

text: pattern

text: with

text: the

text: system

text: of

text: nodes

text: and

text: N

text: Ly

text: cos

text: θ2φN

text: antinodes

text: of

text: the

text: Rashba

text: Then,

text: we

text: switched

text: on

text: light

text: obtaining

text: Rxx

text: that

text: shows

text: a

text: strong

text: where

text: θ

text: is

text: given

text: by

text: θ

text: arctan

text: and

text: φ

text: is

text: the

text: deformation

text: of

text: the

text: previous

text: beating

text: pattern

text: where

text: the

text: gµBB

text: wc

text: nodes

text: and

text: antinodes

text: follow

text: the

text: peaks

text: and

text: valleys

text: of

text: normalized

text: quantum

text: harmonic

text: oscillator

text: wave

text: function,

text: RIRO.

text: We

text: study

text: the

text: dependence

text: on

text: power

text: and

text: frequency

text: i.e.,

text: Landau

text: state,

text: N

text: being

text: the

text: corresponding

text: Landau

text: including

text: the

text: terahertz

text: regime.

text: with

text: RSOI

text: share

text: level

text: index.

text: According

text: to

text: these

text: results

text: the

text: Rashba

text: spin-

text: similar

text: Hamiltonian

text: with

text: topological

text: insulators,

text: then

text: orbit

text: interaction

text: mixes

text: spin-down

text: and

text: spin-up

text: states

text: of

text: we

text: consider

text: that

text: the

text: results

text: that

text: we

text: present

text: in

text: this

text: arti-

text: adjacent

text: Landau

text: levels

text: to

text: give

text: rise

text: to

text: two

text: new

text: energy

text: cle

text: could

text: be

text: of

text: application

text: in

text: the

text: ﬁeld

text: of

text: topological

text: branches

text: of

text: eigenstates

text: of

text: the

text: Hamiltonian

text: insulators

text: under

text: the

text: inﬂuence

text: of

text: radiation.

text: To

text: analyze

text: magnetotransport

text: in

text: with

text: RSOI

text: we

text: II.

text: THEORETICAL

text: MODEL

text: calculate

text: the

text: longitudinal

text: conductivity

text: σxx

text: following

text: the

text: Boltzmann

text: σ

text: is

text: by:

text: xx

text: We

text: consider

text: a

text: in

text: the

text: x

text: y

text: plane

text: with

text: strong

text: Rashba

text: coupling

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text: a

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text: ﬁeld

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text: the

text: x

text: di-

text: σxx

text: dE

text: rection.

text: Using

text: Landau

text: gauge

text: for

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text: potential

text: vector,

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text: Bx,

text: the

text: hamiltonian

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text: H

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text: E

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text: energy,

text: ρ

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text: H

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text: scattering

text: event,

text: HB

text: (x

text: σ0

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text: tchhaatrgaetd

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text: c

text: →p−arallel

text: to

text: the

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text: for

text: the

text: unit

text: matrix,

text: peratures

text: (T)

text: this

text: is

text: the

text: most

text: likely

text: source

text: of

text: scattering

text: σ

text: (σ

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text: σ

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text: the

text: Pauli

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text: w

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text: cyclotron

text: frequency.

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text: 2π

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text: NI

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text: ψi±

text: Ei)

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text: by

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text: N

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text: respectively.

text: V

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text: 5 2. unbalanced orthogonal procrustes problem (UOPP) if and by both the algorithm 1 and the algorithm 2. We randomly only if m > k. Especially when Q serves as a column vector choose the test data matrix with normally distributed singular (k = 1), the problem (16) degenerates to values. Besides, the computer we use is MacBook Air, whose CPU is 1.4 GHz Intel Core i5, RAM is 4 GB 1600 MHz (cid:107) − (cid:107) min Eq g 2 (17) DDR3 and operating system is OS X Yosemite 10.10.5. 2 qTq=1 Case 1: (parameter dependence) Firstly, we try to inves- tigate the GPI method in the algorithm 2 via varying the which is known as the least square problem with a quadratic relaxation parameter γ. Suppose l is the largest eigenvalue equality constraint (LSQE). e − of ET E, then we can let γ = δl such that γI ET E is a To solve the balanced OPP (m = k), we could expand Eq. positive deﬁnite matrix, where δeis an arbitrarymconstant. (16) into 1) From the ﬁgure 1, we can further notice that although the (cid:107) − (cid:107) min EQ G 2 convergence rate for the algorithm 2 is inversely proportional F Q⇒TQ=Ik (cid:107) (cid:107) (cid:107) (cid:107) − to the value of γ, the relaxation parameter γ does not affect min E 2 + G 2 2Tr(QT ET G) the uniform convergence of the GPI method. F F ⇒ QTQ=Ik Case 2:(CPU time comparison for solving UOPP) Sec- max Tr(QT ET G) ondly, we further investigate the proposed GPI method in the QTQ=Ik algorithm 2 by comparing it with ﬁve existing approaches which is same as the problem (6) with treating ET G = M. mentioned in section 1 as EB [7], RSR [8], LSR [9], SP [10] Thus, the balanced OPP has the analytical solution of the and LR [11]. closed form (7). Based on solving LSQE problem, RSR [8] and LSR [9] When m > k, UOPP (16) can be expanded into respectively update the solution row by row and column by column iteratively. EB [7] utilizes the expanded balanced OPP (cid:107) − (cid:107) min EQ G 2 as the objective function. SP [10] employs the projection F Q⇒TQ=Ik − (18) method combined with correction techniques (PMCT) [12]. min Tr(QT ET EQ 2QT ET G). LR [11] solves UOPP by ﬁxing different Lagrangian multipli- QTQ=Ik ers. The proposed GPI method includes two terms as ET E Denote ET E = A and ET G = B, then Eq. (18) is in the outside the loop and A˜W within the loop, whose orders of exact same form as QPSM (2). Based on the algorithm 1, the complexity are m2n and m2k, respectively. Besides, these two algorithm 2 can be proposed to converge to a local minimum terms have the highest orders of complexity for the proposed of UOPP monotonically due to the theoretical supports proved GPI method. Besides, the order of the complexity for each in Section 3. method is shown in the table I. The comparative results are based on ﬁxing E as the square Algorithm 2 GPI for solving UOPP in (16) matrix at ﬁrst hand (Table II) and then extend E to a more 1: Input: The matrix E ∈ Rn×m and the matrix G ∈ Rn×k general case (Table III) afterwards. (Mark − in the table II and where m > k. the table III represents that it takes too much time to record 2: Initialize Q ∈ Rm×k and γ such that QT Q = I and the in the tables.) − k matrix γI ET E is positive deﬁnite, respectively. 1) From the ﬁgure 2, we notice that the existing methods as m 3: While not converge do EB [7], RSR [8], LSR [9], SP [10] LR [11] and the proposed ← − 4: Update matrix M 2(γI ET E)Q + 2ET G. GPI method converge to the same objective value under the 5: Calculate U ∈ Rm×k andmV ∈ Rk×k via the compact same input data. Besides, our proposed GPI method converges SVD of M as M = USV T . faster than other approaches during iteration. ← 6: Update Q UV T . 2) From the table I, the proposed GPI method has the 7: End while lowest order of complexity due to its succinct computational 8: Return Q. process to obtain the optimal solution. During the experiments, we observe that the iteration number t for the LR method is usually very large for the convergence. Thus, the time Generally speaking, QPSM can not be reformulated into consumption for LR method is much larger than that for the UOPP while UOPP could always be rewritten into QPSM. proposed GPI method though orders of complexity for these Therefore, the GPI method is more general than other ap- two approaches seem close. Besides, the GPI method becomes proaches, which can only cope with UOPP. Based on the more efﬁcient when n (the number of data) is large. experimental results involved in the next section, the proposed 3) From the table II, the proposed algorithm 2 (GPI) serves GPI method takes much less time to converge to the solution as the most efﬁcient method under the square matrix case. of UOPP. 4) From the table III, we can observe that LSR [9], SP [10] and RSR [8] are unable to compete with LR [11], EB [7] and GPI due to the complex updating procedures including V. EXPERIMENTAL RESULTS the expanded OPP and solving LSQE. Especially when the In this section, we analyze and report the numerical results dimension increases, the superiority of our proposed GPI of the generalized power iteration method (GPI) represented method would be more obvious. ##LTLine## ##LTLine## ##LTLine##

text: unbalanced

text: orthogonal

text: procrustes

text: problem

text: (UOPP)

text: if

text: and

text: by

text: both

text: the

text: algorithm

text: and

text: the

text: algorithm

text: We

text: randomly

text: only

text: if

text: m

text: k.

text: Especially

text: when

text: Q

text: serves

text: as

text: a

text: column

text: vector

text: choose

text: the

text: test

text: data

text: matrix

text: with

text: normally

text: distributed

text: singular

text: (k

text: the

text: problem

text: degenerates

text: to

text: values.

text: Besides,

text: the

text: computer

text: we

text: use

text: is

text: MacBook

text: Air,

text: whose

text: CPU

text: is

text: GHz

text: Intel

text: Core

text: RAM

text: is

text: GB

text: MHz

text: min

text: Eq

text: g

text: and

text: operating

text: system

text: is

text: OS

text: X

text: Yosemite

text: Case

text: (parameter

text: dependence)

text: Firstly,

text: we

text: try

text: to

text: inves-

text: tigate

text: the

text: GPI

text: method

text: in

text: the

text: algorithm

text: via

text: varying

text: the

text: which

text: is

text: known

text: as

text: the

text: least

text: square

text: problem

text: with

text: a

text: quadratic

text: relaxation

text: parameter

text: γ.

text: Suppose

text: l

text: is

text: the

text: largest

text: eigenvalue

text: equality

text: constraint

text: (LSQE).

text: e

text: of

text: ET

text: E,

text: then

text: we

text: can

text: let

text: γ

text: δl

text: such

text: that

text: γI

text: ET

text: E

text: is

text: a

text: To

text: solve

text: the

text: balanced

text: OPP

text: (m

text: k),

text: we

text: could

text: expand

text: Eq.

text: positive

text: deﬁnite

text: matrix,

text: where

text: δeis

text: an

text: arbitrarymconstant.

text: into

text: From

text: the

text: ﬁgure

text: we

text: can

text: further

text: notice

text: that

text: although

text: the

text: min

text: EQ

text: G

text: convergence

text: rate

text: for

text: the

text: algorithm

text: is

text: inversely

text: proportional

text: F

text: Q⇒TQ=Ik

text: to

text: the

text: value

text: of

text: γ,

text: the

text: relaxation

text: parameter

text: γ

text: does

text: not

text: affect

text: min

text: E

text: G

text: ET

text: G)

text: the

text: uniform

text: convergence

text: of

text: the

text: GPI

text: method.

text: F

text: F

text: QTQ=Ik

text: Case

text: time

text: comparison

text: for

text: solving

text: UOPP)

text: Sec-

text: max

text: Tr(QT

text: ET

text: G)

text: ondly,

text: we

text: further

text: investigate

text: the

text: proposed

text: GPI

text: method

text: in

text: the

text: QTQ=Ik

text: algorithm

text: by

text: comparing

text: it

text: with

text: ﬁve

text: existing

text: approaches

text: which

text: is

text: same

text: as

text: the

text: problem

text: with

text: treating

text: ET

text: G

text: M.

text: mentioned

text: in

text: section

text: as

text: EB

text: RSR

text: LSR

text: SP

text: Thus,

text: the

text: balanced

text: OPP

text: has

text: the

text: analytical

text: solution

text: of

text: the

text: and

text: LR

text: closed

text: form

text: Based

text: on

text: solving

text: LSQE

text: problem,

text: RSR

text: and

text: LSR

text: When

text: m

text: k,

text: UOPP

text: can

text: be

text: expanded

text: into

text: respectively

text: update

text: the

text: solution

text: row

text: by

text: row

text: and

text: column

text: by

text: column

text: iteratively.

text: EB

text: utilizes

text: the

text: expanded

text: balanced

text: OPP

text: min

text: EQ

text: G

text: as

text: the

text: objective

text: function.

text: SP

text: employs

text: the

text: projection

text: F

text: Q⇒TQ=Ik

text: method

text: combined

text: with

text: correction

text: techniques

text: (PMCT)

text: min

text: Tr(QT

text: ET

text: EQ

text: ET

text: G).

text: LR

text: solves

text: UOPP

text: by

text: ﬁxing

text: different

text: Lagrangian

text: multipli-

text: QTQ=Ik

text: ers.

text: The

text: proposed

text: GPI

text: method

text: includes

text: two

text: terms

text: as

text: ET

text: E

text: Denote

text: ET

text: E

text: A

text: and

text: ET

text: G

text: B,

text: then

text: Eq.

text: is

text: in

text: the

text: outside

text: the

text: loop

text: and

text: A˜W

text: within

text: the

text: loop,

text: whose

text: orders

text: of

text: exact

text: same

text: form

text: as

text: QPSM

text: Based

text: on

text: the

text: algorithm

text: the

text: complexity

text: are

text: and

text: respectively.

text: Besides,

text: these

text: two

text: algorithm

text: can

text: be

text: proposed

text: to

text: converge

text: to

text: a

text: local

text: minimum

text: terms

text: have

text: the

text: highest

text: orders

text: of

text: complexity

text: for

text: the

text: proposed

text: of

text: UOPP

text: monotonically

text: due

text: to

text: the

text: theoretical

text: supports

text: proved

text: GPI

text: method.

text: Besides,

text: the

text: order

text: of

text: the

text: complexity

text: for

text: each

text: in

text: Section

text: method

text: is

text: shown

text: in

text: the

text: table

text: I.

text: The

text: comparative

text: results

text: are

text: based

text: on

text: ﬁxing

text: E

text: as

text: the

text: square

text: Algorithm

text: GPI

text: for

text: solving

text: UOPP

text: in

text: matrix

text: at

text: ﬁrst

text: hand

text: (Table

text: II)

text: and

text: then

text: extend

text: E

text: to

text: a

text: more

text: Input:

text: The

text: matrix

text: E

text: Rn×m

text: and

text: the

text: matrix

text: G

text: Rn×k

text: general

text: case

text: (Table

text: III)

text: afterwards.

text: (Mark

text: in

text: the

text: table

text: II

text: and

text: where

text: m

text: k.

text: the

text: table

text: III

text: represents

text: that

text: it

text: takes

text: too

text: much

text: time

text: to

text: record

text: Initialize

text: Q

text: Rm×k

text: and

text: γ

text: such

text: that

text: QT

text: Q

text: I

text: and

text: the

text: in

text: the

text: tables.)

text: k

text: matrix

text: γI

text: ET

text: E

text: is

text: positive

text: deﬁnite,

text: respectively.

text: From

text: the

text: ﬁgure

text: we

text: notice

text: that

text: the

text: existing

text: methods

text: as

text: m

text: While

text: not

text: converge

text: do

text: EB

text: RSR

text: LSR

text: SP

text: LR

text: and

text: the

text: proposed

text: Update

text: matrix

text: M

text: ET

text: E)Q

text: G.

text: GPI

text: method

text: converge

text: to

text: the

text: same

text: objective

text: value

text: under

text: the

text: Calculate

text: U

text: Rm×k

text: andmV

text: Rk×k

text: via

text: the

text: compact

text: same

text: input

text: data.

text: Besides,

text: our

text: proposed

text: GPI

text: method

text: converges

text: SVD

text: of

text: M

text: as

text: M

text: USV

text: T

text: faster

text: than

text: other

text: approaches

text: during

text: iteration.

text: Update

text: Q

text: UV

text: T

text: From

text: the

text: table

text: I,

text: the

text: proposed

text: GPI

text: method

text: has

text: the

text: End

text: while

text: lowest

text: order

text: of

text: complexity

text: due

text: to

text: its

text: succinct

text: computational

text: Return

text: Q.

text: process

text: to

text: obtain

text: the

text: optimal

text: solution.

text: During

text: the

text: experiments,

text: we

text: observe

text: that

text: the

text: iteration

text: number

text: t

text: for

text: the

text: LR

text: method

text: is

text: usually

text: very

text: large

text: for

text: the

text: convergence.

text: Thus,

text: the

text: time

text: Generally

text: speaking,

text: QPSM

text: can

text: not

text: be

text: reformulated

text: into

text: consumption

text: for

text: LR

text: method

text: is

text: much

text: larger

text: than

text: that

text: for

text: the

text: UOPP

text: while

text: UOPP

text: could

text: always

text: be

text: rewritten

text: into

text: QPSM.

text: proposed

text: GPI

text: method

text: though

text: orders

text: of

text: complexity

text: for

text: these

text: Therefore,

text: the

text: GPI

text: method

text: is

text: more

text: general

text: than

text: other

text: ap-

text: two

text: approaches

text: seem

text: close.

text: Besides,

text: the

text: GPI

text: method

text: becomes

text: proaches,

text: which

text: can

text: only

text: cope

text: with

text: UOPP.

text: Based

text: on

text: the

text: more

text: efﬁcient

text: when

text: n

text: (the

text: number

text: of

text: data)

text: is

text: large.

text: experimental

text: results

text: involved

text: in

text: the

text: next

text: section,

text: the

text: proposed

text: From

text: the

text: table

text: II,

text: the

text: proposed

text: algorithm

text: (GPI)

text: serves

text: GPI

text: method

text: takes

text: much

text: less

text: time

text: to

text: converge

text: to

text: the

text: solution

text: as

text: the

text: most

text: efﬁcient

text: method

text: under

text: the

text: square

text: matrix

text: case.

text: of

text: UOPP.

text: From

text: the

text: table

text: III,

text: we

text: can

text: observe

text: that

text: LSR

text: SP

text: and

text: RSR

text: are

text: unable

text: to

text: compete

text: with

text: LR

text: EB

text: and

text: GPI

text: due

text: to

text: the

text: complex

text: updating

text: procedures

text: including

text: V.

text: EXPERIMENTAL

text: RESULTS

text: the

text: expanded

text: OPP

text: and

text: solving

text: LSQE.

text: Especially

text: when

text: the

text: In

text: this

text: section,

text: we

text: analyze

text: and

text: report

text: the

text: numerical

text: results

text: dimension

text: increases,

text: the

text: superiority

text: of

text: our

text: proposed

text: GPI

text: of

text: the

text: generalized

text: power

text: iteration

text: method

text: (GPI)

text: represented

text: method

text: would

text: be

text: more

text: obvious.

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text: 7 Dimension (n,m) 250 GPI(our) 200 PMCT 150 100 50 0 1000 1500 2000 2500 3000 3500 4000 4500 5000 n=m Fig. 4. CPU time comparison under Case 3. CPU time to converge to the optimal solution with a random initial guess but becomes much more efﬁcient especially for the data matrix of large dimension as well. REFERENCES [1] Green, B.: The orthogonal approximation of an oblique simple structure in factor analysis. Psychometrika. 17 (1952) 429-440. [2] Hurley, J., Cattell, R.: The procrustes program: producing direct rotation to test a hypothesized factor structure. Behavioural Science. 6 (1962) 258- 262. [3] Golub, G. H., Van Loan, C. F.: Matrix Computations. The Johns Hopkins University Press (1989). [4] Thomas, V.: Algorithms for the weighted orthogonal procrustes problem and other least squares problems: [D]. (2006) Umea˚ University, Sweden. [5] Souza, P., Leite, C., Borges, H., Fonseca, R.: Online algorithm based on support vectors for orthogonal regression. Pattern Recognition Letters 34 (2013) 1394-1404. [6] Chu, M., Trendaﬁlov, N.: The orthogonally constrained regression re- visted. J. Comput. Graph. Stat. 10 (2001) 746-771. [7] Green, B., Goers, J.: A problem with congruence. The Annual Meeting of the Psychometric Society. Monterey, California (1979). [8] Park, H.: A parallel algorithm for the unbalanced orthogonal procrustes problem. Parallel Computing. 17 (1991) 913-923. [9] Bojanczyk, A., Lutoborski, A.: The procrustes problem for orthogonal stiefel matrices. SIAM. J. Sci. Comput. 21 (1999) 1291-1304. [10] Zhang, Z., Du, K.: Successive projection method for solving the unbal- anced procrustes problem. Science in China: Series A Mathematics. 49 (2006) 971-986. [11] Xia, Y., Han, Y.: Partial lagrangian relaxation for the unbalanced orthogonal procrustes problem. Math. Meth. Oper. Res. 79 (2014) 225- 237. [12] Zhang, Z., Huang, Y.: A projection method for least square prob- lems with quadratic equality constraint. SIAM. J. Matr. Anal. Appl. 25 (2003)188-212. [13] Journe´e, M., Nesterov, Y., Richta´rik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. Journal of Machine Learning Research. 11 (2008) 517-553. [14] Fiori, S.: Formulation and integration of learning differential equations on the Stiefel manifold. IEEE Transactions on Neural Networks. 16 (2005) 1697-1701. [15] Kaneko, T., Fiori, S., Tanaka, T.: Empirical arithmetic averaging over the compact stiefel manifold. IEEE Transactions on Signal Processing. 61 (2013) 883-894. [16] Nie, F., Yuan, J., Huang, H.: Optimal mean robust principal component analysis. in Proc. ICML. (2014) 2755-2763. )s(emit UPC ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Dimension

text: (n,m)

text: GPI(our)

text: PMCT

text: n=m

text: Fig.

text: CPU

label: caption

text: time

text: comparison

text: under

text: Case

text: CPU

text: time

text: to

text: converge

text: to

text: the

text: optimal

text: solution

text: with

text: a

text: random

text: initial

text: guess

text: but

text: becomes

text: much

text: more

text: efﬁcient

text: especially

text: for

text: the

text: data

text: matrix

text: of

text: large

text: dimension

text: as

text: well.

text: REFERENCES

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text: Psychometrika.

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text: University,

text: Sweden.

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text: R.:

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text: Pattern

text: Recognition

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text: M.,

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text: N.:

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text: constrained

text: regression

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text: visted.

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text: Comput.

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text: Computing.

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text: Comput.

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text: unbal-

text: anced

text: procrustes

text: problem.

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text: in

text: China:

text: Series

text: A

text: Mathematics.

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text: Y.,

text: Han,

text: Y.:

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text: relaxation

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text: procrustes

text: problem.

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text: Meth.

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text: Res.

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text: BBU resources. (task) ofﬂoading into account. Figure 1 shows the hybrid deployment of C-RAN with a The cloud resources in the mobile cloud have to be managed mobile cloud for computation ofﬂoading, as opposed to the efﬁciently while also taking delay constraints of compute traditional mobile cloud hosted on the internet. Connected (task) ofﬂoading into consideration. Auto-scaling [12] enables with geographically distributed RRHs and centralised BBUs, cloud administrators to dynamically scale computing resources UEs get access to the VMs (i.e., mobile clones) in a mobile in the cloud for their applications to adapt to workload cloud for computation ofﬂoading. For computation ofﬂoading ﬂuctuations. There are two types of auto-scaling: 1) Horizontal requests, data is ﬁrst transmitted by the base station (RRH scaling adds and removes virtual machines (VM) from an and BBU) Via the uplink. Once processed by a clone (Virtual existing VM pool that serves an application, 2) Vertical scaling Machine) in the mobile cloud, the results will be returned to adds and removes virtual resources from an existing VM. UEs via the downlink. However, our work largely focuses on However, horizontal scaling has been used more often, in com- the mobile cloud side. parison to vertical scaling in literature. Moreover, horizontal As shown in Figure 1, a new kind of resource has been scaling allows the application to achieve higher throughput introduced to the traditional mobile operator’s network. This levels per each addition, but the deployment cost is greater is not only beneﬁcial to the mobile users (UEs) for ofﬂoading than vertical scaling [13]. computationally intensive tasks to the cloud, but there are a When scaling vertically, the resource provisioning intro- number of aspects that operators can beneﬁt from. Operators duces delay, which makes the desired effect arrive late. But are now not only a network pipe provider, but they can also not many previous auto-scaling techniques has taken these offer computing services to the subscribers. Moreover, the op- delays into consideration. [14] stresses the need for future erators may let the subscribers pay more on top of the current work on auto-scaling, taking auto-scaling time delays into price plan for extra computing services that they receive, by consideration. Auto-scaling may scale either virtual disk, introducing new price plans for mobile task ofﬂoading. One memory or CPU resources (virtual CPU is the most used may introduce new components into existing systems, but such auto-scaling resource) or a combination of types of resources. components still have to be managed for utilising the resources Also, one may scale up and scale down VMs depending efﬁciently. on the current amount of workload. Moreover, when scaling Software Deﬁned Networking (SDN) [9] is a centralised resources, one may scale ”continuously” by adding/removing architecture that augments a data plane and a control plane just one resource from the existing VM or may add or remove from traditional networks. Centralised controller nodes have more than one resource from the VM ”non-continuously”. The been introduced into the network for managing resources. aforementioned scenarios should be taken into consideration Such SDN-based architectures and management controllers when designing efﬁcient ofﬂoading techniques. Therefore, it have been introduced to wireless networks in multiple accounts is important to understand the trends of such resize delays [10] [11]. The main focus of above has been on deﬁning before designing auto-scaling algorithms, especially when SDN modules and interfaces (northbound and southbound) auto-scaling for delay sensitive applications. for wireless networks [10], for efﬁcient network service The remainder of this paper is organised as follows. Section deployments. Moreover, SDN has been proposed for Long introduces the proposed task ofﬂoading and resource man- Term Evolution (LTE) wireless network control [10]; e.g. agement, (i.e. uniﬁed) protocol. Then section presents the interference mitigation, network access selection. implemented prototype of the system described above and the It is inescapable that a centralised controller needs to be performance analysis of the computing resource management introduced for managing both computing and resources in the operations using vertical auto-scaling. The conclusions are mobile cloud and communication resources in C-RAN. Such shown in section followed by the acknowledgement in section a controller may still be able to control wireless network . resources, while also managing computing resources when II. UNIFIED PROTOCOL mobile tasks are ofﬂoaded to the cloud. Figure 1 depicts the proposed controller that manages the mobile cloud and Cloud Radio Access Networks allows cellular networks to baseband resources in C-RAN. The dotted lines illustrate the process baseband tasks of multiple cells/RRHs as well as connectivity to both the BBU pool and the mobile cloud to allocate cellular resources to subscribers. In conventional for sending control signals. In this architecture, a signalling mobile task ofﬂoading systems, the ofﬂoading framework and protocol needs to be designed for task ofﬂoading and man- the mobile network are not aware of the ofﬂoading process. aging wireless resources in operator’s network (Communica- The network treats the ofﬂoading data as any other data, tion Manager) and computing resources in the mobile cloud and the ofﬂoading destination resides outside the network (Compute Manager). However, this paper’s focus is mainly operator’s network. The ofﬂoading process could have been on mobile task ofﬂoading, managing mobile cloud computing more efﬁcient if the ofﬂoading network resources and cloud resources and on developing a prototype of such a system resources can be dynamically allocated to ﬁt ofﬂoading re- architecture to show further that the computational resources quirements. ofﬂoaded to the mobile cloud have to be managed efﬁciently First, we design a protocol for communicating between the while at the same time taking delay constraints of compute controller, BBU and the Mobile Cloud for resource allocation.

text: BBU

text: resources.

text: (task)

text: ofﬂoading

text: into

text: account.

text: Figure

text: shows

text: the

text: hybrid

text: deployment

text: of

text: C-RAN

text: with

text: a

text: The

text: cloud

text: resources

text: in

text: the

text: mobile

text: cloud

text: have

text: to

text: be

text: managed

text: mobile

text: cloud

text: for

text: computation

text: ofﬂoading,

text: as

text: opposed

text: to

text: the

text: efﬁciently

text: while

text: also

text: taking

text: delay

text: constraints

text: of

text: compute

text: traditional

text: mobile

text: cloud

text: hosted

text: on

text: the

text: internet.

text: Connected

text: (task)

text: ofﬂoading

text: into

text: consideration.

text: Auto-scaling

text: enables

text: with

text: geographically

text: distributed

text: RRHs

text: and

text: centralised

text: BBUs,

text: cloud

text: administrators

text: to

text: dynamically

text: scale

text: computing

text: resources

text: UEs

text: get

text: access

text: to

text: the

text: VMs

text: (i.e.,

text: mobile

text: clones)

text: in

text: a

text: mobile

text: in

text: the

text: cloud

text: for

text: their

text: applications

text: to

text: adapt

text: to

text: workload

text: cloud

text: for

text: computation

text: ofﬂoading.

text: For

text: computation

text: ofﬂoading

text: ﬂuctuations.

text: There

text: are

text: two

text: types

text: of

text: auto-scaling:

text: Horizontal

text: requests,

text: data

text: is

text: ﬁrst

text: transmitted

text: by

text: the

text: base

text: station

text: (RRH

text: scaling

text: adds

text: and

text: removes

text: virtual

text: machines

text: (VM)

text: from

text: an

text: and

text: BBU)

text: Via

text: the

text: uplink.

text: Once

text: processed

text: by

text: a

text: clone

text: (Virtual

text: existing

text: VM

text: pool

text: that

text: serves

text: an

text: application,

text: Vertical

text: scaling

text: Machine)

text: in

text: the

text: mobile

text: cloud,

text: the

text: results

text: will

text: be

text: returned

text: to

text: adds

text: and

text: removes

text: virtual

text: resources

text: from

text: an

text: existing

text: VM.

text: UEs

text: via

text: the

text: downlink.

text: However,

text: our

text: work

text: largely

text: focuses

text: on

text: However,

text: horizontal

text: scaling

text: has

text: been

text: used

text: more

text: often,

text: in

text: com-

text: the

text: mobile

text: cloud

text: side.

text: parison

text: to

text: vertical

text: scaling

text: in

text: literature.

text: Moreover,

text: horizontal

text: As

text: shown

text: in

text: Figure

text: a

text: new

text: kind

text: of

text: resource

text: has

text: been

text: scaling

text: allows

text: the

text: application

text: to

text: achieve

text: higher

text: throughput

text: introduced

text: to

text: the

text: traditional

text: mobile

text: operator’s

text: network.

text: This

text: levels

text: per

text: each

text: addition,

text: but

text: the

text: deployment

text: cost

text: is

text: greater

text: is

text: not

text: only

text: beneﬁcial

text: to

text: the

text: mobile

text: users

text: (UEs)

text: for

text: ofﬂoading

text: than

text: vertical

text: scaling

text: computationally

text: intensive

text: tasks

text: to

text: the

text: cloud,

text: but

text: there

text: are

text: a

text: When

text: scaling

text: vertically,

text: the

text: resource

text: provisioning

text: intro-

text: number

text: of

text: aspects

text: that

text: operators

text: can

text: beneﬁt

text: from.

text: Operators

text: duces

text: delay,

text: which

text: makes

text: the

text: desired

text: effect

text: arrive

text: late.

text: But

text: are

text: now

text: not

text: only

text: a

text: network

text: pipe

text: provider,

text: but

text: they

text: can

text: also

text: not

text: many

text: previous

text: auto-scaling

text: techniques

text: has

text: taken

text: these

text: offer

text: computing

text: services

text: to

text: the

text: subscribers.

text: Moreover,

text: the

text: op-

text: delays

text: into

text: consideration.

text: stresses

text: the

text: need

text: for

text: future

text: erators

text: may

text: let

text: the

text: subscribers

text: pay

text: more

text: on

text: top

text: of

text: the

text: current

text: work

text: on

text: auto-scaling,

text: taking

text: auto-scaling

text: time

text: delays

text: into

text: price

text: plan

text: for

text: extra

text: computing

text: services

text: that

text: they

text: receive,

text: by

text: consideration.

text: Auto-scaling

text: may

text: scale

text: either

text: virtual

text: disk,

text: introducing

text: new

text: price

text: plans

text: for

text: mobile

text: task

text: ofﬂoading.

text: One

text: memory

text: or

text: CPU

text: resources

text: (virtual

text: CPU

text: is

text: the

text: most

text: used

text: may

text: introduce

text: new

text: components

text: into

text: existing

text: systems,

text: but

text: such

text: auto-scaling

text: resource)

text: or

text: a

text: combination

text: of

text: types

text: of

text: resources.

text: components

text: still

text: have

text: to

text: be

text: managed

text: for

text: utilising

text: the

text: resources

text: Also,

text: one

text: may

text: scale

text: up

text: and

text: scale

text: down

text: VMs

text: depending

text: efﬁciently.

text: on

text: the

text: current

text: amount

text: of

text: workload.

text: Moreover,

text: when

text: scaling

text: Software

text: Deﬁned

text: Networking

text: (SDN)

text: is

text: a

text: centralised

text: resources,

text: one

text: may

text: scale

text: ”continuously”

text: by

text: adding/removing

text: architecture

text: that

text: augments

text: a

text: data

text: plane

text: and

text: a

text: control

text: plane

text: just

text: one

text: resource

text: from

text: the

text: existing

text: VM

text: or

text: may

text: add

text: or

text: remove

text: from

text: traditional

text: networks.

text: Centralised

text: controller

text: nodes

text: have

text: more

text: than

text: one

text: resource

text: from

text: the

text: VM

text: ”non-continuously”.

text: The

text: been

text: introduced

text: into

text: the

text: network

text: for

text: managing

text: resources.

text: aforementioned

text: scenarios

text: should

text: be

text: taken

text: into

text: consideration

text: Such

text: SDN-based

text: architectures

text: and

text: management

text: controllers

text: when

text: designing

text: efﬁcient

text: ofﬂoading

text: techniques.

text: Therefore,

text: it

text: have

text: been

text: introduced

text: to

text: wireless

text: networks

text: in

text: multiple

text: accounts

text: is

text: important

text: to

text: understand

text: the

text: trends

text: of

text: such

text: resize

text: delays

text: The

text: main

text: focus

text: of

text: above

text: has

text: been

text: on

text: deﬁning

text: before

text: designing

text: auto-scaling

text: algorithms,

text: especially

text: when

text: SDN

text: modules

text: and

text: interfaces

text: (northbound

text: and

text: southbound)

text: auto-scaling

text: for

text: delay

text: sensitive

text: applications.

text: for

text: wireless

text: networks

text: for

text: efﬁcient

text: network

text: service

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organised

text: as

text: follows.

text: Section

text: deployments.

text: Moreover,

text: SDN

text: has

text: been

text: proposed

text: for

text: Long

text: introduces

text: the

text: proposed

text: task

text: ofﬂoading

text: and

text: resource

text: man-

text: Term

text: Evolution

text: (LTE)

text: wireless

text: network

text: control

text: e.g.

text: agement,

text: (i.e.

text: uniﬁed)

text: protocol.

text: Then

text: section

text: presents

text: the

text: interference

text: mitigation,

text: network

text: access

text: selection.

text: implemented

text: prototype

text: of

text: the

text: system

text: described

text: above

text: and

text: the

text: It

text: is

text: inescapable

text: that

text: a

text: centralised

text: controller

text: needs

text: to

text: be

text: performance

text: analysis

text: of

text: the

text: computing

text: resource

text: management

text: introduced

text: for

text: managing

text: both

text: computing

text: and

text: resources

text: in

text: the

text: operations

text: using

text: vertical

text: auto-scaling.

text: The

text: conclusions

text: are

text: mobile

text: cloud

text: and

text: communication

text: resources

text: in

text: C-RAN.

text: Such

text: shown

text: in

text: section

text: followed

text: by

text: the

text: acknowledgement

text: in

text: section

text: a

text: controller

text: may

text: still

text: be

text: able

text: to

text: control

text: wireless

text: network

text: resources,

text: while

text: also

text: managing

text: computing

text: resources

text: when

text: II.

text: UNIFIED

text: PROTOCOL

text: mobile

text: tasks

text: are

text: ofﬂoaded

text: to

text: the

text: cloud.

text: Figure

text: depicts

text: the

text: proposed

text: controller

text: that

text: manages

text: the

text: mobile

text: cloud

text: and

text: Cloud

text: Radio

text: Access

text: Networks

text: allows

text: cellular

text: networks

text: to

text: baseband

text: resources

text: in

text: C-RAN.

text: The

text: dotted

text: lines

text: illustrate

text: the

text: process

text: baseband

text: tasks

text: of

text: multiple

text: cells/RRHs

text: as

text: well

text: as

text: connectivity

text: to

text: both

text: the

text: BBU

text: pool

text: and

text: the

text: mobile

text: cloud

text: to

text: allocate

text: cellular

text: resources

text: to

text: subscribers.

text: In

text: conventional

text: for

text: sending

text: control

text: signals.

text: In

text: this

text: architecture,

text: a

text: signalling

text: mobile

text: task

text: ofﬂoading

text: systems,

text: the

text: ofﬂoading

text: framework

text: and

text: protocol

text: needs

text: to

text: be

text: designed

text: for

text: task

text: ofﬂoading

text: and

text: man-

text: the

text: mobile

text: network

text: are

text: not

text: aware

text: of

text: the

text: ofﬂoading

text: process.

text: aging

text: wireless

text: resources

text: in

text: operator’s

text: network

text: (Communica-

text: The

text: network

text: treats

text: the

text: ofﬂoading

text: data

text: as

text: any

text: other

text: data,

text: tion

text: Manager)

text: and

text: computing

text: resources

text: in

text: the

text: mobile

text: cloud

text: and

text: the

text: ofﬂoading

text: destination

text: resides

text: outside

text: the

text: network

text: (Compute

text: Manager).

text: However,

text: this

text: paper’s

text: focus

text: is

text: mainly

text: operator’s

text: network.

text: The

text: ofﬂoading

text: process

text: could

text: have

text: been

text: on

text: mobile

text: task

text: ofﬂoading,

text: managing

text: mobile

text: cloud

text: computing

text: more

text: efﬁcient

text: if

text: the

text: ofﬂoading

text: network

text: resources

text: and

text: cloud

text: resources

text: and

text: on

text: developing

text: a

text: prototype

text: of

text: such

text: a

text: system

text: resources

text: can

text: be

text: dynamically

text: allocated

text: to

text: ﬁt

text: ofﬂoading

text: re-

text: architecture

text: to

text: show

text: further

text: that

text: the

text: computational

text: resources

text: quirements.

text: ofﬂoaded

text: to

text: the

text: mobile

text: cloud

text: have

text: to

text: be

text: managed

text: efﬁciently

text: First,

text: we

text: design

text: a

text: protocol

text: for

text: communicating

text: between

text: the

text: while

text: at

text: the

text: same

text: time

text: taking

text: delay

text: constraints

text: of

text: compute

text: controller,

text: BBU

text: and

text: the

text: Mobile

text: Cloud

text: for

text: resource

text: allocation.


text: TABLE I UOP ATTRIBUTE DEFINITION Field Name Value Type Size (Bytes) Description An integer value that indicates the type of PDU. PDU type Unsigned Integer 4 Refer to Table II for the list of PDU types. An identiﬁer to match requests with replies. The mobile device sets the Request ID in the request PDU and then is copied by the controller and Request ID Unsigned Integer 4 the clone in the response PDU when ofﬂoading. The controller sets this attribute when sending resource management messages. 0 if response packet, 1 if Acknowledgement packet ACK Unsigned Integer 4 and, request rejection packet if the value >1 A set of name-value pairs identifying application Object objects to execute, user data and error messages Object Variable Binding with their corresponding object references. Refer to Table III for object deﬁnition. UE Controller Mobile Cloud C-RAN Make oﬄoading (1) decision DMeackisiniogn OFOFFLFOLAODA_DA_CRCEEQP(T2)(4) Estimavaatiela rbeisliotyu rce( 3) OFFLOAD_ACK(5) MAMNAANGAEG_EC_OCMOPMUPTUET_EACK(6)(8) Allroecsaotuer cCelosne( 7) MANAGE_BBU(6) Resource Allocation MANAGE_BBU_ACK (8) bAanllodcwaidteth(7) OFFLOAD_START(9) APP_REGISTER(10) APP_REQ(11) APP_DATA(12) APP_DATA_ACK(13) Compute Oﬄoading APP_RESPONSE(14) APP_RESPONSE_ACK(15) OFFLOAD_FIN(16) Fig. 3. Uniﬁed Ofﬂoading Protocol: Procedure when successfully allocates resources TABLE II Figure 4 shows the test environment set up for evaluating PDU TYPES (SOME EXAMPLES) the proposed architecture. There are two USRP N210 and one X300 have been set up as RRHs of Amarisoft LTE 100 PDU Type value PDU type 0000 Ofﬂoad Req and OpenAirInterface (OAI) software base stations respec- 0001 Ofﬂoad Accept tively. All nodes in the network are connected via a Gigabit 0002 Ofﬂoad Denied Ethernet switch. The soft base stations are deployed on a 0003 Ofﬂoad Start Dell PowerEdge R210 rack server. OpenStack with Kernel 0004 App Register 0005 App Request Virtual Machine (KVM) as the hypervisor, has been deployed 0006 App Data with no shared storage, for hosting the mobile clones running 0007 App Response Android-x86 operating system. Thinkair [4] has been used as 0008 Ofﬂoad FIN the ofﬂoading framework, of which the server components 0009 Manage Compute 0010 Manage BBU have been installed on the clone in mobile cloud, while the client components are installed on the UE running Android 4.4 operating system. The wireless bandwidth of the base III. EVALUATION stations has been set to 5 MHz. The BBUs are connected to its Mobility Management Entity (MME) via its S1-MME links using S1 Application Protocol (S1AP) that provides signalling A. Testbed Implementation ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I

text: UOP

text: ATTRIBUTE

text: DEFINITION

text: Field

label: table

text: Name

text: Value

text: Type

text: Size

text: (Bytes)

text: Description

text: An

text: integer

text: value

text: that

text: indicates

text: the

text: type

text: of

text: PDU.

text: PDU

text: type

text: Unsigned

text: Integer

text: Refer

text: to

text: Table

text: II

text: for

text: the

text: list

text: of

text: PDU

text: types.

text: An

text: identiﬁer

text: to

text: match

text: requests

text: with

text: replies.

text: The

text: mobile

text: device

text: sets

text: the

text: Request

text: ID

text: in

text: the

text: request

text: PDU

text: and

text: then

text: is

text: copied

text: by

text: the

text: controller

text: and

text: Request

text: ID

text: Unsigned

text: Integer

text: the

text: clone

text: in

text: the

text: response

text: PDU

text: when

text: ofﬂoading.

text: The

text: controller

text: sets

text: this

text: attribute

text: when

text: sending

text: resource

text: management

text: messages.

text: if

text: response

text: packet,

text: if

text: Acknowledgement

text: packet

text: ACK

text: Unsigned

text: Integer

text: and,

text: request

text: rejection

text: packet

text: if

text: the

text: value

text: A

text: set

text: of

text: name-value

text: pairs

text: identifying

text: application

text: Object

text: objects

text: to

text: execute,

text: user

text: data

text: and

text: error

text: messages

text: Object

text: Variable

text: Binding

text: with

text: their

text: corresponding

text: object

text: references.

text: Refer

text: to

text: Table

text: III

text: for

text: object

text: deﬁnition.

text: UE

text: Controller

text: Mobile

text: Cloud

text: C-RAN

text: Make

text: oﬄoading

text: decision

text: DMeackisiniogn

text: Estimavaatiela

text: rbeisliotyu

text: rce(

text: Allroecsaotuer

text: cCelosne(

text: Resource

text: Allocation

text: Compute

text: Oﬄoading

text: Fig.

text: Uniﬁed

text: Ofﬂoading

text: Protocol:

text: Procedure

text: when

text: successfully

text: allocates

text: resources

text: TABLE

text: II

text: Figure

text: shows

text: the

text: test

text: environment

text: set

text: up

text: for

text: evaluating

text: PDU

text: TYPES

text: (SOME

text: EXAMPLES)

text: the

text: proposed

text: architecture.

text: There

text: are

text: two

text: USRP

text: and

text: one

text: have

text: been

text: set

text: up

text: as

text: RRHs

text: of

text: Amarisoft

text: LTE

text: PDU

text: Type

text: value

text: PDU

text: type

text: Ofﬂoad

text: Req

text: and

text: OpenAirInterface

text: (OAI)

text: software

text: base

text: stations

text: respec-

text: Ofﬂoad

text: Accept

text: tively.

text: All

text: nodes

text: in

text: the

text: network

text: are

text: connected

text: via

text: a

text: Gigabit

text: Ofﬂoad

text: Denied

text: Ethernet

text: switch.

text: The

text: soft

text: base

text: stations

text: are

text: deployed

text: on

text: a

text: Ofﬂoad

text: Start

text: Dell

text: PowerEdge

text: rack

text: server.

text: OpenStack

text: with

text: Kernel

text: App

text: Register

text: App

text: Request

text: Virtual

text: Machine

text: (KVM)

text: as

text: the

text: hypervisor,

text: has

text: been

text: deployed

text: App

text: Data

text: with

text: no

text: shared

text: storage,

text: for

text: hosting

text: the

text: mobile

text: clones

text: running

text: App

text: Response

text: operating

text: system.

text: Thinkair

text: has

text: been

text: used

text: as

text: Ofﬂoad

text: FIN

text: the

text: ofﬂoading

text: framework,

text: of

text: which

text: the

text: server

text: components

text: Manage

text: Compute

text: Manage

text: BBU

text: have

text: been

text: installed

text: on

text: the

text: clone

text: in

text: mobile

text: cloud,

text: while

text: the

text: client

text: components

text: are

text: installed

text: on

text: the

text: UE

text: running

text: Android

text: operating

text: system.

text: The

text: wireless

text: bandwidth

text: of

text: the

text: base

text: III.

text: EVALUATION

text: stations

text: has

text: been

text: set

text: to

text: MHz.

text: The

text: BBUs

text: are

text: connected

text: to

text: its

text: Mobility

text: Management

text: Entity

text: (MME)

text: via

text: its

text: links

text: using

text: Application

text: Protocol

text: that

text: provides

text: signalling

text: A.

text: Testbed

text: Implementation

text: ##LTFigure##

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