text: INDUCTION FOR LOCALLY COMPACT QUANTUM GROUPS 9 (cid:98) (cid:98) (cid:48) ⊗ G U = (π id)(V ) (2.5) and (cid:98) (cid:101) ⊗ ⊗ (cid:101) (cid:98) U(Ω a) = αF(Ω)(1 a) (2.6) ∈ F ∈ G for all Ω and a C ( ) (see [15, Deﬁnition A.2 and the following paragraph]). 0 Covariance of Y with respect to π and π in the sense of [15, Remark 3.6] yields the l r bicovariant B-correspondence (cid:101) ∞ G (cid:48) (cid:98) L ( ) (cid:98) F ∞ G ∞ G . L ( ) L ( ) Now [15, Proposition 3.7] shows the existence of a canonically determined Hilbert B-module E ∈ G E Ind( ) together with a unitary representation Ind(X) Rep( , Ind( )) such that (cid:101) (cid:98) L∞(G)(cid:48) (cid:98) ∼ (cid:98) L∞(G)(cid:48) (cid:98) F G ⊗ E ∞ G ∞ G = ∞ G L2( ) Ind( ) ∞ G (2.7) L ( ) L ( ) L ( ) L ( ) (cid:101) · as bicovariant correspondences. In what follows we shall use the symbol Ind ( ) to denote F G (cid:110) ∞ G H V the objects deﬁned above. Note that the fact that is a left L ( / )-module implies in particular existence of a map ∞ G H −(cid:0)→ L (cid:1) E ρ : L ( / ) (Ind ( )) (2.8) V ∗ ⊗ ⊗ ∈ satisfying the covariance relation: Ind(X) 1 ρ(x) Ind(X) = (id ρ)∆G(x) for all x ∞ G H L ( / ). E The Hilbert B-module Ind ( ) is called the Hilbert B-module induced in the sense of Vaes E V from and the unitary representation Ind (X) is called the representation induced by X V (also in the sense of Vaes). 3. Representation-theoretic characterization of open quantum subgroups (cid:98) (cid:98) H In this section we prove that for an open quantum subgroup of a locally compact quantum G ∗ H ⊂ G group we have a canonical C -inclusion Cu( ) Cu( ). We remark that this was previously 0 0 G proved by the ﬁrst three authors under the coamenability assumption on (see [6, Theorem 5.2]). The proof given here is completely diﬀerent from the one in [6]. In fact, in the latter ∞ H we showed through direct computations that every positive deﬁnite function in L ( ) is also ∞ G positive deﬁnite when regarded as an element in L ( ). As in the general, non-coamenable, case the criterion for positive-deﬁniteness used in [6] does not hold (see [4]), that proof cannot be extended to arbitrary locally compact quantum groups. In the present approach we utilize Kustermans’s theory of induced representation(cid:98)s. In view of the 1-1 correspondence between unitary representations of a locally compact G G quantum group and representations of Cu( ), the result mentioned above, together with 0 the existence of a suitable conditional expectation, allows us to apply classical construction ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: INDUCTION

text: FOR

text: LOCALLY

text: COMPACT

text: QUANTUM

text: GROUPS

text: G

text: U

text: (π

text: id)(V

text: and

text: U(Ω

text: a)

text: a)

text: F

text: G

text: for

text: all

text: Ω

text: and

text: a

text: C

text: (see

text: Deﬁnition

text: and

text: the

text: following

text: paragraph]).

text: Covariance

text: of

text: Y

text: with

text: respect

text: to

text: π

text: and

text: π

text: in

text: the

text: sense

text: of

text: Remark

text: yields

text: the

text: l

text: r

text: bicovariant

text: B-correspondence

text: G

text: L

text: F

text: G

text: G

text: L

text: L

text: Now

text: Proposition

text: shows

text: the

text: existence

text: of

text: a

text: canonically

text: determined

text: Hilbert

text: B-module

text: E

text: G

text: E

text: Ind(

text: together

text: with

text: a

text: unitary

text: representation

text: Ind(X)

text: Rep(

text: Ind(

text: such

text: that

text: F

text: G

text: E

text: G

text: G

text: G

text: Ind(

text: G

text: L

text: L

text: L

text: L

text: as

text: bicovariant

text: correspondences.

text: In

text: what

text: follows

text: we

text: shall

text: use

text: the

text: symbol

text: Ind

text: to

text: denote

text: F

text: G

text: G

text: H

text: V

text: the

text: objects

text: deﬁned

text: above.

text: Note

text: that

text: the

text: fact

text: that

text: is

text: a

text: left

text: L

text: )-module

text: implies

text: in

text: particular

text: existence

text: of

text: a

text: map

text: G

text: H

text: L

text: E

text: ρ

text: L

text: (Ind

text: V

text: satisfying

text: the

text: covariance

text: relation:

text: Ind(X)

text: ρ(x)

text: Ind(X)

text: (id

text: ρ)∆G(x)

text: for

text: all

text: x

text: G

text: H

text: L

text: E

text: The

text: Hilbert

text: B-module

text: Ind

text: is

text: called

text: the

text: Hilbert

text: B-module

text: induced

text: in

text: the

text: sense

text: of

text: Vaes

text: E

text: V

text: from

text: and

text: the

text: unitary

text: representation

text: Ind

text: (X)

text: is

text: called

text: the

text: representation

text: induced

text: by

text: X

text: V

text: (also

text: in

text: the

text: sense

text: of

text: Vaes).

text: Representation-theoretic

text: characterization

text: of

text: open

text: quantum

text: subgroups

text: H

text: In

text: this

text: section

text: we

text: prove

text: that

text: for

text: an

text: open

text: quantum

text: subgroup

text: of

text: a

text: locally

text: compact

text: quantum

text: G

text: H

text: G

text: group

text: we

text: have

text: a

text: canonical

text: C

text: -inclusion

text: Cu(

text: Cu(

text: We

text: remark

text: that

text: this

text: was

text: previously

text: G

text: proved

text: by

text: the

text: ﬁrst

text: three

text: authors

text: under

text: the

text: coamenability

text: assumption

text: on

text: (see

text: Theorem

text: The

text: proof

text: given

text: here

text: is

text: completely

text: diﬀerent

text: from

text: the

text: one

text: in

text: In

text: fact,

text: in

text: the

text: latter

text: H

text: we

text: showed

text: through

text: direct

text: computations

text: that

text: every

text: positive

text: deﬁnite

text: function

text: in

text: L

text: is

text: also

text: G

text: positive

text: deﬁnite

text: when

text: regarded

text: as

text: an

text: element

text: in

text: L

text: As

text: in

text: the

text: general,

text: non-coamenable,

text: case

text: the

text: criterion

text: for

text: positive-deﬁniteness

text: used

text: in

text: does

text: not

text: hold

text: (see

text: that

text: proof

text: cannot

text: be

text: extended

text: to

text: arbitrary

text: locally

text: compact

text: quantum

text: groups.

text: In

text: the

text: present

text: approach

text: we

text: utilize

text: Kustermans’s

text: theory

text: of

text: induced

text: In

text: view

text: of

text: the

text: correspondence

text: between

text: unitary

text: representations

text: of

text: a

text: locally

text: compact

text: G

text: G

text: quantum

text: group

text: and

text: representations

text: of

text: Cu(

text: the

text: result

text: mentioned

text: above,

text: together

text: with

text: the

text: existence

text: of

text: a

text: suitable

text: conditional

text: expectation,

text: allows

text: us

text: to

text: apply

text: classical

text: construction

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text: An aﬀective computational model for machine consciousness Rohitash Chandra Artiﬁcial Intelligence and Cybernetics Research Group, Software Foundation, Nausori, Fiji Abstract In the past, several models of consciousness have become popular and have led to the development of models for machine con- sciousness with varying degrees of success and challenges for simulation and implementations. Moreover, aﬀective computing attributes that involve emotions, behavior and personality have not been the focus of models of consciousness as they lacked moti- vation for deployment in software applications and robots. The aﬀective attributes are important factors for the future of machine consciousness with the rise of technologies that can assist humans. Personality and aﬀection hence can give an additional ﬂavor for the computational model of consciousness in humanoid robotics. Recent advances in areas of machine learning with a focus on deep learning can further help in developing aspects of machine consciousness in areas that can better replicate human sensory per- ceptions such as speech recognition and vision. With such advancements, one encounters further challenges in developing models that can synchronize diﬀerent aspects of aﬀective computing. In this paper, we review some existing models of consciousnesses and present an aﬀective computational model that would enable the human touch and feel for robotic systems. Keywords: Machine consciousness, cognitive systems, aﬀective computing, consciousness, machine learning 1. Introduction such as rats seem to express some aspects of consciousnesses, that is not merely for survival. They feature social attributes The deﬁnition of consciousness has been a major challenge such as empathy which is similar to humans [12, 13]. High for simulating or modelling human consciousness [1, 2]. How- level of curiosity and creativity are major attributes of con- soever, a broad deﬁnition of consciousness is the state or quality sciousness which could be a factor that distinguishes humans of awareness which features sentience, subjectivity, the ability from rest of the animals [14, 15, 16]. While intelligence is to experience through sensory perceptions, the state of wakeful- also an underlying aspect of consciousness, it has been shown ness, the sense of ego, and the control of the mind with aware- that intelligence is a necessary, however, not suﬃcient condi- ness of thought processes [3, 4, 5, 6, 7]. tion for creativity [17]. Howsoever, apart from humans, other The challenges in the deﬁnition and models of conscious- animals also show certain levels of creativity [18]. There has ness aﬀects the implementation or simulation study of con- been attempts to enhance existing models in unconventional sciousness. In the past, simulation study has been presented ways through means to incorporate non-materialistic aspects for certain models of consciousnesses, such as the model of of consciousnesses through studies of near-death experiences information ﬂow from global workspace theory[3]. Shanahan [19]. Furthermore, ideas from psychology and quantum me- presented a study where cognitive functions such as anticipa- chanics have also been integrated in a study that challenge the tion and planning were realised through internal simulation of materialistic view of consciousness [20]. interaction with the environment. An implementation based on weightless neurons was used to control a simulated robot [8]. In an attempt to empirically study consciousness, Tononi Further attempts were made to model speciﬁc forms of intel- proposed the information integrated theory of consciousness to ligence through brute-force search heuristics to reproduce fea- quantify the amount of integrated information an entity pos- tures of human perception and cognition, including emotions sesses which determines its level of consciousness [21]. The [7]. theory depends exclusively on the ability of a system to inte- Moreover, small scale implementations can consider mod- grate information, regardless of having a strong sense of self, els based from consciousness in animals that are needed for language, emotion, body, or an environment. Furthermore, it their survival. Although intelligence demonstrated in solving attempts to explain why consciousness requires neither sen- the tasks vary [9, 10], limiting deﬁnitions of consciousness to sory input nor behavioural output in cases such as during the humans is speculative as all living beings tend to have cer- sleeping state. Further work was done with application of tain attributes that overlap with human consciousness. Some integrated information to discrete networks as a function of of the undomesticated animals such as rodents have a history their dynamics and causal architecture [22]. Information in- of survival in challenging and wide range of climate and envi- tegrated theory 3.0 further reﬁned the properties of conscious- ronments [11]. There are some studies that show that animals ness with phenomenological axioms and postulates to lay out Preprint submitted to May 29, 2020 ##LTLine## ##LTLine##

text: An

label: title

text: aﬀective

text: computational

text: model

text: for

text: machine

text: consciousness

text: Rohitash

label: author

text: Chandra

text: Artiﬁcial

text: Intelligence

text: and

text: Cybernetics

text: Research

text: Group,

text: Software

text: Foundation,

text: Nausori,

text: Fiji

text: Abstract

label: paragraph

text: In

label: abstract

text: the

text: past,

text: several

text: models

text: of

text: consciousness

text: have

text: become

text: popular

text: and

text: have

text: led

text: to

text: the

text: development

text: of

text: models

text: for

text: machine

text: con-

text: sciousness

text: with

text: varying

text: degrees

text: of

text: success

text: and

text: challenges

text: for

text: simulation

text: and

text: implementations.

text: Moreover,

text: aﬀective

text: computing

text: attributes

text: that

text: involve

text: emotions,

text: behavior

text: and

text: personality

text: have

text: not

text: been

text: the

text: focus

text: of

text: models

text: of

text: consciousness

text: as

text: they

text: lacked

text: moti-

text: vation

text: for

text: deployment

text: in

text: software

text: applications

text: and

text: robots.

text: The

text: aﬀective

text: attributes

text: are

text: important

text: factors

text: for

text: the

text: future

text: of

text: machine

text: consciousness

text: with

text: the

text: rise

text: of

text: technologies

text: that

text: can

text: assist

text: humans.

text: Personality

text: and

text: aﬀection

text: hence

text: can

text: give

text: an

text: additional

text: ﬂavor

text: for

text: the

text: computational

text: model

text: of

text: consciousness

text: in

text: humanoid

text: robotics.

text: Recent

text: advances

text: in

text: areas

text: of

text: machine

text: learning

text: with

text: a

text: focus

text: on

text: deep

text: learning

text: can

text: further

text: help

text: in

text: developing

text: aspects

text: of

text: machine

text: consciousness

text: in

text: areas

text: that

text: can

text: better

text: replicate

text: human

text: sensory

text: per-

text: ceptions

text: such

text: as

text: speech

text: recognition

text: and

text: vision.

text: With

text: such

text: advancements,

text: one

text: encounters

text: further

text: challenges

text: in

text: developing

text: models

text: that

text: can

text: synchronize

text: diﬀerent

text: aspects

text: of

text: aﬀective

text: computing.

text: In

text: this

text: paper,

text: we

text: review

text: some

text: existing

text: models

text: of

text: consciousnesses

text: and

text: present

text: an

text: aﬀective

text: computational

text: model

text: that

text: would

text: enable

text: the

text: human

text: touch

text: and

text: feel

text: for

text: robotic

text: systems.

text: Keywords:

text: Machine

text: consciousness,

text: cognitive

text: systems,

text: aﬀective

text: computing,

text: consciousness,

text: machine

text: learning

text: Introduction

text: such

text: as

text: rats

text: seem

text: to

text: express

text: some

text: aspects

text: of

text: consciousnesses,

text: that

text: is

text: not

text: merely

text: for

text: survival.

text: They

text: feature

text: social

text: attributes

text: The

text: deﬁnition

text: of

text: consciousness

text: has

text: been

text: a

text: major

text: challenge

text: such

text: as

text: empathy

text: which

text: is

text: similar

text: to

text: humans

text: High

text: for

text: simulating

text: or

text: modelling

text: human

text: consciousness

text: How-

text: level

text: of

text: curiosity

text: and

text: creativity

text: are

text: major

text: attributes

text: of

text: con-

text: soever,

text: a

text: broad

text: deﬁnition

text: of

text: consciousness

text: is

text: the

text: state

text: or

text: quality

text: sciousness

text: which

text: could

text: be

text: a

text: factor

text: that

text: distinguishes

text: humans

text: of

text: awareness

text: which

text: features

text: sentience,

text: subjectivity,

text: the

text: ability

text: from

text: rest

text: of

text: the

text: animals

text: While

text: intelligence

text: is

text: to

text: experience

text: through

text: sensory

text: perceptions,

text: the

text: state

text: of

text: wakeful-

text: also

text: an

text: underlying

text: aspect

text: of

text: consciousness,

text: it

text: has

text: been

text: shown

text: ness,

text: the

text: sense

text: of

text: ego,

text: and

text: the

text: control

text: of

text: the

text: mind

text: with

text: aware-

text: that

text: intelligence

text: is

text: a

text: necessary,

text: however,

text: not

text: suﬃcient

text: condi-

text: ness

text: of

text: thought

text: processes

text: tion

text: for

text: creativity

text: Howsoever,

text: apart

text: from

text: humans,

text: other

text: The

text: challenges

text: in

text: the

text: deﬁnition

text: and

text: models

text: of

text: conscious-

text: animals

text: also

text: show

text: certain

text: levels

text: of

text: creativity

text: There

text: has

text: ness

text: aﬀects

text: the

text: implementation

text: or

text: simulation

text: study

text: of

text: con-

text: been

text: attempts

text: to

text: enhance

text: existing

text: models

text: in

text: unconventional

text: sciousness.

text: In

text: the

text: past,

text: simulation

text: study

text: has

text: been

text: presented

text: ways

text: through

text: means

text: to

text: incorporate

text: non-materialistic

text: aspects

text: for

text: certain

text: models

text: of

text: consciousnesses,

text: such

text: as

text: the

text: model

text: of

text: of

text: consciousnesses

text: through

text: studies

text: of

text: near-death

text: experiences

text: information

text: ﬂow

text: from

text: global

text: workspace

text: Shanahan

text: Furthermore,

text: ideas

text: from

text: psychology

text: and

text: quantum

text: me-

text: presented

text: a

text: study

text: where

text: cognitive

text: functions

text: such

text: as

text: anticipa-

text: chanics

text: have

text: also

text: been

text: integrated

text: in

text: a

text: study

text: that

text: challenge

text: the

text: tion

text: and

text: planning

text: were

text: realised

text: through

text: internal

text: simulation

text: of

text: materialistic

text: view

text: of

text: consciousness

text: interaction

text: with

text: the

text: environment.

text: An

text: implementation

text: based

text: on

text: weightless

text: neurons

text: was

text: used

text: to

text: control

text: a

text: simulated

text: robot

text: In

text: an

text: attempt

text: to

text: empirically

text: study

text: consciousness,

text: Tononi

text: Further

text: attempts

text: were

text: made

text: to

text: model

text: speciﬁc

text: forms

text: of

text: intel-

text: proposed

text: the

text: information

text: integrated

text: theory

text: of

text: consciousness

text: to

text: ligence

text: through

text: brute-force

text: search

text: heuristics

text: to

text: reproduce

text: fea-

text: quantify

text: the

text: amount

text: of

text: integrated

text: information

text: an

text: entity

text: pos-

text: tures

text: of

text: human

text: perception

text: and

text: cognition,

text: including

text: emotions

text: sesses

text: which

text: determines

text: its

text: level

text: of

text: consciousness

text: The

text: theory

text: depends

text: exclusively

text: on

text: the

text: ability

text: of

text: a

text: system

text: to

text: inte-

text: Moreover,

text: small

text: scale

text: implementations

text: can

text: consider

text: mod-

text: grate

text: information,

text: regardless

text: of

text: having

text: a

text: strong

text: sense

text: of

text: self,

text: els

text: based

text: from

text: consciousness

text: in

text: animals

text: that

text: are

text: needed

text: for

text: language,

text: emotion,

text: body,

text: or

text: an

text: environment.

text: Furthermore,

text: it

text: their

text: survival.

text: Although

text: intelligence

text: demonstrated

text: in

text: solving

text: attempts

text: to

text: explain

text: why

text: consciousness

text: requires

text: neither

text: sen-

text: the

text: tasks

text: vary

text: limiting

text: deﬁnitions

text: of

text: consciousness

text: to

text: sory

text: input

text: nor

text: behavioural

text: output

text: in

text: cases

text: such

text: as

text: during

text: the

text: humans

text: is

text: speculative

text: as

text: all

text: living

text: beings

text: tend

text: to

text: have

text: cer-

text: sleeping

text: state.

text: Further

text: work

text: was

text: done

text: with

text: application

text: of

text: tain

text: attributes

text: that

text: overlap

text: with

text: human

text: consciousness.

text: Some

text: integrated

text: information

text: to

text: discrete

text: networks

text: as

text: a

text: function

text: of

text: of

text: the

text: undomesticated

text: animals

text: such

text: as

text: rodents

text: have

text: a

text: history

text: their

text: dynamics

text: and

text: causal

text: architecture

text: Information

text: in-

text: of

text: survival

text: in

text: challenging

text: and

text: wide

text: range

text: of

text: climate

text: and

text: envi-

text: tegrated

text: theory

text: further

text: reﬁned

text: the

text: properties

text: of

text: conscious-

text: ronments

text: There

text: are

text: some

text: studies

text: that

text: show

text: that

text: animals

text: ness

text: with

text: phenomenological

text: axioms

text: and

text: postulates

text: to

text: lay

text: out

text: Preprint

text: submitted

text: to

text: May

text: ##LTLine##

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text: by how the brain computes a model of the body through the With such a breakthrough for development of Watson for body schema and uses it in the control of the body. Hence, they cognitive computing, there remains deep philosophical ques- suggested that a simpliﬁed model of attention through an atten- tions from perspective of natural and artiﬁcial consciousness tion schema would be useful in controlling attention. Lamme [85]. Koch evaluated Watson’s level of consciousnesses from presented deﬁnitions of visual attention and awareness that dis- perspective of integrated information theory of consciousnesses tinguished between them and also explained why they are intri- [86, 22] that views the level of consciousness based on com- cately related. It was suggested that there was overlap between plexity and how integrated the forms of information are in mechanisms of memory and awareness than between those of the system. Watson’s capabilities motivated to further study attention and awareness. Moreover, it was also highlighted that the philosophy, theory, and future of artiﬁcial intelligence phenomenal experience origin from the recurrent interaction based upon Leibniz’s computational formal logic that inspired between groups of neurons [24]. a ’scorecard’ approach to assessing cognitive systems [87]. 2.3. Simulation of aspects of consciousness Metacognition refers to a higher order thinking skills that in- cludes knowledge about when and how to use particular strate- Throughout modern digital history, there have been a num- gies for learning or for problem solving [88]. In relation to ber of developments in areas of artiﬁcial intelligence that mimic metacognition, Watson relied on a skill very similar to hu- aspects or attributes of cognition and consciousness. These de- man self-knowledge as it not only came up with answers but velopments have been made with the hope to replicate and auto- also generated a conﬁdence rating for them. Therefore, Wat- mate some of the tasks that are undertaken by humans given the son possessed elements of metacognition similar to the human industrial demand and constraints of humans on carrying out counterparts in the game of Jeopardy [89]. More recently, Al- demanding tasks in limited time. The replication of some of the phaGo was developed by Google to play the board game Go biological attributes include the feature of learning with ma- which became the ﬁrst program to beat a professional human × chine learning [67] and the attribute of reasoning and planning player without handicaps on a full-sized 19 19 board [90]. with automated reasoning [68]. The attribute which deals with It used deep learning and learned abstract information from vi- sensory perceptions includes the sense of hearing with speech sual board data given by experts. Then it played against itself recognition which covers areas such as voice and speaker iden- across multiple computers through reinforcement learning. Al- tiﬁcation [69]. Moreover, visual perception is covered through though AlphaGo has been very successful, one can argue that it computer vision [70] with speciﬁc cases such as face [71], fa- demonstrated a very constrained aspect of human intelligence cial expression [72], and object recognition [73] . The attributes that may not necessarily display consciousness. of biological motor control have been covered by autonomous Furthermore, ethics and morality are considered as one of movement in humanoid robots [74], while learning to drive has the fundamental aspects of human consciousness. Hence, one been covered through autonomous driving systems [75]. Al- of the future challenges will be to feature attributes of morality though these ﬁelds have emerged, there are a number of chal- in artiﬁcial consciousness. There have been questions about the lenges that include those in computer vision and speech recog- moral aspects of the rise of robotic or digital systems that will nition, especially in dealing with noisy and dynamic environ- have a certain level of consciousness [91, 92, 93]. Colin et. al ments in real-world applications [76, 77]. proposed moral Turing test with the hope to attain moral per- The ﬁeld of natural language processing aims to make com- fection in computational systems [91]. Parthemore and Whitby puter systems understand and manipulate natural languages to questioned the requirements of a moral agent has been also pre- perform the desired tasks [78]. It has been one of the major sented a number of conceptual pre-conditions for being a moral attributes of cognition and consciousness [79]. One of the agent [92]. Arnold and Scheutz argued against moral Turing major breakthroughs that used natural language processing for test and proposed system of veriﬁcation which demands the de- cognitive computing has been the design of Watson, which is sign of transparent, accountable processes of reasoning for the a system capable of answering questions posed in natural lan- performance of autonomous systems [93]. The issues related to guage developed by David Ferrucci [80, 81]. Watson won the morality would need to be integrated with systems that feature game of Jeopardy against human players [82]. It had access artiﬁcial consciousness as it could have a wide range of impli- to 200 million pages of structured and unstructured content in- cations in cases when the system is given tasks, or in charge cluding the full text of Wikipedia. Moreover, IBM Watson was of making decisions that pose a danger to living systems. This not connected to the Internet during the game. There are a num- raises further philosophical questions on the implications of ar- ber of applications of Watson technology that includes various tiﬁcial consciousness. forms of search that have semantic properties. Specially, Wat- 3. An aﬀective computational model son has a high potential for health care for an evidence-based clinical decision support system that aﬀords exploration of a 3.1. Preliminaries broad range of hypotheses and their associated evidence [83]. Humans have long desired a future when advances in Furthermore, it can help in developing breakthrough research in robotics will help solve some of the challenges facing human- medical and life sciences with a further focus on Big Data chal- ity. It is well-known that advances in robotics and artiﬁcial in- lenges . Hence, it was shown that Watson can accelerate the telligence provide potential for advances in health and agricul- identiﬁcation of novel drug candidates and novel drug targets ture. There is hope for addressing some of the most challeng- by harnessing the potential of big data [84]. ing problems such as the need for food, water, and shelter. One 4

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text: Test of PPN gravity with galaxy-scale strong lensing systems 15 Preprint typeset using LATEX style emulateapj v. 12/16/11 Keywords: gravitational lensing: strong - galaxies: structure - cosmology: observations

text: Test

text: of

text: PPN

text: gravity

text: with

text: galaxy-scale

text: strong

text: lensing

text: systems

text: Preprint

label: section

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: Keywords:

text: gravitational

text: lensing:

text: strong

text: galaxies:

text: structure

text: cosmology:

text: observations


text: Rashba spin-orbit interaction enhanced by graphene in-plane deformations by varying the direction of the strains. The dependence of the spin-splitting energy of the conduction band struc- = = + − − ture at with the uniaxial strain is shown i∆n ﬁ(kgure0)7 foEr+d(i0ff)ereEn+t(o0r)ientation angle and (a) weak = s Rashbakstre0ngth and (bε) strong Rashba strength . Changes of thθe spin-splitting R = R = can vary from 17λ%(0)to 509.%00i4n8tthe range of strains between λ(0) a0n.0d578t studied here. 5. Summary and conclusions = − = ε 0.3 ε 0.3 We have developed a model to describe the spectral effects of small in plane deformations on gra- phene, in the vicinity of the K points. We focus on the non-intuitive enhancement of the Rashba SOI due to the stretching of the sigma - overlaps between the A–B sublattices. The deformations considered involve both bond length and sbopnd angle changes. The full Hamiltonian, to lowest order in the lattice momentum and deformation amplitude, taking into account both kinetic energy and spin-orbit coupling effects of deformations, can be cast into a convenient non-Abelian gauge formulation. We × derive the analytical form for the energies as a functionU(o1f)theSUla(t2ti)ce momentum in the vicinity of the K points and its dependence on bond stretching amplitude and angle. Within reasonable lattice deforma- tion strengths of at most 15%, we ﬁnd that the electron/hole bands can be continuously modulated from a semimetal to a semiconductor, involving both shifts in the position and size of the band gap. We also ob- serve how the chiral spin splitting energy can be controlled by changing bonds lengths and bond angles. The results found present an interesting prospect for spintronics application for graphene on deformable substrates or charge induced lattice stretching controlled by the gate voltages [34, 35]. Acknowledgement We are very pleased to have an opportunity to contribute to this special issue of Condensed Matter Physics in honor of our excellent friend Yurko (Yurij Holovatch). EM and BB are respectively grateful to the University of Lorraine and to IVIC for invitations. They also thank the CNRS and FONACIT for support through the “PICS” programme Spin transport and spin manipulations in condensed matter: polarization, spin currents and entanglement. BB and FM thank the DIONICOS programme (FP7 IRSES) for ﬁnancial support and FM acknowledges the project PAPIIT-UNAM IN111317. References 1. Peralta M., Colmenarez L., López A., Berche B., Medina E., Phys. Rev. B, 2016, 94, 235407; doi:10.1103/PhysRevB.94.235407. 2. Calvo H.L., Pastawski H.M., Roche S., Foa Torres L.E.F., Appl. Phys. Lett., 2011, 98, 232103; doi:10.1063/1.3597412. 3. Zhou S., Gweon G.-H., Lanzara A., Ann. Phys., 2006, 321, 1730; doi:10.1016/j.aop.2006.04.011. 4. Kane C.L., Mele E.J., Phys. Rev. Lett., 2005, 95, 226801; doi:10.1103/PhysRevLett.95.226801. 5. Wang X.-F., Chakraborty T., Phys. Rev. B, 2007, 75, 033408; doi:10.1103/PhysRevB.75.033408. 6. Brey L., Fertig H., Phys. Rev. B, 2006, 73, 235411; doi:10.1103/PhysRevB.73.235411. 7. Wang Z., Tang C., Sachs R., Barlas Y., Shi J., Phys. Rev. Lett., 2015, 114, 016603; doi:10.1103/PhysRevLett.114.016603. 8. Chen H., Niu Q., Zhang Z., MacDonald A., Phys. Rev. B, 2013, 87, 144410; doi:10.1103/PhysRevB.87.144410. 9. Carrillo-Bastos R., Faria D., Latgé A., Mireles F., Sandler N., Phys. Rev. B, 2014, 90, 041411(R); doi:10.1103/PhysRevB.90.041411. 10. Ando T., J. Phys. Soc. Jpn., 2000, 69, 1757; doi:10.1143/JPSJ.69.1757. 11. Marchenko D., Varykhalov A., Scholz M.R., Bihlmayer G., Rashba E.I., Rybkin A., Shikin A.M., Rader O., Nat. Commun., 2012, 3, 1232; doi:10.1038/ncomms2227. 12. Huertas-Hernando D., Guinea F., Brataas A., Phys. Rev. B, 2006, 74, 155426; doi:10.1103/PhysRevB.74.155426. 13. Konschuh S., Gmitra M., Fabian J., Phys. Rev. B, 2010, 82, 245412; doi:10.1103/PhysRevB.82.245412. 14. Jin P.Q., Li Y.Q., Zhang F.C., J. Phys. A: Math. Gen., 2006, 39, 7115; doi:10.1088/0305-4470/39/22/022. 15. Medina E., López A., Berche B., EPL, 2008, 83, 47005; doi:10.1209/0295-5075/83/47005. 16. Tokatly I.V., Phys. Rev. Lett., 2008, 101, 106601; doi:10.1103/PhysRevLett.101.106601. 17. Berche B., Chatelain C., Medina E., Eur. J. Phys., 2010, 31, 1267; doi:10.1088/0143-0807/31/5/026. 18. Berche B., Medina E., Eur. J. Phys., 2013, 34, 161; doi:10.1088/0143-0807/34/1/161. 19. Wu Q-P., Liu Z.-F., Chen A.-X., Xiao X.-B., Liu Z.-M., Sci. Rep., 2016, 6, 21590; doi:10.1038/srep21590. 20. Pastawski H.M., Medina E., Rev. Mex. Fis., 2001, 47, 1. 13702-9 ##LTLine## ##LTLine##

text: Rashba

font: CWRNDQ+DroidSans

text: spin-orbit

text: interaction

text: enhanced

text: by

text: graphene

text: in-plane

text: deformations

text: by

font: QOAZIP+DroidSerif

text: varying

text: the

text: direction

text: of

text: the

text: strains.

text: The

text: dependence

text: of

text: the

text: spin-splitting

text: energy

text: of

text: the

text: conduction

text: band

text: struc-

font: UEVWEU+Fourier-Math-Symbols

text: ture

text: at

text: with

text: the

text: uniaxial

text: strain

text: is

text: shown

text: i∆n

text: angle

text: and

text: (a)

text: weak

text: s

font: NCXYKH+Utopia-Italic

text: and

text: (bε)

text: strong

text: Rashba

text: strength

text: Changes

text: of

text: thθe

text: spin-splitting

text: R

text: R

text: can

text: vary

text: from

text: range

text: of

text: strains

text: between

text: studied

text: here.

text: Summary

text: and

text: conclusions

text: ε

text: ε

text: We

text: have

text: developed

text: a

text: model

text: to

text: describe

text: the

text: spectral

text: effects

text: of

text: small

text: in

text: plane

text: deformations

text: on

text: gra-

text: phene,

text: in

text: the

text: vicinity

text: of

text: the

text: K

text: points.

text: We

text: focus

text: on

text: the

text: non-intuitive

text: enhancement

text: of

text: the

text: Rashba

text: SOI

text: due

text: to

text: the

text: stretching

text: of

text: the

text: sigma

text: overlaps

text: between

text: the

text: A–B

text: sublattices.

text: The

text: deformations

text: considered

text: involve

text: both

text: bond

text: length

text: and

text: sbopnd

text: angle

text: changes.

text: The

text: full

text: Hamiltonian,

text: to

text: lowest

text: order

text: in

text: the

text: lattice

text: momentum

text: and

text: deformation

text: amplitude,

text: taking

text: into

text: account

text: both

text: kinetic

text: energy

text: and

text: spin-orbit

text: coupling

text: effects

text: of

text: deformations,

text: can

text: be

text: cast

text: into

text: a

text: convenient

text: non-Abelian

text: gauge

text: formulation.

text: We

text: derive

text: the

text: analytical

text: form

text: for

text: the

text: energies

text: as

text: a

text: momentum

text: in

text: the

text: vicinity

text: of

text: the

text: K

text: points

text: and

text: its

text: dependence

text: on

text: bond

text: stretching

text: amplitude

text: and

text: angle.

text: Within

text: reasonable

text: lattice

text: deforma-

text: tion

text: strengths

text: of

text: at

text: most

text: we

text: ﬁnd

text: that

text: the

text: electron/hole

text: bands

text: can

text: be

text: continuously

text: modulated

text: from

text: a

text: semimetal

text: to

text: a

text: semiconductor,

text: involving

text: both

text: shifts

text: in

text: the

text: position

text: and

text: size

text: of

text: the

text: band

text: gap.

text: We

text: also

text: ob-

text: serve

text: how

text: the

text: chiral

text: spin

text: splitting

text: energy

text: can

text: be

text: controlled

text: by

text: changing

text: bonds

text: lengths

text: and

text: bond

text: angles.

text: The

text: results

text: found

text: present

text: an

text: interesting

text: prospect

text: for

text: spintronics

text: application

text: for

text: graphene

text: on

text: deformable

text: substrates

text: or

text: charge

text: induced

text: lattice

text: stretching

text: controlled

text: by

text: the

text: gate

text: voltages

text: Acknowledgement

text: We

text: are

text: very

text: pleased

text: to

text: have

text: an

text: opportunity

text: to

text: contribute

text: to

text: this

text: special

text: issue

text: of

text: Condensed

text: Matter

text: Physics

text: in

text: honor

text: of

text: our

text: excellent

text: friend

text: Yurko

text: (Yurij

text: Holovatch).

text: EM

text: and

text: BB

text: are

text: respectively

text: grateful

text: to

text: the

text: University

text: of

text: Lorraine

text: and

text: to

text: IVIC

text: for

text: invitations.

text: They

text: also

text: thank

text: the

text: CNRS

text: and

text: FONACIT

text: for

text: support

text: through

text: the

text: “PICS”

text: programme

text: Spin

text: transport

text: and

text: spin

text: manipulations

text: in

text: condensed

text: matter:

text: polarization,

text: spin

text: currents

text: and

text: entanglement.

text: BB

text: and

text: FM

text: thank

text: the

text: DIONICOS

text: programme

text: IRSES)

text: for

text: ﬁnancial

text: support

text: and

text: FM

text: acknowledges

text: the

text: project

text: PAPIIT-UNAM

text: References

text: Peralta

text: M.,

text: Colmenarez

text: L.,

text: López

text: A.,

text: Berche

text: B.,

text: Medina

text: E.,

text: Phys.

text: Rev.

text: B,

text: Calvo

text: H.L.,

text: Pastawski

text: H.M.,

text: Roche

text: S.,

text: Foa

text: Torres

text: L.E.F.,

text: Appl.

text: Phys.

text: Lett.,

text: Zhou

text: S.,

text: Gweon

text: G.-H.,

text: Lanzara

text: A.,

text: Ann.

text: Phys.,

text: Kane

text: C.L.,

text: Mele

text: E.J.,

text: Phys.

text: Rev.

text: Lett.,

text: Wang

text: X.-F.,

text: Chakraborty

text: T.,

text: Phys.

text: Rev.

text: B,

text: Brey

text: L.,

text: Fertig

text: H.,

text: Phys.

text: Rev.

text: B,

text: Wang

text: Z.,

text: Tang

text: C.,

text: Sachs

text: R.,

text: Barlas

text: Y.,

text: Shi

text: J.,

text: Phys.

text: Rev.

text: Lett.,

text: Chen

text: H.,

text: Niu

text: Q.,

text: Zhang

text: Z.,

text: MacDonald

text: A.,

text: Phys.

text: Rev.

text: B,

text: Carrillo-Bastos

text: R.,

text: Faria

text: D.,

text: Latgé

text: A.,

text: Mireles

text: F.,

text: Sandler

text: N.,

text: Phys.

text: Rev.

text: B,

text: Ando

text: T.,

text: J.

text: Phys.

text: Soc.

text: Jpn.,

text: Marchenko

text: D.,

text: Varykhalov

text: A.,

text: Scholz

text: M.R.,

text: Bihlmayer

text: G.,

text: Rashba

text: E.I.,

text: Rybkin

text: A.,

text: Shikin

text: A.M.,

text: Rader

text: O.,

text: Nat.

text: Commun.,

text: Huertas-Hernando

text: D.,

text: Guinea

text: F.,

text: Brataas

text: A.,

text: Phys.

text: Rev.

text: B,

text: Konschuh

text: S.,

text: Gmitra

text: M.,

text: Fabian

text: J.,

text: Phys.

text: Rev.

text: B,

text: Jin

text: P.Q.,

text: Li

text: Y.Q.,

text: Zhang

text: F.C.,

text: J.

text: Phys.

text: A:

text: Math.

text: Gen.,

text: Medina

text: E.,

text: López

text: A.,

text: Berche

text: B.,

text: EPL,

text: Tokatly

text: I.V.,

text: Phys.

text: Rev.

text: Lett.,

text: Berche

text: B.,

text: Chatelain

text: C.,

text: Medina

text: E.,

text: Eur.

text: J.

text: Phys.,

text: Berche

text: B.,

text: Medina

text: E.,

text: Eur.

text: J.

text: Phys.,

text: Wu

text: Q-P.,

text: Liu

text: Z.-F.,

text: Chen

text: A.-X.,

text: Xiao

text: X.-B.,

text: Liu

text: Z.-M.,

text: Sci.

text: Rep.,

text: Pastawski

text: H.M.,

text: Medina

text: E.,

text: Rev.

text: Mex.

text: Fis.,

text: ##LTLine##

text: ##LTLine##


text: Solvable groups of interval exchange transformations Franc¸ois Dahmani, Koji Fujiwara, Vincent Guirardel May 29, 2020. Abstract We prove that any ﬁnitely generated torsion free solvable subgroup of the group IET of all Interval Exchange Transformations is virtually abelian. In contrast, the (cid:111) Z lamplighter groups A k embed in IET for every ﬁnite abelian group A, and we construct uncountably many non pairwise isomorphic 3-step solvable subgroups of IET as semi-direct products of a lamplighter group with an abelian group. (cid:111) Z We also prove that for every non-abelian ﬁnite group F, the group F k does not embed in IET. The group IET and its subgroups The group IET of interval exchange transformations is the group of all bijections of the interval [0, 1) that are piecewise translations with ﬁnitely many discontinuity points. Rather unexpectedly, the recent study of this group has given evidences that it is not as big as one could have thought, in several commonly accepted features. For instance, in [2] we established that IET does not have many free subgroups (if any at all), and that the connected Lie groups that can embed in it are only abelian. Another fact in this direction ∈ is that given any ﬁnitely generated subgroup of IET, and any point x [0, 1) the orbit of x grows (in cardinality) at most polynomially in the word length of the elements of the subgroup ([2, 2.6]). Yet another instance of these evidences is given by the main result of [7] of Juschenko and Monod which implies that certain natural subgroups of this group contain only ∈ R \ Q amenable subgroups. More precisely, given α , the subgroup of IET of transfor- α mations whose translation lengths are all multiples of α modulo 1 is amenable. Indeed, given any ﬁnitely generated subgroup G of IET , G can be viewed as a group of homeo- α morphisms of the cantor set K obtained by blowing up the R -orbit of the discontinuity (cid:55)→ α points of the generators of G, where R is the rotation x x + α mod 1 ([1]). Denoting α by Rˆ the homeomorphism of K induced by R , this embeds G in the full topological α α group of Rˆ , which is amenable by [7]. This has been extended in [6] to subgroups of α ≤ Q Z rational rank 2, ie such that the subgroup of / generated by the translation lengths Z of its elements does not contain 3. Given these evidences, we chose to investigate the possible solvable subgroups of IET. Results In order to describe elementary examples of subgroups of IET, let us enlarge a bit the context, and instead of interval exchange transformations on the interval [0, 1) we D D consider the group IET( ) of interval exchange transformations on a domain consisting of a disjoint union of ﬁnitely many oriented circles, and oriented half-open intervals, closed (cid:39) D on the left (see Section ). This does not make a real diﬀerence as IET IET( ). ∈ N Z R Z Now for all n , n embeds in IET( / ) as a group of rotations. The following gen- eral simple fact then implies that every ﬁnitely generated virtually abelian group embeds in IET. 1

text: Solvable

text: groups

text: of

text: interval

text: exchange

text: transformations

text: Franc¸ois

text: Dahmani,

text: Koji

text: Fujiwara,

text: Vincent

text: Guirardel

text: May

label: date

text: Abstract

text: We

text: prove

text: that

text: any

text: ﬁnitely

text: generated

text: torsion

text: free

text: solvable

text: subgroup

text: of

text: the

text: group

text: IET

text: of

text: all

text: Interval

text: Exchange

text: Transformations

text: is

text: virtually

text: abelian.

text: In

text: contrast,

text: the

text: Z

text: lamplighter

text: groups

text: A

text: k

text: embed

text: in

text: IET

text: for

text: every

text: ﬁnite

text: abelian

text: group

text: A,

text: and

text: we

text: construct

text: uncountably

text: many

text: non

text: pairwise

text: isomorphic

text: solvable

text: subgroups

text: of

text: IET

text: as

text: semi-direct

text: products

text: of

text: a

text: lamplighter

text: group

text: with

text: an

text: abelian

text: group.

text: Z

text: We

text: also

text: prove

text: that

text: for

text: every

text: non-abelian

text: ﬁnite

text: group

text: F,

text: the

text: group

text: F

text: k

text: does

text: not

text: embed

text: in

text: IET.

text: The

text: group

text: IET

text: and

text: its

text: subgroups

text: The

text: group

text: IET

text: of

text: interval

text: exchange

text: transformations

text: is

text: the

text: group

text: of

text: all

text: bijections

text: of

text: the

text: interval

text: that

text: are

text: piecewise

text: translations

text: with

text: ﬁnitely

text: many

text: discontinuity

text: points.

text: Rather

text: unexpectedly,

text: the

text: recent

text: study

text: of

text: this

text: group

text: has

text: given

text: evidences

text: that

text: it

text: is

text: not

text: as

text: big

text: as

text: one

text: could

text: have

text: thought,

text: in

text: several

text: commonly

text: accepted

text: features.

text: For

text: instance,

text: in

text: we

text: established

text: that

text: IET

text: does

text: not

text: have

text: many

text: free

text: subgroups

text: (if

text: any

text: at

text: all),

text: and

text: that

text: the

text: connected

text: Lie

text: groups

text: that

text: can

text: embed

text: in

text: it

text: are

text: only

text: abelian.

text: Another

text: fact

text: in

text: this

text: direction

text: is

text: that

text: given

text: any

text: ﬁnitely

text: generated

text: subgroup

text: of

text: IET,

text: and

text: any

text: point

text: x

text: the

text: orbit

text: of

text: x

text: grows

text: (in

text: cardinality)

text: at

text: most

text: polynomially

text: in

text: the

text: word

text: length

text: of

text: the

text: elements

text: of

text: the

text: subgroup

text: Yet

text: another

text: instance

text: of

text: these

text: evidences

text: is

text: given

text: by

text: the

text: main

text: result

text: of

text: of

text: Juschenko

text: and

text: Monod

text: which

text: implies

text: that

text: certain

text: natural

text: subgroups

text: of

text: this

text: group

text: contain

text: only

text: R

text: Q

text: amenable

text: subgroups.

text: More

text: precisely,

text: given

text: α

text: the

text: subgroup

text: of

text: IET

text: of

text: transfor-

text: α

text: mations

text: whose

text: translation

text: lengths

text: are

text: all

text: multiples

text: of

text: α

text: modulo

text: is

text: amenable.

text: Indeed,

text: given

text: any

text: ﬁnitely

text: generated

text: subgroup

text: G

text: of

text: IET

text: G

text: can

text: be

text: viewed

text: as

text: a

text: group

text: of

text: homeo-

text: α

text: morphisms

text: of

text: the

text: cantor

text: set

text: K

text: obtained

text: by

text: blowing

text: up

text: the

text: R

text: -orbit

text: of

text: the

text: discontinuity

text: α

text: points

text: of

text: the

text: generators

text: of

text: G,

text: where

text: R

text: is

text: the

text: rotation

text: x

text: x

text: α

text: mod

text: Denoting

text: α

text: by

text: Rˆ

text: the

text: homeomorphism

text: of

text: K

text: induced

text: by

text: R

text: this

text: embeds

text: G

text: in

text: the

text: full

text: topological

text: α

text: α

text: group

text: of

text: Rˆ

text: which

text: is

text: amenable

text: by

text: This

text: has

text: been

text: extended

text: in

text: to

text: subgroups

text: of

text: α

text: Q

text: Z

text: rational

text: rank

text: ie

text: such

text: that

text: the

text: subgroup

text: of

text: generated

text: by

text: the

text: translation

text: lengths

text: Z

text: of

text: its

text: elements

text: does

text: not

text: contain

text: Given

text: these

text: evidences,

text: we

text: chose

text: to

text: investigate

text: the

text: possible

text: solvable

text: subgroups

text: of

text: IET.

text: Results

text: In

text: order

text: to

text: describe

text: elementary

text: examples

text: of

text: subgroups

text: of

text: IET,

text: let

text: us

text: enlarge

text: a

text: bit

text: the

text: context,

text: and

text: instead

text: of

text: interval

text: exchange

text: transformations

text: on

text: the

text: interval

text: we

text: D

text: D

text: consider

text: the

text: group

text: IET(

text: of

text: interval

text: exchange

text: transformations

text: on

text: a

text: domain

text: consisting

text: of

text: a

text: disjoint

text: union

text: of

text: ﬁnitely

text: many

text: oriented

text: circles,

text: and

text: oriented

text: half-open

text: intervals,

text: closed

text: D

text: on

text: the

text: left

text: (see

text: Section

text: This

text: does

text: not

text: make

text: a

text: real

text: diﬀerence

text: as

text: IET

text: IET(

text: N

text: Z

text: R

text: Z

text: Now

text: for

text: all

text: n

text: n

text: embeds

text: in

text: IET(

text: as

text: a

text: group

text: of

text: rotations.

text: The

text: following

text: gen-

text: eral

text: simple

text: fact

text: then

text: implies

text: that

text: every

text: ﬁnitely

text: generated

text: virtually

text: abelian

text: group

text: embeds

text: in

text: IET.