text: [29] Dawes RM (1979) The robust beauty of improper linear models in decision making. American psychologist 34: 571. [30] Riedmiller M, Braun H (1992) Rprop - a fast adaptive learning algorithm. Technical report, Proc. of ISCIS VII), Universitat. [31] Igel C, Hu¨sken M (2003) Empirical evaluation of the improved rprop learn- ing algorithms. Neurocomputing 50: 105–123. [32] Samaria F, Harter A (1994) Parameterisation of a stochastic model for human face identiﬁcation. In: Applications of Computer Vision, 1994., Proceedings of the Second IEEE Workshop on. pp. 138-142. doi: 10.1109/ACV.1994.341300. [33] Sohl-Dickstein J, Battaglino PB, DeWeese MR (2011) New method for parameter estimation in probabilistic models: Minimum probability ﬂow. Phys Rev Lett 107: 220601. [34] Roudi Y, Tyrcha J, Hertz J (2009) Ising model for neural data: Model quality and approximate methods for extracting functional connectivity. Phys Rev E 79: 051915. [35] Mattia M, Del Giudice P (2000) Eﬃcient event-driven simulation of large networks of spiking neurons and dynamical synapses. Neural Computation 12: 2305–2329. 27

text: Dawes

text: RM

text: The

text: robust

text: beauty

text: of

text: improper

text: linear

text: models

text: in

text: decision

text: making.

text: American

text: psychologist

text: Riedmiller

text: M,

text: Braun

text: H

text: Rprop

text: a

text: fast

text: adaptive

text: learning

text: algorithm.

text: Technical

text: report,

text: Proc.

text: of

text: ISCIS

text: VII),

text: Universitat.

text: Igel

text: C,

text: Hu¨sken

text: M

text: Empirical

text: evaluation

text: of

text: the

text: improved

text: rprop

text: learn-

text: ing

text: algorithms.

text: Neurocomputing

text: Samaria

text: F,

text: Harter

text: A

text: Parameterisation

text: of

text: a

text: stochastic

text: model

text: for

text: human

text: face

text: identiﬁcation.

text: In:

text: Applications

text: of

text: Computer

text: Vision,

text: Proceedings

text: of

text: the

text: Second

text: IEEE

text: Workshop

text: on.

text: pp.

text: doi:

text: Sohl-Dickstein

text: J,

text: Battaglino

text: PB,

text: DeWeese

text: MR

text: New

text: method

text: for

text: parameter

text: estimation

text: in

text: probabilistic

text: models:

text: Minimum

text: probability

text: ﬂow.

text: Phys

text: Rev

text: Lett

text: Roudi

text: Y,

text: Tyrcha

text: J,

text: Hertz

text: J

text: Ising

text: model

text: for

text: neural

text: data:

text: Model

text: quality

text: and

text: approximate

text: methods

text: for

text: extracting

text: functional

text: connectivity.

text: Phys

text: Rev

text: E

text: Mattia

text: M,

text: Del

text: Giudice

text: P

text: Eﬃcient

text: event-driven

text: simulation

text: of

text: large

text: networks

text: of

text: spiking

text: neurons

text: and

text: dynamical

text: synapses.

text: Neural

text: Computation


text: counting reﬂections) do not belong to the stored patterns, but the clusters of largest mass are not centered on patterns (remember that the Hopﬁeld system is far from the retrieval phase). × In Figure 4, right panel, we compare the inferred J with the real J (N − (N 1) = 2450 elements, taking only P +1 = 5 values); the inference is actually good (the continuous identity line is drawn to guide the eye), and better than the one obtained inferring directly the full J matrix (i.e. not adopting the parametrization of the synaptic matrix in Eq. 8), as shown in Fig. 5, where we − (cid:104)| |(cid:105) compare the relative error ((JInferred J )/ J ) for the two cases. ij ij ij ij 8 6 4 2 0 -1 -0.5 0 0.5 1 Relative Error Figure 5: A posteriori probabilities of inference errors for the cases of the reduced model (red curve) and the full matrix inference (blue curve). Details in the text. From this simple example, we conﬁrm that the adopted parametrization, besides the obvious greater simplicity and lesser computational load, is eﬀective in reducing the eﬀect of noise in the data and makes the inference less prone to overﬁtting, while allowing a good match between model and data. 2.3 Towards a more realistic scenario: a multi-modular network of spiking neurons matching a prescribed spa- tial correlation structure In the previous Section we showed that, from the spatial correlation struc- ture of the conﬁgurations sampled by Monte Carlo, MS clustering was eﬀective in reconstructing the main local energy minima, and allowed for a parsimo- nious parametrization of the synaptic matrix that aﬀorded better inference. In that case, the correspondence between local density maxima in the state space, and local energy minima, was ensured by the existence of a Gibbs equilibrium probability for the Hopﬁeld model. In the perspective of applicability to real electrophysiology data, in the present Section we extend the approach to networks of spiking neurons, and to a situation where the notion of a static energy landscape is no longer strictly applicable. To establish a meaningful benchmark, we want to preserve some a priori knowledge of key features of the state space, to be checked against the found 9 ytisneD ytilibaborP detamitsE ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: counting

label: paragraph

text: reﬂections)

text: do

text: not

text: belong

text: to

text: the

text: stored

text: patterns,

text: but

text: the

text: clusters

text: of

text: largest

text: mass

text: are

text: not

text: centered

text: on

text: patterns

text: (remember

text: that

text: the

text: Hopﬁeld

text: system

text: is

text: far

text: from

text: the

text: retrieval

text: phase).

text: In

text: Figure

text: right

text: panel,

text: we

text: compare

text: the

text: inferred

text: J

text: with

text: the

text: real

text: J

text: (N

text: (N

text: elements,

text: taking

text: only

text: P

text: values);

text: the

text: inference

text: is

text: actually

text: good

text: (the

text: continuous

text: identity

text: line

text: is

text: drawn

text: to

text: guide

text: the

text: eye),

text: and

text: better

text: than

text: the

text: one

text: obtained

text: inferring

text: directly

text: the

text: full

text: J

text: matrix

text: (i.e.

text: not

text: adopting

text: the

text: parametrization

text: of

text: the

text: synaptic

text: matrix

text: in

text: Eq.

text: as

text: shown

text: in

text: Fig.

text: where

text: we

text: compare

text: the

text: relative

text: error

text: ((JInferred

text: J

text: J

text: for

text: the

text: two

text: cases.

text: ij

text: ij

text: ij

text: ij

text: Relative

text: Error

text: Figure

text: A

text: posteriori

text: probabilities

text: of

text: inference

text: errors

text: for

text: the

text: cases

text: of

text: the

text: reduced

text: model

text: (red

text: curve)

text: and

text: the

text: full

text: matrix

text: inference

text: (blue

text: curve).

text: Details

text: in

text: the

text: text.

text: From

text: this

text: simple

text: example,

text: we

text: conﬁrm

text: that

text: the

text: adopted

text: parametrization,

text: besides

text: the

text: obvious

text: greater

text: simplicity

text: and

text: lesser

text: computational

text: load,

text: is

text: eﬀective

text: in

text: reducing

text: the

text: eﬀect

text: of

text: noise

text: in

text: the

text: data

text: and

text: makes

text: the

text: inference

text: less

text: prone

text: to

text: overﬁtting,

text: while

text: allowing

text: a

text: good

text: match

text: between

text: model

text: and

text: data.

text: Towards

text: a

text: more

text: realistic

text: scenario:

text: a

text: multi-modular

text: network

text: of

text: spiking

text: neurons

text: matching

text: a

text: prescribed

text: spa-

text: tial

text: correlation

text: structure

text: In

text: the

text: previous

text: Section

text: we

text: showed

text: that,

text: from

text: the

text: spatial

text: correlation

text: struc-

text: ture

text: of

text: the

text: conﬁgurations

text: sampled

text: by

text: Monte

text: Carlo,

text: MS

text: clustering

text: was

text: eﬀective

text: in

text: reconstructing

text: the

text: main

text: local

text: energy

text: minima,

text: and

text: allowed

text: for

text: a

text: parsimo-

text: nious

text: parametrization

text: of

text: the

text: synaptic

text: matrix

text: that

text: aﬀorded

text: better

text: inference.

text: In

text: that

text: case,

text: the

text: correspondence

text: between

text: local

text: density

text: maxima

text: in

text: the

text: state

text: space,

text: and

text: local

text: energy

text: minima,

text: was

text: ensured

text: by

text: the

text: existence

text: of

text: a

text: Gibbs

text: equilibrium

text: probability

text: for

text: the

text: Hopﬁeld

text: model.

text: In

text: the

text: perspective

text: of

text: applicability

text: to

text: real

text: electrophysiology

text: data,

text: in

text: the

text: present

text: Section

text: we

text: extend

text: the

text: approach

text: to

text: networks

text: of

text: spiking

text: neurons,

text: and

text: to

text: a

text: situation

text: where

text: the

text: notion

text: of

text: a

text: static

text: energy

text: landscape

text: is

text: no

text: longer

text: strictly

text: applicable.

text: To

text: establish

text: a

text: meaningful

text: benchmark,

text: we

text: want

text: to

text: preserve

text: some

text: a

text: priori

text: knowledge

text: of

text: key

text: features

text: of

text: the

text: state

text: space,

text: to

text: be

text: checked

text: against

text: the

text: found

text: ytisneD

text: ytilibaborP

text: detamitsE

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Contents II.ntroduction 2 IWI. KB Method 3 AW. KB Solutions 5 BC. onnection Formulas 6 CB. ohr-Sommerfeld Quantization and Tunneling Rates 15 IEIIx.amples 18 AH. armonic Oscillator 20 BS.chwinger Eﬀect 22 CP.¨oschl-Teller Potential and Quasinormal Modes of A Black Hole 25 DQ. uantum Cosmology 31 ICVo. nclusion 35 Acknowledgments 37 AC.ontours in g(x) = tan(x) Case 37 x References 38 I. INTRODUCTION The WKB approximation, named after Wentzel, Kramers, and Brillouin, is a method for obtaining an approximate solution to a one-dimensional Schrodinger-like diﬀerential equa- tion: (cid:2) (cid:3) −(cid:126) − 2∂2 p2 (ξ) ψ (ξ) = 0, (1) ξ where the real function p2 (ξ) can be either positively or negatively valued. The WKB approximation has a wide range of applications. Its principal applications are in calculating bound-state energies and tunneling rates through potential barriers. 2 ##LTLine##

text: Contents

text: II.ntroduction

text: IWI.

text: KB

text: Method

text: AW.

text: KB

text: Solutions

text: BC.

text: onnection

text: Formulas

text: CB.

text: ohr-Sommerfeld

text: Quantization

text: and

text: Tunneling

text: Rates

text: IEIIx.amples

text: AH.

text: armonic

text: Oscillator

text: BS.chwinger

text: Eﬀect

text: CP.¨oschl-Teller

text: Potential

text: and

text: Quasinormal

text: Modes

text: of

text: A

text: Black

text: Hole

text: DQ.

text: uantum

text: Cosmology

text: ICVo.

text: nclusion

text: Acknowledgments

text: AC.ontours

text: in

text: g(x)

text: tan(x)

text: Case

text: x

text: References

text: I.

label: section

text: INTRODUCTION

text: The

text: WKB

text: approximation,

text: named

text: after

text: Wentzel,

text: Kramers,

text: and

text: Brillouin,

text: is

text: a

text: method

text: for

text: obtaining

text: an

text: approximate

text: solution

text: to

text: a

text: one-dimensional

text: Schrodinger-like

text: diﬀerential

text: equa-

text: tion:

label: equation

text: (ξ)

text: ψ

text: (ξ)

text: ξ

text: where

text: the

text: real

text: function

text: (ξ)

text: can

text: be

text: either

text: positively

text: or

text: negatively

text: valued.

text: The

text: WKB

text: approximation

text: has

text: a

text: wide

text: range

text: of

text: applications.

text: Its

text: principal

text: applications

text: are

text: in

text: calculating

text: bound-state

text: energies

text: and

text: tunneling

text: rates

text: through

text: potential

text: barriers.

text: ##LTLine##


text: 9 0.3 n=1, Parquet 0.2 NRG 1 0.1 0 0.3 n=0.94, Parquet 0.1 0.2 NRG 0.1 0.01 0 Parquet, x = 0.5 0.3 n=0.68, Parquet x = 1.0 NRG Bethe a., x = 0.5 0.2 x = 1.0 0.001 0.1 0 5 10 15 20 0 U -8 -4 0 4 8 ω FIG. 2. (Color online) Spectral function of SIAM for U = 8 FIG. 4. (Color online) Kondo scale calculated from the de- compared with NRG result. nominator of the vertex function K(ω) at the Fermi energy ω = 0 as a function of the interaction strength U compared for two doping parameters x with the exact Bethe ansatz result. 2 Parquet, n Parquet, nT NRG 1.5 1 Parquet Bethe a. 1 0.1 0.5 U = 8 U = 8 U = 12 0 -6 -4 -2 0 2 4 6 0.01 x U = 16 0 0.5 1 1.5 2 FIG. 3. (Color online) Band ﬁllings n and nT as a function − − x of the doping parameter x = E U/2 from the static local d approximation in the reduced parquet equations compared with the NRG result. FIG. 5. (Color online) Kondo scale calculated from the de- nominator of the vertex function K(ω) at the Fermi energy ω = 0 as a function of the doping parameter x compared for a tion reproduce the Kondo scale only in the critical region few interaction strengths with the exact Bethe ansatz result. of the metal-insulator transition where it is exponentially small. We can expect that the farther from half ﬁlling the worse the Kondo scale from the Bethe ansatz will be teraction away from half ﬁlling. The simplest static reproduced. We compared in Fig. 4 the Kondo scale de- approximation is insuﬃcient to achieve a good quanti- pendence on the interaction strength for the ﬁxed doping tative agreement with the exact result there, since the parameter with the Bethe ansatz solution. Although the farther from half ﬁlling the more relevant the dynamical numerical values of the Kondo scale diﬀer, the linearity ﬂuctuations due to multiple electron-electron scatterings of the logarithm of the Kondo scale is correctly repro- become. All ﬂuctuations away from the Kondo regime duced. The shift in the numerical values is caused by a are bounded, of the same order, and the electron-hole diﬀerence in the nonuniversal prefactors that the parquet scatterings lose on dominance. A dynamical approxi- approximation does not reproduce. mation for the irreducible vertex Λ is needed to achieve Similarly we calculated and compared the Kondo more accurate quantitative results. scale as a function of the doping parameter x for a few B. Lattice model ﬁxed interaction strengths in Fig. 5. Kondo scale is no longer exponentially small, we are far from the critical point of the two-particle vertex and only a qualitative The eﬀective interaction is used to generate a mean- behavior of the Kondo scale as a function doping can be ﬁeld approximation, that is, a local approximation to lat- expected. The diﬀerence between our approximation and tice models of interacting electrons. In the mean-ﬁeld ap- the exact solution increases with doping and stronger in- proximation we need to know only the density of states gnilliF noitcnuf lartcepS elacs odnoK elacs odnoK ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Parquet

text: NRG

text: Parquet

text: NRG

text: Parquet,

text: x

text: Parquet

text: x

text: NRG

text: Bethe

text: a.,

text: x

text: x

text: U

text: ω

text: FIG.

text: (Color

text: online)

text: Spectral

text: function

text: of

text: SIAM

text: for

text: U

text: FIG.

text: (Color

text: online)

text: Kondo

text: scale

text: calculated

text: from

text: the

text: de-

text: compared

text: with

text: NRG

text: result.

text: nominator

text: of

text: the

text: vertex

text: function

text: K(ω)

text: at

text: the

text: Fermi

text: energy

text: ω

text: as

text: a

text: function

text: of

text: the

text: interaction

text: strength

text: U

text: compared

text: for

text: two

text: doping

text: parameters

text: x

text: with

text: the

text: exact

text: Bethe

text: ansatz

text: result.

text: Parquet,

text: n

text: Parquet,

text: nT

text: NRG

text: Parquet

text: Bethe

text: a.

text: U

text: U

text: U

text: x

text: U

text: FIG.

text: (Color

text: online)

text: Band

text: ﬁllings

text: n

text: and

text: nT

text: as

text: a

text: function

text: x

text: of

text: the

text: doping

text: parameter

text: x

text: E

text: from

text: the

text: static

text: local

text: d

text: approximation

text: in

text: the

text: reduced

text: parquet

text: equations

text: compared

text: with

text: the

text: NRG

text: result.

text: FIG.

text: (Color

text: online)

text: Kondo

text: scale

text: calculated

text: from

text: the

text: de-

text: nominator

text: of

text: the

text: vertex

text: function

text: K(ω)

text: at

text: the

text: Fermi

text: energy

text: ω

text: as

text: a

text: function

text: of

text: the

text: doping

text: parameter

text: x

text: compared

text: for

text: a

text: tion

text: reproduce

text: the

text: Kondo

text: scale

text: only

text: in

text: the

text: critical

text: region

text: few

text: interaction

text: strengths

text: with

text: the

text: exact

text: Bethe

text: ansatz

text: result.

text: of

text: the

text: metal-insulator

text: transition

text: where

text: it

text: is

text: exponentially

text: small.

text: We

text: can

text: expect

text: that

text: the

text: farther

text: from

text: half

text: ﬁlling

text: the

text: worse

text: the

text: Kondo

text: scale

text: from

text: the

text: Bethe

text: ansatz

text: will

text: be

text: teraction

text: away

text: from

text: half

text: ﬁlling.

text: The

text: simplest

text: static

text: reproduced.

text: We

text: compared

text: in

text: Fig.

text: the

text: Kondo

text: scale

text: de-

text: approximation

text: is

text: insuﬃcient

text: to

text: achieve

text: a

text: good

text: quanti-

text: pendence

text: on

text: the

text: interaction

text: strength

text: for

text: the

text: ﬁxed

text: doping

text: tative

text: agreement

text: with

text: the

text: exact

text: result

text: there,

text: since

text: the

text: parameter

text: with

text: the

text: Bethe

text: ansatz

text: solution.

text: Although

text: the

text: farther

text: from

text: half

text: ﬁlling

text: the

text: more

text: relevant

text: the

text: dynamical

text: numerical

text: values

text: of

text: the

text: Kondo

text: scale

text: diﬀer,

text: the

text: linearity

text: ﬂuctuations

text: due

text: to

text: multiple

text: electron-electron

text: scatterings

text: of

text: the

text: logarithm

text: of

text: the

text: Kondo

text: scale

text: is

text: correctly

text: repro-

text: become.

text: All

text: ﬂuctuations

text: away

text: from

text: the

text: Kondo

text: regime

text: duced.

text: The

text: shift

text: in

text: the

text: numerical

text: values

text: is

text: caused

text: by

text: a

text: are

text: bounded,

text: of

text: the

text: same

text: order,

text: and

text: the

text: electron-hole

text: diﬀerence

text: in

text: the

text: nonuniversal

text: prefactors

text: that

text: the

text: parquet

text: scatterings

text: lose

text: on

text: dominance.

text: A

text: dynamical

text: approxi-

text: approximation

text: does

text: not

text: reproduce.

text: mation

text: for

text: the

text: irreducible

text: vertex

text: Λ

text: is

text: needed

text: to

text: achieve

text: Similarly

text: we

text: calculated

text: and

text: compared

text: the

text: Kondo

text: more

text: accurate

text: quantitative

text: results.

text: scale

text: as

text: a

text: function

text: of

text: the

text: doping

text: parameter

text: x

text: for

text: a

text: few

text: B.

text: Lattice

text: model

text: ﬁxed

text: interaction

text: strengths

text: in

text: Fig.

text: Kondo

text: scale

text: is

text: no

text: longer

text: exponentially

text: small,

text: we

text: are

text: far

text: from

text: the

text: critical

text: point

text: of

text: the

text: two-particle

text: vertex

text: and

text: only

text: a

text: qualitative

text: The

text: eﬀective

text: interaction

text: is

text: used

text: to

text: generate

text: a

text: mean-

text: behavior

text: of

text: the

text: Kondo

text: scale

text: as

text: a

text: function

text: doping

text: can

text: be

text: ﬁeld

text: approximation,

text: that

text: is,

text: a

text: local

text: approximation

text: to

text: lat-

text: expected.

text: The

text: diﬀerence

text: between

text: our

text: approximation

text: and

text: tice

text: models

text: of

text: interacting

text: electrons.

text: In

text: the

text: mean-ﬁeld

text: ap-

text: the

text: exact

text: solution

text: increases

text: with

text: doping

text: and

text: stronger

text: in-

text: proximation

text: we

text: need

text: to

text: know

text: only

text: the

text: density

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text: Figure 2: Graphical representation of the scene generative model and illustration of 3D CAD object models used in this work. (a) RGB image (b) Semantic labels (c) Depth (d) Surface normals (e) Diffuse reﬂections Figure 3: A rendered image sample together with corresponding pixel-level annotations. quired. Please see the supplementary material for details. A ples rendered in this initial setting are shown in Fig 4b. schematic graphical model is as shown in Fig 2 along with 3.3. Sampling and Rendering a few samples of CAD object models used in this work. Although sampling from P(Θ) is easy initially, it even- 3.2. Initialization tually becomes harder as P gets updated iteratively through ← | Bayesian updates: P(Θ) P(Θ)p(. Θ). The reason is that we do not have conjugate relationships between the As shown in Fig 2, our generative model is a physics- classiﬁer’s probabilities and P(.). Hence, these interme- based parametric model whose inputs are a set of scene diate probabbility functions lose their easy-to-sample-from variables Θ such as lighting, weather, geometry and cam- structure. Hence, we use a rejection sampling scheme to era parameters. We assume that all these parameters are sample from P due to its scalability. In general, an open statistically independent of each other, which provides the issue in the use of rejection sampling schemes is to come least expensive option for modeling and sampling. One can up with an optimal scaling factor M, which results in a pro- model dependencies using distributions on these parame- posal distribution that is an envelope to the complicated dis- ters based on an expert’s knowledge for a target domain or tribution that we want to sample from. This issue does not based on additional knowledge such as atmospheric optics, arise in our case as our initial uniform distributions of P geographic and demographic studies. However, in the ab- can behave as envelopes for all intermediate Ps, if they are sence of priors, we use uniform distributions in their per- not re-normalized. However, this ends up increase the prob- missible ranges. For instance, the light source’s intensity in ability of rejecting many samples and therefore generating BLENDER is modeled as uniform(low = 0, high = 6), samples becomes computationally progressively more ex- where an intensity level of 0 can correspond to night while pensive with the number of iterations. We solve this issue 6 corresponds to lighting at noon. With these settings our by normalizing intermediate probability tables with their model was able to render physically plausible and visually respective maximum values. Corresponding labels are ob- realistic images. This scene model was used in our pre- tained through annotation shaders, which we implemented vious work, which is provided as supplementary material. in Blender. An image sample with corresponding labels are Performance of a vision model trained to perform seman- shown in Fig 3. The details about our rendering choices tic segmentation on simulated data was quite good on real- and their impact on the semantic segmentation results can world data. Yet, data-shift was observed due to deviations be found in the supplementary material. between the scene generation statistics and the target real- 3.4. Adversarial Training world domain. Hence, in the present work, we focus on the task of matching generative statistics to those of real-world In a GAN setting, the generator is supplemented with target data such as for instance CityScapes [6]. Some sam- a discriminator D, which is trained to classify samples as 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Graphical

label: caption

text: representation

text: of

text: the

text: scene

text: generative

text: model

text: and

text: illustration

text: of

text: CAD

text: object

text: models

text: used

text: in

text: this

text: work.

text: (a)

text: RGB

text: image

text: (b)

text: Semantic

text: labels

text: (c)

text: Depth

text: (d)

text: Surface

text: normals

text: (e)

text: Diffuse

text: reﬂections

text: Figure

text: A

text: rendered

text: image

text: sample

text: together

text: with

text: corresponding

text: pixel-level

text: annotations.

text: quired.

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text: material

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text: details.

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text: classiﬁer’s

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text: probabbility

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text: easy-to-sample-from

text: variables

text: Θ

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text: prob-

text: missible

text: ranges.

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text: the

text: light

text: source’s

text: intensity

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text: ability

text: of

text: rejecting

text: many

text: samples

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text: therefore

text: generating

text: BLENDER

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text: modeled

text: as

text: uniform(low

text: high

text: samples

text: becomes

text: computationally

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text: intensity

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text: number

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text: intermediate

text: probability

text: tables

text: with

text: their

text: model

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text: able

text: to

text: render

text: physically

text: plausible

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text: visually

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text: maximum

text: values.

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text: labels

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text: ob-

text: realistic

text: images.

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text: scene

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text: annotation

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text: work,

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text: material.

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text: The

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text: segmentation

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text: data

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text: quite

text: good

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text: real-

text: and

text: their

text: impact

text: on

text: the

text: semantic

text: segmentation

text: results

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text: world

text: data.

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text: data-shift

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text: observed

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text: deviations

text: be

text: found

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text: the

text: supplementary

text: material.

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text: the

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text: statistics

text: and

text: the

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text: real-

text: Adversarial

text: Training

text: world

text: domain.

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text: in

text: the

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text: statistics

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text: a

text: GAN

text: setting,

text: the

text: generator

text: is

text: supplemented

text: with

text: target

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text: for

text: instance

text: CityScapes

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text: ##LTFigure##


text: from P(Θ). CamVid is recorded in and around the Cam- column of Fig 4. The structure of these histogram has been bridge region in UK. It provides 701 images along with moved closer to the one of CityScapes through the process high-quality semantic annotations. While tuning the gener- of tuning. Quantitatively, the KL divergence between vir- ative model to CamVid, we randomly sample 500 samples tual data and CityScapes data has been reduced from 0.57 from CamVid in each iteration and set N = 500. before tuning to 0.44 after tuning to CityScapes. A simi- v It is worth highlighting the differences between these lar behavior is observed when the model is trained on the datasets. Each of them has been acquired in a different city CamVid data. Finally we also obtained similar histograms or cities. The camera models used are different. Due to for the ground-truth labels. As with the previous compar- the geographical and demographical differences in weather, isons, on can observe that the label statistics are again closer lighting, object shapes, the statistics of these dataset may to the real datasets after tuning, as shown in the last column differ. For instance, we computed the intensity histograms of Fig 4. This evidence points to the potential usefulness over full CityScapes and CamVid datasets, see Fig 4d and of simulated datasets as virtual proxies for these real world Fig 4j. For better visual comparison, we normalized the his- datasets. tograms with their maximum frequencies. Topologically, 4.2. Generalization of DeepLab these histograms are quite different. Similarly, label statis- In our ﬁrst set of experiments we used CityScapes as the tics also differ, see the histograms of semantic class labels target domain which means that we took the validation set in Fig 4f and Fig 4l. As quantiﬁed in Table 1, these sta- from CityScapes (CS val) for testing. We compared the tistical differences in the training datasets are reﬂected as utility of simulated data generated from the initial model performance shift of DeepLab. For instance, the DeepLab V and the model tuned to CityScapes (V ) in model trained on CityScapes training data (CS train) is init cityscapes terms of generalization of the trained models to CS val. performing at 67.71 IoU points on CS val, a validation V produced good results in classifying the objects such set from CityScapes, i.e. within the same domain. This init as building, vehicles, vegetation, roads, and sky. However, performance is reduced by nearly 13 points instead when pedestrians were poorly recognized due to low frequency the validation set from CamVid (CV val) is used for test- of occurrences and the use of low quality (low polygon ing. Similar behavior is observed when transferring the meshes and textured) CAD models. However, the use of DeepLab model from CityScapes to CamVid. Performance V , which is generated from the model tuned to real degradation when transferring from virtual to real domains cityscapes CityScapes, improved the over-all performance on global is comparable. Similar observations can be found in [22] in IoU by 2.28 points. This time, the per-class IoU measure the context of pedestrian detection with a classiﬁer based on on the pedestrian class also improved to some extent. This HOG and linearSVM. may be credited to the increased number of occurrences af- Virtual reality datasets (V ): To quantify the perfor- ter tuning. This can be discerned in the bar plot of Fig 4, last mance changes due to adversarial tuning, we prepared three column. To measure the statistical signiﬁcance of these im- sets that are simulated from the initial model and the mod- provements, we repeated the training-testing experiment 5 els tuned with the approach discussed in Section to the times and measured the improvement each time. The com- datasets CityScapes and CamVid. We denote them with ± puted mean and standard deviations are 2.28 0.34. V , V and V respectively. Each set has init cityscapes camvid In our second set of experiments we use CamVid as 5000 images along with several annotations along with the target domain and take the validation set from CamVid pixel-wise semantic labels. We ﬁrst compare the perfor- CV val for testing. We compared the utility of the sim- mance statistics of simulated training sets against the target ulated data generated from the initial model V and the datasets used for adversarial tuning. Later in the section, we init model tuned to CamVid V in terms of the generaliza- also compare the generalizations of a vision system on the camvid tion from the trained models to CV val. V already pro- target dataset when it is trained on these sets separately to init duced good results. However, the use of V improved quantify the performance shift due to adversarially trained camvid the overall performance, i.e. the global IoU by 3.42 points. scene generation. 4.1. Statistics of Training sets Interestingly, the DeepLab model trained on Vcityscapes showed improved performance also on the CamVid valida- Though its difﬁcult to appreciate signiﬁcant perfor- tion set, which however was not true the other way around mance changes due to adversarial training by visual in- as seen by a degradation in performance of 6.57%. We con- spection, Figures 4b, 4h, and 4n can be used to obtain jecture that the high number of pedestrians and their diver- insights about how the training affected pixel-level label- sity in the CityScapes set might be one of the reasons. ing. We computed histograms of pixel intensities over the In the ﬁnal set of experiments, we compared the re- full datasets V generated from the initial model, our tar- sults of unsupervised adversarial tuning to those of super- init get data CityScapes and generated the the model tuned to vised domain adaptation, i.e. augmenting the simulated data CityScapes V . These plots are shown in the ﬁrst with 10% labeled samples from the target domain. Clearly, cityscapes 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: from

text: P(Θ).

text: CamVid

text: is

text: recorded

text: in

text: and

text: around

text: the

text: Cam-

text: column

text: of

text: Fig

text: The

text: structure

text: of

text: these

text: histogram

text: has

text: been

text: bridge

text: region

text: in

text: UK.

text: It

text: provides

text: images

text: along

text: with

text: moved

text: closer

text: to

text: the

text: one

text: of

text: CityScapes

text: through

text: the

text: process

text: high-quality

text: semantic

text: annotations.

text: While

text: tuning

text: the

text: gener-

text: of

text: tuning.

text: Quantitatively,

text: the

text: KL

text: divergence

text: between

text: vir-

text: ative

text: model

text: to

text: CamVid,

text: we

text: randomly

text: sample

text: samples

text: tual

text: data

text: and

text: CityScapes

text: data

text: has

text: been

text: reduced

text: from

text: from

text: CamVid

text: in

text: each

text: iteration

text: and

text: set

text: N

text: before

text: tuning

text: to

text: after

text: tuning

text: to

text: CityScapes.

text: A

text: simi-

text: v

text: It

text: is

text: worth

text: highlighting

text: the

text: differences

text: between

text: these

text: lar

text: behavior

text: is

text: observed

text: when

text: the

text: model

text: is

text: trained

text: on

text: the

text: datasets.

text: Each

text: of

text: them

text: has

text: been

text: acquired

text: in

text: a

text: different

text: city

text: CamVid

text: data.

text: Finally

text: we

text: also

text: obtained

text: similar

text: histograms

text: or

text: cities.

text: The

text: camera

text: models

text: used

text: are

text: different.

text: Due

text: to

text: for

text: the

text: ground-truth

text: labels.

text: As

text: with

text: the

text: previous

text: compar-

text: the

text: geographical

text: and

text: demographical

text: differences

text: in

text: weather,

text: isons,

text: on

text: can

text: observe

text: that

text: the

text: label

text: statistics

text: are

text: again

text: closer

text: lighting,

text: object

text: shapes,

text: the

text: statistics

text: of

text: these

text: dataset

text: may

text: to

text: the

text: real

text: datasets

text: after

text: tuning,

text: as

text: shown

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text: the

text: last

text: column

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text: instance,

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text: computed

text: the

text: intensity

text: histograms

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text: Fig

text: This

text: evidence

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text: the

text: potential

text: usefulness

text: over

text: full

text: CityScapes

text: and

text: CamVid

text: datasets,

text: see

text: Fig

text: and

text: of

text: simulated

text: datasets

text: as

text: virtual

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text: these

text: real

text: world

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text: For

text: better

text: visual

text: comparison,

text: we

text: normalized

text: the

text: his-

text: datasets.

text: tograms

text: with

text: their

text: maximum

text: frequencies.

text: Topologically,

text: Generalization

text: of

text: DeepLab

text: these

text: histograms

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text: quite

text: different.

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text: label

text: statis-

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text: our

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text: CamVid

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text: used

text: for

text: test-

text: of

text: occurrences

text: and

text: the

text: use

text: of

text: low

text: quality

text: (low

text: polygon

text: ing.

text: Similar

text: behavior

text: is

text: observed

text: when

text: transferring

text: the

text: meshes

text: and

text: textured)

text: CAD

text: models.

text: However,

text: the

text: use

text: of

text: DeepLab

text: model

text: from

text: CityScapes

text: to

text: CamVid.

text: Performance

text: V

text: which

text: is

text: generated

text: from

text: the

text: model

text: tuned

text: to

text: real

text: degradation

text: when

text: transferring

text: from

text: virtual

text: to

text: real

text: domains

text: cityscapes

text: CityScapes,

text: improved

text: the

text: over-all

text: performance

text: on

text: global

text: is

text: comparable.

text: Similar

text: observations

text: can

text: be

text: found

text: in

text: in

text: IoU

text: by

text: points.

text: This

text: time,

text: the

text: per-class

text: IoU

text: measure

text: the

text: context

text: of

text: pedestrian

text: detection

text: with

text: a

text: classiﬁer

text: based

text: on

text: on

text: the

text: pedestrian

text: class

text: also

text: improved

text: to

text: some

text: extent.

text: This

text: HOG

text: and

text: linearSVM.

text: may

text: be

text: credited

text: to

text: the

text: increased

text: number

text: of

text: occurrences

text: af-

text: Virtual

text: reality

text: datasets

text: (V

text: To

text: quantify

text: the

text: perfor-

text: ter

text: tuning.

text: This

text: can

text: be

text: discerned

text: in

text: the

text: bar

text: plot

text: of

text: Fig

text: last

text: mance

text: changes

text: due

text: to

text: adversarial

text: tuning,

text: we

text: prepared

text: three

text: column.

text: To

text: measure

text: the

text: statistical

text: signiﬁcance

text: of

text: these

text: im-

text: sets

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text: are

text: simulated

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text: the

text: initial

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text: and

text: the

text: mod-

text: provements,

text: we

text: repeated

text: the

text: training-testing

text: experiment

text: els

text: tuned

text: with

text: the

text: approach

text: discussed

text: in

text: Section

text: to

text: the

text: times

text: and

text: measured

text: the

text: improvement

text: each

text: time.

text: The

text: com-

text: datasets

text: CityScapes

text: and

text: CamVid.

text: We

text: denote

text: them

text: with

text: puted

text: mean

text: and

text: standard

text: deviations

text: are

text: V

text: V

text: and

text: V

text: respectively.

text: Each

text: set

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text: init

text: cityscapes

text: camvid

text: In

text: our

text: second

text: set

text: of

text: experiments

text: we

text: use

text: CamVid

text: as

text: images

text: along

text: with

text: several

text: annotations

text: along

text: with

text: the

text: target

text: domain

text: and

text: take

text: the

text: validation

text: set

text: from

text: CamVid

text: pixel-wise

text: semantic

text: labels.

text: We

text: ﬁrst

text: compare

text: the

text: perfor-

text: CV

text: val

text: for

text: testing.

text: We

text: compared

text: the

text: utility

text: of

text: the

text: sim-

text: mance

text: statistics

text: of

text: simulated

text: training

text: sets

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text: the

text: target

text: ulated

text: data

text: generated

text: from

text: the

text: initial

text: model

text: V

text: and

text: the

text: datasets

text: used

text: for

text: adversarial

text: tuning.

text: Later

text: in

text: the

text: section,

text: we

text: init

text: model

text: tuned

text: to

text: CamVid

text: V

text: in

text: terms

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text: the

text: generaliza-

text: also

text: compare

text: the

text: generalizations

text: of

text: a

text: vision

text: system

text: on

text: the

text: camvid

text: tion

text: from

text: the

text: trained

text: models

text: to

text: CV

text: val.

text: V

text: already

text: pro-

text: target

text: dataset

text: when

text: it

text: is

text: trained

text: on

text: these

text: sets

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text: to

text: init

text: duced

text: good

text: results.

text: However,

text: the

text: use

text: of

text: V

text: improved

text: quantify

text: the

text: performance

text: shift

text: due

text: to

text: adversarially

text: trained

text: camvid

text: the

text: overall

text: performance,

text: i.e.

text: the

text: global

text: IoU

text: by

text: points.

text: scene

text: generation.

text: Statistics

text: of

text: Training

text: sets

text: Interestingly,

text: the

text: DeepLab

text: model

text: trained

text: on

text: Vcityscapes

text: showed

text: improved

text: performance

text: also

text: on

text: the

text: CamVid

text: valida-

text: Though

text: its

text: difﬁcult

text: to

text: appreciate

text: signiﬁcant

text: perfor-

text: tion

text: set,

text: which

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text: not

text: true

text: the

text: other

text: way

text: around

text: mance

text: changes

text: due

text: to

text: adversarial

text: training

text: by

text: visual

text: in-

text: as

text: seen

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text: a

text: degradation

text: in

text: performance

text: of

text: We

text: con-

text: spection,

text: Figures

text: and

text: can

text: be

text: used

text: to

text: obtain

text: jecture

text: that

text: the

text: high

text: number

text: of

text: pedestrians

text: and

text: their

text: diver-

text: insights

text: about

text: how

text: the

text: training

text: affected

text: pixel-level

text: label-

text: sity

text: in

text: the

text: CityScapes

text: set

text: might

text: be

text: one

text: of

text: the

text: reasons.

text: ing.

text: We

text: computed

text: histograms

text: of

text: pixel

text: intensities

text: over

text: the

text: In

text: the

text: ﬁnal

text: set

text: of

text: experiments,

text: we

text: compared

text: the

text: re-

text: full

text: datasets

text: V

text: generated

text: from

text: the

text: initial

text: model,

text: our

text: tar-

text: sults

text: of

text: unsupervised

text: adversarial

text: tuning

text: to

text: those

text: of

text: super-

text: init

text: get

text: data

text: CityScapes

text: and

text: generated

text: the

text: the

text: model

text: tuned

text: to

text: vised

text: domain

text: adaptation,

text: i.e.

text: augmenting

text: the

text: simulated

text: data

text: CityScapes

text: V

text: These

text: plots

text: are

text: shown

text: in

text: the

text: ﬁrst

text: with

text: labeled

text: samples

text: from

text: the

text: target

text: domain.

text: Clearly,

text: cityscapes

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Limit density of 2D quantum walk: zeroes of the weight function M. Sˇtefanˇ´ak, I. Bezdˇekov´a, and I. Jex † ‡ Properties of the probability distribution generated by a discrete-time quantum walk, such as the number of peaks it contains, depend strongly on the choice of the initial condition. In the present paper we discuss from this point of view the model of the two-dimensional quantum walk analyzed in K. Watabe et al., Phys. Rev. A 77, 062331, (2008). We show that the limit density can be altered in such a way that it vanishes on the boundary or some line. Using this result one can suppress certain peaks in the probability distribution. The analysis is simpliﬁed considerably by choosing a more suitable basis of the coin space, namely the one formed by the eigenvectors of the coin operator. PACS numbers: ∗ † ‡ ##LTLine##

text: Limit

label: title

text: density

text: of

text: quantum

text: walk:

text: zeroes

text: of

text: the

text: weight

text: function

text: M.

label: author

text: Sˇtefanˇ´ak,

text: I.

text: Bezdˇekov´a,

text: and

text: I.

text: Jex

text: Properties

label: abstract

text: of

text: the

text: probability

text: distribution

text: generated

text: by

text: a

text: discrete-time

text: quantum

text: walk,

text: such

text: as

text: the

text: number

text: of

text: peaks

text: it

text: contains,

text: depend

text: strongly

text: on

text: the

text: choice

text: of

text: the

text: initial

text: condition.

text: In

text: the

text: present

text: paper

text: we

text: discuss

text: from

text: this

text: point

text: of

text: view

text: the

text: model

text: of

text: the

text: two-dimensional

text: quantum

text: walk

text: analyzed

text: in

text: K.

text: Watabe

text: et

text: al.,

text: Phys.

text: Rev.

text: A

text: We

text: show

text: that

text: the

text: limit

text: density

text: can

text: be

text: altered

text: in

text: such

text: a

text: way

text: that

text: it

text: vanishes

text: on

text: the

text: boundary

text: or

text: some

text: line.

text: Using

text: this

text: result

text: one

text: can

text: suppress

text: certain

text: peaks

text: in

text: the

text: probability

text: distribution.

text: The

text: analysis

text: is

text: simpliﬁed

text: considerably

text: by

text: choosing

text: a

text: more

text: suitable

text: basis

text: of

text: the

text: coin

text: space,

text: namely

text: the

text: one

text: formed

text: by

text: the

text: eigenvectors

text: of

text: the

text: coin

text: operator.

text: PACS

text: numbers:

text: ##LTLine##


text: 2 I. INTRODUCTION Quantum walks [1–3] were proposed as extensions of the concept of a classical random walk to the unitary evolution of a quantum particle on a discrete graph or lattice. They have found promising applications in quantum information processing, e.g. in search algorithms [4], graph isomorphism testing [5], ﬁnding structural anomalies in graphs [6], and perfect state transfer [7]. Moreover, quantum walks were shown to be universal tools for quantum computation [8]. Suitable tools for the analysis of homogeneous quantum walks on inﬁnite lattice are the Fourier transformation [9] and the weak-limit theorems [10]. While the properties of many quantum walks on a line are well understood [11–14], less is know about quantum walks on higher-dimensional latices. Indeed, there are many technical diﬃculties, e.g. diagonalization of the evolution operator. One of the few models of 2D quantum walks which is well understood is the one analyzed in [15]. This model is a one-parameter extension of the 2D Grover walk which preserves its key feature, namely the trapping eﬀect (or localization) [16]. The coin parameter controls the area covered by the quantum walk, which in general is an elliptic disc and reduces to a circle for the 2D Grover walk. In the present paper we focus on the role of the initial conditions on the shape of the probability distribution resulting from the 2D quantum walk of [15]. We are interested in initial states which lead to non-generic probability distributions, such as those with reduced number of peaks. In order to ﬁnd them we ﬁrst simplify the results of [15] by converting them to a more suitable basis of the coin space. Following [14] we choose the basis formed by the eigenvectors of the coin operator. We then discuss various initial coin states which result in non-generic probability distribution. In particular, we show that the limit density can be set to zero on some line. This can be used to suppress peaks in the probability distribution. The paper is organized as follows: First, in Section the results of [15] are brieﬂy reviewed. Next, we convert them into more suitable basis to simplify the following analysis. In Section various initial states which lead to non-generic probability distributions are discussed. We conclude and present an outlook in Section . II. 2D QUANTUM WALK Let us ﬁrst brieﬂy review the results of [15]. The authors have considered a quantum walk on a two-dimensional square lattice where the particle can in each step move from its present position (x, y) to the nearest neighbours ± ± | (cid:105) | (cid:105) | (cid:105) | (cid:105) (x 1, y) and (x, y 1). These displacements correspond to the four states R , L , U and D which form the H standard basis of the coin space C. In this standard basis t(cid:112)he coin oper(cid:112)ator is given by the following matrix (cid:112) (cid:112)    (cid:112) − (cid:112) − − −   p 1 p p(1 p) p(1 p)  (cid:112) − (cid:112) − − − 1 p p p(1 p) p(1 p) C = − − − , (1) p(1 p) p(1 p) p 1 p − − − p(1 p) p(1 p) p p 1 × where the parameter p ranges from 0 to 1. For p = 1 the coin operator (1) reduces to the familiar 4 4 Grover 2 matrix. This particular model was analyzed in detail in [16]. Using the Fourier analysis and the weak limit theorem [10] the authors have derived the limit (cid:68)de(cid:16)nsi(cid:17)ty ν(cid:16)(v(cid:17)x, v(cid:69)y) o(cid:90)f the 2D quantum walk. This allows one to evaluate the asymptotic values of all moments of re-scaled position (or pseudo-velocity) through the formula x m y n lim = vmvnν(v , v )dv dv . t→+∞ t t x y x y x y The limit density of the 2D quantum walk is given by [15] M ν(v , v ) = µ(v , v ) (v , v ) + ∆δ (v )δ (v ). (2) x y x y x y 0 x 0 y Here µ(v , v ) denotes the fundamental density which reads [15] x y 2 µ(v , v ) = − − − − 1E, (3) x y π2(1 v + v )(1 + v v )(1 v v )(1 + v + v ) x y x y x y x y ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I.

text: INTRODUCTION

text: Quantum

text: walks

text: were

text: proposed

text: as

text: extensions

text: of

text: the

text: concept

text: of

text: a

text: classical

text: random

text: walk

text: to

text: the

text: unitary

text: evolution

text: of

text: a

text: quantum

text: particle

text: on

text: a

text: discrete

text: graph

text: or

text: lattice.

text: They

text: have

text: found

text: promising

text: applications

text: in

text: quantum

text: information

text: processing,

text: e.g.

text: in

text: search

text: algorithms

text: graph

text: isomorphism

text: testing

text: ﬁnding

text: structural

text: anomalies

text: in

text: graphs

text: and

text: perfect

text: state

text: transfer

text: Moreover,

text: quantum

text: walks

text: were

text: shown

text: to

text: be

text: universal

text: tools

text: for

text: quantum

text: computation

text: Suitable

text: tools

text: for

text: the

text: analysis

text: of

text: homogeneous

text: quantum

text: walks

text: on

text: inﬁnite

text: lattice

text: are

text: the

text: Fourier

text: transformation

text: and

text: the

text: weak-limit

text: theorems

text: While

text: the

text: properties

text: of

text: many

text: quantum

text: walks

text: on

text: a

text: line

text: are

text: well

text: understood

text: less

text: is

text: know

text: about

text: quantum

text: walks

text: on

text: higher-dimensional

text: latices.

text: Indeed,

text: there

text: are

text: many

text: technical

text: diﬃculties,

text: e.g.

text: diagonalization

text: of

text: the

text: evolution

text: operator.

text: One

text: of

text: the

text: few

text: models

text: of

text: quantum

text: walks

text: which

text: is

text: well

text: understood

text: is

text: the

text: one

text: analyzed

text: in

text: This

text: model

text: is

text: a

text: one-parameter

text: extension

text: of

text: the

text: Grover

text: walk

text: which

text: preserves

text: its

text: key

text: feature,

text: namely

text: the

text: trapping

text: eﬀect

text: (or

text: localization)

text: The

text: coin

text: parameter

text: controls

text: the

text: area

text: covered

text: by

text: the

text: quantum

text: walk,

text: which

text: in

text: general

text: is

text: an

text: elliptic

text: disc

text: and

text: reduces

text: to

text: a

text: circle

text: for

text: the

text: Grover

text: walk.

text: In

text: the

text: present

text: paper

text: we

text: focus

text: on

text: the

text: role

text: of

text: the

text: initial

text: conditions

text: on

text: the

text: shape

text: of

text: the

text: probability

text: distribution

text: resulting

text: from

text: the

text: quantum

text: walk

text: of

text: We

text: are

text: interested

text: in

text: initial

text: states

text: which

text: lead

text: to

text: non-generic

text: probability

text: distributions,

text: such

text: as

text: those

text: with

text: reduced

text: number

text: of

text: peaks.

text: In

text: order

text: to

text: ﬁnd

text: them

text: we

text: ﬁrst

text: simplify

text: the

text: results

text: of

text: by

text: converting

text: them

text: to

text: a

text: more

text: suitable

text: basis

text: of

text: the

text: coin

text: space.

text: Following

text: we

text: choose

text: the

text: basis

text: formed

text: by

text: the

text: eigenvectors

text: of

text: the

text: coin

text: operator.

text: We

text: then

text: discuss

text: various

text: initial

text: coin

text: states

text: which

text: result

text: in

text: non-generic

text: probability

text: distribution.

text: In

text: particular,

text: we

text: show

text: that

text: the

text: limit

text: density

text: can

text: be

text: set

text: to

text: zero

text: on

text: some

text: line.

text: This

text: can

text: be

text: used

text: to

text: suppress

text: peaks

text: in

text: the

text: probability

text: distribution.

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: First,

text: in

text: Section

text: the

text: results

text: of

text: are

text: brieﬂy

text: reviewed.

text: Next,

text: we

text: convert

text: them

text: into

text: more

text: suitable

text: basis

text: to

text: simplify

text: the

text: following

text: analysis.

text: In

text: Section

text: various

text: initial

text: states

text: which

text: lead

text: to

text: non-generic

text: probability

text: distributions

text: are

text: discussed.

text: We

text: conclude

text: and

text: present

text: an

text: outlook

text: in

text: Section

text: II.

text: QUANTUM

text: WALK

text: Let

text: us

text: ﬁrst

text: brieﬂy

text: review

text: the

text: results

text: of

text: The

text: authors

text: have

text: considered

text: a

text: quantum

text: walk

text: on

text: a

text: two-dimensional

text: square

text: lattice

text: where

text: the

text: particle

text: can

text: in

text: each

text: step

text: move

text: from

text: its

text: present

text: position

text: (x,

text: y)

text: to

text: the

text: nearest

text: neighbours

text: (x

text: y)

text: and

text: (x,

text: y

text: These

text: displacements

text: correspond

text: to

text: the

text: four

text: states

text: R

text: L

text: U

text: and

text: D

text: which

text: form

text: the

text: H

text: standard

text: basis

text: of

text: the

text: coin

text: space

text: C.

text: In

text: this

text: standard

text: basis

text: coin

text: is

text: given

text: by

text: the

text: following

text: matrix

text: p

text: p

text: p)

text: p)

text: p

text: p

text: p)

text: p)

text: C

text: p)

text: p)

text: p

text: p

text: p)

text: p)

text: p

text: p

text: where

text: the

text: parameter

text: p

text: ranges

text: from

text: to

text: For

text: p

text: the

text: coin

text: operator

text: reduces

text: to

text: the

text: familiar

text: Grover

text: matrix.

text: This

text: particular

text: model

text: was

text: analyzed

text: in

text: detail

text: in

text: Using

text: the

text: Fourier

text: analysis

text: and

text: the

text: weak

text: limit

text: theorem

text: the

text: authors

text: have

text: derived

text: the

text: limit

text: the

text: quantum

text: walk.

text: This

text: allows

text: one

text: to

text: evaluate

text: the

text: asymptotic

text: values

text: of

text: all

text: moments

text: of

text: re-scaled

text: position

text: (or

text: pseudo-velocity)

text: through

text: the

text: formula

text: x

text: m

text: y

text: n

text: lim

text: vmvnν(v

text: v

text: )dv

text: dv

text: t→+∞

text: t

text: t

text: x

text: y

text: x

text: y

text: x

text: y

text: The

text: limit

text: density

text: of

text: the

text: quantum

text: walk

text: is

text: given

text: by

text: M

text: ν(v

text: v

text: µ(v

text: v

text: (v

text: v

text: ∆δ

text: (v

text: )δ

text: (v

text: x

text: y

text: x

text: y

text: x

text: y

text: x

text: y

text: Here

text: µ(v

text: v

text: denotes

text: the

text: fundamental

text: density

text: which

text: reads

text: x

text: y

text: µ(v

text: v

text: x

text: y

text: v

text: v

text: v

text: v

text: v

text: v

text: v

text: v

text: x

text: y

text: x

text: y

text: x

text: y

text: x

text: y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 III. NON-GENERIC PROBABILITY DISTRIBUTIONS Let us now discuss the role of the initial coin state on the shape of the probability distribution. We begin with the | (cid:105) eigenstate σ . In such a case the weight function reduces to + M − v2 − v2 (v , v ) = 1 x −y , (11) x y p 1 p E which vanishes on the boundary ellipse ∂ . Hence, the divergencies of the limit density are suppressed and all propagating peaks will be absent in the resulting probability distribution. We illustrate this eﬀect in Fig. 2, where we choose the coin parameter p = 0.4. | (cid:105) FIG. 2. 2D quantum walk with the initial coin state σ . The coin parameter was chosen as p = 0.4. On the left we display + the probability distribution after 50 steps. Notice the absence of the peaks on ellipse. Indeed, the limit density vanishes at the boundary, which we illustrate on the right. The central peak corresponds to the trapping eﬀect. | (cid:105) Next, we consider the eigenstate σ . For this particular initial coin state the trapping eﬀect vanishes, as was 1 identiﬁed already in [15]. We illustrate this feature in Fig. 3 where we take the coin parameter p = 0.6. | (cid:105) Let us now consider the eigenstate σ as the initial coin state. We ﬁnd that the weight function reduces to 2 M v2 (v , v ) = x . (12) x y p Hence, the limit density vanishes on the line v = 0. This eﬀect is illustrated in Fig. 4 for the coin parameter x p = 0.8. | (cid:105) | (cid:105) In a similar way, the choice of the initial coin state ψ = σ leads to the weight function of the form C 3 M v2 (v , v ) = −y . (13) x y 1 p | (cid:105) Therefore, for σ the density vanishes for v = 0. This feature is depicted Fig. 5. 3 y More generally, when we choose the initial coin state of the form | (cid:105) | (cid:105) | (cid:105) ψ = g σ + g σ , C 2 2 3 3 (cid:12) (cid:12) (cid:12) (cid:12) the weight function reduces into (cid:12) (cid:12) (cid:12) (cid:12) M √g √ g 2 (v , v ) = 2 v + 3− v . x y p x 1 p y ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: III.

text: NON-GENERIC

text: PROBABILITY

text: DISTRIBUTIONS

text: Let

text: us

text: now

text: discuss

text: the

text: role

text: of

text: the

text: initial

text: coin

text: state

text: on

text: the

text: shape

text: of

text: the

text: probability

text: distribution.

text: We

text: begin

text: with

text: the

text: eigenstate

text: σ

text: In

text: such

text: a

text: case

text: the

text: weight

text: function

text: reduces

text: to

text: M

text: (v

text: v

text: x

text: −y

text: x

text: y

text: p

text: p

text: E

text: which

text: vanishes

text: on

text: the

text: boundary

text: ellipse

text: Hence,

text: the

text: divergencies

text: of

text: the

text: limit

text: density

text: are

text: suppressed

text: and

text: all

text: propagating

text: peaks

text: will

text: be

text: absent

text: in

text: the

text: resulting

text: probability

text: distribution.

text: We

text: illustrate

text: this

text: eﬀect

text: in

text: Fig.

text: where

text: we

text: choose

text: the

text: coin

text: parameter

text: p

text: FIG.

text: quantum

text: walk

text: with

text: the

text: initial

text: coin

text: state

text: σ

text: The

text: coin

text: parameter

text: was

text: chosen

text: as

text: p

text: On

text: the

text: left

text: we

text: display

text: the

text: probability

text: distribution

text: after

text: steps.

text: Notice

text: the

text: absence

text: of

text: the

text: peaks

text: on

text: ellipse.

text: Indeed,

text: the

text: limit

text: density

text: vanishes

text: at

text: the

text: boundary,

text: which

text: we

text: illustrate

text: on

text: the

text: right.

text: The

text: central

text: peak

text: corresponds

text: to

text: the

text: trapping

text: eﬀect.

text: Next,

text: we

text: consider

text: the

text: eigenstate

text: σ

text: For

text: this

text: particular

text: initial

text: coin

text: state

text: the

text: trapping

text: eﬀect

text: vanishes,

text: as

text: was

text: identiﬁed

text: already

text: in

text: We

text: illustrate

text: this

text: feature

text: in

text: Fig.

text: where

text: we

text: take

text: the

text: coin

text: parameter

text: p

text: Let

text: us

text: now

text: consider

text: the

text: eigenstate

text: σ

text: as

text: the

text: initial

text: coin

text: state.

text: We

text: ﬁnd

text: that

text: the

text: weight

text: function

text: reduces

text: to

text: M

text: (v

text: v

text: x

text: x

text: y

text: p

text: Hence,

text: the

text: limit

text: density

text: vanishes

text: on

text: the

text: line

text: v

text: This

text: eﬀect

text: is

text: illustrated

text: in

text: Fig.

text: for

text: the

text: coin

text: parameter

text: x

text: p

text: In

text: a

text: similar

text: way,

text: the

text: choice

text: of

text: the

text: initial

text: coin

text: state

text: ψ

text: σ

text: leads

text: to

text: the

text: weight

text: function

text: of

text: the

text: form

text: C

text: M

text: (v

text: v

text: −y

text: x

text: y

text: p

text: Therefore,

text: for

text: σ

text: the

text: density

text: vanishes

text: for

text: v

text: This

text: feature

text: is

text: depicted

text: Fig.

text: y

text: More

text: generally,

text: when

text: we

text: choose

text: the

text: initial

text: coin

text: state

text: of

text: the

text: form

text: ψ

text: g

text: σ

text: g

text: σ

text: C

text: the

text: weight

text: function

text: reduces

text: into

text: M

text: √g

text: g

text: (v

text: v

text: v

text: 3−

text: v

text: x

text: y

text: p

text: x

text: p

text: y

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Z 2-ALGEBRAS AS NONCOMMUTATIVE BLOW-UPS. DENNIS PRESOTTO Abstract. The goal of this note is to ﬁrst prove that for a well behaved Z2- p q “ p q{ p q p q algebra R, the category QGr R : ZGr R Tors R is equivalent to QGr R∆ where R∆ is a diaZgonal-like sub- -algebra. Afterwards we use this result to prove that the 2-algebras as introduced in [12] are QGr-equivalent to a Z diagonal-like sub- -algebra which is a simultaneous noncommutative blowup of a quadratic and a cubic Sklyanin algebra. As such we link the noncommutative Z birational transformation and the associated 2-algebras in [13] and [12] with the noncommutative blowups as in [14] and [19]. Contents 1. Introduction 1 2. Acknowledgements 3 3. Diagonal-like subalgebras 3 4. Summary of the constructions in [13] and [12] 5 Z 5. The 2-algebras as in [12] are Noetherian. 7 6. Ample Z2-sequences and Noetherianity of B˜` 12 7. Proof of Theorem 1.1 17 References 21 1. Introduction Z Throughout this note R is a 2-algebra over some algebraically closed ﬁeld k. § Z (See for the deﬁnition of 2-algebras. We refer the interested reader to [16] for a more thorough introduction to G-algebras). Following tradition [5] we associate p q Z a noncommutative projective scheme Proj R to a 2-algebra R whenever R is Z suﬃciently well behaved (e.g. a Noetherian 2-algebra satisfying some analogue of p q generation in degree 1, see Deﬁnition 3.2). Proj R is deﬁned via its category of “quasicoherent sheaves”: p p qq “ p q “ p q{ p q Qcoh Proj R : QGr R Gr R Tors R p q p q where Gr R is the category of graded right R-modules and Tors R is the full sub- Z category of torsion R-modules. In this way these 2-algebras give non-commutative generalizations of bi-homogeneous algebras. Z ą Multi-homogeneous algebras (i.e. n-graded algebras with with n 1) appear frequently in the literature [6, 7, 9, 17, 18]. These multi-homogeneous algebras S inherit many properties from diagonal subalgebras S . Moreover to each multi- ∆ p q p q homogeneous algebra S one associates a projective scheme Multi Proj S and given 1

text: Z

text: AS

text: NONCOMMUTATIVE

text: BLOW-UPS.

text: DENNIS

text: PRESOTTO

text: Abstract.

text: The

text: goal

text: of

text: this

text: note

text: is

text: to

text: ﬁrst

text: prove

text: that

text: for

text: a

text: well

text: behaved

text: p

text: q

text: p

text: q{

text: p

text: q

text: p

text: q

text: algebra

text: R,

text: the

text: category

text: QGr

text: R

text: ZGr

text: R

text: Tors

text: R

text: is

text: equivalent

text: to

text: QGr

text: R∆

text: where

text: R∆

text: is

text: a

text: diaZgonal-like

text: sub-

text: -algebra.

text: Afterwards

text: we

text: use

text: this

text: result

text: to

text: prove

text: that

text: the

text: as

text: introduced

text: in

text: are

text: QGr-equivalent

text: to

text: a

text: Z

text: diagonal-like

text: sub-

text: -algebra

text: which

text: is

text: a

text: simultaneous

text: noncommutative

text: blowup

text: of

text: a

text: quadratic

text: and

text: a

text: cubic

text: Sklyanin

text: algebra.

text: As

text: such

text: we

text: link

text: the

text: noncommutative

text: Z

text: birational

text: transformation

text: and

text: the

text: associated

text: in

text: and

text: with

text: the

text: noncommutative

text: blowups

text: as

text: in

text: and

text: Contents

text: Introduction

text: Acknowledgements

text: Diagonal-like

text: subalgebras

text: Summary

text: of

text: the

text: constructions

text: in

text: and

text: Z

text: The

text: as

text: in

text: are

text: Noetherian.

text: Ample

text: and

text: Noetherianity

text: of

text: B˜`

text: Proof

text: of

text: Theorem

text: References

text: Introduction

text: Z

text: Throughout

text: this

text: note

text: R

text: is

text: a

text: over

text: some

text: algebraically

text: closed

text: ﬁeld

text: k.

text: Z

text: (See

text: for

text: the

text: deﬁnition

text: of

text: We

text: refer

text: the

text: interested

text: reader

text: to

text: for

text: a

text: more

text: thorough

text: introduction

text: to

text: G-algebras).

text: Following

text: tradition

text: we

text: associate

text: p

text: q

text: Z

text: a

text: noncommutative

text: projective

text: scheme

text: Proj

text: R

text: to

text: a

text: R

text: whenever

text: R

text: is

text: Z

text: suﬃciently

text: well

text: behaved

text: (e.g.

text: a

text: Noetherian

text: satisfying

text: some

text: analogue

text: of

text: p

text: q

text: generation

text: in

text: degree

text: see

text: Deﬁnition

text: Proj

text: R

text: is

text: deﬁned

text: via

text: its

text: category

text: of

text: “quasicoherent

text: sheaves”:

text: p

text: p

text: qq

text: p

text: q

text: p

text: q{

text: p

text: q

text: Qcoh

text: Proj

text: R

text: QGr

text: R

text: Gr

text: R

text: Tors

text: R

text: p

text: q

text: p

text: q

text: where

text: Gr

text: R

text: is

text: the

text: category

text: of

text: graded

text: right

text: R-modules

text: and

text: Tors

text: R

text: is

text: the

text: full

text: sub-

text: Z

text: category

text: of

text: torsion

text: R-modules.

text: In

text: this

text: way

text: these

text: give

text: non-commutative

text: generalizations

text: of

text: bi-homogeneous

text: algebras.

text: Z

text: ą

text: Multi-homogeneous

text: algebras

text: (i.e.

text: n-graded

text: algebras

text: with

text: with

text: n

text: appear

text: frequently

text: in

text: the

text: literature

text: These

text: multi-homogeneous

text: algebras

text: S

text: inherit

text: many

text: properties

text: from

text: diagonal

text: subalgebras

text: S

text: Moreover

text: to

text: each

text: multi-

text: p

text: q

text: p

text: q

text: homogeneous

text: algebra

text: S

text: one

text: associates

text: a

text: projective

text: scheme

text: Multi

text: Proj

text: S

text: and

text: given


text: 20 DENNIS PRESOTTO ´ p´q ` δ 2i δ2i 2 B˜p q p ` ` q B˜p q p ` q 2i,i , 2i 2,i 1 0,3i , 0,3i 3 – – ˚ ˚ τ τ B˜p ` ` q p ` ` q B˜p ` q p ` q B˜p ` q p ` q 2i 2,i 1 , 2j 2,j 1 ´ p´q ` 2,3i 1 , 2,3i 4 ´ p´q “ 0,3i 3 , 0,3i 6 δ 2i δ2i 2 δ 2 δ2 ϕ Next we focus on commutativity of (34). The right vertical arrow in (34) can be 1 1 “ 1 described as follows: write g as a product of elements in Aˇ ` Bˇ ` , 1 “ 1 3i 1 “ 1 3˝i,3˚i 1 1 3i,3i Ñ1 Aˇ ` ` Bˇ ` ` and Aˇ ` ` Bˇ ` ` , apply ϕ τ : Bˇ p ` q 13i 1,3i 2 3i 1,3i 2 3i 2,3i 3 31i 2,3i 3 m,m 1 Bˇ p ` ` q on each of the 3 factors, then g ` is the product of these 3 new m 3,m 4 3i 3 elements. This remark together with (36) shows that the commutativity of (34) reduces to the following claim: “ ¨ ¨ P “ if gp qδp qδp ´ ` q a a a for a A˜p ` q p ` ` q B˜p ` q p ` ` q 2i,i 2i,i 2i 1,i 1 0 1 2“ ˚ n ¨ ˚ 2i,¨i n˚ , 2i,i n 1 2i,i n , 2i,i n 1 then gp ` ` qδp ` ` qδp ` q τ a τ a τ a . 2i 2,i 1 2i 2,i 1 2i 1,i 0 1 2 As each there are embeddings Ñ (37) A˜p q p q ã Aˇ ` ` i,j , m,n i 2j,m 2n P it suﬃces to check the claim in Aˇ. For this denote x for the image of some x A˜ under the embedding (37). Consider the equality ¨ ¨ “ ¨ ¨ (38) gp q δp q δp ´ ` q a a a 2i,i 2i,i 2i 1,i 1 0 1 2 Ñ As δp q, δp ´ ` q, a , a and a lie in Bˇ the 4-periodicity morphism Aˇ 2i,i 2i 1,i 1 0 1˚ 2 ˚ ˚ “ ˚ ˚ “ ˚ m,n Aˇ ` ` sends them to τ δp q, τ δp ´ ` q, τ a τ a , τ a τ a and ˚m 4“,n 4˚ 2i,i 2i 1,i 1 0 0 1 1 τ a τ a . Moreover by (36) 2 2 ˚ “ ˚ “ “ τ δp q τ δp q δp ` ` q δp ` ` q i,j i,j i 2,j 1 i 2,j 1 Finally notice that by construction gp q is the element in Aˇ ` corresponding P 2i,i Ñ 4i,4i 4 to the central element g A , as such Aˇ ` Aˇ ` ` sends it to gp ` ` q. 4 4i,4i 4 4i 4,4i 8 2i 2,i 1 In particular the 4-periodicity of Aˇ turns the equality (38) into ¨ ¨ “ ˚ ¨ ˚ ¨ ˚ gp ` ` q δp ` ` q δp ` q τ a τ a τ a 2i 2,i 1 2i 2,i 1 2i 1,i 0 1 2 proving our claim and hence showing commutativity of (34) and hence of (30). As mentioned above this implies that the inclusion (29) induces an inclusion Ñ 1p q T ã A 3 Our goal is to understand the image of this inclusion. As T is generated in degree 1, it suﬃces to understand the image of – Ý¨Ñ – 1 T A˜p q p q δ2 A˜p q p q A 1 0,0 , 2,1 0,0 , 0,3 3 1 Ñ 1 Ñ 1 { 1 As was mentioned above this image contains g and the image of T A A g 1 3 3 is the same as the image of ¨ Ñ – 1 δp qδp q : B˜p q p q B˜p q p q B 2,1 1,2 0,0 , 2,1 0,0 , 0,3 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: DENNIS

text: PRESOTTO

text: p´q

text: δ

text: B˜p

text: q

text: p

text: q

text: B˜p

text: q

text: p

text: q

text: τ

text: τ

text: B˜p

text: q

text: p

text: q

text: B˜p

text: q

text: p

text: q

text: B˜p

text: q

text: p

text: q

text: p´q

text: p´q

text: δ

text: δ

text: δ2

text: ϕ

text: Next

text: we

text: focus

text: on

text: commutativity

text: of

text: The

text: right

text: vertical

text: arrow

text: in

text: can

text: be

text: described

text: as

text: follows:

text: write

text: g

text: as

text: a

text: product

text: of

text: elements

text: in

text: Aˇ

text: Bˇ

text: Ñ1

text: Aˇ

text: Bˇ

text: and

text: Aˇ

text: Bˇ

text: apply

text: ϕ

text: τ

text: Bˇ

text: p

text: q

text: m,m

text: Bˇ

text: p

text: q

text: on

text: each

text: of

text: the

text: factors,

text: then

text: g

text: is

text: the

text: product

text: of

text: these

text: new

text: m

text: elements.

text: This

text: remark

text: together

text: with

text: shows

text: that

text: the

text: commutativity

text: of

text: reduces

text: to

text: the

text: following

text: claim:

text: P

text: if

text: gp

text: qδp

text: qδp

text: q

text: a

text: a

text: a

text: for

text: a

text: A˜p

text: q

text: p

text: q

text: B˜p

text: q

text: p

text: q

text: 2“

text: n

text: n˚

text: n

text: n

text: n

text: then

text: gp

text: qδp

text: qδp

text: q

text: τ

text: a

text: τ

text: a

text: τ

text: a

text: As

text: each

text: there

text: are

text: embeddings

text: Ñ

text: A˜p

text: q

text: p

text: q

text: ã

text: Aˇ

text: i,j

text: m,n

text: i

text: P

text: it

text: suﬃces

text: to

text: check

text: the

text: claim

text: in

text: Aˇ.

text: For

text: this

text: denote

text: x

text: for

text: the

text: image

text: of

text: some

text: x

text: A˜

text: under

text: the

text: embedding

text: Consider

text: the

text: equality

text: gp

text: q

text: δp

text: q

text: δp

text: q

text: a

text: a

text: a

text: Ñ

text: As

text: δp

text: q,

text: δp

text: q,

text: a

text: a

text: and

text: a

text: lie

text: in

text: Bˇ

text: the

text: morphism

text: Aˇ

text: 1˚

text: m,n

text: Aˇ

text: sends

text: them

text: to

text: τ

text: δp

text: q,

text: τ

text: δp

text: q,

text: τ

text: a

text: τ

text: a

text: τ

text: a

text: τ

text: a

text: and

text: ˚m

text: 4˚

text: τ

text: a

text: τ

text: a

text: Moreover

text: by

text: τ

text: δp

text: q

text: τ

text: δp

text: q

text: δp

text: q

text: δp

text: q

text: i,j

text: i,j

text: i

text: i

text: Finally

text: notice

text: that

text: by

text: construction

text: gp

text: q

text: is

text: the

text: element

text: in

text: Aˇ

text: corresponding

text: P

text: Ñ

text: to

text: the

text: central

text: element

text: g

text: A

text: as

text: such

text: Aˇ

text: Aˇ

text: sends

text: it

text: to

text: gp

text: q.

text: In

text: particular

text: the

text: of

text: Aˇ

text: turns

text: the

text: equality

text: into

text: gp

text: q

text: δp

text: q

text: δp

text: q

text: τ

text: a

text: τ

text: a

text: τ

text: a

text: proving

text: our

text: claim

text: and

text: hence

text: showing

text: commutativity

text: of

text: and

text: hence

text: of

text: As

text: mentioned

text: above

text: this

text: implies

text: that

text: the

text: inclusion

text: induces

text: an

text: inclusion

text: Ñ

text: q

text: T

text: ã

text: A

text: Our

text: goal

text: is

text: to

text: understand

text: the

text: image

text: of

text: this

text: inclusion.

text: As

text: T

text: is

text: generated

text: in

text: degree

text: it

text: suﬃces

text: to

text: understand

text: the

text: image

text: of

text: Ý¨Ñ

text: T

text: A˜p

text: q

text: p

text: q

text: δ2

text: A˜p

text: q

text: p

text: q

text: A

text: Ñ

text: Ñ

text: As

text: was

text: mentioned

text: above

text: this

text: image

text: contains

text: g

text: and

text: the

text: image

text: of

text: T

text: A

text: A

text: g

text: is

text: the

text: same

text: as

text: the

text: image

text: of

text: Ñ

text: δp

text: qδp

text: q

text: B˜p

text: q

text: p

text: q

text: B˜p

text: q

text: p

text: q

text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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