text: 6 TYLER HOFFMAN AND BENJAMIN STEINHURST F Figure 1. An example of a bounding box for [2], α = π. Notice 11 2 that the initial line segment begins at vertex A and that the last line segment is at vertex D. F Proof. Consider the even i case. Curves of the form [i] have ﬁrst and last R 6k+5 vertices on a line in 2 and we wish to show that no other segments cross this line. F It is true for [i] because for any even i it consists of a diagonal walk up, a double 5 segment, and a diagonal walk down of equal length providing a net right turn. This gives that the net change in the vertical coordinate is zero. Notice that, with this F description of [i], it is entirely contained inside of its bounding box. It is true F 5 F F for [i] because it is the assemblage of four copies of [i] and an [i] in the shape 8 5 2 shown in Figure 2. F F F Recall that each curve [i] is made up of curves [i] and [i]− (from the 6k+5 6kF+2 6kF 1 ﬁve-partite structure). Assuming that the claim is true for [i] and [i]− , then it 6k+2 6k 1 holds for their assemblage because of their arrangement in the ﬁve-partite structure. Finally, since the case for odd i introduces only a rotation of π and rotation does 4 (cid:3) not change these properties, it holds for this case as well. F Proposition 8. The bounding boxes of each individual sub-curve of [i] from the n ﬁve-partite structure have disjoint interiors. Proof. From the ﬁve-partite structure, we know that the last vertex of one sub- curve is the ﬁrst vertex of the next sub-curve since the sub-curves are connected and form a continuous path. From Proposition 7, we know that the ﬁrst and last vertices of a sub-curve are also vertices of the bounding box. This means that the bounding boxes of the two sub-curves that share a vertex also share that same ≤ vertex. Due to the fact that we draw with an α π angle and that this vertex is 2 speciﬁcally the ﬁrst or last vertex, no segment from one bounding box, that is one ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: TYLER

text: HOFFMAN

text: AND

text: BENJAMIN

text: STEINHURST

text: F

label: caption

text: Figure

text: An

text: example

text: of

text: a

text: bounding

text: box

text: for

text: α

text: π.

text: Notice

text: that

text: the

text: initial

text: line

text: segment

text: begins

text: at

text: vertex

text: A

text: and

text: that

text: the

text: last

text: line

text: segment

text: is

text: at

text: vertex

text: D.

text: F

text: Proof.

text: Consider

text: the

text: even

text: i

text: case.

text: Curves

text: of

text: the

text: form

text: [i]

text: have

text: ﬁrst

text: and

text: last

text: R

text: vertices

text: on

text: a

text: line

text: in

text: and

text: we

text: wish

text: to

text: show

text: that

text: no

text: other

text: segments

text: cross

text: this

text: line.

text: F

text: It

text: is

text: true

text: for

text: [i]

text: because

text: for

text: any

text: even

text: i

text: it

text: consists

text: of

text: a

text: diagonal

text: walk

text: up,

text: a

text: double

text: segment,

text: and

text: a

text: diagonal

text: walk

text: down

text: of

text: equal

text: length

text: providing

text: a

text: net

text: right

text: turn.

text: This

text: gives

text: that

text: the

text: net

text: change

text: in

text: the

text: vertical

text: coordinate

text: is

text: zero.

text: Notice

text: that,

text: with

text: this

text: F

text: description

text: of

text: [i],

text: it

text: is

text: entirely

text: contained

text: inside

text: of

text: its

text: bounding

text: box.

text: It

text: is

text: true

text: F

text: F

text: F

text: for

text: [i]

text: because

text: it

text: is

text: the

text: assemblage

text: of

text: four

text: copies

text: of

text: [i]

text: and

text: an

text: [i]

text: in

text: the

text: shape

text: shown

text: in

text: Figure

text: F

text: F

text: F

text: Recall

text: that

text: each

text: curve

text: [i]

text: is

text: made

text: up

text: of

text: curves

text: [i]

text: and

text: [i]−

text: (from

text: the

text: ﬁve-partite

text: structure).

text: Assuming

text: that

text: the

text: claim

text: is

text: true

text: for

text: [i]

text: and

text: [i]−

text: then

text: it

text: holds

text: for

text: their

text: assemblage

text: because

text: of

text: their

text: arrangement

text: in

text: the

text: ﬁve-partite

text: structure.

text: Finally,

text: since

text: the

text: case

text: for

text: odd

text: i

text: introduces

text: only

text: a

text: rotation

text: of

text: π

text: and

text: rotation

text: does

text: not

text: change

text: these

text: properties,

text: it

text: holds

text: for

text: this

text: case

text: as

text: well.

text: F

text: Proposition

text: The

text: bounding

text: boxes

text: of

text: each

text: individual

text: sub-curve

text: of

text: [i]

text: from

text: the

text: n

text: ﬁve-partite

text: structure

text: have

text: disjoint

text: interiors.

text: Proof.

text: From

text: the

text: ﬁve-partite

text: structure,

text: we

text: know

text: that

text: the

text: last

text: vertex

text: of

text: one

text: sub-

text: curve

text: is

text: the

text: ﬁrst

text: vertex

text: of

text: the

text: next

text: sub-curve

text: since

text: the

text: sub-curves

text: are

text: connected

text: and

text: form

text: a

text: continuous

text: path.

text: From

text: Proposition

text: we

text: know

text: that

text: the

text: ﬁrst

text: and

text: last

text: vertices

text: of

text: a

text: sub-curve

text: are

text: also

text: vertices

text: of

text: the

text: bounding

text: box.

text: This

text: means

text: that

text: the

text: bounding

text: boxes

text: of

text: the

text: two

text: sub-curves

text: that

text: share

text: a

text: vertex

text: also

text: share

text: that

text: same

text: vertex.

text: Due

text: to

text: the

text: fact

text: that

text: we

text: draw

text: with

text: an

text: α

text: π

text: angle

text: and

text: that

text: this

text: vertex

text: is

text: speciﬁcally

text: the

text: ﬁrst

text: or

text: last

text: vertex,

text: no

text: segment

text: from

text: one

text: bounding

text: box,

text: that

text: is

text: one

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Nobuhiro Maekawa Kobayashi Maskawa Institute, Nagoya University; Department of Physics, Nagoya University, Nagoya 464-8602, Japan Yu Muramatsu School of Physics, KIAS, Seoul 130-722, Korea Recent discovery of neutrino large mixings implies the large mixings in the diagonalizing matrices of ¯5 ﬁelds in SU(5) grand uniﬁed theory (GUT), while the diagonalizing matrices of 10 ﬁelds of SU(5) are expected to have small mixings like Cabibbo-Kobayashi-Maskawa matrix. We calculate the predictions of ﬂavor changing nucleon decays (FCND) in SU(5), SO(10), and E GUT models 6 which have the above features for mixings. We found that FCND can be the main decay mode and play an important role to test GUT models. PACS numbers: I. INTRODUCTION where 10 ﬁelds contain doublet quark q, right-handed up quark uc , and right-handed charged lepton ec , and R R the last term can be obtained from (Y ) ¯5 1 5 + One of the most exciting discoveries in elementary νD ij i j H (M ) 1 1 by integrating the right-handed neutrino particle physics among the latest 20 years is neutrino νR ij i j ﬁelds 1 . Here i = 1, 2, 3 is the index for generation, and oscillation[1, 2], which leads to massive neutrinos and i Y , Y , Y , and M are Yukawa matrix of up type large neutrino mixing angles. Interestingly, this discov- u d,e νD νR quarks, that of down type quarks and charged leptons, ery gives an evidence of grand uniﬁed theory (GUT)[3], Dirac neutrino Yukawa matrix, and right-handed neu- in which uniﬁcation of forces and uniﬁcation of quarks trino mass matrix, respectively. These uniﬁed structures and leptons are realized. This evidence for uniﬁcation of for Yukawa interactions are corresponding to the classi- quarks and leptons makes the idea of GUT quite promis- ﬁcation of the hierarchies of the observed quark and lep- ing, because for uniﬁcation of forces we have already ton masses that up type quark masses have the strongest known an experimental evidence that three gauge cou- hierarchy, neutrino masses have the weakest, and down plings meet at a scale, the GUT scale Λ [4], especially G type quark and charged lepton masses have middle hi- in supersymmetric (SUSY) GUT[5]. Moreover, the large erarchies if neutrino mass hierarchy is normal (not in- neutrino mixing angles imply not only large mixing an- verted). Moreover, if we assume that 10 ﬁelds induce gles of doublet lepton l but also those of right-handed down quark dc in SU(5) GUT because ¯5 ﬁeld of SU(5) stronger hierarchy in Yukawa couplings than ¯5 ﬁelds, contains l andRdc . This suggests an interesting possibil- these various hierarchies for quark and lepton masses R can be explained. Furthermore, this assumption explains ity that the ﬂavor changing processes are seen in nucleon decay which is the most important prediction of GUT 1. that quark mixings are smaller than lepton mixings at the same time if we use a reasonable expectation that the In this paper, we study the ﬂavor changing nucleon decay stronger hierarchy leads to smaller mixings. This bril- and propose that the ﬂavor changing nucleon decay can liant chemistry between the Yukawa structure in SU(5) be a key observation for GUT. GUT and the observed hierarchies of quark and lepton II. QUALITATIVE EVIDENCE FOR SU(5) masses and mixings is quite non-trivial, and therefore it UNIFICATION can be regarded as an experimental signature for uniﬁ- cation of quarks and leptons in SU(5) GUT. First, we explain the qualitative evidence for the uni- III. E UNIFICATION 6 ﬁcation of matters. In SU(5) GUT, Yukawa interactions with Higgs ﬁelds 5 and ¯5 are given by H H E GUT[7–9] is more attractive because the assump- 6 tion in the SU(5) GUT can be derived, and as a result, L various Yukawa matrices can be derived from one basic = (Y ) 10 10 5 + (Y ) 10 ¯5 ¯5 Y u ij i j H d,e ij i j H Yukawa hierarchy[9]. The fundamental representation in − + (Y M 1Y t ) ¯5 ¯5 5 5 , (1) E is divided into SO(10)(SU(5)) representations as νD νR νD ij i j H H 6 (cid:48) 27 = 16(10 + ¯5 + 1) + 10(5 + ¯5 ) + 1(1). (2) This 27 includes one generation quarks and leptons in 1 In Refs. [6], in explicit GUT mod→els which realize large neu- addition to one pair of vector-like ﬁelds 5 + ¯5 and a sin- trino mixings, the importance of P π0µ+ has been discussed. Actually, in some models in Refs. [6], Γ(P → π0µ+) can be glet. If we introduce three 27i (i = 1, 2, 3) for three gen- comparable to Γ(P → π0e+). eration quarks and leptons, we have six ¯5 ﬁelds. Three of ##LTLine##

text: Nobuhiro

label: author

text: Maekawa

text: Kobayashi

text: Maskawa

text: Institute,

text: Nagoya

text: University;

text: Department

text: of

text: Physics,

text: Nagoya

text: University,

text: Nagoya

text: Japan

text: Yu

text: Muramatsu

text: School

text: of

text: Physics,

text: KIAS,

text: Seoul

text: Korea

text: Recent

label: abstract

text: discovery

text: of

text: neutrino

text: large

text: mixings

text: implies

text: the

text: large

text: mixings

text: in

text: the

text: diagonalizing

text: matrices

text: of

text: ¯5

text: ﬁelds

text: in

text: grand

text: uniﬁed

text: theory

text: (GUT),

text: while

text: the

text: diagonalizing

text: matrices

text: of

text: ﬁelds

text: of

text: are

text: expected

text: to

text: have

text: small

text: mixings

text: like

text: Cabibbo-Kobayashi-Maskawa

text: matrix.

text: We

text: calculate

text: the

text: predictions

text: of

text: ﬂavor

text: changing

text: nucleon

text: decays

text: (FCND)

text: in

text: and

text: E

text: GUT

text: models

text: which

text: have

text: the

text: above

text: features

text: for

text: mixings.

text: We

text: found

text: that

text: FCND

text: can

text: be

text: the

text: main

text: decay

text: mode

text: and

text: play

text: an

text: important

text: role

text: to

text: test

text: GUT

text: models.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: where

text: ﬁelds

text: contain

text: doublet

text: quark

text: q,

text: right-handed

text: up

text: quark

text: uc

text: and

text: right-handed

text: charged

text: lepton

text: ec

text: and

text: R

text: R

text: the

text: last

text: term

text: can

text: be

text: obtained

text: from

text: (Y

text: ¯5

text: One

text: of

text: the

text: most

text: exciting

text: discoveries

text: in

text: elementary

text: νD

text: ij

text: i

text: j

text: H

text: (M

text: by

text: integrating

text: the

text: right-handed

text: neutrino

text: particle

text: physics

text: among

text: the

text: latest

text: years

text: is

text: neutrino

text: νR

text: ij

text: i

text: j

text: ﬁelds

text: Here

text: i

text: is

text: the

text: index

text: for

text: generation,

text: and

text: which

text: leads

text: to

text: massive

text: neutrinos

text: and

text: i

text: Y

text: Y

text: Y

text: and

text: M

text: are

text: Yukawa

text: matrix

text: of

text: up

text: type

text: large

text: neutrino

text: mixing

text: angles.

text: Interestingly,

text: this

text: discov-

text: u

text: d,e

text: νD

text: νR

text: quarks,

text: that

text: of

text: down

text: type

text: quarks

text: and

text: charged

text: leptons,

text: ery

text: gives

text: an

text: evidence

text: of

text: grand

text: uniﬁed

text: theory

text: Dirac

text: neutrino

text: Yukawa

text: matrix,

text: and

text: right-handed

text: neu-

text: in

text: which

text: uniﬁcation

text: of

text: forces

text: and

text: uniﬁcation

text: of

text: quarks

text: trino

text: mass

text: matrix,

text: respectively.

text: These

text: uniﬁed

text: structures

text: and

text: leptons

text: are

text: realized.

text: This

text: evidence

text: for

text: uniﬁcation

text: of

text: for

text: Yukawa

text: interactions

text: are

text: corresponding

text: to

text: the

text: classi-

text: quarks

text: and

text: leptons

text: makes

text: the

text: idea

text: of

text: GUT

text: quite

text: promis-

text: ﬁcation

text: of

text: the

text: hierarchies

text: of

text: the

text: observed

text: quark

text: and

text: lep-

text: ing,

text: because

text: for

text: uniﬁcation

text: of

text: forces

text: we

text: have

text: already

text: ton

text: masses

text: that

text: up

text: type

text: quark

text: masses

text: have

text: the

text: strongest

text: known

text: an

text: experimental

text: evidence

text: that

text: three

text: gauge

text: cou-

text: hierarchy,

text: neutrino

text: masses

text: have

text: the

text: weakest,

text: and

text: down

text: plings

text: meet

text: at

text: a

text: scale,

text: the

text: GUT

text: scale

text: Λ

text: especially

text: G

text: type

text: quark

text: and

text: charged

text: lepton

text: masses

text: have

text: middle

text: hi-

text: in

text: supersymmetric

text: (SUSY)

text: Moreover,

text: the

text: large

text: erarchies

text: if

text: neutrino

text: mass

text: hierarchy

text: is

text: normal

text: (not

text: in-

text: neutrino

text: mixing

text: angles

text: imply

text: not

text: only

text: large

text: mixing

text: an-

text: verted).

text: Moreover,

text: if

text: we

text: assume

text: that

text: ﬁelds

text: induce

text: gles

text: of

text: doublet

text: lepton

text: l

text: but

text: also

text: those

text: of

text: right-handed

text: down

text: quark

text: dc

text: in

text: GUT

text: because

text: ¯5

text: ﬁeld

text: of

text: stronger

text: hierarchy

text: in

text: Yukawa

text: couplings

text: than

text: ¯5

text: ﬁelds,

text: contains

text: l

text: andRdc

text: This

text: suggests

text: an

text: interesting

text: possibil-

text: these

text: various

text: hierarchies

text: for

text: quark

text: and

text: lepton

text: masses

text: R

text: can

text: be

text: explained.

text: Furthermore,

text: this

text: assumption

text: explains

text: ity

text: that

text: the

text: ﬂavor

text: changing

text: processes

text: are

text: seen

text: in

text: nucleon

text: decay

text: which

text: is

text: the

text: most

text: important

text: prediction

text: of

text: GUT

text: that

text: quark

text: mixings

text: are

text: smaller

text: than

text: lepton

text: mixings

text: at

text: the

text: same

text: time

text: if

text: we

text: use

text: a

text: reasonable

text: expectation

text: that

text: the

text: In

text: this

text: paper,

text: we

text: study

text: the

text: ﬂavor

text: changing

text: nucleon

text: decay

text: stronger

text: hierarchy

text: leads

text: to

text: smaller

text: mixings.

text: This

text: bril-

text: and

text: propose

text: that

text: the

text: ﬂavor

text: changing

text: nucleon

text: decay

text: can

text: liant

text: chemistry

text: between

text: the

text: Yukawa

text: structure

text: in

text: be

text: a

text: key

text: observation

text: for

text: GUT.

text: GUT

text: and

text: the

text: observed

text: hierarchies

text: of

text: quark

text: and

text: lepton

text: II.

text: QUALITATIVE

text: EVIDENCE

text: FOR

text: masses

text: and

text: mixings

text: is

text: quite

text: non-trivial,

text: and

text: therefore

text: it

text: UNIFICATION

text: can

text: be

text: regarded

text: as

text: an

text: experimental

text: signature

text: for

text: uniﬁ-

text: cation

text: of

text: quarks

text: and

text: leptons

text: in

text: GUT.

text: First,

text: we

text: explain

text: the

text: qualitative

text: evidence

text: for

text: the

text: uni-

text: III.

text: E

text: UNIFICATION

text: ﬁcation

text: of

text: matters.

text: In

text: GUT,

text: Yukawa

text: interactions

text: with

text: Higgs

text: ﬁelds

text: and

text: ¯5

text: are

text: given

text: by

text: H

text: H

text: E

text: is

text: more

text: attractive

text: because

text: the

text: assump-

text: tion

text: in

text: the

text: GUT

text: can

text: be

text: derived,

text: and

text: as

text: a

text: result,

text: L

text: various

text: Yukawa

text: matrices

text: can

text: be

text: derived

text: from

text: one

text: basic

text: (Y

text: (Y

text: ¯5

text: ¯5

text: Y

text: u

text: ij

text: i

text: j

text: H

text: d,e

text: ij

text: i

text: j

text: H

text: Yukawa

text: The

text: fundamental

text: representation

text: in

text: (Y

text: M

text: t

text: ¯5

text: ¯5

text: E

text: is

text: divided

text: into

text: representations

text: as

text: νD

text: νR

text: νD

text: ij

text: i

text: j

text: H

text: H

text: ¯5

text: ¯5

text: This

text: includes

text: one

text: generation

text: quarks

text: and

text: leptons

text: in

text: In

text: Refs.

text: in

text: explicit

text: GUT

text: mod→els

text: which

text: realize

text: large

text: neu-

text: addition

text: to

text: one

text: pair

text: of

text: vector-like

text: ﬁelds

text: ¯5

text: and

text: a

text: sin-

text: trino

text: mixings,

text: the

text: importance

text: of

text: P

text: π0µ+

text: has

text: been

text: discussed.

text: Actually,

text: in

text: some

text: models

text: in

text: Refs.

text: Γ(P

text: can

text: be

text: glet.

text: If

text: we

text: introduce

text: three

text: (i

text: for

text: three

text: gen-

text: comparable

text: to

text: Γ(P

text: eration

text: quarks

text: and

text: leptons,

text: we

text: have

text: six

text: ¯5

text: ﬁelds.

text: Three

text: of

text: ##LTLine##


text: 14 V. NUMERICAL RESULTS This section presents simulation results to evaluate the PSD, complexity, and BER performance of NC-OFDM, TD-NC-OFDM and proposed low-interference schemes. Simulations are performed in a {− − } baseband-equivalent OFDM system with 256 subcarriers mapped onto the subcarrier index set 128, 127, . . . , 127 . 16-QAM digital modulation is employed with a symbol period T = 1/15ms, time-domain oversampling s interval T = T /2048 and CP duration T = 144T . The PSD is evaluated by Welch’s averaged samp s cp samp periodogram method with a 2048-sample Hanning window and 512-sample overlap after observing 105 symbols. To investigate the BER performance, the signal is transmitted through the Extended Vehicular A (EVA) channel model. Figure 7 compares the PSD of NC-OFDM transmit signals with different N and different L. As N increases, the sidelobe suppression performance is further improved in the three methods. Moreover, the low-interference scheme can obtain as good sidelobe suppression performance as the conventional TD- NC-OFDM and NC-OFDM. Figure 7 also shows that with the increase of L, a steeper spectral roll-off can be obtained in the low-interference scheme. With a relatively small L, the sidelobe suppression of the low-interference scheme can approach that of TD-NC-OFDM, such as N=2 and 3 with L=144. 0 Original OFDM NC-OFDM TD-NC-OFDM -20 Low-interference scheme -40 N=0 -60 N=1 N=2 -80 N=3 -100 N=3, L=36 N=3, L=72 -120 -10 -5 0 5 10 Frequency (MHz) Fig. 7. PSDs of the transmit signals of NC-OFDM, TD-NC-OFDM and its low-interference scheme with varying N and varying L. Figure 8 presents the BER performances of NC-OFDM, TD-NC-OFDM and its low-interference scheme with varying N and varying L in the EVA channel. It is shown that the BER performance of the received May 28, 2020 DRAFT )Bd( DSP ##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##

label: paragraph

text: V.

label: section

text: NUMERICAL

text: RESULTS

text: This

text: section

text: presents

text: simulation

text: results

text: to

text: evaluate

text: the

text: PSD,

text: complexity,

text: and

text: BER

text: performance

text: of

text: NC-OFDM,

text: TD-NC-OFDM

text: and

text: proposed

text: low-interference

text: schemes.

text: Simulations

text: are

text: performed

text: in

text: a

text: baseband-equivalent

text: OFDM

text: system

text: with

text: subcarriers

text: mapped

text: onto

text: the

text: subcarrier

text: index

text: set

text: digital

text: modulation

text: is

text: employed

text: with

text: a

text: symbol

text: period

text: T

text: time-domain

text: oversampling

text: s

text: interval

text: T

text: T

text: and

text: CP

text: duration

text: T

text: The

text: PSD

text: is

text: evaluated

text: by

text: Welch’s

text: averaged

text: samp

text: s

text: cp

text: samp

text: periodogram

text: method

text: with

text: a

text: Hanning

text: window

text: and

text: overlap

text: after

text: observing

text: symbols.

text: To

text: investigate

text: the

text: BER

text: performance,

text: the

text: signal

text: is

text: transmitted

text: through

text: the

text: Extended

text: Vehicular

text: A

text: (EVA)

text: channel

text: model.

text: Figure

text: compares

text: the

text: PSD

text: of

text: NC-OFDM

text: transmit

text: signals

text: with

text: different

text: N

text: and

text: different

text: L.

text: As

text: N

text: increases,

text: the

text: sidelobe

text: suppression

text: performance

text: is

text: further

text: improved

text: in

text: the

text: three

text: methods.

text: Moreover,

text: the

text: low-interference

text: scheme

text: can

text: obtain

text: as

text: good

text: sidelobe

text: suppression

text: performance

text: as

text: the

text: conventional

text: TD-

text: NC-OFDM

text: and

text: NC-OFDM.

text: Figure

text: also

text: shows

text: that

text: with

text: the

text: increase

text: of

text: L,

text: a

text: steeper

text: spectral

text: roll-off

text: can

text: be

text: obtained

text: in

text: the

text: low-interference

text: scheme.

text: With

text: a

text: relatively

text: small

text: L,

text: the

text: sidelobe

text: suppression

text: of

text: the

text: low-interference

text: scheme

text: can

text: approach

text: that

text: of

text: TD-NC-OFDM,

text: such

text: as

text: and

text: with

text: Original

text: OFDM

text: NC-OFDM

text: TD-NC-OFDM

text: Low-interference

text: scheme

text: Frequency

text: (MHz)

text: Fig.

text: PSDs

text: of

text: the

text: transmit

text: signals

text: of

text: NC-OFDM,

text: TD-NC-OFDM

text: and

text: its

text: low-interference

text: scheme

text: with

text: varying

text: N

text: and

text: varying

text: L.

text: Figure

text: presents

text: the

text: BER

text: performances

text: of

text: NC-OFDM,

text: TD-NC-OFDM

text: and

text: its

text: low-interference

text: scheme

text: with

text: varying

text: N

text: and

text: varying

text: L

text: in

text: the

text: EVA

text: channel.

text: It

text: is

text: shown

text: that

text: the

text: BER

text: performance

text: of

text: the

text: received

text: May

text: DRAFT

text: )Bd(

text: DSP

text: ##LTFigure##

text: ##LTLine##

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text: 16 with much better BER performance and much lower complexity. In this sense, the low-interference TD- NC-OFDM is a promising alternative to conventional NC-OFDM in future cognitive radio and carrier aggregation combined 5G systems. ACKNOWLEDGMENT This work was supported by the open research fund of National Mobile Communications Research (cid:8) (cid:8) (cid:9)(cid:9) Laboratory, Southeast University (No. 2013D05). APPENDIX: DERIVATION OF Tr E w˜ Hw˜ i i For ease of exposition, Eq. (13) is rewritten as − − b = P 1 (P x − P x ) (25) i f˜ 1 i 1 2 i (cid:8) (cid:8) (cid:9)(cid:9) where P and P include a row of n = 0. 1 2 According to the construction of P and P , Tr E w˜ Hw˜ can be expressed as 1 2 i i (cid:110) (cid:110) (cid:111)(cid:111) (cid:8) (cid:8) (cid:9)(cid:9) (cid:26) (cid:26) (cid:27)(cid:27) (cid:16) (cid:17) Tr E w˜ Hw˜ = Tr E Q b bHQH i i f˜ i i f˜ (cid:26) (cid:26) − (cid:16) −(cid:17) H (cid:27)(cid:27) = Tr E Q P 1P x − xH− PH P 1 QH f˜ f˜ 1 i 1 i 1 1 f˜ f˜ (cid:26) −(cid:16) (cid:17) (cid:27)− H (cid:26) (cid:16) (cid:17) (cid:27) +Tr E Q P 1P x xHPH P 1 QH f˜ f˜ 2 i i 2 f˜ f˜ − − H − − H = Tr P 1P PH P 1 QHQ + Tr P 1P PH P 1 QHQ (.26) f˜ 1 1 f˜ f˜ f˜ f˜ 2 2 f˜ f˜ f˜ { } Then, we rewrite P and P as P = 1/MB and P = 1/MB Φ with B = (j2πk /M)n. 1 2 1 2 2 2 2 n+1,r+1 r We obtain B = ΦHBT , (27) 1 2 { } − with B1 r+1,n+1 = (j2πkr/M)n e jϕkr. According to (27), we arrive at (cid:26) (cid:27) (cid:26) (cid:27) (cid:16) (cid:17) (cid:16) (cid:17) (cid:8) (cid:8) (cid:9)(cid:9) 1 − − H 1 − − H Tr E w˜ Hw˜ = Tr P 1B BH P 1 QHQ + Tr P 1B ΦΦHBH P 1 QHQ i i M2 f˜ 2 2 f˜ f˜ f˜ M2 f˜ 2 2 f˜ f˜ f˜ May 28, 2020 DRAFT ##LTLine## ##LTLine##

text: with

text: much

text: better

text: BER

text: performance

text: and

text: much

text: lower

text: complexity.

text: In

text: this

text: sense,

text: the

text: low-interference

text: TD-

text: NC-OFDM

text: is

text: a

text: promising

text: alternative

text: to

text: conventional

text: NC-OFDM

text: in

text: future

text: cognitive

text: radio

text: and

text: carrier

text: aggregation

text: combined

text: systems.

text: ACKNOWLEDGMENT

text: This

text: work

text: was

text: supported

text: by

text: the

text: open

text: research

text: fund

text: of

text: National

text: Mobile

text: Communications

text: Research

text: Laboratory,

text: Southeast

text: University

text: (No.

text: APPENDIX:

text: DERIVATION

text: OF

text: Tr

text: E

text: w˜

text: Hw˜

text: i

text: i

text: For

text: ease

text: of

text: exposition,

text: Eq.

text: is

text: rewritten

text: as

label: equation

text: b

text: P

text: (P

text: x

text: P

text: x

text: i

text: f˜

text: i

text: i

text: where

text: P

text: and

text: P

text: include

text: a

text: row

text: of

text: n

text: According

text: to

text: the

text: construction

text: of

text: P

text: and

text: P

text: Tr

text: E

text: w˜

text: Hw˜

text: can

text: be

text: expressed

text: as

text: i

text: i

text: Tr

text: E

text: w˜

text: Hw˜

text: Tr

text: E

text: Q

text: b

text: bHQH

text: i

text: i

text: f˜

text: i

text: i

text: f˜

text: H

text: Tr

text: E

text: Q

text: P

text: x

text: xH−

text: PH

text: P

text: QH

text: f˜

text: f˜

text: i

text: i

text: f˜

text: f˜

text: H

text: +Tr

text: E

text: Q

text: P

text: x

text: xHPH

text: P

text: QH

text: f˜

text: f˜

text: i

text: i

text: f˜

text: f˜

text: H

text: H

text: Tr

text: P

text: PH

text: P

text: QHQ

text: Tr

text: P

text: PH

text: P

text: QHQ

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: Then,

text: we

text: rewrite

text: P

text: and

text: P

text: as

text: P

text: and

text: P

text: Φ

text: with

text: B

text: /M)n.

text: r

text: We

text: obtain

text: B

text: ΦHBT

text: with

text: e

text: jϕkr.

text: According

text: to

text: we

text: arrive

text: at

text: H

text: H

text: Tr

text: E

text: w˜

text: Hw˜

text: Tr

text: P

text: BH

text: P

text: QHQ

text: Tr

text: P

text: ΦΦHBH

text: P

text: QHQ

text: i

text: i

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: f˜

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4 TABLE I: Details of DeepLab-LargeFOV network architecture. layer #channel kernel size stride zero-padding size hole size training map size receptive ﬁeld size #weight input image 3 - - - - 513*513 435*435 - conv1 1 64 3*3 1*1 1*1 - 513*513 433*433 1.75K conv1 2 64 3*3 1*1 1*1 - 513*513 431*431 36K pool1 64 3*3 2*2 1*1 - 257*257 215*215 - conv2 1 128 3*3 1*1 1*1 - 257*257 213*213 72K conv2 2 128 3*3 1*1 1*1 - 257*257 211*211 144K pool2 128 3*3 2*2 1*1 - 129*129 105*105 - conv3 1 256 3*3 1*1 1*1 - 129*129 103*103 288K conv3 2 256 3*3 1*1 1*1 - 129*129 101*101 576K conv3 3 256 3*3 1*1 1*1 - 129*129 99*99 576K pool3 256 3*3 2*2 1*1 - 65*65 49*49 - conv4 1 512 3*3 1*1 1*1 - 65*65 47*47 1.13M conv4 2 512 3*3 1*1 1*1 - 65*65 45*45 2.25M conv4 3 512 3*3 1*1 1*1 - 65*65 43*43 2.25M pool4 512 3*3 1*1 1*1 - 65*65 41*41 - conv5 1 512 3*3 1*1 2*2 2*2 65*65 37*37 2.25M conv5 2 512 3*3 1*1 2*2 2*2 65*65 33*33 2.25M conv5 3 512 3*3 1*1 2*2 2*2 65*65 29*29 2.25M pool5 512 3*3 1*1 1*1 - 65*65 27*27 - pool5a 512 3*3 1*1 1*1 - 65*65 25*25 - fc6 1024 3*3 1*1 12*12 12*12 65*65 1*1 4.5M fc7 1024 1*1 1*1 - - 65*65 1*1 1M architecture of SPOP-net and then elaborate on how to train However, due to the possible spatial displacement (e.g. two and apply the SPOP-net. exactly the same objects could appear at different locations in a) Architecture: Our SPOP-net is built upon a an image), accurately predicting the absolute object bounding pre-trained DeepLab-LargeFOV segmentation network [17]. box coordinates is difﬁcult. It is because these two objects have Its architecture is shown in Table I. The receptive ﬁeld of the same visual input for the model, but their locations the × our localization network in the last layer is 435 435. This model needs to learn to predict are totally different. To solve large receptive ﬁeld enables SPOP-net to “see” a large this issue, for each pixel, we change its learning targets from region of the image in its last layer and predict the object the absolute object bounding box coordinates to the offsets bounding boxes effectively. from the pixel to the object bounding box. E.g. for object b) Training: For each pixel, the pixel-wise localization bounding box coordinate x /w, we change the target from − min network aims to predict the bounding box coordinates xmin/w to (xmin xself)/w, here xself is the x coordinate of t = (x /w, y /h, x /w, y /h) of the object that the pixel itself. Changing the coordinates to offsets can be min min max max contains this pixel. Here (x , y ) and (x , y ) conveniently achieved by element-wisely summing the output min min max max denote the coordinates of the top-left and bottom-right of the 2nd last layer and the spatial coordinate map (x or y corners of the object bounding box containing the pixel; h values of all the pixels themselves). Then the absolute object and w represent the height and the width of the image plane bounding box coordinates can be used as learning targets for respectively. Therefore, for a single object, all the pixels the ﬁnal layer. In this way, applying the absolute coordinates inside this object are given the same ground-truth values lea(cid:18)rning targets to the ﬁnal layer is equivalent to applying(cid:19)the (x /w, y /h, x /w, y /h). We train the pixel-wise following object coordinate offsets to the 2nd last layer. min min max max localization network to minimize the following localization − − − − error L that is proportional to the Euclidean distance xmin xself , ymin yself , xmax xself , ymax yself w h w h between the predicted coordinate vector t and the ∗ i ground-truth coordinate vector t for all the foreground Then we can directly obtain the absolute object proposal (cid:88)L i pixels. The loss function is deﬁned as coordinates from the predictions of the ﬁnal layer. After obtaining the output map from the ﬁnal layer having a L ∗(cid:107) − ∗(cid:107) = p t t 2, (1) smaller size than the input image, all the subsequent i i i procedures (e.g. reﬁnement, ranking and NMS) are only i based on the output map of smaller size. Because we just ∗where ti is the predicted 4-d object coordinate vector, and leverage pixel-level prediction of proposals for having higher p is a binary variable indicating whether the pixel i is a chance to hit the ground-truth objects accurately instead of i foreground one: it takes 1 if the pixel i is from a foreground doing pixel-level classiﬁcation as DeepLab. If resizing the ∗ object and 0 otherwise. Such a ﬁltered loss (through p ) smaller output map back into the original size, the i enables the localization network to concentrate on localizing subsequent reﬁnement, ranking and NMS steps will bring foreground objects without being distracted by background much higher computation burden but not signiﬁcant pixels in the training phase. In the practical implementation, performance improvement. as the ﬁnal layer has smaller size than the input image, we resize the ground-truth coordinate map to the same small B. Scale-aware Localization size as the ﬁnal layer. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: JOURNAL

text: OF

text: LATEX

text: CLASS

text: FILES,

text: VOL.

text: NO.

text: AUGUST

text: TABLE

text: I:

text: Details

text: of

text: DeepLab-LargeFOV

text: network

text: architecture.

text: layer

label: table

text: #channel

text: kernel

text: size

text: stride

text: zero-padding

text: size

text: hole

text: size

text: training

text: map

text: size

text: receptive

text: ﬁeld

text: size

text: #weight

text: input

text: image

text: architecture

text: of

text: SPOP-net

text: and

text: then

text: elaborate

text: on

text: how

text: to

text: train

text: However,

text: due

text: to

text: the

text: possible

text: spatial

text: displacement

text: (e.g.

text: two

text: and

text: apply

text: the

text: SPOP-net.

text: exactly

text: the

text: same

text: objects

text: could

text: appear

text: at

text: different

text: locations

text: in

text: a)

text: Architecture:

text: Our

text: SPOP-net

text: is

text: built

text: upon

text: a

text: an

text: image),

text: accurately

text: predicting

text: the

text: absolute

text: object

text: bounding

text: pre-trained

text: DeepLab-LargeFOV

text: segmentation

text: network

text: box

text: coordinates

text: is

text: difﬁcult.

text: It

text: is

text: because

text: these

text: two

text: objects

text: have

text: Its

text: architecture

text: is

text: shown

text: in

text: Table

text: I.

text: The

text: receptive

text: ﬁeld

text: of

text: the

text: same

text: visual

text: input

text: for

text: the

text: model,

text: but

text: their

text: locations

text: the

text: our

text: localization

text: network

text: in

text: the

text: last

text: layer

text: is

text: This

text: model

text: needs

text: to

text: learn

text: to

text: predict

text: are

text: totally

text: different.

text: To

text: solve

text: large

text: receptive

text: ﬁeld

text: enables

text: SPOP-net

text: to

text: “see”

text: a

text: large

text: this

text: issue,

text: for

text: each

text: pixel,

text: we

text: change

text: its

text: learning

text: targets

text: from

text: region

text: of

text: the

text: image

text: in

text: its

text: last

text: layer

text: and

text: predict

text: the

text: object

text: the

text: absolute

text: object

text: bounding

text: box

text: coordinates

text: to

text: the

text: offsets

text: bounding

text: boxes

text: effectively.

text: from

text: the

text: pixel

text: to

text: the

text: object

text: bounding

text: box.

text: E.g.

text: for

text: object

text: b)

text: Training:

text: For

text: each

text: pixel,

text: the

text: pixel-wise

text: localization

text: bounding

text: box

text: coordinate

text: x

text: /w,

text: we

text: change

text: the

text: target

text: from

text: min

text: network

text: aims

text: to

text: predict

text: the

text: bounding

text: box

text: coordinates

text: xmin/w

text: to

text: (xmin

text: xself)/w,

text: here

text: xself

text: is

text: the

text: x

text: coordinate

text: of

text: t

text: (x

text: /w,

text: y

text: /h,

text: x

text: /w,

text: y

text: /h)

text: of

text: the

text: object

text: that

text: the

text: pixel

text: itself.

text: Changing

text: the

text: coordinates

text: to

text: offsets

text: can

text: be

text: min

text: min

text: max

text: max

text: contains

text: this

text: pixel.

text: Here

text: (x

text: y

text: and

text: (x

text: y

text: conveniently

text: achieved

text: by

text: element-wisely

text: summing

text: the

text: output

text: min

text: min

text: max

text: max

text: denote

text: the

text: coordinates

text: of

text: the

text: top-left

text: and

text: bottom-right

text: of

text: the

text: last

text: layer

text: and

text: the

text: spatial

text: coordinate

text: map

text: (x

text: or

text: y

text: corners

text: of

text: the

text: object

text: bounding

text: box

text: containing

text: the

text: pixel;

text: h

text: values

text: of

text: all

text: the

text: pixels

text: themselves).

text: Then

text: the

text: absolute

text: object

text: and

text: w

text: represent

text: the

text: height

text: and

text: the

text: width

text: of

text: the

text: image

text: plane

text: bounding

text: box

text: coordinates

text: can

text: be

text: used

text: as

text: learning

text: targets

text: for

text: respectively.

text: Therefore,

text: for

text: a

text: single

text: object,

text: all

text: the

text: pixels

text: the

text: ﬁnal

text: layer.

text: In

text: this

text: way,

text: applying

text: the

text: absolute

text: coordinates

text: inside

text: this

text: object

text: are

text: given

text: the

text: same

text: ground-truth

text: values

text: targets

text: to

text: the

text: ﬁnal

text: layer

text: is

text: equivalent

text: to

text: (x

text: /w,

text: y

text: /h,

text: x

text: /w,

text: y

text: /h).

text: We

text: train

text: the

text: pixel-wise

text: following

text: object

text: coordinate

text: offsets

text: to

text: the

text: last

text: layer.

text: min

text: min

text: max

text: max

text: localization

text: network

text: to

text: minimize

text: the

text: following

text: localization

text: error

text: L

text: that

text: is

text: proportional

text: to

text: the

text: Euclidean

text: distance

text: xmin

text: xself

text: ymin

text: yself

text: xmax

text: xself

text: ymax

text: yself

text: w

text: h

text: w

text: h

text: between

text: the

text: predicted

text: coordinate

text: vector

text: t

text: and

text: the

text: i

text: ground-truth

text: coordinate

text: vector

text: t

text: for

text: all

text: the

text: foreground

text: Then

text: we

text: can

text: directly

text: obtain

text: the

text: absolute

text: object

text: proposal

text: i

text: pixels.

text: The

text: loss

text: function

text: is

text: deﬁned

text: as

text: coordinates

text: from

text: the

text: predictions

text: of

text: the

text: ﬁnal

text: layer.

text: After

text: obtaining

text: the

text: output

text: map

text: from

text: the

text: ﬁnal

text: layer

text: having

text: a

text: L

text: p

text: t

text: t

text: smaller

text: size

text: than

text: the

text: input

text: image,

text: all

text: the

text: subsequent

text: i

text: i

text: i

text: procedures

text: (e.g.

text: reﬁnement,

text: ranking

text: and

text: NMS)

text: are

text: only

text: i

text: based

text: on

text: the

text: output

text: map

text: of

text: smaller

text: size.

text: Because

text: we

text: just

text: ∗where

text: ti

text: is

text: the

text: predicted

text: object

text: coordinate

text: vector,

text: and

text: leverage

text: pixel-level

text: prediction

text: of

text: proposals

text: for

text: having

text: higher

text: p

text: is

text: a

text: binary

text: variable

text: indicating

text: whether

text: the

text: pixel

text: i

text: is

text: a

text: chance

text: to

text: hit

text: the

text: ground-truth

text: objects

text: accurately

text: instead

text: of

text: i

text: foreground

text: one:

text: it

text: takes

text: if

text: the

text: pixel

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text: from

text: a

text: foreground

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text: pixel-level

text: classiﬁcation

text: as

text: DeepLab.

text: If

text: resizing

text: the

text: object

text: and

text: otherwise.

text: Such

text: a

text: ﬁltered

text: loss

text: (through

text: p

text: smaller

text: output

text: map

text: back

text: into

text: the

text: original

text: size,

text: the

text: i

text: enables

text: the

text: localization

text: network

text: to

text: concentrate

text: on

text: localizing

text: subsequent

text: reﬁnement,

text: ranking

text: and

text: NMS

text: steps

text: will

text: bring

text: foreground

text: objects

text: without

text: being

text: distracted

text: by

text: background

text: much

text: higher

text: computation

text: burden

text: but

text: not

text: signiﬁcant

text: pixels

text: in

text: the

text: training

text: phase.

text: In

text: the

text: practical

text: implementation,

text: performance

text: improvement.

text: as

text: the

text: ﬁnal

text: layer

text: has

text: smaller

text: size

text: than

text: the

text: input

text: image,

text: we

text: resize

text: the

text: ground-truth

text: coordinate

text: map

text: to

text: the

text: same

text: small

text: B.

text: Scale-aware

text: Localization

text: size

text: as

text: the

text: ﬁnal

text: layer.

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text: ≈ features in factor model W FFT are of low rank. After viously logdet(X + δI) was suggested to restrict the rank obtaining W, the recommendation score for user i about an of positive semideﬁnite matrix X (Fazel, Hindi, and Boyd un-purchased/-rated item j is xˆ = xT w , where x = 0, 2003), which is not guaranteed for more general X, and also ij i j ij xT is the i-th row of X, and w is the j-th column of W. δ is required to be small, which leads to signiﬁcantly biased Tihus Xˆ = XW. LorSLIM can mj odel the relations between approximation for small singular values. Compared to some other nonconvex relaxations in the literature (Lu et al. 2014), items even on sparse datasets and thus improves the perfor- our formulation enjoys the simplicity and efﬁcacy. mance. Methodology To further boost the accuracy of Top-N recommender sys- tems, we ﬁrst ﬁll the missing ratings by solving a noncon- Considering that the user-item matrix is often nonnega- ≥ vex optimization problem, based on the assumption that the tive, we add nonnegative constraint X 0, i.e., element- user’ ratings are affected by only a few factors and the re- wise positivity, for easy interpretation of the representation. P sulting rating matrix should be of low rank (Lee et al. 2014), Let be the orthogonal projection operator onto the span and then make the Top-N recommendation. This is differ- of matΩrices vanishing o(cid:26)utside of Ω (i.e., Ωc) so that ent from previous approaches: Middle values of the rating ranges, or the average user or item ratings are commonly uti- ∈ P X , if X Ω; lized to ﬁll out the missing ratings (Breese, Heckerman, and ( (X)) = ij ij ∈ Ω ij 0, if X Ωc. Kadie 1998; Deshpande and Karypis 2004); a more reliable ij approach utilizes content information (Melville, Mooney, Problem (1) can be reformulated as and Nagarajan 2002; Li and Za¨ıane 2004; Degemmis, Lops, and Semeraro 2007), for example, the missing ratings are provided by autonomous agents called ﬁlterbots (Good et al. min logdet((XT X)1/2 + I) + lR (X) 1999), which rate items based on some speciﬁc character- X P P + (2) istics of their content; a low rank matrix factorization ap- s.t. Ω(X) = Ω(M), proach seeks to approximate X by a multiplication of low where lR is the indicator function, deﬁned element- rank factors (Yu et al. 2009). Experimental results demon- wisely as + (cid:26) strate the superior recommendation quality of our approach. Due to the inherent computational complexity of rank ≥ problems, the non-convex rank function is often relaxed 0, if x 0; lR (x) = ∞ to its convex surrogate, i.e. the nuclear norm (Cande`s and + + , otherwise. Recht 2009; Recht, Xu, and Hassibi 2008). However, this substitution is not always valid and can lead to a biased so- Notice that this is a nonconvex optimization problem, lution (Shi and Yu 2011; Kang, Peng, and Cheng 2015b). which is not easy to solve in general. Here we develop Matrix completion with nuclear norm regularization can be an effective optimization strategy based on augmented La- signiﬁcantly hurt when entries of the matrix are sampled grangian multiplier (ALM) method. By introducing an aux- non-uniformly (Srebro and Salakhutdinov 2010). Noncon- iliary variable Y , it has the following equivalent form vex rank approximation has received signiﬁcant attention (Zhong et al. 2015; Kang and Cheng 2015). Thus we use min logdet((XT X)1/2 + I) + lR (Y ) log-determinant (logdet) function to approximate the rank X,Y + (3) P P function and design an effective optimization algorithm. s.t. (X) = (M), X = Y, Ω Ω Problem Formulation which has an augmented Lagrangian function of the form The incomplete user-item purchases/ratings matrix is de- × noted as M of size m n. Mij is 1 or a positive value if user L i has ever purchased/rated item j; otherwise it is 0. Our goal (X, Y, Z) = logdet((XT X)1/2 + I) + lR (Y )+ + is to reconstruct a full matrix X, which is supposed to be µ(cid:107) − Z (cid:107) P P (4) low-rank. Consider the following matrix completion prob- 2 X Y + µ 2F s.t. Ω(X) = Ω(M), lem: where Z is a Lagrange multiplier and µ > 0 is a penalty parameter. min logdet((XT X)1/2 + I) Then, we can apply the alternating minimization idea to X ∈ (1) s.t. X = M , (i, j) Ω, update X, Y , i.e., updating one of the two variables with the ij ij other ﬁxed. where Ω is the set of locations corresponding to the ob- Given the current point Xt, Y t, Zt, we update Xt+1 by served entries and I ∈ Rn×n is an identity matrix. It is easy solving ≤ (cid:107) (cid:107) to show that logdet((XT X)1/2 +I) X ∗, i.e., logdet is a tighter rank approximation function than the nuclear norm. Xt+1 = arg min logdet((XT X)1/2 + I)+ logdet also helps mitigate another inherent disadvantage of X (5) the nuclear norm, i.e., the imbalanced penalization of dif- µt (cid:107) − Zt (cid:107) ferent singular values (Kang, Peng, and Cheng 2015a). Pre- 2 X Y t + µt 2F ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: features

text: in

text: factor

text: model

text: W

text: FFT

text: are

text: of

text: low

text: rank.

text: After

text: viously

text: logdet(X

text: δI)

text: was

text: suggested

text: to

text: restrict

text: the

text: rank

text: obtaining

text: W,

text: the

text: recommendation

text: score

text: for

text: user

text: i

text: about

text: an

text: of

text: positive

text: semideﬁnite

text: matrix

text: X

text: (Fazel,

text: Hindi,

text: and

text: Boyd

text: un-purchased/-rated

text: item

text: j

text: is

text: xˆ

text: xT

text: w

text: where

text: x

text: which

text: is

text: not

text: guaranteed

text: for

text: more

text: general

text: X,

text: and

text: also

text: ij

text: i

text: j

text: ij

text: xT

text: is

text: the

text: i-th

text: row

text: of

text: X,

text: and

text: w

text: is

text: the

text: j-th

text: column

text: of

text: W.

text: δ

text: is

text: required

text: to

text: be

text: small,

text: which

text: leads

text: to

text: signiﬁcantly

text: biased

text: Tihus

text: Xˆ

text: XW.

text: LorSLIM

text: can

text: mj

text: odel

text: the

text: relations

text: between

text: approximation

text: for

text: small

text: singular

text: values.

text: Compared

text: to

text: some

text: other

text: nonconvex

text: relaxations

text: in

text: the

text: literature

text: (Lu

text: et

text: al.

text: items

text: even

text: on

text: sparse

text: datasets

text: and

text: thus

text: improves

text: the

text: perfor-

text: our

text: formulation

text: enjoys

text: the

text: simplicity

text: and

text: efﬁcacy.

text: mance.

text: Methodology

text: To

text: further

text: boost

text: the

text: accuracy

text: of

text: Top-N

text: recommender

text: sys-

text: tems,

text: we

text: ﬁrst

text: ﬁll

text: the

text: missing

text: ratings

text: by

text: solving

text: a

text: noncon-

text: Considering

text: that

text: the

text: user-item

text: matrix

text: is

text: often

text: nonnega-

text: vex

text: optimization

text: problem,

text: based

text: on

text: the

text: assumption

text: that

text: the

text: tive,

text: we

text: add

text: nonnegative

text: constraint

text: X

text: i.e.,

text: element-

text: user’

text: ratings

text: are

text: affected

text: by

text: only

text: a

text: few

text: factors

text: and

text: the

text: re-

text: wise

text: positivity,

text: for

text: easy

text: interpretation

text: of

text: the

text: representation.

text: P

text: sulting

text: rating

text: matrix

text: should

text: be

text: of

text: low

text: rank

text: (Lee

text: et

text: al.

text: Let

text: be

text: the

text: orthogonal

text: projection

text: operator

text: onto

text: the

text: span

text: and

text: then

text: make

text: the

text: Top-N

text: recommendation.

text: This

text: is

text: differ-

text: of

text: matΩrices

text: vanishing

text: of

text: Ω

text: (i.e.,

text: Ωc)

text: so

text: that

text: ent

text: from

text: previous

text: approaches:

text: Middle

text: values

text: of

text: the

text: rating

text: ranges,

text: or

text: the

text: average

text: user

text: or

text: item

text: ratings

text: are

text: commonly

text: uti-

text: P

text: X

text: if

text: X

text: Ω;

text: lized

text: to

text: ﬁll

text: out

text: the

text: missing

text: ratings

text: (Breese,

text: Heckerman,

text: and

text: (X))

text: ij

text: ij

text: Ω

text: ij

text: if

text: X

text: Ωc.

text: Kadie

text: Deshpande

text: and

text: Karypis

text: a

text: more

text: reliable

text: ij

text: approach

text: utilizes

text: content

text: information

text: (Melville,

text: Mooney,

text: Problem

text: can

text: be

text: reformulated

text: as

text: and

text: Nagarajan

text: Li

text: and

text: Za¨ıane

text: Degemmis,

text: Lops,

text: and

text: Semeraro

text: for

text: example,

text: the

text: missing

text: ratings

text: are

text: provided

text: by

text: autonomous

text: agents

text: called

text: ﬁlterbots

text: (Good

text: et

text: al.

text: min

text: logdet((XT

text: I)

text: lR

text: (X)

text: which

text: rate

text: items

text: based

text: on

text: some

text: speciﬁc

text: character-

text: X

text: P

text: P

text: istics

text: of

text: their

text: content;

text: a

text: low

text: rank

text: matrix

text: factorization

text: ap-

text: s.t.

text: Ω(X)

text: Ω(M),

text: proach

text: seeks

text: to

text: approximate

text: X

text: by

text: a

text: multiplication

text: of

text: low

text: where

text: lR

text: is

text: the

text: indicator

text: function,

text: deﬁned

text: element-

text: rank

text: factors

text: (Yu

text: et

text: al.

text: Experimental

text: results

text: demon-

text: wisely

text: as

text: strate

text: the

text: superior

text: recommendation

text: quality

text: of

text: our

text: approach.

text: Due

text: to

text: the

text: inherent

text: computational

text: complexity

text: of

text: rank

text: problems,

text: the

text: non-convex

text: rank

text: function

text: is

text: often

text: relaxed

text: if

text: x

text: lR

text: (x)

text: to

text: its

text: convex

text: surrogate,

text: i.e.

text: the

text: nuclear

text: norm

text: (Cande`s

text: and

text: otherwise.

text: Recht

text: Recht,

text: Xu,

text: and

text: Hassibi

text: However,

text: this

text: substitution

text: is

text: not

text: always

text: valid

text: and

text: can

text: lead

text: to

text: a

text: biased

text: so-

text: Notice

text: that

text: this

text: is

text: a

text: nonconvex

text: optimization

text: problem,

text: lution

text: (Shi

text: and

text: Yu

text: Kang,

text: Peng,

text: and

text: Cheng

text: which

text: is

text: not

text: easy

text: to

text: solve

text: in

text: general.

text: Here

text: we

text: develop

text: Matrix

text: completion

text: with

text: nuclear

text: norm

text: regularization

text: can

text: be

text: an

text: effective

text: optimization

text: strategy

text: based

text: on

text: augmented

text: La-

text: signiﬁcantly

text: hurt

text: when

text: entries

text: of

text: the

text: matrix

text: are

text: sampled

text: grangian

text: multiplier

text: (ALM)

text: method.

text: By

text: introducing

text: an

text: aux-

text: non-uniformly

text: (Srebro

text: and

text: Salakhutdinov

text: Noncon-

text: iliary

text: variable

text: Y

text: it

text: has

text: the

text: following

text: equivalent

text: form

text: vex

text: rank

text: approximation

text: has

text: received

text: signiﬁcant

text: attention

text: (Zhong

text: et

text: al.

text: Kang

text: and

text: Cheng

text: Thus

text: we

text: use

text: min

text: logdet((XT

text: I)

text: lR

text: (Y

text: log-determinant

text: (logdet)

text: function

text: to

text: approximate

text: the

text: rank

text: X,Y

text: P

text: P

text: function

text: and

text: design

text: an

text: effective

text: optimization

text: algorithm.

text: s.t.

text: (X)

text: (M),

text: X

text: Y,

text: Ω

text: Ω

text: Problem

text: Formulation

text: which

text: has

text: an

text: augmented

text: Lagrangian

text: function

text: of

text: the

text: form

text: The

text: incomplete

text: user-item

text: purchases/ratings

text: matrix

text: is

text: de-

text: noted

text: as

text: M

text: of

text: size

text: m

text: n.

text: Mij

text: is

text: or

text: a

text: positive

text: value

text: if

text: user

text: L

text: i

text: has

text: ever

text: purchased/rated

text: item

text: j;

text: otherwise

text: it

text: is

text: Our

text: goal

text: (X,

text: Y,

text: Z)

text: logdet((XT

text: I)

text: lR

text: (Y

text: is

text: to

text: reconstruct

text: a

text: full

text: matrix

text: X,

text: which

text: is

text: supposed

text: to

text: be

text: Z

text: P

text: P

text: low-rank.

text: Consider

text: the

text: following

text: matrix

text: completion

text: prob-

text: X

text: Y

text: µ

text: s.t.

text: Ω(X)

text: Ω(M),

text: lem:

text: where

text: Z

text: is

text: a

text: Lagrange

text: multiplier

text: and

text: µ

text: is

text: a

text: penalty

text: parameter.

text: min

text: logdet((XT

text: I)

text: Then,

text: we

text: can

text: apply

text: the

text: alternating

text: minimization

text: idea

text: to

text: X

text: s.t.

text: X

text: M

text: (i,

text: j)

text: Ω,

text: update

text: X,

text: Y

text: i.e.,

text: updating

text: one

text: of

text: the

text: two

text: variables

text: with

text: the

text: ij

text: ij

text: other

text: ﬁxed.

text: where

text: Ω

text: is

text: the

text: set

text: of

text: locations

text: corresponding

text: to

text: the

text: ob-

text: Given

text: the

text: current

text: point

text: Xt,

text: Y

text: t,

text: Zt,

text: we

text: update

text: by

text: served

text: entries

text: and

text: I

text: Rn×n

text: is

text: an

text: identity

text: matrix.

text: It

text: is

text: easy

text: solving

text: to

text: show

text: that

text: logdet((XT

text: +I)

text: X

text: i.e.,

text: logdet

text: is

text: a

text: tighter

text: rank

text: approximation

text: function

text: than

text: the

text: nuclear

text: norm.

text: arg

text: min

text: logdet((XT

text: I)+

text: logdet

text: also

text: helps

text: mitigate

text: another

text: inherent

text: disadvantage

text: of

text: X

text: the

text: nuclear

text: norm,

text: i.e.,

text: the

text: imbalanced

text: penalization

text: of

text: dif-

text: µt

text: Zt

text: ferent

text: singular

text: values

text: (Kang,

text: Peng,

text: and

text: Cheng

text: Pre-

text: X

text: Y

text: t

text: µt

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: al.,

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text: Phys.

text: I.

text: Hamamoto,

text: Phys.

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text: D.

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text: al.,

text: Nucl.

text: Phys.

text: G.

text: B.

text: Hagemann,

text: private

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text: H.

text: Schnack-Petersen

text: et

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text: 4 Neutron one-particle levels in Woods-Saxon potential R = 3.266 fm (A = 17) a = 0.67 fm 2 0 -2 -4 16 1d3/2 -6 8 -8 2s1/2 1d5/2 -10 2p1/2 -12 2p3/2 -14 -70 -65 -60 -55 -50 -45 -40 depth of Woods-Saxon potential Fig. 6: Calculated neutron one-particle energies as a function of the depth of spherical Woods-Saxon potentials. The (cid:96)=2 one-particle resonant level continues to be well deﬁned up to ε = 2 MeV, while there is no one-particle resonance for (cid:96)=0. The parameters of the Woods-Saxon potential except for the depth are kept constant and are designed approximately for the nucleus − 17C, for which the realistic depth should be around 40 MeV. The ﬁgure is taken from Ref. [47]. 16 ε )VeM( ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Neutron

text: one-particle

text: levels

text: in

text: Woods-Saxon

text: potential

text: R

text: fm

text: (A

text: a

text: fm

text: depth

text: of

text: Woods-Saxon

text: potential

text: Fig.

text: Calculated

text: neutron

text: one-particle

text: energies

text: as

text: a

text: function

text: of

text: the

text: depth

text: of

text: spherical

text: Woods-Saxon

text: potentials.

text: The

text: one-particle

text: resonant

text: level

text: continues

text: to

text: be

text: well

text: deﬁned

text: up

text: to

text: ε

text: MeV,

text: while

text: there

text: is

text: no

text: one-particle

text: resonance

text: for

text: The

text: parameters

text: of

text: the

text: Woods-Saxon

text: potential

text: except

text: for

text: the

text: depth

text: are

text: kept

text: constant

text: and

text: are

text: designed

text: approximately

text: for

text: the

text: nucleus

text: for

text: which

text: the

text: realistic

text: depth

text: should

text: be

text: around

text: MeV.

text: The

text: ﬁgure

text: is

text: taken

text: from

text: Ref.

text: ε

text: )VeM(

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text: strong anharmonicity in both the energy and the E3 tran- sition rate was subsequently analyzed [11]. 3. One-particle motion in deformed and rotating potentials, and yrast spectroscopy in high-spin physics The very basic description of nuclear many-body sys- tems is the (self-consistent) mean-ﬁeld approximation to the many-body problem. In particular, shape is the prop- erty of mean ﬁeld. The ground states of some nuclei are described by densities and mean ﬁelds that are spheri- cal, while others are deformed. In Fig. 3 the even-even nuclei, of which the ground state is observed to be de- formed, are shown. The deformation results from one- particle shell-structure, namely bunching of one-particle levels, in contrast to the prediction of the liquid-drop model. For the nucleus as a quantum many-body sys- tem the presence of deformation is a necessary condition for the occurrence of collective rotation, and the symme- try of the deformation determines the structure of rota- tion. The most important deformation around the ground states of nuclei is the axially-symmetric quadrupole de- formation corresponding to a spheroidal shape (cigar or pancake). Axially-symmetric nuclei cannot perform a col- lective rotation about the symmetry axis. If some nuclei show deformation with a given symmetry it is simplest and most convenient to start with the mean Fig. 2: The upper part expresses the radial transition charge − ﬁeld, which has the same symmetry. One-particle mo- density of the 3 state at 2.614 MeV in 208Pb. The solid line tion in quadrupole-deformed potentials was solved in the shows the calculated value [7], while for the experimental values ﬁfties almost at the same time by three groups [12, 13, 14]. the dashed band was taken from Ref. [8] and the crosspoints Among them, ”Nilsson model” [12] has been used by many were from Ref. [9]. The lower part of the ﬁgure denotes the calculated charge density of the ground state of 208Pb. The people in the analysis of data, presumably because the scale of the x-axis in the lower part is the same as the one in application of the Nilsson model using modiﬁed oscilla- the upper part. The ﬁgure is adapted from Fig. 1 of Ref. [7]. tor potentials to actual nuclei was easiest and very prac- tical. The spectrum of one-particle orbits as a function of deformation, so-called ”Nilsson diagram”, has played − pling) of the 3 state of 208Pb [7] is compared with those an invaluable role in the study of deformed nuclei. (In obtained from electron inelastic scatterings. The large the present article the word, ”Nilsson diagram”, is used peak around the surface can be understood as a feature of for the diagrams of one-particle energies as a function of the shape oscillation. The position of the ﬁrst minimum quadrupole deformation using general one-body potentials around 5 fm coincides closely with the r-value where the such as Woods-Saxon or HF potentials, and not only for ground-state density reaches the maximum, noting that the diagram drawn by using modiﬁed oscillator poten- the radial transition density is approximately proportional tials.) Nilsson diagrams have been extremely useful for to the radial derivative of the static density. The struc- analyzing the data on deformed nuclei. This is mainly be- ∗ · ture of the two small peaks inside the nucleus depends on cause the major part of the (Y Y ) channel of the two- 20 20 the microscopic shell-structure of the octupole vibration body quadrupole-quadrupole interaction is absorbed into in addition to the general quantum-mechanical oscillatory the mean ﬁeld. Consequently, the picture of one-particle structure of the static density. motion in the deformed potential works for deformed nu- Another example in somewhat lighter-mass region, clei much better than that of one-particle motion in spher- which exhibits a very similar coupling between particles ical potentials for spherical nuclei. For example, see the and an octupole vibration, was reported in 1982 [10]. The successful quantitative analysis of various kinds of exper- − septuplet members in 147Gd , (f 3 ) with I = 1/2, imental data of well-deformed (especially odd-A) nuclei 64 83 7/2 I+ − ..., 13/2, in which the octupole vibration 3 of 146Gd at such as 25Mg, 25Al, 159Tb, 169Tm, 175Yb, 175Lu, 177Lu, E = 1.579 MeV with B(E3)=37B (E3) was coupled to 177Hf, 235U, 237Np, 239Pu presented in Ref. [1]. ex W the 83rd odd-neutron in the f orbit, were identiﬁed ex- The ﬁeld of nuclear high-spin physics was activated by 7/2 cept the Iπ = 5/2+ member. Due to the dominance of the the observation of the ”back-bending” phenomena [15], − − proton (h d 1 ) − component in the 3 state, not all which was the ﬁrst band-crossing (between the ground- 11/2 5/2 3 properties of the septuplet members could be analyzed by band and the S-band) along the yrast line in well-deformed perturbation. However, it is unique in the case of the oc- collectively-rotating nuclei. During the seventies the tupole vibration of 146Gd that the two-octupole-phonon possibility of studying nuclei with very large angular- − angular-momentum-stretched state, [(νf ) (3 )2] , momentum was opened up, thanks to the facilities of ac- 7/2 19/2+ in 147Gd was experimentally identiﬁed and the pretty celerating heavy-ions. Collisions between two heavy nuclei 3 ##LTFigure##

text: strong

text: anharmonicity

text: in

text: both

text: the

text: energy

text: and

text: the

text: tran-

text: sition

text: rate

text: was

text: subsequently

text: analyzed

text: One-particle

text: motion

text: in

text: deformed

text: and

text: rotating

text: potentials,

text: and

text: yrast

text: spectroscopy

text: in

text: high-spin

text: physics

text: The

text: very

text: basic

text: description

text: of

text: nuclear

text: many-body

text: sys-

text: tems

text: is

text: the

text: (self-consistent)

text: mean-ﬁeld

text: approximation

text: to

text: the

text: many-body

text: problem.

text: In

text: particular,

text: shape

text: is

text: the

text: prop-

text: erty

text: of

text: mean

text: ﬁeld.

text: The

text: ground

text: states

text: of

text: some

text: nuclei

text: are

text: described

text: by

text: densities

text: and

text: mean

text: ﬁelds

text: that

text: are

text: spheri-

text: cal,

text: while

text: others

text: are

text: deformed.

text: In

text: Fig.

text: the

text: even-even

text: nuclei,

text: of

text: which

text: the

text: ground

text: state

text: is

text: observed

text: to

text: be

text: de-

text: formed,

text: are

text: shown.

text: The

text: deformation

text: results

text: from

text: one-

text: particle

text: shell-structure,

text: namely

text: bunching

text: of

text: one-particle

text: levels,

text: in

text: contrast

text: to

text: the

text: prediction

text: of

text: the

text: liquid-drop

text: model.

text: For

text: the

text: nucleus

text: as

text: a

text: quantum

text: many-body

text: sys-

text: tem

text: the

text: presence

text: of

text: deformation

text: is

text: a

text: necessary

text: condition

text: for

text: the

text: occurrence

text: of

text: collective

text: rotation,

text: and

text: the

text: symme-

text: try

text: of

text: the

text: deformation

text: determines

text: the

text: structure

text: of

text: rota-

text: tion.

text: The

text: most

text: important

text: deformation

text: around

text: the

text: ground

text: states

text: of

text: nuclei

text: is

text: the

text: axially-symmetric

text: quadrupole

text: de-

text: formation

text: corresponding

text: to

text: a

text: spheroidal

text: shape

text: (cigar

text: or

text: pancake).

text: Axially-symmetric

text: nuclei

text: cannot

text: perform

text: a

text: col-

text: lective

text: rotation

text: about

text: the

text: symmetry

text: axis.

text: If

text: some

text: nuclei

text: show

text: deformation

text: with

text: a

text: given

text: symmetry

text: it

text: is

text: simplest

text: and

text: most

text: convenient

text: to

text: start

text: with

text: the

text: mean

text: Fig.

text: The

text: upper

text: part

text: expresses

text: the

text: radial

text: transition

text: charge

text: ﬁeld,

text: which

text: has

text: the

text: same

text: symmetry.

text: One-particle

text: mo-

text: density

text: of

text: the

text: state

text: at

text: MeV

text: in

text: The

text: solid

text: line

text: tion

text: in

text: quadrupole-deformed

text: potentials

text: was

text: solved

text: in

text: the

text: shows

text: the

text: calculated

text: value

text: while

text: for

text: the

text: experimental

text: values

text: ﬁfties

text: almost

text: at

text: the

text: same

text: time

text: by

text: three

text: groups

text: the

text: dashed

text: band

text: was

text: taken

text: from

text: Ref.

text: and

text: the

text: crosspoints

text: Among

text: them,

text: ”Nilsson

text: model”

text: has

text: been

text: used

text: by

text: many

text: were

text: from

text: Ref.

text: The

text: lower

text: part

text: of

text: the

text: ﬁgure

text: denotes

text: the

text: calculated

text: charge

text: density

text: of

text: the

text: ground

text: state

text: of

text: The

text: people

text: in

text: the

text: analysis

text: of

text: data,

text: presumably

text: because

text: the

text: scale

text: of

text: the

text: x-axis

text: in

text: the

text: lower

text: part

text: is

text: the

text: same

text: as

text: the

text: one

text: in

text: application

text: of

text: the

text: Nilsson

text: model

text: using

text: modiﬁed

text: oscilla-

text: the

text: upper

text: part.

text: The

text: ﬁgure

text: is

text: adapted

text: from

text: Fig.

text: of

text: Ref.

text: tor

text: potentials

text: to

text: actual

text: nuclei

text: was

text: easiest

text: and

text: very

text: prac-

text: tical.

text: The

text: spectrum

text: of

text: one-particle

text: orbits

text: as

text: a

text: function

text: of

text: deformation,

text: so-called

text: ”Nilsson

text: diagram”,

text: has

text: played

text: pling)

text: of

text: the

text: state

text: of

text: is

text: compared

text: with

text: those

text: an

text: invaluable

text: role

text: in

text: the

text: study

text: of

text: deformed

text: nuclei.

text: (In

text: obtained

text: from

text: electron

text: inelastic

text: scatterings.

text: The

text: large

text: the

text: present

text: article

text: the

text: word,

text: ”Nilsson

text: diagram”,

text: is

text: used

text: peak

text: around

text: the

text: surface

text: can

text: be

text: understood

text: as

text: a

text: feature

text: of

text: for

text: the

text: diagrams

text: of

text: one-particle

text: energies

text: as

text: a

text: function

text: of

text: the

text: shape

text: oscillation.

text: The

text: position

text: of

text: the

text: ﬁrst

text: minimum

text: quadrupole

text: deformation

text: using

text: general

text: one-body

text: potentials

text: around

text: fm

text: coincides

text: closely

text: with

text: the

text: r-value

text: where

text: the

text: such

text: as

text: Woods-Saxon

text: or

text: HF

text: potentials,

text: and

text: not

text: only

text: for

text: ground-state

text: density

text: reaches

text: the

text: maximum,

text: noting

text: that

text: the

text: diagram

text: drawn

text: by

text: using

text: modiﬁed

text: oscillator

text: poten-

text: the

text: radial

text: transition

text: density

text: is

text: approximately

text: proportional

text: tials.)

text: Nilsson

text: diagrams

text: have

text: been

text: extremely

text: useful

text: for

text: to

text: the

text: radial

text: derivative

text: of

text: the

text: static

text: density.

text: The

text: struc-

text: analyzing

text: the

text: data

text: on

text: deformed

text: nuclei.

text: This

text: is

text: mainly

text: be-

text: ture

text: of

text: the

text: two

text: small

text: peaks

text: inside

text: the

text: nucleus

text: depends

text: on

text: cause

text: the

text: major

text: part

text: of

text: the

text: (Y

text: Y

text: channel

text: of

text: the

text: two-

text: the

text: microscopic

text: shell-structure

text: of

text: the

text: octupole

text: vibration

text: body

text: quadrupole-quadrupole

text: interaction

text: is

text: absorbed

text: into

text: in

text: addition

text: to

text: the

text: general

text: quantum-mechanical

text: oscillatory

text: the

text: mean

text: ﬁeld.

text: Consequently,

text: the

text: picture

text: of

text: one-particle

text: structure

text: of

text: the

text: static

text: density.

text: motion

text: in

text: the

text: deformed

text: potential

text: works

text: for

text: deformed

text: nu-

text: Another

text: example

text: in

text: somewhat

text: lighter-mass

text: region,

text: clei

text: much

text: better

text: than

text: that

text: of

text: one-particle

text: motion

text: in

text: spher-

text: which

text: exhibits

text: a

text: very

text: similar

text: coupling

text: between

text: particles

text: ical

text: potentials

text: for

text: spherical

text: nuclei.

text: For

text: example,

text: see

text: the

text: and

text: an

text: octupole

text: vibration,

text: was

text: reported

text: in

text: The

text: successful

text: quantitative

text: analysis

text: of

text: various

text: kinds

text: of

text: exper-

text: septuplet

text: members

text: in

text: (f

text: with

text: I

text: imental

text: data

text: of

text: well-deformed

text: (especially

text: odd-A)

text: nuclei

text: I+

text: in

text: which

text: the

text: octupole

text: vibration

text: of

text: at

text: such

text: as

text: E

text: MeV

text: with

text: was

text: coupled

text: to

text: presented

text: in

text: Ref.

text: ex

text: W

text: the

text: odd-neutron

text: in

text: the

text: f

text: orbit,

text: were

text: identiﬁed

text: ex-

text: The

text: ﬁeld

text: of

text: nuclear

text: high-spin

text: physics

text: was

text: activated

text: by

text: cept

text: the

text: Iπ

text: member.

text: Due

text: to

text: the

text: dominance

text: of

text: the

text: the

text: observation

text: of

text: the

text: ”back-bending”

text: phenomena

text: proton

text: (h

text: d

text: component

text: in

text: the

text: state,

text: not

text: all

text: which

text: was

text: the

text: ﬁrst

text: band-crossing

text: (between

text: the

text: ground-

text: properties

text: of

text: the

text: septuplet

text: members

text: could

text: be

text: analyzed

text: by

text: band

text: and

text: the

text: S-band)

text: along

text: the

text: yrast

text: line

text: in

text: well-deformed

text: perturbation.

text: However,

text: it

text: is

text: unique

text: in

text: the

text: case

text: of

text: the

text: oc-

text: collectively-rotating

text: nuclei.

text: During

text: the

text: seventies

text: the

text: tupole

text: vibration

text: of

text: that

text: the

text: two-octupole-phonon

text: possibility

text: of

text: studying

text: nuclei

text: with

text: very

text: large

text: angular-

text: angular-momentum-stretched

text: state,

text: [(νf

text: momentum

text: was

text: opened

text: up,

text: thanks

text: to

text: the

text: facilities

text: of

text: ac-

text: in

text: was

text: experimentally

text: identiﬁed

text: and

text: the

text: pretty

text: celerating

text: heavy-ions.

text: Collisions

text: between

text: two

text: heavy

text: nuclei

text: ##LTFigure##