text: could produce meta-stable compound systems with large ing the physics in the yrast region, especially in terms of angular-momenta. Indeed, in heavy-ion collisions in the one-particle motion in rotating potentials, was repeatedly late seventies it was possible to produce compound nuclei emphasized by Bohr and Mottelson during the seventies. with angular momentum all the way up to the limit set The cranking model is easily treated and the rotational by the ﬁssion instability. The maximum value of angular frequency appears as an explicit parameter. Indeed, the momentum which nuclei can accommodate is the order of major part of the analysis of nuclear high-spin phenomena ≈ 100 h¯ for nuclei with A 130 [16]. The study of rapidly has been carried out in terms of cranking models of vari- rotating nuclei provided the opportunity for exploring new ous types. In particular, the so-called Routhian diagram, aspects of nuclear dynamics [17, 18, 19, 20]. namely the diagram plotting one quasiparticle energies in When the angular momentum is accommodated by a the rotating frame as a function of cranking frequency, quantal system such as a nucleus, one recognizes that there played for years a central role in the analysis of nuclear are two fundamental ways of building the angular momen- high-spin data. I remember that around the middle of the tum [21]. The ﬁrst one is collective rotation in the presence seventies Aage Bohr was repeating to say to us working at of a signiﬁcant amount of deformation, while the second the Niels Bohr Institute that one should draw and study one is to build the angular momentum by spin alignments the Routhian diagram for ﬁxed deformation and pairing of individual particles when the angular momentum is di- parameters. (At that time I stupidly thought that such rected along an axis of symmetry of the nucleus. In actual simpliﬁed Routhian diagrams might be qualitatively excel- nuclei we encounter situations in which the increment of lent for getting ideas and ways of understanding physics, angular momentum is achieved partly by increasing the but they would hardly be useful for practical numerical frequency of collective rotation and partly by rearranging applications.) In this connection, I later realized that, the occupation of one-particle orbits (namely, by increas- for example, without plotting such Routhian diagrams it ing the particle alignments). In fact, the collective rota- would have been indeed diﬃcult to notice the fact that tional spectra together with the crossing of the ground the interaction strength between the ground band and the band with the S-band are characteristic of this kind of S-band, namely the sharpness of backbending phenomena, situation. is an oscillating function of the degree of high-j-shell ﬁlling Special interest in nuclear high-spin states attaches to [22]. The extended study with numerical works using re- the region in the neighborhood of the yrast line repre- alistic conﬁguration spaces was eventually carried out by senting the lowest energy for a given angular-momentum. R. Bengtsson and S. Frauendorf [23], and the Routhian di- In this yrast region the nucleus is cold in the sense that agrams of such a simple type turned out to be extremely almost the entire excitation energy of the nucleus is con- useful and have been successfully used in the analysis of sumed in generating the angular momentum. Therefore, high-spin data for the following years. the structure in the yrast region is ordered with simple ex- However, one should keep in mind that the uniform ro- citation modes, and the study of those may be expected to tation of a system, which is a basic assumption of the give important nuclear-structure information on how the cranking model, may become a poor approximation under nucleus responds to the large centrifugal forces associated certain circumstances, because the presence of ﬂuctuations with rotation. The path that the yrast line of actual nuclei in the collective rotational frequency is recognized if one will follow in deformation space with increasing angular- treats more reasonably the exchange of angular momen- momentum will result from the competition between the tum between individual particles and the potential pro- macroscopic centrifugal distortion eﬀect and the quantal duced by the rest of the system [24]. An example outside eﬀects associated with shell-structure. the applicability of the cranking model in its simplest form In the yrast spectroscopy of medium-heavy nuclei an im- is the description of the crossings of the bands with a large portant role has been played by particles in high-j orbits, diﬀerence of spin alignments, irrespective of whether the such as 1g -, 1h -, 1i - and 1j -orbits. This is parameters in the model are chosen self-consistently or 9/2 11/2 13/2 15/2 because the parity of high-j orbits is diﬀerent from that of not. The inapplicability comes from the fact that a mix- other one-particle orbits in respective major shells, there- ing of the two bands for a given rotational frequency is fore, both in quadrupole deformation and under rotation basically considered in the cranking model, while the two particles in high-j orbits do not mix with those in neighbor- bands should interact for a given angular momentum [25]. ing orbits. That means, the wave functions of particles in The experimental front of high-spin physics made a high-j orbits have less ambiguity. Furthermore, as soon as tremendous progress in the decades after the observation rotation sets in, particles in high-j orbits start to align due of ”back-bending”, especially due to the development of to the particularly large Coriolis coupling, while particles multi-gamma-ray detectors, which had a resolution of or- in neighboring other orbits still contribute to collective ro- ders greater than that of the last generation, and 4π or 8π → tation. Consequently, the states consisting of high-j-shell spectrometers; NORDBALL and TESSA EUROGAM → conﬁgurations easily appear in the neighborhood of the and GASP EUROBALL and GAMMASPHERE. yrast line. A typical example is that the intrinsic struc- New frontiers were opened in the nuclear high-spin ture of the S-band contains a large alignment coming from physics, when the superdeformed band in 152Dy was 66 86 the aligned two quasiparticles in the relevant high-j orbit, reported [26] by ﬁnally observing gamma-ray spectra of in contrast to the absence of such alignment in the ground discrete transitions in the decay sequence of the superde- ≈ band, at the angular momentum of the band crossing. formed band of 152Dy extending to I 60 h¯, thanks to the The basis for the analysis of shell-structure eﬀects on development of detectors. The observation of the discrete nuclear high-spin states is the study of one-particle mo- gamma-rays could directly and clearly show the presence tion in rotating potentials. The importance of understand- of the superdeformed band, compared with the informa- 4

text: could

text: produce

text: meta-stable

text: compound

text: systems

text: with

text: large

text: ing

text: the

text: physics

text: in

text: the

text: yrast

text: region,

text: especially

text: in

text: terms

text: of

text: angular-momenta.

text: Indeed,

text: in

text: heavy-ion

text: collisions

text: in

text: the

text: one-particle

text: motion

text: in

text: rotating

text: potentials,

text: was

text: repeatedly

text: late

text: seventies

text: it

text: was

text: possible

text: to

text: produce

text: compound

text: nuclei

text: emphasized

text: by

text: Bohr

text: and

text: Mottelson

text: during

text: the

text: seventies.

text: with

text: angular

text: momentum

text: all

text: the

text: way

text: up

text: to

text: the

text: limit

text: set

text: The

text: cranking

text: model

text: is

text: easily

text: treated

text: and

text: the

text: rotational

text: by

text: the

text: ﬁssion

text: instability.

text: The

text: maximum

text: value

text: of

text: angular

text: frequency

text: appears

text: as

text: an

text: explicit

text: parameter.

text: Indeed,

text: the

text: momentum

text: which

text: nuclei

text: can

text: accommodate

text: is

text: the

text: order

text: of

text: major

text: part

text: of

text: the

text: analysis

text: of

text: nuclear

text: high-spin

text: phenomena

text: h¯

text: for

text: nuclei

text: with

text: A

text: The

text: study

text: of

text: rapidly

text: has

text: been

text: carried

text: out

text: in

text: terms

text: of

text: cranking

text: models

text: of

text: vari-

text: rotating

text: nuclei

text: provided

text: the

text: opportunity

text: for

text: exploring

text: new

text: ous

text: types.

text: In

text: particular,

text: the

text: so-called

text: Routhian

text: diagram,

text: aspects

text: of

text: nuclear

text: dynamics

text: namely

text: the

text: diagram

text: plotting

text: one

text: quasiparticle

text: energies

text: in

text: When

text: the

text: angular

text: momentum

text: is

text: accommodated

text: by

text: a

text: the

text: rotating

text: frame

text: as

text: a

text: function

text: of

text: cranking

text: frequency,

text: quantal

text: system

text: such

text: as

text: a

text: nucleus,

text: one

text: recognizes

text: that

text: there

text: played

text: for

text: years

text: a

text: central

text: role

text: in

text: the

text: analysis

text: of

text: nuclear

text: are

text: two

text: fundamental

text: ways

text: of

text: building

text: the

text: angular

text: momen-

text: high-spin

text: data.

text: I

text: remember

text: that

text: around

text: the

text: middle

text: of

text: the

text: tum

text: The

text: ﬁrst

text: one

text: is

text: collective

text: rotation

text: in

text: the

text: presence

text: seventies

text: Aage

text: Bohr

text: was

text: repeating

text: to

text: say

text: to

text: us

text: working

text: at

text: of

text: a

text: signiﬁcant

text: amount

text: of

text: deformation,

text: while

text: the

text: second

text: the

text: Niels

text: Bohr

text: Institute

text: that

text: one

text: should

text: draw

text: and

text: study

text: one

text: is

text: to

text: build

text: the

text: angular

text: momentum

text: by

text: spin

text: alignments

text: the

text: Routhian

text: diagram

text: for

text: ﬁxed

text: deformation

text: and

text: pairing

text: of

text: individual

text: particles

text: when

text: the

text: angular

text: momentum

text: is

text: di-

text: parameters.

text: (At

text: that

text: time

text: I

text: stupidly

text: thought

text: that

text: such

text: rected

text: along

text: an

text: axis

text: of

text: symmetry

text: of

text: the

text: nucleus.

text: In

text: actual

text: simpliﬁed

text: Routhian

text: diagrams

text: might

text: be

text: qualitatively

text: excel-

text: nuclei

text: we

text: encounter

text: situations

text: in

text: which

text: the

text: increment

text: of

text: lent

text: for

text: getting

text: ideas

text: and

text: ways

text: of

text: understanding

text: physics,

text: angular

text: momentum

text: is

text: achieved

text: partly

text: by

text: increasing

text: the

text: but

text: they

text: would

text: hardly

text: be

text: useful

text: for

text: practical

text: numerical

text: frequency

text: of

text: collective

text: rotation

text: and

text: partly

text: by

text: rearranging

text: applications.)

text: In

text: this

text: connection,

text: I

text: later

text: realized

text: that,

text: the

text: occupation

text: of

text: one-particle

text: orbits

text: (namely,

text: by

text: increas-

text: for

text: example,

text: without

text: plotting

text: such

text: Routhian

text: diagrams

text: it

text: ing

text: the

text: particle

text: alignments).

text: In

text: fact,

text: the

text: collective

text: rota-

text: would

text: have

text: been

text: indeed

text: diﬃcult

text: to

text: notice

text: the

text: fact

text: that

text: tional

text: spectra

text: together

text: with

text: the

text: crossing

text: of

text: the

text: ground

text: the

text: interaction

text: strength

text: between

text: the

text: ground

text: band

text: and

text: the

text: band

text: with

text: the

text: S-band

text: are

text: characteristic

text: of

text: this

text: kind

text: of

text: S-band,

text: namely

text: the

text: sharpness

text: of

text: backbending

text: phenomena,

text: situation.

text: is

text: an

text: oscillating

text: function

text: of

text: the

text: degree

text: of

text: high-j-shell

text: ﬁlling

text: Special

text: interest

text: in

text: nuclear

text: high-spin

text: states

text: attaches

text: to

text: The

text: extended

text: study

text: with

text: numerical

text: works

text: using

text: re-

text: the

text: region

text: in

text: the

text: neighborhood

text: of

text: the

text: yrast

text: line

text: repre-

text: alistic

text: conﬁguration

text: spaces

text: was

text: eventually

text: carried

text: out

text: by

text: senting

text: the

text: lowest

text: energy

text: for

text: a

text: given

text: angular-momentum.

text: R.

text: Bengtsson

text: and

text: S.

text: Frauendorf

text: and

text: the

text: Routhian

text: di-

text: In

text: this

text: yrast

text: region

text: the

text: nucleus

text: is

text: cold

text: in

text: the

text: sense

text: that

text: agrams

text: of

text: such

text: a

text: simple

text: type

text: turned

text: out

text: to

text: be

text: extremely

text: almost

text: the

text: entire

text: excitation

text: energy

text: of

text: the

text: nucleus

text: is

text: con-

text: useful

text: and

text: have

text: been

text: successfully

text: used

text: in

text: the

text: analysis

text: of

text: sumed

text: in

text: generating

text: the

text: angular

text: momentum.

text: Therefore,

text: high-spin

text: data

text: for

text: the

text: following

text: years.

text: the

text: structure

text: in

text: the

text: yrast

text: region

text: is

text: ordered

text: with

text: simple

text: ex-

text: However,

text: one

text: should

text: keep

text: in

text: mind

text: that

text: the

text: uniform

text: ro-

text: citation

text: modes,

text: and

text: the

text: study

text: of

text: those

text: may

text: be

text: expected

text: to

text: tation

text: of

text: a

text: system,

text: which

text: is

text: a

text: basic

text: assumption

text: of

text: the

text: give

text: important

text: nuclear-structure

text: information

text: on

text: how

text: the

text: cranking

text: model,

text: may

text: become

text: a

text: poor

text: approximation

text: under

text: nucleus

text: responds

text: to

text: the

text: large

text: centrifugal

text: forces

text: associated

text: certain

text: circumstances,

text: because

text: the

text: presence

text: of

text: ﬂuctuations

text: with

text: rotation.

text: The

text: path

text: that

text: the

text: yrast

text: line

text: of

text: actual

text: nuclei

text: in

text: the

text: collective

text: rotational

text: frequency

text: is

text: recognized

text: if

text: one

text: will

text: follow

text: in

text: deformation

text: space

text: with

text: increasing

text: angular-

text: treats

text: more

text: reasonably

text: the

text: exchange

text: of

text: angular

text: momen-

text: momentum

text: will

text: result

text: from

text: the

text: competition

text: between

text: the

text: tum

text: between

text: individual

text: particles

text: and

text: the

text: potential

text: pro-

text: macroscopic

text: centrifugal

text: distortion

text: eﬀect

text: and

text: the

text: quantal

text: duced

text: by

text: the

text: rest

text: of

text: the

text: system

text: An

text: example

text: outside

text: eﬀects

text: associated

text: with

text: shell-structure.

text: the

text: applicability

text: of

text: the

text: cranking

text: model

text: in

text: its

text: simplest

text: form

text: In

text: the

text: yrast

text: spectroscopy

text: of

text: medium-heavy

text: nuclei

text: an

text: im-

text: is

text: the

text: description

text: of

text: the

text: crossings

text: of

text: the

text: bands

text: with

text: a

text: large

text: portant

text: role

text: has

text: been

text: played

text: by

text: particles

text: in

text: high-j

text: orbits,

text: diﬀerence

text: of

text: spin

text: alignments,

text: irrespective

text: of

text: whether

text: the

text: such

text: as

text: and

text: -orbits.

text: This

text: is

text: parameters

text: in

text: the

text: model

text: are

text: chosen

text: self-consistently

text: or

text: because

text: the

text: parity

text: of

text: high-j

text: orbits

text: is

text: diﬀerent

text: from

text: that

text: of

text: not.

text: The

text: inapplicability

text: comes

text: from

text: the

text: fact

text: that

text: a

text: mix-

text: other

text: one-particle

text: orbits

text: in

text: respective

text: major

text: shells,

text: there-

text: ing

text: of

text: the

text: two

text: bands

text: for

text: a

text: given

text: rotational

text: frequency

text: is

text: fore,

text: both

text: in

text: quadrupole

text: deformation

text: and

text: under

text: rotation

text: basically

text: considered

text: in

text: the

text: cranking

text: model,

text: while

text: the

text: two

text: particles

text: in

text: high-j

text: orbits

text: do

text: not

text: mix

text: with

text: those

text: in

text: neighbor-

text: bands

text: should

text: interact

text: for

text: a

text: given

text: angular

text: momentum

text: ing

text: orbits.

text: That

text: means,

text: the

text: wave

text: functions

text: of

text: particles

text: in

text: The

text: experimental

text: front

text: of

text: high-spin

text: physics

text: made

text: a

text: high-j

text: orbits

text: have

text: less

text: ambiguity.

text: Furthermore,

text: as

text: soon

text: as

text: tremendous

text: progress

text: in

text: the

text: decades

text: after

text: the

text: observation

text: rotation

text: sets

text: in,

text: particles

text: in

text: high-j

text: orbits

text: start

text: to

text: align

text: due

text: of

text: ”back-bending”,

text: especially

text: due

text: to

text: the

text: development

text: of

text: to

text: the

text: particularly

text: large

text: Coriolis

text: coupling,

text: while

text: particles

text: multi-gamma-ray

text: detectors,

text: which

text: had

text: a

text: resolution

text: of

text: or-

text: in

text: neighboring

text: other

text: orbits

text: still

text: contribute

text: to

text: collective

text: ro-

text: ders

text: greater

text: than

text: that

text: of

text: the

text: last

text: generation,

text: and

text: 4π

text: or

text: 8π

text: tation.

text: Consequently,

text: the

text: states

text: consisting

text: of

text: high-j-shell

text: spectrometers;

text: NORDBALL

text: and

text: TESSA

text: EUROGAM

text: conﬁgurations

text: easily

text: appear

text: in

text: the

text: neighborhood

text: of

text: the

text: and

text: GASP

text: EUROBALL

text: and

text: GAMMASPHERE.

text: yrast

text: line.

text: A

text: typical

text: example

text: is

text: that

text: the

text: intrinsic

text: struc-

text: New

text: frontiers

text: were

text: opened

text: in

text: the

text: nuclear

text: high-spin

text: ture

text: of

text: the

text: S-band

text: contains

text: a

text: large

text: alignment

text: coming

text: from

text: physics,

text: when

text: the

text: superdeformed

text: band

text: in

text: was

text: the

text: aligned

text: two

text: quasiparticles

text: in

text: the

text: relevant

text: high-j

text: orbit,

text: reported

text: by

text: ﬁnally

text: observing

text: gamma-ray

text: spectra

text: of

text: in

text: contrast

text: to

text: the

text: absence

text: of

text: such

text: alignment

text: in

text: the

text: ground

text: discrete

text: transitions

text: in

text: the

text: decay

text: sequence

text: of

text: the

text: superde-

text: band,

text: at

text: the

text: angular

text: momentum

text: of

text: the

text: band

text: crossing.

text: formed

text: band

text: of

text: extending

text: to

text: I

text: h¯,

text: thanks

text: to

text: the

text: The

text: basis

text: for

text: the

text: analysis

text: of

text: shell-structure

text: eﬀects

text: on

text: development

text: of

text: detectors.

text: The

text: observation

text: of

text: the

text: discrete

text: nuclear

text: high-spin

text: states

text: is

text: the

text: study

text: of

text: one-particle

text: mo-

text: gamma-rays

text: could

text: directly

text: and

text: clearly

text: show

text: the

text: presence

text: tion

text: in

text: rotating

text: potentials.

text: The

text: importance

text: of

text: understand-

text: of

text: the

text: superdeformed

text: band,

text: compared

text: with

text: the

text: informa-


text: tion extracted previously from the painstaking analysis of On the other hand, the two phenomena, wobbling exci- complicated continuum gamma-rays. It is noted that the tation modes and chiral bands, are unique in triaxial shape largest angular momentum, which was reported in discrete and were intensively searched for by using the advanced line studies before the discovery of the superdeformed technique developed in high-spin physics. Observation of band, was only 46 ¯h. Because of the large moment of iner- beautiful wobbling excitation modes was reported in the tia of superdeformed nuclei, the large angular-momentum beginning of the 21st century, while, in my opinion, ex- needed for the study of such systems does not necessar- perimental data which clearly pin down chiral bands have ily mean large rotational frequencies. Consequently, the not yet been obtained though several candidates for chiral unique behavior of particles in high-j orbits could largely bands have been reported. In this section I describe the remain also in the analysis of superdeformed bands. The present understanding of these two phenomena. shape of superdeformation in nuclei, where the axis ratio 4.1. Quantized wobbling observed in nuclei of the prolately-deformed shape is approximately 2:1, was known already in the sixties as (low-spin) ﬁssion isomers Nuclei with a triaxial shape can rotate about any of in actinide nuclei [27] and was interpreted as the result the principal axes showing rich spectra of collective rota- of the shell-structure which appears ﬁrst for much larger tion. Though the rotation about the axis with the largest ≈ quadrupole-deformation (β 0.6 in medium-heavy nuclei) moment of inertia is energetically cheapest, while freez- than the one known in nuclear ground states. In subse- ing the intrinsic structure a series of rotational bands can quent years the observation of super- or highly-deformed be built by transferring some angular momentum to the bands was reported also in other mass region than A=150, other two axes. The family of the rotational bands is for- such as A=130 [28] and 190 [29], though the relevant shape mulated in terms of vibrational excitations. The classical was not always close to 2:1. analog of this wobbling motion is the spinning motion of Before 1975 the view might have been accepted that an asymmetric top, but the motion in the nuclear system is there were nuclei that were statically deformed and they quantized and expressed in terms of the wobbling phonon were located far away from closed shells, while there number (nW ). A family of rotational bands with wobbling were nuclei that were spherical and they were located in excitations can be pinned down by speciﬁc electromagnetic the neighborhood of closed-shell regions, though in some decay properties between them. This quantized wobbling nuclei located on the border of spherical and deformed phonon picture was ﬁrst proposed by A. Bohr and B. R. nuclei the simultaneous presence of the low-lying states Mottelson described in [1]. with spherical and deformed shapes was already recog- In 2001 such a wobbling band, a one-phonon wobbling nized. Thanks to the marvelous development of exper- excitation, was discovered in the nucleus 163Lu [32]. One 71 92 imental techniques in high-spin physics, collective rota- year later, in 2002 the next wobbling excitation, a two- tional bands were found at the excitation energy of sev- phonon rotational band, was reported in the same nucleus eral MeV also in some nuclei such as Pb, Sn and Ca [33]. In this experiment high spin states of 163Lu were pop- 82 50 20 isotopes [30], which are at least semi-magic and were sup- ulated using a 29Si beam and a 139La target. The emitted posed to be spherical. While it has been known that for gamma-rays were measured with the EUROBALL detec- a given nucleus Hartree-Fock calculations often predicted tor system. The electromagnetic nature of the decay tran- local energy-minima at several diﬀerent deformations, the sitions from the newly observed band is analyzed and the possible presence of such several shapes in a given nucleus results clearly show the two-phonon nature of the wobbling has been indeed conﬁrmed thanks to the modern experi- motion in good agreement with calculations, in which the mental technique. intrinsic degree of freedom of the rotational bands in 163Lu In the present special issue of Physica Scripta I expect a is represented by one highly-aligned quasiproton in the number of contributions by specialists in the ﬁeld of high- i13/2 shell [34]. spin physics. Therefore, to those contributions I would The state with high-j aligned particles favors a speciﬁc leave to show a tremendous amount of beautiful results (triaxial) shape [35] depending on the degree of the j-shell obtained from all detailed numerical works in comparison ﬁlling. The γ-value favored by the aligned i proton ◦ 13/2 with exciting and impressive experimental data. in 163Lu is around +20 (in the so-called ”Lund conven- 4. Triaxial shape in nuclei tion” [36]). States with large alignments can easily appear in the neighborhood of the yrast line because of the rel- Theoretically the existence of nuclear deformation other atively small rotational angular-momentum (and thereby than axially-symmetric deformation was predicted, and it small rotational energy) needed for building a given to- has been a great challenge to ﬁnd the predicted axially- tal angular-momentum. Furthermore, using a particle- asymmetric (triaxial) deformation of nuclei. In order to rotor model, in which one quasiparticle of i13/2 protons pin down triaxial deformation, it is essential to ﬁnd the is coupled to a triaxial core, it was shown [37] that in the phenomena which are unique in axially-asymmetric shape. presence of one high-j aligned quasiparticle and for the In connection with available experimental data on ”high-j” γ-value strongly favored by the fully-aligned high-j par- conﬁgurations in the yrast spectroscopy, some phenomena ticle the wobbling excitation of the collective rotational → − such as the signature-dependence of B(E2:I I 1) val- angular-momentum of the core appears as the yrare band ues in odd-A nuclei and anomalous signature-splitting of in a certain range of angular momentum and the electro- Routhians in odd-A and odd-odd nuclei, were theoretically magnetic transitions between the yrast and yrare bands suggested as the evidence for triaxial shape [31]. However, show a unique pattern. before 2000 a very clear and ﬁrm evidence for stable tri- The two-phonon wobbling excitation observed in 163Lu axial shape was hardly obtained experimentally. is one of the most exotic properties of the spinning nu- 5

text: tion

label: paragraph

text: extracted

text: previously

text: from

text: the

text: painstaking

text: analysis

text: of

text: On

text: the

text: other

text: hand,

text: the

text: two

text: phenomena,

text: wobbling

text: exci-

text: complicated

text: continuum

text: gamma-rays.

text: It

text: is

text: noted

text: that

text: the

text: tation

text: modes

text: and

text: chiral

text: bands,

text: are

text: unique

text: in

text: triaxial

text: shape

text: largest

text: angular

text: momentum,

text: which

text: was

text: reported

text: in

text: discrete

text: and

text: were

text: intensively

text: searched

text: for

text: by

text: using

text: the

text: advanced

text: line

text: studies

text: before

text: the

text: discovery

text: of

text: the

text: superdeformed

text: technique

text: developed

text: in

text: high-spin

text: physics.

text: Observation

text: of

text: band,

text: was

text: only

text: ¯h.

text: Because

text: of

text: the

text: large

text: moment

text: of

text: iner-

text: beautiful

text: wobbling

text: excitation

text: modes

text: was

text: reported

text: in

text: the

text: tia

text: of

text: superdeformed

text: nuclei,

text: the

text: large

text: angular-momentum

text: beginning

text: of

text: the

text: century,

text: while,

text: in

text: my

text: opinion,

text: ex-

text: needed

text: for

text: the

text: study

text: of

text: such

text: systems

text: does

text: not

text: necessar-

text: perimental

text: data

text: which

text: clearly

text: pin

text: down

text: chiral

text: bands

text: have

text: ily

text: mean

text: large

text: rotational

text: frequencies.

text: Consequently,

text: the

text: not

text: yet

text: been

text: obtained

text: though

text: several

text: candidates

text: for

text: chiral

text: unique

text: behavior

text: of

text: particles

text: in

text: high-j

text: orbits

text: could

text: largely

text: bands

text: have

text: been

text: reported.

text: In

text: this

text: section

text: I

text: describe

text: the

text: remain

text: also

text: in

text: the

text: analysis

text: of

text: superdeformed

text: bands.

text: The

text: present

text: understanding

text: of

text: these

text: two

text: phenomena.

text: shape

text: of

text: superdeformation

text: in

text: nuclei,

text: where

text: the

text: axis

text: ratio

text: Quantized

text: wobbling

text: observed

text: in

text: nuclei

text: of

text: the

text: prolately-deformed

text: shape

text: is

text: approximately

text: was

text: known

text: already

text: in

text: the

text: sixties

text: as

text: (low-spin)

text: ﬁssion

text: isomers

text: Nuclei

text: with

text: a

text: triaxial

text: shape

text: can

text: rotate

text: about

text: any

text: of

text: in

text: actinide

text: nuclei

text: and

text: was

text: interpreted

text: as

text: the

text: result

text: the

text: principal

text: axes

text: showing

text: rich

text: spectra

text: of

text: collective

text: rota-

text: of

text: the

text: shell-structure

text: which

text: appears

text: ﬁrst

text: for

text: much

text: larger

text: tion.

text: Though

text: the

text: rotation

text: about

text: the

text: axis

text: with

text: the

text: largest

text: quadrupole-deformation

text: (β

text: in

text: medium-heavy

text: nuclei)

text: moment

text: of

text: inertia

text: is

text: energetically

text: cheapest,

text: while

text: freez-

text: than

text: the

text: one

text: known

text: in

text: nuclear

text: ground

text: states.

text: In

text: subse-

text: ing

text: the

text: intrinsic

text: structure

text: a

text: series

text: of

text: rotational

text: bands

text: can

text: quent

text: years

text: the

text: observation

text: of

text: super-

text: or

text: highly-deformed

text: be

text: built

text: by

text: transferring

text: some

text: angular

text: momentum

text: to

text: the

text: bands

text: was

text: reported

text: also

text: in

text: other

text: mass

text: region

text: than

text: other

text: two

text: axes.

text: The

text: family

text: of

text: the

text: rotational

text: bands

text: is

text: for-

text: such

text: as

text: and

text: though

text: the

text: relevant

text: shape

text: mulated

text: in

text: terms

text: of

text: vibrational

text: excitations.

text: The

text: classical

text: was

text: not

text: always

text: close

text: to

text: analog

text: of

text: this

text: wobbling

text: motion

text: is

text: the

text: spinning

text: motion

text: of

text: Before

text: the

text: view

text: might

text: have

text: been

text: accepted

text: that

text: an

text: asymmetric

text: top,

text: but

text: the

text: motion

text: in

text: the

text: nuclear

text: system

text: is

text: there

text: were

text: nuclei

text: that

text: were

text: statically

text: deformed

text: and

text: they

text: quantized

text: and

text: expressed

text: in

text: terms

text: of

text: the

text: wobbling

text: phonon

text: were

text: located

text: far

text: away

text: from

text: closed

text: shells,

text: while

text: there

text: number

text: (nW

text: A

text: family

text: of

text: rotational

text: bands

text: with

text: wobbling

text: were

text: nuclei

text: that

text: were

text: spherical

text: and

text: they

text: were

text: located

text: in

text: excitations

text: can

text: be

text: pinned

text: down

text: by

text: speciﬁc

text: electromagnetic

text: the

text: neighborhood

text: of

text: closed-shell

text: regions,

text: though

text: in

text: some

text: decay

text: properties

text: between

text: them.

text: This

text: quantized

text: wobbling

text: nuclei

text: located

text: on

text: the

text: border

text: of

text: spherical

text: and

text: deformed

text: phonon

text: picture

text: was

text: ﬁrst

text: proposed

text: by

text: A.

text: Bohr

text: and

text: B.

text: R.

text: nuclei

text: the

text: simultaneous

text: presence

text: of

text: the

text: low-lying

text: states

text: Mottelson

text: described

text: in

text: with

text: spherical

text: and

text: deformed

text: shapes

text: was

text: already

text: recog-

text: In

text: such

text: a

text: wobbling

text: band,

text: a

text: one-phonon

text: wobbling

text: nized.

text: Thanks

text: to

text: the

text: marvelous

text: development

text: of

text: exper-

text: excitation,

text: was

text: discovered

text: in

text: the

text: nucleus

text: One

text: imental

text: techniques

text: in

text: high-spin

text: physics,

text: collective

text: rota-

text: year

text: later,

text: in

text: the

text: next

text: wobbling

text: excitation,

text: a

text: two-

text: tional

text: bands

text: were

text: found

text: at

text: the

text: excitation

text: energy

text: of

text: sev-

text: phonon

text: rotational

text: band,

text: was

text: reported

text: in

text: the

text: same

text: nucleus

text: eral

text: MeV

text: also

text: in

text: some

text: nuclei

text: such

text: as

text: Pb,

text: Sn

text: and

text: Ca

text: In

text: this

text: experiment

text: high

text: spin

text: states

text: of

text: were

text: pop-

text: isotopes

text: which

text: are

text: at

text: least

text: semi-magic

text: and

text: were

text: sup-

text: ulated

text: using

text: a

text: beam

text: and

text: a

text: target.

text: The

text: emitted

text: posed

text: to

text: be

text: spherical.

text: While

text: it

text: has

text: been

text: known

text: that

text: for

text: gamma-rays

text: were

text: measured

text: with

text: the

text: EUROBALL

text: detec-

text: a

text: given

text: nucleus

text: Hartree-Fock

text: calculations

text: often

text: predicted

text: tor

text: system.

text: The

text: electromagnetic

text: nature

text: of

text: the

text: decay

text: tran-

text: local

text: energy-minima

text: at

text: several

text: diﬀerent

text: deformations,

text: the

text: sitions

text: from

text: the

text: newly

text: observed

text: band

text: is

text: analyzed

text: and

text: the

text: possible

text: presence

text: of

text: such

text: several

text: shapes

text: in

text: a

text: given

text: nucleus

text: results

text: clearly

text: show

text: the

text: two-phonon

text: nature

text: of

text: the

text: wobbling

text: has

text: been

text: indeed

text: conﬁrmed

text: thanks

text: to

text: the

text: modern

text: experi-

text: motion

text: in

text: good

text: agreement

text: with

text: calculations,

text: in

text: which

text: the

text: mental

text: technique.

text: intrinsic

text: degree

text: of

text: freedom

text: of

text: the

text: rotational

text: bands

text: in

text: In

text: the

text: present

text: special

text: issue

text: of

text: Physica

text: Scripta

text: I

text: expect

text: a

text: is

text: represented

text: by

text: one

text: highly-aligned

text: quasiproton

text: in

text: the

text: number

text: of

text: contributions

text: by

text: specialists

text: in

text: the

text: ﬁeld

text: of

text: high-

text: shell

text: spin

text: physics.

text: Therefore,

text: to

text: those

text: contributions

text: I

text: would

text: The

text: state

text: with

text: high-j

text: aligned

text: particles

text: favors

text: a

text: speciﬁc

text: leave

text: to

text: show

text: a

text: tremendous

text: amount

text: of

text: beautiful

text: results

text: (triaxial)

text: shape

text: depending

text: on

text: the

text: degree

text: of

text: the

text: j-shell

text: obtained

text: from

text: all

text: detailed

text: numerical

text: works

text: in

text: comparison

text: ﬁlling.

text: The

text: γ-value

text: favored

text: by

text: the

text: aligned

text: i

text: proton

text: with

text: exciting

text: and

text: impressive

text: experimental

text: data.

text: in

text: is

text: around

text: (in

text: the

text: so-called

text: ”Lund

text: conven-

text: Triaxial

text: shape

text: in

text: nuclei

text: tion”

text: States

text: with

text: large

text: alignments

text: can

text: easily

text: appear

text: in

text: the

text: neighborhood

text: of

text: the

text: yrast

text: line

text: because

text: of

text: the

text: rel-

text: Theoretically

text: the

text: existence

text: of

text: nuclear

text: deformation

text: other

text: atively

text: small

text: rotational

text: angular-momentum

text: (and

text: thereby

text: than

text: axially-symmetric

text: deformation

text: was

text: predicted,

text: and

text: it

text: small

text: rotational

text: energy)

text: needed

text: for

text: building

text: a

text: given

text: to-

text: has

text: been

text: a

text: great

text: challenge

text: to

text: ﬁnd

text: the

text: predicted

text: axially-

text: tal

text: angular-momentum.

text: Furthermore,

text: using

text: a

text: particle-

text: asymmetric

text: (triaxial)

text: deformation

text: of

text: nuclei.

text: In

text: order

text: to

text: rotor

text: model,

text: in

text: which

text: one

text: quasiparticle

text: of

text: protons

text: pin

text: down

text: triaxial

text: deformation,

text: it

text: is

text: essential

text: to

text: ﬁnd

text: the

text: is

text: coupled

text: to

text: a

text: triaxial

text: core,

text: it

text: was

text: shown

text: that

text: in

text: the

text: phenomena

text: which

text: are

text: unique

text: in

text: axially-asymmetric

text: shape.

text: presence

text: of

text: one

text: high-j

text: aligned

text: quasiparticle

text: and

text: for

text: the

text: In

text: connection

text: with

text: available

text: experimental

text: data

text: on

text: ”high-j”

text: γ-value

text: strongly

text: favored

text: by

text: the

text: fully-aligned

text: high-j

text: par-

text: conﬁgurations

text: in

text: the

text: yrast

text: spectroscopy,

text: some

text: phenomena

text: ticle

text: the

text: wobbling

text: excitation

text: of

text: the

text: collective

text: rotational

text: such

text: as

text: the

text: signature-dependence

text: of

text: I

text: val-

text: angular-momentum

text: of

text: the

text: core

text: appears

text: as

text: the

text: yrare

text: band

text: ues

text: in

text: odd-A

text: nuclei

text: and

text: anomalous

text: signature-splitting

text: of

text: in

text: a

text: certain

text: range

text: of

text: angular

text: momentum

text: and

text: the

text: electro-

text: Routhians

text: in

text: odd-A

text: and

text: odd-odd

text: nuclei,

text: were

text: theoretically

text: magnetic

text: transitions

text: between

text: the

text: yrast

text: and

text: yrare

text: bands

text: suggested

text: as

text: the

text: evidence

text: for

text: triaxial

text: shape

text: However,

text: show

text: a

text: unique

text: pattern.

text: before

text: a

text: very

text: clear

text: and

text: ﬁrm

text: evidence

text: for

text: stable

text: tri-

text: The

text: two-phonon

text: wobbling

text: excitation

text: observed

text: in

text: axial

text: shape

text: was

text: hardly

text: obtained

text: experimentally.

text: is

text: one

text: of

text: the

text: most

text: exotic

text: properties

text: of

text: the

text: spinning

text: nu-


text: Proof. Apply Lemmas 2.4, 6.1 and 6.4. Then the theorem follows from ≥ ∗ ≥ | G | | | − p (4, q) g (q) E( ) = q M > q2e γ(logq)3/4. 4 4 M ∗ B Remark 6.6. In the above construction of PHF (4, n,√q, 4), we choose the tangent set of the B { } (cid:100) (cid:101) B hyperedge A(y, m) to be = 0, 2, 5, µ + 5 with µ = 2 logq . This choice of has appeared in B [3], where the authors used such to construct 2-IPP codes. In this paper we choose the same B as they did since in this way we can save the space for proving Lemma 2.4. Actually, when | | B R = 4 there are many choices of satisfying the following conditions ⊆ { (cid:98) − (cid:99)} (a) M 0, 1, . . . , (q 1)/r(R) is R-sum-free, | | − (b) M > q1 o(1), (c) r(R) = o(q(cid:15)) for arbitrary small (cid:15) > 0. | | ≥ B However, for R 5, we do not know whether such exists. 7 Connections to hypergraph Tur´an problems In this section we will study perfect hash families in view of hypergraph Tur´an problems. ∗ − ≤ Theorem 7.1. For arbitrary positive integers t, N, q, it holds that f (q, tN N, t) p (N, q). − ≤ N t Furthermore, N! f (Nq, tN N, t) p (N, q). NN N t Proof. By Lemma 2.2, it suﬃces to prove the ﬁrst statement of the theorem. Recall that if a G G hypergraph is N-uniform N-partite with equal part size q, then E( ) can be represented by × | G | G − an N E( ) q-ary matrix M. If is G(tN N, t)-free, then given any collection of t edges ⊆ G S E( ), it is not hard to verify that in its representation matrix there must exist a row that − G separates S, since otherwise S can contain at most tN N vertices, violating the fact that − is G(tN N, t)-free. Therefore, M can be viewed as the representation matrix of the desired perfect hash family. A direct application of Theorem 7.1 gives the following result. (cid:45) (cid:100) (cid:101)− Corollary 7.2. If 2 N, then for arbitrary (cid:15) > 0, it holds that p (N, q) > q N/2 (cid:15). 3 − − Proof. This corollary follows from the inequality (2), nk o(1) < f (n, 3(r k)+k+1, 3) = o(nk). − r Set N = 2k 1 and t = 3, by Theorem 7.1 one can infer ≥ ∗ ≥ ∗ − ≥ N! − N! (cid:100) (cid:101)− p (N, q) p (N, q) f (q, 3N N, 3) f (Nq, 3N N, 3) > (Nq) N/2 o(1). 3 3 N NN N NN 8 Concluding remarks In this paper we mainly study codes and hash families with the separating property. Several open problems and conjectures concerning the upper or lower bounds are solved. Our two essential methods to study these objects can be summarized as follows. The ﬁrst method is to discover the structural information hidden in the separating property. As an example, our Johnson-type bound (Lemma 3.1) is used to establish Theorem 1.5 and Corollary 5.5. The second one is that we establish a bridge between perfect hash families, graph theory and additive number theory. For example, we solve Conjecture 1.9 by considering a related hypergraph Tur´an problem. We also showed that tools from additive number theory can be used to construct good perfect hash families. As a result, Theorems 5.7, 6.5 and Corollary 7.2 suggest that there may exist a positive answer to Question 1.10. 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Apply

text: Lemmas

text: and

text: Then

text: the

text: theorem

text: follows

text: from

text: G

text: p

text: q)

text: g

text: (q)

text: E(

text: q

text: M

text: M

text: B

text: Remark

text: In

text: the

text: above

text: construction

text: of

text: PHF

text: n,√q,

text: we

text: choose

text: the

text: tangent

text: set

text: of

text: the

text: B

text: B

text: hyperedge

text: A(y,

text: m)

text: to

text: be

text: µ

text: with

text: µ

text: logq

text: This

text: choice

text: of

text: has

text: appeared

text: in

text: B

text: where

text: the

text: authors

text: used

text: such

text: to

text: construct

text: codes.

text: In

text: this

text: paper

text: we

text: choose

text: the

text: same

text: B

text: as

text: they

text: did

text: since

text: in

text: this

text: way

text: we

text: can

text: save

text: the

text: space

text: for

text: proving

text: Lemma

text: Actually,

text: when

text: B

text: R

text: there

text: are

text: many

text: choices

text: of

text: satisfying

text: the

text: following

text: conditions

text: (a)

text: M

text: (q

text: is

text: R-sum-free,

text: (b)

text: M

text: (c)

text: r(R)

text: for

text: arbitrary

text: small

text: B

text: However,

text: for

text: R

text: we

text: do

text: not

text: know

text: whether

text: such

text: exists.

text: Connections

text: to

text: hypergraph

text: Tur´an

text: problems

text: In

text: this

text: section

text: we

text: will

text: study

text: perfect

text: hash

text: families

text: in

text: view

text: of

text: hypergraph

text: Tur´an

text: problems.

text: Theorem

text: For

text: arbitrary

text: positive

text: integers

text: t,

text: N,

text: q,

text: it

text: holds

text: that

text: f

text: (q,

text: tN

text: N,

text: t)

text: p

text: (N,

text: q).

text: N

text: t

text: Furthermore,

text: N!

text: f

text: (Nq,

text: tN

text: N,

text: t)

text: p

text: (N,

text: q).

text: NN

text: N

text: t

text: Proof.

text: By

text: Lemma

text: it

text: suﬃces

text: to

text: prove

text: the

text: ﬁrst

text: statement

text: of

text: the

text: theorem.

text: Recall

text: that

text: if

text: a

text: G

text: G

text: hypergraph

text: is

text: N-uniform

text: N-partite

text: with

text: equal

text: part

text: size

text: q,

text: then

text: E(

text: can

text: be

text: represented

text: by

text: G

text: G

text: an

text: N

text: E(

text: q-ary

text: matrix

text: M.

text: If

text: is

text: G(tN

text: N,

text: t)-free,

text: then

text: given

text: any

text: collection

text: of

text: t

text: edges

text: G

text: S

text: E(

text: it

text: is

text: not

text: hard

text: to

text: verify

text: that

text: in

text: its

text: representation

text: matrix

text: there

text: must

text: exist

text: a

text: row

text: that

text: G

text: separates

text: S,

text: since

text: otherwise

text: S

text: can

text: contain

text: at

text: most

text: tN

text: N

text: vertices,

text: violating

text: the

text: fact

text: that

text: is

text: G(tN

text: N,

text: t)-free.

text: Therefore,

text: M

text: can

text: be

text: viewed

text: as

text: the

text: representation

text: matrix

text: of

text: the

text: desired

text: perfect

text: hash

text: family.

text: A

text: direct

text: application

text: of

text: Theorem

text: gives

text: the

text: following

text: result.

text: Corollary

text: If

text: N,

text: then

text: for

text: arbitrary

text: it

text: holds

text: that

text: p

text: (N,

text: q)

text: q

text: Proof.

text: This

text: corollary

text: follows

text: from

text: the

text: inequality

text: nk

text: f

text: (n,

text: o(nk).

text: r

text: Set

text: N

text: and

text: t

text: by

text: Theorem

text: one

text: can

text: infer

text: N!

text: N!

text: p

text: (N,

text: q)

text: p

text: (N,

text: q)

text: f

text: (q,

text: N,

text: f

text: (Nq,

text: N,

text: (Nq)

text: N

text: NN

text: N

text: NN

text: Concluding

text: remarks

text: In

text: this

text: paper

text: we

text: mainly

text: study

text: codes

text: and

text: hash

text: families

text: with

text: the

text: separating

text: property.

text: Several

text: open

text: problems

text: and

text: conjectures

text: concerning

text: the

text: upper

text: or

text: lower

text: bounds

text: are

text: solved.

text: Our

text: two

text: essential

text: methods

text: to

text: study

text: these

text: objects

text: can

text: be

text: summarized

text: as

text: follows.

text: The

text: ﬁrst

text: method

text: is

text: to

text: discover

text: the

text: structural

text: information

text: hidden

text: in

text: the

text: separating

text: property.

text: As

text: an

text: example,

text: our

text: Johnson-type

text: bound

text: (Lemma

text: is

text: used

text: to

text: establish

text: Theorem

text: and

text: Corollary

text: The

text: second

text: one

text: is

text: that

text: we

text: establish

text: a

text: bridge

text: between

text: perfect

text: hash

text: families,

text: graph

text: theory

text: and

text: additive

text: number

text: theory.

text: For

text: example,

text: we

text: solve

text: Conjecture

text: by

text: considering

text: a

text: related

text: hypergraph

text: Tur´an

text: problem.

text: We

text: also

text: showed

text: that

text: tools

text: from

text: additive

text: number

text: theory

text: can

text: be

text: used

text: to

text: construct

text: good

text: perfect

text: hash

text: families.

text: As

text: a

text: result,

text: Theorems

text: and

text: Corollary

text: suggest

text: that

text: there

text: may

text: exist

text: a

text: positive

text: answer

text: to

text: Question

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [15] S.R. Blackburn. Frameproof codes. SIAM J. Discrete Math., 16(3):499–510 (electronic), 2003. [16] S.R. Blackburn. An upper bound on the size of a code with the k-identiﬁable parent property. J. Combin. Theory Ser. A, 102(1):179–185, 2003. [17] S.R. Blackburn, M. Burmester, Y. Desmedt, and P.R. Wild. Eﬃcient multiplicative sharing schemes. In Advances in cryptology—EUROCRYPT ’96 (Saragossa, 1996), volume 1070 of Lecture Notes in Comput. Sci., pages 107–118. Springer, Berlin, 1996. [18] S.R. Blackburn, T. Etzion, D.R. Stinson, and G.M. Zaverucha. A bound on the size of separating hash families. J. Combin. Theory Ser. A, 115(7):1246–1256, 2008. [19] D. Boneh and J. Shaw. Collusion-secure ﬁngerprinting for digital data. IEEE Trans. Inform. Theory, 44(5):1897–1905, 1998. [20] W.G. Brown, P. Erd˝os, and V.T. S´os. On the existence of triangulated spheres in 3-graphs, and related problems. Period. Math. Hungar., 3(3-4):221–228, 1973. [21] W.G. Brown, P. Erd˝os, and V.T. S´os. Some extremal problems on r-graphs. In New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich, 1971), pages 53–63. Academic Press, New York, 1973. [22] P. Erd˝os, P. Frankl, and V. R¨odl. The asymptotic number of graphs not containing a ﬁxed subgraph and a problem for hypergraphs having no exponent. Graphs Combin., 2(2):113– 121, 1986. [23] P. Erd˝os and D.J. Kleitman. On coloring graphs to maximize the proportion of multicolored k-edges. J. Combinatorial Theory, 5:164–169, 1968. [24] R. Fuji-Hara. Perfect hash families of strength three with three rows from varieties on ﬁnite projective geometries. Des., Codes Cryptogr., to appear. DOI: 10.1007/s10623-015-0052-z. [25] Z. Fu¨redi and M. Ruszink´o. Uniform hypergraphs containing no grids. Adv. Math., 240:302– 324, 2013. [26] C.D. Godsil. Algebraic combinatorics. Chapman and Hall Mathematics Series. Chapman & Hall, New York, 1993. [27] D.L. Hollmann, J.H. van Lint, J.-P. Linnartz, and L.M.G.M. Tolhuizen. On codes with the identiﬁable parent property. J. Combin. Theory Ser. A, 82(2):121–133, 1998. [28] J. K¨orner and K. Marton. New bounds for perfect hashing via information theory. European J. Combin., 9(6):523–530, 1988. [29] S. Martirosyan and T. Trung. Explicit constructions for perfect hash families. Des. Codes Cryptogr., 46(1):97–112, 2008. [30] K. Mehlhorn. Data structures and algorithms. 1. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1984. Sorting and searching. [31] I. Newman and A. Wigderson. Lower bounds on formula size of Boolean functions using hypergraph entropy. SIAM J. Discrete Math., 8(4):536–542, 1995. [32] A. Nilli. Perfect hashing and probability. Combin. Probab. Comput., 3(3):407–409, 1994. [33] I.Z. Ruzsa. Solving a linear equation in a set of integers. I. Acta Arith., 65(3):259–282, 1993. [34] I.Z. Ruzsa and E. Szemer´edi. Triple systems with no six points carrying three triangles. In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, volume 18 of Colloq. Math. Soc. J´anos Bolyai, pages 939–945. North-Holland, Amsterdam-New York, 1978. [35] J.N. Staddon, D.R. Stinson, and R. Wei. Combinatorial properties of frameproof and traceability codes. IEEE Trans. Inform. Theory, 47(3):1042–1049, 2001. 19

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text: subgraph

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text: for

text: hypergraphs

text: having

text: no

text: exponent.

text: Graphs

text: Combin.,

text: P.

text: Erd˝os

text: and

text: D.J.

text: Kleitman.

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text: coloring

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text: to

text: maximize

text: the

text: proportion

text: of

text: multicolored

text: k-edges.

text: J.

text: Combinatorial

text: Theory,

text: R.

text: Fuji-Hara.

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text: strength

text: three

text: with

text: three

text: rows

text: from

text: varieties

text: on

text: ﬁnite

text: projective

text: geometries.

text: Des.,

text: Codes

text: Cryptogr.,

text: to

text: appear.

text: DOI:

text: Z.

text: Fu¨redi

text: and

text: M.

text: Ruszink´o.

text: Uniform

text: hypergraphs

text: containing

text: no

text: grids.

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text: Math.,

text: C.D.

text: Godsil.

text: Algebraic

text: combinatorics.

text: Chapman

text: and

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text: families.

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text: Codes

text: Cryptogr.,

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text: Mehlhorn.

text: Data

text: structures

text: and

text: algorithms.

text: EATCS

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text: on

text: Theoretical

text: Computer

text: Science.

text: Springer-Verlag,

text: Berlin,

text: Sorting

text: and

text: searching.

text: I.

text: Newman

text: and

text: A.

text: Wigderson.

text: Lower

text: bounds

text: on

text: formula

text: size

text: of

text: Boolean

text: functions

text: using

text: hypergraph

text: entropy.

text: SIAM

text: J.

text: Discrete

text: Math.,

text: A.

text: Nilli.

text: Perfect

text: hashing

text: and

text: probability.

text: Combin.

text: Probab.

text: Comput.,

text: I.Z.

text: Ruzsa.

text: Solving

text: a

text: linear

text: equation

text: in

text: a

text: set

text: of

text: integers.

text: I.

text: Acta

text: Arith.,

text: I.Z.

text: Ruzsa

text: and

text: E.

text: Szemer´edi.

text: Triple

text: systems

text: with

text: no

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text: points

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text: Keszthely,

text: Vol.

text: II,

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text: Math.

text: Soc.

text: J´anos

text: Bolyai,

text: pages

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text: Trans.

text: Inform.

text: Theory,


text: A novelty of our proof is that we avoid the use of the grouping coordinates method, which (cid:98) − (cid:99) { } ≥ has appeared in all previous proofs. The constraint C( N/(u 1) , q, w , . . . , w ) u can be ≥ − ≥ 1 t omitted when N u 1 and q u. For the coeﬃcient γ deﬁned in Theorem 1.4, the authors of [7] posed the following question: { } − Question 1.6. ([7]) Is there any type w , . . . , w for which the constant (u 1) in Theorem 1 t − 1.4 can be replaced by another constant strictly smaller than (u 1)? We give a negative answer to their question by presenting the following construction: − − − − Theorem 1.7. There exists a PHF(N; NqN 1, qN 1 + (N 1)qN 2, N + 1) for any integer ≥ ≥ − − q 2 and N 2. As a consequence, γ = u 1 is the minimal real number such that C(u { } ≤ 1, q, w1, . . . , wt ) γq holds for arbitrary q. (cid:80) To see that our construction is actually a negative answer to Question 1.6, one just needs to { (cid:80) } notice that a u-perfect hash family is also w , . . . , w -separating for arbitrary t w = u. If − 1 t − i{=1 i } we set N = u 1 then our construction implies the existence of an SHF(u 1; n, q, w , . . . , w ) − 1 t such that lim →∞ n = u 1 holds for arbitrary t w = u. Therefore, the constant γ can q q − i=1 i never be less than u 1. 1.2 Codes with the identiﬁable parent property We have mentioned 2-IPP codes before and the notion was generalized to codes with the t-identiﬁable parent property (t-IPP codes) in [35]. We postpone the deﬁnition to Section 2 for the sake of saving space. Let i (N, q) denote the maximal cardinality of a t-IPP code of length N over an alphabet t (cid:98) (cid:99) ≤ − (cid:100) − (cid:101) of size q. Let v = (t/2 + 1)2 . One can verify that i (N, q) i (v 1, q N/(v 1) ) (just as the t t case for separating hash families). Thus the problem of bounding i (N, q) can be reduced to − − ≤ −t bounding i (v 1, q). Alon and Stav [6] proved that i (v 1, q) (v 1)q, and they conjectured: t t − Conjecture 1.8. ([6]) There are constructions showing that (v 1) is the best constant in the − ≤ − inequality i (v 1, q) (v 1)q. t Our Theorem 1.7 not only answers Question 1.6 but also veriﬁes this conjecture, since it was observed in [6, 35] that a v-perfect hash family also satisﬁes the t-identiﬁable parent property. 1.3 Perfect hash families (cid:100) − (cid:101) As claimed in [18], the exponent N/(u 1) in the bound of Theorem 1.5 is realistic. We can understand this in two aspects. On the one hand, a probabilistic construction of Blackburn − [13] showed that for any ﬁxed u and any positive real number δ such that δ < N/(u 1), there (cid:98) (cid:99) exists a PHF(N; qδ , q, u) whenever q is suﬃciently large. On the other hand, let p (N, q) t denote the maximal cardinality of a PHF(N; n, q, t), it was respectively observed in [6, 28, 32] ≥ − that p (N, q) (c q)N/(u 1) holds for some constant c . So we can conclude that the exponent (cid:100) u− (cid:101) u − | u N/(u 1) is tight when (u 1) N. − (cid:45) But the problem becomes much more diﬃcult when (u 1) N. It is not known that whether the exponent is tight. Even for the smallest case, u = 3 and N = 3, Walker and Colbourn [39] posed the following conjecture: Conjecture 1.9. ([39]) p (3, q) = o(q2). 3 Note that Theorem 1.5 shows p (3, q) = O(q2). A recent paper [24] showed that p (3, q) = 3 3 Ω(q5/3). Results from ﬁnite geometry were used to construct such families. There is still a huge gap between the upper and lower bounds. For general types of separating hash families, Blackburn et al. [18] asked a similar question: − (cid:45) Question 1.10. ([18]) Let N and w be ﬁxed integers. If (u 1) N, then for suﬃciently i { } large q and arbitrary small (cid:15) > 0, does there exist an SHF(N; n, q, w , . . . , w ) such that ≥ (cid:100) − (cid:101)− 1 t n q N/(u 1) (cid:15)? We prove Conjecture 1.9 in Section 5 (see Theorems 5.4 and 5.7 below). We ﬁnd that perfect hash families are closely related to a hypergraph Tur´an problem. With some transformations, 3 ##LTLine##

text: A

text: novelty

text: of

text: our

text: proof

text: is

text: that

text: we

text: avoid

text: the

text: use

text: of

text: the

text: grouping

text: coordinates

text: method,

text: which

text: has

text: appeared

text: in

text: all

text: previous

text: proofs.

text: The

text: constraint

text: C(

text: N/(u

text: q,

text: w

text: w

text: u

text: can

text: be

text: t

text: omitted

text: when

text: N

text: u

text: and

text: q

text: u.

text: For

text: the

text: coeﬃcient

text: γ

text: deﬁned

text: in

text: Theorem

text: the

text: authors

text: of

text: posed

text: the

text: following

text: question:

text: Question

text: Is

text: there

text: any

text: type

text: w

text: w

text: for

text: which

text: the

text: constant

text: (u

text: in

text: Theorem

text: t

text: can

text: be

text: replaced

text: by

text: another

text: constant

text: strictly

text: smaller

text: than

text: (u

text: We

text: give

text: a

text: negative

text: answer

text: to

text: their

text: question

text: by

text: presenting

text: the

text: following

text: construction:

text: Theorem

text: There

text: exists

text: a

text: PHF(N;

text: NqN

text: qN

text: (N

text: N

text: for

text: any

text: integer

text: q

text: and

text: N

text: As

text: a

text: consequence,

text: γ

text: u

text: is

text: the

text: minimal

text: real

text: number

text: such

text: that

text: C(u

text: q,

text: wt

text: γq

text: holds

text: for

text: arbitrary

text: q.

text: To

text: see

text: that

text: our

text: construction

text: is

text: actually

text: a

text: negative

text: answer

text: to

text: Question

text: one

text: just

text: needs

text: to

text: notice

text: that

text: a

text: u-perfect

text: hash

text: family

text: is

text: also

text: w

text: w

text: -separating

text: for

text: arbitrary

text: t

text: w

text: u.

text: If

text: t

text: i

text: we

text: set

text: N

text: u

text: then

text: our

text: construction

text: implies

text: the

text: existence

text: of

text: an

text: SHF(u

text: n,

text: q,

text: w

text: w

text: t

text: such

text: that

text: lim

text: n

text: u

text: holds

text: for

text: arbitrary

text: t

text: w

text: u.

text: Therefore,

text: the

text: constant

text: γ

text: can

text: q

text: q

text: i

text: never

text: be

text: less

text: than

text: u

text: Codes

text: with

text: the

text: identiﬁable

text: parent

text: property

text: We

text: have

text: mentioned

text: codes

text: before

text: and

text: the

text: notion

text: was

text: generalized

text: to

text: codes

text: with

text: the

text: t-identiﬁable

text: parent

text: property

text: (t-IPP

text: codes)

text: in

text: We

text: postpone

text: the

text: deﬁnition

text: to

text: Section

text: for

text: the

text: sake

text: of

text: saving

text: space.

text: Let

text: i

text: (N,

text: q)

text: denote

text: the

text: maximal

text: cardinality

text: of

text: a

text: t-IPP

text: code

text: of

text: length

text: N

text: over

text: an

text: alphabet

text: t

text: of

text: size

text: q.

text: Let

text: v

text: One

text: can

text: verify

text: that

text: i

text: (N,

text: q)

text: i

text: (v

text: q

text: N/(v

text: (just

text: as

text: the

text: t

text: t

text: case

text: for

text: separating

text: hash

text: families).

text: Thus

text: the

text: problem

text: of

text: bounding

text: i

text: (N,

text: q)

text: can

text: be

text: reduced

text: to

text: −t

text: bounding

text: i

text: (v

text: q).

text: Alon

text: and

text: Stav

text: proved

text: that

text: i

text: (v

text: q)

text: (v

text: and

text: they

text: conjectured:

text: t

text: t

text: Conjecture

text: There

text: are

text: constructions

text: showing

text: that

text: (v

text: is

text: the

text: best

text: constant

text: in

text: the

text: inequality

text: i

text: (v

text: q)

text: (v

text: t

text: Our

text: Theorem

text: not

text: only

text: answers

text: Question

text: but

text: also

text: veriﬁes

text: this

text: conjecture,

text: since

text: it

text: was

text: observed

text: in

text: that

text: a

text: v-perfect

text: hash

text: family

text: also

text: satisﬁes

text: the

text: t-identiﬁable

text: parent

text: property.

text: Perfect

text: hash

text: families

text: As

text: claimed

text: in

text: the

text: exponent

text: N/(u

text: in

text: the

text: bound

text: of

text: Theorem

text: is

text: realistic.

text: We

text: can

text: understand

text: this

text: in

text: two

text: aspects.

text: On

text: the

text: one

text: hand,

text: a

text: probabilistic

text: construction

text: of

text: Blackburn

text: showed

text: that

text: for

text: any

text: ﬁxed

text: u

text: and

text: any

text: positive

text: real

text: number

text: δ

text: such

text: that

text: δ

text: N/(u

text: there

text: exists

text: a

text: PHF(N;

text: qδ

text: q,

text: u)

text: whenever

text: q

text: is

text: suﬃciently

text: large.

text: On

text: the

text: other

text: hand,

text: let

text: p

text: (N,

text: q)

text: t

text: denote

text: the

text: maximal

text: cardinality

text: of

text: a

text: PHF(N;

text: n,

text: q,

text: t),

text: it

text: was

text: respectively

text: observed

text: in

text: that

text: p

text: (N,

text: q)

text: (c

text: q)N/(u

text: holds

text: for

text: some

text: constant

text: c

text: So

text: we

text: can

text: conclude

text: that

text: the

text: exponent

text: u−

text: u

text: u

text: N/(u

text: is

text: tight

text: when

text: (u

text: N.

text: But

text: the

text: problem

text: becomes

text: much

text: more

text: diﬃcult

text: when

text: (u

text: N.

text: It

text: is

text: not

text: known

text: that

text: whether

text: the

text: exponent

text: is

text: tight.

text: Even

text: for

text: the

text: smallest

text: case,

text: u

text: and

text: N

text: Walker

text: and

text: Colbourn

text: posed

text: the

text: following

text: conjecture:

text: Conjecture

text: p

text: q)

text: Note

text: that

text: Theorem

text: shows

text: p

text: q)

text: A

text: recent

text: paper

text: showed

text: that

text: p

text: q)

text: Results

text: from

text: ﬁnite

text: geometry

text: were

text: used

text: to

text: construct

text: such

text: families.

text: There

text: is

text: still

text: a

text: huge

text: gap

text: between

text: the

text: upper

text: and

text: lower

text: bounds.

text: For

text: general

text: types

text: of

text: separating

text: hash

text: families,

text: Blackburn

text: et

text: al.

text: asked

text: a

text: similar

text: question:

text: Question

text: Let

text: N

text: and

text: w

text: be

text: ﬁxed

text: integers.

text: If

text: (u

text: N,

text: then

text: for

text: suﬃciently

text: i

text: large

text: q

text: and

text: arbitrary

text: small

text: does

text: there

text: exist

text: an

text: SHF(N;

text: n,

text: q,

text: w

text: w

text: such

text: that

text: t

text: n

text: q

text: N/(u

text: We

text: prove

text: Conjecture

text: in

text: Section

text: (see

text: Theorems

text: and

text: below).

text: We

text: ﬁnd

text: that

text: perfect

text: hash

text: families

text: are

text: closely

text: related

text: to

text: a

text: hypergraph

text: Tur´an

text: problem.

text: With

text: some

text: transformations,

text: ##LTLine##


text: and the ﬁne mesh. The non-zero entries in the mass and stiﬀness matrix amount to O((2 log(H)/H)d). 4.2 Diagonal preconditioning of the mass matrix This subsection proves that the inverse of the diagonal of the mass matrix is a suitable preconditioner for it. Although this is shown for V with basis functions − H λ Cλ , the arguments and therefore also the result hold as well for the perturbed z z − spaces V (m) of Subsection spanned by the local basis functions λ C λ . H z m z Deﬁne D := dim(S1(T )). Let (Λ ) denote the basis of S1(T ) consist- 0 H k k=1,...,D 0 H − ing of the standard nodal basis functions. Then (λ ) with λ = (1 C)Λ k k=1,...,D k k for C from (2.7) deﬁnes a basis of V . Let M and M denote the mass matrices with respec(cid:88)t to (Λ ) and (λ )H . Let(cid:88)x ∈ RD and let k k=1,...,D k k=1,...,D D D ∈ − ∈ U := x Λ S1(T ) and u := x λ = (1 C)U V H k k 0 H H k k H H k=1 k=1 denote the corresponding functions in S1(T ) and V . Then, 0 H H (cid:62) (cid:107) (cid:107) (cid:107) − (cid:107) ≤ (cid:107) (cid:107) (cid:107) (cid:107) x Mx = u 2 = (1 C)U 2 ( U + CU )2. H H H H ∈ Since CUH Wh, the approximation propert(cid:16)ies (2.5) and the inverse(cid:17) inequal- ity (2.3) imply (cid:16) (cid:17) (cid:107) (cid:107) (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) U + CU U + C(1)H U 1 + C(1)HC (S1(T )) U H H H IH H √ IH inv 0 H H (cid:62) = 1 + C(1)HC (S1(T )) x Mx. IH inv 0 H On the other hand, since U = I u , the L2 stability (2.6) leads to H H H √ √ (cid:62) (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:62) x Mx = U C(0) u = C(0) x Mx. H IH H IH Given A, B ∈ R, let A ≈ B abbreviate that there exist generic constants C > ≤ ≤ 1 0, C > 0 independent of the mesh size, such that A C B C A. Since 2 ≈ − 1 2 C (S1(T )) H 1, the above result reads inv 0 H (cid:62) ≈ (cid:62) x Mx x Mx. Since this result also holds(cid:88)for the unit ve(cid:88)ctors e ∈ RD, this implies j D D (cid:62) (cid:62) ≈ (cid:62) (cid:62) x diag(M)x = x2e Me x2e Me = x diag(M)x. j j j j j j j=1 j=1 (cid:62) ≈ (cid:62) Since x diag(M)x x Mx [30], it follows (cid:62) ≈ (cid:62) ≈ (cid:62) ≈ (cid:62) x diag(M)x x diag(M)x x Mx x Mx. − Therefore, (diag(M)) 1 is a suitable preconditioner for M and the application − of M 1 may be replaced with a few iterations of the preconditioned conjugate gradient method. 15 ##LTLine## ##LTLine## ##LTLine##

text: and

text: the

text: ﬁne

text: mesh.

text: The

text: non-zero

text: entries

text: in

text: the

text: mass

text: and

text: stiﬀness

text: matrix

text: amount

text: to

text: log(H)/H)d).

label: section

text: Diagonal

text: preconditioning

text: of

text: the

text: mass

text: matrix

text: This

text: subsection

text: proves

text: that

text: the

text: inverse

text: of

text: the

text: diagonal

text: of

text: the

text: mass

text: matrix

text: is

text: a

text: suitable

text: preconditioner

text: for

text: it.

text: Although

text: this

text: is

text: shown

text: for

text: V

text: with

text: basis

text: functions

text: H

text: λ

text: Cλ

text: the

text: arguments

text: and

text: therefore

text: also

text: the

text: result

text: hold

text: as

text: well

text: for

text: the

text: perturbed

text: z

text: z

text: spaces

text: V

text: (m)

text: of

text: Subsection

text: spanned

text: by

text: the

text: local

text: basis

text: functions

text: λ

text: C

text: λ

text: H

text: z

text: m

text: z

text: Deﬁne

text: D

text: Let

text: (Λ

text: denote

text: the

text: basis

text: of

text: consist-

text: H

text: k

text: H

text: ing

text: of

text: the

text: standard

text: nodal

text: basis

text: functions.

text: Then

text: (λ

text: with

text: λ

text: C)Λ

text: k

text: k

text: k

text: for

text: C

text: from

text: deﬁnes

text: a

text: basis

text: of

text: V

text: Let

text: M

text: and

text: M

text: denote

text: the

text: mass

text: matrices

text: with

text: to

text: (Λ

text: and

text: (λ

text: )H

text: RD

text: and

text: let

text: k

text: k

text: D

text: D

text: U

text: x

text: Λ

text: and

text: u

text: x

text: λ

text: C)U

text: V

text: H

text: k

text: k

text: H

text: H

text: k

text: k

text: H

text: H

text: denote

text: the

text: corresponding

text: functions

text: in

text: and

text: V

text: Then,

text: H

text: H

text: x

text: Mx

text: u

text: C)U

text: U

text: CU

text: H

text: H

text: H

text: H

text: Since

text: CUH

text: Wh,

text: the

text: approximation

text: and

text: the

text: inequal-

text: ity

text: imply

text: U

text: CU

text: U

text: U

text: U

text: H

text: H

text: H

text: IH

text: H

text: IH

text: inv

text: H

text: H

text: x

text: Mx.

text: IH

text: inv

text: H

text: On

text: the

text: other

text: hand,

text: since

text: U

text: I

text: u

text: the

text: stability

text: leads

text: to

text: H

text: H

text: H

text: x

text: Mx

text: U

text: u

text: x

text: Mx.

text: H

text: IH

text: H

text: IH

text: Given

text: A,

text: B

text: R,

text: let

text: A

text: B

text: abbreviate

text: that

text: there

text: exist

text: generic

text: constants

text: C

text: C

text: independent

text: of

text: the

text: mesh

text: size,

text: such

text: that

text: A

text: C

text: B

text: C

text: A.

text: Since

text: C

text: H

text: the

text: above

text: result

text: reads

text: inv

text: H

text: x

text: Mx

text: x

text: Mx.

text: Since

text: this

text: result

text: also

text: the

text: unit

text: e

text: RD,

text: this

text: implies

text: j

text: D

text: D

text: x

text: diag(M)x

text: Me

text: Me

text: x

text: diag(M)x.

text: j

text: j

text: j

text: j

text: j

text: j

text: Since

text: x

text: diag(M)x

text: x

text: Mx

text: it

text: follows

text: x

text: diag(M)x

text: x

text: diag(M)x

text: x

text: Mx

text: x

text: Mx.

text: Therefore,

text: (diag(M))

text: is

text: a

text: suitable

text: preconditioner

text: for

text: M

text: and

text: the

text: application

text: of

text: M

text: may

text: be

text: replaced

text: with

text: a

text: few

text: iterations

text: of

text: the

text: preconditioned

text: conjugate

text: gradient

text: method.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: very well in the case of locally isolated reﬁnement and essentially two separate spatial discretisation scales. However, adaptive triangulations arising from spatial singularities are typically graded towards the singularity and encounter an ever increasing number of spatial discretisation scales. Although the generalisation of local time stepping to this case with an increasing number of diﬀerent time step sizes is possible [11], its realisation is certainly challenging and its behaviour with regard to stability and computational complexity is still open in such scenarios. Our aim is to provide an alternative approach for the stabilisation of explicit time stepping that is based on reduction of spatial degrees of freedom rather than en- riching the temporal discretisation. Other approaches to overcome a strong CFL condition is the locally implicit method analysed in [19], which combines an explicit method with an implicit method in the region, where the mesh-size is small, and the singular complement method [7], which adds singular functions to the standard ansatz space. The remaining parts of this paper are organised as follows. Section deﬁnes a generalised ﬁnite element space and proves optimal approximation properties and the improved inverse inequality in Lemma 2.1. Section introduces the discretisa- tion of the wave equation and states an error estimate. Section discusses some practical aspects and generalisations of the method, while Section concludes the paper with a numerical experiment. Standard notation on Lebesgue and Sobolev spaces is employed throughout (cid:107) • (cid:107) (cid:107) • (cid:107) • • the paper and := abbreviates the L2 norm over Ω, while ( , ) L2(Ω) • L2(Ω) denotes the L2 scalar product. The notation abbreviates the identity mapping. The space L2(0, T ; X) denotes the space of Bochner square integrable functions ∈ − ∈ from [0, T ] with values in X. The dual pairing between f H 1(Ω) and v H1(Ω) (cid:104) (cid:105) 0 is denoted by f, v − × . The symbol C denotes a generic constant which H 1(Ω) H1(Ω) is independent of the mesh s0ize. 2 Spatial reduction This section recalls the CFL condition for the leapfrog discretisation from Section below in the context of adaptive (spatial) ﬁnite elements and presents our novel reduction technique. 2.1 CFL condition, inverse inequality and approximation Given a shape regular triangulation T , let S1(T ) denote the standard P - h 0 h 1 FEM space of T -piecewise aﬃne and globally continuous functions, which vanish h on ∂Ω. The precise CFL condition for the leapfrog discretisation with underlying ﬁnite element space S1(T ) reads 0 h √ ≤ 2 ∆t , (2.1) C (S1(T )) inv 0 h where C (S1(T )) is the best constant in the inverse inequality in S1(T ), inv 0 h 0 h i.e., (cid:107)∇ (cid:107) ≤ (cid:107) (cid:107) ∈ v C (S1(T )) v for all v S1(T ). h inv 0 h h h 0 h 3 ##LTLine## ##LTLine##

text: very

text: well

text: in

text: the

text: case

text: of

text: locally

text: isolated

text: reﬁnement

text: and

text: essentially

text: two

text: separate

text: spatial

text: discretisation

text: scales.

text: However,

text: adaptive

text: triangulations

text: arising

text: from

text: spatial

text: singularities

text: are

text: typically

text: graded

text: towards

text: the

text: singularity

text: and

text: encounter

text: an

text: ever

text: increasing

text: number

text: of

text: spatial

text: discretisation

text: scales.

text: Although

text: the

text: generalisation

text: of

text: local

text: time

text: stepping

text: to

text: this

text: case

text: with

text: an

text: increasing

text: number

text: of

text: diﬀerent

text: time

text: step

text: sizes

text: is

text: possible

text: its

text: realisation

text: is

text: certainly

text: challenging

text: and

text: its

text: behaviour

text: with

text: regard

text: to

text: stability

text: and

text: computational

text: complexity

text: is

text: still

text: open

text: in

text: such

text: scenarios.

text: Our

text: aim

text: is

text: to

text: provide

text: an

text: alternative

text: approach

text: for

text: the

text: stabilisation

text: of

text: explicit

text: time

text: stepping

text: that

text: is

text: based

text: on

text: reduction

text: of

text: spatial

text: degrees

text: of

text: freedom

text: rather

text: than

text: en-

text: riching

text: the

text: temporal

text: discretisation.

text: Other

text: approaches

text: to

text: overcome

text: a

text: strong

text: CFL

text: condition

text: is

text: the

text: locally

text: implicit

text: method

text: analysed

text: in

text: which

text: combines

text: an

text: explicit

text: method

text: with

text: an

text: implicit

text: method

text: in

text: the

text: region,

text: where

text: the

text: mesh-size

text: is

text: small,

text: and

text: the

text: singular

text: complement

text: method

text: which

text: adds

text: singular

text: functions

text: to

text: the

text: standard

text: ansatz

text: space.

text: The

text: remaining

text: parts

text: of

text: this

text: paper

text: are

text: organised

text: as

text: follows.

text: Section

text: deﬁnes

text: a

text: generalised

text: ﬁnite

text: element

text: space

text: and

text: proves

text: optimal

text: approximation

text: properties

text: and

text: the

text: improved

text: inverse

text: inequality

text: in

text: Lemma

text: Section

text: introduces

text: the

text: discretisa-

text: tion

text: of

text: the

text: wave

text: equation

text: and

text: states

text: an

text: error

text: estimate.

text: Section

text: discusses

text: some

text: practical

text: aspects

text: and

text: generalisations

text: of

text: the

text: method,

text: while

text: Section

text: concludes

text: the

text: paper

text: with

text: a

text: numerical

text: experiment.

text: Standard

text: notation

text: on

text: Lebesgue

text: and

text: Sobolev

text: spaces

text: is

text: employed

text: throughout

text: the

text: paper

text: and

text: abbreviates

text: the

text: norm

text: over

text: Ω,

text: while

text: denotes

text: the

text: scalar

text: product.

text: The

text: notation

text: abbreviates

text: the

text: identity

text: mapping.

text: The

text: space

text: T

text: X)

text: denotes

text: the

text: space

text: of

text: Bochner

text: square

text: integrable

text: functions

text: from

text: T

text: with

text: values

text: in

text: X.

text: The

text: dual

text: pairing

text: between

text: f

text: H

text: and

text: v

text: is

text: denoted

text: by

text: f,

text: v

text: The

text: symbol

text: C

text: denotes

text: a

text: generic

text: constant

text: which

text: H

text: is

text: independent

text: of

text: the

text: mesh

text: Spatial

text: reduction

text: This

text: section

text: recalls

text: the

text: CFL

text: condition

text: for

text: the

text: leapfrog

text: discretisation

text: from

text: Section

text: below

text: in

text: the

text: context

text: of

text: adaptive

text: (spatial)

text: ﬁnite

text: elements

text: and

text: presents

text: our

text: novel

text: reduction

text: technique.

text: CFL

text: condition,

text: inverse

text: inequality

text: and

text: approximation

text: Given

text: a

text: shape

text: regular

text: triangulation

text: T

text: let

text: denote

text: the

text: standard

text: P

text: h

text: h

text: FEM

text: space

text: of

text: T

text: -piecewise

text: aﬃne

text: and

text: globally

text: continuous

text: functions,

text: which

text: vanish

text: h

text: on

text: ∂Ω.

text: The

text: precise

text: CFL

text: condition

text: for

text: the

text: leapfrog

text: discretisation

text: with

text: underlying

text: ﬁnite

text: element

text: space

text: reads

text: h

text: ∆t

text: C

text: inv

text: h

text: where

text: C

text: is

text: the

text: best

text: constant

text: in

text: the

text: inverse

text: inequality

text: in

text: inv

text: h

text: h

text: i.e.,

text: v

text: C

text: v

text: for

text: all

text: v

text: h

text: inv

text: h

text: h

text: h

text: h

text: ##LTLine##

text: ##LTLine##


text: [26] Oswald, P.: On a BPX-preconditioner for P1 elements. Computing 51(2), 125–133 (1993). [27] Peterseim, D.: Variational multiscale stabilization and the exponential decay of ﬁne-scale correctors (2015). Preprint, arXiv:1505.07611 [28] Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990). [29] Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh- Reﬁnement Techniques. Advances in numerical mathematics. Wiley (1996) [30] Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA Journal of Numerical Analysis 7(4), 449–457 (1987). 21

text: Oswald,

text: P.:

text: On

text: a

text: BPX-preconditioner

text: for

text: elements.

text: Computing

text: Peterseim,

text: D.:

text: Variational

text: multiscale

text: stabilization

text: and

text: the

text: exponential

text: decay

text: of

text: ﬁne-scale

text: correctors

text: Preprint,

text: Scott,

text: L.R.,

text: Zhang,

text: S.:

text: Finite

text: element

text: interpolation

text: of

text: nonsmooth

text: functions

text: satisfying

text: boundary

text: conditions.

text: Math.

text: Comp.

text: Verfürth,

text: R.:

text: A

text: Review

text: of

text: a

text: Posteriori

text: Error

text: Estimation

text: and

text: Adaptive

text: Mesh-

text: Reﬁnement

text: Techniques.

text: Advances

text: in

text: numerical

text: mathematics.

text: Wiley

text: Wathen,

text: A.J.:

text: Realistic

text: eigenvalue

text: bounds

text: for

text: the

text: Galerkin

text: mass

text: matrix.

text: IMA

text: Journal

text: of

text: Numerical

text: Analysis


text: and the L2 stability (cid:107) (cid:107) ≤ (cid:107) (cid:107) ∈ I v C(0) v for all v V. (2.6) H IH While (2.5) is a standard property of quasi-interpolations, the L2-stability (2.6) is not, e.g., the Scott-Zhang quasi-interpolation [28] is not L2 stable. For an admissible projective quasi-interpolation, which satisﬁes both (2.5) and (2.6), one may think of the L2 projection onto S1(T ), which is H1 stable on quasi-uniform 0 H meshes [2]. Another example for this Clément-type quasi-interpolation is given in | ⊆ Subsection below. Denote the kernel of I as W := ker(I ) S1(T ). ∈ H ∈ h H S01(Th) 0 h Given v S1(T ), deﬁne the projection Cv W of v onto W by H 0 H H h H h ∇ ∇ ∇ ∇ ∈ ( w , Cv ) = ( w , v ) for all w W . (2.7) h H L2(Ω) h H L2(Ω) h h Section below discusses the eﬃcient computation of this projection, e.g., by − localisation. Deﬁne the space V by V := (1 C)S1(T ), which implies H H 0 H ⊕ S1(T ) = W V (2.8) 0 h h H ∇• ∇• and the sum is orthogonal with respect to ( , ) . The following lemma L2(Ω) ≤ − proves that the inverse inequality (2.3) holds with constant C (V ) CH 1 inv H independent of the minimal mesh size in T . Moreover, a direct consequence of h this lemma is that the approximation property (2.4) is preserved in the coarse space V . H ∈ ∈ Lemma 2.1. There exists a constant C such that for all u V with ∆u ∈ ∇ ∇appr − ∈ L2(Ω) and all u S1(T ) with ( u , v ) = ( ∆u, v ) for all v h 0 h h h L2(Ω) h L2(Ω) h S1(T ), it holds 0 h (cid:107) − (cid:107) ≤ (cid:107) (cid:107) inf u v C H ∆u . (2.9) ∈ h H H1(Ω) appr vH VH ≤ − Furthermore, the constant C (V ) from (2.3) satisﬁes C (V ) CH 1. inv H inv H ∈ Proof. Let u V be the Galerkin projection of u onto V , i.e., H H h H ∇ ∇ ∇ ∇ ∈ ( u , v ) = ( u , v ) for all v V . H H L2(Ω) h H L2(Ω) H H − ∈ Set e := u u S1(T ). The Galerkin orthogonality h h H 0 h ∇ ∇ ∈ ( e , v ) = 0 for all v V h H L2(Ω) H H ∈ and the orthogonality of the subspace decomposition (2.8) imply that e W . h h The approximation properties (2.5) of I therefore lead to H (cid:107) (cid:107) (cid:107) − (cid:107) ≤ (cid:107)∇ (cid:107) (1) e = e I e C H e h h H h IH h 5

text: and

text: the

text: stability

label: equation

text: I

text: v

text: v

text: for

text: all

text: v

text: V.

text: H

text: IH

text: While

text: is

text: a

text: standard

text: property

text: of

text: quasi-interpolations,

text: the

text: is

text: not,

text: e.g.,

text: the

text: Scott-Zhang

text: quasi-interpolation

text: is

text: not

text: stable.

text: For

text: an

text: admissible

text: projective

text: quasi-interpolation,

text: which

text: satisﬁes

text: both

text: and

text: one

text: may

text: think

text: of

text: the

text: projection

text: onto

text: which

text: is

text: stable

text: on

text: quasi-uniform

text: H

text: meshes

text: Another

text: example

text: for

text: this

text: Clément-type

text: quasi-interpolation

text: is

text: given

text: in

text: Subsection

text: below.

text: Denote

text: the

text: kernel

text: of

text: I

text: as

text: W

text: ker(I

text: H

text: h

text: H

text: h

text: Given

text: v

text: deﬁne

text: the

text: projection

text: Cv

text: W

text: of

text: v

text: onto

text: W

text: by

text: H

text: H

text: H

text: h

text: H

text: h

text: w

text: Cv

text: w

text: v

text: for

text: all

text: w

text: W

text: h

text: H

text: h

text: H

text: h

text: h

text: Section

text: below

text: discusses

text: the

text: eﬃcient

text: computation

text: of

text: this

text: projection,

text: e.g.,

text: by

text: localisation.

text: Deﬁne

text: the

text: space

text: V

text: by

text: V

text: which

text: implies

text: H

text: H

text: H

text: W

text: V

text: h

text: h

text: H

text: and

text: the

text: sum

text: is

text: orthogonal

text: with

text: respect

text: to

text: The

text: following

text: lemma

text: proves

text: that

text: the

text: inverse

text: inequality

text: holds

text: with

text: constant

text: C

text: (V

text: CH

text: inv

text: H

text: independent

text: of

text: the

text: minimal

text: mesh

text: size

text: in

text: T

text: Moreover,

text: a

text: direct

text: consequence

text: of

text: h

text: this

text: lemma

text: is

text: that

text: the

text: approximation

text: property

text: is

text: preserved

text: in

text: the

text: coarse

text: space

text: V

text: H

text: Lemma

text: There

text: exists

text: a

text: constant

text: C

text: such

text: that

text: for

text: all

text: u

text: V

text: with

text: ∆u

text: ∇appr

text: and

text: all

text: u

text: with

text: u

text: v

text: ∆u,

text: v

text: for

text: all

text: v

text: h

text: h

text: h

text: h

text: h

text: h

text: it

text: holds

text: h

text: inf

text: u

text: v

text: C

text: H

text: ∆u

text: h

text: H

text: appr

text: vH

text: VH

text: Furthermore,

text: the

text: constant

text: C

text: (V

text: from

text: satisﬁes

text: C

text: (V

text: CH

text: inv

text: H

text: inv

text: H

text: Proof.

text: Let

text: u

text: V

text: be

text: the

text: Galerkin

text: projection

text: of

text: u

text: onto

text: V

text: i.e.,

text: H

text: H

text: h

text: H

text: u

text: v

text: u

text: v

text: for

text: all

text: v

text: V

text: H

text: H

text: h

text: H

text: H

text: H

text: Set

text: e

text: u

text: u

text: The

text: Galerkin

text: orthogonality

text: h

text: h

text: H

text: h

text: e

text: v

text: for

text: all

text: v

text: V

text: h

text: H

text: H

text: H

text: and

text: the

text: orthogonality

text: of

text: the

text: subspace

text: decomposition

text: imply

text: that

text: e

text: W

text: h

text: h

text: The

text: approximation

text: properties

text: of

text: I

text: therefore

text: lead

text: to

text: H

text: e

text: e

text: I

text: e

text: C

text: H

text: e

text: h

text: h

text: H

text: h

text: IH

text: h