text: could be used in a local interpolation along the adiabatic connection that preserves size consistency.32 Finally, our study also provides reference data for the anions of the He isoelectronic series close to and at the quantum phase transition that can be valuable to test the accuracy of new DFT approximations (see, e.g., Ref. 63 which presents correlation potentials from RPA approaches that are good approximations to the true correlation potential). ACKNOWLEDGMENTS We are very grateful to Erik van Lenthe for his assistance with the ADF package, and to A. J. Cohen and P. Mori-Sanchez for pointing out an error in Fig. 10 concerning the Hartree-Fock curve. This work was supported by the Netherlands Organization for Scientiﬁc Research (NWO) through a Vidi grant and by the NSF through grant DMR-0908653. REFERENCES afxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@url [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty P. Hohenberg and W. Kohn, noop Phys. Rev. 136, B 864 (1964). W. Kohn and L. J. Sham, noop Phys. Rev. 140, A 1133 (1965). A. J. Cohen, P. Mori-Sánchez, and W. Yang, noop Chem. Rev. 112, 289 (2012). C. J. Cramer and D. G. Truhlar, noop Phys. Chem. Chem. Phys. 11, 10757 (2009). A. D. Becke, noop J. Chem. Phys. 138, 074109 (2013). A. D. Becke, noop J. Chem. Phys. 138, 161101 (2013). A. D. Becke, noop J. Chem. Phys. 139, 021104 (2013). J. P. Perdew, A. Ruzsinszky, J. Tao, V. N. Staroverov, G. E. Scuseria, and G. I. Csonka, noop J. Chem. Phys. 123, 062201 (2005). R. Peverati and D. G. Truhlar, noop Phil. Trans. R. Soc. A 372, 20120476 (2014). M. Seidl, P. Gori-Giorgi, and A. Savin, noop Phys. Rev. A 75, 042511 (2007). P. Gori-Giorgi, G. Vignale, and M. Seidl, noop J. Chem. Theory Comput. 5, 743 (2009). P. Gori-Giorgi, M. Seidl, and G. Vignale, noop Phys. Rev. Lett. 103, 166402 (2009). 27

text: could

text: be

text: used

text: in

text: a

text: local

text: interpolation

text: along

text: the

text: adiabatic

text: connection

text: that

text: preserves

text: size

text: Finally,

text: our

text: study

text: also

text: provides

text: reference

text: data

text: for

text: the

text: anions

text: of

text: the

text: He

text: isoelectronic

text: series

text: close

text: to

text: and

text: at

text: the

text: quantum

text: phase

text: transition

text: that

text: can

text: be

text: valuable

text: to

text: test

text: the

text: accuracy

text: of

text: new

text: DFT

text: approximations

text: (see,

text: e.g.,

text: Ref.

text: which

text: presents

text: correlation

text: potentials

text: from

text: RPA

text: approaches

text: that

text: are

text: good

text: approximations

text: to

text: the

text: true

text: correlation

text: potential).

text: ACKNOWLEDGMENTS

text: We

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text: Erik

text: van

text: Lenthe

text: for

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text: ADF

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text: error

text: in

text: Fig.

text: concerning

text: the

text: Hartree-Fock

text: curve.

text: This

text: work

text: was

text: supported

text: by

text: the

text: Netherlands

text: Organization

text: for

text: Scientiﬁc

text: Research

text: (NWO)

text: through

text: a

text: Vidi

text: grant

text: and

text: by

text: the

text: NSF

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text: Message Encoding for Spread and Orbit Codes Anna-Lena Trautmann Department of Electrical and Computer Systems Engineering Monash University, Clayton, Australia Department of Electrical and Electronic Engineering University of Melbourne, Australia Email: anna-lena.trautmann@unimelb.edu.au Abstract— Spread codes and orbit codes are special families of II. PRELIMINARIES constant dimension subspace codes. These codes have been well- F studied for their error correction capability and transmission We denote the ﬁnite ﬁeld with q elements by . The set F P q rate, but the question of how to encode messages has not been of all subspaces of n is denoted by (n) and the set of all investigated. In this work we show how the message space can subspaces of Fn of dqimension k, calledqthe Grassmannian, is be chosen for a given code and how message en- and decoding G q U ∈ G denoted by (k, n). We represent a vector space (k, n) can be done. q ∈ F × q by a matrix U k n such that the row space of U, denoted q U by rs(U), is equal to . A subspace code is simply a subset of I. INTRODUCTION P G (n) and a constant dimension code is a subset of (k, n). q P q Subspace codes are deﬁned to be sets of subspaces of some A metric on (n) is given by the subspace distance ([11]) F q given ambient space n of dimension n over the ﬁnite ﬁeld U V U V − U ∩ V q with q elements. When we talk about constant dimension dS( , ) := dim( ) + dim( ) 2 dim( ) codes, we restrict ourselves to subspace codes, whose code- U V ∈ P C for any , (n). The minimum distance d ( ) of a words all have the same constant dimension. Subspace codes C ⊆qP S subspace code (n) is the minimum of all the pairwise in general, and constant dimension codes in particular, have q distances of the codewords. Since the dual of a subspace code received much attention since it was shown in [11] how these C C has the same minimum distance as (see e.g. [11]), it codes can be used for random network coding. ≤ is customary to restrict oneself to k n/2, which we will In that same paper [11] a class of Reed-Solomon-like codes assume throughout the paper. is proposed, which was shown to be equivalent to the lifting of G A spread code [12] in (k, n) is deﬁned as a set of maximum rank distance codes [18]. For theses codes one can G q M elements of (k, n) that pairwise intersect only trivially and easily ﬁnd a suitable message space (or message set) and q F | M cover the whole space n. They exist if and only if k n, an encoding map, that maps injectively to the subspace q − − have minimum distance 2k and cardinality (qn 1)/(qk 1). code. For more information on different constructions and decoding During the last years other constructions of subspace codes algorithms of spread codes see [8], [12], [13], [20]. We will use were developed, e.g. in [1], [2], [3], [4], [6], [7], [10], [12], the following construction, which gives rise to a Desarguesian [19], [24]. Some of these constructions have the mere purpose G spread code in (k, n) ([20]): of giving an improved transmission rate (i.e. larger cardinality q G 1) Let m := n/k and consider (1, m), which has of the code for the same parameters), while others also have − − − · · · qk − − qk(m 1)+qk(m 2)+qk(m 3)+ +1 = (qn 1)/(qk 1) some structure that can be used e.g. for decoding. The problem elements. Trivially, all these lines intersect only trivially. of message encoding has been addressed in almost none of 2) Let P be the companion matrix of an irreducible these papers and is hence an open question for most of these F codes. We want to study this problem for two classes of pFolyn∼omFial over q of degree k. Then it holds that = [P] and we can use this isomorphism in any subspace codes, namely spread codes and orbit codes. qk q G element of (1, m) (i.e. we replace any coordinate The paper is organized as follows: In the following section qk with the respective matrix) to receive a spread code in we will give some preliminaries, among others the spread G (k, n). code and orbit code construction. In Section we investigate q a natural message space and encoding map for Desarguesian Example 1. Le∼t α be a ro(cid:18)ot of x2 (cid:19)+ x + 1, i.e. a primitive F F spread codes, which we then extend to an encoding map on element of = [α]. The respective companion matrix is 22 2 a set of integer numbers. In Section we do the same for orbit codes. In Section we propose a hybrid encoding method, 0 1 P = . combining two encoding and decoding algorithms for spread 1 1 codes. We conclude this work in Section . G { } Then 22(1, 2) = rs(1, 0), rs(1, α), rFs(1, α∼2), Frs(1, 1), rs(0, 1) The author was supported by Swiss National Science Foundation Fellow- and substituting all elemenFts of 22 = 2[α] Gwith its ship No. 147304. corresponding element from [P] gives a spread in (2, 4). 2 2

text: Message

label: title

text: Encoding

text: for

text: Spread

text: and

text: Orbit

text: Codes

text: Anna-Lena

label: author

text: Trautmann

text: Department

label: paragraph

text: of

text: Electrical

text: and

text: Computer

text: Systems

text: Engineering

text: Monash

text: University,

text: Clayton,

text: Australia

text: Department

text: of

text: Electrical

text: and

text: Electronic

text: Engineering

text: University

text: of

text: Melbourne,

text: Australia

text: Email:

text: anna-lena.trautmann@unimelb.edu.au

text: Abstract—

text: Spread

label: abstract

text: codes

text: and

text: orbit

text: codes

text: are

text: special

text: families

text: of

text: II.

label: section

text: PRELIMINARIES

text: constant

text: dimension

text: subspace

text: codes.

text: These

text: codes

text: have

text: been

text: well-

text: F

text: studied

text: for

text: their

text: error

text: correction

text: capability

text: and

text: transmission

text: We

text: denote

text: the

text: ﬁnite

text: ﬁeld

text: with

text: q

text: elements

text: by

text: The

text: set

text: F

text: P

text: q

text: rate,

text: but

text: the

text: question

text: of

text: how

text: to

text: encode

text: messages

text: has

text: not

text: been

text: of

text: all

text: subspaces

text: of

text: n

text: is

text: denoted

text: by

text: (n)

text: and

text: the

text: set

text: of

text: all

text: investigated.

text: In

text: this

text: work

text: we

text: show

text: how

text: the

text: message

text: space

text: can

text: subspaces

text: of

text: Fn

text: of

text: dqimension

text: k,

text: calledqthe

text: Grassmannian,

text: is

text: be

text: chosen

text: for

text: a

text: given

text: code

text: and

text: how

text: message

text: en-

text: and

text: decoding

text: G

text: q

text: U

text: G

text: denoted

text: by

text: (k,

text: n).

text: We

text: represent

text: a

text: vector

text: space

text: (k,

text: n)

text: can

text: be

text: done.

text: q

text: F

text: q

text: by

text: a

text: matrix

text: U

text: k

text: n

text: such

text: that

text: the

text: row

text: space

text: of

text: U,

text: denoted

text: q

text: U

text: by

text: rs(U),

text: is

text: equal

text: to

text: A

text: subspace

text: code

text: is

text: simply

text: a

text: subset

text: of

text: I.

text: INTRODUCTION

text: P

text: G

text: (n)

text: and

text: a

text: constant

text: dimension

text: code

text: is

text: a

text: subset

text: of

text: (k,

text: n).

text: q

text: P

text: q

text: Subspace

text: codes

text: are

text: deﬁned

text: to

text: be

text: sets

text: of

text: subspaces

text: of

text: some

text: A

text: metric

text: on

text: (n)

text: is

text: given

text: by

text: the

text: subspace

text: distance

text: F

text: q

text: given

text: ambient

text: space

text: n

text: of

text: dimension

text: n

text: over

text: the

text: ﬁnite

text: ﬁeld

text: U

text: V

text: U

text: V

text: U

text: V

text: q

text: with

text: q

text: elements.

text: When

text: we

text: talk

text: about

text: constant

text: dimension

text: dS(

text: dim(

text: dim(

text: dim(

text: codes,

text: we

text: restrict

text: ourselves

text: to

text: subspace

text: codes,

text: whose

text: code-

text: U

text: V

text: P

text: C

text: for

text: any

text: (n).

text: The

text: minimum

text: distance

text: d

text: of

text: a

text: words

text: all

text: have

text: the

text: same

text: constant

text: dimension.

text: Subspace

text: codes

text: C

text: ⊆qP

text: S

text: subspace

text: code

text: (n)

text: is

text: the

text: minimum

text: of

text: all

text: the

text: pairwise

text: in

text: general,

text: and

text: constant

text: dimension

text: codes

text: in

text: particular,

text: have

text: q

text: distances

text: of

text: the

text: codewords.

text: Since

text: the

text: dual

text: of

text: a

text: subspace

text: code

text: received

text: much

text: attention

text: since

text: it

text: was

text: shown

text: in

text: how

text: these

text: C

text: C

text: has

text: the

text: same

text: minimum

text: distance

text: as

text: (see

text: e.g.

text: it

text: codes

text: can

text: be

text: used

text: for

text: random

text: network

text: coding.

text: is

text: customary

text: to

text: restrict

text: oneself

text: to

text: k

text: which

text: we

text: will

text: In

text: that

text: same

text: paper

text: a

text: class

text: of

text: Reed-Solomon-like

text: codes

text: assume

text: throughout

text: the

text: paper.

text: is

text: proposed,

text: which

text: was

text: shown

text: to

text: be

text: equivalent

text: to

text: the

text: lifting

text: of

text: G

text: A

text: spread

text: code

text: in

text: (k,

text: n)

text: is

text: deﬁned

text: as

text: a

text: set

text: of

text: maximum

text: rank

text: distance

text: codes

text: For

text: theses

text: codes

text: one

text: can

text: G

text: q

text: M

text: elements

text: of

text: (k,

text: n)

text: that

text: pairwise

text: intersect

text: only

text: trivially

text: and

text: easily

text: ﬁnd

text: a

text: suitable

text: message

text: space

text: (or

text: message

text: set)

text: and

text: q

text: F

text: M

text: cover

text: the

text: whole

text: space

text: n.

text: They

text: exist

text: if

text: and

text: only

text: if

text: k

text: n,

text: an

text: encoding

text: map,

text: that

text: maps

text: injectively

text: to

text: the

text: subspace

text: q

text: have

text: minimum

text: distance

text: and

text: cardinality

text: (qn

text: code.

text: For

text: more

text: information

text: on

text: different

text: constructions

text: and

text: decoding

text: During

text: the

text: last

text: years

text: other

text: constructions

text: of

text: subspace

text: codes

text: algorithms

text: of

text: spread

text: codes

text: see

text: We

text: will

text: use

text: were

text: developed,

text: e.g.

text: in

text: the

text: following

text: construction,

text: which

text: gives

text: rise

text: to

text: a

text: Desarguesian

text: Some

text: of

text: these

text: constructions

text: have

text: the

text: mere

text: purpose

text: G

text: spread

text: code

text: in

text: (k,

text: n)

text: of

text: giving

text: an

text: improved

text: transmission

text: rate

text: (i.e.

text: larger

text: cardinality

text: q

text: G

text: Let

text: m

text: n/k

text: and

text: consider

text: m),

text: which

text: has

text: of

text: the

text: code

text: for

text: the

text: same

text: parameters),

text: while

text: others

text: also

text: have

text: qk

text: qk(m

text: (qn

text: some

text: structure

text: that

text: can

text: be

text: used

text: e.g.

text: for

text: decoding.

text: The

text: problem

text: elements.

text: Trivially,

text: all

text: these

text: lines

text: intersect

text: only

text: trivially.

text: of

text: message

text: encoding

text: has

text: been

text: addressed

text: in

text: almost

text: of

text: Let

text: P

text: be

text: the

text: companion

text: matrix

text: of

text: an

text: irreducible

text: these

text: papers

text: and

text: is

text: hence

text: an

text: open

text: question

text: for

text: most

text: of

text: these

text: F

text: codes.

text: We

text: want

text: to

text: study

text: this

text: problem

text: for

text: two

text: classes

text: of

text: pFolyn∼omFial

text: over

text: q

text: of

text: degree

text: k.

text: Then

text: it

text: holds

text: that

text: [P]

text: and

text: we

text: can

text: use

text: this

text: isomorphism

text: in

text: any

text: subspace

text: codes,

text: namely

text: spread

text: codes

text: and

text: orbit

text: codes.

text: qk

text: q

text: G

text: element

text: of

text: m)

text: (i.e.

text: we

text: replace

text: any

text: coordinate

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: In

text: the

text: following

text: section

text: qk

text: with

text: the

text: respective

text: matrix)

text: to

text: receive

text: a

text: spread

text: code

text: in

text: we

text: will

text: give

text: some

text: preliminaries,

text: among

text: others

text: the

text: spread

text: G

text: (k,

text: n).

text: code

text: and

text: orbit

text: code

text: construction.

text: In

text: Section

text: we

text: investigate

text: q

text: a

text: natural

text: message

text: space

text: and

text: encoding

text: map

text: for

text: Desarguesian

text: Example

text: Le∼t

text: α

text: be

text: a

text: of

text: x

text: i.e.

text: a

text: primitive

text: F

text: F

text: spread

text: codes,

text: which

text: we

text: then

text: extend

text: to

text: an

text: encoding

text: map

text: on

text: element

text: of

text: [α].

text: The

text: respective

text: companion

text: matrix

text: is

text: a

text: set

text: of

text: integer

text: numbers.

text: In

text: Section

text: we

text: do

text: the

text: same

text: for

text: orbit

text: codes.

text: In

text: Section

text: we

text: propose

text: a

text: hybrid

text: encoding

text: method,

text: P

text: combining

text: two

text: encoding

text: and

text: decoding

text: algorithms

text: for

text: spread

text: codes.

text: We

text: conclude

text: this

text: work

text: in

text: Section

text: G

text: Then

text: α),

text: The

text: author

text: was

text: supported

text: by

text: Swiss

text: National

text: Science

text: Foundation

text: Fellow-

text: and

text: substituting

text: all

text: elemenFts

text: of

text: Gwith

text: its

text: ship

text: No.

text: corresponding

text: element

text: from

text: [P]

text: gives

text: a

text: spread

text: in


text: [24] A.-L. Trautmann and J. Rosenthal. New improvements on the echelon- Ferrers construction. In Proceedings of the 19th International Sympo- sium on Mathematical Theory of Networks and Systems – MTNS, pages 405–408, Budapest, Hungary, 2010.

label: reference

text: A.-L.

text: Trautmann

text: and

text: J.

text: Rosenthal.

text: New

text: improvements

text: on

text: the

text: echelon-

text: Ferrers

text: construction.

text: In

text: Proceedings

text: of

text: the

text: International

text: Sympo-

text: sium

text: on

text: Mathematical

text: Theory

text: of

text: Networks

text: and

text: Systems

text: MTNS,

text: pages

text: Budapest,

text: Hungary,


text: Compatible ﬁnite element methods for numerical weather prediction C. J. Cotter and A. T. T. McRae May 26, 2020 Abstract This article takes the form of a tutorial on the use of a particular class of mixed ﬁnite element methods, which can be thought of as the ﬁnite element extension of the C-grid staggered ﬁnite diﬀerence method. The class is often referred to as compatible ﬁnite ele- ments, mimetic ﬁnite elements, discrete diﬀerential forms or ﬁnite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores. 1 Introduction Mixed ﬁnite element methods are a generalisation of staggered ﬁnite diﬀerence methods, and are intended to address the same problem: spurious pressure mode(s) observed in the ﬁnite diﬀerence A-grid and are also observed in ﬁnite element methods when the same ﬁnite ﬁnite element method, diﬀerent ﬁnite element spaces are selected for diﬀerent variables. The vast range of available ﬁnite element spaces is both a blessing and a curse, and many diﬀerent combinations have been proposed, analysed and used for large scale geophysical ﬂuid dynamics applications, particularly in the ocean modelling community Le Roux et al. (2005, 2007); Le Roux and Pouliot (2008); Danilov (2010); Cotter et al. (2009); Cotter and Ham (2011); Rostand and Le Roux (2008); Le Roux (2012); Comblen et al. (2010), whilst many other combinations have been used in engineering applications where diﬀerent scales and modelling aspects are important. In this article we limit the scope considerably by discussing only one particular family of mixed ﬁnite element methods, known variously as compatible ﬁnite elements (the term that we shall use here), mimetic ﬁnite elements, discrete diﬀerential forms or ﬁnite element exterior calculus. These ﬁnite element methods have the important property that diﬀerential operators such as grad and curl map from one ﬁnite element space; these embedding properties lead to discrete versions of the div-curl and curl-grad identities of vector calculus. This echoes the important “mimetic” properties of the C-grid ﬁnite diﬀerence method, discussed in full generality on unstructured grids, and shown to be highly relevant and useful in geophysical ﬂuid dynamics applications, in Thuburn et al. (2009); Ringler et al. (2010). This generalisation of the C-grid method is very useful since it allows (i) use arbitrary grids, with no requirement of orthogonal grids, without loss of consistency/convergence rate; (ii) extra ﬂexibility in the choice of discretisation to optimize the ratio between global velocity degrees of freedom (DoFs) and global pressure DoFs to eliminate spurious mode branches; and (iii) the option to increase the consistency/convergence order. Compatible ﬁnite element methods were ﬁrst identiﬁed in the 1970s, and quickly became very popular amongst numerical analysts since this additional mathematical structure facilitated proofs of stability and convergence, and provided powerful insight. These results were collected and uniﬁed in Brezzi and Fortin (1991), an excellent book which has been out of print for a long 1

text: Compatible

text: ﬁnite

text: element

text: methods

text: for

text: numerical

text: weather

text: prediction

text: C.

text: J.

text: Cotter

text: and

text: A.

text: T.

text: T.

text: McRae

text: May

text: Abstract

text: This

text: article

text: takes

text: the

text: form

text: of

text: a

text: tutorial

text: on

text: the

text: use

text: of

text: a

text: particular

text: class

text: of

text: mixed

text: ﬁnite

text: element

text: methods,

text: which

text: can

text: be

text: thought

text: of

text: as

text: the

text: ﬁnite

text: element

text: extension

text: of

text: the

text: C-grid

text: staggered

text: ﬁnite

text: diﬀerence

text: method.

text: The

text: class

text: is

text: often

text: referred

text: to

text: as

text: compatible

text: ﬁnite

text: ele-

text: ments,

text: mimetic

text: ﬁnite

text: elements,

text: discrete

text: diﬀerential

text: forms

text: or

text: ﬁnite

text: element

text: exterior

text: calculus.

text: We

text: provide

text: an

text: elementary

text: introduction

text: in

text: the

text: case

text: of

text: the

text: one-dimensional

text: wave

text: equation,

text: before

text: summarising

text: recent

text: results

text: in

text: applications

text: to

text: the

text: rotating

text: shallow

text: water

text: equations

text: on

text: the

text: sphere,

text: before

text: taking

text: an

text: outlook

text: towards

text: applications

text: in

text: three-dimensional

text: compressible

text: dynamical

text: cores.

text: Introduction

text: Mixed

text: ﬁnite

text: element

text: methods

text: are

text: a

text: generalisation

text: of

text: staggered

text: ﬁnite

text: diﬀerence

text: methods,

text: and

text: are

text: intended

text: to

text: address

text: the

text: same

text: problem:

text: spurious

text: pressure

text: mode(s)

text: observed

text: in

text: the

text: ﬁnite

text: diﬀerence

text: A-grid

text: and

text: are

text: also

text: observed

text: in

text: ﬁnite

text: element

text: methods

text: when

text: the

text: same

text: ﬁnite

text: ﬁnite

text: element

text: method,

text: diﬀerent

text: ﬁnite

text: element

text: spaces

text: are

text: selected

text: for

text: diﬀerent

text: variables.

text: The

text: vast

text: range

text: of

text: available

text: ﬁnite

text: element

text: spaces

text: is

text: both

text: a

text: blessing

text: and

text: a

text: curse,

text: and

text: many

text: diﬀerent

text: combinations

text: have

text: been

text: proposed,

text: analysed

text: and

text: used

text: for

text: large

text: scale

text: geophysical

text: ﬂuid

text: dynamics

text: applications,

text: particularly

text: in

text: the

text: ocean

text: modelling

text: community

text: Le

text: Roux

text: et

text: al.

text: Le

text: Roux

text: and

text: Pouliot

text: Danilov

text: Cotter

text: et

text: al.

text: Cotter

text: and

text: Ham

text: Rostand

text: and

text: Le

text: Roux

text: Le

text: Roux

text: Comblen

text: et

text: al.

text: whilst

text: many

text: other

text: combinations

text: have

text: been

text: used

text: in

text: engineering

text: applications

text: where

text: diﬀerent

text: scales

text: and

text: modelling

text: aspects

text: are

text: important.

text: In

text: this

text: article

text: we

text: limit

text: the

text: scope

text: considerably

text: by

text: discussing

text: only

text: one

text: particular

text: family

text: of

text: mixed

text: ﬁnite

text: element

text: methods,

text: known

text: variously

text: as

text: compatible

text: ﬁnite

text: elements

text: (the

text: term

text: that

text: we

text: shall

text: use

text: here),

text: mimetic

text: ﬁnite

text: elements,

text: discrete

text: diﬀerential

text: forms

text: or

text: ﬁnite

text: element

text: exterior

text: calculus.

text: These

text: ﬁnite

text: element

text: methods

text: have

text: the

text: important

text: property

text: that

text: diﬀerential

text: operators

text: such

text: as

text: grad

text: and

text: curl

text: map

text: from

text: one

text: ﬁnite

text: element

text: space;

text: these

text: embedding

text: properties

text: lead

text: to

text: discrete

text: versions

text: of

text: the

text: div-curl

text: and

text: curl-grad

text: identities

text: of

text: vector

text: calculus.

text: This

text: echoes

text: the

text: important

text: “mimetic”

text: properties

text: of

text: the

text: C-grid

text: ﬁnite

text: diﬀerence

text: method,

text: discussed

text: in

text: full

text: generality

text: on

text: unstructured

text: grids,

text: and

text: shown

text: to

text: be

text: highly

text: relevant

text: and

text: useful

text: in

text: geophysical

text: ﬂuid

text: dynamics

text: applications,

text: in

text: Thuburn

text: et

text: al.

text: Ringler

text: et

text: al.

text: This

text: generalisation

text: of

text: the

text: C-grid

text: method

text: is

text: very

text: useful

text: since

text: it

text: allows

text: (i)

text: use

text: arbitrary

text: grids,

text: with

text: no

text: requirement

text: of

text: orthogonal

text: grids,

text: without

text: loss

text: of

text: consistency/convergence

text: rate;

text: (ii)

text: extra

text: ﬂexibility

text: in

text: the

text: choice

text: of

text: discretisation

text: to

text: optimize

text: the

text: ratio

text: between

text: global

text: velocity

text: degrees

text: of

text: freedom

text: (DoFs)

text: and

text: global

text: pressure

text: DoFs

text: to

text: eliminate

text: spurious

text: mode

text: branches;

text: and

text: (iii)

text: the

text: option

text: to

text: increase

text: the

text: consistency/convergence

text: order.

text: Compatible

text: ﬁnite

text: element

text: methods

text: were

text: ﬁrst

text: identiﬁed

text: in

text: the

text: and

text: quickly

text: became

text: very

text: popular

text: amongst

text: numerical

text: analysts

text: since

text: this

text: additional

text: mathematical

text: structure

text: facilitated

text: proofs

text: of

text: stability

text: and

text: convergence,

text: and

text: provided

text: powerful

text: insight.

text: These

text: results

text: were

text: collected

text: and

text: uniﬁed

text: in

text: Brezzi

text: and

text: Fortin

text: an

text: excellent

text: book

text: which

text: has

text: been

text: out

text: of

text: print

text: for

text: a

text: long


text: long time evolution of this state accurately, it is essential that the numerical discretisation exactly reproduces steady geostrophic states, otherwise the errors in representing this balance will lead to spurious motions that are much larger than the slow evolution when Rossby waves or nonlinear evolution are introduced, and the forecast will be useless. The C-grid staggering exactly reproduces steady geostrophic states, if the Coriolis term is represented correctly (see Thuburn et al. (2009); Thuburn and Cotter (2012) for how to do this on very general grids); this accounts for the popularity and success of the C-grid in numerical weather prediction. (cid:82) We now demonstrate that compatible ﬁnite element methods also have steady geostrophic ∇· states under the same conditions (this was ﬁrst shown in Cotter and Shipton (2012)). If u = 0 ∈ − ∇ · ∈ ∇⊥ and Ω u d x = 0 for ∈u V1, th(cid:90)en ht = H(cid:90) u = 0. Next, we can ﬁnd ψ V0 with u = ψ. Then we solve for h V from the equation 2 ∀ ∈ φgh d x = φfψ d x, φ V , (20) 2 Ω Ω (cid:90) (cid:90) (cid:90) i.e., gh is the projection of fψ into V . Then 2 · −(cid:90) · ⊥ (cid:90) ∇ · w u d x = fw u d x + wgh d x, t Ω Ω Ω (cid:90) · ∇ (cid:90)∇ · = fw ψ d x + wgh d x, Ω Ω − ∇ · ∇ · = wfψ d x + wgh d x, Ω Ω = 0, where we may integrate by parts in the third line since w has continuous normal component and ψ is continuous which means that the integration by parts is exact, and where we note in ∇ · ∈ ∇ · the fourth line that w V , and so we may use Equation (20) with φ = w, meaning that 2 we obtain 0 in the ﬁnal line. Hence, we have an exact steady state. The extension to the nonlinear shallow water equations makes use of the vector invariant form, (cid:18) (cid:19) ⊥ ∇ 1| | u + qhu + gh + u 2 = 0, (21) t 2 ∇ · h + (hu) = 0, (22) t where q is the shallow water potential vorticity ∇⊥ · u + f ∇⊥ · −∂u ∂u q = , u = 1 + 2 , (23) h ∂y ∂x · ∇ ⊥ ∇ | | ∇⊥ and we have used the split (u )u = qu + ( u 2/2). If we apply to Equation (21) and substitute Equation (23) we obtain ∇ · (qh) + (qhu) = 0, (24) t which is the conservation law for q. This takes an important role in predicting the large scale balanced ﬂow. Three issues need to be addressed when discretising these equations with compatible ﬁnite ∈ ∈ | | element methods. First, if u V and h V , then gh + u 2/2 has discontinuities and the 1 2 ∇ · gradient is not globally deﬁned. Second, although we can evaluate u globally, we cannot ∇ · evaluate (hu) globally, as h is discontinuous. Third, we need a way of calculating q. The 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: long

text: time

text: evolution

text: of

text: this

text: state

text: accurately,

text: it

text: is

text: essential

text: that

text: the

text: numerical

text: discretisation

text: exactly

text: reproduces

text: steady

text: geostrophic

text: states,

text: otherwise

text: the

text: errors

text: in

text: representing

text: this

text: balance

text: will

text: lead

text: to

text: spurious

text: motions

text: that

text: are

text: much

text: larger

text: than

text: the

text: slow

text: evolution

text: when

text: Rossby

text: waves

text: or

text: nonlinear

text: evolution

text: are

text: introduced,

text: and

text: the

text: forecast

text: will

text: be

text: useless.

text: The

text: C-grid

text: staggering

text: exactly

text: reproduces

text: steady

text: geostrophic

text: states,

text: if

text: the

text: Coriolis

text: term

text: is

text: represented

text: correctly

text: (see

text: Thuburn

text: et

text: al.

text: Thuburn

text: and

text: Cotter

text: for

text: how

text: to

text: do

text: this

text: on

text: very

text: general

text: grids);

text: this

text: accounts

text: for

text: the

text: popularity

text: and

text: success

text: of

text: the

text: C-grid

text: in

text: numerical

text: weather

text: prediction.

text: We

text: now

text: demonstrate

text: that

text: compatible

text: ﬁnite

text: element

text: methods

text: also

text: have

text: steady

text: geostrophic

text: states

text: under

text: the

text: same

text: conditions

text: (this

text: was

text: ﬁrst

text: shown

text: in

text: Cotter

text: and

text: Shipton

text: If

text: u

text: and

text: Ω

text: u

text: d

text: x

text: for

text: ∈u

text: ht

text: u

text: Next,

text: we

text: can

text: ﬁnd

text: ψ

text: with

text: u

text: ψ.

text: Then

text: we

text: solve

text: for

text: h

text: V

text: from

text: the

text: equation

text: φgh

text: d

text: x

text: φfψ

text: d

text: x,

text: φ

text: V

text: Ω

text: Ω

text: i.e.,

text: gh

text: is

text: the

text: projection

text: of

text: fψ

text: into

text: V

text: Then

text: w

text: u

text: d

text: x

text: fw

text: u

text: d

text: x

text: wgh

text: d

text: x,

text: t

text: Ω

text: Ω

text: Ω

text: fw

text: ψ

text: d

text: x

text: wgh

text: d

text: x,

text: Ω

text: Ω

text: wfψ

text: d

text: x

text: wgh

text: d

text: x,

text: Ω

text: Ω

text: where

text: we

text: may

text: integrate

text: by

text: parts

text: in

text: the

text: third

text: line

text: since

text: w

text: has

text: continuous

text: normal

text: component

text: and

text: ψ

text: is

text: continuous

text: which

text: means

text: that

text: the

text: integration

text: by

text: parts

text: is

text: exact,

text: and

text: where

text: we

text: note

text: in

text: the

text: fourth

text: line

text: that

text: w

text: V

text: and

text: so

text: we

text: may

text: use

text: Equation

text: with

text: φ

text: w,

text: meaning

text: that

text: we

text: obtain

text: in

text: the

text: ﬁnal

text: line.

text: Hence,

text: we

text: have

text: an

text: exact

text: steady

text: state.

text: The

text: extension

text: to

text: the

text: nonlinear

text: shallow

text: water

text: equations

text: makes

text: use

text: of

text: the

text: vector

text: invariant

text: form,

text: u

text: qhu

text: gh

text: u

text: t

text: h

text: (hu)

text: t

text: where

text: q

text: is

text: the

text: shallow

text: water

text: potential

text: vorticity

text: u

text: f

text: −∂u

text: ∂u

text: q

text: u

text: h

text: ∂y

text: ∂x

text: and

text: we

text: have

text: used

text: the

text: split

text: (u

text: )u

text: qu

text: u

text: If

text: we

text: apply

text: to

text: Equation

text: and

text: substitute

text: Equation

text: we

text: obtain

text: (qh)

text: (qhu)

text: t

text: which

text: is

text: the

text: conservation

text: law

text: for

text: q.

text: This

text: takes

text: an

text: important

text: role

text: in

text: predicting

text: the

text: large

text: scale

text: balanced

text: ﬂow.

text: Three

text: issues

text: need

text: to

text: be

text: addressed

text: when

text: discretising

text: these

text: equations

text: with

text: compatible

text: ﬁnite

text: element

text: methods.

text: First,

text: if

text: u

text: V

text: and

text: h

text: V

text: then

text: gh

text: u

text: has

text: discontinuities

text: and

text: the

text: gradient

text: is

text: not

text: globally

text: deﬁned.

text: Second,

text: although

text: we

text: can

text: evaluate

text: u

text: globally,

text: we

text: cannot

text: evaluate

text: (hu)

text: globally,

text: as

text: h

text: is

text: discontinuous.

text: Third,

text: we

text: need

text: a

text: way

text: of

text: calculating

text: q.

text: The

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) integrating the h term by parts in the ﬁnite element approximation, and we obtain x (cid:90) L − ∀ ∈ wu w h d x = 0, w V , t x 0 0 L ∀ ∈ φ(h + u ) d x = 0, φ V . t x 1 0 There is no boundary term arising from integration by parts due to the periodic boundary conditions. Three out of the four terms in these two equations involve trivial projections that do nothing, the u , h and u terms. This means that they introduce no further errors beyond t t x approximating the initial conditions in the ﬁnite element spaces. The only term that we have to worry about is the discretised h term, where we would like to convince ourselves that there are x no spurious modes. This is done by showing that the following mathematical condition holds. Deﬁnition 2 (inf-sup condition). The s(cid:12)(cid:12)p(cid:82)aces V0 an(cid:12)(cid:12)d V1 satisfy the inf-sup condition3 if there exists a constant C > 0, independent of t(cid:12)he choice o(cid:12)f mesh, such that L w h d x 0 x ≥ (cid:107) (cid:107) sup (cid:107) (cid:107) C h , (12) ∈ (cid:54) w L2 w V0,w=0 x L2 ∈ for all non-constant h V . 1 This prevents spurious modes because it says that for any non-constant h, there exists at least one w such that the integral is reasonably large in magnitude compared to the size of w x and h. In general, proving the inf-sup condition for mixed ﬁnite element methods is a fairly technical business (see Auricchio et al. (2004) for a review). However, for our compatible ﬁnite element discretisation it is completely straightforward. Proposition 3 (inf-sup condition for compatible ﬁnite elements). Let V and V be chosen such ∈ ∈ 0 1 that if h V is non-constant then we can ﬁnd w V such that w = h. Then the inf-sup 1 0 x condition is satisﬁed, with C = 1. (cid:48) (cid:48) Proof. For any non-consta(cid:12)nt h, take w(cid:12) su(cid:12)ch that w =(cid:12) h (which is possible by the assumption (cid:12)(cid:82) (cid:12) (cid:12)(cid:82) x (cid:12) of the proposition). Then (cid:12) (cid:12) (cid:12) (cid:12) L L (cid:48) (cid:107) (cid:107) s∈up 0(cid:107)wwx(cid:107)h d x ≥ 0(cid:107)ww(cid:48)x(cid:107)h d x = (cid:107)hh(cid:107)2L2 = (cid:107)h(cid:107)L2. (13) w V0 x L2 x L2 L2 − The assumption of the proposition is true whenever V = CG(p), V = DG(p 1). This means 0 1 that our compatible ﬁnite element methods will be free from spurious modes. This concludes our discussion of the one-dimensional case; in the next section we discuss the construction of two-dimensional compatible ﬁnite element spaces and their application to the shallow water equations. 3 Application to linear and nonlinear shallow water equa- tions on the sphere In one dimension, a compatible ﬁnite element method was described for the wave equation, ∈ ∈ by choosing two ﬁnite element spaces V and V , such that (i) if u V , then u V ; and (ii) 0 1 0 x 1 3Here sup is short for supremum, which is the maximum value, roughly speaking. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: integrating

text: the

text: h

text: term

text: by

text: parts

text: in

text: the

text: ﬁnite

text: element

text: approximation,

text: and

text: we

text: obtain

text: x

text: L

text: wu

text: w

text: h

text: d

text: x

text: w

text: V

text: t

text: x

text: L

text: φ(h

text: u

text: d

text: x

text: φ

text: V

text: t

text: x

text: There

text: is

text: no

text: boundary

text: term

text: arising

text: from

text: integration

text: by

text: parts

text: due

text: to

text: the

text: periodic

text: boundary

text: conditions.

text: Three

text: out

text: of

text: the

text: four

text: terms

text: in

text: these

text: two

text: equations

text: involve

text: trivial

text: projections

text: that

text: do

text: nothing,

text: the

text: u

text: h

text: and

text: u

text: terms.

text: This

text: means

text: that

text: they

text: introduce

text: no

text: further

text: errors

text: beyond

text: t

text: t

text: x

text: approximating

text: the

text: initial

text: conditions

text: in

text: the

text: ﬁnite

text: element

text: spaces.

text: The

text: only

text: term

text: that

text: we

text: have

text: to

text: worry

text: about

text: is

text: the

text: discretised

text: h

text: term,

text: where

text: we

text: would

text: like

text: to

text: convince

text: ourselves

text: that

text: there

text: are

text: x

text: no

text: spurious

text: modes.

text: This

text: is

text: done

text: by

text: showing

text: that

text: the

text: following

text: mathematical

text: condition

text: holds.

text: Deﬁnition

text: (inf-sup

text: condition).

text: The

text: satisfy

text: the

text: inf-sup

text: if

text: there

text: exists

text: a

text: constant

text: C

text: independent

text: of

text: choice

text: mesh,

text: such

text: that

text: L

text: w

text: h

text: d

text: x

text: x

text: sup

text: C

text: h

text: w

text: w

text: x

text: for

text: all

text: non-constant

text: h

text: V

text: This

text: prevents

text: spurious

text: modes

text: because

text: it

text: says

text: that

text: for

text: any

text: non-constant

text: h,

text: there

text: exists

text: at

text: least

text: one

text: w

text: such

text: that

text: the

text: integral

text: is

text: reasonably

text: large

text: in

text: magnitude

text: compared

text: to

text: the

text: size

text: of

text: w

text: x

text: and

text: h.

text: In

text: general,

text: proving

text: the

text: inf-sup

text: condition

text: for

text: mixed

text: ﬁnite

text: element

text: methods

text: is

text: a

text: fairly

text: technical

text: business

text: (see

text: Auricchio

text: et

text: al.

text: for

text: a

text: review).

text: However,

text: for

text: our

text: compatible

text: ﬁnite

text: element

text: discretisation

text: it

text: is

text: completely

text: straightforward.

text: Proposition

text: (inf-sup

text: condition

text: for

text: compatible

text: ﬁnite

text: elements).

text: Let

text: V

text: and

text: V

text: be

text: chosen

text: such

text: that

text: if

text: h

text: V

text: is

text: non-constant

text: then

text: we

text: can

text: ﬁnd

text: w

text: V

text: such

text: that

text: w

text: h.

text: Then

text: the

text: inf-sup

text: x

text: condition

text: is

text: satisﬁed,

text: with

text: C

text: Proof.

text: For

text: any

text: h,

text: take

text: that

text: w

text: h

text: (which

text: is

text: possible

text: by

text: the

text: assumption

text: x

text: of

text: the

text: proposition).

text: Then

text: L

text: L

text: s∈up

text: d

text: x

text: d

text: x

text: w

text: x

text: x

text: The

text: assumption

text: of

text: the

text: proposition

text: is

text: true

text: whenever

text: V

text: CG(p),

text: V

text: DG(p

text: This

text: means

text: that

text: our

text: compatible

text: ﬁnite

text: element

text: methods

text: will

text: be

text: free

text: from

text: spurious

text: modes.

text: This

text: concludes

text: our

text: discussion

text: of

text: the

text: one-dimensional

text: case;

text: in

text: the

text: next

text: section

text: we

text: discuss

text: the

text: construction

text: of

text: two-dimensional

text: compatible

text: ﬁnite

text: element

text: spaces

text: and

text: their

text: application

text: to

text: the

text: shallow

text: water

text: equations.

text: Application

text: to

text: linear

text: and

text: nonlinear

text: shallow

text: water

text: equa-

text: tions

text: on

text: the

text: sphere

text: In

text: one

text: dimension,

text: a

text: compatible

text: ﬁnite

text: element

text: method

text: was

text: described

text: for

text: the

text: wave

text: equation,

text: by

text: choosing

text: two

text: ﬁnite

text: element

text: spaces

text: V

text: and

text: V

text: such

text: that

text: (i)

text: if

text: u

text: V

text: then

text: u

text: V

text: and

text: (ii)

text: x

text: sup

text: is

text: short

text: for

text: supremum,

text: which

text: is

text: the

text: maximum

text: value,

text: roughly

text: speaking.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 4: Top: Diagram illustrating V = CG 1, V = RT 0, V = DG 0, on quadrilaterals. On 0 1 2 squares, CG 1 means bilinear, i.e. linear function of x multiplied by linear function of y. RT 0 functions are vector-valued, with the x-component being linear in x and constant in y, and the y-component being constant in y and linear in x. Bottom: Diagram illustrating V = CG 2, 0 V = RT 1, V = DG 1, where CG 2 are biquadratic functions (product of quadratic function 1 2 of x and quadratic function of y), RT 1 have x-component quadratic in x and linear in y, and y-component linear in x and quadratic in y. (cid:90) (cid:90) (cid:90) · · ⊥ (cid:90) − ∇ · ∇ ∀ ∈ w u d x + fw u d x g w h d x = 0, w V , (16) t 1 Ω Ω Ω ∇ · ∀ ∈ φ (h + H u) d x = 0, φ V . (17) t 2 Ω ∇· ∈ Since h +H u V , the projection of Equation (15) is trivial, i.e. Equation (15) is satisﬁed t 2 exactly under this discretisation. This means that we only need to scrutinise the discretisation of the Coriolis and the pressure gradient terms. In the previous section, we discussed the inf-sup condition for one dimensional compa(cid:12)t(cid:82)ible ﬁnite ele(cid:12)ment methods. The equivalent condition in (cid:12) (cid:12) the two dimensional (and in fact, three dimensional) case is ∇ · wh d x ≥ (cid:107) (cid:107) sup Ω(cid:107)∇ · (cid:107) C h , (18) ∈ w L2 w V1 L2 ∈ for all non-constant h V . In the compatible ﬁnite element case we again obtain C = 1, 2 and can provably avoid spurious pressure modes. (cid:82) Regarding the Coriolis term, the crucial condition for large scale balanced ﬂow (i.e., large scale numerical weather prediction) is that the numerical discretisation supports exact geostrophic ∇ · balance. In the model equations (14-15), if u = 0 and u d x = 0, then (15) implies that ∇⊥ Ω h = 0. We have u = ψ for some streamfunction ψ. If we choose gh = fψ then t − ⊥ − ∇ ∇ − u = fu g h = (fψ gh) = 0, (19) t and we have a steady state, which we call geostrophic balance. If we allow f to vary with y, leading to Rossby waves, or introduce nonlinear terms, then this state of geostrophic balance starts to evolve on a slow timescale relative to the rapidly oscillating gravity waves. In large scale ﬂow, the weather system stays close to this balanced state. If we wish to predict the 10 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: Top:

label: caption

text: Diagram

text: illustrating

text: V

text: CG

text: V

text: RT

text: V

text: DG

text: on

text: quadrilaterals.

text: On

text: squares,

text: CG

text: means

text: bilinear,

text: i.e.

text: linear

text: function

text: of

text: x

text: multiplied

text: by

text: linear

text: function

text: of

text: y.

text: RT

text: functions

text: are

text: vector-valued,

text: with

text: the

text: x-component

text: being

text: linear

text: in

text: x

text: and

text: constant

text: in

text: y,

text: and

text: the

text: y-component

text: being

text: constant

text: in

text: y

text: and

text: linear

text: in

text: x.

text: Bottom:

text: Diagram

text: illustrating

text: V

text: CG

text: V

text: RT

text: V

text: DG

text: where

text: CG

text: are

text: biquadratic

text: functions

text: (product

text: of

text: quadratic

text: function

text: of

text: x

text: and

text: quadratic

text: function

text: of

text: y),

text: RT

text: have

text: x-component

text: quadratic

text: in

text: x

text: and

text: linear

text: in

text: y,

text: and

text: y-component

text: linear

text: in

text: x

text: and

text: quadratic

text: in

text: y.

text: w

text: u

text: d

text: x

text: fw

text: u

text: d

text: x

text: g

text: w

text: h

text: d

text: x

text: w

text: V

text: t

text: Ω

text: Ω

text: Ω

text: φ

text: (h

text: H

text: u)

text: d

text: x

text: φ

text: V

text: t

text: Ω

text: Since

text: h

text: +H

text: u

text: V

text: the

text: projection

text: of

text: Equation

text: is

text: trivial,

text: i.e.

text: Equation

text: is

text: satisﬁed

text: t

text: exactly

text: under

text: this

text: discretisation.

text: This

text: means

text: that

text: we

text: only

text: need

text: to

text: scrutinise

text: the

text: discretisation

text: of

text: the

text: Coriolis

text: and

text: the

text: pressure

text: gradient

text: terms.

text: In

text: the

text: previous

text: section,

text: we

text: discussed

text: the

text: inf-sup

text: condition

text: for

text: one

text: dimensional

text: ﬁnite

text: methods.

text: The

text: equivalent

text: condition

text: in

text: the

text: two

text: dimensional

text: (and

text: in

text: fact,

text: three

text: dimensional)

text: case

text: is

text: wh

text: d

text: x

text: sup

text: C

text: h

text: w

text: w

text: for

text: all

text: non-constant

text: h

text: V

text: In

text: the

text: compatible

text: ﬁnite

text: element

text: case

text: we

text: again

text: obtain

text: C

text: and

text: can

text: provably

text: avoid

text: spurious

text: pressure

text: modes.

text: Regarding

text: the

text: Coriolis

text: term,

text: the

text: crucial

text: condition

text: for

text: large

text: scale

text: balanced

text: ﬂow

text: (i.e.,

text: large

text: scale

text: numerical

text: weather

text: prediction)

text: is

text: that

text: the

text: numerical

text: discretisation

text: supports

text: exact

text: geostrophic

text: balance.

text: In

text: the

text: model

text: equations

text: if

text: u

text: and

text: u

text: d

text: x

text: then

text: implies

text: that

text: Ω

text: h

text: We

text: have

text: u

text: ψ

text: for

text: some

text: streamfunction

text: ψ.

text: If

text: we

text: choose

text: gh

text: fψ

text: then

text: t

text: u

text: fu

text: g

text: h

text: (fψ

text: gh)

text: t

text: and

text: we

text: have

text: a

text: steady

text: state,

text: which

text: we

text: call

text: geostrophic

text: balance.

text: If

text: we

text: allow

text: f

text: to

text: vary

text: with

text: y,

text: leading

text: to

text: Rossby

text: waves,

text: or

text: introduce

text: nonlinear

text: terms,

text: then

text: this

text: state

text: of

text: geostrophic

text: balance

text: starts

text: to

text: evolve

text: on

text: a

text: slow

text: timescale

text: relative

text: to

text: the

text: rapidly

text: oscillating

text: gravity

text: waves.

text: In

text: large

text: scale

text: ﬂow,

text: the

text: weather

text: system

text: stays

text: close

text: to

text: this

text: balanced

text: state.

text: If

text: we

text: wish

text: to

text: predict

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Abstract We study dark matter that inelastically scatters and de-excites in direct detection experiments, as an interpretation of the CDMS-Si events in light of the recent LUX data. The constraints from LUX and XENON10 require the mass-splitting between the DM excited and de-excited states to | | be δ (cid:38) 50 keV. At the same time, the CDMS-Si data itself do not allow for a consistent DM | | ∼ interpretation for mass splittings larger than δ 200 keV. We ﬁnd that a low threshold analysis will be needed to rule out this interpretation of the CDMS-Si events. In a simple model with a kinetically mixed dark photon, we show that the CDMS-Si rate and the thermal relic abundance can both be accommodated. Preprint: CP3-Origins-2013-053 DNRF90, DIAS-2013-53. 1

text: Abstract

font: IIYDCR+URWPalladioL-Roma

text: We

text: study

text: dark

text: matter

text: that

text: inelastically

text: scatters

text: and

text: de-excites

text: in

text: direct

text: detection

text: experiments,

text: as

text: an

text: interpretation

text: of

text: the

text: CDMS-Si

text: events

text: in

text: light

text: of

text: the

text: recent

text: LUX

text: data.

text: The

text: constraints

text: from

text: LUX

text: and

text: require

text: the

text: mass-splitting

text: between

text: the

text: DM

text: excited

text: and

text: de-excited

text: states

text: to

font: EIDHMY+Pxsy

text: be

text: δ

font: XBLLNJ+Rpxmi

font: AWMJVC+Pxsya

text: keV.

text: At

text: the

text: same

text: time,

text: the

text: CDMS-Si

text: data

text: itself

text: do

text: not

text: allow

text: for

text: a

text: consistent

text: DM

text: interpretation

text: for

text: mass

text: splittings

text: larger

text: than

text: δ

text: keV.

text: We

text: ﬁnd

text: that

text: a

text: low

text: threshold

text: analysis

text: will

text: be

text: needed

text: to

text: rule

text: out

text: this

text: interpretation

text: of

text: the

text: CDMS-Si

text: events.

text: In

text: a

text: simple

text: model

text: with

text: a

text: kinetically

text: mixed

text: dark

text: photon,

text: we

text: show

text: that

text: the

text: CDMS-Si

text: rate

text: and

text: the

text: thermal

text: relic

text: abundance

text: can

text: both

text: be

text: accommodated.

text: Preprint:


text: This can happen if this conversion process stays in equilibrium below temperatures of order the mass-splitting δ. To see that this does not occur, let us brieﬂy sketch the thermal history of the model. After chemical decoupling, the DM density becomes ﬁxed but it retains kinetic equilibrium with the SM by elastically scattering on electrons. Estimating the rate of this scattering as (cid:39) α α Γ ε2 X EM T5 (10) X e i m4 φ we ﬁnd that this fails to keep up with Hubble expansion and hence decouples around ∼ ∗ T 20 MeV. After this point, the DM sector, consisting of X and X , has a temperature 1 2 ∗ T = T2/T where T is the SM temperature. Now we can estimate the temperature X γ γ → (cid:104) (cid:105) at which X X X X freezes out by equating H(T ) = n (T ) σ → v . With 1 1 2 2 γ X X X X X X (cid:104) (cid:105) (cid:39) 1 1 1 2 2 σ → v πα2Xm2X we ﬁnd that the conversion process falls out of equilibrium also X1X1 X2X2 m4 φ around 20 MeV. Thus we expect no large depletion of excited states and roughly estimate ≈ n n . Therefore for consistency one should rescale the cross sections in Fig. 2 by a X X 1 2 factor of 2 since the density of X is half that of the total DM density. 1 V. CONCLUSIONS We have discussed dark matter that inelastically scatters and de-excites in direct detec- tion experiments (“Exothermic” DM) as an explanation for the CDMS-Si events, consistent with all null searches. The constraints from LUX and XENON10 require the mass-splitting | | to be δ (cid:38) 50 keV as already discussed in [5]. At the same time, the CDMS-Si data itself | | ∼ only allow splittings up to δ 200 keV since for larger splittings one cannot simultane- ously account for the observed low and high energy events. In a simple model with a kinetically mixed dark photon, we have shown that the CDMS-Si rate and the thermal relic abundance can be accommodated. To rule out this scenario with a Xenon-based experiment will require an S2-only analysis (or other low- threshold technique [32]) with high exposure. Note Added While this manuscript was being prepared for upload the paper [33] appeared which also considers LUX constraints on exothermic DM as an explanation of CDMS-Si data. The authors perform a global ﬁt to the 3 parameters σ , m and δ. We ﬁnd similar limits p X in parameter space, see e.g. Fig. 3. 8 ##LTLine## ##LTLine##

text: This

text: can

text: happen

text: if

text: this

text: conversion

text: process

text: stays

text: in

text: equilibrium

text: below

text: temperatures

text: of

text: order

text: the

text: mass-splitting

text: δ.

text: To

text: see

text: that

text: this

text: does

text: not

text: occur,

text: let

text: us

text: brieﬂy

text: sketch

text: the

text: thermal

text: history

text: of

text: the

text: model.

text: After

text: chemical

text: decoupling,

text: the

text: DM

text: density

text: becomes

text: ﬁxed

text: but

text: it

text: retains

text: kinetic

text: equilibrium

text: with

text: the

text: SM

text: by

text: elastically

text: scattering

text: on

text: electrons.

text: Estimating

text: the

text: rate

text: of

text: this

text: scattering

text: as

text: α

text: α

text: Γ

text: ε2

text: X

text: EM

text: X

text: e

text: i

text: φ

text: we

text: ﬁnd

text: that

text: this

text: fails

text: to

text: keep

text: up

text: with

text: Hubble

text: expansion

text: and

text: hence

text: decouples

text: around

text: T

text: MeV.

text: After

text: this

text: point,

text: the

text: DM

text: sector,

text: consisting

text: of

text: X

text: and

text: X

text: has

text: a

text: temperature

text: T

text: where

text: T

text: is

text: the

text: SM

text: temperature.

text: Now

text: we

text: can

text: estimate

text: the

text: temperature

text: X

text: γ

text: γ

text: at

text: which

text: X

text: X

text: X

text: X

text: freezes

text: out

text: by

text: equating

text: H(T

text: n

text: (T

text: σ

text: v

text: With

text: γ

text: X

text: X

text: X

text: X

text: X

text: X

text: σ

text: v

text: we

text: ﬁnd

text: that

text: the

text: conversion

text: process

text: falls

text: out

text: of

text: equilibrium

text: also

text: φ

text: around

text: MeV.

text: Thus

text: we

text: expect

text: no

text: large

text: depletion

text: of

text: excited

text: states

text: and

text: roughly

text: estimate

text: n

text: n

text: Therefore

text: for

text: consistency

text: one

text: should

text: rescale

text: the

text: cross

text: sections

text: in

text: Fig.

text: by

text: a

text: X

text: X

text: factor

text: of

text: since

text: the

text: density

text: of

text: X

text: is

text: half

text: that

text: of

text: the

text: total

text: DM

text: density.

text: V.

text: CONCLUSIONS

text: We

text: have

text: discussed

text: dark

text: matter

text: that

text: inelastically

text: scatters

text: and

text: de-excites

text: in

text: direct

text: detec-

text: tion

text: experiments

text: (“Exothermic”

text: DM)

text: as

text: an

text: explanation

text: for

text: the

text: CDMS-Si

text: events,

text: consistent

text: with

text: all

text: searches.

text: The

text: constraints

text: from

text: LUX

text: and

text: require

text: the

text: mass-splitting

text: to

text: be

text: δ

text: keV

text: as

text: already

text: discussed

text: in

text: At

text: the

text: same

text: time,

text: the

text: CDMS-Si

text: data

text: itself

text: only

text: allow

text: splittings

text: up

text: to

text: δ

text: keV

text: since

text: for

text: larger

text: splittings

text: one

text: cannot

text: simultane-

text: ously

text: account

text: for

text: the

text: observed

text: low

text: and

text: high

text: energy

text: events.

text: In

text: a

text: simple

text: model

text: with

text: a

text: kinetically

text: mixed

text: dark

text: photon,

text: we

text: have

text: shown

text: that

text: the

text: CDMS-Si

text: rate

text: and

text: the

text: thermal

text: relic

text: abundance

text: can

text: be

text: accommodated.

text: To

text: rule

text: out

text: this

text: scenario

text: with

text: a

text: Xenon-based

text: experiment

text: will

text: require

text: an

text: analysis

text: (or

text: other

text: low-

text: threshold

text: technique

text: with

text: high

text: exposure.

text: Note

text: Added

text: While

text: this

text: manuscript

text: was

text: being

text: prepared

text: for

text: upload

text: the

text: paper

text: appeared

text: which

text: also

text: considers

text: LUX

text: constraints

text: on

text: exothermic

text: DM

text: as

text: an

text: explanation

text: of

text: CDMS-Si

text: data.

text: The

text: authors

text: perform

text: a

text: global

text: ﬁt

text: to

text: the

text: parameters

text: σ

text: m

text: and

text: δ.

text: We

text: ﬁnd

text: similar

text: limits

text: p

text: X

text: in

text: parameter

text: space,

text: see

text: e.g.

text: Fig.

text: ##LTLine##

text: ##LTLine##


text: 12 F. FEO, M.R. POSTERARO, C. ROBERTO − − ∈ (cid:52) where βp = α1−p = F 1(p/2), p [0, 1] are deﬁned in the previous section and stands for the symmetric diﬀerence between sets. Remark 4.1. To help the reader in many computations throughout all this section, we − − ≤ ≤ observe that the set E =−( βa, βb), with 0 a b, has perimeter P(E) = J(b/2) + J(a/2) and measure µ(E) = (b a)/2. The next lemma summarizes some basic properties on the asymmetry λ(E). ∈ F Lemma 4.2. Let µ and E, F be two sets with ﬁnite µ-perimeter. Then, (cid:52) (cid:52) i) E F = Ec Fc, ii) λ(E) = λ(Ec), ≤ ≤ − iii) 0 λ(E) 2 min(µ(E), 1 µ(E)), where Ec denotes the complement of E. Proof. The assertions are easy and left to the reader. (cid:3) The next result is a ﬁrst reduction to ﬁnd the sets with minimal perimeter and given ≤ ≤ measure and asymmetry. We ﬁrst consider the case 0 < µ(E) 1. The other case 1 2 2 µ(E) < 1 can be obtained using complementary sets and Lemma 4.2. {− ∈ } Given a set E, we set E¯ = x, x E for its symmetric with respect to the origin. We will show, in Proposition 4.3 below, that the minimal sets among all sets of given measure p and given asymmetry λ have to be found among the following sets and their (cid:16) (cid:17) (cid:16) (cid:17) symmetric: (cid:16) (cid:17) (cid:16) (cid:17) − − ∪ ≤ ≤ (4.2) E1 = (cid:16) βλ , βp+λ (cid:17) βp+λ , βλ , if 0 λ 2p, −∞2 − 2 ∪ 2 ∞2 ≤ ≤ (4.3) E2 = , βp+λ βp−λ, + and E¯2, if 0 λ p, − − ≤ ≤ (4.4) E3 = βλ−p, βλ+p and E¯3, if p λ 2p, ≤ ≤ (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) and, if 0 λ 2p, −∞ − ∪ − ∪ ∞ (4.5) E4 = , βp−λ β1−λ , β1−λ βp−λ , + . 2 2 2 2 Observe that E2 and E¯2 are not deﬁned when λ > p and that E3 and E¯3 are not deﬁned when λ < p. ∈ F ∈ Proposition 4.3. Let µ and E be a Borel set with measure µ(E) = p (0, 1] and ∈  2 asymmetry λ(E) = λ [0, 2p]. Then,  ≤ ≤ ≥ min P(Ei) if 0 λ p (4.6) P(E) i=1,2,4 ≤ ≤ min P(Ei) if p λ 2p. i=1,3,4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: FEO,

text: M.R.

text: POSTERARO,

text: C.

text: ROBERTO

font: WEUJDN+txsy

text: where

text: βp

font: KYVUXQ+rtxmi

font: FXMWMG+rtxr

text: α1−p

text: F

text: p

text: are

text: deﬁned

text: in

text: the

text: previous

text: section

text: and

text: stands

text: for

text: the

text: symmetric

text: diﬀerence

text: between

text: sets.

text: Remark

text: To

text: help

text: the

text: reader

text: in

text: many

text: computations

text: throughout

text: all

text: this

text: section,

text: we

text: observe

text: that

text: the

text: set

text: E

text: βa,

text: βb),

text: with

text: a

text: b,

text: has

text: perimeter

text: P(E)

text: and

text: measure

text: µ(E)

text: (b

text: The

text: next

text: lemma

text: summarizes

text: some

text: basic

text: properties

text: on

text: the

text: asymmetry

text: λ(E).

text: F

text: Lemma

text: Let

text: µ

text: and

text: E,

text: F

text: be

text: two

text: sets

text: with

text: ﬁnite

text: µ-perimeter.

text: Then,

text: i)

text: E

text: F

text: Ec

text: Fc,

text: ii)

text: λ(E)

text: λ(Ec),

text: iii)

text: λ(E)

text: min(µ(E),

text: µ(E)),

text: where

text: Ec

text: denotes

text: the

text: complement

text: of

text: E.

text: Proof.

text: The

text: assertions

text: are

text: easy

text: and

text: left

text: to

text: the

text: reader.

text: The

text: next

text: result

text: is

text: a

text: ﬁrst

text: reduction

text: to

text: ﬁnd

text: the

text: sets

text: with

text: minimal

text: perimeter

text: and

text: given

text: measure

text: and

text: asymmetry.

text: We

text: ﬁrst

text: consider

text: the

text: case

text: µ(E)

text: The

text: other

text: case

text: µ(E)

text: can

text: be

text: obtained

text: using

text: complementary

text: sets

text: and

text: Lemma

text: Given

text: a

text: set

text: E,

text: we

text: set

text: E¯

text: x,

text: x

text: E

text: for

text: its

text: symmetric

text: with

text: respect

text: to

text: the

text: origin.

text: We

text: will

text: show,

text: in

text: Proposition

text: below,

text: that

text: the

text: minimal

text: sets

text: among

text: all

text: sets

text: of

text: given

text: measure

text: p

text: and

text: given

text: asymmetry

text: λ

text: have

text: to

text: be

text: found

text: among

text: the

text: following

text: sets

text: and

text: their

text: symmetric:

text: βλ

text: βp+λ

text: βp+λ

text: βλ

text: if

text: λ

text: −∞2

text: ∞2

text: βp+λ

text: βp−λ,

text: and

text: if

text: λ

text: p,

text: βλ−p,

text: βλ+p

text: and

text: if

text: p

text: λ

text: and,

text: if

text: λ

text: βp−λ

text: β1−λ

text: β1−λ

text: βp−λ

text: Observe

text: that

text: and

text: are

text: not

text: deﬁned

text: when

text: λ

text: p

text: and

text: that

text: and

text: are

text: not

text: deﬁned

text: when

text: λ

text: p.

text: F

text: Proposition

text: Let

text: µ

text: and

text: E

text: be

text: a

text: Borel

text: set

text: with

text: measure

text: µ(E)

text: p

text: and

text: asymmetry

text: λ(E)

text: λ

text: Then,

text: min

text: P(Ei)

text: if

text: λ

text: p

text: P(E)

text: min

text: P(Ei)

text: if

text: p

text: λ

text: ##LTLine##

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text: 18 F. FEO, M.R. POSTERARO, C. ROBERTO ≤ ≥ ≤ ≤ − since λ 2p. This implies that P(E1) P(E3) (whe−n p λ 1 p). − Cons−ider no−w the fu−nction L14(p,−λ) =≤:= P≤(E1) P(E4) (cid:55)→= 2J(λ/4) + 2J((2p + λ)/4) 2J((2p λ)/4) 2J(≥(2 λ)/4) w−ith 1 p λ −2p. Sin−ce λ L1−4(p, λ) is non-decreasing, we have L14(p, λ) L14(p, 1 p) = −2J((1≥ p)/4) 2J((3p 1)/4). Last function is non-increasing (in≥p). Hence, L14(−p, 1 ≤ p)≤ L14(1/2, 1/2) = 0. This guarantees that, as announced P(E1) P(E4) when 1 p λ 2p. ≤ ≤ At this point is remains to−compare P(E3) and P(E4) when p λ 2p, considering the function L43(p, λ) := P(E4) P≤(E3).≤We will dist−inguish between two sub-cases. − We start by−dealing−with p λ −min(2p−, 1 p). In that case, L43(p, λ) = 2J((2p λ)/4) + 2J((2 λ)/4) J((p + λ)/2) J((λ p)/2) is obviously non-increasing in λ. In order to deduce the sign o−f L43 we need to study the extreme points H(p) := L43(p, p) and G(p) := L43(p, min(2p, 1 p)): − λ p min(2p, 1 p) H(p) (cid:55)→ (cid:64) λ L43(p, λ) (cid:64) (cid:64)(cid:82) G(p) − − First, we observe that H(p) := L43(p, p) ≥= 2J(p/4) + 2J((2 p)/4) −J(p) is non- increasing (take the derivative) so that H(p) H(1/2) = 2J(1/8) + 2J(3/8) J(1/2) > 0 (cid:55)→ − thanks to (4.10). Then, we notice that p G(p) := L43(p, min(2p, 1 p)) is obviously non-increasing on [0, 1/3] and non-decreasing on [1/3, 1/2]. S−ince G(0) = L43(0, 0) = 2J(1/2) > 0, G(1/2) = L43(1/2, 1/2) = 2J(1−/8) + 2J(−3/8) J(1/2) > 0 thanks to (4.10), −and G(1/3) = L43(1/3, 2/3) = 2J(1/3) J(1/6) J(1/2) < 0−(since the slope [J(1/2) J(1/3)]/(1/6) is larger, by convexity of J, than the slope [J(1/3) J(1/6)]/(1/6)), we end up with the following diagram: p 0 p1 1/3 p2 1/2 G(0) > 0(cid:72) (cid:8)(cid:42)G(1/2) > 0 (cid:72) (cid:8) (cid:72)(cid:106) (cid:8) G 0 0 (cid:72) (cid:8)(cid:42) (cid:72) (cid:8) (cid:72)(cid:106) (cid:8) G(1/3) < 0 ∈ ∈ ≥ for some p1 ∈(0, 1/3)∪and some p2 (1/3, 1/2)−. From this we conclude that P(E4) P(E3) when p [0, p1] [p2∈, 1/2], and that P(E4) P(E3) changes sign (at a unique point λ0(p)) when λ varies and p (p1, p2≤) is≤ﬁxed.{This l−ead}s to the existence of the function λ0. This completes the picture f−or p≤ λ≤ min 2p, 1 p . ∈ Consider ﬁnally the range 1 p λ 2p (which exists only if p [1/3, 1/2]). In that case, the function L43(p, λ) reads − − − − − − L43(p, λ) = 2J((2p λ)/4) + 2J((2 λ)/4) J(1 (p + λ)/2) J((λ p)/2). Here we used one again the symmetry of J about 1/2 in order to deal only with variables ≥ (cid:55)→ belonging to [0, 1/2] (observe in particular that (p+λ)/2 1/2). The map p L43(p, λ) i−s clearly increasing. Hence, we need to study the extremal points H(λ) := L43(max(λ/2, 1 λ), λ) and G(λ) := L43(1/2, λ): ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: FEO,

text: M.R.

text: POSTERARO,

text: C.

text: ROBERTO

text: since

text: λ

text: This

text: implies

text: that

text: (whe−n

text: p

text: λ

text: p).

text: Cons−ider

text: no−w

text: the

text: fu−nction

text: w−ith

text: p

text: λ

text: Sin−ce

text: λ

text: λ)

text: is

text: non-decreasing,

text: we

text: have

text: λ)

text: p)

text: Last

text: function

text: is

text: non-increasing

text: (in≥p).

text: Hence,

text: p)≤

text: This

text: guarantees

text: that,

text: as

text: announced

text: when

text: p

text: λ

text: At

text: this

text: point

text: is

text: remains

text: to−compare

text: and

text: when

text: p

text: λ

text: considering

text: the

text: function

text: λ)

text: will

text: dist−inguish

text: between

text: two

text: sub-cases.

text: We

text: start

text: by−dealing−with

text: p

text: λ

text: p).

text: In

text: that

text: case,

text: λ)

text: J((p

text: J((λ

text: is

text: obviously

text: non-increasing

text: in

text: λ.

text: In

text: order

text: to

text: deduce

text: the

text: sign

text: o−f

text: we

text: need

text: to

text: study

text: the

text: extreme

text: points

text: H(p)

text: p)

text: and

text: G(p)

text: p)):

text: λ

text: p

text: p)

text: H(p)

text: λ

text: λ)

text: G(p)

text: First,

text: we

text: observe

text: that

text: H(p)

text: p)

text: −J(p)

text: is

text: non-

text: increasing

text: (take

text: the

text: derivative)

text: so

text: that

text: H(p)

text: thanks

text: to

text: Then,

text: we

text: notice

text: that

text: p

text: G(p)

text: p))

text: is

text: obviously

text: non-increasing

text: on

text: and

text: non-decreasing

text: on

text: S−ince

text: thanks

text: to

text: −and

text: the

text: slope

text: is

text: larger,

text: by

text: convexity

text: of

text: J,

text: than

text: the

text: slope

text: we

text: end

text: up

text: with

text: the

text: following

text: diagram:

text: p

text: G

text: for

text: some

text: some

text: From

text: this

text: we

text: conclude

text: that

text: when

text: p

text: and

text: that

text: changes

text: sign

text: (at

text: a

text: unique

text: point

text: when

text: λ

text: varies

text: and

text: p

text: is≤ﬁxed.{This

text: l−ead}s

text: to

text: the

text: existence

text: of

text: the

text: function

text: This

text: completes

text: the

text: picture

text: f−or

text: p≤

text: λ≤

text: min

text: p

text: Consider

text: ﬁnally

text: the

text: range

text: p

text: λ

text: (which

text: exists

text: only

text: if

text: p

text: In

text: that

text: case,

text: the

text: function

text: λ)

text: reads

text: λ)

text: (p

text: J((λ

text: Here

text: we

text: used

text: one

text: again

text: the

text: symmetry

text: of

text: J

text: about

text: in

text: order

text: to

text: deal

text: only

text: with

text: variables

text: belonging

text: to

text: (observe

text: in

text: particular

text: that

text: The

text: map

text: p

text: λ)

text: i−s

text: clearly

text: increasing.

text: Hence,

text: we

text: need

text: to

text: study

text: the

text: extremal

text: points

text: H(λ)

text: λ),

text: λ)

text: and

text: G(λ)

text: λ):

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text: QUANTITATIVE ISOPERIMETRIC INEQUALITIES 23 (cid:90) (cid:90) (cid:18) (cid:19) (cid:18) (cid:19) (cid:48)(cid:48) leading to Equation (4.16), we have, by monotonicity of J , δ(E2) = 2p+λ2 −t J(cid:48)(cid:48)(u)dudt ≥ J(cid:48)(cid:48) 2p + λ2 × λ2 2 ≥ J(cid:48)(cid:48)(14/2−)λ2, p t λ 2 2 ≤ ∇ where in the last inequality(cid:55)→we used that p + λ 1. Finally, thanks to (4.17), the 2- condition and the fact that t J(t)/t2 is non-increasing (a consequence of the assumption (cid:48) J concave), we have  δ(E3) ≥ J(p) − 2J(p/2) ≥ εJ(p/2) ≥ εJ(1/2)p2 ≤ ≤ ≤ ≥ λ2 if p 1/3 and p λ 2p ≥ εJ(1/2) εJ(1/2) 4 ≥ ≤ ≤ − λ2, λ2 if p 1/3 and p λ 1 p 4 9 ≤ ≤ ≤ − where in the last line we used that λ (cid:32) 2/3 (in the range p λ (cid:33)1 p). Collecting the previous computations leads to (cid:48) (cid:48)(cid:48) − ≥ J (1/6) J (1/2 ) J(1/2) i=m1,i2n,3 δ(Ei) ε min 2 , 4 , 4 λ2. (cid:48) ≥ (cid:16) (cid:17) (cid:16) (cid:17) Using that xJ (x) J(x) (a consequence of the fact that J(0) = 0 and that J is convex), and (cid:55)→ (cid:48) (cid:48) (cid:48) (cid:48) ≥ that t J (t)/t is non-increasing (since J is concave and J (0+) = 0), we have 1 J (1/6) (cid:48) ≥ (cid:48) (cid:48)(cid:48) − (cid:48)(cid:48) − 2 1 J (1/2) 1 J(1/2). Hence, min J (1/6), J (1/2 ), J(1/2) = min J (1/2 ), J(1/2) . Then, 6 (cid:48) 3 (cid:48) (cid:48)2(cid:48) (cid:16)≤ 4(cid:48) 4∈ (cid:17) 4 4 (cid:55)→ since J is concave and J (0+) = 0, xJ (x) J (x), x (0, 1/2). Also, since t J(t)/t2 (cid:48) ≤ ∈ (cid:48)(cid:48) ≤ is non-increasing, we have xJ (x) 2J(x), x (0, 1/2). In turn, J (x) 2J(x)/x2 so that ≥ (cid:48)(cid:48) − (cid:48)(cid:48) − ≥ (cid:48)(cid:48) − J(1/2) J (1/2 )/8. As a conclusion, min J (1/2 ), J(1/2) J (1/2 )/32. 4 3 This ends the proof of the theorem. (cid:3) 5. Functional forms. As it is well known, isoperimetric inequalities have often equivalent functional forms, see e.g. [32, 16]. In this section, using the results of the preceding sections, we shall derive some weak embedding properties, and also some weak Cheeger inequalities, in quantitative forms. We need ﬁrst to introduce some notations, and in particular the notion of rearrangement of a function with respect to a probability measure. We refer to [7] for more on this topic. ⊂ → Let Ω be a measurable set and u : Ω R R be a measurable function. The level + sets of u are the sets (cid:16) (cid:17) { ∈ } ∈ Ehu := x Ω: u(x) > h , h R+. ≥ Then, we d(cid:55)→eﬁne the distribution function of u as µu(h) = µ Ehu for every∞h 0. The mapping h µu(h) is non-negative{, de∈creasing and r}ight continuous on [0, + [. Moreover µ∗u has a jump at h if and only if µu( x Ω : u(x) = h ) (cid:44) 0. The decreasing rearrangement u of u is the generalized inverse of µu(h), namely ∗ { ≥ } ∈ u (s) = sup h 0 : µu(h) > s for(cid:16)s (0, µ((cid:16)Ω)) .(cid:17)(cid:17) (cid:16) (cid:16) (cid:17) (cid:17) Now, let E be a measurable subset of R. We denote by E# the complement of the (unique) −∞ − ∪ − − ∞ symmetric interval such that µ(E#) = µ(E), that is E# = , F 1 (cid:32)µ(E) (cid:33) F 1 µ(E) , + . 2 2 Finally, the rearrangement of the function u with respect to µ is the function − ∗ − ∞ − µ(Ω) u#(x) = u#( x) = u (2F(x)) for < x < F 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: QUANTITATIVE

text: ISOPERIMETRIC

text: INEQUALITIES

font: YDMQJV+txex

text: leading

text: to

text: Equation

text: we

text: have,

text: by

text: monotonicity

text: of

text: J

text: −t

text: λ2

text: λ2

text: p

text: t

text: λ

text: where

text: in

text: the

text: last

text: used

text: that

text: p

text: λ

text: Finally,

text: thanks

text: to

text: the

text: condition

text: and

text: the

text: fact

text: that

text: t

text: is

text: non-increasing

text: (a

text: consequence

text: of

text: the

text: assumption

text: J

text: concave),

text: we

text: have

text: J(p)

text: λ2

text: if

text: p

text: and

text: p

text: λ

text: λ2

text: if

text: p

text: and

text: p

text: λ

text: p

text: where

text: in

text: the

text: last

text: line

text: we

text: used

text: that

text: λ

text: (in

text: the

text: range

text: p

text: λ

text: p).

text: Collecting

text: the

text: previous

text: computations

text: leads

text: to

text: J

text: J

text: δ(Ei)

text: ε

text: min

text: Using

text: that

text: xJ

text: (x)

text: J(x)

text: (a

text: consequence

text: of

text: the

text: fact

text: that

text: and

text: that

text: J

text: is

text: convex),

text: and

text: that

text: t

text: J

text: (t)/t

text: is

text: non-increasing

text: (since

text: J

text: is

text: concave

text: and

text: J

text: we

text: have

text: J

text: J

text: Hence,

text: min

text: J

text: J

text: min

text: J

text: Then,

text: 4∈

text: since

text: J

text: is

text: concave

text: and

text: J

text: xJ

text: (x)

text: J

text: (x),

text: x

text: Also,

text: since

text: t

text: is

text: non-increasing,

text: we

text: have

text: xJ

text: (x)

text: x

text: In

text: turn,

text: J

text: (x)

text: so

text: that

text: J

text: As

text: a

text: conclusion,

text: min

text: J

text: J

text: This

text: ends

text: the

text: proof

text: of

text: the

text: theorem.

text: Functional

text: forms.

text: As

text: it

text: is

text: well

text: known,

text: isoperimetric

text: inequalities

text: have

text: often

text: equivalent

text: functional

text: forms,

text: see

text: e.g.

text: In

text: this

text: section,

text: using

text: the

text: results

text: of

text: the

text: preceding

text: sections,

text: we

text: shall

text: derive

text: some

text: weak

text: embedding

text: properties,

text: and

text: also

text: some

text: weak

text: Cheeger

text: inequalities,

text: in

text: quantitative

text: forms.

text: We

text: need

text: ﬁrst

text: to

text: introduce

text: some

text: notations,

text: and

text: in

text: particular

text: the

text: notion

text: of

text: rearrangement

text: of

text: a

text: function

text: with

text: respect

text: to

text: a

text: probability

text: measure.

text: We

text: refer

text: to

text: for

text: more

text: on

text: this

text: topic.

text: Let

text: Ω

text: be

text: a

text: measurable

text: set

text: and

text: u

text: Ω

text: R

text: R

text: be

text: a

text: measurable

text: function.

text: The

text: level

text: sets

text: of

text: u

text: are

text: the

text: sets

text: Ehu

text: x

text: Ω:

text: u(x)

text: h

text: h

text: R+.

text: Then,

text: we

text: the

text: distribution

text: function

text: of

text: u

text: as

text: µu(h)

text: µ

text: Ehu

text: for

text: every∞h

text: The

text: mapping

text: h

text: µu(h)

text: is

text: non-negative{,

text: de∈creasing

text: and

text: r}ight

text: continuous

text: on

text: Moreover

text: µ∗u

text: has

text: a

text: jump

text: at

text: h

text: if

text: and

text: only

text: if

text: µu(

text: x

text: Ω

text: u(x)

text: h

text: The

text: decreasing

text: rearrangement

text: u

text: of

text: u

text: is

text: the

text: generalized

text: inverse

text: of

text: µu(h),

text: namely

text: u

text: (s)

text: sup

text: h

text: µu(h)

text: s

text: Now,

text: let

text: E

text: be

text: a

text: measurable

text: subset

text: of

text: R.

text: We

text: denote

text: by

text: E#

text: the

text: complement

text: of

text: the

text: (unique)

text: symmetric

text: interval

text: such

text: that

text: µ(E#)

text: µ(E),

text: that

text: is

text: E#

text: F

text: F

text: µ(E)

text: Finally,

text: the

text: rearrangement

text: of

text: the

text: function

text: u

text: with

text: respect

text: to

text: µ

text: is

text: the

text: function

text: µ(Ω)

text: u#(x)

text: u#(

text: x)

text: u

text: for

text: x

text: F

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 28 F. FEO, M.R. POSTERARO, C. ROBERTO (cid:16) (cid:17) (cid:16) (cid:17) (cid:20) (cid:18) (cid:16) (cid:17) (cid:19) (cid:18) (cid:16) (cid:17) (cid:19)(cid:21) Hence, − λ(Ehu+) + λ(Ehu ) ≥ µ Eu+ + µ Eu− − µ Eu+ ∩ E|u| # + µ Eu− ∩ E|u| # 2 2 h h h h h h | | | | − | | ∩ | | λ(E u ) = µ(E u ) µ(E u (E u )#) = h h h h 2 which is the expected result. (cid:3) (cid:90) (cid:90) (cid:32)(cid:90) (cid:33) (cid:16) (cid:16) (cid:17)(cid:17) (cid:12) (cid:12) (cid:12) (cid:12) Proof of Lemma 5.6. By Jensen’s inequality we h(cid:12) ave (cid:12) 1 (cid:52) (cid:32)1(cid:90) (cid:90) − (cid:33) 0 Ψ µ Ehu Ehv dh = 0 Ψ R (cid:12)(cid:12)(cid:12)χEhu(x) χEhv(x)(cid:12)(cid:12)(cid:12) dµ(x) dh ≥ Ψ (cid:32)(cid:90) 1(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) χ(cid:104) Eu(x) − χEv(x) (cid:105)dµ(x(cid:12)(cid:12)(cid:12)(cid:12) )dh (cid:33) 0 (cid:12)(cid:12) R h h (cid:12)(cid:12) 1 − = Ψ χEu(x) χEv(x) dh dµ(x) R 0 h h which leads to the expected result. (cid:3) Acknowledgement This work was partially done during the visits made by the ﬁrst two authors to MODAL’X, Universite´ Paris Ouest Nanterre la De´fense. Hospitality and support of this institution is gratefully acknowledged. References [1] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. [2] T. Aubin. Proble`mes isope´rime´triques et espaces de Sobolev. Journal of Diﬀerential Geometry, 11(4):573– 598, 1976. [3] G. I. Barenblatt. On some unsteady motions of a liquid and gas in a porous medium. Akad. Nauk SSSR. Prikl. Mat. Meh., 16:67–78, 1952. [4] F. Barthe. Log-concave and spherical models in isoperimetry. Geom. Funct. Anal., 12(1):32–55, 2002. [5] F. Barthe, P. Cattiaux, and C. Roberto. Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express, (2):39–60, 2005. [6] F. Barthe, P. Cattiaux, and C. Roberto. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam., 22(3):993–1067, 2006. [7] C. Bennett and R. Sharpley. Interpolation of operators, volume 129 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1988. [8] S. Bobkov. Extremal properties of half-spaces for log-concave distributions. Ann. Probab., 24(1):35–48, 1996. [9] S. Bobkov, N. Gozlan, C. Roberto, and P.-M. Samson. Bounds on the deﬁcit in the log-sobolev inequality. Preprint, 2013. [10] S. G. Bobkov. Large deviations and isoperimetry over convex probability measures with heavy tails. Elec- tron. J. Probab., 12:1072–1100 (electronic), 2007. [11] S. G. Bobkov and C. Houdre´. Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc., 129(616):viii+111, 1997. [12] C. Borell. Convex measures on locally convex spaces. Ark. Mat., 12:239–252, 1974. [13] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30(2):207–216, 1975. ##LTLine## ##LTLine## ##LTLine##

text: F.

text: FEO,

text: M.R.

text: POSTERARO,

text: C.

text: ROBERTO

text: Hence,

text: λ(Ehu+)

text: λ(Ehu

text: µ

text: Eu+

text: µ

text: Eu−

text: µ

text: Eu+

text: E|u|

text: µ

text: Eu−

text: E|u|

text: h

text: h

text: h

text: h

text: h

text: h

text: λ(E

text: u

text: µ(E

text: u

text: µ(E

text: u

text: (E

text: u

text: h

text: h

text: h

text: h

text: which

text: is

text: the

text: expected

text: result.

text: Proof

text: of

text: Lemma

text: By

text: Jensen’s

text: inequality

text: we

text: ave

text: Ψ

text: µ

text: Ehu

text: Ehv

text: dh

text: Ψ

text: R

text: dµ(x)

text: dh

text: Ψ

text: Eu(x)

text: χEv(x)

text: )dh

text: R

text: h

text: h

text: Ψ

text: χEu(x)

text: χEv(x)

text: dh

text: dµ(x)

text: R

text: h

text: h

text: which

text: leads

text: to

text: the

text: expected

text: result.

text: Acknowledgement

text: This

text: work

text: was

text: partially

text: done

text: during

text: the

text: visits

text: made

text: by

text: the

text: ﬁrst

text: two

text: authors

text: to

text: MODAL’X,

text: Universite´

text: Paris

text: Ouest

text: Nanterre

text: la

text: De´fense.

text: Hospitality

text: and

text: support

text: of

text: this

text: institution

text: is

text: gratefully

text: acknowledged.

text: References

text: L.

text: Ambrosio,

text: N.

text: Fusco,

text: and

text: D.

text: Pallara.

text: Functions

text: of

text: bounded

text: variation

text: and

text: free

text: discontinuity

text: problems.

text: Oxford

text: Mathematical

text: Monographs.

text: The

text: Clarendon

text: Press

text: Oxford

text: University

text: Press,

text: New

text: York,

text: T.

text: Aubin.

text: Proble`mes

text: isope´rime´triques

text: et

text: espaces

text: de

text: Sobolev.

text: Journal

text: of

text: Diﬀerential

text: Geometry,

text: G.

text: I.

text: Barenblatt.

text: On

text: some

text: unsteady

text: motions

text: of

text: a

text: liquid

text: and

text: gas

text: in

text: a

text: porous

text: medium.

text: Akad.

text: Nauk

text: SSSR.

text: Prikl.

text: Mat.

text: Meh.,

text: F.

text: Barthe.

text: Log-concave

text: and

text: spherical

text: models

text: in

text: isoperimetry.

text: Geom.

text: Funct.

text: Anal.,

text: F.

text: Barthe,

text: P.

text: Cattiaux,

text: and

text: C.

text: Roberto.

text: Concentration

text: for

text: independent

text: random

text: variables

text: with

text: heavy

text: tails.

text: AMRX

text: Appl.

text: Math.

text: Res.

text: Express,

text: F.

text: Barthe,

text: P.

text: Cattiaux,

text: and

text: C.

text: Roberto.

text: Interpolated

text: inequalities

text: between

text: exponential

text: and

text: Gaussian,

text: Orlicz

text: hypercontractivity

text: and

text: isoperimetry.

text: Rev.

text: Mat.

text: Iberoam.,

text: C.

text: Bennett

text: and

text: R.

text: Sharpley.

text: Interpolation

text: of

text: operators,

text: volume

text: of

text: Pure

text: and

text: Applied

text: Mathematics.

text: Academic

text: Press

text: Inc.,

text: Boston,

text: MA,

text: S.

text: Bobkov.

text: Extremal

text: properties

text: of

text: half-spaces

text: for

text: log-concave

text: distributions.

text: Ann.

text: Probab.,

text: S.

text: Bobkov,

text: N.

text: Gozlan,

text: C.

text: Roberto,

text: and

text: P.-M.

text: Samson.

text: Bounds

text: on

text: the

text: deﬁcit

text: in

text: the

text: log-sobolev

text: inequality.

text: Preprint,

text: S.

text: G.

text: Bobkov.

text: Large

text: deviations

text: and

text: isoperimetry

text: over

text: convex

text: probability

text: measures

text: with

text: heavy

text: tails.

text: Elec-

text: tron.

text: J.

text: Probab.,

text: (electronic),

text: S.

text: G.

text: Bobkov

text: and

text: C.

text: Houdre´.

text: Some

text: connections

text: between

text: isoperimetric

text: and

text: Sobolev-type

text: inequalities.

text: Mem.

text: Amer.

text: Math.

text: Soc.,

text: C.

text: Borell.

text: Convex

text: measures

text: on

text: locally

text: convex

text: spaces.

text: Ark.

text: Mat.,

text: C.

text: Borell.

text: The

text: Brunn-Minkowski

text: inequality

text: in

text: Gauss

text: space.

text: Invent.

text: Math.,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:2) (cid:3) 6 F. FEO, M.R. POSTERARO, C. ROBERTO (cid:2) (cid:3) ∈ F(cid:16) (cid:17) ∈ ∈(cid:16) − (cid:17) Lemma 3.5. Let µ . For p (0, 1)−and t 0,≥1 p , set ψp(t) = J(t) + J(p + t). T−hen, [i)]ψp is symmet−ric about 12p, if p− 1/−2, ψp is−(cid:16)strictly c(cid:17)onvex(cid:16) on 0, 1(cid:17) −p , dec(cid:16)reasing o(cid:17)n 0, 1 p , increasin(cid:16)g on 1 (cid:17)p, 1 p , 1 p is a(cid:16) minimum(cid:17)and 0, 1 p 2 2 2 − − are(cid:16)maxi−m(cid:17)a. if p < 1/2, ψp i(cid:16)s stri−ctly(cid:17)convex on 0, 12 p−, on− 12 p, 12 and on 1, 1 p , it is increasing on 0, 1 p , decreasing on 1 p, 1 p , increasing 2− 2 − −2 2 on 1 p, 1 and decreasing on 1, 1 p . Moreover 1 p, 1 are maxima and − 2 −2 2 2 2 1 p, 0, 1 p are minima. 2 p> p< 12−p 12 (((123))) to 1−p t to 1−p t (cid:55)→ ≥ Figure 1. The shape of the function t ψp(t) for p 1/2 and p < 1/2. Proof. The proof is elementary and left to the reader. (cid:3) We end this section with a converse of Proposition 3.3, namely that the shifting property of Proposition 3.3 (to be really precise only a weaker form is needed) implies that µ is log- convex. Proposition 3.6. Let µ be a probability measure on the line, symmetric with respect to a point (say the origin for simplicity) with density f. Assume that f is continuous, positive − ∈ ∞ on ( α, α) for some α (0, ] and such that the following shifting property holds: (cid:0) (cid:1) ≥ ≤ 1) if (a, b) is an interval with µ ((a, b)) < 1, a < 0, b > 0 and a + b 0 (risp. a + b 0) 2 then ≥ (cid:48) (cid:48) P ((a, b)) P (a , b ) (cid:48) (cid:48) ≥ (cid:48) (cid:48) ≤ for any left-shifted (risp. right-shifted) of (a, b) w(cid:0) ith a +(cid:1) b 0 (risp. a + b 0); ≥ ≥ ≤ 2) if (a, b) is an interval with µ ((a, b)) 1and a + b 0 (risp. a + b 0) then 2 ≥ (cid:48) (cid:48) P ((a, b)) P (a , b ) (cid:48) (cid:48) ≥ (cid:48) (cid:48) ≤ for any left-shifted (risp. right-shifted) of (a, b) with a + b 0 (risp. a + b 0). Then µ is log-convex. (cid:16) (cid:17) (cid:16) (cid:17) Proof. By Proposition 2.2, continuity and symmetry of J(t) we only have to prove that, for ∈ ± ∈ all t 0, 1 and all d so that t d 0, 1 , it holds 2 2 ≥ 1 − (3.1) J(t) (J(t d) + J(t + d)) . 2 ##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##

text: F.

text: FEO,

text: M.R.

text: POSTERARO,

text: C.

text: ROBERTO

label: list

text: Lemma

text: Let

text: µ

text: For

text: p

text: t

text: p

text: set

text: ψp(t)

text: J(t)

text: J(p

text: t).

text: T−hen,

text: [i)]ψp

text: is

text: symmet−ric

text: about

text: if

text: p−

text: ψp

text: on

text: −p

text: p

text: on

text: p

text: p

text: is

text: p

text: if

text: p

text: ψp

text: on

text: p−,

text: on−

text: p,

text: and

text: on

text: p

text: it

text: is

text: increasing

text: on

text: p

text: decreasing

text: on

text: p,

text: p

text: increasing

text: 2−

text: −2

text: on

text: p,

text: and

text: decreasing

text: on

text: p

text: Moreover

text: p,

text: are

text: maxima

text: and

text: −2

text: p,

text: p

text: are

text: minima.

text: p>

text: p<

text: to

text: t

text: to

text: t

text: Figure

text: The

text: shape

text: of

text: the

text: function

text: t

text: ψp(t)

text: for

text: p

text: and

text: p

text: Proof.

text: The

text: proof

text: is

text: elementary

text: and

text: left

text: to

text: the

text: reader.

text: We

text: end

text: this

text: section

text: with

text: a

text: converse

text: of

text: Proposition

text: namely

text: that

text: the

text: shifting

text: property

text: of

text: Proposition

text: (to

text: be

text: really

text: precise

text: only

text: a

text: weaker

text: form

text: is

text: needed)

text: implies

text: that

text: µ

text: is

text: log-

text: convex.

text: Proposition

text: Let

text: µ

text: be

text: a

text: probability

text: measure

text: on

text: the

text: line,

text: symmetric

text: with

text: respect

text: to

text: a

text: point

text: (say

text: the

text: origin

text: for

text: simplicity)

text: with

text: density

text: f.

text: Assume

text: that

text: f

text: is

text: continuous,

text: positive

text: on

text: α,

text: α)

text: for

text: some

text: α

text: and

text: such

text: that

text: the

text: following

text: shifting

text: property

text: holds:

text: if

text: (a,

text: b)

text: is

text: an

text: interval

text: with

text: µ

text: ((a,

text: b))

text: a

text: b

text: and

text: a

text: b

text: (risp.

text: a

text: b

text: then

text: P

text: ((a,

text: b))

text: P

text: (a

text: b

text: for

text: any

text: left-shifted

text: (risp.

text: right-shifted)

text: of

text: (a,

text: b)

text: ith

text: a

text: b

text: (risp.

text: a

text: b

text: if

text: (a,

text: b)

text: is

text: an

text: interval

text: with

text: µ

text: ((a,

text: b))

text: a

text: b

text: (risp.

text: a

text: b

text: then

text: P

text: ((a,

text: b))

text: P

text: (a

text: b

text: for

text: any

text: left-shifted

text: (risp.

text: right-shifted)

text: of

text: (a,

text: b)

text: with

text: a

text: b

text: (risp.

text: a

text: b

text: Then

text: µ

text: is

text: log-convex.

text: Proof.

text: By

text: Proposition

text: continuity

text: and

text: symmetry

text: of

text: J(t)

text: we

text: only

text: have

text: to

text: prove

text: that,

text: for

text: all

text: t

text: and

text: all

text: d

text: so

text: that

text: t

text: d

text: it

text: holds

text: J(t)

text: (J(t

text: d)

text: J(t

text: d))

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##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##