text: 9 images in [110] direction of selected samples reinforce of the magnetic anisotropy energy of the antiferromagnet- pc these results. ically ordered high-spin Co3+ ions. We presented a quan- titative model including the anisotropic magnetoresistance The magnetic properties were investigated with∼SQUID and weak anti-localization effects to explain the opposite magnetometry. The AFM ordering temperature T 43(10)K behavior and its magnitude of the low-temperature mag- N of the thick SCIO∼thin ﬁlms is reduced in comparison to the netoresistance in the tensile strained from the compressive bulk value (T 70K22). Using polarization dependent strained SCIO thin ﬁlms. N x-ray absorption spectroscopy at the Co L edges, we re- ACKNOWLEDGMENTS 2,3 vealed that the Co 3d orbital occupation strongly depends on the strain in the SCIO thin ﬁlms induced by the sub- The authors like to thank S. Meir for developing the strates. The tensile strained SCIO thin ﬁlms stabilize the shadow mask argon ion etching method at Augsburg Uni- occupation of the minority orbital state with x/y charac- versity and B. Meir for the support with it. We thank R. ter, whereas the compressive strained SCIO thin ﬁlms favor Pentcheva and P. Seiler for stimulating discussions. The the occupation of the minority orbital state with z charac- work at Augsburg was supported by the Deutsche Forschungs- ter. Together with the calculations using the well proven gemeinschaft (DFG) through SPP 1666 and TRR 80. The conﬁguration interaction cluster model, we were able to de- research in Dresden was partially supported by the DFG termine the magnetic easy axis change due to the induced through SFB 1143 and the study in Göttingen through SFB strain from the substrates, and the sign and the magnitude 1073 (TP A02). ∗ 27 See supplemental material for details concerning the in-situ lithographical prozess, additional structural information, the foreign phase 1 S. Vasala and M. Karppinen, Prog. Solid State Ch.43, (2015) 1. determination, details of the zero ﬁeld transport properties as well as the magnetoresistance modell and additional XAS spectra. 2 A. Ohtomo, S. Chakraverty, H. Mashiko, T. Oshima, and M. Kawasak28i, MJ..MNa.tIelri.eRv,eMs..2V8. A(2b0ra1s3h)e5v,.A. P. Litvinchuk, V. G. Hadjiev, H. Guo, and A. Gupta, Phys. Rev. B 75 (2007) 104118. 3 M. T. Anderson, K. B. Greenwood, G. A. Taylor, and K. R. Poeppelme29ierK,.PDro.gT.rSuoolnidg,SJt.aLteavCehrdemièr.e2,2M(.1P9.9S3in)g1h9,7S.. 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text: images

text: in

text: direction

text: of

text: selected

text: samples

text: reinforce

text: of

text: the

text: magnetic

text: anisotropy

text: energy

text: of

text: the

text: antiferromagnet-

text: pc

text: these

text: results.

text: ically

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text: high-spin

text: ions.

text: We

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text: magnetoresistance

text: The

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text: were

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text: and

text: weak

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text: ACKNOWLEDGMENTS

text: vealed

text: that

text: the

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text: the

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text: S.

text: Vasala

text: and

text: M.

text: Karppinen,

text: Prog.

text: Solid

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text: details

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text: the

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text: and

text: additional

text: XAS

text: spectra.

text: A.

text: Ohtomo,

text: S.

text: Chakraverty,

text: H.

text: Mashiko,

text: T.

text: Oshima,

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text: P.

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text: “On

text: Kruskal’s

text: uniqueness

text: condi-

text: tion

text: for

text: the

text: CAMDECOMP/PARAFAC

text: decomposition,”

text: Linear

text: Algebra

text: Appl.,

text: vol.

text: no.

text: pp.

text: Jan.


text: 3 can be rewritten as (cid:0) (cid:80) (cid:1) In this context, the broader focus of the present paper is on identifying the GSO S that is considered to be the best − y = L 1 h Sl x = Hx, (2) possible description of the structure of a non-stationary signal l=0 l y = Hx [cf. (2), where x is not white]. For generic (non- identity) input covariance matrix C , we face the challenge ≤ Y x for some particular h and L N. Given a set := that the signal covariance [cf. (3)] { } y(p) P of P independent samples of a non-stationary p=1 random signal y adhering to the network diffusion model (2), C = HC H (5) y x the problem is to identify the GSO S which is optimal in some sense as described in Section . Fundamental to the topology inference approach developed is no longer simultaneously diagonalizable with S. This rules out using the eigenvectors of the sample covariance here is to note that because H is a polynomial on S, then: i) all such graph ﬁlters (spanned by the unknown coefﬁcients h) Cˆ y as spectral templates of S. Still, as argued following (2) the eigenvectors of the GSO coincide with those of the have the same eigenvectors; and ii) such eigenvectors are the same as those of the shift, namely V. In other words, while the graph ﬁlter H that governs the underlying diffusion dynamics. This motivates using realizations of observed signals together diffusion implicit in H obscures the eigenvalues of the GSO, the eigenvectors V are preserved as spectral templates of the with additional information on the excitation inputs x (either realizations of the graph signals, sparsity assumptions, or the underlying network topology. Next, Section describes how to leverage (2) to obtain the GSO eigenbasis from a set of nodal covariance matrix Cx [25]) to identify the ﬁlter H, with the observations Y, by ﬁrst estimating the unknown graph ﬁlter H. ultimate goal of estimating its eigenvectors V. This system Y identiﬁcation task in the graph setting is the subject dealt with We show that the information in is in general not enough in Sections and , but before we close the loop showing how to uniquely recover H. Hence, we will resort to additional knowledge on the input signal x (either realizations, sparsity to recover S given its estimated eigenbasis Vˆ . B. Using the spectral templates to recover the shift properties, or, second-order statistical information) and also possibly on the structure of the graph ﬁlter H. Section outlines Given estimates Vˆ of the ﬁlter eigenvectors, recovery of how to use the spectral templates V to recover the desired S amounts to selecting its eigenvalues Λ and to that end we GSO by estimating its eigenvalues Λ and, as byproduct, the assume that the shift of interest is optimal in some sense. graph shift S = VΛVT itself. At the same time, we should account for the discrepancies A. Stationary versus non-stationary observations between Vˆ and the actual eigenvectors of S, due to ﬁnite Y sample size constraints and unavoidable errors in estimating Consider that the P observations in correspond to inde- the ﬁlter H. Accordingly, we build on [11] and seek for the pendent realizations of a process y adhering to the generative Y shift operator S that: (a) is optimal with respect to (often model in (2). The goal is to use to estimate the spectral S convex) criteria f(S); (b) belongs to a convex set that templates V of the ﬁlter H that governs the diffusion in (2). speciﬁes the desired type of shift operator (e.g., the adjacency To gain insights, suppose ﬁrst that x is white so that C = x A or Laplacian L); and (c) is close to Vˆ ΛVˆ T as measured I [11]. Then the covariance matrix of y = Hx is · · N by a convex matrix distance d( , ). Formally, one can solve Cy := E[yyT ] = E[Hx(Hx)T ] = HE[xxT ]H = H2. (3) S∗ := argmin f(S), s. to d(S, Vˆ ΛVˆ T ) ≤ (cid:15) (6) ∈S Λ,S In obtaining the third equality with used that H is sym- which is a convex optimization problem provided f(S) (cid:80) (cid:80) metric, because it is a polynomial in the symmetric GSO S. is convex, and (cid:15) is a tuning parameter chosen based on a Using the spectral decomposition of S = VΛVT to express priori information on the imperfections. Within the scope of − − the ﬁlter as H = L 1 h (VΛVT )l = V( L 1 h Λl)VT , the signal model (1), the formulation (6) entails a general l=0 (cid:16)l (cid:17) l=0 l we can diagonalize the cov(cid:80)ariance matrix as class of network topology inference problems parametrized by the choices in (a)-(c) above. The spectrum of alternatives − 2 C = V L 1 h Λl VT . (4) is brieﬂy outlined next, while concrete choices are made for y l=0 l the numerical tests in Section . Criteria. The selection of f(S) allows to incorporate physical Such a covariance expression is precisely the requirement characteristics of the desired graph into the formulation, while for a graph signal to be stationary in S [8, Def. 2.b]. Remark- being consistent with the spectral templates Vˆ . For instance, (cid:107) (cid:107) ably, if y is graph stationary, or equivalently if x is white, (4) the matrix (pseudo-)norm f(S) = S which counts the 0 shows that the eigenvectors of the shift S, the ﬁlter H, and number of nonzero entries in S can be used to minimize (cid:80) the covariance C are all the same. As a result, to estimate the number of edges towards identifying sparse graphs (e.g., y { } (cid:107) (cid:107) V from the observations y(p) P it sufﬁces to form the of direct relations among signal elements); f(S) = S p=1 1 sample covariance Cˆ = 1 P y(p)(y(p))T and use its is a convex proxy for the aforementioned edge cardinality y P p=1 (cid:107) (cid:107) eigenvectors as spectral templates to recover S [11], [29]; see function. Alternatively, the Frobenius norm f(S) = S can F also Section . be adopted to minimize the energy of the edges in the graph, or ##LTLine##

text: can

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text: the

text: eigenvalues

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text: GSO,

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text: underlying

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text: matrix

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text: identify

text: the

text: ﬁlter

text: H,

text: with

text: the

text: observations

text: Y,

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text: ﬁrst

text: estimating

text: the

text: unknown

text: graph

text: ﬁlter

text: H.

text: ultimate

text: goal

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text: estimating

text: its

text: eigenvectors

text: V.

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text: setting

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text: close

text: the

text: loop

text: showing

text: how

text: to

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text: H.

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text: we

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text: additional

text: knowledge

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text: the

text: input

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text: statistical

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text: and

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text: the

text: structure

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text: the

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text: ﬁlter

text: H.

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text: estimates

text: Vˆ

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text: the

text: ﬁlter

text: eigenvectors,

text: recovery

text: of

text: how

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text: the

text: desired

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text: amounts

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text: eigenvalues

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text: graph

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text: S

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text: the

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text: the

text: discrepancies

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text: versus

text: non-stationary

text: observations

text: between

text: Vˆ

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text: the

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text: Y

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text: the

text: P

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text: the

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text: we

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text: for

text: the

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text: process

text: y

text: adhering

text: to

text: the

text: generative

text: Y

text: shift

text: operator

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text: that:

text: (a)

text: is

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text: respect

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text: (often

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text: (b)

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text: L);

text: and

text: (c)

text: is

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text: Vˆ

text: ΛVˆ

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text: the

text: covariance

text: matrix

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text: y

text: Hx

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text: by

text: a

text: convex

text: matrix

text: distance

text: d(

text: Formally,

text: one

text: can

text: solve

text: Cy

text: E[yyT

text: E[Hx(Hx)T

text: HE[xxT

text: ]H

text: S∗

text: argmin

text: f(S),

text: s.

text: to

text: d(S,

text: Vˆ

text: ΛVˆ

text: T

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text: obtaining

text: the

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text: f(S)

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text: tuning

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text: the

text: spectral

text: decomposition

text: of

text: S

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text: priori

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text: the

text: imperfections.

text: Within

text: the

text: scope

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text: the

text: ﬁlter

text: as

text: H

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text: h

text: (VΛVT

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text: l

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text: can

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text: the

text: matrix

text: as

text: class

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text: network

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text: problems

text: parametrized

text: by

text: the

text: choices

text: in

text: (a)-(c)

text: above.

text: The

text: spectrum

text: of

text: alternatives

text: C

text: V

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text: h

text: Λl

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text: brieﬂy

text: outlined

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text: The

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text: instance,

text: ably,

text: if

text: y

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text: stationary,

text: or

text: equivalently

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text: x

text: is

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text: the

text: matrix

text: (pseudo-)norm

text: f(S)

text: S

text: which

text: counts

text: the

text: shows

text: that

text: the

text: eigenvectors

text: of

text: the

text: shift

text: S,

text: the

text: ﬁlter

text: H,

text: and

text: number

text: of

text: nonzero

text: entries

text: in

text: S

text: can

text: be

text: used

text: to

text: minimize

text: the

text: covariance

text: C

text: are

text: all

text: the

text: same.

text: As

text: a

text: result,

text: to

text: estimate

text: the

text: number

text: of

text: edges

text: towards

text: identifying

text: sparse

text: graphs

text: (e.g.,

text: y

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text: from

text: the

text: observations

text: y(p)

text: P

text: it

text: sufﬁces

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text: of

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text: relations

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text: elements);

text: f(S)

text: S

text: sample

text: covariance

text: Cˆ

text: P

text: y(p)(y(p))T

text: and

text: use

text: its

text: is

text: a

text: convex

text: proxy

text: for

text: the

text: aforementioned

text: edge

text: cardinality

text: y

text: P

text: eigenvectors

text: as

text: spectral

text: templates

text: to

text: recover

text: S

text: see

text: function.

text: Alternatively,

text: the

text: Frobenius

text: norm

text: f(S)

text: S

text: can

text: F

text: also

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text: be

text: adopted

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text: 2 (cid:90) (cid:88) for a well deﬁned temperature. In this paper we describe a novel, third interpretation of the LBM. This new interpretation is based on the Prin- ≈ b f(x , v, t )φ(v) dv f (x , t ) φ(v ) (3) ciple of Minimum Discrimination Information (or Mini- j n α j n α i=1 mum Cross Entropy) MinxEnt. We will call this LBM based on MinxEnt, “MinxEnt-LBM.” where φ(v) is some function of v. For example, to This paper is organized in the following way: the LBM, calculate the local density, (cid:88)φ(v) = 1, resulting in, and some of its current variations are described in Section II. Section III outlines the foundation of our method, b MinxEnt and describes the MinxEnt-LBM framework. ρ(x , t ) = f (x , t ) j n α j n Section IV gives some numerical results and Section V i=1 oﬀers ﬁnal thoughts and conclusions. and local momentum, φ(v) =(cid:88)v, resulting in, II. THE LATTICE BOLTZMANN METHOD b In kinetic theory [54], the evolution of macroscopic ρ(x , t )u(x , t ) = f (x , t ) v . j n j n α j n α properties often involves understanding the behaviour of α=1 a mass expectation density, f, (hereafter called a “distri- Henceforth, to ease notation we will omit the arguments bution”), in the local density, ρ and macroscopic velocity u. B. Collision Rules f = f(x, v, t). The LBM aims to understand f by tracking a related 1. Single Relaxation Time, SRT-LBM sampled version of it. This section aims to explain how f is sampled via discretization and how its evolution is The collision step accounts for changes to the compo- simulated. nents of the distribution arising from collisions between A. LBM Simulation Procedure ﬂuid particles as speciﬁed by a collision rule. The most popular collision rule is based the lineariza- tion of the kinetic collision term of the LGA [3, 4] and Consider the discretization of time by ﬁnite time step ∈ N further approximation by assuming a single relaxation of length δ such that t = t + δ , n . Further t n+1 n t time τ [5]: consider a discrete set of D dimensional velocities, V = { | ∈ R } v , v , . . . , v v D , and a discrete set of positions 1 2 b α organized into a regular∈lattice, Λ. The positions on the 1 − lattice are s∈uch that xj ∈ Λ if and only if xi = xj + v δt ∆ = f(xj, tn) + τ (feq(xj, tn) f(xj, tn)) . (4) for some v V and x Λ. j Having discretized velocity, space and time we deﬁne where τ is some predetermined relaxation time that is the vector-valued distribution, f(x , t ) (denoted f here- j n related to the ﬂuid viscosity and feq is appropriately cho- after), by, § sen “equilibrium” distribution (see ). The relationship between τ and viscosity is lattice dependent and is shown f(x , v , t ) ∈ { } for a speciﬁc lattice in (23). This single relaxation time fα(xj, tn) = Wα ωj(vα) n i 1, 2, . . . , b . (1) approach was made more popular in [8] and assumed the α name Lattice BGK (LBGK) owing to its similarity to the Bhatnagar-Gross-Krook kinetic equation [55]. For this where W and ω(v ) are quadrature weights corre- reason it is common in the literature to refer to LBMs α α sponding to the particular choice of V. A particular ex- using the single relaxation time collision step as “LBGK.” § ample of W and ω(v ) is given in . 2. Multiple Relaxation Time, MRT-LBM α α Given a distribution f at time t we approximate f at n time tn+1 with a two-step process; an instantaneous local Because of its single adjustable parameter, τ, diﬀerent “collision step” followed by a “streaming step.” This ﬂuid properties (such as viscosity and the Prandtl num- process can be summarized by the equation, ber) cannot be independently speciﬁed in SRT-LBM sim- ulations. In an attempt to rectify this issue and improve numerical stability researchers returned to the more gen- ∀ ∈ { } f (x + v δt, t ) = ∆ [f(x , t )] i 1, . . . , b eral linearized collision term of [3, 4] which allowed for α j α n+1 α j n (2) multiple relaxation times during the collision step [46– 53]. This is accomplished via the collision rule: for some choice of local collision rule ∆. With knowledge of f, macroscopic quantities at xj are ∆ = f(x , t ) + T−1BT (feq(x , t ) − f(x , t )) (5) calculated via, j n j n j n ##LTLine## ##LTLine##

text: for

text: a

text: well

text: deﬁned

text: temperature.

text: In

text: this

text: paper

text: we

text: describe

text: a

text: novel,

text: third

text: interpretation

text: of

text: the

text: LBM.

text: This

text: new

text: interpretation

text: is

text: based

text: on

text: the

text: Prin-

text: b

text: f(x

text: v,

text: t

text: )φ(v)

text: dv

text: f

text: (x

text: t

text: φ(v

text: ciple

text: of

text: Minimum

text: Discrimination

text: Information

text: (or

text: Mini-

text: j

text: n

text: α

text: j

text: n

text: α

text: mum

text: Cross

text: Entropy)

text: MinxEnt.

text: We

text: will

text: call

text: this

text: LBM

text: based

text: on

text: MinxEnt,

text: “MinxEnt-LBM.”

text: where

text: φ(v)

text: is

text: some

text: function

text: of

text: v.

text: For

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text: This

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text: following

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text: LBM,

text: calculate

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text: density,

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text: described

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text: Section

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text: Section

text: III

text: outlines

text: the

text: foundation

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text: our

text: method,

text: b

text: MinxEnt

text: and

text: describes

text: the

text: MinxEnt-LBM

text: framework.

text: ρ(x

text: t

text: f

text: (x

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text: j

text: n

text: α

text: j

text: n

text: Section

text: IV

text: gives

text: some

text: numerical

text: results

text: and

text: Section

text: V

text: oﬀers

text: ﬁnal

text: thoughts

text: and

text: conclusions.

text: and

text: local

text: momentum,

text: φ(v)

text: resulting

text: in,

text: II.

text: THE

text: LATTICE

text: BOLTZMANN

text: METHOD

text: b

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text: kinetic

text: theory

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text: evolution

text: of

text: macroscopic

text: ρ(x

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text: understanding

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text: density,

text: f,

text: (hereafter

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text: “distri-

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text: in

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text: local

text: density,

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text: macroscopic

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text: u.

text: B.

text: Collision

text: Rules

text: f

text: f(x,

text: v,

text: t).

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text: understand

text: f

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text: how

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text: The

text: collision

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text: accounts

text: for

text: changes

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text: the

text: compo-

text: simulated.

text: nents

text: of

text: the

text: distribution

text: arising

text: from

text: collisions

text: between

text: A.

text: LBM

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text: Procedure

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text: by

text: a

text: collision

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text: The

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text: tion

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text: LGA

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text: Consider

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text: discretization

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text: approximation

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text: single

text: relaxation

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text: length

text: δ

text: such

text: that

text: t

text: t

text: δ

text: n

text: Further

text: t

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text: time

text: τ

text: consider

text: a

text: discrete

text: set

text: of

text: D

text: dimensional

text: velocities,

text: V

text: R

text: v

text: v

text: v

text: v

text: D

text: and

text: a

text: discrete

text: set

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text: positions

text: b

text: α

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text: Λ.

text: The

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text: lattice

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text: s∈uch

text: that

text: xj

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text: xi

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text: tn)

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text: tn)

text: f(xj,

text: tn))

text: for

text: some

text: v

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text: x

text: Λ.

text: j

text: Having

text: discretized

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text: space

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text: time

text: we

text: deﬁne

text: where

text: τ

text: is

text: some

text: predetermined

text: relaxation

text: time

text: that

text: is

text: the

text: vector-valued

text: distribution,

text: f(x

text: t

text: (denoted

text: f

text: here-

text: j

text: n

text: related

text: to

text: the

text: ﬂuid

text: viscosity

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text: feq

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text: appropriately

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text: after),

text: by,

text: sen

text: “equilibrium”

text: distribution

text: (see

text: The

text: relationship

text: between

text: τ

text: and

text: viscosity

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text: lattice

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text: is

text: shown

text: f(x

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text: t

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text: a

text: speciﬁc

text: lattice

text: in

text: This

text: single

text: relaxation

text: time

text: fα(xj,

text: tn)

text: Wα

text: ωj(vα)

text: n

text: i

text: b

text: approach

text: was

text: made

text: more

text: popular

text: in

text: and

text: assumed

text: the

text: α

text: name

text: Lattice

text: BGK

text: (LBGK)

text: owing

text: to

text: its

text: similarity

text: to

text: the

text: Bhatnagar-Gross-Krook

text: kinetic

text: equation

text: For

text: this

text: where

text: W

text: and

text: ω(v

text: are

text: quadrature

text: weights

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text: reason

text: it

text: is

text: common

text: in

text: the

text: literature

text: to

text: refer

text: to

text: LBMs

text: α

text: α

text: sponding

text: to

text: the

text: particular

text: choice

text: of

text: V.

text: A

text: particular

text: ex-

text: using

text: the

text: single

text: relaxation

text: time

text: collision

text: step

text: as

text: “LBGK.”

text: ample

text: of

text: W

text: and

text: ω(v

text: is

text: given

text: in

text: Multiple

text: Relaxation

text: Time,

text: MRT-LBM

text: α

text: α

text: Given

text: a

text: distribution

text: f

text: at

text: time

text: t

text: we

text: approximate

text: f

text: at

text: n

text: time

text: with

text: a

text: two-step

text: process;

text: an

text: instantaneous

text: local

text: Because

text: of

text: its

text: single

text: adjustable

text: parameter,

text: τ,

text: diﬀerent

text: “collision

text: step”

text: followed

text: by

text: a

text: “streaming

text: step.”

text: This

text: ﬂuid

text: properties

text: (such

text: as

text: viscosity

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text: the

text: Prandtl

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text: process

text: can

text: be

text: summarized

text: by

text: the

text: equation,

text: ber)

text: cannot

text: be

text: independently

text: speciﬁed

text: in

text: SRT-LBM

text: sim-

text: ulations.

text: In

text: an

text: attempt

text: to

text: rectify

text: this

text: issue

text: and

text: improve

text: numerical

text: stability

text: researchers

text: returned

text: to

text: the

text: more

text: gen-

text: f

text: (x

text: v

text: δt,

text: t

text: [f(x

text: t

text: i

text: b

text: eral

text: linearized

text: collision

text: term

text: of

text: which

text: allowed

text: for

text: α

text: j

text: α

text: α

text: j

text: n

text: multiple

text: relaxation

text: times

text: during

text: the

text: collision

text: step

text: This

text: is

text: accomplished

text: via

text: the

text: collision

text: rule:

text: for

text: some

text: choice

text: of

text: local

text: collision

text: rule

text: With

text: knowledge

text: of

text: f,

text: macroscopic

text: quantities

text: at

text: xj

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text: f(x

text: t

text: (feq(x

text: t

text: f(x

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text: via,

text: j

text: n

text: j

text: n

text: j

text: n

text: ##LTLine##

text: ##LTLine##


text: (cid:88) − | | Since If p˜ = 0, then u (r˜, p˜) = X , and consequently, n n n − | | − − | | · · · | | ≤ − (c Xi ) = (i(cid:63) 1)c ( X1 + + Xi(cid:63)−1 ), m∈in ui(r˜, p˜) un(r˜, p˜) = m∈in ui(r, p) = k c. i∈N\S(cid:63) In thei cNase (ii), min ∈ u (r, pi) N= k − c. Suppose that i N i ∈ then | | · · · | | − m| in|i−∈N−ui(r˜, p˜) > mini∈N ui(r, p). |Let|−j −N be−such that X2 + + Xi(cid:63) > (i(cid:63) 1)c. |Xj|−p˜−j ≥= |min|i−∈N−ui(r˜, p˜). T∈hus, Xj |p˜j |>− k− c. Si−nce This is, however, a contradiction since c ≥ |Xi| for all i ∈ TXhuis, p˜i (cid:88) Xj p˜(cid:88)j for all i (cid:88)N, then Xi p˜i > k c. [i(cid:63)] (by assumption), and so, − ≥ | | · · · | | (i(cid:63) 1)c(cid:88) X + + X . | |− − (cid:88)| |− 2 i(cid:63) X p˜ = X p˜ i i i N Thus, ∈ ∈ ∈ i N i N i N (cid:88)i∈S(cid:63) p−i ≤(cid:88)p(cid:63)S(cid:63). = i(cid:88)∈N|Xi|−r˜N Moreover, | (cid:88)|− = X r − ≤ − i N p p ∈ i i i N ∈ ∈ − | |− i S i S(cid:63) = nk X r ⊆ i N since S S(cid:63) (by de(cid:88)ﬁnition). By combining the above ∈ − i N arguments, it then follows that (cid:80) > nk nc, − ≤ ∈ pi r˜S, or equivalently, ( ∈ |X |+r )/n (cid:80)< c. Since c = i S | | − i N i (cid:80)N ≥ | | min ∈ ( X +p ) (by deﬁnition) and c X (by assump- as was to be shown. i N i i n| | tion), then it is easy to see that c = ( ∈ X +r )/n. | i| N i N Lemma 8. (r, p) is optimal. This is a contradiction since ( ∈ X +r )/n < c. ≤ i N i N Thus, min ∈ u (r˜, p˜) min ∈ u (r, p). This completes Proof: The proof follows from the same argument as in i N i i N i the proof. the proof of Lemma 5, and hence omitted to(cid:80)avoid repetition. (cid:80) REFERENCES Lemma 9. For any optimal (r˜, p˜), we≥have i∈N ui(r, p) = [1] S. El Rouayheb, M. Chaudhry, and A. Sprintson, “On the Minimum ∈ u (r˜, p˜) and min ∈ u (r, p) min ∈ u (r˜, p˜). Number of Transmissions in Signle-Hop Wireless Coding Networks,” i N i i N i i N i in Proc. IEEE ITW’07, Sep. 2007, pp. 120–125. Proof: The proof of the ﬁrst part (i.e., maximum sum- [2] S. El Rouayheb, A. Sprintson, and P. Sadeghi, “On Coding for u(cid:88)tility) is straig(cid:88)htforward. Take(cid:88)an arbitrary op(cid:88)timal (r˜, p˜). Cooperative Data Exchange,” in Proc. IEEE ITW’10, Jan. 2010. ∈ P [3] T. A. Courtade and R. D. Wesel, “Efﬁcient Universal Recovery in Since r˜N = rN and p, p˜ N, then Broadcast Networks,” in Proc. 48th Annu. Allerton Conf. Commun., | |− | |− Control, Comput., Oct. 2010, pp. 1542–1549. u (r˜, p˜) = X r˜ = X r = u (r, p). [4] A. Sprintson, P. Sadeghi, G. Booker, and S. El Rouayheb, “A Ran- i i N i N i ∈ ∈ ∈ ∈ domized Algorithm and Performance Bounds for Coded Cooperative i N i N i N i N Data Exchange,” in Proc. IEEE ISIT’10, Jun. 2010, pp. 1888–1892. For the proof of the second part (i.e., maximum min-utility), [5] ——, “Deterministic Algorithm for Coded Cooperative Data Ex- we need to show that change,” in Proc. ICST QShine, Nov. 2010. [6] M. Gonen and M. Langberg, “Coded Cooperative Data Exchange ≤ Problem for General Topologies,” in Proc. IEEE ISIT’12, Jul. 2012, min u (r˜, p˜) min u (r, p) ∈ i ∈ i pp. 2606–2610. i N i N [7] M. Yan, A. Sprintson, and I. Zelenko, “Weakly Secure Data Exchange for any optimal (r˜, p˜). Take an arbitrary optimal (r˜, p˜). Since with Generalized Reed Solomon Codes,” in Proc. IEEE ISIT’14, Jun. ≥ p˜+ = r˜ (otherwise, p˜ > r = p since p˜+ r˜ 2014, pp. 1366–1370. i i N N N i i [8] T. A. Courtade and R. D. Wesel, “Coded Cooperative Data Exchange (by rationality of (r˜, p˜)), and so, (r˜, p˜) cannot be optimal), | |− − (cid:44) in Multihop Networks,” IEEE Trans. Inf. Theory, vol. 60, no. 2, pp. then u (r˜, p˜) = X p˜ . Note that p˜ = p . Let c 1136–1158, Feb. 2014. i | | − i i | |− − N − N ≥ | | min ∈ ( X +p ). Note that X p = k c if c X , [9] T. A. Courtade and T. R. Halford, “Coded Cooperative Data Exchange i| N |−i− i | | − |i | i | | i for a Secret Key,” in Proc. IEEE ISIT’14, Jun. 2014, pp. 776–780. and X p = X = k X if c < X . Thus, i −i { |i |} i∈ | i |≥ | | [10] N. Milosavljevic, S. Pawar, S. E. Rouayheb, M. Gastpar, and K. Ram- u (r, p) = k max c, X for all i N. Since X X chandran, “Efﬁcient Algorithms for the Data Exchange Problem,” i ∈ i n i for all i N (by assumption), then it follows that IEEE Trans. Inf. Theory, vol. 62, no. 4, pp. 1878–1896, Apr. 2016. [11] C. Chan, A. Al-Bashabsheh, Q. Zhou, N. Ding, T. Liu, and − { | |} A. Sprintson, “Successive Omniscience,” IEEE Trans. Inf. Theory, min u (r, p) = k max c, X . ∈ i n vol. 62, no. 6, pp. 3270–3289, Jun. 2016. i N | | ≥ | | [12] N. Ding, C. Chan, T. Liu, R. Kennedy, and P. Sadeghi, “A Game- We consider two cases: (i) c < X , and (ii) c X . Theoretic Perspective on Communication for Omniscience,” in Proc. n − | | | n | In the case (i), min ∈ u (r, p) = k X = X . If 2016 Australian Communications Theory Workshop (AusCTW’16), − i N i n n Jan. 2016, pp. 95–100. p˜ > 0, then n [13] A. Heidarzadeh and A. Sprintson, “Cooperative Data Exchange: A | |− − | | Coalition Game Perspective,” in Proc. 54rd Annu. Allerton Conf. un(r˜, p˜) = Xn p˜n < Xn = m∈in ui(r, p). Commun., Control, Comput., Sep. 2016. i N 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Since

text: If

text: p˜

text: then

text: u

text: (r˜,

text: p˜)

text: X

text: and

text: consequently,

text: n

text: n

text: n

text: (c

text: Xi

text: m∈in

text: ui(r˜,

text: p˜)

text: un(r˜,

text: p˜)

text: m∈in

text: ui(r,

text: p)

text: k

text: c.

text: In

text: thei

text: cNase

text: (ii),

text: min

text: u

text: (r,

text: pi)

text: N=

text: k

text: c.

text: Suppose

text: that

text: i

text: N

text: i

text: then

text: m|

text: in|i−∈N−ui(r˜,

text: p˜)

text: mini∈N

text: ui(r,

text: p).

text: |Let|−j

text: −N

text: be−such

text: that

text: |Xj|−p˜−j

text: |min|i−∈N−ui(r˜,

text: p˜).

text: T∈hus,

text: Xj

text: |p˜j

text: k−

text: c.

text: Si−nce

text: This

text: is,

text: however,

text: a

text: contradiction

text: since

text: c

text: |Xi|

text: for

text: all

text: i

text: TXhuis,

text: p˜i

text: Xj

text: for

text: all

text: i

text: then

text: Xi

text: p˜i

text: k

text: c.

text: (by

text: assumption),

text: and

text: so,

text: X

text: X

text: X

text: p˜

text: X

text: p˜

text: i

text: i

text: i

text: N

text: Thus,

text: i

text: N

text: i

text: N

text: i

text: N

text: p−i

text: Moreover,

text: X

text: r

text: i

text: N

text: p

text: p

text: i

text: i

text: i

text: N

text: i

text: S

text: i

text: nk

text: X

text: r

text: i

text: N

text: since

text: S

text: (by

text: By

text: combining

text: the

text: above

text: i

text: N

text: arguments,

text: it

text: then

text: follows

text: that

text: nk

text: nc,

text: pi

text: r˜S,

text: or

text: equivalently,

text: |X

text: |+r

text: )/n

text: c.

text: Since

text: c

text: i

text: S

text: i

text: N

text: i

text: min

text: X

text: +p

text: (by

text: deﬁnition)

text: and

text: c

text: X

text: (by

text: assump-

text: as

text: was

text: to

text: be

text: shown.

text: i

text: N

text: i

text: i

text: n|

text: tion),

text: then

text: it

text: is

text: easy

text: to

text: see

text: that

text: c

text: X

text: +r

text: )/n.

text: i|

text: N

text: i

text: N

text: Lemma

text: (r,

text: p)

text: is

text: optimal.

text: This

text: is

text: a

text: contradiction

text: since

text: X

text: +r

text: )/n

text: c.

text: i

text: N

text: i

text: N

text: Thus,

text: min

text: u

text: (r˜,

text: p˜)

text: min

text: u

text: (r,

text: p).

text: This

text: completes

text: Proof:

text: The

text: proof

text: follows

text: from

text: the

text: same

text: argument

text: as

text: in

text: i

text: N

text: i

text: i

text: N

text: i

text: the

text: proof.

text: the

text: proof

text: of

text: Lemma

text: and

text: hence

text: omitted

text: repetition.

text: REFERENCES

text: Lemma

text: For

text: any

text: optimal

text: (r˜,

text: p˜),

text: we≥have

text: i∈N

text: ui(r,

text: p)

text: S.

text: El

text: Rouayheb,

text: M.

text: Chaudhry,

text: and

text: A.

text: Sprintson,

text: “On

text: the

text: Minimum

text: u

text: (r˜,

text: p˜)

text: and

text: min

text: u

text: (r,

text: p)

text: min

text: u

text: (r˜,

text: p˜).

text: Number

text: of

text: Transmissions

text: in

text: Signle-Hop

text: Wireless

text: Coding

text: Networks,”

text: i

text: N

text: i

text: i

text: N

text: i

text: i

text: N

text: i

text: in

text: Proc.

text: IEEE

text: Sep.

text: pp.

text: Proof:

text: The

text: proof

text: of

text: the

text: ﬁrst

text: part

text: (i.e.,

text: maximum

text: sum-

text: S.

text: El

text: Rouayheb,

text: A.

text: Sprintson,

text: and

text: P.

text: Sadeghi,

text: “On

text: Coding

text: for

text: is

text: arbitrary

text: (r˜,

text: p˜).

text: Cooperative

text: Data

text: Exchange,”

text: in

text: Proc.

text: IEEE

text: Jan.

text: P

text: T.

text: A.

text: Courtade

text: and

text: R.

text: D.

text: Wesel,

text: “Efﬁcient

text: Universal

text: Recovery

text: in

text: Since

text: r˜N

text: rN

text: and

text: p,

text: p˜

text: N,

text: then

text: Broadcast

text: Networks,”

text: in

text: Proc.

text: Annu.

text: Allerton

text: Conf.

text: Commun.,

text: Control,

text: Comput.,

text: Oct.

text: pp.

text: u

text: (r˜,

text: p˜)

text: X

text: r˜

text: X

text: r

text: u

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text: Sample

font: YUUTUO+NimbusSanL-BoldItal

text: et

text: al.

text: Reverberation

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text: Mapping

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